Feynman-Kac-Type Theorems and Gibbs Measures on Path Space: Volume 1 Feynman-Kac-Type Formulae and Gibbs Measures [2nd rev. ed.] 9783110330397, 9783110330045

Table of contents :
Contents
Preface to the second edition
Preface to the first edition
1. Heuristics and history
2. Brownian motion
3. Lévy processes
4. Feynman–Kac formulae
5. Gibbs measures associated with Feynman–Kac semigroups
6. Notes and references
Bibliography
Index

Citation preview

József Lőrinczi, Fumio Hiroshima, and Volker Betz Feynman–Kac-Type Theorems and Gibbs Measures on Path Space

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

Volume 34/1

József Lőrinczi, Fumio Hiroshima, and Volker Betz

Feynman–Kac-Type Theorems and Gibbs Measures on Path Space |

Volume 1: Feynman–Kac-Type Formulae and Gibbs Measures 2nd edition

Mathematics Subject Classification 2010 Primary: 47D08, 81Q10, 35J10; Secondary: 47-01, 60-01 Authors Prof. Dr. József Lőrinczi Loughborough University Department of Mathematical Sciences Schofield Building Loughborough, LE11 3TU United Kingdom [email protected]

Prof. Dr. Volker Betz Technische Universität Darmstadt Fachbereich Mathematik Schloßgartenstr. 7 64289 Darmstadt Germany [email protected]

Prof. Dr. Fumio Hiroshima Kyushu University Faculty of Mathematics 744 Motooka Nishiku 819-0395 Japan [email protected]

ISBN 978-3-11-033004-5 e-ISBN (PDF) 978-3-11-033039-7 e-ISBN (EPUB) 978-3-11-038993-7 ISSN 0179-0986 Library of Congress Control Number: 2019947690 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Preface to the second edition | IX Preface to the first edition | XI 1 1.1 1.2

Heuristics and history | 1 Feynman path integrals and Feynman–Kac formulae | 1 Plan and scope of the second edition | 5

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4

Brownian motion | 9 Concepts and facts of general measure theory and probability | 9 Elements of general measure theory | 9 Probability measures and limit theorems | 16 Random variables | 29 Conditional expectation and regular conditional probability measures | 38 Random processes | 45 Basic concepts and facts | 45 Martingale properties | 50 Stopping times and optional sampling | 53 Markov properties | 67 Feller transition kernels and generators | 72 Invariant measures | 74 Brownian motion and Wiener measure | 77 Construction of Brownian motion | 77 Two-sided Brownian motion | 84 Conditional Wiener measure | 88 Martingale properties of Brownian motion | 89 Markov properties of Brownian motion | 92 Local path properties of Brownian motion | 97 Global path properties of Brownian motion | 103 Stochastic calculus based on Brownian motion | 107 The classical integral and its extensions | 107 Stochastic integrals | 108 Extension of stochastic integrals | 115 Itô formula | 119 Stochastic differential equations | 128 Brownian bridge | 134 Weak solution and time change | 136 Girsanov theorem and Cameron–Martin formula | 140

2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.4.7 2.4.8

VI | Contents 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.5.3 3.6 3.6.1 3.6.2

Lévy processes | 143 Lévy processes and the Lévy–Khintchine formula | 143 Infinitely divisible random variables | 143 Lévy–Khintchine formula | 149 Lévy processes | 154 Martingale properties of Lévy processes | 160 Markov properties of Lévy processes | 161 Sample path properties of Lévy processes | 165 Càdlàg version | 165 Two-sided Lévy processes | 169 Random measures and Lévy–Itô decomposition | 178 Poisson random measures | 178 Lévy–Itô decomposition | 186 Itô formula for semimartingales | 188 Point processes | 188 Itô formula for semimartingales | 194 Exponentials of Lévy processes and recurrence properties | 201 Exponential functionals of Lévy processes | 201 Capacitary measures | 203 Recurrence properties of Lévy processes | 204 Subordinators and Bernstein functions | 206 Subordinators and subordinate Brownian motion | 206 Bernstein functions | 209

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2

Feynman–Kac formulae | 217 Schrödinger semigroups | 217 Schrödinger equation and path integral solutions | 217 Linear operators and their spectra | 218 Spectral resolution | 223 Compact operators and trace ideals | 227 Schrödinger operators | 232 Schrödinger operators through quadratic forms | 236 Confining potentials and decaying potentials | 239 Strongly continuous operator semigroups | 243 Feynman–Kac formula for Schrödinger operators | 246 Bounded smooth external potentials | 246 Derivation through the Trotter product formula | 249 Kato-class potentials | 251 Feynman–Kac formula for Kato-decomposable potentials | 264 Properties of Schrödinger operators and semigroups | 270 Kernel of the Schrödinger semigroup | 270 Positivity improving and uniqueness of ground state | 271

Contents | VII

4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.7 4.7.1 4.7.2 4.7.3 4.7.4 4.7.5 4.8 4.8.1 4.8.2 4.8.3 4.9 4.9.1 4.9.2 4.9.3 4.9.4 4.9.5 4.9.6 4.9.7 4.9.8 4.9.9 4.9.10

Degenerate ground state and Klauder phenomenon | 275 Existence and non-existence of ground states | 277 Sojourn times and existence of bound states | 282 The number of eigenfunctions with negative eigenvalues | 289 Application to canonical commutation relations | 307 Exponential decay of eigenfunctions | 314 Feynman–Kac formula for Schrödinger operators with vector potentials | 320 Feynman–Kac–Itô formula | 320 Alternative proof of the Feynman–Kac–Itô formula | 324 Extension to singular external and vector potentials | 327 Kato-class potentials and Lp -Lq boundedness | 333 Feynman–Kac formula for unbounded semigroups and Stark effect | 335 Feynman–Kac formula for relativistic Schrödinger operators | 339 Relativistic Schrödinger operator | 339 Relativistic Kato-class potentials | 344 Decay of eigenfunctions | 351 Non-relativistic limit | 356 Feynman–Kac formula for Schrödinger operators with spin | 359 Schrödinger operators with spin 21 | 359 A jump process | 361 Feynman–Kac formula for the jump process | 363 Extension to singular external potentials and singular vector potentials | 367 Decay of eigenfunctions and martingale properties | 371 Feynman–Kac formula for relativistic Schrödinger operators with spin | 375 Relativistic Schrödinger operator with spin 21 | 375 Martingale properties | 381 Decay of eigenfunctions | 384 Feynman–Kac formula for nonlocal Schrödinger operators | 388 Nonlocal Schrödinger operators | 388 Vector potentials | 389 Ψ-Kato-class potentials | 392 Fractional Kato-class potentials | 401 Generalized spin | 406 Recurrence properties and existence of bound states | 412 The number of eigenfunctions with negative eigenvalues | 413 Decay of eigenfunctions | 423 Massless relativistic harmonic oscillator | 432 Embedded eigenvalues | 436

VIII | Contents 5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.4 5.5 5.5.1 5.5.2 5.5.3 6

Gibbs measures associated with Feynman–Kac semigroups | 449 Ground state transform and related processes | 449 Ground state transform and the intrinsic semigroup | 449 Ground state-transformed processes as solutions of SDE | 454 P(ϕ)1 -processes with continuous paths | 458 Dirichlet principle | 464 Mehler’s formula | 467 P(ϕ)1 -processes with càdlàg paths | 476 Gibbs measures on path space | 481 From Feynman–Kac formulae to Gibbs measures | 481 Gibbs measures on Brownian paths | 485 Gibbs measures on càdlàg paths | 492 Gibbs measures for external potentials | 494 Existence | 494 Uniqueness | 497 Gibbs measures for external and pair interaction potentials: direct method | 503 Gibbs measures for external and pair interaction potentials: cluster expansion | 511 Cluster representation | 511 Basic estimates and convergence of cluster expansion | 516 Further properties of the Gibbs measure | 518 Notes and references | 521 Notes to the Preface | 521 Notes to Chapter 1 | 522 Notes to Chapter 2 | 522 Notes to Chapter 3 | 525 Notes to Chapter 4 | 526 Notes to Chapter 5 | 535

Bibliography | 539 Index | 553

Preface to the second edition Since the publication of the first edition of this book in 2011 we have received much feedback from our colleagues and students, which kept encouraging us to consider a second edition. In this new edition, we set ourselves the task to amplify the text both by an account of the latest developments in functional integration applied to quantum theory, and to make the perusal of the book more comfortable to our readers by having most of the background material closer at hand. This meant to accept that the length of this book will be more manageable by splitting the material in two parts, each of a volume of its own. This first volume offers then a detailed discussion of Feynman–Kac type path integral representations and their uses in a study of progressively more elaborate models ranging from classical to relativistic and more generally non-local Schrödinger operators with electrostatic and/or magnetic potentials, to further including spin. The second volume titled “Applications in Rigorous Quantum Field Theory” makes then the passage from applications in quantum mechanics to applications in quantum fields. The original Feynman–Kac formula, and its many subsequent variants, is a mathematical relation connecting an evolution equation, an operator, and a random process, offering a probabilistic representation of solutions obtained as space-time transformations of an initial function under a semigroup, by running a suitable random process and averaging over its paths. This theory is then a fascinating meeting point of functional analysis, stochastic processes, and stochastic analysis, and it is in continuing development. Those who are new to these topics and wish to learn about Feynman–Kac techniques may appreciate that in this edition Chapters 2 and 3 are now devoted to a discussion including detailed proofs of essential background material of concepts and facts of measure theory, probability, the theory of continuous-time stochastic processes and martingales, jump processes, and stochastic analysis. The first section of the main Chapter 4 similarly reviews fundamentals of linear operator theory in general, and Schrödinger operators in particular. The bulk of the chapter then continues with a systematic presentation of Feynman–Kac formulae for operators primarily of interest in mathematical quantum theory. Chapter 5 is about the “right hand side” of the Feynman–Kac formula including now a discussion of the related random processes with continuous paths or paths with jump discontinuities, and their distributions obtained as Gibbs measures. Experts will notice that new material includes Feynman–Kac representations of semigroups generated by relativistic and other nonlocal Schrödinger operators, a use of sojourn times, explicit solutions, eigenfunctions corresponding to embedded eigenvalues, and new results involving jump processes. The bibliography has also been updated, and we recommend the reading of this book with reference to the notes and comments in Chapter 6. https://doi.org/10.1515/9783110330397-201

X | Preface to the second edition We thank our collaborators Anup Biswas, Takashi Ichinose, Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki, Itaru Sasaki, Xiaochuan Yang, with whom we had the opportunity to further develop functional integration methods in the recent years, and who have greatly enriched our experience. We also thank Gonzalo A. Bley, Thomas Norman Dam, Batu Güneysu, Christian Jäh, Nikolai Leonenko, Oliver Matte, Jacob Schach Møller, René Schilling, Stefan Steinerberger, Renming Song, Yuma Takabayashi, Bruno Toaldo, Zoran Vondraček for reflections, suggestions or just interest in our book. The first named author gratefully thanks IHES, Bures-sur-Yvette, for visiting fellowships through several years, where part of this book has been written, and Professor David Ruelle for sustained interest and discussions. The second named author thanks the kind hospitality of Aarhus University and the International Network Program of the Danish Agency for Science, Technology and Innovation, where part of this work has been done. His work was also financially supported by JSPS KAKENHI Grant Number JP16H03942 and JSPS KAKENHI Grant Number JP16K17612. Finally, we also express our appreciation to Sabina Dabrowski of Walter de Gruyter, and Ina Talandienė and her team at VTeX UAB, Lithuania, for their professionalism and support. József Lőrinczi, Fumio Hiroshima, Volker Betz

August 2019

Preface to the first edition For a long time disciplines such as astronomy, celestial mechanics, the theory of heat, and the theory of electromagnetism counted among the most important sources in the development of mathematics, continually providing new problems and challenges. The main mathematical facts resulting from these fields typically crystallized in linear ordinary or partial differential equations. In the early 20th century Heisenberg’s discovery of a rule showing that the measurement of the position and momentum of a quantum particle give different answers, depending on the order of their measurement, led to the concept of commutation relations, and it made matrix theory find an unexpected application in the natural sciences. This resulted in an explosion of interest in linear structures developed by what we call today linear algebra and linear analysis, among them functional analysis, operator theory, C ∗ -algebras, and distribution theory. Similarly, quantum mechanics inspired much of group theory, lattice theory, and quantum logic. Following this initial boom Wigner’s beautiful phrase talking of the “unreasonable effectiveness of mathematics” marks a point of reflection on the fact that it is by no means obvious why this kind of abstract and sophisticated mathematics can be expected at all to make faithful descriptions and reliable predictions of natural phenomena. If indeed this can be appreciated as a miracle, as he said, the degree of flexibility that mathematics allows in its uses is perhaps at least a remarkable fact. Feynman has pointed to the different cultures of using mathematics when he remarked that “if all of mathematics disappeared, physics would be set back by exactly one week.” While mathematics cannot compete with physics in discovering new phenomena and offering explanations of them, physics continues to depend on the terminology, arsenal, and discipline of mathematics. Mathematical physics, which is part of mathematics and therefore operates by its rules and standards, has set the goal to understand the models of physics in a rigorous way. Mathematical physicists thus find themselves at the borderline, listening to physics and speaking mathematics, at best able to use these functionalities interchangeably. If one of the imports of early quantum mechanics has been the realization that the basic laws on the atomic scale can be formulated by linear superposition rules, another was a probabilistic interpretation of the wave function. This connection with chance, amplified and strongly advocated by the Copenhagen school, was no less revolutionary than the commutation relations. Richard Feynman has discovered a second connection with probability when he offered a representation of the state of a particle in terms of averages over all of its possible histories from one point in space-time to another. While this was an instance seriously questioning his dictum quoted above and his use of mathematics was in this case very problematic, the potential lying in the description advanced in his work turned out to be far reaching. The mathematihttps://doi.org/10.1515/9783110330397-202

XII | Preface to the first edition cian Mark Kac was the first to show that this method could be made sense of and was eminently viable. Our project to write the present monograph has grown out of the thematic program At the Interface of PDE, Self-Adjoint Operators and Stochastics: Models with Exclusion organized by the first named author at the Wolfgang Pauli Institute, Vienna, in 2006. The initial concept was a smaller-scale but up-to-date account of Feynman–Kactype formulae and their uses in quantum field theory, which we wanted to dedicate to the person all three of us have much to thank to, both scientifically and personally, Herbert Spohn. At that time we secretly meant this as a present for his 60th birthday, and we are glad that we are now able to make this hopefully more mature tribute on the occasion of his 65th birthday! Apart from the inspiring and pleasant environment at Zentrum Mathematik, Munich University of Technology, where our collaboration has started, and Wolfgang Pauli Institut, University of Vienna, hosting the thematic program, we are indebted to a number of other individuals and institutions. We are thankful for the joint work and friendship of our closest collaborators in this direction, Asao Arai, Massimiliano Gubinelli, Masao Hirokawa, and Robert Adol’fovich Minlos. We also thank Cristian Gérard, Takashi Ichinose, Kamil Kaleta, Jacek Małecki, Annalisa Panati, and Akito Suzuki for ongoing related joint work. It is furthermore a pleasure to thank Norbert Mauser, Director of the Wolfgang Pauli Institute, for supporting the program, and Sergio Albeverio, Rodrigo Bañuelos, Krzysztof Bogdan, Carlo Boldrighini, Thierry Coulhon, Laure Coutin, Cécile DeWitt-Morette, Aernout van Enter, Bill Faris, Gero Friesecke, Jürg Fröhlich, Mohammud Foondun, Hans-Otto Georgii, Alexander Grigor’yan, Martin Hairer, Takeru Hidaka, Wataru Ichinose, Keiichi Itô, Niels Jacob, Hiroshi Kawabi, Tadeusz Kulczycki, Kazuhiro Kuwae, Terry Lyons, Tadahiro Miyao, Hirofumi Osada, Habib Ouerdiane, Sylvie Roelly, Itaru Sasaki, Tomoyuki Shirai, Toshimitsu Takaesu, Setsuo Taniguchi, Josef Teichmann, Daniel Ueltschi, and Jakob Yngvason for many useful discussions over the years. We thank especially Erwin Schrödinger Institut, Vienna, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, and Università La Sapienza, Rome, for repeatedly providing excellent environments to further the project and where parts of the manuscript have been written. Also, we thank the hospitality of Kyushu University, Loughborough University, and Warwick University, for accommodating mutual visits and extended stays. We are particularly pleased to thank the professionalism, assistance, and endless patience of Robert Plato, Simon Albroscheit, and Friederike Dittberner of Walter de Gruyter. Also, we are grateful to Max Lein for technical support in the initial phase of this project. József Lőrinczi, Fumio Hiroshima, Volker Betz

February 2011

1 Heuristics and history 1.1 Feynman path integrals and Feynman–Kac formulae In Feynman’s work concepts such as paths and actions, just discarded by the new-born quantum mechanics, witnessed a comeback in the description of the time evolution of quantum particles, proposing an alternative to Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. It is worthwhile to briefly review how this turned out to be a viable project and led to a mathematically meaningful approach to quantum mechanics and quantum field theory. A basic tool in the hands of the quantum physicist is the Schrödinger equation iℏ

𝜕 ℏ2 ψ(t, x) = − Δψ(t, x) + V(x)ψ(t, x), 𝜕t 2m ψ(0, x) = ϕ(x), (t, x) ∈ ℝ+ × ℝ3

(1.1.1)

with the Hamilton operator H=−

ℏ2 Δ+V 2m

(1.1.2)

and a complex-valued wave function ψ describing a quantum particle of mass m at position x in space at time t. Here ℏ is Planck’s constant, V is a potential in whose force field the particle is moving, and ϕ describes the initial state of the quantum particle. The Schrödinger equation can be solved only under exceptional choices of V. However, formally the solution of the equation can be written as ψ(t, x) = (e−(it/ℏ)H ϕ)(x) for any V. The difficulty is that for a general choice of V we do not know how to calculate the right-hand side of the equality. The action of the complex exponential operator on ϕ can be evaluated by using an integral kernel G allowing to write the wave function of the particle at position y at time t based on its knowledge in x at s < t, i. e., ψ(t, y) = ∫ G(x, y; s, t)ψ(s, x)dx. ℝ3

This representation can then in principle be iterated from one interval [s, t] to another, and thus our knowledge of the wave function “propagates” from the initial time to any time point in the future. This observation suggests that at least a qualitative analysis of the solution might be possible for a wide choice of V by using this integral kernel, and the key to this approach is the information content of the kernel. https://doi.org/10.1515/9783110330397-001

2 | 1 Heuristics and history Feynman proposed a method of constructing this integral kernel in the following way. Consider first a classical Hamiltonian function H (p, q) =

1 2 p + V(q), 2m

(p, q) ∈ ℝ3 × ℝ3 ,

(1.1.3)

describing a particle at position q moving with momentum p under the same potential V as above. The corresponding Lagrangian is obtained through the Legendre transform ̇ − H (p, q) = L (q, q)̇ = qp

m 2 q̇ − V(q), 2

where q̇ =

𝜕H p = . 𝜕p m

Classical mechanics says that the particle is equivalently described by the action functional t

̇ A (q, q;̇ 0, t) = ∫ L (q(s), q(s))ds, 0

where q(s) is the path of the particle as a function of the time s. In particular, the equation of motion can be derived by the principle of least action by using variational calculus, and it is found that the particle follows a route from a fixed initial to a fixed endpoint which minimizes the action. In the late 1920s Dirac worked on a Lagrangian formulation of quantum mechanics and arrived at the conclusion that the ideas of the classical Lagrangian theory can be taken over, though not its equations. Specifically, he arrived at a rule he called “transformation theory” allowing to connect the positions at s and t of the quantum ̇ particle through the phase factors e(i/ℏ)A (q,q;s,t) . By dividing up the interval [s, t] into small pieces, Dirac noticed [89–91] that due to the fact that ℏ is of the order of 10−34 in standard units, this is so small that the integrands in the exponent are rapidly oscillating functions around zero. Therefore, as a result of the many cancellations the only significant contribution into the integral is given by those q whose large variations over the subintervals involve small differences in the Lagrangian. Feynman, while a Princeton research assistant during 1940–41, turned Dirac’s somewhat vague observations into a more definite formulation. He argued that the propagators G(x, y; s, t), which were Dirac’s “contact transformations,” can be obtained as limits over products of propagators at intermediary time points. This means that with a partition of [s, t] to subintervals [tj , tj+1 ] with tj = s + jb, j = 0, . . . , n − 1, b = (t − s)/n, the equality n−1

n−1

G(x, y; s, t) = lim ∏ ∫ G(xj , xj+1 ; tj , tj+1 ) ∏ dxj n→∞

j=0

ℝ3

j=1

1.1 Feynman path integrals and Feynman–Kac formulae

| 3

holds, where G(xj , xj+1 ; tj , tj+1 ) = √

xj+1 − xj m ib exp ( L (xj+1 , )). 2bπiℏ ℏ b

From here it is possible to conclude that i (e−(it/ℏ)H )(x, y) = G(x, y; 0, t) = const ∫ exp ( A (q, q;̇ 0, t)) ∏ dq(s), ℏ 0≤s≤t

(1.1.4)

Ωxy

giving a formal expression of the integral kernel of the operator e−(it/ℏ)H for all t. Here dq(s) is the Lebesgue measure, and Ωxy is the set of continuous paths q(s) such that q(0) = x and q(t) = y. The right-hand side of the expression is Feynman’s integral. These results have been obtained first in his 1942 thesis, which remained unpublished; the paper [117] published in 1948, however, reproduces the crucial statements (see also [120]). Feynman was able to apply his approach also to quantum field theory [118, 119]. For all its intuitive appeal and potential, Feynman’s method did not win immediate acclaim, not even in the ranks of theoretical physicists. Mathematically viewed, the object Feynman proposed was by no means easy to define. Mathematicians hitherto have not found a satisfactory meaning of Feynman integrals covering a sufficiently wide class of choices of V. For instance, there is no countably additive measure on Ωxy allocating equal weight to every path, and A (q, q;̇ 0, t) fails to be defined on much ̇ of Ωxy since it involves q(s). Various ways of trying to cope with these difficulties have been explored. Several authors considered purely imaginary potentials V and masses m leading to a complex measure to define Feynman’s integral, but these attempts had to face the problem of infinite total variation [56, 71, 274]. Another suggestion was to define the Feynman integral in terms of Fresnel distributions [87, 4, 88, 57, 61, 358]. Other attempts of constructing solutions of Schrödinger’s equation from Feynman’s integral were made through piecewise classical paths or broken lines or polygonal paths [127, 181, 185, 175, 360, 193]. In Feynman’s days integration theory on the space of continuous functions had already been in existence due to Wiener’s work on Brownian motion, initiated in 1923. It was Kac in 1949 who first realized that this was a suitable framework, however, not for the path integral (1.1.4) directly; see also [195, 196]. In contrast to Feynman’s expression, the Feynman–Kac formula offers an integral representation on path space for the semigroup e−(1/ℏ)tH instead of e−(i/ℏ)tH , obtained upon replacing t by −it. Why this should actually improve the situation can be seen by the following argument. Aṅ 2 󳨃→ −q(s) ̇ 2 in (1.1.4) alytic continuation s 󳨃→ −is, ds 󳨃→ −ids, and the replacement q(s) lead to the kernel t

t

2

(e−tH )(x, y) = const ∫ e−(1/ℏ) ∫0 V(q(s))ds e−(m/2ℏ) ∫0 q(s) ds ∏ dq(s). Ωxy

̇

0≤s≤t

(1.1.5)

4 | 1 Heuristics and history Since defining the latter factor as a measure remains to be a problem as ∫ ∏ dq(s) = ∞ Ωxy

(1.1.6)

0≤s≤t

for every t, one might boldly hope that its support 𝒞xy is a set on which the exponential weights typically vanish. If q̇ were almost surely plus or minus infinite on this set, then the situation t

1 m ̇ 2 ds)[ exp (− ∫ q(s) [ =0 ℏ 2 [ 0 [𝒞xy

(1.1.7)

would occur; 𝒞xy would thus be required to consist of continuous but nowhere differentiable paths. Once this can be allowed, zero in (1.1.7) can possibly cancel infinity in (1.1.6), and it would in principle be possible that t

1 m ̇ 2 ds) ∏ dq(s) = 0 ⋅ ∞ exp (− ∫ q(s) ℏ 2 0≤s≤t 0

is a well-defined measure. As we will explain below, the miracle happens: The paths of a random process (Bt )t≥0 called Brownian motion just have the required properties and the Feynman–Kac formula t

(e−tH )(x, y) =

∫ C(ℝ+ ,ℝ3 )

x,y

e− ∫0 V(Bs (ω))ds d𝒲[0,t] (ω),

t ≥ 0, x, y ∈ ℝ3 ,

(1.1.8)

rigorously holds provided some technical but highly satisfying conditions hold on V. In particular, there is a probability measure supported on the space 𝒞xy identified with the space C(ℝ+ , ℝ3 ) of continuous functions ℝ+ → ℝ3 , and it can be identified as Wiener measure 𝒲 conditional on paths leaving from x at time 0 and ending in y at time t. In other words, the kernel of the semigroup at the left-hand side (the object corresponding to G(x, y; 0, t) above) can be studied by running a Brownian motion weighted by the potential V. Kac has actually proven [194] that the heat equation with dissipation 𝜕f 1 = Δf − Vf , 𝜕t 2

ψ(x, 0) = ϕ(x),

(1.1.9)

is solved by the function f (x, t) =

t

∫ C(ℝ+ ,ℝ3 )

e− ∫0 V(Bs (ω)) ds ϕ(Bt (ω)) d𝒲 x (ω).

(1.1.10)

1.2 Plan and scope of the second edition

| 5

The above heat equation with dissipation is actually the same as the Schrödinger equation (1.1.1) when analytic continuation t 󳨃→ −it is made, a fact that has been noticed by Ehrenfest as early as in 1927. In particular, on setting V = 0 and formally replacing ϕ with a δ-distribution, the fundamental solution Π(x, t) of the standard heat equation is obtained. The relation of this so-called heat kernel Π(x, t) =

1 |x|2 exp (− ), 3/2 2t (2πt)

t > 0,

(1.1.11)

is once again close with random processes as it gives the transition probabilities of Brownian motion. The Feynman–Kac formula thus provides a link between an operator semigroup, the heat equation, and Brownian motion. This observation has opened up a whole new conceptual and technical framework pointing to and exploring the rich interplay between self-adjoint operators, probability, and partial differential equations.

1.2 Plan and scope of the second edition The Feynman–Kac formula for the operator H = − 21 Δ + V derived from the heat equation with dissipation can be written in any dimension d as (e−tH f )(x) =

t

∫

e− ∫0 V(Bs (ω)) ds f (Bt (ω)) d𝒲 x (ω),

C(ℝ+ ,ℝd )

where (Bt )t≥0 is accordingly an ℝd -valued Brownian motion. Due to the connections outlined above, the operator H is called a Schrödinger operator with potential V. If the potential is such that H admits an eigenfunction ψ at eigenvalue E, then e−tH ψ = e−Et ψ and thus this eigenfunction can be represented in terms of an average over the paths of Brownian motion given by the right-hand side. This makes it possible to obtain information on the spectral properties of H by probabilistic means. In particular, the Feynman–Kac formula can be used to study the ground states (i. e., eigenfunctions at the bottom of the spectrum) of Schrödinger operators to a great detail. It is natural to ask whether it is possible to derive representations similar to the Feynman–Kac formula for any other operator. In a companion volume we will show that also other operators, used in quantum field theory, allow to obtain Feynman– Kac-type formulae with appropriately modified random processes. On the one hand, this relation will allow to ask and answer questions relevant to quantum field theory in a rigorous way. On the other hand, we will face situations in which new concepts and methods need to be developed, which thus generates new mathematics. We split the material of this monograph in two volumes of comparable size, each focussing on one aspect explained above. In the present Volume 1 we discuss Feynman-Kac type representations for classical and nonlocal (relativistic and other)

6 | 1 Heuristics and history Schrödinger operators, their related random processes, and explore the wide-ranging applications of this relationship. In Volume 2 we carry out a similar programme for specific operators in quantum field theory. The present volume consists of six chapters, the last being a section of notes and commentary of the literature. In Chapter 1 we have outlined above the origins of the idea of path integrals applied in quantum mechanics and quantum field theory, due to the physicist Richard Ph. Feynman, and formulated in a mathematically rigorous way by the mathematician Mark Kac. In Chapter 2 we make preparatory steps to a rigorous development of these ideas, first by offering background material from probability, the theory of random processes, and stochastic analysis. We start from general measure theory, martingales and Markov processes, and a review of Brownian motion. We conclude by a section devoted to basic concepts of Itô integration theory. In Chapter 3 we give a similar survey of Lévy processes and discuss related concepts such as infinite divisibility, the Lévy–Khintchine representation formula, and Markov and martingale properties of jump processes. We continue with sample path properties, Poisson random measures, the canonical Lévy–Itô decomposition, and the Itô formula for semimartingales. Finally, we discuss exponentials of Lévy processes, capacitary measures, subordinate Brownian motion and related Bernstein functions. In Chapter 4 we start by a review of background material of the theory of selfadjoint operators and their semigroups, with a special focus on Schrödinger operators. We present established material in the light of our present goals and framework. The remainder of this main chapter deals then with a systematic presentation of variants of increasing complexity of the Feynman–Kac formula. We cover cases starting from an external scalar potential, and continue by including vector potentials describing interactions with magnetic fields, spin, and relativistic effects. Mathematically this means a range of operators consisting of the sum of the negative Laplacian and a multiplication operator of varying regularity, further including gradient operator terms and pseudodifferential operators such as the square root of the Laplacian and related operators. From the perspective of random processes involved, we proceed from Brownian motion to diffusions (P(ϕ)1 -processes and Itô diffusions) and subordinate Brownian motion (jump Lévy processes such as relativistic stable and other cases). Apart from first establishing the variants of the Feynman–Kac formula for these choices of operators and processes, we also address their applications in the study of analytic and spectral properties of these operators (properties of integral kernels, positivity improving properties, Lp -smoothing properties, intrinsic ultracontractivity, existence, uniqueness and multiplicity of eigenfunctions, decay of eigenfunctions, number of eigenfunctions with negative eigenvalues, diamagnetic inequalities, spectral comparison inequalities, etc). From the perspective of quantum mechanics, these results translate into the bound state properties of nonrelativistic or relativistic quantum particles under scalar or magnetic potentials, with or without spin. Finally, we present a class of

1.2 Plan and scope of the second edition

| 7

nonlocal Schrödinger operators given by Bernstein functions of the Laplacian, possibly including magnetic fields and generalized spin, and study their Feynman–Kac formulae and related properties. In Chapter 5 we focus on the “right-hand side” of the Feynman–Kac formula and its variants. In all of the cases that we consider, the kernel of the operator semigroup can be expressed in terms of an expectation over the paths of a random process with respect to a measure weighted by exponential densities dependent on functionals additive in the time intervals. This structure suggests to view this as a new probability measure which can be interpreted as a Gibbs measure on path space for the potential(s) appearing directly in or derived from the generator of the process. Gibbs measures are well understood in discrete stochastic models and originate from statistical mechanics, where they have been used to model thermodynamic equilibrium states. In our context this interpretation has another relevance, and our main focus here is to make use of the so obtained Gibbs measures in studying analytic and spectral properties of the operators at the left-hand side of the Feynman–Kac formula. In this chapter we define and prove the existence and properties of Gibbs measures on path space. First we present this for the case of external potentials and define the framework of P(ϕ)1 -processes. Then we establish existence and further fundamental results (uniqueness, almost sure path behavior, mixing properties, etc) for the case when the densities of the Gibbs measure also contain a pair interaction potential. In the concluding Chapter 6 we offer a commented guide to the selected bibliography, which we have extended and updated from the previous edition. Undoubtedly, our list of references could be further increased, and we trust that the interested reader will be able to find their way through further literature by relying on this list.

2 Brownian motion 2.1 Concepts and facts of general measure theory and probability 2.1.1 Elements of general measure theory For a more self-contained presentation of the material, we now present some fundamental facts often used in measure theory. Let Ω be a set. 2Ω denotes the power-set of Ω, i. e., the set of all its subsets. F ⊂ 2Ω is called a σ-field if it satisfies (1) Ω ∈ F ; (2) A ∈ F implies Ac ∈ F ; (3) An ∈ F , n ∈ ℕ, implies ⋃∞ n=1 An ∈ F . A pair (Ω, F ) consisiting of a non-empty set Ω and σ-field F is called a measurable space. When Ω is equipped with a topology, its Borel σ-field is defined by the minimal σ-field including all open sets of Ω and is denoted by ℬ(Ω). The pair (Ω, ℬ(Ω)) is called a Borel measurable space. A measurable space (Ω, F ) is separable if F is generated by a countable collection of subsets in Ω. In particular, if Ω is a locally compact space with a countable basis, then (Ω, ℬ(Ω)) is separable. We will assume throughout that Borel measurable spaces are separable. Let (Ω, F ) be a measurable space. A set function μ : F → [0, ∞) is called a measure if ∞

∞

n=1

n=1

μ( ⋃ An ) = ∑ μ(An ) holds for all sets An ∈ F , n ∈ ℕ, such that Ai ∩ Aj = 0 whenever i ≠ j. A set function μ : F → [0, ∞) is called a finitely additive measure if it satifies m

m

n=1

n=1

μ( ⋃ An ) = ∑ μ(An ) for any finitely many mutually disjoint sets An ∈ F , n = 1, . . . , m, m ∈ ℕ. A measurable space equipped with a measure μ is a measure space, denoted by (Ω, F , μ). If μ(Ω) < ∞, then μ is called a finite measure. If there exists a sequence An ∈ F , n = 1, 2, . . ., such that Ω = ⋃∞ n=1 An and μ(An ) < ∞ for all n ∈ ℕ, then μ is a σ-finite measure. Let μ be a measure on a Borel measurable space (Ω, ℬ(Ω)). The measure μ is called locally finite if for every point ω ∈ Ω there exists an open neighborhood U ∋ ω such that μ(U) < ∞. If A ⊂ 2Ω satisfies that (1) Ω ∈ A ; (2) A ∈ A implies Ac ∈ A ; (3) A1 , . . . , An ∈ A implies ⋃nm=1 Am ∈ A , https://doi.org/10.1515/9783110330397-002

10 | 2 Brownian motion then A is called a set-algebra or field. Whenever M ⊂ 2Ω satisfies that (1) for every inreasing sequence (An )n∈ℕ ⊂ M it follows that ⋃∞ n=1 An ∈ M ; (2) for every decreasing sequence (Bn )n∈ℕ ⊂ M it follows that ⋂∞ n=1 Bn ∈ M ,

then M is called a monotone family. Denote by σ(A ) the σ-field generated by A ⊂ 2Ω , and by m(A ) the monotone family generated by A . Proposition 2.1. If A is a set-algebra, then m(A ) = σ(A ). Proof. Since σ(A ) is a monotone family, σ(A ) ⊃ m(A ) follows. To prove that σ(A ) ⊂ m(A ), it is sufficient to show that m(A ) is a σ-field. We show that A, B ∈ m(A ) implies that A ∪ B, A \ B, B \ A ∈ m(A ). Define DA = {B ∈ 2Ω | A ∪ B, A \ B, B \ A ∈ m(A )}. It is easily seen that DA is a monotone family and m(A ) ⊂ DA for all A ∈ A . This implies that B ∈ DA for every B ∈ m(A ) and A ∈ A . Note that B ∈ DA if and only if A ∈ DB . Hence B ∈ DA for every A ∈ m(A ) and B ∈ A . We see that A ⊂ DA for every A ∈ m(A ), and m(A ) ⊂ m(DA ) = DA for all A ∈ m(A ). This shows that A, B ∈ m(A ) implies that A ∪ B, A \ B, B \ A ∈ m(A ). Hence we furthermore have that A ∈ m(A ) implies Ac ∈ m(A ), and Aj ∈ m(A ), j ∈ ℕ, implies that ⋃nj=1 Aj ∈ m(A ) for every ∞ n ∈ ℕ. Since ⋃nj=1 Aj ↑ ⋃∞ j=1 Aj as n → ∞, we have ⋃j=1 Aj ∈ m(A ). Thus m(A ) is a σ-field. A ⊂ 2Ω is called a π-system if n

A1 , . . . , An ∈ A implies ⋂ Am ∈ A . m=1

Furthermore, L ⊂ 2Ω is called a λ-system if (1) ϕ ∈ L ; (2) A ∈ L implies Ac ∈ L ; (3) An ∈ L , n ∈ ℕ, and An ∩ Am = 0 for n ≠ m implies ⋃∞ n=1 An ∈ L . Let λ(I ) denote the λ-system generated by I ⊂ 2Ω . Proposition 2.2 (π-λ theorem). Let I be a π-system. Then the following hold: (1) σ(I ) = λ(I ). (2) If L is a λ-system and L ⊃ I , then L ⊃ σ(I ). Proof. Since σ(I ) is a λ-system, σ(I ) ⊃ λ(I ) follows. To show σ(I ) ⊂ λ(I ), it suffices to see that λ(I ) is a σ-field. Let D ⊂ 2Ω be a λ-system. It is straightforward to check that D is a π-system if and only if D is a σ-field. Thus it is sufficient to show that λ(I ) is a π-system, i. e., that A, B ∈ λ(I ) implies that A ∩ B ∈ λ(I ). Define the subset

2.1 Concepts and facts of general measure theory and probability | 11

DB = {A ∈ λ(I ) | A ∩ B ∈ λ(I )}. We see that λ(I ) is a π-system if and only if λ(I ) ⊂ DB for every B ∈ λ(I ). We note that DE is a λ-system for every E ∈ λ(I ). Since I is a π-system, we have I ⊂ DE for all E ∈ I . This implies λ(I ) ⊂ λ(DE ) = DE for every E ∈ I . From this it follows that B ∩ E ∈ λ(I ) for every B ∈ λ(I ) and E ∈ I . Hence I ⊂ DB and λ(I ) ⊂ DB for every B ∈ λ(I ), which proves (1). Let L be a λ-system and L ⊃ I . We have L = λ(L ) ⊃ λ(I ) = σ(I ). Thus (2) follows. Part (1) of Proposition 2.2 is known as the π-λ theorem. The following is a useful extension theorem of finitely additive measures. Proposition 2.3 (Hopf’s extension theorem). Let A ⊂ 2Ω be a set-algebra, and the set function μ : A → [0, ∞) be a finitely aditive measure. Then there exists a measure μ̄ on σ(A ) such that ̄ μ(A) = μ(A),

A∈A,

if and only if An ∈ A , n = 1, 2, . . ., are mutually disjoint, and ⋃∞ n=1 An ∈ A implies that ∞

∞

n=1

n=1

μ( ⋃ An ) = ∑ μ(An ). In addition, if μ is σ-finite, then the extension μ̄ is unique. We note that the condition An ∈ A , n = 1, 2, . . ., are mutually disjoint, and ⋃∞ n=1 An ∈ A ∞ ∞ implies that μ(⋃n=1 An ) = ∑n=1 μ(An ) above is equivalent to having limn→∞ μ(An ) = 0 for every decreasing sequence (An )n∈ℕ ⊂ A such that ⋂∞ n=1 An = 0. A ⊂ 2Ω is called a semi-ring if (1) 0 ∈ A ; (2) for every A, B ∈ A , the set B \ A is a finite union of mutually disjoint sets in A ; (3) A, B ∈ A implies A ∩ B ∈ A . Let A be a semi-ring and A ∈ σ(A ). Lemma 2.4 below shows that any set from σ(A ) can be well approximated by sets obtained from finite operations involving sets in A . Denote by A △ B = (A \ B) ∪ (B \ A) the symmetric difference of A and B. Lemma 2.4 (Approximation of measures). Let Ω be a given set, A ⊂ 2Ω be a semi-ring, and μ a σ-finite measure on σ(A ). Then the following hold. (1) For every A ∈ σ(A ) with μ(A) < ∞ and ε > 0, there exist k ∈ ℕ and mutually disjoint sets A1 , . . . , Ak ∈ A such that μ(A △ ⋃kn=1 An ) < ε. (2) For every A ∈ σ(A ) and ε > 0, there exist mutually disjoint sets {An }n∈ℕ ⊂ A such ∞ that A ⊂ ⋃∞ n=1 An and μ(⋃n=1 An \ A) < ε.

12 | 2 Brownian motion Proof. (1) Let ε > 0. By the definition of the measure μ, there exists a covering ∞ B1 , B2 , . . . ∈ A such that A ⊂ ⋃∞ n=1 Bn and μ(A) ≥ ∑n=1 μ(Bn ) − ε/2. Since μ(A) < ∞, there ∞ exists m such that ∑n=m+1 μ(Bn ) < ε/2. In general, for any sets C, D, and E, we have ∞ C △ D ⊂ (C △ (D ∪ E)) ∪ E. Choosing C = A, D = ⋃m n=1 Bn and E = ⋃n=m+1 Bn , we have m

∞

∞

∞

n=1

n=1

n=m+1

n=1

μ(A △ ⋃ Bn ) ≤ μ(A △ ⋃ Bn ) + μ( ⋃ Bn ) ≤ μ( ⋃ Bn ) − μ(A) +

ε ≤ ε. 2

m n−1 n−1 We see that ⋃m n=1 Bn = B1 ∪ ⋃n=2 ⋂j=1 (Bn \ Bj ). Note that Cn = ⋂j=1 (Bn \ Bj ), n = 2, . . . , m, are disjoint sets. Since A is a semi-ring, the set Bn \Bj can be obtained as a finite union of mutually disjoint sets of A , and there exist a finite number k ∈ ℕ and mutually k k disjoint sets A1 , . . . , Ak ∈ A such that ⋃m n=1 Bn = ⋃n=1 An . Then μ(A △ ⋃n=1 An ) < ε follows.

(2) Let A ∈ σ(A ) and En ↑ Ω such that En ∈ σ(A ) and μ(En ) < ∞, for all n ∈ ℕ. For each n n we can choose a covering (Bn,m )m∈ℕ of A ∩ En with μ(A ∩ En ) ≥ ∑∞ m=1 μ(Bn,m ) − ε/2 . Moreover, it is also possible to show that there exist An ∈ A , n ∈ ℕ, such that A1 , A2 , . . . ∞ are mutually disjoint and ⋃∞ n,m=1 Bn,m = ⋃n=1 An . Thus ∞

∞ ∞

∞ ∞

n=1

n=1 m=1 ∞ ∞

n=1 m=1

μ( ⋃ An \ A) = μ( ⋃ ⋃ Bn,m \ A) ≤ μ( ⋃ ⋃ (Bn,m \ (A ∩ En ))) ≤ ∑ (( ∑ μ(Bn,m )) − μ(A ∩ En )) ≤ ε. n=1

m=1

Hence (2) is proven. With measurable spaces (Ω, F ) and (S, S ) we write f ∈ F /S to indicate that the function f : Ω → S is measurable with respect to F and S , i. e., f −1 (E) ∈ F ,

E ∈ S.

Whenever f ∈ F /ℬ(ℝd ), f is called a Borel measurable function. Two measurable spaces (Ω, F ) and (Ω󸀠 , F 󸀠 ) are said to be Borel isomorphic if there exists a bijection f : Ω → Ω󸀠 such that f ∈ F /F 󸀠 and f −1 ∈ F 󸀠 /F . A measurable space (Ω, F ) is called a standard measurable space if it is Borel isomorphic to one of the measurable spaces ({1, . . . , n}, ℬ({1, . . . , n})),

(ℕ, ℬ(ℕ)),

(𝕄, ℬ(𝕄)),

where 𝕄 = {0, 1}ℕ = {ω = {ω1 , ω2 , . . .} | ωj = 0, 1}. The sets {1, . . . , n} and ℕ are understood to be endowed with the discrete topology, and 𝕄 with the product topology. We define the integral of a Borel measurable function f on (Ω, F , μ), which will be denoted by any of the symbols ∫ fdμ = ∫ f (x)dμ(x) = ∫ f (x)dμ. Ω

Ω

Ω

2.1 Concepts and facts of general measure theory and probability | 13

Next we review the convergence properties of measures on a metric space. Let S be a metric space. A σ-finite Borel measure μ on (S, ℬ(S)) is called a Radon measure if it is locally finite and inner regular, i. e., μ(A) = sup{μ(K) | K ⊂ A and K is compact} for all A ∈ ℬ(S). Note that since μ is locally finite, μ(K) < ∞ for every compact set K. Definition 2.5 (Weak convergence and vague convergence). Let S be a metric space. (1) A sequence of finite measures (μn )n∈ℕ on (S, ℬ(S)) is weakly convergent to a finite measure μ whenever lim ∫ fdμn = ∫ fdμ,

n→∞

S

(2.1.1)

S

for every bounded continuous function f on S. (2) A sequence of Radon measures (μn )n∈ℕ on (S, ℬ(S)) is vaguely convergent to a Radon measure μ whenever lim ∫ fdμn = ∫ fdμ,

n→∞

S

(2.1.2)

S

for every continuous function f with compact support on S. In general, a sequence of finite measures (μn )∈ℕ may not converge to a finite measure in vague topology. A sufficient condition is the following. Suppose that supn∈ℕ μn (S) ≤ m and μn vaguely converges to μ as n → ∞. Since μ is inner regular, for every ε > 0 there exists a compact set Kε such that μ(S \ Kε ) < ε. We have μ(Kε ) = limn→∞ μn (Kε ) ≤ m and μ(S) ≤ m + ε, for every ε > 0. In particular, μ is finite and μ(S) ≤ m. We note that even though μn (S) = m for all n, in general μ(S) ≠ m. Let Pf (S) be the set of finite measures on (S, ℬ(S)), and PR (S) the set of Radon measures on (S, ℬ(S)). Weak convergence induces the weak topology τw on Pf (S), which is the weakest topology such that for every bounded continuous function f , the map Pf (S) ∋ Q 󳨃→ ∫S fdQ ∈ ℝ is continuous. It is known that if S is separable, (Pf (S), τw ) is metrizable. Similarly, vague convergence induces the vague topology τv on PR (S), which is the weakest topology such that for every continuous function f with compact support, the map PR (S) ∋ Q 󳨃→ ∫S fdQ ∈ ℝ is continuous. It is also known that if S is locally compact, then (PR (S), τv ) is a Hausdorff space. The following is an example of a result on vague convergence, known as Helly’s second theorem or Helly’s selection theorem. Consider a sequence of finite measures (μn )n∈ℕ on the measurable space (ℝd , ℬ(ℝd )). If supn∈ℕ μn (ℝd ) < ∞, then (μn )n∈ℕ is called uniformly bounded.

14 | 2 Brownian motion Proposition 2.6 (Helly’s second theorem). Let (μn )n∈ℕ be uniformly bounded on the measurable space (ℝd , ℬ(ℝd )). Then there exist a subsequence (μnk )nk ∈ℕ and a measure μ on (ℝd , ℬ(ℝd )) such that lim ∫ f (x)μnk (dx) = ∫ f (x)μ(dx),

nk →∞

ℝd

ℝd

for every continuous function f with compact support. Proof. We give an outline of the proof. Let I = {(a1 , b1 ) × ⋅ ⋅ ⋅ × (ad , bd ) | aj , bj ∈ ℚ, j = 1, . . . , d} be the Cartesian product of d intervals whose end-points are rational numbers. Let M = ℚd , which is clearly a countable set and its elements can be enumerated as M = {m1 , m2 , . . .}. Define the function Fn on ℝd by Fn (x) = Fn (x1 , . . . , xd ) = ∫ 1(−∞,x1 ]×⋅⋅⋅×(−∞,xd ] μn (dx). ℝd

Fn is called the distribution function of μn . Consider the restriction of Fn on M. Since {Fn (m1 )}n∈ℕ is a bounded set of ℝ, there exists a subsequence Fn1 (m1 ) convergent to the real number F(m1 ). From Fn1 (m2 ) we can select a subsequence Fn2 (m2 ) which converges to F(m2 ) and so on. Repeating this procedure, by a diagonal argument we see that Fnn (mk ) converges to the real number F(mk ) as n → ∞, for every mk ∈ M. For x ∈ ℝd , define ̄ F(x) =

sup

ξ1 ≤x1 ,...,ξd ≤xd ,(ξ1 ,...,ξd )∈M

F(ξ ).

Thus it is seen that there exists a measure μ such that ̄ F(x) = ∫ 1(−∞,x1 ]×⋅⋅⋅×(−∞,xd ] μ(dx) ℝd

and μnn (A) → μ(A) for every A ∈ I. This is proven in a similar way to Corollary 2.46 below. Thus it follows that ∫ 1A (x)μnn (dx) → ∫ 1A (x)μ(dx), ℝd

A ∈ I,

ℝd

as n → ∞. The proof can be completed by a limiting argument. Recall that a complete and separable metric space is called a Polish space.

2.1 Concepts and facts of general measure theory and probability | 15

Example 2.7 (Compact case). Let (M, d) be a compact metric space, and C(M) denote the set of continuous functions on M. Define the metric ρ(f , g) = max{|f (m) − g(m) | m ∈ M} on C(M). Then (C(M), ρ) is a Polish space. Example 2.8 (Non-compact case). Let C(ℝ) (resp. C([0, ∞))) be the set of continuous functions on ℝ (resp. [0, ∞)). We introduce the locally uniform topology given by the metric 1 {( sup |f (x) − g(x)|) ∧ 1} . k 2 0≤|x|≤k k=0 ∞

d(f , g) = ∑

(2.1.3)

C(ℝ) and C([0, ∞)) are Polish spaces under this metric. We discuss some properties of measures on Polish spaces, which will be useful in the study of path measures. In general it is known that a σ-finite measure ν on a metric space S endowed with a Borel σ-field ℬ(S) is a regular measure, i. e., for arbitrary A ∈ ℬ(S) and ε > 0 there exist a closed set K and an open set U such that K ⊂ A ⊂ U and ν(U \ K) < ε. Lemma 2.9. Let (S, d) be a Polish space with metric d, and μ a measure on (S, ℬ(S)). Take A ∈ ℬ(S) such that μ(A) < ∞. Then for every ε > 0 there exists a compact set K ⊂ A such that μ(A \ K) < ε. Proof. Let μA (⋅) = μ(⋅∩A). The measure μA is finite and thus σ-finite. Hence there exists a closed set K ⊂ A such that μA (A \ K) = μ(A \ K) < ε. Restricting the metric d on K, we see that (K, d) is also a Polish space. Let {an }n∈ℕ ⊂ K ̄ be a countable dense subset of K. Hence for every k ≥ 1 we have ⋃∞ n=1 B1/k (an ) = K, ̄ where Br (a) denotes the closed ball centered in a with radius r. It can be seen that limN→∞ μ(K \ ⋃Nn=1 B̄ 1/k (an )) = 0. There exists Nk such that N

k ε μ(K \ ⋃ B̄ 1/k (an )) < k . 2 n=1

Nk ̄ Let F = ⋂∞ k=1 ⋃n=1 B1/k (an ) ⊂ K, which is closed and precompact. Since S is complete, so is F. This yields that F is compact and ∞

Nk

k=1

n=1

μ(K \ F) ≤ ∑ μ(K \ ⋃ B̄ 1/k (an )) ≤ ε. It is then straightforward that μ(A \ F) ≤ μ(A \ K) + μ(K \ F) ≤ 2ε.

16 | 2 Brownian motion We complete this section by recalling Fourier transform, which is an important tool in the analysis of measures. The Schwartz space S (ℝd ) is the set of infinitely differentiable complex-valued functions f (x) on ℝd for which sup |x α Dβ f (x)| < ∞

x∈ℝd

for all α = (α1 , . . . , αd ) and β = (β1 , . . . , βd ). Here αi and βj are nonnegative integers, and α

α

xα = x1 1 ⋅ ⋅ ⋅ xdd and Dβ =

𝜕|β|

β β 𝜕x1 1 ⋅⋅⋅𝜕xdd

with |β| = ∑dj=1 βj . Schwartz space is a dense subspace

of L2 (ℝd ). The Fourier transform F : S (ℝd ) → S (ℝd ) is a bijective linear map defined by Ff (k) =

1 ∫ f (x)e−ik⋅x dx. (2π)d/2 ℝd

For convenience, we denote the Fourier transform Ff (k) of f by f ̂(k). The inverse Fourier transform is given by F −1 f (x) =

1 ∫ f (k)e+ik⋅x dk, (2π)d/2 ℝd

which we will denote by f ̌(x). We have FF −1 = F −1 F = I on S (ℝd ) and (Ff , Fg) = (f , g), where (f , g) denotes the scalar product on L2 (ℝd ). Since S (ℝd ) is dense in L2 (ℝd ), Fourier transform F can be extended to a unitary operator F̄ on L2 (ℝd ). We use the same notation F instead of F̄ in what follows.

2.1.2 Probability measures and limit theorems A probability space is a measure space (Ω, F , P) such that P(Ω) = 1, in which Ω is called sample space, F algebra of events, and P a probability measure on (Ω, F ). The probability P({ω ∈ Ω | conditions}) of an event defined in terms of some conditions will be simply denoted as P(conditions). An event is said to be almost sure with respect to P if it has probability one under P. If a property holds for every ω in an almost sure event, then we say that it holds almost surely and we write P-a. s., or simply a. s.

2.1 Concepts and facts of general measure theory and probability | 17

Let (Ω, F ) be a measurable space. F is called countably determined if there exists a countable subset G ⊂ F such that if any two probability measures agree on G , then they coincide. Two events A, B ∈ F are called independent if P(A ∩ B) = P(A)P(B). Two sub-σ-fields F1 , F2 ⊂ F are independent if all A ∈ F1 and B ∈ F2 are pairwise independent. We can formalize the concept of an event occurring infinitely often by the so-called Borel–Cantelli lemma. Theorem 2.10 (Borel–Cantelli lemma). Let (Ω, F , P) be a probability ∞ A1 , A2 , . . . ∈ F , and write Ā = ⋂∞ m=1 ⋃n=m An . ∞ ̄ (1) If ∑n=1 P(An ) < ∞, then P(A) = 0. ̄ (2) If ∑∞ n=1 P(An ) = ∞ and {An }n∈ℕ are independent events, then P(A) = 1.

space,

Proof. Note that BN = ⋂Nm=1 ⋃∞ n=m An , N = 1, 2, . . ., are decreasing sets as N increases and limN→∞ BN = A.̄ Thus ∞

∞

P(A)̄ = lim P(BN ) = lim P( ⋃ An ) ≤ lim ∑ P(An ) = 0. N→∞

N→∞

N→∞

n=N

n=N

c ⋃Nm=1 ⋂∞ n=m An ,

Hence (1) follows. On the other hand, CN = N = 1, 2, . . ., are increasing sets as N increases and limN→∞ CN = Ā c . Thus we see that ∞

P(Ā c ) = lim P(CN ) = lim P( ⋂ Acn ). m→∞

N→∞

n=m

By the independence of An , we furthermore have ∞

k

k

k

P( ⋂ Acn ) = lim ∏ (1 − P(An )) = lim e∑n=m log(1−P(An )) ≤ lim e− ∑n=m P(An ) = 0. n=m

k→∞ n=m

k→∞

k→∞

Thus (2) follows. The Borel–Cantelli lemma provides a zero–one law for independent events. Next we discuss another zero–one law for independent σ-fields, using the following concept. Definition 2.11 (Tail σ-field). Let I be a countable infinite index set and {Fn }n∈I a family of σ-fields. The σ-field 𝒯 ({Fn }n∈I ) = ⋂ σ ( ⋃ Fj ) J⊂I #J 0. By Lemma 2.4 there exist mutually disjoint sets F1 , . . . , FN ∈ B such that

P(A △ (F1 ∪ ⋅ ⋅ ⋅ ∪ FN )) < ε.

(2.1.8)

There exists n ∈ ℕ such that F1 , . . . , FN ∈ Bn and hence F1 ∪ ⋅ ⋅ ⋅ ∪ FN ∈ σ(F1 ∪ ⋅ ⋅ ⋅ ∪ Fn ). Obviously, A ∈ σ(⋃∞ m=n+1 Fm ), hence A is independent of F1 ∪ ⋅ ⋅ ⋅ ∪ Fn . Thus ε > P(A \ (F1 ∪ ⋅ ⋅ ⋅ ∪ Fn )) = P(A ∩ (F1 ∪ ⋅ ⋅ ⋅ ∪ Fn )c )

= P(A)(1 − P(F1 ∪ ⋅ ⋅ ⋅ ∪ Fn )) ≥ P(A)(1 − P(A) − ε).

Letting ε ↓ 0 yields P(A)(1 − P(A)) = 0, and the theorem follows. There are useful characterizations of weak convergence of probability measures on a metric space (S, d). Let Cb (S) and Cu (S) be the set of bounded continuous functions on S and the set of uniformly continuous functions on S, respectively. Lemma 2.14. Let S be a metric space and P, P1 , P2 , . . . be a sequence of probability measures on (S, ℬ(S)). Then the following are equivalent.

2.1 Concepts and facts of general measure theory and probability | 19

(1) (2) (3) (4)

(Pn )n∈ℕ is weakly convergent to P as n → ∞. limn→∞ 𝔼Pn [f ] = 𝔼P [f ], for all f ∈ Cb (S) ∩ Cu (S). For every closed set F ⊂ S, lim supn→∞ Pn (F) ≤ P(F). For every open set U ⊂ S, lim infn→∞ Pn (U) ≥ P(U).

Proof. (1) 󳨐⇒ (2) is trivial. We show (2) 󳨐⇒ (3). Let f ∈ Cb (ℝ) be defined by 1, { { f (t) = {1 − t, { {0,

t ≤ 0, 0 < t ≤ 1, t > 1.

For a closed set F ⊂ S and k > 0, define ϕk (x) = f (kd(x, F)), where d(x, F) denotes the distance between x and F. We see that limk→∞ ϕk (x) = 1F (x). Hence ϕk ∈ Cb (S) ∩ Cu (S) and ϕk ≥ 1F . It follows then that 𝔼Pn [ϕk ] ≥ Pn (F). Letting n → ∞ yields 𝔼P [ϕk ] ≥ lim supn→∞ Pn (F) by (2). Letting k → ∞ again yields (3). It is clear that (3) and (4) are equivalent. We show (3) 󳨐⇒ (1). Let f ∈ Cb (S) and suppose that 0 ≤ f (x) < 1 without loss of generality. Fix k ∈ ℕ. Define sets Fi = {x ∈ S | f (x) ≥ i/k} for 0 ≤ i ≤ k; Fi is closed and a decreasing sequence. It is direct to check that k

∑ i=1

k i−1 i Pn (Fi−1 \ Fi ) ≤ 𝔼Pn [f ] ≤ ∑ Pn (Fi−1 \ Fi ). k k i=1

From this we deduce that 1 k 1 1 k ∑ Pn (Fi ) ≤ 𝔼Pn [f ] ≤ + ∑ Pn (Fi ). k i=1 k k i=1

(2.1.9)

Letting n → ∞ on both sides, we see that by (3), lim sup 𝔼Pn [f ] ≤ n→∞

1 1 k + ∑ P(Fi ). k k i=1

(2.1.10)

Since (2.1.9) is also valid for Pn replaced by P, we have 1 1 k 1 + ∑ P(Fi ) ≤ + 𝔼P [f ]. k k i=1 k By (2.1.10)–(2.1.11) we then have lim sup 𝔼Pn [f ] ≤ n→∞

1 + 𝔼P [f ]. k

Taking k → ∞, we have lim sup 𝔼Pn [f ] ≤ 𝔼P [f ]. n→∞

Replacing f by 1 − f , we obtain (1).

(2.1.11)

20 | 2 Brownian motion Let (M, d) be a compact metric space. Define P (M) = {P | P is a probability measure on (M, ℬ(M))}.

Let {fn }n∈ℕ be a dense subset of C(M)+ , i. e., the set of positive functions in C(M). We define τ : P (M) × P (M) → [0, ∞) by 1 󵄨󵄨 󵄨 ( 𝔼 [f ] − 𝔼Q [fn ]󵄨󵄨󵄨 ∧ 1) . n 󵄨󵄨 P n 2 n=1 ∞

τ(P, Q) = ∑ This gives a metric on P (M).

Lemma 2.15. Let (P (M), τ) be as above. Then (P (M), τ) is a compact metric space. Proof. We show compactness. Let f ∈ C(M) and {fn }n∈ℕ be a dense subset of C(M)+ . Let Pn ∈ P (M), write 𝔼Pn [f ] = ln (f ), and consider an,m = ln (fm ). Fix m ≥ 1. 0 ≤ an,m ≤ ‖fm ‖, where ‖f ‖ = ρ(f , 0). Then (an,m )n∈ℕ is a bounded sequence for each m. By a diagonalization argument we can find a subsequence n1 < n2 < . . . such that ank ,m → am as nk → ∞ for each m. We have |lni (f ) − lnj (f )| ≤ |lni (f ) − lni (fm )| + |lni (fm ) − lnj (fm )| + |lnj (fm ) − lnj (f )| ≤ 2‖f − fm ‖ + |ani ,m − anj ,m |.

(2.1.12)

For ε > 0 we can take m such that ‖f − fm ‖ ≤ ε. By (2.1.12) it follows that (lni (f ))∞ i=1 is a Cauchy sequence, and thus there exists limni →∞ lni (f ) = l(f ). Hence l(f ) satisfies l(1) = 1 and |l(f )| ≤ ‖f ‖. We conclude that l : C(M)+ → ℝ is a positive linear functional. By the Riesz–Markov representation theorem there exists a Baire measure P on M such that l(f ) = ∫ f (m)dP. M d

Furthermore, P ∈ P (M) and Pni → P follow. Definition 2.16 (Tightness). Let S be a metric space and P a probability measure on (S, ℬ(S)). (1) A family P of probability measures on (S, ℬ(S)) is relatively compact whenever every sequence of elements of P contains a weakly convergent subsequence. (2) A family P of probability measures on (S, ℬ(S)) is tight whenever for every ε > 0 there exists a compact set Kε ⊂ S such that P(Kε ) > 1 − ε, for every P ∈ P . Theorem 2.17 (Prokhorov’s theorem). Let S be a Polish space. A family of probability measures P on (S, ℬ(S)) is relatively compact if and only if it is tight.

2.1 Concepts and facts of general measure theory and probability | 21

Proof. Suppose that P is tight. Since the metric space (S, d) is separable, there exists a metric d̃ on S such that d and d̃ give the same topology, and (S, d)̃ is totally bounded. Thus we may assume that (S, d) is totally bounded. We denote the completion of (S, d) ̂ i. e., we have a map i : S → Ŝ such that by (S,̂ d), ̂ (1) d(i(x), i(y)) = d(x, y); (2) i(S) is dense in S;̂ (3) (S,̂ d)̂ is complete. (S,̂ d)̂ is totally bounded and complete, which is equivalent to compactness. Define the image measure on Ŝ by P̂ = P ∘ i−1 , P ∈ P , and denote the set of image measures induced by i on Ŝ by P̂ . We denote then the set of probability measures on Ŝ by Q . P̂ ⊂ Q . (Q , τ) is a compact metric space by Lemma 2.15. Hence P̂ is precompact. d Thus there exist (Pn )n∈ℕ ⊂ P and Q ∈ Q such that P̂ n → Q as n → ∞. Since P is tight, for arbitrary N ≥ 1 there exists a compact set KN ⊂ S such that Pn (KN ) ≥ 1 − 1/N holds for all n ≥ 1. Note that i(KN ) ⊂ Ŝ is also compact since i is a continuous map. We can see by Lemma 2.14 that 1 Q(i(KN )) ≥ lim sup P̂ n (i(KN )) = lim sup Pn (KN ) ≥ 1 − , N n→∞ n→∞

(2.1.13)

where we used P̂ n (i(KN )) = Pn (KN ), which follows by the fact that i is injective. Write S0 = ⋃∞ N=1 KN . It follows from (2.1.13) that Q(i(S0 )) ≥ 1, which implies that Q(i(S0 )) = 1. Define the probability measure P on S by P(A) = Q(i(A ∩ S0 )). For a bounded and uniformly continuous function f on S, we have ̂ ∫ f (ω)dP = ∫ f (ω)dP = ∫ f ̂(ω)dQ. S

S0

(2.1.14)

Ŝ

Here f ̂ is a continuous function on Ŝ defined in the following way. Let ω̂ ∈ S.̂ There exists a sequence (ωn )n∈ℕ ⊂ S such that i(ωn ) → ω̂ as n → ∞. Since f is uniformly continuous, (f (ωn ))n∈ℕ is a Cauchy sequence. Thus there exists the limit a = limn→∞ f (ωn ) ∈ ℂ. We define f ̂(ω)̂ = a. In particular f ̂(i(ω)) = f (ω) for all ω ∈ S. It is also seen that f ̂ is continuous and f ̂ satisfying (2.1.14) is unique. By (2.1.14) we have ̂ P̂ n → ∫ f (ω)dP = ∫ f ̂(ω)dQ. ̂ ∫ f (ω)dPn = ∫ f ̂(ω)d S

Ŝ

S

Ŝ

Hence P is relatively compact. Next suppose that P is relatively compact. Let {ωn }n∈ℕ ⊂ S be a countable dense set. Define An,N = ⋃Ni=1 B̄ 1/n (ωi ), where B̄ 1/n (ω) denotes the closed ball of radius 1/n centered at ω. An,N ↑ S as N → ∞ for all n. Define δ = sup inf sup P(Acn,N ). n∈ℕ N∈ℕ P∈P

22 | 2 Brownian motion Thus there exists a natural number n such that for every N there is PN ∈ P , satisfying PN (Acn,N ) ≥ δ/2. Since S is precompact, (PN )N∈ℕ contains a weakly convergent subsequence (PNk )k∈ℕ , whose weak limit we denote by P. We have by Lemma 2.14, P(Acn,N ) ≥ lim inf PNk (Acn,N ) ≥ lim inf PNk (Acn,Nk ) ≥ k→∞

k→∞

δ . 2

Since Acn,N ↓ 0 as N → ∞, limN→∞ P(Acn,N ) = 0 follows. Hence δ = 0. Fix ε > 0; then δ = 0 implies that for every n we can choose natural number Nn󸀠 such that P(Acn,N 󸀠 ) < n ε/2n , for all P ∈ P . Note that A = ⋂∞ An,N 󸀠 ⊂ S is totally bounded. So the closure Ā of A is compact. For every P ∈ P ,

n=1

n

∞

P((A)̄ c ) ≤ P(Ac ) ≤ ∑ P(Acn,N 󸀠 ) ≤ ε. n=1

n

Hence P(A)̄ = P(S) − P((A)̄ c ) ≥ 1 − ε and P is tight. We introduce on C(ℝ) or C([0, ∞)) the locally uniform topology given by the metric (2.1.3). We will be mainly interested in the cases when S is one of these two function spaces. Since both are Polish spaces under the locally uniform topology generated by (2.1.3), any family of probability measures on (S, ℬ(S)) has a weakly convergent sequence if it is tight. Take S = C(ℝ). Recall that for T, δ > 0 the modulus of continuity of ω ∈ S is defined by mT (ω, δ) = max |ω(s) − ω(t)|. |s−t|≤δ −T≤s,t≤T

(2.1.15)

For S = C([0, ∞)) the modulus of continuity is defined by replacing −T by 0. Lemma 2.18. A set A ⊂ C(ℝ) has a compact closure in the locally uniform topology if and only if the following two conditions hold: sup |ω(0)| < ∞,

(2.1.16)

lim sup mT (ω, δ) = 0.

(2.1.17)

ω∈A

δ↓0 ω∈A

Proof. First we show that (1) ω 󳨃→ mT (ω, δ) is continuous in the locally uniform topology; (2) δ 󳨃→ mT (ω, δ) is non-decreasing; (3) limδ↓0 mT (ω, δ) = 0. Suppose that on the interval [−T, T] the sequence (ωn )n∈ℕ is uniformly convergent to ω as n → ∞. Let sups∈[−T,T] |ωn (s) − ω(s)| ≤ ε. We see that mT (ω, δ) ≤ max (|ω(s) − ωn (s)| + |ωn (s) − ωn (t)| + |ωn (t) − ω(t)|) ≤ 2ε + mT (ωn , δ). |s−t|≤δ −T≤s,t≤T

2.1 Concepts and facts of general measure theory and probability | 23

Similarly, mT (ωn , δ) ≤ 2ε + mT (ω, δ). limn→∞ mT (ωn , δ) = mT (ω, δ) follows. Part (2) is trivial, and (3) follows from the fact that t 󳨃→ ω(t) is uniformly continuous in [−T, T]. Assume that the closure Ā of A is compact. Since Ā is contained in the union of the open sets Gn = {ω ∈ C(ℝ) | |ω(0)| < n}, n = 1, 2, . . ., A is contained in a specific Gn . Thus (2.1.16) follows. For ε > 0, let Kδ = {ω ∈ Ā | mT (ω, δ) ≥ ε}. By the continuity of mT (ω, δ) on ω, each Kδ is closed and Kδ ⊂ A.̄ Thus Kδ is compact. Since limδ↓0 mT (ω, δ) = 0, we have ⋂δ>0 Kδ = 0. This implies Kδε = 0 for some δε > 0, and thus (2.1.17) follows. Next assume (2.1.16)–(2.1.17). We prove that Ā is compact; it suffices to show that every sequence (ωn )n∈ℕ ⊂ A has a convergent subsequence. Fix T > 0, and note that for some δ1 > 0 we have mT (ω, δ1 ) ≤ 1, for all ω ∈ A by (2.1.17). For a fixed integer m ≥ 1 and t ∈ [−T, T] \ {0} with (m − 1)δ1 < t ≤ mδ1 ∧ T, we have m−1

|ω(t)| ≤ |ω(0)| + ∑ |ω(kδ1 ) − ω((k − 1)δ1 )| + |ω(t) − ω((m − 1)δ1 )| ≤ |ω(0)| + m. k=1

If r ∈ ℚ∩[−T, T], (ωn (r))n∈ℕ is bounded for every r. Let {r0 , r1 , . . .} be the enumeration of (0) ℚ ∩ [−T, T]. Choose (ω(0) n )n∈ℕ such that ωn (r0 ) converges to a limit denoted by ω(r0 ). (1) (1) From (ω(0) n )n∈ℕ , choose a further subsequence (ωn )n∈ℕ such that ωn (r1 ) converges to a limit denoted by ω(r1 ). Continuing this process we obtain a diagonal sequence (n) (n) (ω(n) n )n∈ℕ . Hence ωn (r) → ω(r) for all r ∈ ℚ ∩ [−T, T]. Put ωn = ω̃ n . By (2.1.17), for every ε > 0 there exists δ(ε) > 0 such that |ω̃ n (s) − ω̃ n (t)| ≤ ε, whenever −T ≤ s, t ≤ T, and |s − t| ≤ δ(ε). The same inequality holds for ω with |ω(s) − ω(t)| ≤ ε, whenever s, t ∈ [−T, T] ∩ ℚ and |s − t| ≤ δ(ε). Then ω can be extended to a continuous function on [−T, T], denoted by ω.̄ We have ω̄ ∈ C([−T, T]). Furthermore, it follows ̄ − ω(t)| ̄ that |ω(s) ≤ ε whenever −T ≤ s, t ≤ T, and |s − t| ≤ δ(ε). For n sufficiently large, whenever t ∈ [−T, T] there is some rk ∈ ℚ such that k ≤ n and |t − rk | ≤ δ(ε). For ̄ j )| ≤ ε, for all j = 0, 1, 2, . . . , n and m ≥ M. sufficiently large M ≥ n we have |ω̃ m (rj ) − ω(r We then conclude that ̄ ̄ k )| + |ω(r ̄ k ) − ω(t)| ̄ |ω̃ m (t) − ω(t)| ≤ |ω̃ m (t) − ω̃ m (rk )| + |ω̃ m (rk ) − ω(r < 3ε, for all m ≥ M and t ∈ [−T, T]. Thus ω̃ n converges uniformly on bounded intervals to the function ω̄ ∈ C([−T, T]). Theorem 2.19 (Tightness of probability measures). A sequence (Pn )n∈ℕ of probability measures on (C(ℝ), ℬ(C(ℝ))) is tight if and only if lim sup Pn (|ω(0)| > λ) = 0,

(2.1.18)

λ↑∞ n∈ℕ

lim sup Pn (mT (ω, δ) > ε) = 0, δ↓0 n∈ℕ

T > 0, ε > 0.

(2.1.19)

Proof. Suppose that (Pn )n∈ℕ is tight. Given ε > 0, there exists a compact set K with Pn (K) ≥ 1 − ε for every n. According to Lemma 2.18, for sufficiently large λ > 0 we have

24 | 2 Brownian motion |ω(0)| ≤ λ for all ω ∈ K. Thus Pn (|ω(0)| ≤ λ) = Pn (|ω(0)| ≤ λ, ω ∈ K) + Pn (|ω(0)| ≤ λ, ω ∈ K c ) ≤ Pn (|ω(0)| ≤ λ, ω ∈ K) + ε

and we see that limλ↑∞ supn∈ℕ Pn (|ω(0)| > λ) ≤ ε. Since ε is arbitrary, (2.1.18) follows. According to Lemma 2.18 again, if T and ε are given, then there exists δ0 such that mT (ω, δ) ≤ ε for 0 < δ < δ0 and ω ∈ K. This gives (2.1.19). Next suppose (2.1.18)–(2.1.19) hold. Given a positive integer T and ε > 0, choose λ > 0 such that supn∈ℕ Pn (|ω(0)| > λ) ≤ ε/2T+1 . Also, choose δk > 0, k = 1, 2, . . ., such that sup Pn (mT (ω, δk ) > n∈ℕ

1 ε ) ≤ T+k+1 . k 2

Define 󵄨󵄨 1 AT = {ω ∈ C(ℝ) 󵄨󵄨󵄨 |ω(0)| ≤ λ, mT (ω, δk ) ≤ , k = 1, 2, . . .} 󵄨 k and A = ⋂∞ T=1 AT . By Lemma 2.18 the set A is compact. We have ∞

Pn (AT ) = 1 − Pn (AcT ) ≥ 1 − Pn (|ω(0)| > λ) − ∑ Pn (mT (ω, δk ) > 1/k) ≥1−

ε

2T+1

∞

−∑ k=1

ε

2T+k+1

ε =1− T 2

k=1

and ∞

∞

∞

T=1

T=1

T=1

Pn (A) = Pn ( ⋂ AT ) = 1 − Pn ( ⋃ AcT ) ≥ 1 − ∑ Pn (AcT ) ≥ 1 − ε follows for all n, and (Pn )n∈ℕ is tight. In the remaining part of this subsection we discuss a key extension theorem due to Kolmogorov. Let Ω be a Polish space, J an index set, and Λ ⊂ J, |Λ| < ∞, with | ⋅ | denoting the cardinality of a finite set. Consider the family of product probability spaces (ΩΛ , ℬ(ΩΛ ), μΛ ), where ΩΛ = ×|Λ| Ω. An element ω ∈ ΩΛ is regarded as a map Λ 󳨃→ Ω. Take now Λ1 ⊂ Λ2 ⊂ J and let πΛ1 Λ2 : ΩΛ2 → ΩΛ1

(2.1.20)

denote the projection defined by πΛ1 Λ2 (ω) = ω⌈Λ1 . The equality μΛ1 (E) = μΛ2 (πΛ−11 Λ2 (E))

(2.1.21)

for E ∈ ℬ(ΩΛ1 ) and Λ1 ⊂ Λ2 with |Λ1 |, |Λ2 | < ∞, is called Kolmogorov consistency relation.

2.1 Concepts and facts of general measure theory and probability | 25

Theorem 2.20 (Kolmogorov extension theorem). Let (ΩΛ , ℬ(ΩΛ ), μΛ ), Λ ⊂ J, with |Λ| < ∞, be a family of probability spaces with underlying Polish space Ω. Suppose that μΛ satisfies the Kolmogorov consistency relation (2.1.21), and define J

𝒜 = {πΛJ (E) ⊂ Ω | E ∈ ℬ(Ω ), Λ ⊂ J, |Λ| < ∞} , −1

|Λ|

−1 where πΛJ (E) = {ω ∈ ΩJ | ω⌈Λ ∈ E}. Then there exists a unique probability measure μ on J (Ω , σ(𝒜)) such that μΛ (E) = μ(πΛ−1 (E)), for all E ∈ ℬ(ΩΛ ) with |Λ| < ∞.

Proof. Define the set function μ̃ : 𝒜 → [0, ∞) by −1 ̃ ΛJ μ(π (E)) = μΛ (E),

E ∈ 𝒜.

(2.1.22)

−1 We show that (2.1.22) is well defined. When πΛJ (E) = πΛ−1󸀠 J (E 󸀠 ), it is seen for Λ󸀠󸀠 ⊃ Λ󸀠 , Λ that −1 (πΛΛ󸀠󸀠 ∘ πΛ󸀠󸀠 J )−1 (E) = πΛJ (E) = πΛ−1󸀠 J (E 󸀠 ) = (πΛ󸀠 Λ󸀠󸀠 ∘ πΛ󸀠󸀠 J )−1 (E 󸀠 ) −1 −1 −1 󸀠 and thus πΛ−1󸀠󸀠 J ∘ πΛΛ 󸀠󸀠 (E) = πΛ󸀠󸀠 J ∘ πΛ󸀠 Λ󸀠󸀠 (E ). From this it follows that −1 −1 󸀠 πΛΛ 󸀠󸀠 (E) = πΛ󸀠 Λ󸀠󸀠 (E ).

Hence −1 −1 󸀠 󸀠 μΛ (E) = μΛ󸀠󸀠 (πΛΛ 󸀠󸀠 (E)) = μΛ󸀠󸀠 (πΛ󸀠 Λ󸀠󸀠 (E )) = μΛ󸀠 (E )

follows by Kolmogorov consistency, and thus μ̃ is well-defined. Next we extend the set function μ̃ to a probability measure μ on σ(𝒜) by Hopf’s ̃ n ) = 0 for extension theorem. In order to do that it suffices to show that limn→∞ μ(A −1 every sequence (An )n∈ℕ ⊂ 𝒜 such that A1 ⊃ A2 ⊃ . . . and ⋂∞ A = 0. Let A n = πΛn J (En ) j=1 j and suppose that it is decreasing, i. e., An+1 = {ω | ω⌈Λn+1 ∈ En+1 } ⊂ An = {ω | ω⌈Λn ∈ En }. 󸀠 󸀠 We have An ⊂ {ω | ω⌈Λ󸀠n+1 ∈ En+1 }, where Λ󸀠n+1 = Λn+1 ∪ Λn and En+1 = En+1 ∪ En . Hence we may assume that En ⊂ En+1 and Λn ⊂ Λn+1 . Suppose that there exists ε > 0 such ̃ n ) > ε for all n. Since Ω is a Polish space and μΛn (En ) < ∞, by Lemma 2.9 that μ(A there exists a non-empty compact set Kn ⊂ En such that μΛn (En \ Kn ) < ε/2n+1 . Let Bn = πΛ−1n (Kn ) ⊂ An . We obtain

̃ n \ Bn ) = μ(π ̃ Λ−1 (En \ Kn )) = μΛn (En \ Kn ) < μ(A n Let Cn = ⋂nj=1 Bj . Cn ⊂ An and n

n

̃ n ) ≥ μ(A ̃ n ) − μ( ̃ ⋃ (Ak \ Bk )) ≥ ε − ∑ μ(C k=1

k=1

ε

2k+1

ε . 2n+1

ε > . 2

Hence Cn ≠ 0, and we will show in the next lemma that D = ⋂∞ n=1 Cn ≠ 0. Since D ⊂ ∞ ̃ A , this contradicts A = 0. Thus lim μ(A ) = 0. ⋂∞ ⋂ n→∞ n j=1 j j=1 j

26 | 2 Brownian motion Lemma 2.21. It follows that D = ⋂∞ n=1 Cn ≠ 0. Proof. First observe that C1 = B1 = πΛ−11 J (K̃ 1 ), where K̃ 1 = K1 is compact, and C2 = B1 ∩ B2 = πΛ−11 J (K̃ 1 ) ∩ πΛ−12 J (K2 ) = {ω | ω⌈Λ2 ∈ K2 ∩ πΛ−11 Λ2 (K̃ 1 )} = πΛ−12 J (K̃ 2 ), where K̃ 2 = K2 ∩ πΛ−11 Λ2 (K̃ 1 ) is a non-empty compact set. Repeating this procedure, we have Cn = B1 ∩ ⋅ ⋅ ⋅ ∩ Bn = πΛ−1n J (K̃ n ) and K̃ n = Kn ∩ πΛ−1n−1 Λn (K̃ n−1 ) is a non-empty compact set. It is straightforward to check that πΛm Λn (K̃ n ) ⊂ K̃ m for m < n. Thus Fn = πΛ1 Λn (K̃ n ) ⊂ K̃ 1 and Fn is closed and so compact. Since Fn+1 = πΛ1 Λn (πΛn Λn+1 (K̃ n+1 )) ⊂ πΛ1 Λn (K̃ n ) = Fn , Fn , n = 1, 2, . . ., is a non-increasing sequence of compact sets in Ω. Hence ⋂∞ n=1 Fn ≠ 0 and there exists x1 such that ∞

x1 ∈ ⋂ Fn ⊂ K̃ 1 . n=1

Similarly, Fn󸀠 = πΛ2 Λn (K̃ n ) ∩ πΛ−11 Λ2 (x1 ) ⊂ K̃ 2 is a non-increasing sequence of compact sets. Note that πΛ1 Λ2 (Fn󸀠 ) = πΛ1 Λn (K̃ n ) ∩ {x1 } = Fn ∩ {x1 } = {x1 } ≠ 0. 󸀠 Thus Fn󸀠 ≠ 0, which implies that K̃ 2 ⊃ ⋂∞ n=2 Fn ≠ 0 and there exists x2 such that ∞

x2 ∈ ⋂ Fn󸀠 ⊂ K̃ 2 . n=2

Furthermore, Fn󸀠󸀠 = πΛ3 Λn (K̃ n ) ∩ πΛ−12 Λ3 (x2 ) ⊂ K̃ 3 is also a non-increasing sequence of 󸀠󸀠 non-empty compact sets. Thus K̃ 3 ⊃ ⋂∞ n=3 Fn ≠ 0 and there also exists x3 such that ∞

x3 ∈ ⋂ Fn󸀠󸀠 ⊂ K̃ 3 . n=3

2.1 Concepts and facts of general measure theory and probability | 27

Repeating this procedure, we find a sequence xm , m = 1, 2, . . ., such that xm ∈ πΛm Λn (K̃ n ) ⊂ K̃ m for all n ≥ m. Define ω∗ ∈ ΩJ by ω∗ = {

ω∗ ⌈Λk = xk , ω∗ ⌈J\⋃∞ , k=1 Λk

k = 1, 2, . . . , arbitrary.

ω∗ is well defined by the construction of the sequence (xk )∞ k=1 . Hence πΛn J (ω∗ ) = xn ∈ −1 ̃ ̃ Kn and ω∗ ∈ πΛn J (Kn ) = Cn for all n ≥ 1, and we conclude that ω∗ ∈ ⋂∞ n=1 Cn . One application of Kolmogorov’s extension theorem is in the construction of countable direct products of probability measures. Example 2.22 (Countable product of probability measures). Let (ℝ, ℬ(ℝ), μk ), k ∈ ℕ, be a family of probability measures. Take Λ1 ⊂ Λ2 ⊂ ℕ and let πΛ1 Λ2 : ℝΛ2 → ℝΛ1 be the projection defined by πΛ1 Λ2 (x) = x⌈Λ1 for x ∈ ℝΛ2 . Let Λ1 = {k1 , . . . , kn } and Λ2 = {k1 , . . . , kn , kn+1 , . . . , kn+m } for simplicity. Also, let E = E1 × ⋅ ⋅ ⋅ × En ∈ ℬ(ℝn ). Thus we see that πΛ−11 Λ2 (E) = E × ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℝ × ⋅⋅⋅ × ℝ. m

Hence it follows that μΛ1 (E) = μΛ2 (πΛ−11 Λ2 (E)) and Kolmogorov consistency holds. Thus by the Kolmogorov extension theorem there exists a unique probability measure μ on (ℝℕ , ℬ(ℝℕ )) such that μ(∏ Ej × ∏ ℝ) = ∏ μj (Ej ). j∈Λ1

k∈ℕ\Λ1

j∈Λ1

The measure μ can be denoted by ∞

μ = ∏ μk . k=1

Countable direct products of Gaussian measures will be important in Chapters 2–4 in Volume 2. We can also construct an uncountable product of probability measures. Example 2.23 (Uncountable product of probability measures). Let ℝ[0,∞) denote the set of all ℝ-valued functions on [0, ∞). An n-dimensional cylinder set in ℝ[0,∞) is defined by C = {ω ∈ ℝ[0,∞) | (ω(t1 ), . . . , ω(tn )) ∈ E1 × ⋅ ⋅ ⋅ × En , tj ∈ [0, ∞), Ej ∈ ℬ(ℝ), j = 1, . . . , n}. (2.1.23)

28 | 2 Brownian motion Let A denote the family of all finite dimensional cylinder sets, and F = σ(A ). Let Λ be the set of finite sequences t = (t1 , . . . , tn ) of distinct, non-negative numbers, where the length n of these sequences ranges over the set of positive integers. For every t = (t1 , . . . , tn ) ∈ Λ with length n, we assume that there exists a probability measure Pt on the measurable space (ℝn , ℬ(ℝn )) such that (1) if s = (ti1 , . . . , tin ) is a permutation of t, then Pt (E1 × ⋅ ⋅ ⋅ × En ) = Ps (Ei1 × ⋅ ⋅ ⋅ × Ein ); (2) if s = (t1 , . . . , tm ) for m < n, then Pt (E1 × ⋅ ⋅ ⋅ × Em × ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℝ × ⋅ ⋅ ⋅ × ℝ) = Ps (E1 × ⋅ ⋅ ⋅ × Em ). n−m

By the Kolmogorov extension theorem there exists a probability measure P on the measurable space (ℝ[0,∞) , F ) such that P(C) = Pt (E1 × ⋅ ⋅ ⋅ × En ) for C in (2.1.23). Finally, by introducing the locally uniform topology on the subset C(ℝ+ ; ℝ) ⊂ ℝ , with the standard notation ℝ+ = [0, ∞), we obtain the relationship between the σ-field generated by cylinder sets and the Borel σ-field. For later use, here we extend C(ℝ+ ; ℝ) to [0,∞)

d

X = C(ℝ ; ℝ ), +

the space of ℝd -valued continuous functions on ℝ+ , equipped with the metric (2.1.3). As seen before, this metric induces the locally uniform topology in which fn → f if and only if fn → f uniformly on every compact set in ℝ+ . The so obtained topology gives rise to the Borel σ-field ℬ(X ). Lemma 2.24. The Borel σ-field ℬ(X ) coincides with the σ-field generated by cylinder sets of the form E ,...,E

At11,...,tn n = {ω ∈ X | (ω(t1 ), . . . , ω(tn )) ∈ E1 × ⋅ ⋅ ⋅ × En }

(2.1.24)

with E1 , . . . , En ∈ ℬ(ℝd ), 0 ≤ t1 < t2 < . . . < tn , and n ∈ ℕ. Proof. Let C be the σ-field generated by cylinder sets of the form (2.1.24). We show that ℬ(X ) = C . First consider the inclusion C ⊂ ℬ(X ). Note that C coincides with the σ-field generated by cylinder sets of the form (2.1.24) with open sets E1 , . . . , En . For s ≥ 0 and open set E ⊂ ℝd let H = H(s, E) = {ω ∈ X | ω(s) ∈ E}. For ω0 ∈ H, define B(ω0 , ε) = {ω ∈ X | dist(ω, ω0 ) < ε}, where dist(⋅, ⋅) is the metric on X , given by (2.1.3). Let ε > 0 be sufficiently small. For every ω ∈ B(ω0 , ε), we see that ω(s) ∈ E. Hence B(ω0 , ε) ⊂ H and H is open. This implies that H ∈ ℬ(X ). Thus the cylinder set E ,...,E At11,...,tn n = ⋂nj=1 H(tj , Ej ) is also open for open sets Ej . Then C ⊂ ℬ(X ) follows since C is the minimal σ-field containing all cylinder sets.

2.1 Concepts and facts of general measure theory and probability | 29

Next consider the converse inclusion C ⊃ ℬ(X ). Let A ∈ ℬ(X ). Since X is separable, we have A = ⋃∞ j=1 B(ωj , εj ). Note that 󵄨󵄨 ∞ (sup 0≤t≤n, t∈ℚ |ω0 (t) − ω(t)|) ∧ 1 󵄨 B(ω0 , ε) = {ω ∈ X 󵄨󵄨󵄨 ∑ < ε} . 󵄨󵄨 n=1 2n We obtain B(ω0 , ε) ∈ C by a simple limiting argument, and thus C ⊃ ℬ(X ).

2.1.3 Random variables Consider a given probabilty space (Ω, F , P). The integral with respect to P of a Borel measurable function f is called expectation of f and will be denoted by any of the symbols ∫ f (ω)dP(ω) = ∫ fdP = 𝔼P [f ]. Ω

(2.1.25)

Ω

We omit the subscript P of 𝔼P unless any confusion may arise, and write simply 𝔼. An F /S -measurable function X mapping a probability space (Ω, F , P) to a measurable space (S, S ) is called an S-valued random variable. The inverse map identifies the sample points in Ω on which the observation of event A depends. For an S-valued random variable X the probability measure PX on (S, S ) defined by PX (A) = P(X −1 (A)),

A ∈ S,

is called the image measure or the distribution of P under X : Ω → S. Using this, we have 𝔼[f (X)] = ∫ f (x)dPX (x)

(2.1.26)

S

for a bounded Borel measurable function f : S → ℝd . The covariance of two ℝd -valued random variables X and Y on the same probability space is defined by cov(X, Y) = 𝔼[(X − mX )(Y − mY )],

(2.1.27)

where mX = 𝔼[X] and mY = 𝔼[Y] are the expectations of X and Y, respectively. For two S-valued random variables X and Y, not necessarily defined on the same probability d

space, we use the notation X = Y whenever X and Y are identically distributed, i. e., PX = PY .

30 | 2 Brownian motion Example 2.25 (Gaussian random variable). Let m ∈ ℝ and σ > 0. A random variable X : Ω → ℝ with probability distribution dPX (x) =

1

√2πσ 2

e

− (x−m) 2 2σ

2

dx d

is called a Gaussian random variable and the notation X = N(m, σ 2 ) is used. A calculation gives that m is the mean, and σ 2 is the variance of X. An N(0, 1)-distributed random variable is a standard Gaussian random variable. A random vector X = (X1 , . . . , Xd ) : Ω → ℝd is a multivariate Gaussian random variable if its probability distribution is given by dPX (x) =

1

(2π)d/2 (det C)1/2

1

e− 2 (x−m)⋅C

−1

(x−m)

dx,

where x = (x1 , . . . , xd ) ∈ ℝd , C = (Cij )1≤i,j≤d is a d × d symmetric, strictly positive definite matrix, and m = (m1 , . . . , md ) ∈ ℝd is a vector. The matrix element Cij gives the covariance cov(Xi , Xj ) of the multivariate Gaussian random variable, and mj = 𝔼[Xj ]. Let (S, S ) be a measurable space, and X be an S-valued random variable on (Ω, F , P). The notation σ(X) is used for the minimal σ-field such that X is measurable, i. e., σ(X) = {X −1 (U) | U ∈ S }.

(2.1.28)

Let Y be also a random variable on (Ω, F , P). If the sub-σ-fields σ(X) and σ(Y) are independent, then X and Y are said to be independent. We consider the distribution of the sum of random variables. Let P and Q be probability measures on (ℝd , ℬ(ℝd )). The convolution of P and Q is a probability measure on (ℝd , ℬ(ℝd )) defined by (P ∗ Q)(A) = ∫ Q(A − x)dP(x),

A ∈ ℬ(ℝd ),

ℝd

where A−x = {y −x | y ∈ A}. Note that P ∗Q = Q∗P. Let X and Y be ℝd -valued independent random variables and P and Q be distributions on ℝd of X and Y, respectively. It follows that 𝔼P [f (X + Y)] = ∫ dP(x) ∫ f (x + y)dQ(y) = ∫ f (z)(P ∗ Q)(dz) ℝd

ℝd

ℝd

for all bounded Borel measurable functions f . Then the distribution of the sum of two independent random variables X + Y is given by P ∗ Q. If X and Y are independent, then cov(f (X), g(Y)) = 0 for all bounded Borel measurable functions f and g. For two Gaussian random variables the converse is also true, i. e., if their covariance is zero, then they are independent.

2.1 Concepts and facts of general measure theory and probability | 31

Lemma 2.26. Let (Ω, F , P) be a probability space and Xi , i = 1, . . . , n, real-valued random variables on it. Then Xi , i = 1, . . . , n, are independent if and only if n

𝔼[f1 (X1 ) ⋅ ⋅ ⋅ f (Xn )] = ∏ 𝔼[fi (Xi )],

(2.1.29)

i=1

for all bounded Borel measurable functions fi , i = 1, . . . , n. Proof. Suppose that Xi , i = 1, . . . , n, are independent. Let fi (x) = 1Ei (x) with Ei ∈ ℬ(ℝ). We see that n

n

n

i=1

i=1

i=1

𝔼 [∏ 1Ei (Xi )] = P(X1 ∈ E1 , . . . , Xn ∈ En ) = ∏ P(Xi ∈ Ei ) = ∏ 𝔼[1Ei (Xi )]. By a limiting argument (2.1.29) follows. Conversely, suppose (2.1.29). Take Ak ∈ σ(Xk ), k = 1, . . . , n. We see that P(A1 ∩ ⋅ ⋅ ⋅ ∩ An ) = 𝔼[∏nk=1 1Ak (Xk )] = ∏nk=1 𝔼[1Ak (Xk )] = ∏nk=1 P(Ak ). Thus Xi , i = 1, . . . , n, are independent. We have a second useful criterion to show independence of random variables. Lemma 2.27. Let (Ω, F , P) be a probability space and Xi , i = 1, . . . , n, real-valued random variables on it. Then Xi , i = 1, . . . , n, are independent if and only if for every (ξ1 , . . . , ξn ) ∈ ℝn , n

n

𝔼[ei ∑j=1 ξj Xj ] = ∏ 𝔼[eiξj Xj ]. j=1

(2.1.30)

Proof. The necessity part is directly obtained from Lemma 2.26. To obtain the sufficiency part, suppose that (2.1.30) holds and let fj ∈ S (ℝ), for j = 1, . . . , n. Using that f (x) = (2π)−1/2 ∫ f ̂ (k)eikx dk, we have j

ℝ j

n

n

j=1

j=1

𝔼 [∏ fj (Xj )] = ∏ 𝔼[fj (Xj )].

(2.1.31)

Let C0 (ℝ) be the set of continuous functions on ℝ with compact support. For f ∈ C0 (ℝ) there exists a sequence (gn )n∈ℕ ⊂ S (ℝ) such that gn → f , uniformly as n → ∞. Equality (2.1.31) holds for every fj ∈ C0 (ℝ) by a limiting argument. Fix f2 , . . . , fn ∈ C0 (ℝ), and define 󵄨󵄨 n n 󵄨󵄨 A = {A ∈ ℬ(ℝ) 󵄨󵄨󵄨 𝔼 [1A (X1 ) ∏ fj (Xj )] = 𝔼[1A (X1 )]𝔼 [∏ fj (Xj )] } . 󵄨󵄨 j=2 j=2 󵄨 Denote the family of finite unions of semi-open intervals in ℝ by I. If A = (a, b], then by approximating the indicator function 1A pointwise by a function in C0 (ℝ) we see that (a, b] ∈ A by (2.1.31). Hence I ⊂ A follows. Since A is a monotone family, we get n n A = ℬ(ℝ). Repeating this procedure we see that 𝔼[∏j=1 1Ej (Xj )] = ∏j=1 𝔼[1Ej (Xj )] for arbitrary Ej ∈ ℬ(ℝ), and thus the independence of X1 , . . . , X n is proven.

32 | 2 Brownian motion In what follows a real-valued random variable will be simply called a random variable. We summarize the notions of convergence of random variables next. Definition 2.28 (Convergence of random variables). Let (Xn )n∈ℕ be a sequence of random variables and X another random variable, all on a given probability space (Ω, F , P). a. s. (1) We write Xn → X, i. e., Xn is convergent almost surely to X if P ( lim Xn = X) = 1. n→∞

Lp

(2) We write Xn → X, i. e., Xn is convergent in Lp -sense to X for 1 ≤ p < ∞, if lim 𝔼 [|Xn − X|p ] = 0.

n→∞ P

(3) We write Xn → X, i. e., Xn is convergent in probability to X if for every ε > 0, lim P (|Xn − X| ≥ ε) = 0.

n→∞ d

(4) We write Xn → X, i. e., Xn is convergent in distribution or weakly or in law to X if lim 𝔼[f (Xn )] = 𝔼[f (X)]

n→∞

for every bounded continuous function f . Note that for any given sequence (Xn )n∈ℕ of random variables on a probability d

space (Ω, F , P) with values in a metric space (S, ℬ(S)), the convergence Xn → X is equivalent to weak convergence of P ∘ Xn−1 to P ∘ X −1 . To discuss the various types of convergence of measurable functions, first we study a criterion for relative sequential compactness in L1 (Ω) = L1 (Ω, dμ) in terms of the following concept.

Definition 2.29 (Uniform integrability). Let (Ω, F , μ) be a measure space. A set of functions A ⊂ L1 (Ω, dμ) is uniformly integrable if inf sup ∫(|f (x)| − g(x))+ dμ(x) = 0,

0≤g∈L1 f ∈A

(2.1.32)

Ω

where h(x)+ = max{h(x), 0} denotes the non-negative part of h. Lemma 2.30. Let (Ω, F , μ) be a finite measure space. Then A ⊂ L1 (Ω) is uniformly integrable if and only if inf sup ∫ |f |1{|f |>a} dμ = 0.

a∈[0,∞) f ∈A

Ω

(2.1.33)

2.1 Concepts and facts of general measure theory and probability | 33

Proof. Assuming that (2.1.33) holds, we have inf sup ∫(|f | − a)+ dμ = 0.

a∈[0,∞) f ∈A

(2.1.34)

Ω

Equality (2.1.34) furthermore implies uniform integrability of A since the infimum is taken over the smaller set of constant functions. Next suppose that A is uniformly integrable. There exists 0 ≤ gε ∈ L1 (Ω) such that supf ∈A ∫Ω (|f | − gε )+ dμ ≤ ε. Define g̃ε = 2gε/2 . For f ∈ A , ∫ |f |1{|f |>g̃ε } dμ ≤ ∫(|f | − gε/2 )+ 1{|f |>g̃ε } dμ + ∫ gε/2 1{|f |>g̃ε } dμ.

Ω

Ω

Ω

We see that ∫Ω (|f | − gε/2 )+ 1{|f |>g̃ε } dμ ≤ ε/2 and gε/2 1{|f |>g̃ε } ≤ (|f | − gε/2 )+ 1{|f |>g̃ε } . Hence ε ∫ gε/2 1{|f |>g̃ε } dμ ≤ ∫(|f | − gε/2 )+ 1{|f |>g̃ε } dμ ≤ . 2

Ω

Ω

Summing up we have sup ∫ |f |1{|f |>g̃ε } dμ ≤ ε,

f ∈A

and thus

Ω

ε sup ∫ |f |1{|f |>g̃ε/2 } dμ ≤ . 2 f ∈A Ω

Choose a sufficiently large aε such that ∫Ω g̃ε/2 1{g̃ε/2 >aε } dμ ≤ ε/2. We see that ∫ |f |1{|f |>aε } dμ ≤ ∫ |f |1{|f |>aε } 1{|f |>g̃ε/2 } dμ + ∫ |f |1{|f |>aε } 1{|f |≤g̃ε/2 } dμ

Ω

Ω

Ω

≤ ∫ |f |1{|f |>g̃ε/2 } dμ + ∫ gε/2 1{|f |>aε } dμ ≤ ∫ |f |1{|f |>g̃ε/2 } dμ + ∫ gε/2 1{g̃ε/2 >aε } dμ < ε. Ω

Ω

Ω

Ω

The relationships of the various types of convergence can be summarized as follows. Theorem 2.31. Let (Xn )n∈ℕ be a sequence of random variables and X another random variable, all on a probability space (Ω, F , P). a. s.

Lp

P

(1) If Xn → X or Xn → X for some p ≥ 1, then Xn → X as n → ∞.

34 | 2 Brownian motion P

d

(2) If Xn → X, then Xn → X as n → ∞. P

a. s.

(3) If Xn → X, then there exists a subsequence Xnk , k ≥ 1, such that Xnk → X as k → ∞. P

(4) (Dominated convergence theorem) Let Xn → X as n → ∞ and suppose that there is a random variable Y such that |Xn | ≤ Y for all n ∈ ℕ, and 𝔼[Y] < ∞. Then L1

X ∈ L1 (Ω, dP) and Xn → X.

a. s.

(5) (Monotone convergence theorem) Let Xn → X as n → ∞. Suppose that the sequence (Xn )n∈ℕ is monotone increasing and there exists M > 0 such that 𝔼[Xn ] < M for all n. Then X ∈ L1 (Ω, dP) and 𝔼[Xn ] → 𝔼[X] as n → ∞. a. s.

Proof. (1) Let ε > 0. Suppose that Xn → X. Let An = {ω ∈ Ω | |Xn (ω) − X(ω)| > ε},

E = {ω ∈ Ω | lim Xn (ω) = X(ω)}. n→∞

Thus P(An ) = P(An ∩ E) + P(An ∩ E c ) = P(An ∩ E), since P(E c ) = 0. Thus P(An ) = P

P(An ∩ E) → 0 as n → ∞, and hence Xn → X as n → ∞. L

p

Next suppose that Xn → X. We see that P(An ) ≤ ε−p ‖Xn − X‖pLp P

and limn→∞ P(An ) = 0 follows. Thus Xn → X as n → ∞. (2) Let f be a bounded and uniformly continuous function, and choose ε > 0. We see that |f (x) − f (y)| ≤ ε for all x, y ∈ ℝ such that |x − y| ≤ δ, with some δ > 0. Let Kn = {ω ∈ Ω | |Xn (ω) − X(ω)| ≤ δ}. Hence we see that |𝔼[f (Xn ) − f (X)]| ≤ 𝔼[|f (Xn ) − f (X)|1Kn ] + 𝔼[|f (Xn ) − f (X)|1Knc ] ≤ ε + M𝔼[1Knc ], where M = supx∈ℝ |f (x)|. Thus lim |𝔼[f (Xn ) − f (X)]| ≤ ε.

n→∞

(2.1.35)

By a simple limiting argument, (2.1.35) holds for any bounded continuous function f . P

(3) Since Xn → X, there exists a subsequence nk such that P(|X − Xnk | > 1/k) ≤ 2−k . Thus ∑∞ k=1 P(|X − Xnk | > 1/k) < ∞. By the Borel–Cantelli lemma we then have ∞

∞

P( ⋃ ⋂ {|X − Xnk | ≤ 1/k}) = 1. m=1 k=m

2.1 Concepts and facts of general measure theory and probability | 35

Thus Xnk → X as nk → ∞, almost surely. (4) By Fatou’s lemma we have 𝔼[|X|] ≤ lim inf 𝔼[|Xn |] ≤ 𝔼[|Y|] < ∞. n→∞

Thus X ∈ L1 (Ω, dP). Since 𝔼[|Xn |1{|Xn |>a} ] ≤ 𝔼[Y1{Y>a} ], we have inf sup 𝔼[|Xn |1{|Xn |>a} ] ≤

a∈[0,∞) n∈ℕ

inf 𝔼[Y1{Y>a} ] = 0,

a∈[0,∞)

and hence (Xn )n∈ℕ is uniformly integrable by Lemma 2.30. Suppose, to the contrary, that Xn does not converge to X in L1 (Ω, dP), i. e., there exist ε > 0 and a subsequence nk , k ∈ ℕ, such that ‖X − Xnk ‖L1 > 2ε.

(2.1.36)

We reset the counter and write Xnk as Xn for notational simplicity. Note that (X −Xn )n∈ℕ is also uniformly integrable. There exists 0 ≤ g ∈ L1 (Ω, dP) such that 𝔼[(|X − Xn | − g)+ ] < ε. This implies that lim sup 𝔼[1{|X−Xn |>g} |X − Xn |] < ε.

(2.1.37)

n→∞

Define gn (ω) = |X(ω) − Xn (ω)| ∧ g(ω). It follows that gn ≤ g and gn → 0 almost surely as n → ∞. Fatou’s lemma again gives lim sup 𝔼[gn ] = 𝔼[g] − lim inf 𝔼[g − gn ] ≤ 𝔼[g] − 𝔼[ lim (g − gn )] = 0. n→∞

n→∞

n→∞

Since {|X − Xn | = gn } = {|X − Xn | ≤ g}, together with (2.1.37) this implies that lim sup ‖X − Xn ‖L1 = lim sup (𝔼[1{|X−Xn |>g} |X − Xn |] + 𝔼[1{|X−Xn |≤g} |X − Xn |]) n→∞

n→∞

≤ lim sup 𝔼[1{|X−Xn |>g} |X − Xn |] + lim sup 𝔼[gn ] ≤ ε. n→∞

n→∞

This contradicts (2.1.36). (5) This is a fundamental fact in measure theory. Corollary 2.32. Suppose that {Xn }n∈ℕ is a uniformly integrable set of random variables on a probability space (Ω, F , P). Then lim supn→∞ 𝔼[|Xn |] < ∞. If, in addition, Xn → X L1

almost surely or in probability, then Xn → X. Proof. Note that for every λ > 0 it follows that 𝔼[|Xn |] ≤ λ + 𝔼[|Xn |1{|Xn |>λ} ]. Thus we have lim supn→∞ 𝔼[|Xn |] < ∞. For the second statement, in the same way as in the L1

proof of (4) of Theorem 2.31, we see that Xn → X as n → ∞.

36 | 2 Brownian motion In probability theory we often have to deal with sums of independent, identically distributed random variables {Xn }n∈ℕ and related (e. g., Cesàro) means. One notable property of ∑ni=1 Xi /n is that it converges to the non-random number 𝔼[X0 ], almost surely as n → ∞. This is the celebrated (strong) law of large numbers. If almost sure convergence is replaced by convergence in probability, the proof of is more straightforward. Lemma 2.33. Let (Ω, F , P) be a probability space and (Xn )n∈ℕ a sequence of independent random variables such that 𝔼[Xn2 ] < ∞ and 𝔼[Xn ] = 0, for all n ∈ ℕ. Let a > 0. Then 󵄨󵄨 k 󵄨󵄨 1 n 󵄨󵄨 󵄨󵄨 P (max 󵄨󵄨󵄨∑ Xi 󵄨󵄨󵄨 > a) ≤ 2 ∑ σi2 , 󵄨 1≤k≤n 󵄨󵄨 a i=1 󵄨i=1 󵄨󵄨 where σi2 = 𝔼[Xi2 ]. Proof. Let A = {max1≤k≤n |Sk | > a} = ⋃nj=1 Aj , where k

Sk = ∑ Xi , i=1

Aj = {|Si | < a, i = 1, . . . , j − 1, and |Sj | ≥ a}.

Note that Ai ∩ Aj = 0 for i ≠ j. We then have n

n

n

j=1

j=1

∑ σi2 ≥ 𝔼[Sn2 1A ] = ∑ 𝔼[Sn2 1Aj ] = ∑ 𝔼[(Sn − Sj + Sj )2 1Aj ] i=1

n

n

n

j=1

j=1

= ∑ 𝔼[((Sn − Sj )2 − 2Sj (Sn − Sj ) + Sj2 )1Aj ] ≥ ∑ 𝔼[Sj2 1Aj ] ≥ ∑ a2 P(Aj ) = a2 P(A). j=1

Hence P(A) ≤ ∑ni=1 σi2 /a2 follows. Lemma 2.34. Let (Ω, F , P) be a probability space and (Xn )n∈ℕ a sequence of indepenn 2 dent random variables such that ∑∞ n=1 𝔼[Xn ] < ∞. Then Sn = ∑i=1 Xi converges almost surely as n → ∞. Proof. It is sufficient to show that (Sn )n∈ℕ is a Cauchy sequence, almost surely. Let ε > 0 be fixed. Write 𝔼[Xn2 ] = σn2 . Define AN (ε) = {ω ∈ Ω | ∃i, j > N such that |Si (ω) − Sj (ω)| ≥ 2ε} ∞ and write A = ⋃∞ n=m ⋂N=1 AN (1/n), where 1/m < ε; clearly, ω ∈ A implies that there exists n = n(ω) ∈ ℕ such that for all N there exist i, j > N with |Si (ω) − Sj (ω)| ≥ 2/n. On the other hand, ω ∈ Ω \ A implies that for every n there exists N = N(ω) ∈ ℕ such that for all i, j > N it follows that |Si (ω) − Sj (ω)| < 2/n. Thus it is sufficient to show P(A) = 0. Let BN = {ω ∈ Ω | ∃i > Nsuch that|Si (ω) − SN (ω)| ≥ ε}. We have AN (ε) ⊂ BN , since 2ε ≤ |Si − Sj | ≤ |Si − SN | + |SN − Sj |. Furthermore, we consider

2.1 Concepts and facts of general measure theory and probability | 37

BN,n = {ω ∈ Ω | there exists n ≥ i > N such that |Si (ω) − SN (ω)| ≥ ε} = {ω ∈ Ω | max |XN+1 + ⋅ ⋅ ⋅ + Xi | ≥ ε} . N C = ∫ |x|dF(x) ≥ ∑ ℝ

n=0

∫ n≤|x| n}. We have ∞

∞

∞

n=0

n=0

n=0

∑ P(An ) = ∑ P(|X0 | > n) = ∑ ∫ dF(x) ∞

= ∑ (k + 1) k=0

|x|>n ∞

∫ k n, Y (ω) ∑m ∑m (X (ω) − Yi (ω)) ∑m i=0 Xi (ω) = i=0 i + i=0 i m m m Y (ω) ∑n (X (ω) − Yi (ω)) ∑m = i=0 i + i=0 i →a m m as m → ∞. Thus the statement follows. 2.1.4 Conditional expectation and regular conditional probability measures Let X be a random variable on a probability space (Ω, F , P) and G ⊂ F a sub-σ-field. Recall that the conditional expectation Z = 𝔼[X|G ]

2.1 Concepts and facts of general measure theory and probability | 39

of X with respect to G is defined as the unique G -measurable random variable such that 𝔼[1A X] = 𝔼[1A Z],

A ∈ G.

(2.1.41)

The left-hand side of (2.1.41) defines a probability measure PX (A) = 𝔼[1A X] on G and 𝔼[X|G ] is in fact the Radon–Nikodým derivative dPX /dP. Let P(A|B) = P(A ∩ B)/P(B) for A, B ∈ F such that P(B) ≠ 0. Then F ∋ A 󳨃→ PB (A) = P(A|B)

defines a probability measure on (Ω, F ). Suppose that An ∈ F , n = 1, . . . , N ≤ ∞, satisfy that An ∩Am = 0 for n ≠ m and ⋃Nn=1 An = Ω. Let G = σ({An }Nn=1 ) be the sub-σ-field generated by {An }Nn=1 . We have N

𝔼[X|G ] = ∑ 𝔼PA [X]1An . n=1

n

From this it can be said that the conditional expectation 𝔼[X|G ] gives an approximation of X in terms of G -measurable functions. We summarize some basic properties of conditional expectation below. Proposition 2.37 (Properties of conditional expectation). Let (Ω, F , P) be a probability space, and X and Y be random variables on it such that X, Y ∈ L1 (Ω). Let G ⊂ H ⊂ F be sub-σ-fields. Then the following hold. (1) (Linearity) 𝔼[aX + bY|G ] = a𝔼[X|G ] + b𝔼[Y|G ], for a, b ∈ ℂ. (2) (Monotonicity) If X ≥ Y a. s., then 𝔼[X|G ] ≥ 𝔼[Y|G ] a. s. (3) If Y is measurable with respect to G and 𝔼[|XY|] < ∞, then 𝔼[XY|G ] = Y𝔼[X|G ]. (4) (Tower property) 𝔼[𝔼[X|H ]|G ] = 𝔼[𝔼[X|G ]|H ] = 𝔼[X|G ]. (5) (Triangle inequality) |𝔼[X|G ]| ≤ 𝔼[|X||G ]. (6) (Independence) If σ(X) and G are independent, then 𝔼[X|G ] = 𝔼[X]. Proof. (1) The right-hand side is G -measurable, hence for A ∈ G we have 𝔼[1A (a𝔼[X|G ] + b𝔼[Y|G ])] = a𝔼[1A X] + b𝔼[1A Y] = 𝔼[1A (aX + bY)]. Thus (1) follows. (2) Let A = {𝔼[X|G ] < 𝔼[Y|G ]} ∈ G . Since from the assumption it follows that 0 ≤ 𝔼[1A (X − Y)] = 𝔼[1A X|G ] − 𝔼[1A Y|G ], we have P(A) = 0. This gives (2). (3) First we assume that X, Y ≥ 0. For every n ∈ ℕ define Yn = 2−n [2n Y], where −n [z] is the integer part of z. We have Yn = ∑∞ and Yn ↑ Y as n → ∞. Let k=1 1{Yn =k2−n } k2 A ∈ G . By the monotone convergence theorem it can be seen that 𝔼[1A Yn 𝔼[X|G ]] → 𝔼[1A Y𝔼[X|G ]] as n → ∞. On the other hand, since {Yn = k2−n } ∈ G , we see that ∞

∞

k=1

k=1

𝔼[1A Yn 𝔼[X|G ]] = ∑ 𝔼[1A 1{Yn =k2−n } k2−n 𝔼[X|G ]] = ∑ 𝔼[1A 1{Yn =k2−n } k2−n X] = 𝔼[1A Yn X] → 𝔼[1A YX]

40 | 2 Brownian motion as n → ∞. Hence (3) follows for X ≥ 0 and Y ≥ 0. In the genral case, write X = X + − X − and Y = Y + − Y − and use linearity of the conditional expecation. Then (3) follows. (4) The second equality follows from (3) with Y = 𝔼[X|H ] and X = 1. Let A ∈ G . Then A ∈ H and 𝔼[1A 𝔼[𝔼[X|G ]|H ]] = 𝔼[1A 𝔼[X|G ]] = 𝔼[1A X] = 𝔼[1A 𝔼[X|H ]]. Thus (4) follows. (5) This follows from (1) and (2). (6) Trivially 𝔼[X] is G -measurable. Let A ∈ G . Since X and 1A are independent, we see that 𝔼[𝔼[X|G ]1A ] = 𝔼[X1A ] = 𝔼[X]𝔼[1A ] = 𝔼[𝔼[X]1A ]. Then (6) is proven. The next lemma is of key importance. Lemma 2.38. Let (Ω, F , P) be a probability space and (S, S ) a measurable space. Let, moreover, X be an S-valued random variable on Ω, and Y a real-valued σ(X)-measurable random variable on Ω. Then there exists an S -measurable function f on S such that Y = f ∘ X a. s. Proof. It suffices to show the lemma for non-negative Y. Let L be the set of real-valued σ(X)-measurable random variables on Ω satisfying the assumption of the lemma. If A ∈ σ(X), then 1A ∈ L , and all σ(X)-measurable step functions belong to L . Since Y is σ(X)-measurable, there exists a sequence (ϕn )n∈ℕ of non-decreasing step functions such that ϕn → Y almost surely. Since ϕn ∈ L , there exists a sequence (fn )n∈ℕ of S -measurable functions such that ϕn = fn ∘ X. Define the function f on S by f (x) = lim supn→∞ fn (x) for x ∈ S. f is S -measurable and f (X(ω)) = lim sup fn ∘ X(ω) = lim fn ∘ X(ω) = Y(ω), n→∞

n→∞

and hence Y ∈ L . Let X be a real-valued random variable on (Ω, F , P) and F a measurable function on it. The random variable 𝔼[F|σ(X)] is a σ(X)-measurable function and by Lemma 2.38 it can be expressed as 𝔼[F|σ(X)] = f (X), with a Borel measurable function f . This function is usually denoted by f (x) = 𝔼[F|X = x].

(2.1.42)

Note that if f (X) = k(X) = 𝔼[F|σ(X)] a. s., then f (x) = k(x) for P ∘ X −1 -a. e. x ∈ ℝ, so the ambiguity in the expression (2.1.42) has measure zero under P ∘ X −1 . Let Y be another random variable on (Ω, F , P), and h, g Borel measurable functions on ℝ. Since 𝔼[h(Y)g(X)] = 𝔼[h(Y)𝔼[g(X)|σ(Y)]] by the tower property of conditional expectations, using the notation (2.1.42) we have 𝔼[h(Y)g(X)] = ∫ h(x)𝔼[g(X)|Y = x]μY (dx), ℝd

where μY (dx) denotes the distribution of Y on ℝ.

(2.1.43)

2.1 Concepts and facts of general measure theory and probability | 41

The conditional expectation 𝔼[1A |G ] is P(A|G ) = 𝔼[1A |G ], i. e., the conditional probability with respect to G . It is in general not obvious that P(⋅|G )(ω) defines a probability measure for almost every ω, since P(A|G )(⋅) is defined only almost surely for each A. Let P(A|G )(ω) be defined on ω ∈ Ω \ NA with P(NA ) = 0. Hence P(⋅|G )(ω) : F → [0, 1] is defined only for ω ∈ Ω \ ⋃A∈F NA . It is, however, in general not clear whether P (⋃A∈F NA ) = 0. This justifies the following definition. Definition 2.39 (Regular conditional probability measure). Let (Ω, F , P) be a probability space and G a sub-σ-field of F . A function Q : Ω × F → [0, 1] is called a regular conditional probability measure for F given G if (1) for every ω ∈ Ω, Q(ω, ⋅) is a probability measure on (Ω, F ); (2) for every A ∈ F , Q(⋅, A) is G -measurable; (3) for every A ∈ F , Q(ω, A) = P(A|G )(ω), P-a. s. For the purposes below, we will be particularly interested in the case where the sub-σ-field is generated by a random variable X, and give another variant of this definition. We write 𝔼[1A |X = x] = P(A|X = x). Definition 2.40 (Regular conditional probability measure). Let (Ω, F , P) be a probability space and X a random variable on it with values in a measure space (S, S ). A function Q : S × F → [0, 1] is called a regular conditional probability measure for F given X if (1) for every x ∈ S, Q(x, ⋅) is a probability measure on (Ω, F ); (2) for every A ∈ F , Q(⋅, A) is S -measurable; (3) for every A ∈ F , Q(x, A) = P(A|X = x), P ∘ X −1 -a. s. Theorem 2.41 (Existence of regular conditional probability measures). Let (Ω, F ) be a standard measurable space and P a probability measure on (Ω, F ). (1) Let G be a sub-σ-field of F . Then there exists a regular conditional probability measure Q(ω, A) for F given G , and it is unique, i. e., if Q󸀠 is another regular conditional probability measure, then there exists M ∈ G with P(M) = 0, such that Q(ω, A) = Q󸀠 (ω, A), for A ∈ F and ω ∈ Ω \ M. (2) Let Q(ω, ⋅) be a regular conditional probability measure for F given G . In addition to (1), suppose that H is a countably determined sub-σ-field of G . Then there exists N ∈ G with P(N) = 0, such that Q(ω, A) = 1A (ω), for A ∈ H and ω ∈ Ω \ N. (3) Let X be a random variable on (Ω, F ) with values in a measurable space (S, S ). Then there exists a regular conditional probability measure Q(x, A) for F given X, and it is unique, i. e., if Q󸀠 is another regular conditional probability measure, then there exists M ∈ S with P ∘ X −1 (M) = 0, such that Q(x, A) = Q󸀠 (x, A), for A ∈ F and x ∈ S \ M.

42 | 2 Brownian motion (4) In addition to (3) suppose that S is countably determined and {x} ∈ S for every x ∈ S. Then there exists N ∈ S with P ∘ X −1 (N) = 0, such that Q(x, X −1 (B)) = 1B (x) for B ∈ S and x ∈ S \ N. In particular, Q(x, X −1 (x)) = 1, for x ∈ S \ N. Proof. (1) We suppose that (Ω, F ) is Borel isomorphic to (𝕄, ℬ(𝕄)); the other cases are more straightforward. Choose Ω = 𝕄 and F = ℬ(𝕄). Let 𝕄n = {0, 1}n and πn : 𝕄 ∋ ω = {ωn }n∈ℕ 󳨃→ {ω1 , . . . , ωn } ∈ 𝕄n be a projection. Define Fn = πn−1 (ℬ(𝕄n )). We see that σ(⋃∞ n=1 Fn ) = ℬ (𝕄). Notice that Fn is an increasing family of finite σ-fields. Note that ∫ 1A (ω)dP(ω) = ∫ P(A|G )(ω)dP(ω) B

B

for every B ∈ G . We set Pn (ω, A) = P(A|G )(ω) for A ∈ Fn . The G -measurable function ω 󳨃→ Pn (ω, A) satisfying above identity is uniquely determined for ω ∉ N(A, n) ∈ G , where P(N(A, n)) = 0. Let Nn = ⋃A∈Fn N(A, n) ∈ G . We see that P(Nn ) = 0 and Pn (ω, A) is uniquely determined for all ω ∉ Nn and for all A ∈ Fn . In particular, Pn (ω, ⋅) is a probability measure on Fn for ω ∉ Nn . Let N = ⋃∞ n=1 Nn ∈ G ; then P(N) = 0. We then construct the family of probability spaces (𝕄n , Fn , Pn (ω, ⋅)), n ∈ ℕ, for every ω ∉ N, which satisfies the consistency relation Pn (ω, A) = Pn−1 (ω, A),

n ≥ 2,

(2.1.44)

for A ∈ Fn−1 . We define the set function P(ω, ⋅) on ⋃∞ k=1 Fk by P(ω, A) = Pn (ω, A),

A ∈ Fn .

P(ω, ⋅) is well defined by (2.1.44). Fix ω ∉ N. We extend P(ω, ⋅) to a measure on ∞ ∞ σ(⋃∞ k=1 Fk ). Let An ∈ ⋃k=1 Fk be such that An ⊃ An+1 ⊃ . . . and ⋂n=1 An = 0. By Hopf’s extension theorem, it is sufficient to show that limn→∞ P(ω, An ) = 0 in order to see the existence of extension of P(ω, ⋅). Since {An }n∈ℕ is a family of closed sets in a compact space 𝕄, ⋂∞ n=1 An = 0 implies that Am = 0 for all m ≥ N with some N. Thus ̄ limn→∞ P(ω, An ) = 0. By Hopf’s extension theorem there exists a measure P(ω, ⋅) on ∞ ̄ σ(⋃∞ F ) such that P(ω, A) = P(ω, A) for any A ∈ F . Let μ be a probability ⋃ k=1 k k=1 k measure on (𝕄, ℬ(𝕄)). Define Q(ω, A) = {

̄ P(ω, A), μ(A),

ω ∉ N, ω ∈ N.

Q(ω, ⋅) is a probability measure on (𝕄, ℬ(𝕄)) for every ω ∈ 𝕄. For A ∈ ⋃∞ k=1 Fk , Q(ω, A) = P(A|G )(ω) for ω ∈ 𝕄 \ N, and Q(⋅, A) is G -measurable. These facts can be extended to A ∈ ℬ(𝕄) by a limiting argument. Then Q(⋅, ⋅) is a regular conditional probability measure for ℬ(𝕄) given G . Next we show its uniqueness. Let Q󸀠 (⋅, ⋅) be a regular conditional probability measure for ℬ(𝕄) given G , and consider

2.1 Concepts and facts of general measure theory and probability | 43

∞

M = {ω | Q(ω, A) ≠ Q󸀠 (ω, A) for some A ∈ ⋃ Fk } ∈ G . k=1

We have P(M) = 0, since for A ∈ ⋃∞ k=1 Fk and ω ∉ N, P(A|G )(ω) is unique and it follows that P(A|G )(ω) = Q(ω, A) = Q󸀠 (ω, A). Hence we can conclude that Q(ω, A) = Q󸀠 (ω, A) for A ∈ ℬ(𝕄) and for ω ∉ M by a limiting argument, and uniqueness follows. (2) Let H0 ⊂ H be a countable set. Since ∫ 1A (ω)dP(ω) = ∫ P(A|G )(ω)dP(ω) = ∫ Q(ω, A)dP(ω) B

B

B

for B ∈ G , for every A ∈ H0 there exists NA ∈ G with P(NA ) = 0 such that Q(ω, A) = 1A (ω) for ω ∉ NA . Both Q(ω, A) and 1A (ω) define probability measures on H and Q(ω, A) = 1A (ω) on H0 for ω ∈ ̸ NA . We have Q(ω, A) = 1A (ω) on H for ω ∈ ̸ NA . Let N = ⋃A∈H0 NA . Then P(N) = 0 and P(ω, A) = 1A (ω), for all A ∈ H and ω ∉ N. (3) The proof is a minor modification of part (1). Choose Ω = 𝕄, F = ℬ(𝕄), and note that ∫ 1A (ω)dP(ω) = ∫ P(A|σ(X))(ω)dP(ω) = ∫ P(A|X = x)dP ∘ X −1 (x) B

B

X(B)

for B ∈ σ(X). Write Pn (x, A) = P(A|X = x) for A ∈ Fn . An S -measurable function x 󳨃→ Pn (x, A) is uniquely determined for x ∉ N(A, n) ∈ S , where P ∘ X −1 (N(A, n)) = 0. Let Nn = ⋃A∈Fn N(A, n) ∈ S . We see that P ∘ X −1 (Nn ) = 0 and Pn (x, A) is uniquely determined for all x ∉ Nn and for all A ∈ Fn . In particular, Pn (x, ⋅) is a probability −1 measure on Fn for x ∉ Nn . Let N = ⋃∞ n=1 Nn ∈ S , and then P ∘ X (N) = 0. We then construct the family of probability spaces (𝕄n , Fn , Pn (x, ⋅)), n ∈ ℕ, for each x ∉ N, which satisfies the consistency relation Pn (x, A) = Pn−1 (x, A),

n ≥ 2,

(2.1.45)

for A ∈ Fn−1 . Define the set function P(x, ⋅) on ⋃∞ k=1 Fk by P(x, A) = Pn (x, A),

A ∈ Fn .

P(x, ⋅) is well defined by (2.1.45). Fix x ∉ N. We can extend P(x, ⋅) to a measure on σ(⋃∞ k=1 Fk ) by Hopf’s extension theorem in a similar manner to (1). Let μ be a probability measure on (𝕄, ℬ(𝕄)). Define ̄ A), P(x, Q(x, A) = { μ(A),

x ∉ N, x ∈ N.

The set function Q(x, ⋅) is a probability measure on (𝕄, ℬ(𝕄)) for every x ∈ 𝕄. For A ∈ ⋃∞ k=1 Fk , we have Q(x, A) = P(A|X = x) for x ∈ ̸ N and Q(⋅, A) is S -measurable. These facts can be extended for A ∈ ℬ(𝕄) by a limiting argument. Then Q(⋅, ⋅) is a regular

44 | 2 Brownian motion conditional probability measure for ℬ(𝕄) given X, and we can show its uniqueness in the same way as in (1). (4) The proof is similar to part (2). Note that ∫ 1A (X(ω))dP(ω) = ∫ 1X −1 (A) (ω)dP(ω) = ∫ Q(x, X −1 (A))dP ∘ X −1 (x) X −1 (B)

B

X −1 (B)

for B ∈ S . On the other hand, we have ∫ 1A (X(ω))dP(ω) = ∫ 1A (x)dP ∘ X −1 (x) X −1 (B)

B

for B ∈ S . Let S0 ⊂ S be a countable set. If A ∈ S0 , then there exists NA ∈ S with P ∘ X −1 (NA ) = 0 such that Q(x, X −1 (A)) = 1A (x) for x ∈ ̸ NA . Both Q(x, X −1 (A)) and 1A (x) define probability measures on S , and Q(x, X −1 (A)) = 1A (x) on S0 for x ∉ NA . We see that Q(x, X −1 (A)) = 1A (x) on H for x ∈ ̸ NA . Set N = ⋃A∈H0 NA . Then P ∘ X −1 (N) = 0 and Q(x, X −1 (A)) = 1A (x) for all A ∈ S and x ∉ N. Intuitively, (2) and (4) of Theorem 2.41 suggest that the supports of Q(ω, ⋅) and Q(x, ⋅) are {ω} and X −1 (x), respectively. In what follows the regular conditional probability measures Q(ω, A) and Q(x, A) will be denoted by P(A|G ) and P(A|X = x), respectively, unless any confusion may arise. Remark 2.42. (1) It is well known that a Polish space with a Borel σ-field is a standard measurable space, and every measurable subset of a standard measurable space with the induced σ-field is also a standard measurable space. (2) If (Ω, F ) is a standard measurable space, then F is countably determined, and {x} ∈ F for every x ∈ Ω. Corollary 2.43. Let (Ω, F , P) be a probability space such that Ω is a Polish space, and F = ℬ(Ω). Let G be a sub-σ-field of F , S also a Polish space, and S = ℬ(S). Consider the random variable X : Ω → S. The following hold. (1) (a) There exists a regular conditional probability measure Q(ω, A) for F given G , and it is unique. (b) There exists N ∈ G with P(N) = 0, such that Q(ω, A) = 1A (ω) for A ∈ G and ω ∈ Ω \ N. (2) (a) There exists a unique regular conditional probability measure Q(x, A) for F given X, such that Q(x, X −1 (B)) = 1B (x) for B ∈ S and x ∈ S \ N. In particular, Q(x, X −1 (x)) = 1, for x ∈ S \ N. Proof. Since (Ω, F ) is a standard measurable space, x ∈ S implies {x} ∈ S , and both F and S are countably determined by Remark 2.42, thus the corollary follows directly from Theorem 2.41.

2.2 Random processes | 45

Example 2.44. Let Ω and S be Polish spaces, and (Ω, F ) and (S, S ) be measurable spaces, where both F and S are Borel σ-fields. Consider a probability measure P on (Ω, F ) and a random variable X : Ω → S. Let Q(x, ⋅), x ∈ S, be the related regular conditional probability measure, and dPX the distribution of X on S. We see that ∫ 1A (ω)dP(ω) = ∫ Q(x, A)dPX (x) = ∫ dPX (x) ∫ 1A (ω)Q(x, dω). Ω

S

S

Ω

Write Q(x, dω) = P x (dω), i. e., 𝔼xP [. . .] = ∫Ω . . . Q(x, dω). Hence 𝔼P [f ] = ∫ 𝔼xP [f ]dPX (x) S

and 𝔼xP [f ] = 𝔼P [f |X = x], for bounded Borel measurable functions f on S.

2.2 Random processes 2.2.1 Basic concepts and facts A random process is a mathematical model of a phenomenon evolving in time in a way which cannot be predicted with certainty. Let (Ω, F , P) be a probability space and J be a given set. A family of S-valued random variables (Xt )t∈J is called a random process with index set J. When S = ℝd we refer to it as a real-valued random process, no matter the dimension. If J is uncountable, then (Xt )t≥0 is called a continuous-time random process, and if J is countable, then (Xn )n∈ℕ or (Xn )n∈ℕ̇ is called a discrete-time random process. Here we recall ℕ̇ = ℕ ∪ {0}. For any fixed t ∈ J the map ω 󳨃→ Xt (ω) is an S-valued random variable on Ω, while for any fixed ω ∈ Ω the map t 󳨃→ Xt (ω) is a function called (sample) path. This distinction justifies for any random process to address distributional properties on the one hand and sample path properties on the other. Let (Ω, F , P) be a probability space and X = (X1 , . . . , Xd ) an ℝd -valued dimensional random valiable on it. Define the measurable function F : ℝd → ℝ by F(x1 , . . . , xd ) = P(X1 ≤ x1 , . . . , Xd ≤ xd ) = P ∘ X −1 ((−∞, x1 ] × ⋅ ⋅ ⋅ × (−∞, xd ]).

(2.2.1)

This is called the d-dimensional distribution function of X. Recall the more general definition. Definition 2.45 (Distribution functions). The function F : ℝd → ℝ is called a d-dimensional distribution function if

46 | 2 Brownian motion (1) F(x1 , . . . , xn ) is right-continuous in each variable when the others are fixed; (2) if (a1 , . . . , ad ) and (b1 , . . . , bd ) are points in ℝd with ak < bk , then d

∏ Δ(ak , bk )F(x) ≥ 0, k=1 kth

kth

where Δ(ak , bk )F(x) = F(x1 , . . . , bk , . . . , xd ) − F(x1 , . . . , ak , . . . , xd ); (3) if xj ↓ −∞ for some j, then F(x1 , . . . , xd ) ↓ 0; (4) if xj ↑ +∞ for all 1 ≤ j ≤ d, then F(x1 , . . . , xd ) ↑ 1. By this definition, the function F of X defined by (2.2.1) is a distribution function. Conversely, for any d-dimensional distribution function F we define d

P̃ F ((a1 , b1 ] × ⋅ ⋅ ⋅ (ad , bd ]) = ∫ ∏ Δ(ak , bk )F(x)dx ℝd

k=1

and it is known that P̃ F can be extended to a measure PF on (ℝd , ℬ(ℝd )). The probability measure PF on ℝd is a Lebesgue–Stieltjes measure, denoted by dF(x) instead of PF (dx). Let PX be the distribution of X on ℝd and F be defined by (2.2.1). We have 𝔼[f (X)] = ∫ f (x)PX (dx) = ∫ f (x)dF(x). ℝd

(2.2.2)

ℝd

Conversely, let F be a distribution function. It is known that there is a probability space and a d-dimensional random variable X such that its distribution function is F. A stronger statement is as follows. A sequence (Fn (x1 , . . . , xn )n )n∈ℕ of distribution functions is said to satisfy a consistency relation if lim Fn (x1 , . . . , xn ) = Fn−1 (x1 , . . . , xn−1 ),

xn ↑∞

n ≥ 2.

(2.2.3)

Corollary 2.46. Let (Fn (x1 , . . . , xn ))n∈ℕ be a sequence of distribution functions satisfying the consistency relation (2.2.3). Then there exists a probability space (Ω, F , P) and a discrete-time random process (Xn )n∈ℕ such that P(X1 ≤ x1 , . . . , Xn ≤ xn ) = Fn , (x1 , . . . , xn ) for all n ≥ 1. Proof. For Λ1 ⊂ Λ2 ⊂ ℕ let πΛ1 Λ2 : ℝ|Λ2 | → ℝ|Λ1 | be the projection defined as (2.1.20) with J replaced by the countable set ℕ. Let Ω = ℝℕ = {ω : ℕ → ℝ} and −1

A = {πΛℕ (E) ⊂ ℝ

ℕ

| E ∈ ℬ(ℝ|Λ| ), Λ ⊂ ℕ, |Λ| < ∞}.

For a finite index set Λ = {j1 , . . . , jn } we define the probability measure μΛ (E) = ∫ 1E (x1 , . . . , xn )dFn ℝn

2.2 Random processes |

47

for E ∈ ℬ(ℝn ). By consistency of (Fn )n∈ℕ it is straightforward to check that for Λ1 ⊂ Λ2 , μΛ2 (πΛ−11 Λ2 (E)) = μΛ1 (E)

(2.2.4)

̃ −1 (E)) = μΛ (E) for E ∈ ℬ(ℝ|Λ| ). Since for E ∈ ℬ(ℝ|Λ1 | ). Define P̃ : A → [0, ∞) by P(π Λℕ (2.2.4) implies that the set of measures {μΛ }Λ⊂ℕ,|Λ| 0 by the assumption. This implies by the Borel–Cantelli lemma that ∞

∞

P( ⋃ ⋂ { maxn |Xk/2n − X(k−1)/2n | ≥ 2−γn }c ) = 1. m=1 n=m 1≤k≤2

There exists Ω0 ∈ F such that P(Ω0 ) = 1, and for all ω ∈ Ω0 there exists n0 = n0 (ω) such that max |Xk/2n (ω) − X(k−1)/2n (ω)| < 2−γn

(2.2.9)

1≤k≤2n

for all n > n0 . For each n let Dn = {k/2n | k = 0, 1, . . . , 2n } and D = ⋃∞ n=1 Dn ⊂ [0, 1] be the set of dyadic rationals in [0, 1]. We fix ω ∈ Ω0 and n > n0 , and show that for every m>n m

|Xt (ω) − Xs (ω)| ≤ 2 ∑ 2−γj

(2.2.10)

j=n+1

for arbitrary s, t ∈ Dm with 0 < t − s < 2−n . The estimate (2.2.10) can be proven as follows. For m = n + 1 we only have t = k/2m and s = (k − 1)/2m , and thus it follows from (2.2.9). For m > n + 1 it can be shown by induction. Suppose that (2.2.10) is valid for m = n + 1, . . . , M − 1. Take s, t ∈ DM such that s < t and consider the numbers t 1 = max{u ∈ DM | u ≤ t} and s1 = min{u ∈ DM−1 | u ≥ s}. Note that s ≤ s1 ≤ t 1 ≤ t, s1 − s ≤ 2−M and t − t 1 ≤ 2−M . From (2.2.9) we have |Xs1 (ω) − Xs (ω)| ≤ 2−γM ,

|Xt (ω) − Xt 1 (ω)| ≤ 2−γM ,

and from (2.2.10) with m = M − 1 we have M−1

|Xt 1 (ω) − Xs1 (ω)| ≤ 2 ∑ 2−γj . j=n+1

Thus M

|Xt (ω) − Xs (ω)| ≤ |Xt (ω) − Xt 1 (ω)| + |Xt 1 (ω) − Xs1 (ω)| + |Xs1 (ω) − Xs (ω)| ≤ 2 ∑ 2−γj j=n+1

leading to (2.2.10). For every s, t ∈ D with 0 < t − s < 2−n0 (ω) , choose n ≥ n0 (ω) such that 2−(n+1) ≤ t − s ≤ 2−n . We have from (2.2.10) |Xt (ω) − Xs (ω)| ≤

2 2 2−γ(n+1) ≤ |t − s|γ 1 − 2−γ 1 − 2−γ

(2.2.11)

for 0 < t − s < 2−n0 (ω) . This implies that (Xt )t∈D is uniformly Hölder continuous for every ω ∈ Ω0 . Define 0, { { X̃ t (ω) = {Xt (ω), { {limn→∞ Xsn (ω),

ω ∉ Ω0 , ω ∈ Ω0 , t ∈ D, ω ∈ Ω0 , t ∉ D.

(2.2.12)

50 | 2 Brownian motion Here (sn )n∈ℕ ⊂ D is a sequence of dyadic rationals such that sn → t as n → ∞. Note that Xsn (ω) is a Cauchy sequence by (2.2.11), and there exists limn→∞ Xsn (ω), which is independent of the choice of sn . Hence we have |X̃ t (ω) − X̃ s (ω)| ≤

2 |t − s|γ 1 − 2−γ

for |t − s| < 2−n0 (ω) . Thus (X̃ t )t≥0 is a continuous random process. Note that Xt (ω) = X̃ t (ω) for t ∈ D and ω ∈ Ω0 , and Xsn → Xt in probability, which implies that there exists a subsequence s󸀠n such that Xs󸀠n converges almost surely to Xt . On the other hand, Xs󸀠n → X̃ t by the definition. Hence Xt = X̃ t almost surely for every t ∈ [0, 1], and (X̃ t )t∈[0,1] is a continuous version of (Xt )t∈[0,1] . 2.2.2 Martingale properties Martingales are a class of random processes with particularly convenient properties. In this section we briefly review some basic definitions and facts. Definition 2.50 (Filtration). Let (Ω, F ) be a measurable space and (Ft )t≥0 a family of sub-σ-fields of F . The collection (Ft )t≥0 is called a filtration whenever Fs ⊂ Ft for s ≤ t. The given measurable space endowed with a filtration is called a filtered space. A filtration (Ft )t≥0 is called right-continuous if Ft = ⋂s>t Fs . A filtered space is denoted by (Ω, F , (Ft )t≥0 ). Intuitively, Ft contains the information known to an observer at time t. Let (Xt )t≥0 be an S-valued random process. A convenient choice of filtration is the natural filtration given by FtX = σ(Xs , 0 ≤ s ≤ t), containing the information accumulated by observing Xs up to time s = t. Definition 2.51 (Adapted process). Let (Ω, F , (Ft )t≥0 ) be a filtered space and (Xt )t≥0 a random process on this space. The process is called (Ft )s≥0 -adapted whenever Xt is Ft -measurable for every t ≥ 0. If a process (Xt )t≥0 is (Ft )t≥0 -adapted it means that the process does not depend on events at time t other than contained in Ft , in particular, it does not depend on the future. Obviously, (Xt )t≥0 is adapted to its natural filtration (FtX )t≥0 , or equivalently, the natural filtration (FtX )t≥0 is the smallest filtration making (Xt )t≥0 adapted. Definition 2.52 (Martingale, supermartingale, and submartingale). Consider a filtered probability space (Ω, F , (Ft )t≥0 , P). The random process (Xt )t≥0 is called an (Ft )t≥0 -martingale (resp. supermartingale, submartingale) whenever (1) 𝔼[|Xt |] < ∞, for every t ≥ 0; (2) (Xt )t≥0 is (Ft )t≥0 -adapted; (3) 𝔼[Xt |Fs ] = Xs (resp. ≤, ≥), almost surely for every 0 ≤ s ≤ t.

2.2 Random processes |

51

The martingale property intuitively means that the best prediction of the future mean of a random process is the current value, i. e., the are no embedded biases and trends in the evolution of the process. We recall a useful inequality next, which is an extension of the Schwarz inequality 𝔼[X]2 ≤ 𝔼[X 2 ], X ∈ L2 (Ω, dP), to arbitrary convex functions beyond the quadratic power function. Proposition 2.53 (Jensen’s inequality). Let (Ω, F , P) be a probability space and X a random variable on it. Let ϕ : ℝ → ℝ be convex, and suppose that X, ϕ(X) ∈ L1 (Ω). Then ϕ(𝔼[X|G ]) ≤ 𝔼[ϕ(X)|G ], for every sub-σ-field G ⊂ F . Proof. By the definition of a convex function, the set A = {(x, y) ∈ ℝ2 | y ≥ ϕ(x)} is a convex subset of ℝ2 . Therefore we can find a straight line ℓx (y) = ϕ(x) + a(y − x) through every boundary point (x, ϕ(x)) of A such that {ℓx (y) | y ∈ ℝ} does not intersect the interior of A. This means that ℓx (y) ≤ ϕ(y), for all y ∈ ℝ. Choosing x = 𝔼[X|G ] and y = X, we obtain the pointwise inequality ℓ𝔼[X|G ] (X) ≤ ϕ(X). Taking conditional expectations, we find 𝔼[ϕ(X)|G ] ≥ 𝔼[ℓ𝔼(X|G ) (X)|G ] = 𝔼[ϕ(𝔼[X|G ]) + a(X − 𝔼[X|G ])|G ] = ϕ(𝔼[X|G ]). This shows the claim. Given a martingale (Xt )t≥0 , by Jensen’s inequality it is immediate that (ϕ(Xt ))t≥0 is a submartingale for any convex function ϕ if ϕ(X) ∈ L1 (Ω, dP). In particular, (Xt2 )t≥0 is a submartingale. The following observation on conditional expectation will be useful below. Let (Ω, F , P) be a probability space, A ⊂ F a sub-σ-field, and N = {A ∈ F | P(A) = 0}. In the next lemma we show that 𝔼[X|A ] coincides with 𝔼[X|σ(A ∪ N )]. Lemma 2.54. Let (Ω, F , P) be a probability space. Suppose that A , B ⊂ F are independent sub-σ-fields. Let X be a random variable on it, and also suppose that σ(σ(X) ∪ A ) and B are independent. Then 𝔼[X|σ(A ∪ B )] = 𝔼[X|A ]. In particular 𝔼[X|σ(A ∪ N )] = 𝔼[X|A ], where N = {A ∈ F | P(A) = 0} denotes the family of all null sets. Proof. Let Y = 𝔼[X|A ]. The random variable Y is A -measurable, and hence σ(A ∪ B )-measurable. Let C = {A ∩ B | A ∈ A , B ∈ B }. Since A , B ⊂ C , we can see σ(A ∪ B ) ⊂ σ(C ). On the other hand, C is a π-system. Hence the π–λ theorem yields σ(C ) ⊂ σ(A ∪ B ), and thus σ(C ) = σ(A ∪ B ). Since D = {C ∈ F | 𝔼[1C X] = 𝔼[1C Y]}

is a σ-field, in order to show the lemma it is sufficient to show that 𝔼[1A∩B X] = 𝔼[1A∩B Y],

A ∩ B ∈ C.

(2.2.13)

52 | 2 Brownian motion Indeed, (2.2.13) implies C ⊂ D , which yields that σ(C ) ⊂ D and 𝔼[1C X] = 𝔼[1C Y] for all C ∈ σ(C ). By the independence of A and B , it follows that 𝔼[1A∩B Y] = 𝔼[1A 1B Y] = 𝔼[1B ]𝔼[1A Y]. Since 𝔼[1A Y] = 𝔼[1A X], we see that 𝔼[1A∩B Y] = 𝔼[1B ]𝔼[1A X]. By the independence of B and σ(σ(X) ∪ A ) we also get 𝔼[1B ]𝔼[1A X] = 𝔼[1B 1A X] = 𝔼[1A∩B X]. Thus 𝔼[1A∩B X] = 𝔼[1A∩B Y] follows. Proposition 2.55. Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space. Define +

Ft = ⋂Fs s>t

and

F t = σ(Ft ∪ N ),

where N = {A ∈ F | P(A) = 0} is the family of all null sets. Then we have the following: (1) F t = {A ∈ F | ∃B ∈ Ft such that A ≠ B, P(AΔB) = 0}; (2) Ft ⊂ Ft+ ⊂ F t ; (3) F t is right-continuous; (4) 𝔼[X|Ft ] = 𝔼[X|F t ], for every random variable X ∈ L1 (Ω). Proof. Let Gt = {A ∈ F | there exists B ∈ Ft such that A ≠ B, P(AΔB) = 0}. We show that Gt = F t . Since Gt contains both Ft and N , and is a σ-field, we have F t ⊂ Gt . To show the inverse inclusion, let A ∈ Gt . There exists B ∈ Ft such that P(AΔB) = 0. Note that N1 = A \ B ∈ N and N2 = B \ A ∈ N . By A = (B ∪ N1 ) \ N2 , we see that A ∈ σ(Ft ∪ N ) = F t . Thus Gt ⊂ F t and (1) is proven. Next consider (2). The first inclusion is trivial, we thus show the second. Let A ∈ + Ft . It follows that 𝔼[1A |Ft ] = 𝔼[1A |Ft+ ]. Consider g = 𝔼[1A |Ft ], which is an Ft -measurable function and g = 𝔼[1A |Ft+ ] = 1A . The range of g is {0, 1}. Let B = g −1 ({1}). Clearly, B is Ft -measurable and 1A = g = 1B almost surely, which implies that P(AΔB) = 0. Then (2) is proven by (1). Consider (3). It is sufficient to show ⋂r>s Fr ⊂ Fs . Take A ∈ ⋂r>s Fr , and consider a sequence (rn )n∈ℕ such that rn ↓ s. Since A ∈ Frn , by part (1) there exists Bn ∈ Frn such ∞ ∞ ∞ + that P(AΔBn ) = 0. Let C = ⋂∞ n=1 ⋃m=n Bm . Since ⋃m=n Bm ∈ Frn , C ∈ ⋂n=1 Frn = Fs ⊂ F s by (2). Thus there exists D ∈ Fs such that P(CΔD) = 0.

(2.2.14)

∞ Since AΔC ⊂ AΔ(⋃∞ n=1 Bn ) = ⋃n=1 (AΔBn ), we have ∞

P(AΔC) ≤ ∑ P(A △ Bn ) = 0. n=1

(2.2.15)

Hence by (2.2.14) and (2.2.15), P(AΔD) = 0 and D ∈ Fs follows, and A ∈ F s is proven. Finally (4) follows from Lemma 2.54.

2.2 Random processes | 53

We have seen above that the filtration (Ft )t≥0 is not right-continuous in general, while the filtration (Ft+ )t≥0 is right-continuous by definition. However, we will see in the study of stopping times in the next section that it is desirable to work with a rightcontinuous filtration including all null sets. The family (F t )t≥0 defined below has all the required properties. Definition 2.56 (Augmented filtration). Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space. Define F t = σ(Ft ∪ N ), where N = {A ∈ F | P(A) = 0} denotes the family of all null sets. Then (F t )t≥0 is called an augmented filtration.

2.2.3 Stopping times and optional sampling There is a useful weakening of the martingale property which removes the condition of integrability in Definition 2.52. To introduce and apply it below, we first need a concept of random times. Definition 2.57 (Stopping time). Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space. A random variable τ : Ω → [0, ∞] such that {τ ≤ t} ∈ Ft for all t ≥ 0 is called a stopping time. For a stopping time τ, we consider Fτ = {A ∈ F | A ∩ {τ ≤ t} ∈ Ft for all t ≥ 0}.

Intuitively, Fτ contains the events occurring before or at the given random time τ. Lemma 2.58. Suppose the filtration (Ft )t≥0 is right-continuous. Then τ is a stopping time with respect to (Ft )t≥0 if and only if {τ < t} ∈ Ft , for every t ≥ 0. Proof. Suppose that {τ ≤ t} ∈ Ft . It follows that {τ < t} = ⋃∞ n=1 {τ ≤ t − 1/n} ∈ Ft . Next ∞ we suppose that {τ < t} ∈ Ft . Since {τ ≤ t} = ⋂n=m {τ < t + 1/n} ∈ Ft+1/m , we have {τ ≤ t} ∈ ⋂∞ m=1 Ft+1/m = Ft . Thus τ is a stopping time. Example 2.59. Let (Ft )t≥0 be right-continuous. If (τn )n∈ℕ is a sequence of stopping times with respect to (Fn )n∈ℕ and τn ↓ τ as n → ∞, then τ is also a stopping time since {τ < t} = ⋃∞ n=1 {τn < t} ∈ Ft . Similarly, if a sequence of stopping times (τn )n∈ℕ satisfies that τn ↑ τ as n → ∞, then again τ is a stopping time since {τ ≤ t} = ⋂∞ n=1 {τn < t} ∈ Ft . The first hitting time by an adapted process of an open or a closed set plays an important role in martingale theory. First hitting times are stopping times with respect to the augmented filtration. Lemma 2.60 (First hitting time). Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space such that (Ft )t≥0 is right-continuous and contains all null sets. Also, let E be a Polish space and consider the measurable space (E, ℬ(E)). Let (Xt )t≥0 be an E-valued continuous (Ft )-adapted process, and A ⊂ E an open or closed set. The first hitting time of A by

54 | 2 Brownian motion (Xt )t≥0 defined by τA = inf{t > 0 | Xt ∈ A} is a stopping time with respect to (Ft )t≥0 . Proof. There exists N ∈ F such that P(N) = 0 and the paths in Ω \ N ∋ t 󳨃→ Xt (ω) ∈ E are continuous. Let Yt (ω) = Xt (ω), ω ∉ N, and Yt (ω) = e, ω ∈ N. Here e ∈ E is arbitrary, and we fix it. t 󳨃→ Yt (ω) is continuous for every ω ∈ Ω. Define τA󸀠 = inf{t > 0 | Yt ∈ A}, and let C = {τA ≤ t} and C 󸀠 = {τA󸀠 ≤ t}. Since Ft ∋ N, we see that Ft ∋ C if and only if C \ N ∈ Ft , and moreover C \ N ∈ Ft , C 󸀠 \ N ∈ Ft , and C 󸀠 ∈ Ft are equivalent to each other. It follows that {τA ≤ t} ∈ Ft ⇐⇒ {τA󸀠 ≤ t} ∈ Ft . Hence we may assume that t 󳨃→ Xt (ω) is continuous for every ω ∈ Ω in the remainder of the proof. By the right-continuity of (Ft )t≥0 , {τA < t} ∈ Ft if and only if {τA ≤ t} ∈ Ft . This is due to Lemma 2.58. Suppose that A is open. It is immediate to see that {τA < t} =

⋃ {Xq ∈ A}.

q∈ℚ,q N, + X∞ ≤ N.

+ N We see that (X∞ ) ∈ L2 (Ω) for each N. Take ε > 0. By the dominated convergence + N + theorem, |𝔼[(X∞ ) ] − 𝔼[X∞ ]| < ε for all N > N0 , with some N0 . Since P(Xn+ > a) ≤ + 𝔼[Xn ]/a ≤ 𝔼[X∞ ]/a, it also follows that + N + 2N 1/2 lim 𝔼[(X∞ ) 1{Xn+ >a} ] ≤ 𝔼[(X∞ ) ] P(Xn+ > a)1/2 ≤ ε

a→∞

for all a > a0 with some a0 . We see that + + N + N + 𝔼[X∞ 1{Xn+ >a} ] ≤ 𝔼[(X∞ ) 1{Xn+ >a} ] + |𝔼[(X∞ ) ] − 𝔼[X∞ ]| ≤ 2ε

follows for all a > a0 uniformly in n. This implies by (2.2.25) that sup 𝔼[Xn+ 1{Xn+ >a} ] < 2ε n∈ℕ̇

and thus infa≥0 supn∈ℕ̇ 𝔼[Xn+ 1{Xn+ >a} ] = 0 is obtained, showing that (Xn+ )n∈ℕ̇ is uniformly integrable. Next we suppose that (Xn+ )n∈ℕ̇ is uniformly integrable. By Corol+ lary 2.32, Xn+ → X∞ in L1 (Ω). Take a > 0. Then Xn ∨ (−a) → X∞ ∨ (−a), as n → ∞ in L1 (Ω), and (Xn ∨ (−a))n∈ℕ̇ is a submartingale. We have 𝔼[X∞ ∨ (−a)|Fn ] = lim 𝔼[Xm ∨ (−a)|Fn ] ≥ Xn ∨ (−a), m→∞

and letting a ↑ ∞ we obtain 𝔼[X∞ |Fn ] ≥ Xn . Proposition 2.68 (Lévy’s upward theorem). Let (Ω, F , (Fn )n∈ℕ̇ , P) be a filtered probability space, and Y ∈ L1 (Ω). Consider Xn = 𝔼[Y|Fn ]. Then (Xn )n∈ℕ̇ is a uniformly integrable martingale and X∞ = limn→∞ Xn exists almost surely and X∞ ∈ L1 (Ω). Furthermore, X∞ = 𝔼[Y|F∞ ], where F∞ = σ(⋃∞ n=0 Fn ).

(2.2.26)

2.2 Random processes | 59

Proof. Since |Xn | ≤ 𝔼[|Y||Fn ], it can be proven that (Xn )n∈ℕ̇ is uniformly integrable in a similar manner to the proof of Proposition 2.67. Also, we see that 𝔼[Xm |Fn ] = 𝔼[𝔼[Y|Fm ]|Fn ] = 𝔼[Y|Fn ] = Xn for m > n, i. e., (Xn )n∈ℕ̇ is a martingale. We see thus that X∞ = limn→∞ Xn exists a. s., X∞ ∈ L1 (Ω), and 𝔼[X∞ |Fn ] = Xn by Proposition 2.67. Note that X∞ is F∞ -measurable. Let C = {B ∈ F | 𝔼[X∞ 1B ] = 𝔼[Y1B ]}.

If B ∈ Fn , then 𝔼[Y1B ] = 𝔼[Xn 1B ] = 𝔼[X∞ 1B ]. Thus ⋃∞ n=0 Fn ⊂ C . Clearly, C is a ∞ λ-system. By the π–λ theorem it follows that C ⊃ σ(⋃n=0 Fn ). This proves (2.2.26). The martingale (Xn )n∈ℕ̇ defined by Xn = 𝔼[Y|Fn ] is called Doob’s martingale. Further assuming uniform integrability of (Xn )nℕ̇ , Theorem 2.63 can be extended to unbounded stopping times. Theorem 2.69 (Discrete optional sampling theorem: unbounded case). Let (Xn )n∈ℕ̇ be a uniformly integrable and discrete-time martingale on a filtered probability space ̇ (Ω, F , (Fn )n∈ℕ̇ , P), and let τ, ρ be ℕ-valued stopping times such that τ(ω) ≤ ρ(ω), for all ω ∈ Ω. Then 𝔼[Xρ |Fτ ] = Xτ .

(2.2.27)

Proof. Note that supn∈ℕ̇ 𝔼[|Xn |] < ∞ by Corollary 2.32. In the same way as the proof of Propositions 2.68, X∞ = limn→∞ Xn exists almost surely in L1 (Ω), and Xn = 𝔼[X∞ |Fn ]. Theorem 2.63 gives 𝔼[X∞ |Fτ∧n ] = Xτ∧n = 𝔼[Xn |Fτ∧n ].

(2.2.28)

Letting n → ∞ on both sides above, by Proposition 2.68 we have 𝔼[X∞ |G ] = Xτ ,

(2.2.29)

where G = σ(⋃∞ n=0 Fτ∧n ). We need to show that G can be replaced by Fτ . Take A ∈ Fτ . We have 𝔼[1A X∞ ] = 𝔼[1A∩{τa} ],

m ≤ n.

(2.2.33)

Furthermore, let Mn− = max{−Mn , 0}. Since the function x 󳨃→ max{x, 0} is convex, it − follows from Jensen’s inequality that 𝔼[Mn− |Fm ] ≥ Mm for n ≤ m, and it is also true − − that 𝔼[Mn ] ≥ 𝔼[Mm ] for n ≤ m. Hence 𝔼[|Mn |] = 𝔼[Mn ] + 2𝔼[Mn− ] ≤ sup 𝔼[Mn ] + 2𝔼[M0− ] n∈ℕ̇

and supn∈ℕ̇ 𝔼[|Mn |] < C follows with some C. We conclude that P(|Mn | > a) ≤

1 C sup 𝔼[|Mn |] = . a n∈ℕ̇ a

We truncate Mm by N, { { N Mm = {Mm , { {−N,

Mm > N, −N ≤ Mm ≤ N, Mm < −N.

(2.2.34)

2.2 Random processes |

61

N Note that Mm ∈ L2 (Ω) for each N. Let ε > 0 be given. By the dominated convergence N theorem, |𝔼[|Mm |] − 𝔼[|Mm |]| < ε for all N > N0 and some N0 . By (2.2.34) it also follows that N N 1/2 𝔼[|Mm |1{|Mn |>a} ] ≤ 𝔼[|Mm |] P(|Mn | > a)1/2 ≤ ε

for all a > a0 with some a0 , uniformly in n. It follows that N N 𝔼[|Mm |1{|Mn |>a} ] ≤ 𝔼[|Mm |1{|Mn |>a} ] + |𝔼[|Mm |] − 𝔼[|Mm |]| ≤ 2ε

for all a > a0 , uniformly in n. This implies by (2.2.33) that sup 𝔼[|Xn |1{|Mn |>a} ] = sup 𝔼[|Xm |1{|Mn |>a} ] < 2ε n∈ℕ̇

n∈ℕ̇

and then infa≥0 supn∈ℕ̇ 𝔼[|Mn |1{|Mn |>a} ] = 0 is obtained. We thus see that {Xτn }n∈ℕ̇ is uniformly integrable. Similarly, we can show that {Xρn }n∈ℕ̇ is uniformly integrable. Hence Xτn and Xρn converge in L1 (Ω) by Corollary 2.32. By right-continuity Xτn → Xτ and Xρn → Xρ follow almost surely, and we see that Xτn → Xτ and Xρn → Xρ in L1 (Ω). Now take A ∈ Fτ , by (2.2.30), 𝔼[Xτn 1A ] = 𝔼[Xρn 1A ]. By L1 convergence, this yields 𝔼[Xτ 1A ] = 𝔼[Xρ 1A ], if we let n tend to infinity. Corollary 2.71. Let (Xt )t≥0 be a right-continuous martingale on a filtered probability space (Ω, F , (Ft )t≥0 , P), and τ, ρ stopping times such that τ(ω) ≤ ρ(ω), for all ω ∈ Ω. Then 𝔼[Xt∧ρ |Fτ ] = Xt∧τ . Proof. Fix t ≥ 0 and define the martingale by (YT )T≥0 = (Xt∧T )T≥0 . Write Y∞ = Xt . It is sufficient to show that (YT )T≥0 is uniformly integrable by Theorem 2.70. Let a > 0. We estimate 𝔼[|YT |1{|YT |>a} ]. Note that 𝔼[Y∞ |FT ] = YT . In the same way as (2.2.32), we have the identity 𝔼[|YT |1{|YT |>a} ] = 𝔼[YT ] − 𝔼[YT 1{YT ≤a} ] − 𝔼[YT 1{YT a} ] = 𝔼[|Y∞ |1{|YT |>a} ]. In the same way as in the proof of uniform integrability of {Xτn }n∈ℕ̇ in Theorem 2.70, we have supT≥0 𝔼[|YT |] < ∞ and infa≥0 supT≥0 𝔼[|YT |1{|YT |>a} ] = 0 follows. Thus (YT )T≥0 is uniformly integrable.

62 | 2 Brownian motion Theorem 2.72 (Martingale inequality). Let (Xt )t≥0 be a right-continuous martingale on a filtered probability space (Ω, F , (Ft )t≥0 , P). (1) If XT ∈ Lp (Ω) for some p ≥ 1, then for all λ > 0, P ( sup |Xt | ≥ λ) ≤ 0≤t≤T

1 𝔼[|XT |p ]. λp

(2.2.36)

(2) If XT ∈ Lp (Ω) for some p > 1, then 𝔼[ sup |Xt |p ] ≤ ( 0≤t≤T

p p ) 𝔼[|XT |p ]. p−1

(2.2.37)

Proof. Fix λ > 0. Let (In )n∈ℕ be an increasing sequence of finite subsets of I = [0, T] such that ⋃∞ n=1 In = [0, T] ∩ ℚ ∩ {T}. Let (Yt )t≥0 be a right-continuous submartingale with respect to (Ft )t≥0 . Since t 󳨃→ Yt (ω) is right-continuous, we see that ∞

{sup Yt ≥ λ} = ⋃ {max Yt ≥ λ}. t∈I

t∈In

n=1

Let In = {j1 , . . . , jN } such that j1 < . . . < jN , and consider (Yjm )jm ∈In . Define τ={

min{jn | Yjn ≥ λ} jN

if {jn | Yjn ≥ λ} ≠ 0, if {jn | Yjn ≥ λ} = 0.

Note that τ is a bounded stopping time such that τ ≤ jN . Since (Yt )t≥0 is a submartingale, by Theorem 2.63 we have 𝔼[YjN ] = 𝔼[𝔼[YjN |ℱτ ]] ≥ 𝔼[Yτ ] = 𝔼[Yτ 1{max1≤n≤N Yj ≥ λP( max Yjn ≥ λ) + 𝔼[YjN 1{max1≤n≤N Yj n=1,...,N

n

n

≥λ} ]

+ 𝔼[YjN 1{max1≤n≤N Yj

n

(M + 1)/n} ∈ M(M+1)/n . i=1

It then follows that 𝔼[1A∩{τ>(M+1)/n} Xt ] = 𝔼[𝔼[1A∩{τ>(M+1)/n} Xt |M(M+1)/n ]] = 𝔼[1A∩{τ>(M+1)/n} X(M+1)/n ], since (Xt )t≥0 is a martingale. By (2.2.41) we have M−1

𝔼[1A∩{τ>(j+1)/n} Xt∧τn ] = ∑ 𝔼[1A∩{l/nM/n} X(M+1)/n ]. (2.2.42) l=j+1

Proceeding in this manner step by step, we arrive at 𝔼[1A∩{τ>(j+1)/n} Xt∧τn ] = 𝔼[1A∩{τ>(j+1)/n} X(j+2)/n ] = 𝔼[1A∩{τ>(j+1)/n} X(j+1)/n ] = 𝔼[1A∩{τ>(j+1)/n} Xs ] = 𝔼[1A∩{τ>(j+1)/n} Xs∧τn ],

since s = (j + 1)/n, A ∩ {τ > (j + 1)/n} = ⋂m i=1 {Xti ∈ Bi } ∩ {τ > (j + 1)/n} is in M(j+1)/n and s ∧ τn = s on this set. Recalling (2.2.40), we have 𝔼[1A Xs∧τn ] = 𝔼[1A Xt∧τn ].

(2.2.43)

Next we consider the case of j/n < s < (j + 1)/n for some j. We modify τn by taking 0, { { { { { i+1 , τn = { n { s, { { { j+1 { n ,

τ = 0, i s, since |Xt | is a submartingale. Taking A = {|Xs∧τn | > λ} = {Xs∧τn ∈ [−λ, λ]c },

t = N + 1,

2.2 Random processes |

65

we get from (2.2.45) 𝔼[1{|Xs∧τ

n

|>λ} |Xs∧τn |]

≤ 𝔼[1{|Xs∧τ

n

|>λ} |Xτn |].

(2.2.46)

Since |Xt | is a submartingale, we see that 𝔼[1{|Xs∧τ

n

|>λ}∩{τn =sni } |Xτn |]

≤ 𝔼[𝔼[1{|Xs∧τ

n

= 𝔼[1{|Xs∧τ

n

|>λ}∩{τn =sni } |Xsni |]

|>λ}∩{τn =sni } |XN+1 ||Msni ]]

= 𝔼[1{|Xs∧τ

n

|>λ}∩{τn =sni } |XN+1 ].

(2.2.47)

Summing over i, we have 𝔼[1{|Xs∧τ

n

|>λ} |Xτn |]

≤ 𝔼[1{|Xs∧τ

n

|>λ} |XN+1 |].

From (2.2.46) it follows that 𝔼[1{|Xs∧τ

n

|>λ} |Xs∧τn |]

≤ 𝔼[1{|Xs∧τ

n

|>λ} |XN+1 |].

(2.2.48)

Consider supn∈ℕ sups≥0 |Xs∧τn |. Since τ is bounded and τn ↓ τ as n ↑ ∞, we have sup sup |Xs∧τn | = sup sup |Xr | ≤ sup |Xr | ≤ sup |Xr |. n∈ℕ s≥0

n∈ℕ 0≤r≤τn

0≤r≤τ1

0≤r≤N

Since (Xt )t≥0 is a right-continuous martingale, we have P(sup sup |Xs∧τn | > λ) ≤ P( sup |Xr | > λ) ≤ n∈ℕ s≥0

0≤r≤N

𝔼[|XN |] λ

by the martingale inequality quoted in Theorem 2.72. Hence P(sup sup |Xs∧τn | > λ) → 0 n∈ℕ s≥0

as λ → ∞. The right-hand side of (2.2.48) can be estimated as 𝔼[1{|Xs∧τ

n

|>λ} |XN+1 |]

≤ 𝔼[1{sup0≤r≤N |Xr |>λ} |XN+1 |] → 0

as λ → ∞. Thus inf sup 𝔼[1{|Xs∧τ

λ≥0 n∈ℕ

n

|>λ} |Xs∧τn |]

≤ inf 𝔼[1{sup0≤r≤N |Xr |>λ} |XN+1 |] = 0 λ≥0

and {Xs∧τn }n∈ℕ is uniformly integrable and Xs∧τn converges in L1 (Ω) as n → ∞. Similarly, we can show that {Xt∧τn }n∈ℕ is also uniformly integrable. Since t 󳨃→ Xt is rightcontinuous and τn ↓ τ as n ↑ ∞, it can be seen that limn→∞ Xt∧τn = Xt∧τ in L1 (Ω). It follows that ∞ > lim supn→∞ 𝔼[|Xt∧τn |] = 𝔼[|Xt∧τ |] from Corollary 2.32. Moreover, from (2.2.43) it follows that 𝔼[1A Xs∧τ ] = 𝔼[1A Xt∧τ ].

(2.2.49)

66 | 2 Brownian motion Since this is true for any set A of the form (2.2.39), (Xt∧τ )t≥0 is a martingale and rightcontinuous in t. Finally, suppose that τ is unbounded. The stopping time τ ∧ N is bounded. Thus with N > t we get 𝔼[|Xt∧(τ∧N) |] = 𝔼[|Xt∧τ |] < ∞, and 𝔼[1A Xs∧τ∧N ] = 𝔼[1A Xt∧τ∧N ]

(2.2.50)

is obtained. Taking N > t, the relation (2.2.49) follows. To conclude this section, we discuss the Doob–Meyer decomposition of submartingales. Let (Xn )n∈ℕ̇ be a discrete submartingale with respect to (Fn )n∈ℕ̇ . Define An = An−1 + 𝔼[Xn − Xn−1 |Fn−1 ] for n ≥ 1 with A0 = 0. We have Xn = Mn + An , where Mn = Xn − An . It is direct to see that An is adapted, An ≤ An+1 , and Mn is a martingale. We can extend this to the continuous case. Definition 2.74 (Increasing process). An adapted process (At )t≥0 on a filtered probability space (Ω, F , (Ft )t≥0 , P) is called an increasing process if for a. e. ω ∈ Ω (1) A0 (ω) = 0; (2) t 󳨃→ At (ω) is a nondecreasing, right-continuous function; (3) 𝔼[At ] < ∞, for every t ≥ 0 hold. Furthermore, if 𝔼[A∞ ] < ∞ with A∞ = limt→∞ At , an increasing process (At )t≥0 is called an integrable increasing process. Definition 2.75 (Natural process). An increasing process (At )t≥0 is called a natural process if for every bounded martingale (Xt )t≥0 , 𝔼[ ∫ Xs dAs ] = 𝔼[ ∫ Xs− dAs ] (0,t]

(0,t]

holds, for every t ≥ 0. Remark 2.76. It is known that an integrable increasing process is natural if and only if it is predictable (see Definition 3.64). If an integrable increasing process is continuous, then it is natural. Definition 2.77 (Classes D and DL). Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space. Let S (resp. Sa ) be the set of all stopping times τ satisfying P(τ < ∞) = 1 (respectively, P(τ ≤ a) = 1 for a given finite number a > 0). An adapted process (Xt )t≥0 with respect to (Ft )t≥0 is said to be of class D if the family {Xτ }τ∈S is uniformly integrable, and of class DL if the family {Xτ }τ∈Sa is uniformly integrable for every 0 < a < ∞.

2.2 Random processes |

67

Every right-continuous martingale (Xt )t≥0 is of class DL, since 𝔼[|Xσ |1{|Xσ |>c} ] ≤ 𝔼[|Xa |1{|Xσ |>c} ] for σ ∈ Sa by Theorem 2.70. Here we do not need to assume that (Xt )t≥0 is uniformly integrable since σ is bounded. By the inequality sup P(|Xσ | > c) ≤ sup

σ∈Sa

σ∈Sa

𝔼[|Xσ |] 𝔼[|Xa |] ≤ , c c

it is seen that the family of random variables {Xσ }σ∈Sa is uniformly integrable. Similarly, we can show that every right-continuous uniformly integrable martingale (Xt )t≥0 is of class D. If a right-continous submartingale X = (Xt )t≥0 is represented as Xt = Mt + At , where (Mt )t≥0 is a martingale and A = (At )t≥0 an increasing process, then (Xt )t≥0 is of class DL. The converse statement is known as Doob–Meyer decomposition. We state this without proof. Theorem 2.78 (Doob–Meyer decomposition). Let (Ω, F , (Ft )t≥0 , P) be a filtered probability space. Suppose that (Ft )t≥0 is a right-continuous and complete filtration, and (Xt )t≥0 is a right-continuous submartingale of class DL. Then it has the decomposition Xt = Mt + At ,

0 ≤ t < ∞,

(2.2.51)

where (Mt )t≥0 is a right-continuous martingale and (At )t≥0 an increasing process. The random process (At )t≥0 can be chosen to be a natural process, and with this extra condition (Mt )t≥0 and (At )t≥0 are uniquely determined. Furthermore, if (Xt )t≥0 is of class D, then (Xt )t≥0 is uniformly integrable and (At )t≥0 is integrable. 2.2.4 Markov properties The Markov property of a random process is a formalised expression of independence of the future of the process from its past. In this subsection we discuss some equivalent statements of the Markov property of a random process. Definition 2.79 (Markov process). Let (Xt )t≥0 be an adapted process on a filtered probability space (Ω, F , (Ft )t≥0 , P). The process (Xt )t≥0 is said to be a Markov process with respect to (Ft )t≥0 whenever 𝔼 [f (Xt )|Fs ] = 𝔼 [f (Xt )|σ(Xs )] ,

0 ≤ s ≤ t,

(2.2.52)

for all bounded Borel measurable functions f . Markov processes can be characterized by their probability transition kernels.

68 | 2 Brownian motion Definition 2.80 (Probability transition kernel). A map p : ℝ+ × ℝ+ × ℝd × ℬ(ℝd ) → ℝ, (s, t, x, A) 󳨃→ p(s, t, x, A), is called a probability transition kernel if (1) p(s, t, x, ⋅) is a probability measure on ℬ(ℝd ); (2) p(s, t, ⋅, A) is Borel measurable; (3) for every 0 ≤ r ≤ s ≤ t, the Chapman–Kolmogorov identity ∫ p(s, t, y, A)P(r, s, x, dy) = P(r, t, x, A)

(2.2.53)

ℝd

holds. Let (Xt )t≥0 be a Markov process. We define pX (s, t, x, A) = 𝔼[1A (Xt )|Xs = x]

(2.2.54)

for A ∈ ℬ(ℝd ), 0 ≤ s ≤ t < ∞, and x ∈ ℝd . Recall the notation at the right hand side of (2.2.54). The random variable 𝔼[1A (Xt )|σ(Xs )] is σ(Xs )-measurable and by Lemma 2.38 it can be expressed as 𝔼[1A (Xt )|σ(Xs )] = f (Xs ) with a Borel measurable function f , and we use the notation f (x) = 𝔼[1A (Xt )|Xs = x]. By (2.2.54) we then see that pX (s, t, Xs , A) = 𝔼[1A (Xt )|σ(Xs )]. Lemma 2.81. Suppose that (Xt )t≥0 is a Markov process on a filtered probability space (Ω, F , (Ft )t≥0 , P). Then pX is a probability transition kernel. Proof. Properties (1)–(2) of Definition 2.79 are straightforward to check, therefore we only show (3). Since ∫ℝd 1A (y)pX (r, s, x, dy) = 𝔼[1A (Xs )|Xr = x], we have the equality ∫ℝd f (y)pX (r, s, x, dy) = 𝔼[f (Xs )|Xr = x] for all bounded Borel measurable functions f . Hence ∫ pX (s, t, y, A)pX (r, s, x, dy) = 𝔼[pX (s, t, Xs , A)|Xr = x].

(2.2.55)

ℝd

Here pX (s, t, Xs , A) is given by pX (s, t, y, A) evaluated at y = Xs . Since pX (s, t, Xs , A) = 𝔼[1A (Xt )|σ(Xs )], we have pX (s, t, Xs , A) = 𝔼[1A (Xt )|Fs ] by the Markov property. Inserting this into (2.2.55) we obtain ∫ pX (s, t, y, A)pX (r, s, x, dy) = 𝔼[𝔼[1A (Xt )|Fs ]|Xr = x].

(2.2.56)

ℝd

Since σ(Xr ) ⊂ Fs , we have 𝔼[𝔼[1A (Xt )|Fs ]|σ(Xr )] = 𝔼[1A (Xt )|σ(Xr )], and therefore 𝔼[𝔼[1A (Xt )|Fs ]|Xr = x] = pX (r, t, x, A). Combining this with (2.2.56), ∫ℝd pX (s, t, y, A)pX (r, s, x, dy) = pX (r, t, x, A) is obtained.

2.2 Random processes |

69

The probability measure P 0 (A) = P(X0 ∈ A)

(2.2.57)

is the initial distribution of the Markov process describing the random variable at t = 0. The family of finite-dimensional distributions of (Xt )t≥0 is determined by its probability transition kernel and initial distribution through the formula P(Xt0 ∈ A0 , . . . , Xtn ∈ An ) =

∫ (ℝd )(n+1)

n

n

i=0

i=1

(∏ 1Ai (xi )) (∏ p(ti−1 , ti , xi−1 , dxi )) 1A0 (x0 )P 0 (dx0 ), (2.2.58)

for n = 1, 2, . . ., and where t0 = 0. Proposition 2.82. Let (Xt )t≥0 be an ℝd -valued random process on a probability space (Ω, F , P), P 0 a probability measure on ℝd , and p(s, t, x, A) a probability transition kernel. Suppose that the finite-dimensional distributions of the process are given by (2.2.58). Then (Xt )t≥0 is a Markov process under the natural filtration with probability transition kernel p(s, t, x, A). Proof. It suffices to show that n

n

j=0

j=0

𝔼 [1A (Xr ) ∏ 1Aj (Xtj )] = 𝔼 [𝔼 [1A (Xr )|σ(Xt )] ∏ 1Aj (Xtj )]

(2.2.59)

for every 0 = t0 < t1 ≤ . . . ≤ tn ≤ t ≤ r, and every A, Aj ∈ ℬ(ℝd ), j = 0, 1, . . . , n. From the identity 𝔼[1A (Xt )𝔼[f (Xr )|σ(Xt )]] = 𝔼[1A (Xt )f (Xr )], it follows that ∫ 1A (y)𝔼[f (Xr )|Xt = y]Pt (dy) = ∫ 1A (y)f (y󸀠 )p(t, r, y, dy󸀠 )Pt (dy), ℝd

ℝd

where Pt (dy) = ∫ℝd P 0 (dy0 )p(0, t, y0 , dy) and Pt (dy) is the distribution of Xt on ℝd . Thus it is seen that 𝔼[f (Xr )|Xt = y] = ∫ f (y󸀠 )p(t, r, y, dy󸀠 ).

(2.2.60)

ℝd

By the definition of 𝔼[f (Xr )|Xt = y] we have 𝔼[f (Xr )|σ(Xt )] = ∫ f (y)p(t, r, Xt , dy)

(2.2.61)

ℝd

and thus 𝔼[1A (Xr )|σ(Xt )] = p(t, r, Xt , A).

(2.2.62)

Using (2.2.62), (2.2.58) and the Chapman–Kolmogorov identity, (2.2.59) follows.

70 | 2 Brownian motion Furthermore, we have the following strong description of the converse situation. Proposition 2.83. Suppose that a map p satisfying the properties in Definition 2.80 and a probability distribution P 0 on ℝd are given. Then there exist a filtered probability space (Ω, F , (Ft )t≥0 , P) and a Markov process (Xt )t≥s on it such that p is its probability transition kernel and 𝔼[1A (Xt )|Xs = x] = p(s, t, x, A)

(2.2.63)

holds, and moreover P ∘ X0−1 = P 0 . Proof. Let Ω = (ℝd )[0,∞) be the set of all functions ω : [0, ∞) → ℝd , let G = σ(A ) be the σ-field generated by cylinder sets A defined by (2.2.6) with ℝ replaced by ℝd . Consider the measurable space (Ω, G ). For I ⊂ [0, ∞) we define a sub σ-field GI by GI = σ({ω ∈ Ω | ω(u), u ∈ I}), and Gs = G[0,s] . We also define a family of finite-dimensional distributions by choosing arbitrary 0 = t0 < t1 < . . . < tn , and xi ∈ ℝd for 0 ≤ i ≤ n, and every n ∈ ℕ, by the expressions n

n

i=0

i=1

∫ (∏ 1(−∞,xi ] (yi )) ∏ p(ti−1 , ti , yi−1 , dyi )P 0 (dy0 ),

F{t0 ,t1 ,...,tn } (x0 , x1 , . . . , xn ) =

(ℝd )n+1

n ∈ ℕ,

F{t0 } (x0 ) = ∫ 1(−∞,x0 ] (y0 )P 0 (dy0 ). ℝd

The Chapman–Kolmogorov identity gives for 0 ≤ k ≤ n lim F{t0 ,t1 ,...,tn } (x0 , x1 , . . . , xn ) = F{t0 ,...,tk−1 ,tk+1 ,...,tn } (x0 , . . . , xk−1 , xk+1 , . . . , xn ).

xk ↑∞

Thus the family of distribution functions (FΛ )Λ⊂[0,∞),|Λ| 0 and define fα = ∫0 αe−αt Pt f (x)dt. fα ∈ C∞ (ℝd ), since ∞

f ∈ C∞ (ℝd ). Let M be the closed subspace of C∞ (ℝd ) spanned by f such that (2.2.73) ∞ holds. Take f ≥ 0 and define fα,h (x) = ∫h αe−αt Pt f (x)dx; then fα,h (x) ↑ fα (x) as h ↓ 0. This convergence is uniform as the sequence of continuous monotone functions fα,h , h ≥ 0, converges to fα pointwise on the compact set ℝd ∪ {∞}. Since M is closed, fα ∈ M follows for any nonnegative f ∈ C∞ (ℝd ) and therefore for all f ∈ C∞ (ℝd ). Let l ∈ C∞ (ℝd )∗ be a linear functional on C∞ (ℝd ). Since ‖fα ‖∞ ≤ ‖f ‖∞ and, for every x, limα→∞ fα (x) = f (x), it follows that fα → f weakly as α → ∞ and hence l(fα ) → l(f ). Suppose l(g) = 0 for all g ∈ M . We see that l(f ) = limα→∞ l(fα ) = 0 for every f ∈ C∞ (ℝd ). Thus the Hahn–Banach theorem gives C∞ (ℝd ) = M . The converse statement of Proposition 2.86 also holds. Proposition 2.87. For every nonnegative C0 -semigroup on C∞ (ℝd ) there exists a unique Feller transition kernel such that (2.2.72) holds. Proof. Let {Pt : t ≥ 0} be a nonnegative C0 -semigroup. It will be shown in Lemma 2.88 below that there exists a unique kernel p(t, x, dy) such that Pt f (x) = ∫ p(t, x, dy)f (y),

f ∈ C∞ (ℝd ).

ℝd

From ‖Pt ‖ ≤ 1 it follows that p(t, x, ℝd ) ≤ 1. By continuity of x 󳨃→ Pt f (x) we get measurability of x 󳨃→ p(t, x, B). The Chapman–Kolmogorov identity can be shown by the

74 | 2 Brownian motion semigroup property Pt Ps = Ps+t . Finally, strong continuity of t 󳨃→ Pt f yields (3) of Definition 2.84. Lemma 2.88. Let l be a linear functional on C∞ (ℝd ). Then there exists a finite signed measure ρ such that l(f ) = ∫ℝd f (x)dρ(x) for all f ∈ C∞ (ℝd ). Proof. Let K = (ℝḋ ) = ℝd ∪ {∞} be the one-point compactification of ℝd . For every f ∈ C∞ (ℝd ) we define f ̃ ∈ C(K) by f ̃(x) = f (x) for x ∈ ℝd , and f ̃(∞) = 0. Any g ∈ C(K) can be represented by g = c + f ̃, where c is a constant and f ∈ C∞ (ℝd ). The map i : C(K) ∋ g 󳨃→ f ∈ C∞ (ℝd ) is linear and ‖ig‖ ≤ 2‖g‖. We define a linear functional l ̃ on ̃ = l(ig). Hence l ̃ is a continuous linear functional on C(K) and there exists C(K) by l(g) a measure on (K, ℬ(K)) such that ̃ = ∫ g(x)dρ(x) l(g) K

by the Riesz theorem. It follows that l(f ) = ∫K f ̃(x)ρ(dx) = ∫ℝd f (x)ρ(dx). Let (Xt )t≥0 be a Markov process and consider its Feller transition kernel. Then {Pt : t ≥ 0} in (2.2.72) is a C0 -semigroup by Proposition 2.86 and there exists a closed operator L on C∞ (ℝd ) by the Hille–Yoshida theorem (Proposition 4.80 below) such that 1 lim (Pt f − f ) = Lf , t

t→0

where L is called the generator of the Markov process and is formally written as Pt = etL . The domain of L is defined by 󵄩󵄩 P f − f 󵄩󵄩 󵄨󵄨 D(L) = {f ∈ C∞ (ℝd ) 󵄨󵄨󵄨 ∃ g such that lim 󵄩󵄩󵄩󵄩 t − g 󵄩󵄩󵄩󵄩 = 0} . 󵄨 t↓0 󵄩 t 󵄩∞ We will see in Section 3.1.5 below that Feller transition kernels form a rich class and further examples of generators will be given.

2.2.6 Invariant measures Finally we discuss time-invariant measures μ on (ℝd , ℬ(ℝd )) related to Markov processes. As seen below, formally an invariant measure is the eigenfunction associated with zero-eigenvalue for the adjoint of the generator L of the Markov process. Furthermore, by using the invariant measure we can extend the semigroup {Pt : t ≥ 0} on C∞ (ℝd ) to Lp (ℝd , dμ) with suitable p > 0. Definition 2.89 (Invariant measure). Let (Xt )t≥0 be an ℝd -valued stationary Markov process on a filtered probability space (Ω, F , (Ft )t≥0 , P), with probability transition

2.2 Random processes |

75

kernel p(t, x, A). A probability measure μ on (ℝd , ℬ(ℝd )) is an invariant measure for (Xt )t≥0 if it satisfies ∫ p(t, x, A)μ(dx) = μ(A) ℝd

for every t ≥ 0 and A ∈ ℬ(ℝd ). Lemma 2.90. Let (Xt )t≥0 be a d-dimensional stationary Markov process on a filtered probability space (Ω, F , (Ft )t≥0 , P) with a Feller transition kernel p(t, x, A). Define the semigroup {Pt : t ≥ 0} on C∞ (ℝd ) by (2.2.72). Then a measure μ is invariant for (Xt )t≥ if and only if for all f ∈ C∞ (ℝd ) and all t ≥ 0, ∫ (Pt f )(x)dμ(x) = ∫ f (x)dμ(x). ℝd

(2.2.74)

ℝd

Proof. By the definition of the invariant measure, μ is invariant if and only if (2.2.74) holds for all f (x) = 1A (x) with A ∈ ℬ(ℝd ). By a limiting argument the claim follows. To verify directly that a given μ is an invariant measure is often difficult. Therefore the following criterion is useful. Theorem 2.91. Let the assumptions of Lemma 2.90 hold, and L be the generator of the semigroup {Pt : t ≥ 0}. Then μ is an invariant measure for (Xt )t≥0 if and only if for all f ∈ D(L), ∫ (Lf )(x)dμ(x) = 0.

(2.2.75)

ℝd

Proof. Let μ be an invariant measure and f ∈ D(L). We have t

Pt f − f = ∫ 0

t

d P fds = ∫ Ps Lfds. ds s 0

It follows that the family { 1t (Pt f − f )}t>0 is uniformly bounded, and thus by the dominated convergence theorem, ∫ (Lf )(x)dμ(x) = lim

t→0

ℝd

1 ∫ (Pt f − f )dμ(x) = 0. t ℝd

Conversely, let (2.2.75) hold for all f ∈ D(L). It follows from general theory that for every t > 0 and g ∈ C∞ (ℝd ) there exists f ∈ D(L) with f − tLf = g. It follows that ∫ℝd f (x)dμ(x) = ∫ℝd g(x)dμ(x), and by iteration we get ∫ (1 − tL)−n g(x)dμ(x) = ∫ f (x)dμ(x) ℝd

ℝd

76 | 2 Brownian motion for all n ∈ ℕ. The integrand at the left-hand side is uniformly bounded and converges to Pt g, and by the dominated convergence theorem the claim follows. Theorem 2.92. Choose p ≥ 1. Let the assumptions of Lemma 2.90 hold, and suppose that C∞ (ℝd ) is dense in Lp (ℝd , dμ). Then {Pt : t ≥ 0} can be extended to a C0 -semigroup on Lp (ℝd , dμ). Proof. Let f ∈ C∞ (ℝd ). We have Pt f (x) = 𝔼[f (Xt )|X0 = x] and |Pt f (x)|p ≤ 𝔼[|f (Xt )|p |X0 = x] = Pt (|f |p )(x). Then ∫ |Pt f (x)|p dμ(x) ≤ ∫ Pt (|f |p )(x)dμ(x) = ∫ |f (x)|p dμ(x). ℝd

ℝd

ℝd

Here we used that μ is an invariant measure. Thus ‖Pt f ‖Lp (ℝd ,dμ) ≤ ‖f ‖Lp (ℝd ,dμ) and

C∞ (ℝd ) is dense in Lp (ℝd , dμ). We can extend Pt to a contraction operator on Lp , which we denote by P̄ t . It is straightforward to see that P̄ t P̄ s = P̄ t+s for s, t ≥ 0, and P̄ 0 = I. We show strong continuity of t 󳨃→ P̄ t ; it is sufficient to show it at t = 0. Since p(t, x, dy) is a Feller transition kernel, Pt f (x) is continuous at t = 0 for every x ∈ ℝd and all f ∈ C∞ (ℝd ). As C∞ (ℝd ) is dense in Lp (ℝd , dμ), Lp continuity follows from the dominated convergence theorem. Finally, we introduce a martingale associated with the Markov process (Xt )t≥0 and its generator. Theorem 2.93. Let (Xt )t≥0 be a d-dimensional stationary Markov process on a filtered probability space (Ω, F , (Ft )t≥0 , P) with a Feller transition kernel p(t, x, A). Define the semigroup {Pt : t ≥ 0} on C∞ (ℝd ) by (2.2.72), and let L be the generator of Pt and f ∈ D(L). Then t

Mt = f (Xt ) − f (X0 ) − ∫(Lf )(Xs )ds,

t ≥ 0,

0

is a martingale on (Ω, F , (Ft )t≥0 , P). Proof. It is straightforward to see that 𝔼[|Mt |] < ∞, t ≥ 0. By the Markov property 𝔼[f (Xt )|Fs ] = Pt−s f (Xs ). It is trivial to see that t

󵄨󵄨 󵄨 𝔼[Mt |Fs ] = Ms + 𝔼[f (Xt ) − f (Xs ) − ∫ Lf (Xr )dr 󵄨󵄨󵄨Fs ]. 󵄨󵄨 s

(2.2.76)

2.3 Brownian motion and Wiener measure

Since f ∈ D(L), we have

d Pf dt t

| 77

t

= LPt f = Pt Lf and Pt f − f = ∫0 LPs fds. It follows that

t

t

󵄨󵄨 󵄨 𝔼[f (Xt ) − f (Xs ) − ∫ Lf (Xr )dr 󵄨󵄨󵄨Fs ] = Pt−s f (Xs ) − f (Xs ) − ∫ LPr−s f (Xs )dr 󵄨󵄨 s s t−s

= Pt−s f (Xs ) − f (Xs ) − ∫ LPr f (Xs )dr = 0. 0

Hence the second term at the right-hand side of (2.2.76) is zero and the martingale property is proven. t

The asymptotic behavior of 1t ∫0 (Lf )(Xs )ds for given (classes of) functions as t → ∞, is an interesting object to study, leading to functional central limit theorems, extending the standard central limit theorem.

2.3 Brownian motion and Wiener measure 2.3.1 Construction of Brownian motion After this extensive preparation of background material, we turn now to discussing Brownian motion, which is a random process of fundamental importance in pure and applied probability, and will play a central role throughout below. While Brownian motion is one of the best studied random processes, we offer here a brief summary of the aspects that we will need for our purposes in this book, and refer to the literature for many interesting properties which we do not even mention. Since Brownian motion lies at the intersection of many desirable properties a random process may have, by being a process with independent increments, a Gaussian process, a (strong) Markov process, a martingale, and the only Lévy process with almost surely continuous paths, many of the general facts discussed before can now be applied by particularising the underlying distribution. Definition 2.94 (Brownian motion). A real-valued random process (Bt )t≥0 on a probability space (Ω, F , P x ), x ∈ ℝ, is called Brownian motion or Wiener process starting at x, whenever the following conditions hold: (1) P x (B0 = x) = 1; (2) the increments (Bti − Bti−1 )1≤i≤n are independent Gaussian random variables for every collection 0 = t0 < t1 < . . . < tn , n ∈ ℕ, of time-points, with d

Bti − Bti−1 = N(0, ti − ti−1 ); (3) the function t 󳨃→ Bt (ω) is continuous for a. e. ω ∈ Ω.

78 | 2 Brownian motion In particular, (Bt )t≥0 is called standard Brownian motion when x = 0. A d-component vector of independent Brownian motions (B1t , . . . , Bdt )t≥0 is called d-dimensional Brownian motion. Brownian motion is thus a real-valued continuous-time random process with index set ℝ+ = [0, ∞). Consider d

X = C(ℝ ; ℝ ) +

(2.3.1)

equipped with the metric given by (2.1.3). First we discuss the canonical representation of Brownian motion on the probability space (X , ℬ(X ), 𝒲 x ), where 𝒲 x is Wiener measure to be defined below. In this representation a realization of the process (Bt )t≥0 is given by Bt (ω) = ω(t) ∈ ℝd ,

ω∈X,

(2.3.2)

called coordinate process. For each fixed ω ∈ X we think of ℝ+ ∋ t 󳨃→ Bt (ω) ∈ ℝd as a Brownian path. We write Πt (x) =

1 |x|2 exp (− ), 2t (2πt)d/2

t > 0, x ∈ ℝd ,

(2.3.3)

called d-dimensional heat kernel. Definition 2.95 (Wiener measure). Let n ∈ ℕ be an arbitrary natural number. Consider an arbitrary collection of time-points 0 = t0 < t1 < . . . < tn , and an arbitrary collection of Borel sets Ej ∈ ℬ(ℝd ), j = 1, . . . , n. The unique probability measure 𝒲 x on (X , ℬ(X )) whose family of finite-dimensional distributions is given by x

𝒲 (B0 = x) = 1

(2.3.4)

and n

n

n

j=1

j=1

j=1

x

𝒲 (Bt1 ∈ E1 , . . . , Btn ∈ En ) = ∫ (∏ 1Ej (xj )) (∏ Πtj −tj−1 (xj−1 − xj )) ∏ dxj ℝdn

(2.3.5)

where x0 = x and Bt (ω) = ω(t) for ω ∈ X , is called Wiener measure starting at x ∈ ℝd . We write 𝒲 = 𝒲 0 , and for expectation with respect to Wiener measure we will use the notations 𝔼x [F] = 𝔼x𝒲 [F] = 𝔼𝒲 x [F],

𝔼0 [F] = 𝔼0𝒲 [F] = 𝔼𝒲 0 [F] = 𝔼[F].

(2.3.6)

By the definition above we have n

n

n

n

j=1

j=1

j=1

j=1

𝔼x [∏ fj (Btj )] = ∫ (∏ fj (xj )) (∏ Πtj −tj−1 (xj−1 − xj )) ∏ dxj ℝdn

(2.3.7)

2.3 Brownian motion and Wiener measure

| 79

for all bounded Borel measurable functions f1 , . . . , fn . In particular, 𝔼x [f (Bt )] = ∫ f (y)Πt (x − y)dy.

(2.3.8)

ℝd

A prime application of this formula is to obtain the moments of Brownian motion. Proposition 2.96 (Moments). Let (Bt )t≥0 be one-dimensional standard Brownian motion, and s, t ≥ 0. Then we have the following expressions. (1) Characteristic function: ϕ(u) = 𝔼[eiu(Bt −Bs ) ] = e−

|t−s|u2 2

u ∈ ℝ.

,

(2) Moment generating function: ψ(u) = 𝔼[eu(Bt −Bs ) ] = e

|t−s|u2 2

u ∈ ℝ.

,

(3) Moments: k

{ 2k k! |t − s| 𝔼[(Bt − Bs ) ] = { {0 (2k)!

n

if n = 2k, k ∈ ℕ̇ if n = 2k + 1, k ∈ ℕ.̇

Proof. The characteristic and moment generating functions are obtained by completion of the square in the exponent and Gaussian integration, i. e., we have ϕ(u) = ∫ eiux Πt−s (x)dx = e−

|t−s|u2 2

.

ℝ

In the same way we can see (2). The moments can be obtained from either of these functions. Write X = Bt − Bs . By (1) we see that ϕ ∈ C ∞ (ℝ) and 2n

∞ 1 dk ϕ 1 k (0)u + R = |t − s|k u2k , ∑ 2n+1 k k! k! dx k=0 k=0

ϕ(u) = ∑

1 k 2k where R2n+1 is the remainder term and it gives R2n+1 = ∑∞ by the k=2n+1 k! |t − s| u uniquenes of the Taylor expansion. Hence R2n+1 → 0 as n → ∞ follows. Since k

k

dk ϕ

d d iuX k iuX 𝔼[| dx |] ≤ 𝔼[|uk |] < ∞, we can exchange 𝔼 and dx ]. ke k , and dx k (u) = 𝔼[(iu) e Notice that the odd moments vanish for symmetry reasons. We have ∞ 1 k (−1)n i 𝔼[X k ]uk = ∑ 𝔼[X 2n ]u2n . k! (2n)! n=0 k=0 ∞

𝔼[eiuX ] = ∑

Using the right-hand side of the expression of the characteristic function, we obtain 1

2

∞ n 1 1 1 (−1)n (− |t − s|u2 ) = ∑ |t − s|n u2n . n n! 2 n! 2 n=0 n=0 ∞

e− 2 |t−s|u = ∑

Comparing the coefficients on the two sides yields the required result.

80 | 2 Brownian motion Our next goal is to discuss the existence of Wiener measure. The main challenge is to show that there is no conflict between the Gaussian increments and path continuity. In Chapter 3 below we will see that path continuity is exceptional, and in a large canonical class of processes it only occurs for Brownian motion. Theorem 2.97 (Existence of Wiener measure). Wiener measure 𝒲 x exists on the measurable space (X , ℬ(X )), and the coordinate process Bt (ω) = ω(t), t ≥ 0, on the probability space (X , ℬ(X ), 𝒲 x ) is Brownian motion starting at x ∈ ℝd . Proof. We assume d = 1, for d ≥ 2 the proof is similar. The idea of the proof is first to construct a probability measure on all real-valued functions on the positive semi-axis, and next show that this measure is actually supported on the subspace of continuous functions. Let Λ1 ⊂ Λ2 ⊂ [0, ∞) with |Λ1 | < ∞ and |Λ2 | < ∞. Without restricting generality, we chose Λ1 = {t1 , . . . , tn } and Λ2 = {t1 , . . . , tn , s1 , . . . , sm }, and define the projections πΛ1 Λ2 : ℝ|Λ2 | → ℝ|Λ1 | by the map (xt1 , . . . , xtn , xs1 , . . . , xsm ) 󳨃→ (xt1 , . . . , xtn ). We define the projection πΛ : ℝ[0,∞) → ℝ|Λ| similarly. Also, define a probability measure μΛ on ℝ|Λ| for Λ = {t1 , . . . , tn } by n

n

n

j=1

j=1

j=1

μΛ (E1 × ⋅ ⋅ ⋅ × En ) = ∫ (∏ 1Ej (xj )) (∏ Πtj −tj−1 (xj−1 − xj )) ∏ dxj ℝn

with x0 = 0. It is direct to check that πΛ−12 Λ1 (E1 × ⋅ ⋅ ⋅ × En ) = E1 × ⋅ ⋅ ⋅ × En × ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℝ × ⋅ ⋅ ⋅ × ℝ and m

μΛ1 (E1 × ⋅ ⋅ ⋅ × En ) = μΛ2 (E1 × ⋅ ⋅ ⋅ × En × ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ℝ × ⋅ ⋅ ⋅ × ℝ). m

(2.3.9)

Equality (2.3.9) is the Kolmogorov consistency relation (2.1.21) for μΛ above. Define the set function μ on −1

[0,∞)

A = {πΛ[0,∞) (E) ⊂ ℝ

| E ∈ ℬ(ℝ|Λ| , Λ ⊂ [0, ∞), |Λ| < ∞}

−1 ̃ (πΛ[0,∞) by μ (E)) = μΛ (E). In a first step of the proof we show that μ̃ is well defined and can be uniquely extended to a measure μ on (ℝ[0,∞) , G ) satisfying (2.3.4) and (2.3.5), where G is the σ-field generated by the cylinder sets in ℝ[0,∞) , i. e., G = σ(A ). This results by an application of the Kolmogorov extension theorem (Theorem 2.20). Let (Yt )t≥0 be the coordinate process on (ℝ[0,∞) , G , μ), i. e., Yt (ω) = ω(t) for ω ∈ ℝ[0,∞) . We then see that (1) μ(Y0 = 0) = 1; (2) the increments {Ytj − Ytj−1 }nj=1 are independent for every division 0 = t0 < t1 < . . . < d

tn , and Ytj − Ytj−1 = N(0, tj − tj−1 ).

2.3 Brownian motion and Wiener measure

| 81

Next in a second step we prove that there exists a continuous version (Ỹ t )t≥0 of (Yt )t≥0 by using 𝔼μ [|Yt − Ys |4 ] ≤ C|t − s|2

(2.3.10)

and the Kolmogorov–Čentsov theorem (Theorem 2.49). Note that (Ỹ t )t≥0 is not the coordinate process. Although Ỹ t (ω) ≠ Yt (ω) in general, it is true that Ỹ t (ω) = Yt (ω) for ω ∈ ̸ Nt , where the null set Nt depends on t. The process (Ỹ t )t≥0 satisfies (1)–(3) holding for (Yt )t≥0 . With the so constructed process (Ỹ t )t≥0 on (ℝ[0,∞) , G , μ), let X⋅ : (ℝ[0,∞) , G , μ) → (X , ℬ(X )) be defined by X⋅ (ω) = Ỹ ⋅ (ω) ∈ X for ω ∈ ℝ[0,∞) . It can be checked directly that X −1 (E) ∈ G for any cylinder set E, and for E = {ω ∈ X | ω(tj ) ∈ Ej , Ej ∈ ℬ(ℝ), j = 1, . . . , n} we can indeed see that X −1 (E) = {ω ∈ ℝ[0,∞) | Xtj (ω) ∈ Ej , j = 1, . . . , n} = {ω ∈ ℝ[0,∞) | Ỹ tj (ω) ∈ Ej , j = 1, . . . , n} N

⊂ {ω ∈ ℝ[0,∞) | Ytj (ω) ∈ Ej , j = 1, . . . , n} ⋃ Ntj j=1

n

N

j=1

j=1

= (⋂{ω ∈ ℝ[0,∞) | ω(tj ) ∈ Ej }) ⋃ Ntj ∈ G . Here Ntj , j = 1, . . . , n, are null sets such that Ytj = Ỹ tj on ℝ[0,∞) \ Ntj . Thus X⋅ ∈ G /ℬ(X ), since ℬ(X ) coincides with the σ-field generated by the cylinder sets. The image measure μX of μ under X on (X , ℬ(X )) is denoted by 𝒲 0 . It is seen that 𝒲 0 is Wiener measure with x = 0, and the coordinate process Bt : ω ∈ X 󳨃→ ω(t) is Brownian motion on (X , ℬ(X ), 𝒲 0 ) starting at 0. Indeed the finite-dimensional distributions are n

𝔼0 [∏ 1Ej (Btj )] = μ ∘ X −1 ({ω̄ ∈ X | Btj (ω)̄ ∈ Ej , j = 1, . . . , n}), j=1

n ∈ ℕ.

Note that X ̄ = X(ℝ[0,∞) ) ⊂ X satisfies 𝒲 0 (X ̄ ) = 1. The paths ω̄ ∈ X ̄ \ N̄ can be ̄ = 0. Hence we have ̄ = Xt (ω) with some ω ∈ ℝ[0,∞) , where 𝒲 0 (N) described as ω(t) n

̄ j ) ∈ Ej , j = 1, . . . , n}) 𝔼0 [∏ 1Ej (Btj )] = μ ∘ X −1 ({ω̄ ∈ X | Btj (ω)̄ = ω(t j=1

= μ ∘ X −1 ({X⋅ (ω) ∈ X | Xtj (ω) ∈ Ej , j = 1, . . . , n}) = μ({ω ∈ ℝ[0,∞) | Xtj (ω) ∈ Ej , j = 1, . . . , n}

82 | 2 Brownian motion = μ({ω ∈ ℝ[0,∞) | Ỹ tj (ω) ∈ Ej , j = 1, . . . , n}) = μ({ω ∈ ℝ[0,∞) | Ytj (ω) ∈ Ej , j = 1, . . . , n}). d Here we used that Ỹ t = Yt almost surely. We see that Bt = Yt , and thus 0

𝒲 (B0 = 0) = μ(Y0 = 0) = 1,

𝔼0 [Bt Bs ] = 𝔼μ [Yt Ys ] = t ∧ s, d

{Btj − Btj−1 }, j = 1, . . . , n, are independent, and Btj − Btj = N(0, tj − tj−1 ) follows. Hence 1 (Bt )t≥0 is a Brownian motion. Define Z⋅ : (X , ℬ(X ), 𝒲 0 ) → (X , ℬ(X )) by Zt (ω) = Bt (ω)+x. It can be seen that Z ∈ ℬ(X )/ℬ(X ). We denote the image measure of 𝒲 0 under Z on (X , ℬ(X )) by 𝒲 x . This is then Wiener measure starting at x, and the coordinate process Bt (ω) = ω(t), t ≥ 0, under 𝒲 x is Brownian motion starting at x. Now we turn to studying some fundamental distributional properties of Brownian motion. By the definition of standard Brownian motion we see that Bt = Bt − B0 a. s., and hence d

Bt = N(0, t),

t ≥ 0.

Thus Bt is a Gaussian random variable for every t > 0, with 𝔼[Bt ] = 0 and variance 𝔼[B2t ] = t. In the next result we show that this structure goes over also to process level. Theorem 2.98 (Gaussian property). Let s, t ≥ 0. Standard Brownian motion (Bt )t≥0 , Bt = (B1t , . . . , Bdt ), is a Gaussian process with mean and covariance j

𝔼[Bt ] = 0,

𝔼[Bjs Bkt ]

j = 1, . . . , d,

= (s ∧ t) δjk ,

j, k = 1, . . . , d,

where δjk is the Kronecker symbol. Notice that for standard Brownian motion the last expression coincides with the covariance. For one-dimensional Brownian motion starting at x, the mean is 𝔼x [Bt ] = x and the covariance becomes cov(Bs , Bt ) = 𝔼x [(Bs − 𝔼x [Bs ])(Bt − 𝔼x [Bt ])] = 𝔼x [Bs Bt ] − x 2 . Proof. For simplicity, we discuss the proof for one-dimensional standard Brownian motion only, the general case causes no special difficulty. Recall that an ℝ-valued random process (Xt )t≥0 is called a Gaussian process if for all 0 ≤ t1 ≤ . . . ≤ tn , every linear combination ∑ni=1 ci Xti , c1 , . . . , cn ∈ ℝ, is a Gaussian random variable, for all n ∈ ℕ.

2.3 Brownian motion and Wiener measure

| 83

Let n ∈ ℕ be arbitrary, c1 , . . . , cn ∈ ℝ, and consider any 0 ≤ t1 ≤ . . . ≤ tn . Then notice that we can write by telescoping n

n

i=1

i=1

∑ ci Bti = ∑ (ci + ⋅ ⋅ ⋅ + cn )(Bti − Bti−1 ). Since the increments Bti −Bti−1 are all independent, and constant multiples of Gaussian random variables are again Gaussian random variables, the right hand side above is the sum of independent Gaussian random variables, hence another Gaussian random variable. Next we calculate the mean and covariance. The first equality is immediate. For the second choose first s < t. Since Bt − Bs and Bs are independent, we have for the covariance 𝔼[(Bt − 𝔼[Bt ])(Bs − 𝔼[Bs ])] = 𝔼[Bt Bs ] = 𝔼[(Bt − Bs )Bs ] + 𝔼[B2s ] = s, and similarly obtain t in case t < s. This property allows a characterization of Brownian motion and we have the following converse statement, which we state for convenience for the one-dimensional case only. Theorem 2.99. Let (Xt )t≥0 be a Gaussian process such that X0 = 0 a. s. and (1) 𝔼[Xt ] = 0 and 𝔼[Xs Xt ] = s ∧ t, for all s, t ≥ 0; (2) t 󳨃→ Xt is a. s. continuous. Then (Xt )t≥0 is standard Brownian motion. Proof. We only need to prove that the increments (Xti − Xti−1 )1≤i≤n are independent Gaussian random variables distributed by N(0, ti − ti−1 ), for any partition 0 = t0 < t1 < . . . < tn , n ∈ ℕ. Let 0 < r < s < t. Then 𝔼[(Xr − X0 )(Xt − Xs )] = 𝔼[Xr (Xt − Xs )] = 𝔼[Xr Xt ] − 𝔼[Xr Xs ] = r ∧ t − r ∧ s = 0. On the other hand, also 𝔼[Xr − X0 ]𝔼[Xt − Xs ] = 0, and since they are Gaussian random variables, this implies that Xr − X0 and Xt − Xs are independent. Furthermore, we have 𝔼[Xti − Xti−1 ] = 0 and var(Xti − Xti−1 ) = 𝔼[(Xti − Xti−1 )2 ] = 𝔼[Xt2i ] − 2𝔼[Xti−1 Xti ] + 𝔼[Xt2i−1 ]

= ti − 2ti−1 ∧ ti + ti−1 = ti − 2ti−1 + ti−1 = ti − ti−1 .

Since all other conditions are satisfied, we have that (Xt )t≥0 is a standard Brownian motion. The following are basic invariance properties of Brownian motion. Theorem 2.100 (Symmetry properties). Let (Bt )t≥0 be standard Brownian motion. Then the following symmetry properties hold:

84 | 2 Brownian motion d

(1) (spatial homogeneity) if (Bxt )t≥0 is Brownian motion starting at x, then Bt + x = Bxt ; d

(2) (reflection symmetry) −Bt = Bt ; d

(3) (self-similarity) √cBt/c = Bt for all c > 0; 0, if t = 0, d (4) (time inversion) Let Zt (ω) = { then Zt = Bt ; tB1/t (ω), if t > 0, d

(5) (time reversibility) Bt−s − Bt = Bs for all 0 ≤ s ≤ t. Proof. These equalities follow easily from (2.3.7). The fact that Z0 = 0 a. s. in (4) is an application of the strong law of large numbers discussed in Section 2.3.7. 2.3.2 Two-sided Brownian motion For our purposes below it will be sometimes of interest to have Brownian motion extended over the full time-line ℝ instead of running it on [0, ∞) only. This can be done on the measurable space (X, ℬ(X)) with path space X = C(ℝ; ℝd ).

(2.3.11)

Consider X ̃ = X × X , with X as given by (2.3.1), and μx = 𝒲 x × 𝒲 x . Let ω = (ω1 , ω2 ) ∈ X ̃ and define ω (t) B̃ t (ω) = { 1 ω2 (−t)

if t ≥ 0, if t < 0.

Since B̃ t (ω) is continuous in t ∈ ℝ under μx , X : (X ̃ , ℬ(X ̃ )) → (X, ℬ(X)) can be defined by X. (ω) = B̃ . (ω). It can be seen that X ∈ ℬ(X ̃ )/ℬ(X) by showing that X −1 (E) ∈ ℬ(X ̃ ) for any cylinder set E ∈ ℬ(X). Thus X is an X-valued random variable on X ̃ . Denote the image measure of μx under X on (X, ℬ(X)) by 𝒲̃ x . The coordinate process denoted by B̃ t : X ∋ ω 󳨃→ ω(t) ∈ ℝd

(2.3.12)

is Brownian motion over ℝ on (X, ℬ(X), 𝒲̃ x ), with B̃ 0 = x almost surely. The properties of Brownian motion indexed by the real line ℝ can be summarized as follows. Proposition 2.101 (Two-sided Brownian motion). (B̃ t )t∈ℝ satisfies the following: (1) 𝒲̃ x (B̃ 0 = x) = 1; (2) the increments (B̃ ti − B̃ ti−1 )1≤i≤n are independent Gaussian random variables for every d collection 0 = t0 < t1 < . . . < tn , n ∈ ℕ, with B̃ t − B̃ s = N(0, t − s) for t > s ≥ 0; (3) the increments (B̃ −ti−1 − B̃ −ti )1≤i≤n are independent Gaussian random variables for

d every collection 0 = −t0 > −t1 > . . . > −tn , n ∈ ℕ, with B̃ −t − B̃ −s = N(0, s − t) for 0 ≥ −t > −s;

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| 85

(4) the function ℝ ∋ t 󳨃→ B̃ t (ω) ∈ ℝ is continuous for a. e. ω; d

(5) B−t = Bt for t ∈ ℝ, and Bt , Bs for t > 0 > s are independent. We use the previous notation Bt and 𝒲 x for B̃ t and 𝒲̃ x , respectively, unless any confusion may arise. Next we consider time-shifts for two-sided Brownian motion. It can be checked directly through the finite-dimensional distributions (2.3.7) that all joint distributions of B̃ t0 , . . . , B̃ tn for arbitrary time-points −∞ < t0 < t1 < . . . < tn < ∞, n ∈ ℕ, with respect to dx ⊗ d𝒲̃ x are invariant under time-shift under some conditions. We highlight this through some simple cases. Let 0 < s ≤ t. We see that ∫ 𝔼x𝒲 [f (Bs )g(Bt )]dx = ∫ f (x)g(y)Πs (x)Πt−s (y − x)dxdy. ℝd

ℝd

On the other hand, ∫ 𝔼x𝒲 [f (B0 )g(Bt−s )]dx = ∫ f (x)g(y)Πt−s (y − x)dxdy, ℝd

ℝd

and thus we conclude that in general ∫ 𝔼x𝒲 [f (Bs )g(Bt )]dx ≠ ∫ 𝔼x𝒲 [f (B0 )g(Bt−s )]dx. ℝd

ℝd

It follows that ∫ℝd 𝔼x𝒲 [f (Bs )g(Bt )]dx for 0 < s ≤ t is not time-shift invariant. Now let −s ≤ 0 ≤ t. We see that ∫ 𝔼x𝒲̃ [f (B̃ −s )g(B̃ t )]dx = ∫ 𝔼x𝒲̃ [f (B̃ −s )]𝔼x𝒲 [g(Bt )]dx ℝd

ℝd

by the independence of B̃ −s and B̃ t , and further by reflection symmetry we have = ∫ 𝔼x𝒲 [f (Bs )]𝔼x𝒲 [g(Bt )]dx = ∫ f (z)g(y)Πs (z − x)Πt (z − y)dxdydz ℝd

ℝd

= ∫ f (z)g(y)Πs+t (x − y)dxdy = ∫ 𝔼x𝒲 [f (B0 )g(Bs+t )]dx. ℝd

ℝd

Hence ∫ 𝔼x𝒲̃ [f (B̃ −s )g(B̃ t )]dx = ∫ 𝔼x𝒲 [f (B0 )g(Bs+t )]dx ℝd

ℝd

follows. Furthermore, by a similar computation we see that ∫ 𝔼x𝒲̃ [f (B̃ −s )g(B̃ t )]dx = ∫ 𝔼x𝒲̃ [f (B̃ −s+r )g(B̃ t+r )]dx ℝd

ℝd

86 | 2 Brownian motion for −t ≤ r ≤ s. Thus ∫ℝd 𝔼x𝒲̃ [f (B̃ −s )g(B̃ t )]dx is r-shift invariant for −t ≤ r ≤ s. We emphasize again that for 0 ≤ s ≤ t, ∫ 𝔼x𝒲̃ [f (B̃ 0 )g(B̃ t )]dx ≠ ∫ 𝔼x𝒲̃ [f (B̃ s )g(B̃ s+t )]dx ℝd

ℝd

while ∫ 𝔼x𝒲̃ [f (B̃ 0 )g(B̃ t )]dx = ∫ 𝔼x𝒲̃ [f (B̃ −s )g(B̃ t−s )]dx. ℝd

ℝd

Remark 2.102. In Section 4.2.1 below we will see that 𝔼x [f (Bt )] = (e−th f )(x), where h = −(1/2)Δ, and Δ is the Laplacian in d-dimensions. Thus we have (f , e−th g)L2 (ℝd ) = ∫ 𝔼x𝒲̃ [f ̄(B0 )g(Bt )]dx. ℝd

The left-hand side is equal to (f , e−th g)L2 (ℝd ) = (e−sh f , e−(t−s)h g)L2 (ℝd ) ,

0 ≤ s ≤ t,

(2.3.13)

and the right-hand side above is given by (e−sh f , e−(t−s)h g)L2 (ℝd ) = ∫ 𝔼x𝒲 [f ̄(Bs )]𝔼x𝒲 [g(Bt−s )]dx ℝd

= ∫ 𝔼x𝒲̃ [f ̄(B̃ −s )]𝔼x𝒲 [g(Bt−s )]dx = ∫ 𝔼x𝒲̃ [f ̄(B̃ −s )g(B̃ t−s )]dx. ℝd

ℝd

We have thus identically ∫ 𝔼x𝒲̃ [f ̄(B0 )g(Bt )]dx = ∫ 𝔼x𝒲̃ [f ̄(B̃ −s )g(B̃ t−s )]dx, ℝd

0 ≤ s ≤ t.

ℝd

This allows then to view time-shift invariance as a semigroup property given by (2.3.13). We conclude by the following result. Proposition 2.103 (Time-shift invariance). Let m, n ∈ ℕ, t1 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m be arbitrary time-points, f1 , fn+m ∈ L2 (ℝd ), and fi , 2 ≤ i ≤ n + m − 1, be bounded Borel measurable functions. Then n+m

n+m

∫ 𝔼x𝒲̃ [ ∏ fi (B̃ ti )] dx = ∫ 𝔼x𝒲 [ ∏ fi (Bti −t1 )] dx. ℝd

i=1

ℝd

i=1

(2.3.14)

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| 87

In particular, n+m

n+m

∫ 𝔼x𝒲̃ [ ∏ fi (B̃ ti )] dx = ∫ 𝔼x𝒲̃ [ ∏ fi (B̃ ti +s )] dx i=1

ℝd

i=1

ℝd

(2.3.15)

for all −tn+m ≤ s ≤ −t1 . Proof. Both sides of (2.3.14) are finite by the assumptions on the functions fj . We prove (2.3.14). Let −t1 ≤ . . . ≤ −tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . By independence of B̃ −t and B̃ s for −t ≤ 0 ≤ s and reflection symmetry, we have n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∫ 𝔼x𝒲̃ [∏ fi (B̃ −ti ) ∏ fj (B̃ tj )] dx = ∫ 𝔼x𝒲̃ [∏ fi (B̃ −ti )] 𝔼x𝒲̃ [ ∏ fj (B̃ tj )] dx ℝd

ℝd

n

n+m

i=1

j=n+1

= ∫ 𝔼x𝒲 [∏ fi (Bti )] 𝔼x𝒲 [ ∏ fj (Btj )] dx. ℝd

Hence it is seen that = ∫ dx ℝd

n+m

n

2

i=1

i=1

∏ dxi (∏ fi (xi )) Πtn (x − xn ) (∏ Πti−1 −ti (xi−1 − xi ))

∫ ℝd(n+m) n+m

i=n

n+m

× ( ∏ fj (xj )) Πtn+1 (xn+1 − x) ∏ Πtj −tj−1 (xj − xj−1 ). j=n+1

j=n+2

Since ∫ℝd Πtn (x − xn )Πtn+1 (xn+1 − x)dx = Πtn +tn+1 (xn+1 − xn ), we have n+m

n

i=1

i=2

∫ ∏n+m i=1 dxi ( ∏ fi (xi )) (∏ Πti−1 −ti (xi−1 − xi ))

=

ℝd(n+m)

n+m

× Πtn +tn+1 (xn+1 − xn ) ∏ Πtj −tj−1 (xj − xj−1 ). j=n+2

On the other hand, we have n

n+m

i=1

i=n+1

∫ 𝔼x𝒲 [∏ fi (Bt1 −ti ) ∏ fi (Bt1 +tj )] dx ℝd

= ∫ dx ℝd

∫ ℝd(n+m)

n+m

n+m

n

i=1

i=1

i=2

∏ dxi ( ∏ fi (xi )) Π0 (x, x1 ) (∏ Πti−1 −ti (xi−1 − xi )) n+m

× Πtn +tn+1 (xn+1 − xn ) ∏ Πtj −tj−1 (xj − xj−1 ) j=n+2

=

∫ ℝd(n+m)

n+m

n+m

n

i=1

i=1

i=2

∏ dxi ( ∏ fi (xi )) (∏ Πti−1 −ti (xi−1 − xi ))

(2.3.16)

88 | 2 Brownian motion n+m

× Πtn +tn+1 (xn+1 − xn ) ∏ Πtj −tj−1 (xj − xj−1 ).

(2.3.17)

j=n+2

Here we used ∫ℝd Π0 (x, x1 )dx = 1. Comparing (2.3.16) and (2.3.17) we obtain (2.3.14), and hence (2.3.15) follows. 2.3.3 Conditional Wiener measure Conditionals of the Wiener path measure and related conditioned processes have important applications, and we will often use them below. We discuss next Brownian bridge measure obtained by pinning down paths at a starting and an ending position. Definition 2.104 (Brownian bridge measure). Let (Bt )t≥0 be Brownian motion, A1 , . . . , An Borel sets of ℝd , and n ∈ ℕ arbitrary. Given a division 0 < T1 ≤ t1 < . . . < tn ≤ T2 and x, y ∈ ℝd , the probability measure on (X , ℬ(X )) defined by x,y

𝒲[T ,T ] (Bt1 ∈ A1 , . . . , Btn ∈ An ) 1

2

= ∫

Πt1 −T1 (x − x1 )Πt2 −t1 (x1 − x2 ) ⋅ ⋅ ⋅ ΠT2 −tn (xn − y) ΠT2 −T1 (x − y)

ℝnd

n

n

i=1

i=1

(∏ 1Ai (xi )) ∏ dxi

(2.3.18)

is called Brownian bridge measure starting in x at t = T1 and ending in y at t = T2 . Furthermore, the measure x,y

ΠT2 −T1 (x − y)𝒲[T ,T ] 1

(2.3.19)

2

is called conditional Wiener measure. Note that the Brownian bridge measure is a probability measure, while the conditional Wiener measure is not. x,y x,y We write 𝔼[T ,T ] for expectation with respect to 𝒲[T ,T ] . From the definition it is 1

2

x,y

directly seen that 𝒲 x and 𝒲[T ,T ] are related through 1

1

2

2

n

n

x,y

𝔼x [∏ fj (Btj )] = ∫ ΠT2 −T1 (x − y)𝔼[T ,T ] [∏ fj (Btj )] dy, j=1

1

ℝd

2

j=1

(2.3.20)

for all Borel measurable bounded function f1 , . . . , fn . In particular, n

n

x,y

𝔼x [f (BT1 ) (∏ fj (Btj )) g(BT2 )] = ∫ ΠT2 −T1 (x − y)f (x)g(y)𝔼[T ,T ] [∏ fj (Btj )] dy. j=1

1

ℝd

2

j=1

Definition 2.105 (Full Wiener measure). We refer to the measure d𝒲̄ = d𝒲 x ⊗ dx as full Wiener measure, carrying infinite mass.

on X × ℝd

(2.3.21)

2.3 Brownian motion and Wiener measure

| 89

Although full Wiener measure is not a probability measure on X × ℝd , we use the notation 𝔼𝒲̄ [. . .] for ∫X ×ℝd . . . d𝒲̄ . Full Wiener measure d𝒲 is expressed in terms of the Brownian bridge measure as x,y

d𝒲̄ = ( ∫ ΠT2 −T1 (x − y)d𝒲[T ,T ] dy) dx, 1

ℝd

2

or in other words, n

∫ ∏ fi (Bti )d𝒲̄ = X

×ℝd

i=1

n

x,y

∫ ΠT2 −T1 (x − y)𝔼[T ,T ] [∏ fi (Bti )] dxdy. 1

ℝd ×ℝd

2

i=1

(2.3.22)

For T1 ≤ t1 ≤ . . . ≤ tn ≤ T2 define 𝔼 [. . . | BT2 = x, BT1 = y] by n n 󵄨󵄨 x,y 𝔼 [∏ fj (Btj ) 󵄨󵄨󵄨 BT2 = x, BT1 = y] = ΠT2 −T1 (x − y)𝔼[T ,T ] [∏ fj (Btj )] . 1 2 󵄨 j=1

j=1

(2.3.23)

Thus for T1 ≤ t1 ≤ . . . ≤ tn ≤ T2 we have n

𝔼𝒲̄ [f (BT1 ) (∏ fj (Btj )) g(BT2 )] = j=1

n

󵄨󵄨 ∫ g(x)f (y)𝔼 [∏ fj (Btj ) 󵄨󵄨󵄨 BT1 = x, BT2 = y] dxdy. 󵄨

ℝd ×ℝd

j=1

2.3.4 Martingale properties of Brownian motion Brownian motion is a martingale of primary importance in the class of continuoustime random processes. In fact, the martingale property of a process with almost surely continuous paths, and of a related quadratic process, forces it to be a Brownian motion. Proposition 2.106. Let (Bt )t≥0 be Brownian motion on a filtered probability space (Ω, F , (Ft )t≥0 , P x ), where (Ft )t≥0 is the natural filtration of (Bt )t≥0 . Then both (Bt )t≥0 and (B2t − t)t≥0 are (Ft )-martingales. Proof. Adaptedness of the two processes and absolute integrability of Bt and B2t − t are straightforward. Let t ≥ s. We have 𝔼x [Bt |Fs ] = 𝔼x [(Bt − Bs ) + Bs |Fs ] = 𝔼x [Bt − Bs |Fs ] + 𝔼x [Bs |Fs ] = 𝔼x [Bt − Bs ] + Bs = Bs

(2.3.24)

by the independence of Bt − Bs and Br for r ≤ s. By a similar calculation it is easy to show also the second claim. Remarkably, there is a strong converse of this proposition.

90 | 2 Brownian motion Proposition 2.107 (Lévy martingale characterization theorem). Let (Xt )t≥0 be a random process on a probability space (Ω, F , P), and (Ft )t≥0 the natural filtration of (Xt )t≥0 . Suppose that (1) X0 = 0 almost surely, (2) t 󳨃→ Xt is almost surely continuous, and (3) both (Xt )t≥0 and (Xt2 − t)t≥0 are (Ft )-martingales. Then (Xt )t≥0 is standard Brownian motion. Proof. Let 0 ≤ s ≤ t. It is sufficient to show that Xt − Xs is independent of Fs and d

Xt − Xs = N(0, t − s). This follows from

𝔼[eiξ (Xt −Xs ) |Fs ] = e−(t−s)ξ

2

/2

,

ξ ∈ ℝ.

(2.3.25)

2

Indeed (2.3.25) yields that 𝔼[eiξ (Xt −Xs ) ] = e−(t−s)ξ /2 and 𝔼[eiξ (Xt −Xs ) 1A ] = 𝔼[eiξ (Xt −Xs ) ]P(A) for A ∈ Fs . This proves that Xt − Xs is independent of Fs by Lemma 2.27. In order to prove (2.3.25) we use Taylor expansion. Set ΔX = Xt − Xs and Δ = t − s. We have eiξ ΔX − e−ξ

2

Δ/2

= iξ ΔX +

ξ2 ξ2 ξ4 iξ 3 ξ6 Δ − (ΔX)2 − Δ2 − (ηΔX)3 + (θΔ)3 , 2 2 8 6 48

where η ∈ (0, 1) and θ ∈ (0, 1). Taking conditional expectation with respect to Fs , we have 𝔼[eiξ ΔX − e−ξ

2

Δ/2

ξ2

|Fs ] = −

ξ 4 2 iξ 3 ξ6 Δ − 𝔼[(ηΔX)3 |Fs ] + (θΔ)3 . 8 6 48

ξ2

Here we used 𝔼[iξ ΔX + 2 Δ − 2 (ΔX)2 |Fs ] = 0, which is obtained from the fact that Xt2 − t is a martingale. By the Hölder inequality we find 𝔼[(ηΔX)3 |Fs ] ≤ 𝔼[|ΔX|3 |Fs ] ≤ (𝔼[(ΔX)4 |Fs ])3/4 ≤ 43/4 Δ3/2 . The latter estimate is obtained in Lemma 2.108 below. Therefore we have 𝔼[eiξ ΔX − e−ξ

2

Δ/2

|Fs ] ≤ cξ (Δ2 + Δ3 + Δ3/2 )

with constant cξ . Suppose that Δ < 1. Then 𝔼[eiξ ΔX − e−ξ

2

Δ/2

|Fs ] ≤ cξ󸀠 Δ3/2 .

(2.3.26)

󵄨 󵄨 Let tj = s + (t − s)j/n. Using the bound 󵄨󵄨󵄨󵄨∏nj=1 aj − ∏nj=1 bj 󵄨󵄨󵄨󵄨 ≤ ∑nj=1 |aj − bj | for aj , bj ∈ ℂ and |aj |, |bj | ≤ 1, we find 󵄨󵄨 n n 󵄨󵄨󵄨 1 2 1 2 󵄨󵄨 󵄨 󵄨 󵄨󵄨𝔼[eiξ (Xt −Xs ) − e− 2 ξ (t−s) |Fs ]󵄨󵄨󵄨 ≤ 𝔼 [󵄨󵄨󵄨∏ eiξ (Xtj −Xtj−1 ) − ∏ e− 2 ξ (tj −tj−1 ) 󵄨󵄨󵄨 |Fs ] 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 󵄨󵄨 j=1 n 1 2 󵄨󵄨 iξ (X −X ) 󵄨󵄨 ≤ ∑ 𝔼 [󵄨󵄨󵄨e tj tj−1 − e− 2 ξ (tj −tj−1 ) 󵄨󵄨󵄨 |Fs ] . 󵄨 󵄨 j=1

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| 91

If n is sufficiently large, we get tj − tj−1 < 1. Hence we have by (2.3.26) n c󸀠 (t − s) 1 2 󵄨󵄨 󵄨 󵄨󵄨𝔼[eiξ (Xt −Xs ) − e− 2 ξ (t−s) |Fs ]󵄨󵄨󵄨 ≤ c󸀠 ∑(tj − tj−1 )3/2 = ξ . ξ 󵄨󵄨 󵄨󵄨 √n j=1

The right-hand side converges to zero as n → ∞. Lemma 2.108. We have that 𝔼[(Xt − Xs )4 |Fs ] ≤ 4|t − s|2 . Proof. Let tj = s + (t − s)j/n and ΔXj = Xtj − Xtj−1 , j = 1, . . . , n. By Fatou’s lemma we have n

n

n

𝔼 [lim inf ∑ ΔXj2 ] ≤ lim inf 𝔼 [∑ ΔXj2 ] = lim inf ∑(tj − tj−1 ) = t − s, n→∞

n→∞

j=1

n→∞

j=1

j=1

where we used 𝔼[(Xt − Xs )|Fs ] = t − s for s ≤ t. Therefore lim infn→∞ ∑nj=1 ΔXj2 is almost surely finite, and from path continuity we obtain n

n

lim inf ∑ |ΔXj |2+ε ≤ ( lim max |ΔXk |ε ) lim inf ∑ ΔXj2 = 0. n→∞

n→∞ 1≤k≤n

j=1

n→∞

j=1

Thus 4

n

(Xt − Xs )4 = (∑ ΔXj ) j=1

4

n

n

n

n

j=1

j=1

k=1

n

n

n

j=1

k=1

= lim inf {(∑ ΔXj ) + 3 ∑ ΔXj4 − 4 (∑ ΔXj3 ) ( ∑ ΔXk )} n→∞

j=1

4

n

n

k

≤ lim inf {(∑ ΔXj ) + 3 ∑ ΔXj4 − 4 (∑ ΔXj3 ) ( ∑ ΔXk ) +2 ∑ ∑ ΔXj2 ΔXk2 } n→∞

j=1

j=1

k=1 j=1

2

2

n k { n } = lim inf {2 ∑ ΔXj (∑ ΔXk ) + 4 ( ∑ ∑ ΔXj ΔXk ) } . n→∞ k =j̸ k=1 j=1 { j=1 }

By Fatou’s lemma again, 2

n

n

2 󵄨󵄨

k

𝔼[(Xt − Xs )4 |Fs ] ≤ lim inf 𝔼 [ 2 ∑ ΔXj (∑ ΔXk ) + 4 ( ∑ ∑ ΔXj ΔXk ) n→∞ k =j̸ k=1 j=1 [ j=1

󵄨󵄨 󵄨󵄨 F ] . 󵄨󵄨 s 󵄨󵄨 󵄨 ]

We expand the squares on the right-hand side and obtain 2

n

n

2

k

2 ∑ ΔXj (∑ ΔXk ) + 4 ( ∑ ∑ ΔXj ΔXk ) j=1

k =j̸

n

= c∑

∑

j=1 k =l,k ̸ =j,l ̸ =j̸

k=1 j=1

ΔXj2 ΔXk ΔXl + c󸀠

k

∑ j 0, and n ∈ ℕ. Moreover, its initial distribution P 0 = δx is a Dirac measure with mass at x, while its probability transition kernel pBM is given by pBM (t, x, A) = ∫ 1A (z)Πt (x − z)dz

(2.3.28)

ℝd

or, equivalently, pBM (t, x, dy) = Πt (x − y)dy.

(2.3.29)

Proof. The Markov property with probability transition kernel (2.3.29) follows by the definition of the finite-dimensional distributions of Brownian motion and Proposition 2.82. Note that pBM (s, t, x, A) = ∫ℝd 1A (z)Πt−s (x − z)dz. We show (2.3.27). Let h(y) = 𝔼x [∏nj=1 f (Bs+tj )|Bs = y]. By the identity n

n

j=1

j=1

󵄨󵄨 𝔼x [1A (Bs ) ∏ fj (Bs+tj )] = 𝔼x [1A (Bs )𝔼x [∏ fj (Bs+tj ) 󵄨󵄨󵄨 σ(Bs )]] 󵄨 we have n

n

n

j=1

j=2

j=1

∫ 1A (y)Πs (y − x) (∏ fj (xj )) Πt1 (x1 − y) (∏ Πtj −tj−1 (xj − xj−1 )) dy ∏ dxj (ℝd )n+1

= ∫ 1A (y)h(y)Πs (y − x)dy. ℝd

Comparing both sides above, h(y) =

n

n

n

j=1

j=2

j=1

∫ (∏ fj (xj )) Πt1 (x1 − y) (∏ Πtj −tj−1 (xj − xj−1 )) ∏ dxj (ℝd )n+1

is obtained, while the right-hand side equals 𝔼y [∏nj=1 fj (Btj )]. Hence n n 󵄨󵄨 𝔼x [∏ f (Bs+tj ) 󵄨󵄨󵄨 σ(Bs )] = 𝔼Bs [∏ fj (Btj )] 󵄨 j=1

and (2.3.27) follows.

j=1

94 | 2 Brownian motion From the proof of Theorem 2.112 it follows that the Markov property (2.2.52) can be reformulated as 𝔼x [f (Bt )|Fs ] = 𝔼Bs [f (Bt−s )],

t ≥ s.

(2.3.30)

In what follows, the Markov property will be considered with respect to the natural filtration, unless specified otherwise. Corollary 2.113. Let (Bt )t≥0 be d-dimensional Brownian motion starting at x ∈ ℝd on the filtered probability space (Ω, F , (FtBM )t≥0 , P x ). Let ξ ∈ ℝd . Then the random process 1

2

(eξ ⋅Bt − 2 |ξ | t )t≥0 is an (FtBM )t≥0 -martingale. Proof. By (2.3.30) we have

1 1 2 󵄨 2 󵄨 𝔼x [eξ ⋅Bt − 2 |ξ | t 󵄨󵄨󵄨 Fs ] = 𝔼Bs [eξ ⋅Bt−s − 2 |ξ | t ] . 󵄨

The right-hand side above can be computed as 1

1

2

1

2

2

𝔼Bs [eξ ⋅Bt−s − 2 |ξ | t ] = e− 2 |ξ | t ∫ eξ ⋅y Πt−s (Bs − y)dy = eξ ⋅Bs − 2 |ξ | s ℝd

and the corollary follows. Brownian motion has a yet stronger Markov property than (2.3.27) since the same equality holds even for random times. Theorem 2.114 (Strong Markov property). Let (Bt )t≥0 be d-dimensional Brownian motion on a filtered probability space (Ω, F , (FtBM )t≥0 , P), and τ a stopping time such that P(τ < ∞) = 1. Then Wt = Bt+τ − Bτ , t ≥ 0, is also a Brownian motion, which is independent of FτBM . Moreover, 𝔼x [f (Bt+τ )|Fτ ] = 𝔼Bτ [f (Bt )], for every bounded continuous function f . Proof. Fix 0 = t0 ≤ t1 ≤ . . . ≤ tn and ξ ∈ ℝd . Let τn be an approximation of τ, i. e., 0, τn = { i+1 n

,

τ = 0, i 0 | Bt = a} be the first hitting time of level a, and (B)̃ t≥0 be defined by (2.3.31). Denote S̃t = sup0≤s≤t B̃ s . We have P(St ≥ a, Bt ≤ a − y) = P(S̃t ≥ a, B̃ t ≤ a − y) = P(t ≥ τa , Bt ≥ a + y). Since P(Bt ≥ a + y) = P(t ≥ τa , Bt ≥ a + y) + P(t < τa , Bt ≥ a + y) = P(t ≥ τa , Bt ≥ a + y), equality (2.3.32) follows. By (2.3.32) we also see that P(St ≥ a) = P(St ≥ a, Bt ≤ a) + P(St ≥ a, Bt > a) = 2P(St ≥ a, Bt ≥ a) = 2P(Bt ≥ a).

We conclude by briefly discussing the transition semigroup of Brownian motion, and its generator. Proposition 2.117. The probability transition kernel pBM of Brownian motion is a Feller transition kernel, and its generator is 21 Δ on C0∞ (ℝd ).

2.3 Brownian motion and Wiener measure

| 97

Proof. We have already shown that pBM (t, x, A) = ∫ℝd 1A (z)Πt (x − z)dz in (2.3.28). Thus 2

pBM (t, x, ℝd ) = 1 identically holds. Since PtBM f (x) = (2πt)−d/2 ∫ℝd e−|z| /2t f (x + z)dz for f ∈ C∞ (ℝd ), by dominated convergence it is clear that PtBM f (x) is continuous in x and lim|x|→∞ PtBM f (x) = 0. Thus PtBM f ∈ C∞ (ℝd ). Next we check that lim PtBM f (x) = f (x).

(2.3.33)

t→0

Notice that f is uniformly continuous on the compact set ℝd ∪{∞}. Thus for every ε > 0 there exists δ > 0 such that |f (x + y) − f (x)| ≤ ε, for all x ∈ ℝd and |y| < δ. Hence |PtBM f (x) − f (x)| ≤ ∫ |f (x + y) − f (x)|Πt (y)dy + ∫ |f (x + y) − f (x)|Πt (y)dy |y| 0 it follows that p

𝒲 (Vn > ε) ≤

1 2 𝔼[|Vnp |2 ] ≤ 2 (𝔼[|Vnp − 2n(1−(p/2)) |2 ] + 22n(1−(p/2)) ) . 2 ε ε

(2.3.42)

p We have ∑∞ n=1 𝒲 (Vn > ε) < ∞ and from the Borel–Cantelli lemma it follows again that ∞ ∞ p 𝒲 (⋃m=1 ⋂n=m {Vn ≤ ε}) = 1. Hence we see that Vnp → 0 as n → ∞, almost surely. By (2.3.41)and (2.3.42) both convergences hold also in probability.

102 | 2 Brownian motion (2) Next we show Lq -convergence. It can be seen that 𝔼[|Vnp |q ] ≤ 2nq 𝔼[|B1/2n |qp ] ≤ 2nq(1−(1/2)p) . Thus in the case of p > 2, Vnp → 0 in Lq . Suppose next p = 2. Write for simplicity Xj = n

n

|Bj/2n − B(j−1)/2n |, and choose q = 2. We see that 𝔼[|Vn2 − 1|2 ] = 𝔼[(∑2j=1 Xj2 )2 − 2 ∑2j=1 Xj2 + 1]. We note that {Xn }n∈ℕ are independent random variables and 𝔼[Xn ] = 0. We obtain 2n

2n

2n

𝔼[|Vn2 − 1|2 ] = ∑ 𝔼[Xj2󸀠 Xj2 ] + ∑ 𝔼[Xj4 ] − 2 ∑ 𝔼[Xj2 ] + 1 j=1

j󸀠 =j̸ 2n

j=1

2n

2n

= ∑ 2−2n + ∑ 2−2n c4 − 2 ∑ 2−n + 1 j󸀠 =j̸ n

j=1

n

= 2 (2 − 1)2

−2n

j=1

n

+2 ⋅2

−2n

c4 − 2 ⋅ 2n ⋅ 2−n + 1.

Thus Vn2 → 1 as n → ∞ in L2 . Next suppose q = 4. In the same way as for L2 we see that 𝔼[|Vn2

4

− 1| ] =

2n

𝔼 [(∑ Xj2 ) j=1

4

2n

−

4(∑ Xj2 ) j=1

3

2n

+

6(∑ Xj2 ) j=1

2

2n

− 4(∑ Xj2 ) + 1] . j=1

We compute the fourth order term and get 2n

𝔼[(∑ Xj2 ) j=1

4

2n

]= ∑ i=j̸ =k̸ =l̸ 2

n

𝔼[Xi2 Xj2 Xk2 Xl2 ]

2n

+ ∑ 𝔼[Xi2 Xj2 Xk4 ] i=j̸ =k̸

n

2n

i=j̸

i=1

2

+ ∑ 𝔼[Xi4 Xj4 ] + ∑ 𝔼[Xi6 Xj2 ] + ∑ 𝔼[Xi8 ], i=j̸

n

where ∑2i=j̸ =k̸ =l̸ denotes summation over all 1 ≤ i, j, k, l ≤ 2n such that i, j, k, l are all different from each other (the other sums are similarly understood). We get 2n

∑ 𝔼[Xi2 Xj2 Xk2 Xl2 ] = 2−4n 2n (2n − 1)(2n − 2)(2n − 3) → 1

i=j̸ =k̸ =l̸

as n → ∞, and 2n

∑ 𝔼[Xi2 Xj2 Xk4 ] = 2−4n c4 2n (2n − 1)(2n − 2) → 0

i=j̸ =k̸

as n → ∞. Similarly, 2n

∑ 𝔼[Xi4 Xj4 ] = 2−4n c42 2n (2n − 1) → 0, i=j̸

2.3 Brownian motion and Wiener measure

| 103

2n

∑ 𝔼[Xi6 Xj2 ] = 2−4n 2n (2n − 1) → 0, i=j̸ 2n

∑ 𝔼[Xi8 ] = 2−4n c8 2n → 0. i=1

n

n

A further computation of the same kind yields that 𝔼[(∑2j=1 Xj2 )3 ] → 1, 𝔼[(∑2j=1 Xj2 )2 ] → n

1, and 𝔼[(∑2j=1 Xj2 )] → 1. Putting together these results we obtain that Vn2 → 1 in L4 . Take now q = 2m. In this case 𝔼[|Vn2

n

2m

m󸀠

2 󸀠 2m − 1] ] = ∑ ( 󸀠 )(−1)2m−m 𝔼[(∑ Xj2 ) ] m j=1 m󸀠 =0 2m

n

and we see that limn→∞ 𝔼[(∑2j=1 Xj2 )m ] = 1 similarly as above. Hence 󸀠

2m

lim 𝔼[|Vn2 − 1]2m ] = ∑ (

n→∞

m󸀠 =0

󸀠 2m )(−1)2m−m = 0, m󸀠

which implies Vn2 → 1 as n → ∞ in L2m .

2.3.7 Global path properties of Brownian motion Next we discuss the typical long time behaviour of Brownian motion. Proposition 2.121. Let (Bt )t≥0 be standard Brownian motion. Then lim inf Bt = −∞ t→∞

and

lim sup Bt = ∞ t→∞

hold almost surely. Proof. Fix a sequence (tn )n≥1 such that tn → ∞ as n → ∞. Due to the self-similarity d

of Brownian motion we have Btn = √tn B1 , and thus P(Btn > K) = P(B1 > K/√tn ) → 2 ∞ (2π)−1/2 ∫0 e−|x| /2 dx

∞

= 1/2 as n → ∞, for every K > 0. By Fatou’s lemma we get

∞

P( ⋃ ⋂ {Btn > K}) = 𝔼 [lim sup 1{Bt m=1 n=m

n→∞

n

>K} ]

≥ lim sup 𝔼[1{Bt

Hence P(lim supn→∞ Btn > K) ≥ 1/2. Therefore 1 P(lim sup Btn = ∞) ≥ . 2 n→∞

n→∞

n

>K} ]

1 = . 2

104 | 2 Brownian motion Let t0 = 0 and Zn = Btn − Btn−1 . The random variables (Zn )n∈ℕ are independent and Btn = ∑nj=1 Zj . We have ∞

I = {lim sup Btn = ∞} = {lim sup(Btn − Btm ) = ∞} ∈ σ( ⋃ σ(Zn )) n→∞

n→∞

n=m+1

∞ ∞ ∞ for each m. Since I ∈ ⋂∞ m=1 σ(⋃n=m+1 σ(Zn )) and ⋂m=1 σ(⋃n=m+1 σ(Zn )) is a tail-σ-field, Kolmogorov’s zero–one law yields P(I) ∈ {0, 1}. Since this probability is not less than 1/2 by the above, it must then be 1. Hence it follows that lim supn→∞ Btn = ∞ almost surely. By reflection symmetry of Brownian motion we immediately obtain lim infn∈ℕ Btn = −∞ almost surely.

Proposition 2.122. Let (Bt )t≥0 be standard Brownian motion. Then lim inf t→∞

Bt = −∞ and √t

lim sup t→∞

Bt =∞ √t

hold almost surely. Proof. Let K > 0 be arbitrary and consider the stopping time τ = inf{t > 0 | Bt ≥ K √t}. Take the events As = {τ < s} and A = {τ = 0}. It is directly seen that A = ⋂ As ∈ ⋂ σ(Br , 0 ≤ r ≤ t). s>0

t>0

d

By Blumenthal’s zero–one law it follows that P(A) ∈ {0, 1}. Using Bt /√t = B1 , we obtain that P(A) = inf P(As ) ≥ inf P(Bs ≥ K √s) = P(B1 ≥ K) > 0 s>0

s>0

and thus P(A) = 1. This means, in particular, that for every ε > 0 there exists 0 < t < ε such that Bt /√t ≥ K almost surely, for every K > 0. Thus lim supt↓0 Bt /√t = ∞ d

almost surely. Since tB1/t = Bt , we also see that lim supt→∞ Bt /√t = ∞ almost surely. It follows similarly that lim inft→∞ Bt /√t = −∞. Proposition 2.123 (Strong Law of Large Numbers). Let (Bt )t≥0 be standard Brownian motion. Then Bt =0 t→∞ t lim

almost surely. Proof. By the martingale inequality shown in Theorem 2.72, we see that 𝔼[ sup |Bt /t|2 ] ≤ S≤t≤T

1 4 4 𝔼[ sup |Bt |2 ] ≤ 2 𝔼[|BT |2 ] = 2 T. 2 S S S S≤t≤T

2.3 Brownian motion and Wiener measure

| 105

This implies that P( sup |Bt /t| > ε) ≤ S≤t≤T

1 1 4T 𝔼[ sup |B /t|2 ] ≤ 2 2 . ε2 S≤t≤T t ε S

Put S = 2n and T = 2n+1 . We have P(sup2n ≤t≤2n+1 |Bt /t| > ε) ≤

8 −n 2 ε2

and

∞

∑ P( sup |Bt /t| > ε) < ∞.

n=1

2n ≤t≤2n+1

Hence the Borel–Cantelli lemma yields ∞

∞

P ( ⋃ ⋂ { sup |Bt /t| ≤ ε}) = 1 m=1 n=m 2n ≤t≤2n+1

which implies the statement. The last two statements above over- and underestimated the typical growth profile of Brownian motion. The following result identifies the almost sure growth rate precisely. Proposition 2.124 (Law of Iterated Logarithm). Let (Bt )t≥0 be standard Brownian motion. Then (1) lim supt→∞ Bt /√2t log log t = 1; (2) lim inft→∞ Bt /√2t log log t = −1; (3) lim supt↓0 Bt /√2t log log(1/t) = 1; (4) lim inft↓0 Bt /√2t log log(1/t) = −1 hold almost surely. Proof. First we consider (1) and divide the proof into two steps. Step 1: (Upper bound) Fix ε > 0 and r > 1. Let 𝒜n = { sup Bt ≥ (1 + ε)√2r n log log r n }. 0≤t≤r n

By the Reflection Principle obtained in Corollary 2.116 we have P(𝒜n ) ≤ 2P(Brn ≥ (1 + ε)√2r n log log r n ) = 2P(Brn /√r n ≥ (1 + ε)√2 log log r n ). d

Since Brn /√r n = B1 , we obtain P(𝒜n ) ≤ 2P (B1 ≥ (1 + ε)√2 log log r n ) .

(2.3.43)

It can be directly seen that (2π)−1/2

2 x 1 2 e−x /2 ≤ P(B1 > x) ≤ (2π)−1/2 e−x /2 , x 1 + x2

x > 0.

(2.3.44)

106 | 2 Brownian motion By (2.3.43) and (2.3.44) we have P(𝒜n ) ≤

2 n 2 1 e−(1+ε) log log r ≤ c(n log r)−(1+ε) . (1 + ε)√π √log log r n

∞ ∞ c Thus ∑∞ n=1 P(𝒜n ) < ∞, and hence P(⋃m=1 ⋂n=m 𝒜n ) = 1 follows from the Borel–Cantelli lemma, which implies that

lim sup n→∞

sup0≤t≤rn Bt

√2r n log log r n

≤ (1 + ε)

almost surely. For every t > 1, there exists n such that t ∈ [r n−1 , r n ]. Note that t 󳨃→ √2t log log t is increasing. For sufficiently large t we have Bt

√2t log log t

≤

sup0≤s≤rn Bs

√2r n log log r n

√2r n log log r n

√2r n−1 log log r n−1

.

Thus we conclude that lim sup t→∞

Bt ≤ (1 + ε)√r. √2t log log t

Since ε > 0 and r > 1 are arbitrary, we obtain lim sup t→∞

Bt

√2t log log t

≤ 1.

(2.3.45)

Step 2: (Lower bound) Let ℬn = {Br n − Br n−1 ≥ √2(r n − r n−1 ) log log r n }

for n ≥ 1. The events ℬn , n = 1, 2, . . ., are independent due to the independence of the increments of (Bt )t≥0 . Moreover, it follows that P(ℬn ) = P ( Since

Brn −rn−1 d = √r n −r n−1

P(

B n n−1 Brn − Brn−1 ≥ √2 log log r n ) = P ( r −r ≥ √2 log log r n ) . n n−1 √r − r √r n − r n−1

B1 , we have

n Brn −rn−1 1 1 c ≥ √2 log log r n ) ≥ e− log log r ≥ . √r n − r n−1 √8π √2 log log r n √ n log n

Here we used the lower bound in (2.3.44), and 2x/(1 + x 2 ) ≥ 1/x for all x > 1. Hence ∑∞ n=1 P(ℬn ) = ∞, and thus it follows by the Borel–Cantelli lemma that ∞ P(⋂m=1 ⋃∞ n=m ℬn ) = 1. Hence for infinitely many n ≥ 1, Brn ≥ √2(r n − r n−1 ) log log r n + Brn−1 .

2.4 Stochastic calculus based on Brownian motion

|

107

Using the estimate in Step 1 for (−Bt )t≥0 we get that −Brn−1 (ω) ≤ 2√2r n−1 log log r n−1 ≤

2 √2r n log log r n , √r

(2.3.46)

almost surely for every n ≥ n0 (ω), with some n0 (ω). We then obtain Brn ≥ √2(r n − r n−1 ) log log r n −

2 √2r n log log r n √r

and conclude that lim sup t→∞

Bt

√2t log log t

≥ lim sup n→∞

Br n n √2r log log r n

≥ √1 −

1 2 − . r √r

(2.3.47)

Letting r → ∞ along a countable sequence completes the proof of (1). Since (−Bt )t≥0 and (tB1/t )t≥0 are again Brownian motions, parts (2), (3), and (4) follow from (1).

2.4 Stochastic calculus based on Brownian motion 2.4.1 The classical integral and its extensions Classical analysis provides a working notion of integral called after Riemann. In this concept a real-valued function f on an interval I = [a, b] is considered, I is divided into subintervals Δn = {a = x0 < . . . < xn = b}, and limits of sums of the type n−1

∑ f (ξj )(xj+1 − xj ),

j=0

ξj ∈ [xj , xj+1 ),

are considered when limn→∞ m(Δn ) = 0, where m(Δn ) is the mesh of the partition. When such a limit exists independent of the choice of the partition and the values ξj , b

we call it the Riemann integral ∫a f (x)dx of f . A sufficient condition for the Riemann integral of f to exist is that f is continuous. If for some purpose one is interested in integrating a function against the increments of another, it is possible to extend the concept of Riemann integral to cover this b case. This extension is achieved by the Stieltjes integral ∫a f (x)dg(x), obtained as the limit of sums of the type n−1

∑ f (ξj )(g(xj+1 ) − g(xj )),

j=0

ξj ∈ [xj , xj+1 ).

The Riemann integral then follows when the integrator is the identity function, g(x) = x. Two sufficient conditions for the Stieltjes integral of f with respect to g

108 | 2 Brownian motion to exist are (1) f and g have no discontinuities at the same point x, (2) f is continuous and g is of bounded variation. However, (2) is not a necessary condition. Surprisingly, there are few results on weaker conditions on the existence of the Riemann–Stieltjes integral. Young has shown as early as in 1936 that if f is ρ-Hölder b

continuous and g is γ-Hölder continuous such that γ + ρ > 1, then ∫a f (x)dg(x) exists. A nearly optimal (sufficient and “very nearly” also necessary) condition is that

(1) f and g are not both discontinuous at the same point x; (2) f has bounded p-variation and g has bounded q-variation for some p, q > 0 such that 1/p + 1/q = 1. b

Our interest is to define integrals of the type ∫a Xt dBt where we would use Brownian motion as integrator and a random process Xt as integrand. Clearly, because Brownian motion is nowhere differentiable, the usual Riemann–Stieltjes integral will not work. However, since Brownian motion is of bounded p-variation for p > 2, see Proposition 2.120, provided the integrand is of bounded q-variation with suitable q < 2, Young’s criterion can be applied. This is possible, in particular, when (Xt )t≥0 has bounded variation, i. e., q = 1. A sufficient condition for this is that t 󳨃→ Xt is differentiable and has a bounded derivative. However, the approach offered by Young or the conditions above quickly show their limitations as they would fail to cover the simple b integral ∫a Bt dBt . In order to include this and more interesting cases a genuinely new type of integral is needed. 2.4.2 Stochastic integrals Recall the basic theorem of calculus: If f ∈ C 1 (ℝd , ℝ) and g ∈ C 1 (ℝ, ℝd ), then t

f (g(t)) − f (g(0)) = ∫ ∇f (g(s)) ⋅ (dg(s)/ds) ds. 0

Since Brownian paths are nowhere differentiable, we cannot substitute Bs for g(s) in the formula above. In spite of that, a similar formula still exists with an extra term appearing as a correction. Indeed, we will see that almost surely we have t

f (Bt ) − f (B0 ) = ∫ ∇f (Bs ) ⋅ dBs + 0

t

1 ∫ Δf (Bs ) ds, 2 0

with an expression formally written as dBs “=” (dBs /ds)ds, which will be explained below. The first term at the right-hand side above leads to a new concept of integral. Following the conventional concept of Riemann–Stieltjes integrals, in order to construct the integral with respect to Brownian motion we divide the interval into n subintervals and look at approximating sums

2.4 Stochastic calculus based on Brownian motion n

Sn = ∑ f (Bsj )(Btj+1 − Btj ), j=1

|

109

sj ∈ [tj , tj+1 ).

The pathwise (that is, in X pointwise) limit of the random variable Sn does not exist due to the unbounded variation of Brownian paths, thus it cannot be defined as a Stieltjes integral. However, as will be seen below, when Sn is regarded as an element of L2 (X , d𝒲 ), it does have a limit. But even in this situation we face the peculiar feature that, unlike in the case of the Riemann–Stieltjes integral, the limit depends on how the point sj is chosen and a different object is obtained when sj = tj or when sj = tj+1 . First we consider one-dimensional standard Brownian motion. Example 2.125. Take a division of the interval [0, T] into subintervals of equal length, i. e., choose tj = jT/n. Then a direct calculation gives limn→∞ ∑n−1 j=0 Btj (Btj+1 − Btj ) =

1 2 2 − T) and limn→∞ ∑n−1 j=0 Btj+1 (Btj+1 − Btj ) = 2 (BT + T) in L -sense. Indeed, a rearrangement of the terms gives for the first sum 1 2 (B 2 T

n−1

1 1 n−1 ∑ Btj (Btj+1 − Btj ) = B2T − ∑ (Btj+1 − Btj )2 . 2 2 j=0 j=0

By Proposition 2.120 the latter sum converges in L2 sense to the quadratic variation VT2 = T, thus the first claim follows. The other limit can be computed similarly. To remove the problem of dependence on the intermediary point, we restrict here to the choice sj = tj , i. e., we choose the integrand to be adapted to the natural filtration (FtBM )t≥0 of standard Brownian motion. We call (ϕ(t))t≥0 a random step process if there is a sequence 0 = t0 < . . . < tn and random variables f0 , . . . , fn ∈ L2 (X , d𝒲 ), fj are ℱtBM -adapted and j n−1

ϕ(t, ω) = ∑ fj (ω)1[tj ,tj+1 ) (t). j=0

(2.4.1)

2 The notation Mstep will be used for the space of random step processes.

Definition 2.126. We denote by M 2 (S, T) the class of functions f : ℝ+ × X → ℝ such that (1) f ∈ ℬ(ℝ+ ) × ℬ(X )/ℬ(ℝ); (2) f (t, ω) is (FtBM )-adapted; T

(3) 𝔼[∫S |f (t, ω)|2 dt] < ∞.

We note that the filtration (FtBM )t≥0 is right-continuous and contains all the null sets, while the natural filtration (Ft )t≥0 of Brownian motion is not. There are general reasons in stochastic analysis how the filtration is selected. For instance, first hitting times of an open or a closed set are stopping times under (FtBM )t≥0 , see Proposition 2.60, and thus makes this filtration more convenient to use.

110 | 2 Brownian motion Let ϕ(t, ω) be (2.4.1) with S = t0 < t1 < . . . < tn = T, and (Bt )t≥0 one-dimensional Brownian motion. Define T

n−1

∫ ϕ(t, ω)dBt = ∑ fj (ω)(Btj+1 (ω) − Btj (ω)). j=0

S

(2.4.2)

Lemma 2.127. Let ϕ(t, ω) be as in (2.4.2). Then 2

󵄨󵄨 T 󵄨󵄨 T 󵄨 󵄨󵄨 [󵄨󵄨 󵄨] 𝔼 [󵄨󵄨󵄨∫ ϕ(t, ω)dBt 󵄨󵄨󵄨 ] = 𝔼x [∫ |ϕ(t, ω)|2 dt ] . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 [S ] [󵄨 S ] x

(2.4.3)

T

Proof. Denote in shorthand I(ϕ) = ∫S ϕ(t, ω)dBt , ΔBj = Btj+1 − Btj , and Δtj = tj+1 − tj . Clearly 𝔼x [ΔBj ] = 0, 𝔼x [(ΔBj )2 ] = Δtj . We have n−1

|I(ϕ)|2 = ∑ fj2 (ΔBj )2 + 2 j=0

∑ 0≤j n.

Step 4: If f ∈ M 2 (S, T), by (2.4.7)–(2.4.9) we can choose a sequence of random step T 2 functions (ϕn )n∈ℕ ⊂ Mstep such that 𝔼x [∫S |f (t, ω) − ϕn (t, ω)|2 dt] → 0 as n → ∞. T

Hence by Lemma 2.4.6 and the Itô isometry (∫S ϕn dBt )n∈ℕ is a Cauchy sequence in T

L2 (X , d𝒲 x ). Thus ∫S ϕn dBt converges to a random variable as n → ∞ in L2 (X , d𝒲 x ). Now we are in the position to define the Itô integral. Definition 2.129 (Itô integral / stochastic integral). Let f ∈ M 2 (S, T) and a sequence T 2 (ϕn )n∈ℕ ⊂ Mstep be such that 𝔼x [∫S |f (t, ω) − ϕn (t, ω)|2 dt] → 0 as n → ∞. We define the Itô integral of f by T

T

∫ f (t, ω)dBt = lim ∫ ϕn (t, ω)dBt . S

n→∞

S

(2.4.10)

112 | 2 Brownian motion T

Note that the definition of ∫S f (t, ω)dBs is independent of the choice of random step functions ϕn . t

Example 2.130. We compute ∫0 Bs dBs as follows. Take a division 0 = s0 < . . . < sn = t t

of the interval, with sj = jt/n. Write ϕn (t) = ∑n−1 j=0 Bsj 1[sj ,sj+1 ) . The integrals ∫0 ϕn (s, ω)dBs t

approximate ∫0 Bs dBs in L2 -sense and we see by Example 2.125 that t

t

0

0

n−1 1 t ∫ Bs dBs = lim ∫ ϕn (t)dBs = lim ∑ Bsj (Bsj+1 − Bsj ) = B2t − . n→∞ n→∞ 2 2 j=0

Next we present some basic properties of the Itô integral. Theorem 2.131 (Itô isometry). Let f ∈ M 2 (S, T). The expectation of any Itô integral is T

𝔼x [∫ f (s, ω)dBs ] = 0, [S ] and its covariance is given by 󵄨󵄨 T 󵄨󵄨2 T 󵄨 󵄨󵄨 [󵄨󵄨󵄨 󵄨] 𝔼 [󵄨󵄨∫ f (s, ω)dBs 󵄨󵄨󵄨 ] = 𝔼x [∫ |f (s, ω)|2 ds] . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 [S ] [󵄨 S ] x

The latter equality is called Itô isometry for this class of functions. T

Proof. This follows from the definition of ∫S f (t, ω)dBt and (2.4.3). Theorem 2.132 (Martingale property). Let f ∈ M 2 (0, t) for all t ≥ 0. The random prot cess (It )t≥0 , It = ∫0 f (s, ω)dBs , is an (FtBM )-martingale, i. e., (1) It is FtBM -measurable, (2) 𝔼x [|It |] < ∞, (3) 𝔼x [It |FsBM ] = Is for s ≤ t. t

Proof. Statements (1) and (2) are immediate; we show (3). Let It (n) = ∫0 ϕn (s, ω)dBs , (n) where ϕn = ∑n−1 j=0 fj (ω)1[t (n) −t (n) ) (t); It (n) is an approximating sequence of It . We have j+1

𝔼

x

[It (n)|FsBM ]

j

s

󵄨󵄨 t 󵄨󵄨 󵄨 [ = ∫ ϕn (u, ω)dBu + 𝔼 ∫ ϕn (r, ω)dBr 󵄨󵄨󵄨 FsBM ] . 󵄨󵄨 󵄨󵄨 0 [s ] x

The claim follows by showing that the second term at the right-hand side above vanishes. We have 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨 [ 𝔼 ∫ ϕn (r, ω)dBr 󵄨󵄨󵄨 FsBM ] = 𝔼x [ fj(n) (Bt (n) − Bt (n) )󵄨󵄨󵄨 FsBM ] , ∑ 󵄨󵄨 󵄨 j+1 j (n) ≤t 󵄨󵄨 [s ] s≤tj(n) ≤tj+1 x

2.4 Stochastic calculus based on Brownian motion

| 113

where fj(n) (Bt (n) − Bt (n) ) is independent of FsBM and thus 𝔼[fj(n) (Bt (n) − Bt (n) )|FsBM ] = j+1

j

j+1

𝔼[fj(n) (Bt (n) − Bt (n) )] = 0. Therefore, j+1

j

j

󵄨󵄨 t 󵄨󵄨 󵄨 [ 𝔼 ∫ ϕn (r, ω)dBr 󵄨󵄨󵄨 FsBM ] = 0. 󵄨󵄨 󵄨󵄨 [s ] x

Theorem 2.133 (Continuous version). Let f ∈ M 2 (0, T). The random process (It )0≤t≤T , t It = ∫0 f (s, ω)dBs , has a continuous version (It̃ )0≤t≤T , i. e., It̃ is almost surely continuous in t and 𝒲 x (It̃ = It ) = 1 for all 0 ≤ t ≤ T. 2 Proof. Without restricting generality we set x = 0. Let (ϕn (t))n∈ℕ ⊂ Mstep be such that t

t

𝔼[∫0 |f (t, ω) − ϕn (t, ω)|2 dt] → 0 as n → ∞. Write It (n) = ∫0 ϕn (s, ω)dBs . Note that It (n) is continuous in t for all n and is an FtBM -martingale. This implies that It (n) − It (m) is also an FtBM -martingale, thus by the martingale inequality (see Theorem 2.72) we have 𝒲 ( sup |It (n) − It (m)| > ε) ≤ 0≤t≤T

1 𝔼[|IT (n) − IT (m)|2 ] ε2 T

=

1 [ 𝔼 ∫ |ϕn (s) − ϕm (s)|2 ds] → 0 ε2 [0 ]

as m, n → ∞. Hence we can choose a subsequence nk → ∞ such that P(Ak ) ≤ 2−k , where Ak = {ω ∈ X | sup |It (nk+1 , ω) − It (nk , ω)| > 2−k }. 0≤t≤T

Since ∑∞ k=1 P(Ak )

∞ < ∞, we get 𝒲 (⋂∞ N=1 ⋃k=N Ak ) = 0 by the Borel–Cantelli lemma. Thus for almost every ω ∈ X it follows that

sup |It (nk+1 , ω) − It (nk , ω)| ≤ 2−k

0≤t≤T

for all k > N(ω), with some N(ω). Therefore (It (nk , ω))k∈ℕ is a Cauchy sequence and so it converges uniformly in t ∈ [0, T] as k → ∞ for almost every ω ∈ X , which we denote by It̃ (ω). Note that It̃ (ω) is continuous in t for almost every ω ∈ X since the convergence is uniform in t. Since It (nk ) → It in L2 (X ), we conclude that It̃ = It almost surely. From now on we will consider the continuous version of the Itô integral of a random process without explicitly stating it. Next we consider d-dimensional Brownian motion (Bt )t≥0 and f = (f1 , . . . , fd ) such that fμ ∈ M 2 (0, t), μ = 1, . . . , d. We define the Itô integral t

d

t

∫ f (s, ω) ⋅ dBs = ∑ ∫ fμ (s, ω)dBμs . 0

μ=1 0

(2.4.11)

114 | 2 Brownian motion t

It is direct to see that the basic properties 𝔼x [∫0 f (s, ω) ⋅ dBs ] = 0 and t

x

t

t

𝔼 [∫ fμ (s, ω)dBμs ∫ fν (s, ω)dBνs ] = δμν 𝔼x [∫ fμ (s, ω)fν (s, ω)ds] 0 [0 ] [0 ] similarly hold. Let Cb (ℝ × ℝd ) denote the set of bounded continuous functions on ℝ × ℝd and n Cb (ℝ × ℝd ) the set of bounded and n times continuously differentiable functions on ℝ × ℝd . Consider the following specific cases. Corollary 2.134. Let (Bt )t≥0 be d-dimensional Brownian motion. Write tj = tj/2m and μ μ μ ΔBj = Btj − Btj−1 , 1 ≤ j ≤ 2m . (1) Let f ∈ Cb (ℝd ). Then

2n

μ lim ∑ f (Btj−1 )ΔBj n→∞ j=1

t

= ∫ f (Bs )dBμs .

(2.4.12)

0

(2) Let f ∈ Cb2 (ℝd ). Then 2n

lim ∑

n→∞

j=1

t

t

1 1 μ (f (Btj−1 ) + f (Btj )) ΔBj = ∫ f (Bs )dBμs + ∫ 𝜕μ f (Bs )ds. 2 2 0

(2.4.13)

0

The limits in both expressions above are understood in L2 (X , d𝒲 x ) sense. Proof. Statement (1) is directly obtained from the definition of Itô integral. For (2) we write d

f (Btj ) − f (Btj−1 ) = ∑ 𝜕ν f (Btj−1 )ΔBνj + ν=1

1 d ρ ∑ 𝜕 𝜕 f (ξ )ΔBj ΔBνj 2 ρ,ν=1 ρ ν

with some ξ = ξ (ω) by the Taylor expansion. We see that 󵄨󵄨 2n 󵄨󵄨󵄨 d 2n n→∞ μ μ 󵄨󵄨 󵄨 ∑ 󵄨󵄨󵄨(f (Btj ) − f (Btj−1 )) ΔBj − ∑ 𝜕ν f (Btj−1 )ΔBνj ΔBj 󵄨󵄨󵄨 ≤ C ∑ |Btj − Btj−1 |3 󳨀→ 0 󵄨󵄨 󵄨 󵄨 ν=1 j=1 󵄨 j=1 󵄨 in L2 (X , d𝒲 x ), where C is a constant independent of ω. Let μ be fixed. By writing f (Btj−1 ) + f (Btj ) = 2f (Btj−1 ) + (f (Btj ) − f (Btj−1 )) and making a Taylor expansion we have 2n

μ

l. h. s. (2.4.13) = lim ∑ f (Btj−1 )ΔBj + n→∞ t

=

∫ f (Bs )dBμs 0

yielding (2.4.13).

j=1

t

1 + ∫ 𝜕μ f (Bs )ds, 2 0

n

2 d 1 μ lim ∑ ∑ 𝜕ν f (Btj−1 )ΔBνj ΔBj 2 n→∞ j=1 ν=1

| 115

2.4 Stochastic calculus based on Brownian motion

Definition 2.135 (Stratonovich integral). The right-hand side of (2.4.13) defines the Stratonovich integral, i. e., for f = (f1 , . . . , fd ) ∈ (Cb2 (ℝd ))d , t

t

t

∫ f (Bs ) ∘ dBs = ∫ f (Bs ) ⋅ dBs + 0

0

1 ∫ ∇ ⋅ f (Bs )ds. 2

(2.4.14)

0

2.4.3 Extension of stochastic integrals Finally, we define a more general stochastic integral for a wider class of integrands by T weakening the conditions of M 2 (0, T). Condition 𝔼[∫0 |f (s, ω)|2 ds] < ∞ is too strong to derive the so-called Feynman–Kac–Itô formula in Chapter 4. We then want to deT fine the stochastic integral ∫0 f (s, ω)dBs for f such that it does not necessarily satisfy T

𝔼[∫0 |f (s, ω)|2 ds] < ∞.

Lemma 2.136. Let f , g ∈ M 2 (0, t), and let τ and ρ be stopping times with respect to (FtBM )t≥0 such that τ ≤ ρ a. s. Then we have the following: t∧ρ

(1) 𝔼[∫0

t∧τ

fdBs |FτBM ] = ∫0

t (∫0

2

(2) Xt = fdBs ) − (3) it follows that

t ∫0

2

fdBs ;

|f | ds, t ≥ 0, is an (FtBM )t≥0 -martingale;

t∧ρ

t∧τ

t∧ρ

t∧τ

t∧ρ

0

0

0

0

t∧τ

󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝔼[( ∫ fdBs − ∫ fdBs )( ∫ gdBs − ∫ gdBs )󵄨󵄨󵄨FτBM ] = 𝔼[ ∫ fgds󵄨󵄨󵄨FτBM ]; 󵄨󵄨 󵄨󵄨 in particular, we have t∧ρ

t∧τ

0

0

t∧ρ

2󵄨

󵄨󵄨 󵄨󵄨 󵄨 𝔼[( ∫ fdBs − ∫ fdBs ) 󵄨󵄨󵄨FτBM ] = 𝔼[ ∫ |f |2 ds󵄨󵄨󵄨FτBM ]; 󵄨󵄨 󵄨󵄨 t∧τ

(4) ∫0

t∧τ

t

fdBs = ∫0 f 1{s≤τ} dBs . t

Proof. Since (∫0 fdBs )t≥0 is a continuous martingale, statement (1) follows from Corollary 2.71. For (2) we see that s

2

s

t

t

0

s

s

2

t

Xt = (∫ fdBr ) + 2 (∫ fdBr ) (∫ fdBr ) + (∫ fdBr ) − ∫ |f |2 dr. 0

0 t

It is straightforward to show 𝔼[Xt |FsBM ] = Xs . We show (3). Write It (f ) = ∫0 fdBs . We have 𝔼[(It∧ρ (f ) − It∧τ (f ))2 |FτBM ] = It∧τ (f )2 − 2It∧τ (f )𝔼[It∧ρ (f )|FτBM ] + 𝔼[It∧ρ (f )2 |FτBM ]. (2.4.15)

116 | 2 Brownian motion Since (It )t≥0 and (Xt )t≥0 are continuous martingales by (2), we have 𝔼[Xt∧ρ |FτBM ] = Xt∧τ ,

𝔼[It∧ρ |FτBM ] = It∧τ

by Corollary 2.71. Inserting this into the right-hand side of (2.4.15), we have 𝔼[(It∧ρ (f ) − It∧τ (f ))

2

|FτBM ]

󵄨󵄨 󵄨󵄨 t∧ρ t∧ρ t∧τ 󵄨 󵄨 2 󵄨󵄨󵄨 BM ] 2 2 󵄨󵄨󵄨 [ [ = 𝔼 ∫ |f | ds󵄨󵄨 Fτ − ∫ |f | ds = 𝔼 ∫ |f | ds󵄨󵄨 FτBM ] . 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 [0 ] 0 [ t∧τ ]

Replacing f in this equality first by f +g and then by f −g, and subtracting the resulting equation, we obtain (3). Finally, we prove (4). Let f ̃(s, ω) = f (s, ω)1{s≤τ(ω)} . We have It∧τ (f ) − It (f ̃) = It∧τ (f − f ̃) − (It (f ̃) − It∧τ (f ̃)). By (3) we see that 󵄨󵄨 t∧τ 󵄨󵄨 󵄨 BM ] 2 BM 2 ̃ ̃ ̃ 𝔼 [(It∧τ (f − f ) − Is∧τ (f − f )) |Fs∧τ ] = 𝔼 [ ∫ |f − f | ds󵄨󵄨󵄨 Fs∧τ =0 󵄨󵄨 󵄨 s∧τ 󵄨 [ ]

(2.4.16)

for arbitrary s ≥ 0, and then It∧τ (f − f ̃) = 0 follows from taking the expectation on both sides of (2.4.16). Moreover, 󵄨󵄨 t 󵄨 2 BM 2 󵄨󵄨󵄨 BM ] ̃ ̃ ̃ [ 𝔼[(It (f ) − It∧τ (f )) |Ft∧τ ] = 𝔼 ∫ |f | ds󵄨󵄨 Ft∧τ =0 󵄨󵄨 󵄨󵄨 [t∧τ ] is derived. Then 𝔼[(It∧ρ (f ) − It (f ̃))2 ] = 0 follows, which gives the proof of (4). Definition 2.137. Let 𝕄2 (S, T) be the class of f : ℝ+ × X → ℝ such that (1) f ∈ ℬ(ℝ+ ) × ℬ(X )/ℬ(ℝ); (2) f (t, ⋅) is (FtBM )-adapted; T

(3) 𝒲 (∫S |f (s, ω)|2 ds < ∞) = 1.

Let f ∈ 𝕄2 (0, T) and define a stopping time with respect to (FtBM )t≥0 by 󵄨 t 2 {inf {t ≥ 0 󵄨󵄨󵄨 ∫0 f (s, ω) ds ≥ n} τn = { {∞

󵄨 t if {t ≥ 0 󵄨󵄨󵄨 ∫0 f (s, ω)2 ds ≥ n} ≠ 0, 󵄨 t if {t ≥ 0 󵄨󵄨󵄨 ∫0 f (s, ω)2 ds ≥ n} = 0.

(2.4.17)

t

Note that (∫0 f (s, ⋅)2 ds)t≥0 is a continuous adapted process with respect to the filtrat

tion (FtBM )t≥0 . The first hitting time τn of (∫0 f (s, ⋅)2 ds)t≥0 to the closed set [n, ∞) is a stopping time by Proposition 2.60. It follows that τn ↑ ∞ as n ↑ ∞, since we assume T 𝒲 (∫0 |f (s, ω)|2 ds < ∞) = 1. Let f ∈ 𝕄2 (0, t). Although f ∈ M 2 (0, t) is not necessarily

2.4 Stochastic calculus based on Brownian motion

| 117

satisfied, we see that f ̃(s, ω) = f (s, ω)1{s≤τn (ω)} ∈ M 2 (0, t) for each n. We can define the t stochastic integral ∫ f ̃(s, ω)dB . Define s

0

Itn (f )

t∧τn

= ∫ f ̃(s, ω)dBs . 0

y

More precisely, Itn (f ) is defined by ∫0 f ̃(s, ω)dBs evaluated as y = t ∧ τn . Lemma 2.138. Let f ∈ 𝕄2 (0, T), and the stopping time τn be defined by (2.4.17). Then for m < n, Itn (f ) = Itm (f ),

0 ≤ t ≤ τm .

Proof. By (3) of Lemma 2.136 we see that 𝔼[(Itn (f )

−

Itm (f ))2 |FτBM ] m

󵄨󵄨 󵄨󵄨 󵄨 ]. = 𝔼 [ ∫ |f (s, ω)| ds󵄨󵄨󵄨󵄨 FτBM 󵄨󵄨 m 󵄨 t∧τ [ m ] 󵄨 t∧τn

2

Thus t∧τn

𝔼[(Itn (f ) − Itm (f ))2 1A ] = 𝔼 [ ∫ |f (s, ω)|2 ds1A ] [t∧τm ]

(2.4.18)

for any A ∈ FτBM . Take A = {t ≤ τm } ∈ FmBM . From (2.4.18) we have m 𝔼[(Itn (f )

−

Itm (f ))2 1A ]

t

= 𝔼 [∫ |f (s, ω)|2 ds1A ] = 0, [t

]

which implies that Itn (f ) = Itm (f ) if t ≤ τm . Definition 2.139 (Extension of Itô/stochastic integral). Let f ∈ 𝕄2 (0, t) and define the t stopping time τn by (2.4.17). Then the stochastic integral ∫0 f (s, ω)dBs is defined by t∧τn

t

∫ f (s, ω)dBs = ∫ f (s, ω)1{s≤τn } dBs , 0

0 ≤ t ≤ τn .

(2.4.19)

0

Remark 2.140. (1) The extension of the stochastic integral (2.4.19) is well defined by Lemma 2.138. Let f ∈ 𝕄2 (0, t). Then by (4) of Lemma 2.136 we have t∧τn

t

∫ f (s, ω)1{s≤τn } dBs = ∫ f (s, ω)1{s≤τn } dBs . 0

0

(2.4.20)

118 | 2 Brownian motion (2) We can define the stochastic integral not only for Brownian motion but also for a continuous local martingale (Xt )t≥0 . A random process (Xt )t≥0 on a filtered probability space (Ω, F , (Ft )t≥0 , P) is said to be a local martingale whenever there exists a nondecreasing sequence (τn )n∈ℕ̇ of stopping times such that (Xt∧τn )t≥0 is a martingale and P(limn→∞ τn = ∞) = 1. The resulting integral is denoted by T ∫0 f (s, ω)dXs . (3) Let f ∈ 𝕄2 (0, t). Note that the integral in Definition 2.139 is not an (FtBM )t≥0 -martingale and the Itô isometry is also not automatically satisfied. By (1) of Lemma 2.136 it follows that t∧τn

󵄨󵄨 s∧τn 󵄨󵄨 󵄨 BM 𝔼 [ ∫ f 1{r≤τn } dBr 󵄨󵄨󵄨 Fs ] = ∫ f 1{r≤τn } dBr 󵄨󵄨 󵄨󵄨 0 [ 0 ] t

for all n and 0 ≤ s ≤ t. Then (∫0 fdBr )t≥0 is a local martingale. Example 2.141. Let f ∈ L2loc (ℝ). It is not necessarily true that f ∈ M 2 (0, t); however, it t

is true that f ∈ 𝕄2 (0, t). Therefore ∫0 f (Bs )dBs is well defined in the sense of (2.4.19).

Using random processes given by Itô integrals we have the following important class. Definition 2.142 (Itô process). A random process (Xt )t≥0 on the probability space (X , ℬ(X ), 𝒲 ) is called Itô process if it has the form t

t

Xt = X0 + ∫ bs ds + ∫ σs ⋅ dBs , 0

t ≥ 0,

(2.4.21)

0

where t (1) (bt )t≥0 is an (FtBM )t≥0 -adapted process with ∫0 |bs |ds < ∞; (2) σs = (σs1 , . . . , σsd ) and σ⋅i ∈ 𝕄2 (0, t), i = 1, . . . , d. t

The stochastic integral ∫0 σs ⋅ dBs is understood in the sense of Definition 2.139, i. e., t

t∧τn

∫ σs ⋅ dBs = ∫ 1{s≤τn } σs ⋅ dBs , 0

0 ≤ t ≤ τn .

0

In concise differential notation the above integral equality is written as dXt = bt dt + σt ⋅ dBt .

(2.4.22)

The process (bt )t≥0 is called drift term and (σt )t≥0 diffusion term. It should be stressed that the stochastic differentials dXt resp. dBt have no meaning other than the shorthand notation explained above.

2.4 Stochastic calculus based on Brownian motion

| 119

2.4.4 Itô formula Another landmark result of stochastic analysis is the Itô formula which is the counterpart of the fundamental change of variable rule in classical analysis. In particular, this provides a tool for computing Itô integrals without directly having to use the definition. Recall that C 1,2 (ℝ+ × ℝ) denotes the space of functions on ℝ+ × ℝ, which are C 1 with respect to the first variable and C 2 with respect to the second. Theorem 2.143 (Itô formula for Brownian motion). Let h ∈ C 1,2 (ℝ+ × ℝ) and define Xt = h(t, Bt ). Then (Xt )t≥0 is an Itô process and we have t

t

̇ B ) + 1 𝜕2 h(s, B )) ds + ∫ 𝜕 h(s, B )dB , Xt − X0 = ∫ (h(s, s s x s s 2 x 0

(2.4.23)

0

with ḣ = 𝜕s h(s, x) or in equivalent differential notation ̇ B ) + 1 𝜕2 h(t, B )) dt + 𝜕 h(t, B )dB . dXt = (h(t, t t x t t 2 x

(2.4.24)

The following more general formula is readily obtained. Theorem 2.144 (Itô formula for Itô process). Let (Bt )t≥0 be d-dimensional Brownian motion and (Xt )t≥0 = (Xti )t≥0,1≤i≤D an ℝD -valued Itô process on a filtered probability space (X , ℬ(X ), (FtBM )t≥0 , 𝒲 ) such that dXti = bit dt + σti ⋅ dBt ,

i = 1, . . . , D,

(2.4.25)

where (bit )t≥0 is (FtBM )-adapted and bit (ω) ∈ L1loc (ℝ+ , dt), σti

=

iμ (σt )1≤μ≤d

∀ω ∈ X ,

⊂ ⋃ 𝕄2 (0, T). T≥0

Let h = (h1 , . . . , hm ) ∈ C 1,2 (ℝ+ × ℝD ; ℝm ). Then Ytk = hk (t, Xt ) is also an Itô process with t

t

D D 1 D Ytk − Y0k = ∫ (ḣ k + ∑ hki bis + ∑ hkij (σs σsT )ij ) ds + ∫ ∑ hki σsi ⋅ dBs , 2 i,j=1 i=1 i=1 0

0

1 ≤ k ≤ m. (2.4.26)

Here ḣ k = 𝜕t hk , hki = 𝜕xi hk and hkij = 𝜕xi 𝜕xj hk , evaluated at (s, Xs ). In differential notation, D D 1 D dYtk = (ḣ k + ∑ hki bit + ∑ hkij (σt σtT )ij ) dt + ∑ hki σti ⋅ dBt . 2 i,j=1 i=1 i=1

(2.4.27)

120 | 2 Brownian motion Proof. Step 1: Suppose that h ∈ C ∞ (ℝ+ × ℝD ; ℝm ) in Step 1–Step 6 below. We define a (D + 1)-dimensional stochastic process by i

X, X̂ ti = { t t,

1 ≤ i ≤ D, i = D + 1,

which satisfies dX̂ ti = b̂ it dt + σ̂ ti ⋅ dBt ,

i = 1, . . . , D + 1,

(2.4.28)

where bi , b̂ it = { t 1,

1 ≤ i ≤ D, i = D + 1,

iμ σ̂ t = {

iμ

σt , 0,

1 ≤ i ≤ D, i = D + 1.

Let ĥ = (ĥ 1 , . . . , ĥ m ) ∈ C ∞ (ℝD+1 ; ℝm ) and define Ŷ t = h(̂ X̂ t ). (2.4.27) is reduced to D+1 D+1 1 D+1 dŶ tk = ( ∑ ĥ ki b̂ it + ∑ ĥ kij (σ̂ t σ̂ tT )ij ) dt + ∑ ĥ ki σ̂ ti ⋅ dBt . 2 i,j=1 i=1 i=1

(2.4.29)

iμ iμ We show (2.4.29) and reset D + 1 by D, and write h, bit and σt for h,̂ b̂ it and σ̂ t , respectively, for simplicity. Furthermore, we assume h ∈ C ∞ (ℝD ; ℝm ). t

t

Step 2: Let Mti = ∫0 σsi ⋅ dBs and Ait = ∫0 bis ds, and Mt = (Mt1 , . . . , MtD ) and At = (A1t , . . . , ADt ). For every t ≥ 0 and every n we define the set In (t) by In (t) = {|X0 (ω)| > n} ∪ {|Mt (ω)| > n} ∪ {|At (ω)| > n}. Let us define the hitting time τn by τn = {

inf{t > 0 | In (t) ≠ 0} ∞

if {t > 0 | In (t) ≠ 0} ≠ 0, if {t > 0 | In (t) ≠ 0} = 0.

(2.4.30)

τn is a stopping time with respect to (FtBM )t≥0 . We define the stopped process by Xt(n) = Xt∧τn . We see that Xt(n) (ω) → Xt (ω) as n → ∞. For a continuous function f , we define t

the stopping time ρm = inf{t ≥ 0 | ∫0 |f (Xs )|2 ds ≥ m}. Since we have

󵄨󵄨 τn ∧ρm 󵄨󵄨2 τn ∧ρm τn ∧ρm 󵄨 󵄨󵄨 [󵄨󵄨󵄨 󵄨󵄨 ] (n) [ 𝔼 [󵄨󵄨 ∫ f (Xs )dBs − ∫ f (Xs )dBs 󵄨󵄨 ] = 𝔼 ∫ |f (Xs(n) ) − f (Xs )|2 ds] = 0, 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 0 [ 0 ] [󵄨 0 ] t

t

we see that ∫0 f (Xs(n) )dBs = ∫0 f (Xs )dBs for 0 ≤ t ≤ τn ∧ ρm . We also see that t

t

∫0 f (Xs(n) )ds → ∫0 f (Xs )ds as n → ∞ almost surely. So it is sufficient to show (2.4.29) for Xt replaced by Xt(n) . Furthermore, since hk (Xt(n) ), hki (Xt(n) ) and hkij (Xt(n) ) are bounded,

2.4 Stochastic calculus based on Brownian motion

| 121

we can assume that hk ∈ Cb∞ (ℝD ), i. e., hk , hki and hkij are continuous and bounded for all i, j in what follows. Step 3: We reset Xt(n) to Xt . Hence it is sufficient to prove D

dYtk = (∑ hki bit + i=1

D 1 D k ∑ hij (σt σtT )ij ) dt + ∑ hki σti ⋅ dBt 2 i,j=1 i=1

(2.4.31) iμ

for hk ∈ Cb∞ (ℝD ). Let 0 = s0 < s1 < . . . < sJ = t. We first consider the case where σs is a step function of the form J

iμ

σsiμ (ω) = ∑ σj (ω)1[sj−1 ,sj ) (s), j=1

iμ

where σj is measurable with respect to FtBM . Notice that j J

hk (Xt ) − hk (X0 ) = ∑ (hk (Xsm ∧t ) − hk (Xsm−1 ∧t )) . m=1

In order to show (2.4.31) it is sufficient to prove u

hk (Xu ) − hk (Xs ) = ∫ hki (Xr )bir dr + s

u

u

s

s

1 ∫ hkij (Xr )σriμ σrjμ dr + ∫ hki (Xr )σriμ dBμr . 2

(2.4.32)

Here sp−1 ≤ s < u < sp with some p, and the summation over repeated indices is understood. In what follows we fix k and write h for hk for notational simplicity. Since s, u ∈ [sp−1 , sp ), Xui is of the form Xui

=

Xsi

+

σpiμ (Bμu

−

Bμs )

u

+ ∫ bir dr,

i = 1, . . . , D.

(2.4.33)

s

Hence u 󳨃→ Xu is continuous. Step 4: We divide the interval [s, u] into n intervals and set tl = s + (u − s)l/n, 0 ≤ l ≤ n. Let i ΔXl−1 = Xtil − Xtil−1 ,

l = 1, . . . , n.

By the Taylor expansion we have n

i h(Xu ) − h(Xs ) = ∑ hi (Xtl−1 )ΔXl−1 + l=1

1 n j i ΔXl−1 + R, ∑ h (X )ΔXl−1 2 l=1 ij tl−1

(2.4.34)

122 | 2 Brownian motion where

n

j

i k R = ∑ hijk (ξl−1 )ΔXl−1 ΔXl−1 ΔXl−1 l=1

with some ξl−1 = θl Xtl−1 + (1 − θl )Xtl and θl = θl (ω) ∈ (0, 1). Let Δil−1 B

=

μ σpiμ (Btl

−

μ Btl−1 ),

Δil−1 b

tl

= ∫ bir dr,

1 ≤ l ≤ n.

tl−1

j

We have limn→∞ 𝔼[| ∑nl=1 Δil−1 BΔl−1 BΔkl−1 B|2 ] = 0. Taking a subsequence n󸀠 , we also see j

that limn󸀠 →∞ 𝔼[| ∑nl=1 Δil−1 bΔl−1 bΔkl−1 b|2 ] = 0. Hence limn󸀠 →∞ 𝔼[|R|2 ] = 0 follows. In what follows we reset n󸀠 to n. Inserting (2.4.33) into (2.4.34), we can rewrite (2.4.34) as 󸀠

5

h(Xt ) − h(Xs ) = ∑ In(j) + R,

(2.4.35)

j=1

where

n

In(1) = ∑ hi (Xtl−1 )Δil−1 B, l=1

In(3)

1 n j = ∑ hij (Xtl−1 )Δil−1 BΔl−1 B, 2 l=1

In(5) =

n

and

In(2) = ∑ hi (Xtl−1 )Δil−1 b,

and

In(4) = ∑ hij (Xtl−1 )Δil−1 BΔl−1 b,

1 n j ∑ h (X )Δi bΔ b. 2 l=1 ij tl−1 l−1 l−1

l=1 n

j

l=1

Step 5: We prove the following: u iμ σp ∫ hi (Xr )dBμr , s u a. s. In(2) → ∫ hi (Xr )bir dr, s u 2 (3) L 1 iμ jμ In → σp σp ∫ hij (Xr )dr, 2 s a. s. (4) In → 0, a. s. In(5) → 0. L2 In(1) →

We have In(1)

=

u iμ σp ∫ Φi,n (r)dBμr , s

(2.4.36) (2.4.37) (2.4.38) (2.4.39) (2.4.40)

2.4 Stochastic calculus based on Brownian motion

| 123

where Φi,n (r) = ∑nl=1 hi (Xtl−1 )1[tl−1 ,tl ) (r). Since hi is bounded, by the dominated convergence theorem we see that u

lim 𝔼 [∫ |Φi,n (r) − hi (Xr )|2 dr ] = 0. n→∞ [s ] Hence (2.4.36) follows; (2.4.37) is straightforward. Next we show (2.4.38). Fix μ and μ󸀠 , iμ jμ󸀠

and denote Ψ(Xtl−1 ) = Ψμμ (Xtl−1 ) = hij (Xtl−1 )σp σp . It is sufficient to show that 󸀠

n

μ

μ

μ󸀠

u

L2

μ󸀠

∑ Ψ(Xtl−1 )(Btl − Btl−1 )(Btl − Btl−1 ) → δμμ󸀠 ∫ Ψ(Xr )dr. l=1

μμ󸀠

We set Zl = Zl

s

μ

μ

μ󸀠

μ󸀠

= (Btl − Btl−1 )(Btl − Btl−1 ) − (tl − tl−1 )δμμ󸀠 . We have

󵄨󵄨 n 󵄨󵄨2 n 󵄨󵄨 󵄨 μ μ μ󸀠 μ󸀠 󵄨 𝔼 [󵄨󵄨∑ Ψ(Xtl−1 )(Btl − Btl−1 )(Btl − Btl−1 ) − δμμ󸀠 ∑ Ψ(Xtl−1 )(tl − tl−1 )󵄨󵄨󵄨󵄨 ] 󵄨󵄨l=1 󵄨󵄨 l=1 󵄨󵄨 n 󵄨󵄨2 󵄨 󵄨 = 𝔼 [󵄨󵄨󵄨󵄨∑ Ψ(Xtl−1 )Zl 󵄨󵄨󵄨󵄨 ] = Jn1 + Jn2 . 󵄨󵄨l=1 󵄨󵄨 Here Jn1 = ∑nl=1 𝔼[|Ψ(Xtl−1 )Zl |2 ] and Jn2 = ∑nl=l̸ 󸀠 𝔼[Ψ(Xtl−1 )Zl Ψ(Xt 󸀠 )Zl󸀠 ]. It can be directly l −1

seen that 0 ≤ Jn1 ≤ supx∈ℝd Ψ(x)2 ∑nl=1 𝔼[Zl2 ] and limn→∞ ∑nl=1 𝔼[Zl2 ] = 0. Hence it follows that limn→∞ Jn1 = 0. On the other hand, we see that for l < l󸀠 𝔼[Ψ(Xtl−1 )Zl Ψ(Xt 󸀠 )Zl󸀠 ] = 𝔼[Ψ(Xtl−1 )Ψ(Xt 󸀠 )Zl󸀠 ]𝔼[Zl ] = 0. l −1

l −1

L2

u

Hence Jn2 = 0. Together with ∑nl=1 Ψ(Xtl−1 )(tl − tl−1 ) → ∫s Ψ(Xr )dr, (2.4.38) follows. We see that μ

μ

u

|In(4) | ≤ sup |hij (x)||σpiμ | max |Btl − Btl−1 | ∫ |bjr |dr. 1≤l≤n

x∈ℝd

s

Since t 󳨃→ Bt is almost surely continuous, the right-hand side above almost surely converges to zero as n → ∞. In a similar way we see that (2.4.40) converges to zero as n → ∞. t

iμ

Step 6: Suppose that σ⋅iμ ∈ 𝕄2 (0, T). The integral ∫0 σr dBr is defined by t

∫ σriμ dBr 0

t

= ∫ σriμ 1{r≤ρm } dBr , 0

t ≤ ρm ,

124 | 2 Brownian motion where ρm is the stopping time defined by ρm = {

󵄨 t iμ inf {t ≥ 0 󵄨󵄨󵄨 ∫0 |σs |2 ds ≥ n}

󵄨 t iμ if {t ≥ 0 󵄨󵄨󵄨 ∫0 |σs |2 ds ≥ m} ≠ 0, 󵄨 t iμ if {t ≥ 0 󵄨󵄨󵄨 ∫0 |σs |2 ds ≥ m} = 0.

∞ iμ

There exists a sequence of step functions σt (n) such that t

𝔼 [∫ |σriμ (n) − σriμ 1{r≤ρm } |2 dr ] → 0 [0 ] iμ

(n → ∞).

iμ

Set σ̃ r = σr 1{r≤ρm } . Define Xti (n)

=

X0i (n)

t

+

∫ bir dr 0

t

+ ∫ σsiμ (n)dBμs . 0

iμ

By Steps 1–5, (2.4.32) with s = 0 and u = t is valid with Xti and σt replaced by Xti (n) and iμ σt (n), respectively, i. e., h(Xt (n)) − h(X0 (n)) t

=

∫ hi (Xr (n))bir dr 0

t

t

0

0

1 + ∫ hij (Xr (n))σriμ (n)σrjμ (n)dr + ∫ hi (Xr (n))σriμ (n)dBμr . 2

(2.4.41)

We see that 󵄨󵄨 t 󵄨󵄨2 t 󵄨 󵄨 [󵄨󵄨󵄨 iμ μ iμ μ 󵄨󵄨󵄨 ] 𝔼 [󵄨󵄨∫ hi (Xr (n))σr (n)dBr − ∫ hi (Xr )σ̃ r dBr 󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 0 [󵄨 0 ] 2 󵄨󵄨 󵄨󵄨 t 󵄨󵄨2 󵄨󵄨󵄨 t 󵄨 󵄨󵄨 󵄨󵄨 [󵄨󵄨󵄨 [ 󵄨 󵄨] iμ iμ μ 󵄨󵄨󵄨 ] iμ μ ≤ 2𝔼 [󵄨󵄨∫ hi (Xr (n))(σr (n) − σ̃ r )dBr 󵄨󵄨 ] + 2𝔼 [󵄨󵄨󵄨∫(hi (Xr (n)) − hi (Xr ))σ̃ r dBr 󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 [󵄨 0 ] [󵄨 0 ] 2

t

t

≤ 2 sup |hi (x)| 𝔼 [∫ |σriμ (n) − σ̃ riμ |2 dr ] + 2𝔼 [∫ |(hi (Xr (n)) − hi (Xr ))σ̃ riμ |2 dr ] . x∈ℝd [0 ] [0 ] The right-hand side above converges to zero as n → ∞. There exists a subsequence t t iμ μ iμ μ n󸀠 such that ∫0 hi (Xr (n󸀠 ))σr (n󸀠 )dBr almost surely converges to ∫0 hi (Xr )σ̃ r dBr . Note that Xt (n) → Xt uniformly in t ∈ [0, T] almost surely. Thus h(Xt (n)) − h(X0 (n)) → h(Xt ) − h(X0 ) almost surely as n → ∞. It follows that 󵄨󵄨 t 󵄨󵄨2 󵄨 󵄨 [󵄨󵄨󵄨 i 󵄨󵄨󵄨 ] lim 𝔼 [󵄨󵄨∫(hi (Xr (n)) − hi (Xr ))br dr 󵄨󵄨 ] = 0 n→∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 [󵄨 0 ]

2.4 Stochastic calculus based on Brownian motion

| 125

from the dominated convergence theorem. Similarly we see that 󵄨󵄨 t 󵄨󵄨2 t 󵄨 󵄨 [󵄨󵄨󵄨 iμ jμ iμ jμ 󵄨󵄨󵄨 ] 𝔼 [󵄨󵄨∫ hij (Xr (n))σr (n)σr (n)dr − ∫ hij (Xr )σ̃ r σ̃ r dr 󵄨󵄨 ] → 0 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 0 [󵄨 0 ] as n → ∞. Then the theorem follows by taking the limit as n → ∞ on both sides of (2.4.41). Step 7: Finally we show (2.4.26) for h ∈ C 1,2 (ℝ+ × ℝd ; ℝm ). Take a sequence (hn )n∈ℕ ⊂ C ∞ (ℝ+ × ℝd ; ℝm ) such that hn , ḣ n , hni , hnij locally uniformly converges to h, h,̇ hi , hij , respectively as n → ∞. (2.4.26) is valid for hn for a. e. ω ∈ X \ Nn , where Nn is a null set and depends on hn . Let N = ⋃∞ n=1 Nn . By a simple limiting argument (if necessary, taking a subsequence) it is seen that (2.4.26) is valid for h for a. e. ω ∈ X \ N. Remark 2.145 (Rules of Itô differential calculus). For practical purposes the following formal rules of Itô differential calculus are useful: dtdt = 0,

μ

dtdBt = 0,

μ

dBt dt = 0,

μ

dBt ⋅ dBνt = δμν dt

(2.4.42)

for all μ, ν = 1, . . . , d. The last property intuitively corresponds to the fact that the quadratic variation of Brownian motion grows linearly in time, that is, Bt has large fluctuations on a small scale, dBt = O(√dt). Using the Itô rules we can express the Itô formula in Theorem 2.144 in differential form as D 1 D j dYtk = ḣ k dt + ∑ hki dXti + ∑ hkij dXti dXt , 2 i=1 i,j=1

1 ≤ k ≤ m.

(2.4.43)

An application of the Itô formula gives the following useful result. Corollary 2.146 (Product formula). Let (Xt )t≥0 and (Yt )t≥0 be two Itô processes. Then the product formula d(Xt Yt ) = dXt ⋅ Yt + Xt ⋅ dYt + dXt ⋅ dYt

(2.4.44)

holds. Proof. Choose g(x, y) = xy, x = Xt , and y = Yt . Then the formula follows directly from the Itô formula. Example 2.147. When the integrand f (s) is continuous and of bounded variation but nonrandom, i. e., independent of ω, an application of the Itô formula gives t

t

∫ f (s)dBs = f (t)Bt − ∫ Bs df (s). 0

0

126 | 2 Brownian motion Example 2.148. Let Zt = g(t, Xt ) with g(t, x) ∈ C 1,2 (ℝ+ × ℝ), where dXt = V(Bt )dt + a(Bt ) ⋅ dBt and a = (a1 , . . . , ad ). Then 1 dZt = (ġ + (a ⋅ a)Δx g + V𝜕x g) dt + (𝜕x g)a ⋅ dBt . 2

(2.4.45)

This formula shows the relationship between Stratonovich integral and Itô integral. In the particular case g(t, x) = g(x) and Xt = Bt , t

g(Bt ) = g(B0 ) + ∫(𝜕x g)(Bs ) ∘ dBs .

(2.4.46)

0

We conclude this section by two further applications of the Itô formula. 1 2 Example 2.149 (Wick product). Put h(t, x) = eαx− 2 α t ; notice that 21 Δh + ḣ = 0 is satisfied. The Itô formula yields dh(t, Bt ) = αhdBt . Hence,

t

1 (h(t, Bt ) − 1). α

(2.4.47)

1 :exp(αBt ): = exp (αBt − α2 t) , 2

(2.4.48)

∫ h(s, Bs )dBs = 0

Define the Wick product of eαBt by

giving t

t

t

1 :exp (∫ f (s, Bs )dBs ): = exp (∫ f (s, Bs )dBs − 𝔼x [∫ |f (s, Bs )|2 ds]) . 2 0 0 [0 ] This definition will be useful in the discussion of the Cameron–Martin formula (2.4.97) below. The Wick product of Bnt can then be written as :Bnt : = (

dn h ) (t, Bt )⌈α=0 , dαn

and as a result we obtain by (2.4.47) t

∫:Bns :dBs = 0

1 :Bn+1 :. n+1 t

The following inequality offers useful bounds on the moments of the martingale T ∫0 f (s, Bs ) ⋅ dBs .

| 127

2.4 Stochastic calculus based on Brownian motion

Proposition 2.150 (Burkholder–Davis–Gundy (BDG) inequality). Let m ∈ ℕ and supT pose that 𝔼[∫0 |f (t, Bt )|2m dt] < ∞. Then 󵄨󵄨 T 󵄨󵄨2m T 󵄨 󵄨󵄨 [󵄨󵄨󵄨 󵄨 ] 𝔼 [󵄨󵄨∫ f (t, Bt ) ⋅ dBt 󵄨󵄨󵄨 ] ≤ (m(2m − 1))m T m−1 𝔼 [∫ |f (t, Bt )|2m dt ] . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 [0 ] [󵄨 0 ]

(2.4.49)

Proof. We apply the Itô formula to |Xt |2m , where dXt = f (t, Bt ) ⋅ dBt , and obtain d|Xt |2m = m(2m − 1)|Xt |2m−2 |f (t, Bt )|2 dt + 2m|Xt |2m−1 f (t, Bt ) ⋅ dBt . Hence T

2m

𝔼 [|Xt | ] ≤ m(2m − 1) ∫ 𝔼 [|Xs |2m−2 |f (s, Bs )|2 ] ds.

(2.4.50)

0

By Jensen’s inequality we have 𝔼[|Xt |2m |Fs ] ≥ |Xs |2m for t ≥ s. Thus 𝔼 [|Xs |2m ] ≤ 𝔼 [|XT |2m ] . By (2.4.50) and the Hölder inequality it follows that 2m

𝔼 [|XT | ] ≤ m(2m − 1) (𝔼[|XT |

(2m−2)p

1/p

])

q

T

1/q

2

(𝔼 [(∫ |f (s, Bs )| ds) ]) [ 0 ]

for 1/p + 1/q = 1. Putting p = m/(m − 1) and q = m we obtain m

T

1/m

(𝔼[|XT |2m ])1/m ≤ m(2m − 1)𝔼 [(∫ |f (s, Bs )|2 ds) ] [ 0 ]

.

Taking the mth power of both sides, by the Schwarz inequality T

𝔼[|XT |2m ] ≤ [m(2m − 1)]m T m−1 𝔼 [∫ |f (s, Bs )|2m ds] . [0 ] t

Since (∫0 f (s, Bs ) ⋅ dBs )t≥0 is a continuous martingale provided that f ∈ M 2 (0, t) for all t ≥ 0, combining the BDG inequality and the martingale inequality (2.2.52) further useful inequalities can be derived.

128 | 2 Brownian motion 2.4.5 Stochastic differential equations In this section we consider a class of stochastic differential equations (SDEs) and their solutions. An SDE can be thought of as arising from an ordinary differential equation Ẋ t = at Xt in which at is replaced by a factor consisting of a nonrandom and a noise term. The solution of such an equation is a random process. The theory of SDEs makes it possible to implement the noise term mathematically and allows to show that under sufficient regularity a solution exists, which is continuous in t and has the Markov property. Consider the system of SDEs Xti

t

=

∫ bis (Xs )ds 0

t

+ ∫ σsi (Xs ) ⋅ dBs ,

X0 = x,

i = 1, . . . , n,

(2.4.51)

0

or in differential form dXti = bit (Xt )dt + σti (Xt ) ⋅ dBt ,

X0 = x,

i = 1, . . . , n.

(2.4.52)

Here (Bt )t≥0 is d-dimensional Brownian motion, bt = (b1t , . . . , bnt ) : ℝn → ℝn , and σt = (σt1 , . . . , σtn ) : ℝn → ℝd×n . Definition 2.151 (Strong solution). A strong solution of SDE (2.4.51) or (2.4.52) on (X , ℬ(X ), 𝒲 ) is a process (Xtx )t≥0 with the following properties: (1) t 󳨃→ Xtx is almost surely continuous; (2) (Xtx )t≥0 is adapted with respect to (FtBM )t≥0 ; (3) 𝒲 (X0x = x) = 1; t

(4) 𝒲 (∫0 (|bis (Xs )| + |σ ij (Xs )|2 )ds < ∞) = 1 holds for 1 ≤ i ≤ n, 1 ≤ j ≤ d, and 0 ≤ t < ∞; (5) (2.4.51) holds almost surely. Theorem 2.152 (Strong solution of SDEs). Suppose that bt = (b1t , . . . , bnt ) : ℝn → ℝn and σt = (σt1 , . . . , σtn ) : ℝn → ℝd×n satisfy, for t ∈ [0, T], |bt (x)| + |σt (x)| ≤ C(1 + |x|),

(2.4.53)

|bt (x) − bt (y)| + |σt (x) − σt (y)| ≤ D|x − y|,

(2.4.54)

with some constants C and D independent of x, y ∈ ℝn and t ∈ [0, T], where we write ij |σt (x)| = (∑ni=1 ∑dj=1 |σt (x)|2 )1/2 and |bt (x)| = (∑ni=1 |bit (x)|2 )1/2 . Then there exists a unique x strong solution (Xt )t∈[0,T] of dXti = bit (Xt )dt + σti (Xt ) ⋅ dBt ,

X0i = x i ∈ ℝ,

i = 1, . . . , n,

(2.4.55)

as an n-dimensional random process on a probability space (X , ℬ(X ), 𝒲 ) such that (1) (Xtx )t≥0 is (FtBM )-adapted;

2.4 Stochastic calculus based on Brownian motion

| 129

T

(2) 𝔼 [∫0 |Xtx |2 dt] < ∞; (3) Xtx is almost surely continuous in t ∈ [0, T]. Proof. Let Xt = (Xt1 , . . . , Xtn ), bt = (b1t , . . . , bnt ), and σt = (σt1 , . . . , σtn ). The SDE is represented as t

t

Xt − X0 = ∫ bs (Xs )ds + ∫ σs (Xs ) ⋅ dBs . 0

(2.4.56)

0

We first show the uniqueness of the solution. Suppose that Xt and Zt are solutions of (2.4.56). Put as = bs (Xs ) − bs (Zs ) and γs = σs (Xs ) − σs (Zs ). We have 󵄨󵄨 t t t t 󵄨󵄨󵄨2 󵄨 󵄨󵄨 ] [󵄨󵄨󵄨 𝔼[|Xt − Zt | ] = 𝔼 [󵄨󵄨∫ as ds + ∫ γs dBs 󵄨󵄨󵄨 ] ≤ 2t𝔼 [∫ |as |2 ds] + 2𝔼 [∫ |γs |2 ds] 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 0 [0 ] [0 ] [󵄨 0 ] 2

t

≤ 2(1 + t)D2 ∫ 𝔼[|Xs − Zs |2 ]ds.

(2.4.57)

0 t

Defining v(t) = 𝔼[|Xt − Zt |2 ], we have v(t) ≤ A ∫0 v(s)ds, where A = 2(1 + T)D2 . By the Gronwall inequality quoted in Proposition 2.153 we conclude that v(t) = 0 for all t ≥ 0. Thus 𝒲 (Xt − Zt = 0 for all t ≥ 0) = 1. Next we prove the existence of a solution. Define Yt(0) = X0 = x and Yt(k) inductively by t

Yt(k+1)

= X0 +

t

∫ bs (Ys(k) )ds 0

+ ∫ σs (Ys(k) ) ⋅ dBs .

(2.4.58)

0

In the same way as in (2.4.57) we have 𝔼[|Yt(k+1)

−

Yt(k) |2 ]

2

t

≤ (1 + T)2D ∫ 𝔼[|Ys(k) − Ys(k−1) |2 ]ds 0

for k ≥ 1 and t ≤ T, and 𝔼[|Yt(1) − Yt(0) |2 ] ≤ 2C 2 t(1 + 𝔼[|X0 |2 ]) + 2C 2 t 2 (1 + 𝔼[|X0 |2 ]) ≤ A1 t, where A1 = 2C 2 (1 + T)(1 + 𝔼[|X0 |2 ]) only depends on C, T and 𝔼[|X0 |2 ]. By induction we obtain 𝔼[|Yt(k+1) − Yt(k) |2 ] ≤

(A2 t)k+1 (k + 1)!

(2.4.59)

130 | 2 Brownian motion for k ≥ 0 and t ≤ T. Here A2 only depends on C, T, and 𝔼[|X0 |2 ]. We have T

sup

t∈[0,T]

|Yt(k+1)

−

Yt(k) |

≤

∫ |bs (Ys(k) )

−

0

bs (Ys(k−1) )|ds

󵄨󵄨 t 󵄨󵄨 󵄨 󵄨 + sup 󵄨󵄨󵄨∫(σs (Ys(k) ) − σs (Ys(k−1) )) ⋅ dBs 󵄨󵄨󵄨. 󵄨 󵄨󵄨 t∈[0,T]󵄨 0

By T

{ sup

t∈[0,T]

|Yt(k+1)

−

Yt(k) |

> 2 } ⊂ {∫ |bs (Ys(k) ) − bs (Ys(k−1) )|ds > −k

0

2−k } 2

󵄨󵄨 t 󵄨󵄨 2−k 󵄨 󵄨 ∪ { sup 󵄨󵄨󵄨∫(σs (Ys(k) ) − σs (Ys(k−1) )) ⋅ dBs 󵄨󵄨󵄨 > }, 󵄨 󵄨󵄨 2 󵄨 t∈[0,T] 0

we have 𝒲 ( sup |Yt

(k+1)

t∈[0,T]

󵄨󵄨 T 󵄨󵄨󵄨2 󵄨 − Yt(k) | > 2−k ) ≤ 𝒲 (󵄨󵄨󵄨∫ |bs (Ys(k) ) − bs (Ys(k−1) )|ds󵄨󵄨󵄨 > 2−2k−2 ) 󵄨󵄨 󵄨󵄨 0

󵄨󵄨 t 󵄨󵄨 󵄨 󵄨 + 𝒲 ( sup 󵄨󵄨󵄨∫(σs (Ys(k) ) − σs (Ys(k−1) )) ⋅ dBs 󵄨󵄨󵄨 > 2−k−1 ). 󵄨󵄨 t∈[0,T]󵄨󵄨 0

By the martingale inequality we also have 𝒲 ( sup |Yt

(k+1)

t∈[0,T]

2k+2

≤2

− Yt(k) | > 2−k )

T

T

∫ 𝔼[|bs (Ys(k) ) 0

T

≤ 22k+2 D2 (T + 1) ∫ 0

−

bs (Ys(k−1) )|2 ]ds

+2

2k+2

T

∫ 𝔼[|σs (Ys(k) ) − σs (Ys(k−1) )|2 ]ds 0

(A2 t)k (4A2 T)k+1 dt ≤ , k! (k + 1)!

2

(k+1) if A2 ≥ D (T + 1). Since ∑∞ − Yt(k) | > 2−k ) < ∞, by the Borel– k=0 𝒲 (supt∈[0,T] |Yt Cantelli lemma, we see that ∞

∞

𝒲 ( ⋃ ⋂ { sup |Yt

(k+1)

m=1 k=m t∈[0,T]

− Yt(k) | ≤ 2−k }) = 0.

For a. e. ω there exists k0 = k0 (ω) such that sup |Yt(k+1) (ω) − Yt(k) (ω)| ≤ 2−k

t∈[0,T]

for all k ≥ k0 . Therefore, n−1

Yt(n) (ω) = Yt(0) (ω) + ∑ (Yt(k+1) (ω) − Yt(k) (ω)) k=0

2.4 Stochastic calculus based on Brownian motion

| 131

uniformly converges in [0, T] almost surely. Denote the limit by Xt (ω) = limn→∞ Yt(n) (ω). It is seen that t → Xt is continuous, since Yt(n) is continuous and the convergence is uniform in t. Moreover Xt is Ft -measurable, since Yt(n) is Ft -measurable for all n. Next note that by (2.4.59) (𝔼[|Yt(m) − Yt(n) |2 ])

1/2

m−1

1/2

≤ ∑ (𝔼[|Yt(k+1) − Yt(k) |2 ]) k=n

∞

≤ ∑( k=n

(A2 t)k+1 ) (k + 1)!

1/2

→0

as n → ∞ and (Yt(n) )n∈ℕ is a Cauchy sequence in the Hilbert space L2 (X , d𝒲 ). Thus 󸀠

a subsequence Yt(n ) almost surely converges to some Yt . We reset n󸀠 by n. We have Yt = Xt a. s. In particular ∞ > 𝔼[|Yt |2 ] = 𝔼[|Xt |2 ] follows. It remains to show that Xt satisfies (2.4.56). For all n we have t

Yt(n+1)

= X0 +

t

∫ bs (Ys(n) )ds 0

+ ∫ σs (Ys(n) ) ⋅ dBs .

(2.4.60)

0

It follows that limn→∞ Yt(n+1) = Xt a. s. We note that T

T

1/2

2

∞ (A t)k+1 𝔼 [∫ |Xt − Yt(n) |2 dt ] ≤ lim sup 𝔼 [∫ |Yt(m) − Yt(n) |2 dt ] ≤ T( ∑ ( 2 ) ) (k + 1)! m→∞ k=n [0 ] [0 ] (2.4.61)

by Fatou’s lemma. By (2.4.61) and the Itô isometry, taking a subsequence n󸀠 , we have t

t

󸀠

lim ∫ σs (Ys(n ) ) ⋅ dBs = ∫ σs (Xs ) ⋅ dBs , 󸀠

n →∞

0

(2.4.62)

0

and by the Hölder inequality, t

lim

n󸀠 →∞

󸀠 ∫ bs (Ys(n ) )ds

0

t

= ∫ bs (Xs )ds

(2.4.63)

0

in L2 (X , d𝒲 ). Therefore taking the limit of both sides of (2.4.60) we can obtain the desired results. Proposition 2.153 (Gronwall inequality). Let v(t) be a nonnegative function such that t v(t) ≤ C + A ∫0 v(s)ds for t ∈ [0, T] with some constants A, C ≥ 0. Then it follows that v(t) ≤ CeAt for t ∈ [0, T]. t

Proof. We may suppose A ≠ 0. Let f (t) = w(t)e−At and w(t) = ∫0 v(s)ds. Since f (t) = t

∫0 f 󸀠 (s)ds ≤ − AC (e−At − 1), we have w(t) ≤ follows.

C At (e A

− 1). Hence v(t) ≤ C + aw(t) ≤ CeAt

132 | 2 Brownian motion An important class of solutions of SDEs is the following. Definition 2.154 (Diffusion process). A random process (Xt )t≥0 on a filtered probability space (Ω, F , (Ft )t≥0 , P) is a diffusion process whenever (1) (Xt )t≥0 is a Markov process; (2) paths t 󳨃→ Xt (ω) are continuous for a. e. ω ∈ X . We now consider (2.4.55) with bt and σt replaced by time-independent b and σ, respectively. We present a result without a proof. Theorem 2.155 (Itô diffusion). Suppose that the functions b = (b1 , . . . , bn ) : ℝn → ℝn and σ = (σ 1 , . . . , σ n ) : ℝn → ℝd×n satisfy |b(x) − b(y)| + |σ(x) − σ(y)| ≤ D|x − y| with some constant D independent of x, y ∈ ℝn . Then the unique solution (Xtx )t≥0 , Xtx = (Xt1,x , . . . , Xtn,x ), of dXti = bi (Xt )dt + σ i (Xt ) ⋅ dBt ,

X0i = x i ∈ ℝ,

i = 1, . . . , n,

(2.4.64)

satisfies the Markov property Xx

x 𝔼[f (Xs+t )|FsBM ] = 𝔼[f (Xt s )] Xx

y

for every bounded Borel function f , where 𝔼[f (Xt s )] = 𝔼[f (Xt )] evaluated at y = Xsx . In x x particular, 𝔼[f (Xs+t )|FsBM ] = 𝔼[f (Xs+t )|σ(Xsx )] holds. The diffusion process in Theorem 2.155 is called Itô diffusion. Let (Xtx )t≥0 , Xtx = (Xt1,x , . . . , Xtn,x ), be the solution of SDE (2.4.51) and let Ut : L∞ (ℝd ) → L∞ (ℝd ) be given by x

Ut f (x) = 𝔼[f (Xt )].

(2.4.65)

Definition 2.156 (Infinitesimal generator). The infinitesimal generator HX of the solution (Xtx )t≥0 is defined by 1 HX f (x) = lim (Ut f (x) − f (x)). t↓0 t

(2.4.66)

Theorem 2.157. Let (Xtx )t≥0 be the solution of dXti = bi (Xt )dt + σ i (Xt ) ⋅ dBt ,

X0 = x,

i = 1, . . . , n,

(2.4.67)

where b = (b1 , . . . , bn ) and σ = (σ 1 , . . . , σ n ) are independent of t and suppose (2.4.53) and (2.4.54). Then its generator is given by the second-order partial differential operator HX = acting on C02 (ℝd ).

n 1 n ∑ (σσ T )ij (x)𝜕i 𝜕j + ∑ bi (x)𝜕i 2 i,j=1 i=1

(2.4.68)

2.4 Stochastic calculus based on Brownian motion

| 133

Proof. Let (Yti )t≥0 be Itô process, dYti = uit dt + vti ⋅ dBt , with Y0i = x i . The Itô formula (2.4.31) yields t

t

f (Yt ) − f (Y0 ) = ∫ ( 0

n n 1 n ∑ (vs vsT )ij 𝜕i 𝜕j f (Ys ) + ∑ uis 𝜕i f (Ys )) ds + ∫ ∑(𝜕i f (Ys ))vsi ⋅ dBs . 2 i,j=1 i=1 i=1 0

t

Taking expectations, the martingale part ∫0 ∑ni=1 (𝜕i f )vsi ⋅ dBs vanishes and we are left with t

n

n

1 𝔼[f (Yt )] = f (x) + 𝔼 [∫ ( ∑ (vs vsT )ij 𝜕i 𝜕j f (Ys ) + ∑ uis 𝜕i f (Ys )) ds] . 2 i,j=1 i=1 [0 ] Hence t

Ut f (x) − f (x) = 𝔼 [∫ (

[0

n 1 n ∑ (σσ T )ij (Xsx )𝜕i 𝜕j f (Xsx ) + ∑ bi (Xsx )𝜕i f (Xsx )) ds] . 2 i,j=1 i=1 ]

Since Xsx is continuous in s, the theorem now follows directly. The last equality is in fact a simple version of Dynkin’s formula. Example 2.158 (Linear SDE). Assume that the n × n, n × 1, and n × d matrices At , at , and σt , respectively, are nonrandom and locally bounded. Let Φt be the n × n matrix function solving the nonrandom equation Φ̇ t = At Φt . Then the solution of the linear SDE dXt = (At ⋅ Xt + at ) dt + σt ⋅ dBt

(2.4.69)

is given by t

t

−1 Xt = Φt (X0 + ∫ Φ−1 s as ds + ∫ Φs σs ⋅ dBs ) , 0

(2.4.70)

0

where Φ−1 t is the inverse matrix of Φt . This is easily checked by an application of the Itô formula. Example 2.159 (Langevin equation). Let d = 1 and a, σ > 0. Consider the SDE dXt = −aXt dt + σdBt ,

X0 = x ∈ ℝ.

(2.4.71)

By (2.4.70) the solution of (2.4.71) is given by Xtx

t

= xe

−at

+ σ ∫ e−a(t−s) dBs . 0

(2.4.72)

134 | 2 Brownian motion The random process (Xtx )t≥0 is called Ornstein–Uhlenbeck process. We have by (2.4.72) 𝔼[Xtx ] = xe−at ,

cov[Xtx Xsx ] =

σ2 (1 − e−2a(s∧t) ). 2a

(2.4.73)

Moreover, the generator of Xtx is given by L=

σ2 Δ − ax ⋅ ∇. 2

(2.4.74)

2.4.6 Brownian bridge Brownian bridge is a continuous Gaussian process starting in a at t = T1 and ending in b at t = T2 . Definition 2.160 (Brownian bridge). A Brownian bridge on the interval [T1 , T2 ] starting in a ∈ ℝd and ending in b ∈ ℝd is a continuous multivariate Gaussian random process (Xt )T1 ≤t≤T2 on a probability space (Ω, F , P) such that (1) XT1 = a and XT2 = b a. s.; (2) 𝔼[Xt ] = a(1 −

t−T1 ) T2 −T1

(3) cov(Xs , Xt ) = s ∧ t −

1 + b Tt−T ; −T 2

st . T2 −T1

1

There are several representations of Brownian bridge of which now we present three. Example 2.161 (SDE for Brownian bridge). The Brownian bridge starting in a at t = 0 and ending in b at t = T solves the SDE dXt =

b − Xt dt + dBt , T −t

0 ≤ t < T,

X0 = a.

(2.4.75)

As a linear SDE, it can be solved exactly through (2.4.70), giving t

Xt = a (1 −

t t 1 ) + b + (T − t) ∫ dB , T T T −s s

0 ≤ t < T.

(2.4.76)

0

Let

t

αt = (T − t) ∫ 0

1 dB , T −r r

0 ≤ t < T.

Note that αt is a Gaussian random variable with 𝔼[αt ] = 0 and t∧s

𝔼[αs αt ] = (T − t)(T − s) ∫ 0

1 st dr = s ∧ t − . T (T − r)2

(2.4.77)

2.4 Stochastic calculus based on Brownian motion

| 135

Thus 𝔼[Xt ] = a (1 −

t t )+b , T T

cov(Xt , Xs ) = s ∧ t −

st . T

It is easily seen that Xt → b as t → T in L2 (X , d𝒲 ). Furthermore, it can also be shown that Xt (ω) → b as t → T for a. e. ω ∈ X . Define X, X̃ t = { t b,

0 ≤ t < T, t = T.

(2.4.78)

Hence (X̃ t )0≤t≤T is Brownian bridge starting in a at t = 0 and ending b at t = T. Example 2.162 (Brownian bridge by time change). Let (Bt )t≥0 be one-dimensional Brownian motion. Let βt = (T − t)B

t (T−t)T

,

0 ≤ t < T.

It is seen that βt is a Gaussian random variable with 𝔼[βt ] = 0 and 𝔼[βs βt ] = s ∧ t − Let Xt = a (1 −

t t ) + b + βt , T T

st . T

0 ≤ t < T.

It can also be shown that Xt → b as t → T a. s. Define X̃ t by (2.4.78) with αt replaced by βt . Hence (X̃ t )0≤t≤T is Brownian bridge starting in a at t = 0 and ending b at t = T. Example 2.163. Let (Bt )t≥0 be one-dimensional Brownian motion starting from 0, and define γt = B t −

t B , T T

0 ≤ t ≤ T.

It is seen that γt is a Gaussian random variable with 𝔼[γt ] = 0 and 𝔼[γs γt ] = s ∧ t − Let Xt = a (1 −

t t ) + b + γt , T T

st . T

0 ≤ t ≤ T.

Hence (Xt )0≤t≤T is Brownian bridge starting in a at t = 0 and ending in b at t = T. x,y

We finally state the relationship between Brownian bridge measure 𝒲[0,T] and the Brownian bridge process. Theorem 2.164. Let (Xt )0≤t≤T be Brownian bridge starting in a at t = 0 and ending in b at t = T on a probability space (Ω, F , P). Let 0 = t0 < t1 < . . . < tn ≤ T and Zj = Then we have the following:

X tj

T − tj

−

Xtj−1

T − tj−1

.

136 | 2 Brownian motion (1) {Zj }nj=1 are independent. (2) {Zj }nj=1 is a collection of Gaussian random variables such that 𝔼[Zj ] =

b(tj − tj−1 )

(T − tj )(T − tj−1 )

,

cov(Zj , Zj ) =

(tj − tj−1 )

(T − tj )(T − tj−1 )

.

Proof. Both statements follow directly from above. Using this theorem we can compute the finite-dimensional distributions of Brownian bridge. Corollary 2.165. Let (Xt )0≤t≤T be Brownian bridge starting in a at t = 0 and ending in b at t = T on a probability space (Ω, F , P). Let 0 = t0 < t1 < . . . < tn ≤ T and Aj ∈ ℬ(ℝ), j = 1, . . . , n. Then n

𝔼 [∏ 1Aj (Xtj )] = ∫ j=1

ΠT−tn (xn , b) ∏nj=1 Πtj −tj−1 (xj−1 − xj ) ΠT (b − a)

ℝn

n

n

j=1

j=1

(∏ 1Aj (xj )) ∏ dxj .

(2.4.79)

Let 𝒲 be full Wiener measure on X × ℝ. We recall that n

x,y

n

𝔼𝒲 [(∏ fj (Btj ))] = ∫ ΠT (x − y)𝔼[0,T] [∏ fj (Btj )] dxdy j=1

j=1

ℝ×ℝ

for 0 ≤ t1 ≤ . . . ≤ tn ≤ T. Combining this with Corollary 2.165 we obtain the following corollary. x,y

Corollary 2.166 (Brownian bridge and Brownian bridge measure). Let (Xt )0≤t≤T be Brownian bridge starting in x at t = 0 and ending in y at t = T on a probability space (X , ℬ(X ), 𝒲 0 ). Let 𝒲 = dx ⊗ 𝒲 x be full Wiener measure. Then n

n

x,y

𝔼𝒲 [(∏ fj (Btj ))] = ∫ ΠT (x − y)𝔼𝒲 0 [∏ fj (Xtj )] dxdy j=1

ℝ×ℝ

j=1

(2.4.80)

for 0 ≤ t1 ≤ . . . ≤ tn ≤ T. 2.4.7 Weak solution and time change In the previous section we studied strong solutions of SDEs. In this section we briefly study the notion of weak solutions. Definition 2.167 (Weak solution). Let (Bt )t≥0 be d-dimensional Brownian motion, and bt = (b1t , . . . , bnt ) : ℝn → ℝn and σt = (σt1 , . . . , σtn ) : ℝn → ℝd×n . A weak solution of Xti

t

=

∫ bis (Xs )ds 0

t

+ ∫ σsi (Xs ) ⋅ dBs , 0

X0 = x,

i = 1, . . . , n,

(2.4.81)

2.4 Stochastic calculus based on Brownian motion

| 137

is a triplet ((X̃ t )t≥0 , (B̃ t )t≥0 ),

(Ω, F , P),

(Ft )t≥0 ,

(2.4.82)

where (1) (Ω, F , P) is a probability space; (2) (Ft )t≥0 is an augmented filtration on (Ω, F , P); (3) (B̃ t )t≥0 is an n-dimensional Brownian motion on (Ω, F , P); (4) (X̃ t )t≥0 is an ℝn -valued continuous and adapted process with respect to (Ft )t≥0 ; (5) P(X̃ 0 = x) = 1; (6) ((X̃ t )t≥0 , (B̃ t )t≥0 ) satisfies t

t

0

0

X̃ ti = ∫ bis (X̃ s )ds + ∫ σsi (X̃ s ) ⋅ dB̃ s ,

i = 1, . . . , n.

(2.4.83)

The concept of weak solutions of SDEs is convenient for mathematical reasons, since there exist SDEs which have no strong solutions but still have a weak solution. Example 2.168 (Tanaka equation). Consider the one-dimensional SDE with initial condition X0 = 0 dXt = sgn(Xt )dBt ,

(2.4.84)

where sgn(x) = {

1,

x > 0,

−1,

x ≤ 0.

Equation (2.4.84) is called the Tanaka equation. Note that σt (x) = σ(x) = sgn(x) does not satisfy the Lipschitz condition given by (2.4.54) and thus Theorem 2.152 can not be applied. Indeed it is known that (2.4.84) has no strong solution. A weak solution of (2.4.84) can, however, be obtained as follows. Let (X̃ t )t≥0 be a Brownian motion on a probability space (Ω, F , P) and (Ft )t≥0 the natural filtration of (X̃ t )t≥0 . Define (Wt )t≥0 by t

Wt = ∫ sgn(X̃ s )dX̃ s .

(2.4.85)

0

Note that we take the continuous version of (Wt )t≥0 . Thus (Wt )t≥0 is a continuous t (Ft )-martingale with covariance 𝔼P [|Wt |2 ] = ∫ 𝔼P [|sgn(X̃ s )|2 ]ds = t. Hence (W 2 −t)t≥0 0

t

is a martingale, which implies that (Wt )t≥0 is a Brownian motion by Proposition 2.107. Hence the triplet ((X̃ t )t≥0 , (Wt )t≥0 ), (Ω, F , P), (Ft )t≥0 is a weak solution of (2.4.84).

138 | 2 Brownian motion Example 2.169 (Bessel process). Let (Bt )t≥0 be d-dimensional Brownian motion on a probability space (Ω, F , P). Consider the Euclidean distance from the origin Rt = |Bt | = √(B1t )2 + ⋅ ⋅ ⋅ + (Bdt )2 .

(2.4.86)

The random process (Rt )t≥0 is called Bessel process. The Bessel process starting at r ≥ 0 is a weak solution of the SDE t

Rt = r + ∫ 0

d−1 ds + B̂ t , 2Rs

(2.4.87)

t where (B̂ t )t≥0 is a one-dimensional Brownian motion. Note that since 𝔼[∫0 |Bis /Rs |2 ds] ≤ t

t < ∞, the random process (∫0 Bis /Rs dBis )t≥0 is a continuous martingale for every i, t

moreover, we have 𝔼[(∑di=1 ∫0 Bis /Rs dBis )2 ] = t. Hence t

d

i

B B̃ t = ∑ ∫ s dBis R s i=1

(2.4.88)

0

is a one-dimensional Brownian motion by Theorem 2.107. Heuristically one can apply the Itô formula to derive (2.4.87) for Rt = f (B1t , . . . , Bdt ), where f (x1 , . . . , xd ) = √∑di=1 xi2 . The difficulty here is that f is not differentiable at the origin, and the Itô formula cannot be applied directly to f . Let Yt = R2t . We have d

t

Yt = r 2 + 2 ∑ ∫ Bis dBis + td. i=1 0

Let g(y) = √y and define gε (y) by 3 √ε + gε (y) = { 8 √y,

3 y 4√ε

−

1 y2 , 8ε√ε

y < ε, y ≥ ε.

It follows that gε ∈ C 2 (ℝ) and limε↓0 gε (y) = g(y). The Itô formula yields d

gε (Yt ) = gε (r 2 ) + ∑ Iti (ε) + Jt (ε) + Kt (ε), i=1

where

t

Iti (ε) = ∫ (1{Ys ≥ε} 0

t

Kt (ε) = ∫ 1{Ys 0 be fixed, and N ∈ ℕ. Suppose that ε (1 + 2‖f ‖∞ )(μ1 + μ2 )(ℝd \ [−N, N]d ) < . 2 Let C 󸀠 be the algebra of the linear hull of {f2πm | m ∈ ℤd }, and consider the subset of functions CN = {g⌈[−N,N]d | g ∈ C 󸀠 }. The algebra CN is closed under complex conjugation. Hence the Stone–Weierstrass theorem yields that CN is dense in Cb ([−N, N]d ; ℂ). Thus there exists g ∈ C 󸀠 such that δ=

sup

x∈[−N,N]d

|g(x) − f (x)| < min {1,

ε }. 2(μ1 + μ2 )(ℝd )

Since g(x) = g(x − kN) for all k ∈ ℤd , we have ‖g‖∞ = supx∈[−N,N]d |g(x)| ≤ 1 + ‖f ‖∞ and ‖g − f ‖∞ ≤ ‖f ‖∞ + ‖g‖∞ ≤ 1 + 2‖f ‖∞ . Thus we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε ε 󵄨󵄨 󵄨 󵄨󵄨 ∫ fdμ1 − ∫ fdμ2 󵄨󵄨󵄨 ≤ ‖f − g‖∞ (μ1 + μ2 ) (ℝd \ [−N, N]d ) + δ (μ1 + μ2 ) (ℝd ) ≤ + = ε 󵄨󵄨 󵄨󵄨 2 2 󵄨󵄨 d 󵄨 󵄨󵄨 󵄨ℝ ℝd

3.1 Lévy processes and the Lévy–Khintchine formula

|

147

and ∫ℝd fdμ1 = ∫ℝd fdμ2 for f ∈ Cb (ℝd ). Let A ⊂ ℝd be closed. Define ρε : ℝd → [0, 1] by ρε (x) = 1 − φ(d(x, A)/ε), where d(x, A) denotes the distance between x and A, and φ : ℝ → [0, 1] by φ(t) = (t ∨ 0) ∧ 1. Thus ρε (x) = 1 for x ∈ A, and ρε (x) = 0 for x such that d(x, A) ≥ ε. Hence we have ∫ℝd ρε (x)μ1 (dx) = ∫ℝd ρε (x)μ2 (dx). Taking the limit ε ↓ 0 on both sides, we have μ1 (A) = μ2 (A), and thus μ1 = μ2 follows. Next we discuss some technicalities concerning characteristic functions. Lemma 3.5. Let μ and (μn )n∈ℕ be probability measures on ℝd , and write μ̂ and (μ̂ n )n∈ℕ for their characteristic functions, respectively. Then we have the following: 2 2 ̂̄ ̂ ̂ ̂ (1) We have |μ(u)| = μ̂ ∗ μ,̄ where μ(u) = μ(−u). In particular, |μ(u)| is also a characteristic function of the probability measure μ ∗ μ.̄ ̂ (2) If μn → μ weakly as n → ∞, then μ̂ n (u) → μ(u) uniformly on any compact set as n → ∞. (3) Suppose that μ̂ n (u) → ϕ(u) as n → ∞ for every u, and ϕ(u) is continuous at u = 0. Then ϕ(u) is a characteristic function of a probability measure μ, and μn weakly converges to μ as n → ∞. Proof. Statements (1) and (2) are straightforward. We prove (3). By Bochner’s theorem it is sufficient to show that ϕ(u) is uniformly continuous. For every s, t ∈ ℝ we have |ϕ(t) − ϕ(s)|2 ≤ 4ϕ(0)|ϕ(0) − ϕ(t − s)|.

(3.1.6)

To see this, note that (ϕ(ti − tj ))1≤i,j≤n is a Hermitian positive definite matrix. In particular, choosing (t1 , t2 , t3 ) = (t, s, 0), gives that the matrix ϕ(0) M = (ϕ(t − s) ϕ(t)

ϕ(t − s) ϕ(0) ϕ(s)

ϕ(t) ϕ(s) ) ϕ(0)

has a non-negative determinant. By |ϕ(t)| ≤ ϕ(0) and |ϕ(s)| ≤ ϕ(0), we have det M ≤ ϕ(0)3 − ϕ(0)(|ϕ(t) − ϕ(s)|2 + |ϕ(t − s)|2 ) + 2ϕ(0)2 |ϕ(0) − ϕ(t − s)|. On rearranging terms we then obtain |ϕ(t) − ϕ(s)|2 ≤ ϕ(0)2 − |ϕ(t − s)|2 + 2ϕ(0)|ϕ(0) − ϕ(t − s)| ≤ 4ϕ(0)|ϕ(0) − ϕ(t − s)|. This proves (3.1.6). Thus continuity at u = 0 implies uniform continuity of ϕ(u). Hence ̂ there exists a probability measure μ such that μ(u) = ϕ(u). Next we show that μn → μ. Let ε > 0. We claim that the sequence (μn )n∈ℕ is tight, i. e., there exists a compact set K ⊂ ℝd such that μn (K) > 1 − ε for sufficiently large n. We see that 1 1 ) μ (dy) ∫ 1{|y|≥2/t} μn (dy) = 2 ∫ 1{|y|≥2/t} μn (dy) ≤ 2 ∫ 1{|y|≥2/t} (1 − 2 t|y| n

ℝd

ℝd

ℝd

148 | 3 Lévy processes ≤ 2 ∫ 1{|y|≥2/t} (1 − ℝd

t

= ∫ μn (dy) ∫ ℝd

−t

t

ℝd

1 − eiξy 1 dξ = ∫(1 − ϕn (ξ ))dξ . t t

Thus lim supn→∞ ∫ℝd 1{|y|≥2/t} μn (dy) ≤ d

sin y sin y ) μn (dy) ≤ 2 ∫ (1 − ) μn (dy) t|y| t|y|

−t

1 t ∫ (1 t −t

− ϕ(ξ ))dξ ≤ ε/2 for sufficiently small

0 < t. This gives μn ({y ∈ ℝ | |y| > 2/t}) < ε, for small enough t and sufficiently large n, implying that μn ({y ∈ ℝd | |y| ≤ 2/t}) > 1 − ε. Thus (μn )n∈ℕ is tight, and thus there exists a subsequence (μ󸀠n )n∈ℕ weakly convergent to a probability measure μ󸀠 , and μ󸀠 = μ. Thus every subsequence (μ󸀠n )n∈ℕ of (μn )n∈ℕ contains a subsequence (μ󸀠󸀠 n )n∈ℕ such that μn󸀠󸀠 weakly converges to μ. Hence μn weakly converges to μ. Lemma 3.6. If μ is an infinitely divisible probability measure, then μ̂ has no zeroes. ̂ Proof. For every n ∈ ℕ there exists a characteristic function μ̂ n such that μ(u) = μ̂ n (u)n . 2 2/n ̂ We have |μ̂ n (u)| = |μ(u)| , which implies that lim |μ̂ n (n)|2 = ϕ(u) = {

n→∞

1, 0,

̂ μ(u) ≠ 0, ̂ μ(u) = 0.

̂ ̂ is continuous, ϕ(u) = 1 in a neighborhood of u = 0. We see that Since μ(0) = 1 and μ(u) |μ̂ n (u)|2 is a characteristic function and the limit ϕ(u) is also a characteristic function by Lemma 3.5. Hence ϕ is continuous and ϕ(u) = 1, for all u ∈ ℝd . Hence there is no ̂ point u such that μ(u) = 0. More generally, it can be seen that for a continuous function c : ℝd → ℂ such that c(0) = 1 and c has no zero, there exists a unique continuous function f : ℝd → ℂ such that c(u) = exp(f (u)) and f (0) = 0. Moreover, for every positive integer k there exists a unique continuous function gk such that gk (0) = 0 and c(u) = gk (u)k . Also, it can be shown that gk (u) = exp(f (u)/k). In particular, if c(u) is a characteristic function and so it has no zero, by Bochner’s theorem it follows that gk (u) is also a characteristic function. Let μ be an infinitely divisible probability measure. Then μ̂ has no zero and ̂ ̂ 1/k are defined by η(u) and it has the representation μ(u) = eη(u) . Thus log μ̂ and μ(u) eη(u)/k , respectively. Furthermore, the probability measure with characteristic function ̂ 1/k∗ = μ̂ 1/k . ̂ 1/k will be denoted by μ1/k∗ , i. e., μ μ(u) Lemma 3.7. Let (μn )n∈ℕ be a sequence of infinitely divisible probability measures, and μn → μ weakly as n → ∞. Then μ is also an infinitely divisible probability measure. 2/k ̂ Proof. Since |μ̂ n (u)|2/k → |μ(u)| and |μ̂ n (u)|2/k = |μ̂ n (u)1/k |2 , both |μ̂ n (u)|2/k and 2/k ̂ ̂ ̂ 1/k can be defined. Fur|μ(u)| are characteristic functions. Thus μ(u) ≠ 0 and μ(u) ̂ 1/k uniformly on every compact set. In particular, μ(u) ̂ 1/k is thermore, μ̂ n (u)1/k → μ(u) 1/k ̂ continuous in a neighborhood of u = 0. Thus μ(u) is also a characteristic function and μ̂ is infinitely divisible.

3.1 Lévy processes and the Lévy–Khintchine formula

|

149

Theorem 3.8. Let μ be an infinitely divisible probability measure and t ≥ 0. Then μt∗ is well-defined and is an infinitely divisible probability measure. ̂ 1/n = (μ(u) ̂ 1/kn )k and Proof. Let t = 1/n with n ∈ ℕ. It is trivial to see that μ(u) 1/kn 1/n∗ μ̂ is a characteristic function. Thus μ is infinitely divisible, and so is μm/n∗ = 1/n∗ 1/n∗ μ ∗ ⋅⋅⋅ ∗ μ , for all n, m ∈ ℕ. There exists rn ∈ ℚ such that rn → t as n → ∞. ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m

̂ rn → μ(u) ̂ t uniformly on every compact set and μ(u) ̂ t is continuous in Hence μ(u) t t∗ rn ∗ ̂ u = 0. Thus μ(u) is a characteristic function of μ and μ → μt∗ weakly by (3) of t∗ Lemma 3.5. By Lemma 3.7 it follows that μ is infinitely divisible. 3.1.2 Lévy–Khintchine formula Next we will see that there is a general formula describing the characteristic function of any infinitely divisible random variable. Definition 3.9 (Lévy measure). A σ-finite Borel measure ν on ℬ(ℝd \ {0}) satisfying ∫ (|y|2 ∧ 1)ν(dy) < ∞

(3.1.7)

ℝd \{0}

is called a Lévy measure. An infinitely divisible distribution is uniquely characterized by three parameters called Lévy triplet, as given by the following fundamental result. Theorem 3.10 (Lévy–Khintchine formula). (1) If a probability distribution μ on the measurable space (ℝd , ℬ(ℝd )) is infinitely divisible, then there exist a vector b ∈ ℝd , a non-negative definite symmetric d × d ̂ matrix A, and a Lévy measure ν such that μ(u) = eη(u) for all u ∈ ℝd , where 1 η(u) = ib ⋅ u − u ⋅ Au + ∫ (eiu⋅y − 1 − iu ⋅ y 1{|y|2√d/h ρn (dx) < ε for all n > n0 . Thus (3.1.12) fol0 lows. Together with Helly’s second theorem, (3.1.11) and (3.1.12) imply that there exists a subsequence ρnk weakly convergent to a probability measure ρ. Define the probability measure ν by ν(dx) = (|x|2 ∧ 1)−1 ρ(dx) for |x| > 0 and ν({0}) = 0. We have 1 log μ̂ n (u) = ibn ⋅ u − u ⋅ An,ε u + I1 + I2 , 2 where I1 = ∫ g(u, y) |y|≥ε

(3.1.15)

ρn (dy) , |y|2 ∧ 1

I2 = ∫ (g(u, y) + (u ⋅ y)2 /2) |y| 0 | ∫|x|=ε ρ(dx) = 0}. We have lim I n→∞ 1

= ∫ g(u, y) |y|>ε

ρ(dy) ρ(dy) ρ(dy) + ∫ g(u, y) 2 = ∫ g(u, y) 2 |y|2 ∧ 1 |y| ∧ 1 |y| ∧ 1 |y|=ε

|y|>ε

for ε ∈ E, and lim lim I1 =

E∋ε↓0 n→∞

(3.1.16)

∫ g(u, y) ℝd \{0}

ρ(dy) . |y|2 ∧ 1

(3.1.17)

By (3.1.12) we also see that lim sup |I2 | = 0. ε↓0 n∈ℕ

(3.1.18)

By considering the real and the imaginary parts of (3.1.15), we have lim lim sup u ⋅ An,ε u = lim lim inf u ⋅ An,ε u < ∞,

E∋ε↓0

E∋ε↓0 n→∞

n→∞

lim sup bn ⋅ u = lim inf bn ⋅ u < ∞. n→∞

n→∞

(3.1.19) (3.1.20)

154 | 3 Lévy processes Thus (3.1.19) is represented by u ⋅ Au with a non-negative definite symmetric matrix A, and (3.1.20) by b ⋅ u with a suitable b. In (3.1.19) we may drop the restriction ε since u ⋅ An,ε u is monotone. Thus ̂ ℜ log μ(u) = −u ⋅ Au/2 + ℜ ∫ g(u, y)ν(dy), ℝd \{0}

̂ ℑ log μ(u) = b ⋅ u + ℑ ∫ g(u, y)ν(dy). ℝd \{0}

̂ This means that μ(u) is represented by (3.1.10). By monotonicity we can remove limE∋ε↓0 in (3.1.19). Furthermore, by the uniqueness of the representation of μ,̂ we 󸀠 can choose a subsequence (ρ󸀠󸀠 n )n∈ℕ from any subsequence (ρn )n∈ℕ of (ρn )n∈ℕ such that ρ󸀠󸀠 n converges to ρ. Hence ρn → ρ without choosing a subsequence. Then (1) follows, and from this (2) and (3) follow without going to a subsequence. Next we assume (1)–(3). Define ρn (dy) = (|y|2 ∧ 1)νn (dy) and ρ(dy) = (|y|2 ∧ 1)ν(dy). By (1) we can derive (3.1.16) and (3.1.17). By (1) and (2) we have (3.1.11), implying (3.1.18). Thus we have lim μ̂ (u) n→∞ n

1 = ib ⋅ u − u ⋅ Au + ∫ g(u, y)ν(dy), 2 ℝd \{0}

whose right-hand side equals η.

3.1.3 Lévy processes We consider the following large class of random processes. Definition 3.14 (Lévy process). A random process (Xt )t≥0 on a probability space (Ω, F , P) is called a Lévy process if (1) X0 (ω) = 0 almost surely, (2) the increments (Xti − Xti−1 )1≤i≤n are independent random variables for every choice of time-points 0 = t0 < t1 < . . . < tn , and all n ∈ ℕ, d

(3) Xt − Xs = Xt−s , for every 0 ≤ s ≤ t, (4) the function t 󳨃→ Xt is stochastically continuous, i. e., for all t ≥ 0 and ε > 0 lim P(|Xs − Xt | > ε) = 0. s→t

A Lévy process (Xt )t≥0 can also be defined to start at an arbitrary x ∈ ℝ by replacing (1) of Definition 3.14 with X0 (ω) = x almost surely. There is a close relationship between infinitely divisible distributions and Lévy processes.

3.1 Lévy processes and the Lévy–Khintchine formula

| 155

Theorem 3.15. (1) Let (Xt )t≥0 be a Lévy process. Then for every t > 0, the distribution μt of Xt is infinitely ̂t (u) = μ(u) ̂ t , where μ is the distribution of X1 . divisible and μt = μt∗ , i. e., μ (2) Conversely, for every infinitely divisible distribution μ on ℝd with a triplet (b, A, ν), where b ∈ ℝd , A is a d × d nonnegative definite symmetric matrix, and ν is a Lévy measure, there exists a Lévy process (Xt )t≥0 on (Ω, F , P) such that the distribution of X1 is μ and the characteristic triplet of Xt is (tb, tA, tν), i. e., 𝔼[eiuXt ] = etη(u) , where η is given by the Lévy–Khintchine formula (3.1.8). Proof. (1) Let tk = tk/n. By telescoping Xt = (Xt1 − Xt0 ) + ⋅ ⋅ ⋅ + (Xtn − Xtn−1 ), the random variable Xt is obtained as a sum of independent and identically distributed random variables. Thus Xt is an infinitely divisible random variable for every t ≥ 0. By ̂ ̂ 1/n . We X1 = (X1/n − X0 ) + ⋅ ⋅ ⋅ + (Xn/n − Xn−1/n ) we have μ̂ 1/n (u)n = μ(u), i. e., μ̂ 1/n (u) = μ(u) also have Xm/n = (X1/n − X0 ) + ⋅ ⋅ ⋅ + (Xm/n − X(m−1)/n ) ̂ m/n also follows. Thus the distribution of Xm/n is μm/n∗ . In and hence μ̂ m/n (u) = μ(u) general, if a random processes Yn converges to Y in probability as n → ∞, then the distribution μn of Yn weakly converges to the distribution μ of Y. Let rn → t with rn ∈ ℚ. Since (Xt )t≥0 is a Lévy process, Xrn → Xt in probability, and thus μrn ∗ → μt weakly. On rn ∗ = μ̂ rn → μ̂ t . Thus μ̂ = μ̂ t , which implies μ = μt∗ . ̂ the other hand, μ t t (2) Suppose that μ is an infinitely divisible probability measure. Note that μt∗ ∗ μs∗ = μ(t+s)∗ . Let Ω = (ℝd )[0,∞) , A the set of cylinder sets, and Xt (ω) = ω(t), ω ∈ Ω, be the coordinate process. For Ej ∈ ℬ(ℝd ), j = 0, 1, . . . , n, define μ{t0 ,...,tn } (E0 × ⋅ ⋅ ⋅ × En ) =

n

1E0 (y0 )1E1 (y0 + y1 ) ⋅ ⋅ ⋅ 1En (y0 + ⋅ ⋅ ⋅ + yn ) ∏ μ(ti −ti−1 )∗ (dyi ),

∫

i=0

(ℝd )(n+1)

where we understand t−1 = 0. In the same way as in the proof of Theorem 2.97, existence of a probability measure on (Ω, σ(A )) follows. In fact, the family of measures {μΛ }Λ⊂[0,∞),|Λ| ε) = 0 for every ε > 0. We have P(|Xt | > ε) = ∫ 1{|y|>ε} μt∗ (dy). ℝd

̂ t∗ (u) = μ(u) ̂ t → 1. Thus we have ∫ℝd f (y)μt∗ (dy) → f (0) as t → 0, Taking t → 0 we get μ for every bounded function f , continuous at u = 0. So limε→0 P(|Xt | > ε) = 0, and ̂ t∗ (u) = (Xt )t≥0 is a Lévy process. By (3.1.22) the distribution of Xt under P is μt∗ , and μ t η(u) ̂ ̂ μ(u) , where μ is the distribution of X1 . Since μ(u) = e , where η(u) is given by (3.1.8), t tη(u) ̂ t∗ ̂ we have μ (u) = μ(u) =e . Thus the characteristic triplet for Xt is (tb, tA, tν). The vector b is called drift term, the matrix A diffusion matrix , the measure ν Lévy (jump) measure , and the triplet (b, A, ν) the characteristics of the Lévy process (Xt )t≥0 , while η given by (3.1.8) is called its Lévy symbol. By Theorem 3.15 the processes in Example 3.2 are all Lévy processes. Example 3.16 (Brownian motion). The Lévy triplet of Brownian motion with constant drift is (b, A, 0) and its Lévy symbol is η(u) = ib ⋅ u − 21 u ⋅ Au.

| 157

3.1 Lévy processes and the Lévy–Khintchine formula

Example 3.17 (Poisson process). A non-negative integer-valued Lévy process (Nt )t≥0 is called a Poisson process with intensity λ > 0 if it has Lévy triplet (0, 0, λδ1 ), where δ1 is a one-dimensional Dirac point measure, and Lévy symbol η(u) = λ(eiu − 1). It has independent, identically distributed increments (Nti − Nti−1 )0≤i≤n , for every choice of time-points 0 = t0 < t1 < . . . < tn , and every n ∈ ℕ, with Poisson distribution d

Nti − Nti−1 = Poi(λ(ti − ti−1 )), i = 1, . . . , n. In particular, we have 𝔼[1{n} (Nt )] = and characteristic function

(λt)n −λt e , n!

𝔼[eiuXt ] = eλt(e

iu

−1)

,

n≥0

u ∈ ℝ.

Example 3.18 (Compound Poisson processes). Let (Zn )n∈ℕ be independent and iden̂ tically distributed random variables with characteristic function σ(u), and (Nt )t≥0 a d

Poisson process such that Nt = Poi(λt). Then Nt

Yt = ∑ Zn n=1

is called a compound Poisson process. Its characteristic function is ̂ μt̂ (u) = exp(λ(σ(u) − 1)t), ̂ and η(u) = λ(σ(u) − 1). The Cauchy process is a specific case of a class of special interest, which we describe in the next example. Example 3.19 (Stable processes). A random variable X is called stable if for every d

n ∈ ℕ there exist an ≥ 0 and bn ∈ ℝd such that X1 + ⋅ ⋅ ⋅ + Xn = an X + bn , and strictly stable if bn = 0, where X1 , . . . , Xn are independent, identically distributed copies of X. (Stability under taking sums explains the name of this class.) A Lévy process (Xt )t≥0 is called a stable process if all random variables Xt , t > 0, are stable. For a stable process d

we have that for every a > 0 there exist b > 0 and c ∈ ℝd such that Xat = bXt + ct, which holds for a strictly stable process with c = 0. This is an extension of the scaling property of Brownian motion; see Proposition 2.100 (5) below. From the definition it can be seen that for a stable process there exist 0 < α ≤ 2 and bt ∈ ℝd such that d

Xt = t 1/α X1 + bt ; for a strictly stable process bt = 0. Here α is called the stability index of the stable process. For a stable process either α = 2 and its Lévy triplet is (b, A, 0), i. e., it is a Gaussian process with mean b and covariance matrix A, or 0 < α < 2 and its Lévy triplet is (b, 0, ν) with Lévy measure ∞

ν(E) = ∫ λ(dθ) ∫ 1E (rθ) Sd−1

0

dr , r 1+α

(3.1.23)

158 | 3 Lévy processes where Sd−1 is the (d − 1)-dimensional unit sphere centered in the origin. The above integral can be computed; for d = 1 the calculation gives ν(dy) = (

c1 c2 1(0,∞) + 1(−∞,0) ) dy, y1+α/2 (−y)1+α/2

c1 , c2 ≥ 0.

(3.1.24)

Stable processes with index 1 ≤ α ≤ 2 are type-C Lévy processes, while those with 0 < α < 1 are type-B. A stable process with index 0 < α ≤ 2 is rotationally invariant d

if R Xt = Xt , for all R ∈ O(ℝd ), i. e., under the action of the orthogonal group. In this case the Lévy symbol of the process becomes η(u) = −c|u|α , with some c > 0, and for 0 < α < 2 the Lévy measure has a uniform distribution on Sd−1 . In particular, the rotationally invariant 1-stable process is the Cauchy process. Let (Xt )t≥0 be a d-dimensional rotationally invariant α-stable process on a probability space (Ω, F , P). The characteristic function of (Xt )t≥0 is given by α

𝔼[eiξ ⋅Xt ] = e−t|ξ | ,

ξ ∈ ℝd , t ≥ 0.

(3.1.25)

The probability transition (or heat) kernel pXt (x) of (Xt )t≥0 is a smooth real-valued function on ℝd given by pXt (x) =

α 1 ∫ e−t|ξ | e−ix⋅ξ dξ , d (2π)

x ∈ ℝd , t > 0.

ℝd

We have the scaling property pXt (x) = t −d/α pX1 (t −1/α x), and thus the generic bound pXt (x) ≤ t −d/α

α 1 ∫ e−|ξ | dξ . d (2π)

(3.1.26)

ℝd

The asymptotic behavior of pX1 (x) can be computed as lim |x|d+α pX1 (x) = α2α−1

|x|→∞

1 απ d+α α sin( )Γ( )Γ( ). 2 2 2 π (d+2)/2

Hence for every ε > 0 there exist a1 (ε), a2 (ε) > 0 such that a1 (ε)t a (ε)t ≤ pXt (x) ≤ 2 d+α , d+α |x| |x|

|x| > ε.

For every ε > 0, together with (3.1.26) we have the kernel estimates b1 (ε) (

t t ∧ t −d/α ) ≤ pXt (x) ≤ b2 (ε) ( d+α ∧ t −d/α ) , |x|d+α |x|

|x| > ε

(3.1.27)

3.1 Lévy processes and the Lévy–Khintchine formula

| 159

with positive constants b1 (ε) and b2 (ε). By a more precise calculation, these estimates improve to 1 t t ( ∧ t −d/α ) ≤ pXt (x) ≤ C ( d+α ∧ t −d/α ) C |x|d+α |x|

(3.1.28)

with a constant C > 0. The Lévy measure of (Xt )t≥0 is given by ν(dx) = 𝒜d,−α |x|−d−α dx, where

−γ −d/2 Γ((d

− γ)/2) . |Γ(γ/2)|

𝒜d,γ = 2 π

Next we discuss some properties of Lévy processes. In general, their sums are not necessarily Lévy processes again, but independent terms give rise to Lévy processes. Theorem 3.20 (Sums of Lévy processes). Let (Xt )t≥0 and (Yt )t≥0 be Lévy processes on a probability space (Ω, F , P). If (Xt )t≥0 and (Yt )t≥0 are independent, then (Zt )t≥0 , Zt = Xt + Yt , is also a Lévy process. Proof. By independence 𝔼[eiu⋅(Zt −Zs ) ] = 𝔼[eiu⋅(Xt −Xs ) ]𝔼[eiu⋅(Yt −Ys ) ] = 𝔼[eiu⋅Xt−s ]𝔼[eiu⋅Yt−s ] = 𝔼[eiu⋅Zt−s ]. d

Thus Zt − Zs = Zt−s follows, and the independent increments property of the process (Zt )t≥0 is similarly shown. We have P(|Zt − Zs | > ε) ≤ P(|Xt − Xs | > ε/2) + P(|Yt − Ys | > ε/2), showing stochastic continuity. Theorem 3.21 (Convergence of Lévy processes). Let (Xt )t≥0 be a random process on a probability space (Ω, F , P) and (Xt(n) )t≥0 , n ∈ ℕ, be a sequence of Lévy processes on it. Suppose that Xt(n) converges to Xt in probability as n → ∞ for every t ≥ 0, and limn→∞ lim supt→∞ P(|Xt(n) − Xt | > a) = 0, for all a > 0. Then (Xt )t≥0 is also a Lévy process. Proof. It is direct to see that X0 = 0. Since convergence in probability implies convergence of the characteristic functions, we have 𝔼[eiu⋅(Xt −Xs ) ] = lim 𝔼[eiu⋅(Xt

(n)

n→∞

−Xt(n) )

] = lim 𝔼[eiu⋅Xt−s ] = 𝔼[eiu⋅Xt−s ], (n)

n→∞

d

thus Xt − Xs = Xt−s . Similarly, we can show that n

𝔼[∏ e j=1

iu⋅(Xtj −Xtj−1 )

n

] = ∏ 𝔼[e j=1

iu⋅(Xtj −Xtj−1 )

],

160 | 3 Lévy processes thus the increments of (Xt )t≥0 are independent. Furthermore, we have P(|Xt | > a) ≤ P(|Xt − Xt(n) | > a/2) + P(|Xt(n) | > a/2) and hence lim sup P(|Xt | > a) ≤ lim sup P(|Xt − Xt(n) | > a/2) + lim sup P(|Xt(n) | > a/2). t→0

t→0

t→0

Note that lim supt→0 P(|Xt(n) | > a/2) = limt→0 P(|Xt(n) | > a/2) = 0, which also means that limt→0 P(|Xt | > ε) ≤ lim supt→0 P(|Xt | > ε) = 0, giving stochastic continuity. Theorem 3.22 (Versions of Lévy processes). Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) and (Yt )t≥0 a version of (Xt )t≥0 . Then (Yt )t≥0 is also a Lévy process. Proof. The fact Y0 = 0 is obvious. Fix 0 ≤ s < t. Let Ds,t = {ω ∈ Ω | Xs (ω) = Ys (ω) and Xt (ω) = Yt (ω)}. Note that P(Dcs,t ) ≤ P(Xs ≠ Ys ) + P(Xt ≠ Yt ) = 0, implying P(Ds,t ) = 1. Also, we have P(Yt − Ys ∈ A) = P(Yt − Ys ∈ A, Ds,t ) + P(Yt − Ys ∈ A, Dcs,t )

= P(Yt − Ys ∈ A, Ds,t ) = P(Xt − Xs ∈ A, Ds,t )

≤ P(Xt − Xs ∈ A) = P(Xt−s ∈ A) = P(Yt−s ∈ A). d

The reverse inequality is proven in a similar fashion, and thus Yt − Ys = Yt−s follows. It follows similarly that the increments of (Yt )t≥0 are independent. Stochastic continuity is implied by P(|Yt − Ys | > ε) ≤ P(|Xt − Xs | > ε). 3.1.4 Martingale properties of Lévy processes As in the case of Brownian motion, more general Lévy processes also have martingale properties, which are very helpful whenever available. Proposition 3.23. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) with Lévy symbol η, and take a fixed u ∈ ℝd . Then (Yt )t≥0 , Yt = eiu⋅Xt −tη(u) , is a martingale with respect to the natural filtration (Ft )t≥0 of (Xt )t≥0 . Proof. 𝔼[|Yt |] < ∞ is straightforward. Also, 𝔼[Yt |Fs ] = Ys 𝔼[eiu⋅(Xt −Xs )−(t−s)η(u) ] = Ys , for all 0 ≤ s ≤ t. Below the following martingales constructed by subtracting a term called compensator will be important. Let (Nt )t≥0 be a Poisson process with intensity λ. The process (Ñ t )t≥0 = (Nt − λt)t≥0 is called a compensated Poisson process.

3.1 Lévy processes and the Lévy–Khintchine formula

|

161

Lemma 3.24 (Martingales for Poisson processes). Let (Nt )t≥0 be a Poisson process with intensity λ on a probability space (Ω, F , P). Then (Ñ t )t≥0 and (Ñ t2 − λt)t≥0 are martingales with respect to the natural filtration (Ft )t≥0 of (Nt )t≥0 . Proof. The first statement follows by 𝔼[Ñ t |Fs ] = 𝔼[Nt − Ns + Ns − λt|Fs ] = 𝔼[Nt − Ns ] + 𝔼[Ns − λt|Fs ] = λ(t − s) − λt + Ns = Ñ s . The proof of the second statement is completely similar.

3.1.5 Markov properties of Lévy processes In this section we show that Lévy processes are strong Markov processes possessing Feller transition kernels, and derive their generators. Let (Xt )t≥0 be a Lévy process on (Ω, F , P). Let μt be the infinitely divisible distribution of Xt and define p(s, t, x, A) by p(s, t, x, A) = μt−s (A − x)

(3.1.29)

for 0 ≤ s ≤ t, x ∈ ℝd , and A ∈ ℬ(ℝd ), where A − x = {y − x | y ∈ A}. Note that μt (A − x) = P(Xt +x ∈ A). By the definition of a Lévy process we have p(s, t, x, A) = P(Xt −Xs ∈ A−x) and p(s, t, x, A) is a probability transition kernel. The Chapman–Kolmogorov identity results from the semigroup property μt−s ∗ μu−t = μu−s ,

s < t < u,

(3.1.30)

i. e., ∫ μt−s (A − y)μu−t (dy) = μu−s (A),

A ∈ ℬ(ℝd ).

(3.1.31)

ℝd

Write p(s, t, x, A) = p(t − s, x, A).

(3.1.32)

Proposition 3.25. A Lévy process (Xt )t≥0 is a stationary Markov process and its probability transition kernel is given by (3.1.32). Proof. By independence of the increments, the finite-dimensional distributions of the process are P(X0 ∈ A0 , . . . , Xn ∈ An ) =

∫ (ℝd )(n+1)

n

n

i=0

i=1

(∏ 1Ai (xi )) ∏ p(ti − ti−1 , xi−1 , dxi )P 0 (dx0 ), (3.1.33)

where P 0 (dx) = δ(x) is Dirac point measure with mass at x = 0. Thus by Proposition 2.82, (Xt )t≥0 is a Markov process with probability transition kernel p(t, x, A).

162 | 3 Lévy processes Theorem 3.26 (C0 -semigroup on C∞ (ℝd )). The probability transition kernel p(t, x, A) of a Lévy process is a Feller transition kernel. In particular, f 󳨃→ Pt f (x) = ∫ℝd f (y)p(t, x, dy) defines a C0 -semigroup on C∞ (ℝd ).

Proof. The fact p(t, x, A) ≤ 1 is trivial. We show that Pt f ∈ C∞ (ℝd ) for f ∈ C∞ (ℝd ). Let xn → x as n → ∞. By the dominated convergence theorem we have lim P f (xn ) n→∞ t

= ∫ lim f (xn + y)p(t, 0, dy) = Pt f (x) ℝd

n→∞

and lim Pt f (x) = ∫ lim f (x + y)p(t, 0, dy) = 0.

|x|→∞

ℝd

|x|→∞

Thus Pt f ∈ C∞ (ℝd ). To conclude, we show that limt→0 Pt f = f for f ∈ C∞ (ℝd ) in the sup-norm. Since f is uniformly continuous, for every ε > 0 there exists an r > 0 such that supx∈ℝd |f (x + y) − f (x)| < ε for all |y| < r. Moreover, by stochastic continuity of the Lévy process, ∫|y|>r p(t, 0, dy) = P(|Xt | > r) → 0 as t → 0. Thus there exists t0 such that ∫|y|>r p(t, 0, dy) < ε for 0 < t < t0 . Hence 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 sup |Pt f (x) − f (x)| = sup 󵄨󵄨󵄨 ∫ (f (x + y) − f (x))p(t, 0, dy)󵄨󵄨󵄨󵄨 󵄨󵄨 x∈ℝd x∈ℝd 󵄨󵄨󵄨 d 󵄨󵄨 󵄨ℝ 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = sup 󵄨󵄨󵄨 ∫ (f (x + y) − f (x))p(t, 0, dy) + ∫ (f (x + y) − f (x))p(t, 0, dy)󵄨󵄨󵄨󵄨 󵄨󵄨 x∈ℝd 󵄨󵄨󵄨 |y|>r 󵄨|y|≤r 󵄨󵄨 ≤ ε ∫ p(t, 0, dy) + ε‖f ‖∞ |y| ε for j = 1, . . . , n. Define B(p, ε, M) = {ω ∈ Ω | Xt (ω) has ε-oscillation p times in M}. Also, Xt (ω) has ε-oscillation infinitely often in M if for every n, Xt (ω) has ε-oscillation n times in M. Put B(∞, ε, M) = {ω ∈ Ω | Xt (ω) has ε-oscillation infinitely often in M}. Let 󵄨󵄨 K = {ω ∈ Ω 󵄨󵄨󵄨 ∃ lim Xs (ω) and ∃ lim Xs (ω) for all t > 0} . 󵄨 s∈ℚ,s↓t s∈ℚ,s↑t Define AN,k = Ω \ B(∞, 1/k, [0, N] ∩ ℚ) and ∞ ∞

K 󸀠 = ⋂ ⋂ AN,k . N=1 k=1

We have K 󸀠 ∈ F , since ℚ is countable. Lemma 3.31. It follows that K 󸀠 = K. Proof. It is trivial to see that K 󸀠 ⊃ K. We show that K 󸀠 ⊂ K. Let ω ∈ K 󸀠 and tn ∈ ℚ be such that tn ↓ t as n → ∞. For every k there exists n0 such that |Xtn (ω) − Xt0 (ω)| ≤ 1/k for all n ≥ n0 . Hence limn→∞ Xtn (ω) exists and thus so does lims∈ℚ,s↓t Xs (ω). Similarly, we can show that lims∈ℚ,s↑t Xs (ω) exists. Next we show that both càdlàg and càglàd versions exist. Lemma 3.32. Suppose that a random process (Xt )t≥0 is stochastically continuous on a probability space (Ω, F , P) and P(K 󸀠 ) = 1. Then there exist (Yt )t≥0 and (Zt )t≥0 such that (1) P(Xt = Yt ) = 1 and t 󳨃→ Yt (ω) is càdlàg, (2) P(Xt = Zt ) = 1 and t 󳨃→ Zt (ω) is càglàd, for every ω ∈ Ω.

166 | 3 Lévy processes Proof. For ω ∈ K 󸀠 we can define Yt (ω) = lims∈ℚ,s↓t Xs (ω) by Lemma 3.31. For ω ∈ ̸ K 󸀠 we set Yt (ω) = 0. Thus t → Yt (ω) is càdlàg for every ω ∈ Ω. By the stochastic continuity of t 󳨃→ Xt there exists a sequence ℚ ⊃ (sn )n∈ℕ such that Xsn → Xt in probability as ℚ ∋ sn ↓ t, while Xsn → Yt almost surely due to P(K 󸀠 ) = 1. Hence P(Xt = Yt ) = 1 follows for t ≥ 0. In the same way as (Yt )t≥0 , for ω ∈ K 󸀠 we define Zt (ω) = lims∈ℚ,s↑t Xs (ω), and for ω ∈ ̸ K 󸀠 we set Zt (ω) = 0. Thus t → Zt (ω) is càglàd for every ω ∈ Ω and P(Xt = Zt ) = 1 follows for t ≥ 0. Denote by Bε (x) the open ball centered at x ∈ ℝd with radius ε. Let (Xt )t≥0 be a Markov process on a probability space (Ω, F , P) and pX (s, t, x, A) the probability transition kernel of (Xt )t≥0 . Define αε,T (u) = sup {pX (s, t, x, Bε (x)c ) | x ∈ ℝd , s, t ∈ [0, T] with t − s ∈ [0, u]} ,

(3.2.1)

where pX (s, t, x, Bε (x)c ) = P(Xt − Xs ∈ Bε (x)c \ {x})

= P(Xt − Xs ∈ Bε (0)c \ {0}) = P(|Xt − Xs | ≥ ε)

and then αε,T (u) = sups,t∈[0,T],t−s∈[0,u] P(|Xt − Xs | ≥ ε). Lemma 3.33. Let (Xt )t≥0 be a Markov process on a probability space (Ω, F , P). Let p be a positive integer and 0 ≤ u ≤ t1 < . . . < tn ≤ v ≤ T and M = {t1 , . . . , tn }. Then 𝔼[1B(p,4ε,M) ] ≤ (2αε,T (v − u))p . Proof. We prove this by induction on p. First, let p = 1 and consider Ck = {ω ∈ Ω | max |Xtj (ω) − Xu (ω)| ≤ 2ε and |Xtk (ω) − Xu (ω)| > 2ε}, 1≤j≤k−1

Dk = {ω ∈ Ω| |Xv (ω) − Xtk (ω)| > ε}. The sets C1 , . . . , Cn are pairwise disjoint and n

n

k=1

k=1

B(1, 4ε, M) ⊂ ⋃ Ck ⊂ {ω ∈ Ω | |Xv (ω) − Xu (ω)| ≥ ε} ∪ ⋃ (Ck ∩ Dk ). Hence n

n

k=1

k=1

𝔼[1B(1,4ε,M) ] ≤ 𝔼[1{|Xv −Xu |≥ε} ] + ∑ 𝔼[1Ck 1Dk ] ≤ αε,T (v − u) + ∑ 𝔼[1Ck ]αε,T (v − u) by the Markov property. Thus 𝔼[1B(1,4ε,M) ] ≤ 2αε,T (v − u)

(3.2.2)

3.2 Sample path properties of Lévy processes |

167

follows, and the statement holds for p = 1. Next assume that it is true for p − 1. Let Fk = B(p − 1, 4ε, {t1 , . . . , tk }) \ B(p − 1, 4ε, {t1 , . . . , tk−1 }),

Gk = B(1, 4ε, {tk , . . . , tn }).

Note that B(p − 1, 4ε, {t1 , . . . , tk }) = 0 if p − 1 < k. Then F1 , . . . , Fn are disjoint and B(p − 1, 4ε, M) = ⋃nk=1 Fk , and furthermore we see that B(p, 4ε, M) ⊂ ⋃nk=1 (Fk ∩ Gk ). Consequently we have n

n

k=1

k=1

𝔼[1B(p,4ε,M) ] ≤ ∑ 𝔼[1Fk 1Gk ] ≤ 2 ⋃ 𝔼[1Fk ]αε,T (v − u) by the Markov property and (3.2.2). Thus 𝔼[1B(p,4ε,M) ] ≤ 𝔼[1B(p−1,4ε,M) ]2αε,T (v − u) ≤ (2αε,T (v − u))p and the lemma follows. The following result has a similar role for general Lévy processes than the Kolmogorov–Čentsov theorem for Brownian motion. The so-called Dynkin–Kinney condition given by (3.2.3) below replaces the criterion in Theorem 2.49 for path continuity, and establishes a càdlàg and a càglàd property of sample paths instead. Theorem 3.34 (Dynkin–Kinney condition). Let (Xt )t≥0 be a Markov process on a probability space (Ω, F , P) such that lim αε,T (u) = 0 u↓0

(3.2.3)

for every ε > 0 and T > 0, where αε,T is given by (3.2.1). Then there exist (Yt )t≥0 and (Zt )t≥0 such that (1) P(Xt = Yt ) = 1 and t 󳨃→ Yt (ω) is càdlàg, (2) P(Xt = Zt ) = 1 and t 󳨃→ Zt (ω) is càglàd, for every ω ∈ Ω. Proof. Condition (3.2.3) implies stochastic continuity of (Xt )t≥0 . Thus it is sufficient to prove that P(K 󸀠 ) = 1 by Lemma 3.32, and actually it suffices to show that P(AcN,k ) = 0 for every fixed N and k. Choose l large enough so that 2α1/(4k) (N/l) < 1. We have l 1 1 j−1 j P(AcN,k ) = P (B (∞, , [0, N] ∩ ℚ)) ≤ ∑ lim P (B (p, , [ N, N] ∩ ℚ)) . p→∞ k k l l j=1

Enumerating the elements of [ j−1 N, jl N] ∩ ℚ as t1 , t2 , . . ., we have l 1 j−1 j 1 P (B (p, , [ N, N] ∩ ℚ)) = lim P (B (p, , {t1 , . . . , tn })) . n→∞ k l l k Since 1 P (B (p, , {t1 , . . . , tn })) ≤ (2α1/(4k),N (N/l))p k

168 | 3 Lévy processes by Lemma 3.33, we get on letting n → ∞ that 1 j−1 j P (B (p, , [ N, N] ∩ ℚ)) ≤ (2α1/(4k),N (N/l))p . k l l The right-hand side goes to zero as p → ∞. Hence P(AcN,k ) = 0 follows. Corollary 3.35 (Càdlàg and càglàd versions). Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P). Then a càdlàg version and a càglàd version of (Xt )t≥0 exist. Proof. Let pX be the probability transition kernel of (Xt )t≥0 and Bε (x) be the ball centered at x with radius ε > 0. We have pX (s, t, x, Bε (x)c ) = P(|Xt − Xs | ≥ ε). We can show that P(|Xt − Xs | ≥ ε) uniformly converges to zero as |s − t| → 0 for s, t ∈ [0, T] in Lemma 3.36. Thus limu↓0 αε,T (u) = 0 and the statement follows by Lemma 3.32. The following gives a strengthened version of stochastic continuity. Lemma 3.36 (Uniform stochastic continuity). Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P). Then for every T > 0 we have that P(|Xt − Xs | ≥ ε) uniformly converges to zero as |s − t| → 0, for s, t ∈ [0, T]. Proof. Fix η > 0. By stochastic continuity, for every t > 0 there exists δt > 0 such that P(|Xt − Xs | ≥ ε/2) ≤ η/2, for |s − t| < δt . Let It = (t − δt /2, t + δt /2). There is a finite covering Itj , j = 1, . . . , n, such that ⋃nj=1 Itj ⊃ [0, T]. Take δ = min1≤j≤n δtj . If |s − t| < δ and s, t ∈ [0, T], then t ∈ Itj for some j; hence |s − tj | < δtj and P(|Xt − Xs | > ε) ≤ P(|Xt − Xtj | > ε/2) + P(|Xs − Xtj | > ε/2) < η. To conclude the discussion of sample path properties, we finally point out the topological structure of the space of càdlàg paths. We will denote the set of ℝd -valued càdlàg paths on [0, ∞) by d

XD = D([0, ∞); ℝ ).

(3.2.4)

To introduce a topology on XD , let I 󸀠 be the collection of strictly increasing functions h : [0, ∞) → [0, ∞). In particular, we have h(0) = 0, limt→∞ h(t) = ∞, and t 󳨃→ h(t) is continuous. Let I be the set of Lipschitz continuous functions h ∈ I 󸀠 such that 󵄨󵄨 h(t) − h(s) 󵄨󵄨󵄨 󵄨󵄨 < ∞. γ(h) = sup 󵄨󵄨󵄨󵄨log t − s 󵄨󵄨 s,t≥0 󵄨 s=t̸

For X, Y ∈ XD we define the metric ∞

dD (X, Y) = inf (γ(h) ∨ ∫ e−u (sup |Xt∧u − Yh(u)∧u | ∧ 1)du) . h∈I

0

t≥0

(3.2.5)

The properties of the so induced topological space are summarized below without proof.

3.2 Sample path properties of Lévy processes |

169

Proposition 3.37 (Skorokhod topology). (1) The space (XD , dD ) is a separable and complete metric (i. e., a Polish) space. (2) Let B be the smallest σ-field containing all finite-dimensional cylinder sets, and B (XD ) be the Borel σ-field with the topology induced by dD . Then B = B (XD ). 3.2.2 Two-sided Lévy processes Next we describe how Lévy processes can be extended to run over the whole real line. Let (Xt )t≥0 be the coordinate process of a Lévy process on a probability space (XD , ℬ(XD ), P x ), and (X̄ t )t≥0 the coordinate process of a Lévy process on a probability space (XD̄ , ℬ(XD̄ ), P̄ x ), such that t 󳨃→ Xt is càdlàg and t 󳨃→ X̄ t is càglàd. Consider the product probability space (XD × XD̄ , ℬ(XD ) × ℬ(XD̄ ), P x ⊗ P̄ x ) and define for ω = (ω1 , ω2 ) ∈ XD × XD̄ , ̂t (ω) = {ω1 (t), X ω2 (−t),

t ≥ 0, t < 0.

̂t (ω) is càdlàg for t ∈ ℝ. Let Then t 󳨃→ X XD = D(ℝ; ℝd )

(3.2.6)

be the space of càdlàg paths on ℝ, and ℬ(XD ) be the family of cylinder sets on XD . ̂⋅ by Denote the image measure of P x ⊗ P̄ x on (XD , ℬ(XD )) with respect to X ̂−1 , Qx = (P x ⊗ P̄ x ) ∘ X ⋅ and the coordinate process on (XD , ℬ(XD ), Qx ) by (Yt )t∈ℝ , which is a Lévy process indexed by ℝ on this space. The properties of the so obtained process can be summarized as follows. Proposition 3.38 (Two-sided Lévy process). The following hold: (1) Qx (Y0 = x) = 1; (2) the increments {Yti − Yti−1 }1≤i≤n are independent for every 0 = t0 < t1 < . . . < tn with d

Yt − Ys = Yt−s for t > s; (3) the increments {Y−ti−1 − Y−ti }1≤i≤n are independent for every 0 = −t0 > −t1 > . . . > −tn d

with Y−t − Y−s = Ys−t for −t > −s; (4) the function ℝ ∋ t 󳨃→ Yt (ω) ∈ ℝ is almost surely càdlàg; d

(5) Y−t = Yt for t ∈ ℝ, and Yt and Ys for t > 0 and s < 0 are independent.

170 | 3 Lévy processes It can be checked through the finite-dimensional distributions that the joint distributions of Yt0 , . . . , Ytn , −∞ < t0 < t1 < . . . < tn < ∞, n ∈ ℕ, are invariant with respect to time-shift under some conditions. Remark 3.39. As seen in Remark 2.102 for Brownian motion, time-shift invariance of an ℝd -valued two-sided Lévy process (Yt )t∈ℝ is also related to the symmetry of the semigroup generated by process. Consider Pt f (x) = 𝔼x [f (Yt )], satisfying the semigroup property Pt+s = Pt Ps for t, s ≥ 0, by the Markov property of the process. Then we have ∫ 𝔼x [f ̄(Y0 )g(Yt )]dx = (f , Pt g)L2 (ℝd ) . ℝd

When Pt is symmetric, i. e., (f , Pt g)L2 (ℝd ) = (Pt f , g)L2 (ℝd ) holds, we also have (f , Pt g)L2 (ℝd ) = (Ps f , Pt−s g)L2 (ℝd ) for 0 ≤ s ≤ t, which can equally be represented as ∫ 𝔼x [f ̄(Y0 )g(Yt )]dx = ∫ 𝔼x [f ̄(Ys )]𝔼x [g(Yt−s )]dx. ℝd

ℝd

The right-hand side above can be computed as ∫ 𝔼x [f ̄(Y0 )g(Yt )]dx = ∫ 𝔼x [f ̄(Y−s )]𝔼x [g(Yt−s )]dx = ∫ 𝔼x [f ̄(Y−s )g(Yt−s )]dx. ℝd

ℝd

ℝd

Hence ∫ℝd 𝔼x [f ̄(Y0 )g(Yt )]dx is time-shift invariant for all 0 ≤ s ≤ t. The crucial assumption under which shift invariance is obtained is that Pt is symmetric. Let μt be the distribution of Yt for t ≥ 0. If μt (A) = μt (−A) for A ∈ ℬ(ℝd ), then as seen in Theorem 3.27, it follows that (f , Pt g)L2 (ℝd ) = ∫ f ̄(x)g(x + y)μt (dy)dx = ∫ f ̄(x − y)g(x)μt (dy)dx. ℝd

ℝd

Since μt (A) = μt (−A), replacing y for −y, we see that (f , Pt g)L2 (ℝd ) = ∫ f ̄(x + y)g(x)μt (dy)dx = (Pt f , g)L2 (ℝd ) . ℝd

Hence Pt is symmetric. Thus the condition μt (A) = μt (−A) ensures that Pt is symmetric. Proposition 3.40 (Time-shift invariance). Let (Yt )t∈ℝ be an ℝd -valued two-sided Lévy process, with Y0 = 0 almost surely, and μt be the distribution of Yt for t ≥ 0. Suppose that μt (A) = μt (−A), for every t ≥ 0 and A ∈ ℬ(ℝd ). Let λ be a shift invariant measure on ℝd , i. e., λ(A + a) = λ(A) for every A ∈ ℬ(ℝd ) and a ∈ A , where A = ⋃t≥0 Ran Yt . Let

3.2 Sample path properties of Lévy processes |

171

t1 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . For bounded functions fi , i = 2, . . . , n + m − 1 and f1 , fn+m ∈ L2 (ℝd , λ(dx)), it follows that n+m

n+m

∫ 𝔼x [ ∏ fi (Yti )] λ(dx) = ∫ 𝔼x [ ∏ fi (Yti +s )] λ(dx) i=1

ℝd

(3.2.7)

i=1

ℝd

for −tn+m ≤ s ≤ −t0 . Here n

n

∫ 𝔼x [∏ fi (Yti )] λ(dx) = ∫ 𝔼 [∏ fi (Yti + x)] λ(dx). i=1

ℝd

i=1

ℝd

Proof. Let Pt (x, A) = 𝔼[1A (Yt + x)], for A ∈ ℬ(ℝd ). We have n

∫ 𝔼x [∏ fj (Ytj )] λ(dx) = j=1

ℝd

n

n

j=1

j=1

(∏ fj (xj )) ∏ Ptj −tj−1 (xj−1 , dxj )λ(dx),

∫ ℝd ×ℝdn

where t0 = 0 and x0 = x. Let −t1 ≤ . . . ≤ −tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . We show that n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∫ 𝔼x [∏ fi (Y−ti ) ∏ fj (Ytj )] λ(dx) = ∫ 𝔼x [∏ fi (Yt1 −ti ) ∏ fj (Yt1 +tj )] λ(dx). (3.2.8) ℝd

ℝd

Thus (3.2.7) follows from (3.2.8). By independence of Y−t and Ys for −t ≤ 0 ≤ s and reflection symmetry we have n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∫ 𝔼x [∏ fi (Y−ti ) ∏ fj (Ytj )] λ(dx) = ∫ 𝔼x [∏ fi (Y−ti )] 𝔼x [ ∏ fj (Ytj )] λ(dx) ℝd

n

n+m

i=1

j=n+1

ℝd

= ∫ 𝔼x [∏ fi (Yti )] 𝔼x [ ∏ fj (Ytj )] λ(dx). ℝd

(3.2.9)

Hence we can compute the left-hand side of (3.2.9) as n+m

2

i=1

i=n

l. h. s. (3.2.9) = ∫ λ(dx) ∫ ( ∏ fi (xi )) Ptn (x, dxn ) (∏ Pti−1 −ti (xi , dxi−1 )) ℝd

ℝd(n+m)

n+m

× Ptn+1 (x, dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

By Lemma 3.41 we see the equality of measures on ℝd × ℝd , i. e., ∫ Ptn (x, dxn )Ptn+1 (x, dxn+1 )λ(dx) = Ptn +tn+1 (xn , dxn+1 )λ(dxn ). ℝd

172 | 3 Lévy processes Thus we have l. h. s. (3.2.9) =

n+m

n

i=1

i=2

∫ ( ∏ fi (xi )) (∏ Pti−1 −ti (xi , dxi−1 )) λ(dxn ) ℝd(n+m)

n+m

× Ptn +tn+1 (xn , dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

(3.2.10)

On the other hand, the right-hand side of (3.2.9) can be computed as n+m

n

r. h. s. (3.2.9) = ∫ λ(dx) ∫ ( ∏ fi (xi )) P0 (x, dx1 ) (∏ Pti−1 −ti (xi−1 , dxi )) ℝd

ℝd(n+m)

i=1

i=2

n+m

× Ptn +tn+1 (xn , dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ) j=n+2

n+m

n

∫ ( ∏ fi (xi )) λ(dx1 ) (∏ Pti−1 −ti (xi−1 , dxi ))

=

ℝd(n+m)

i=1

i=2

n+m

× Ptn +tn+1 (xn , dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

(3.2.11)

Here we used ∫ℝd P0 (x, dx1 )λ(dx) = λ(dx1 ) as measures, due to ∫A ∫ℝd P0 (x, dx1 )λ(dx) = ∫A P0 (x, ℝd )λ(dx) = ∫A λ(dx) = λ(A). By Lemma 3.42 we also see that n

n

i=2

i=2

(∏ Pti−1 −ti (xi , dxi−1 )) λ(dxn ) = λ(dx1 ) (∏ Pti−1 −ti (xi−1 , dxi ))

(3.2.12)

as measures. Comparing (3.2.10) and (3.2.11) with (3.2.12) we obtain (3.2.8), and hence (3.2.7) follows for every s ∈ ℝ. To complete the proof of Proposition 3.40, it remains to show Lemmas 3.41–3.42. Lemma 3.41. Let the assumptions of Proposition 3.40 hold, and consider Pt (x, A) = 𝔼[1A (Yt + x)]. Then for s, t ≥ 0, ∫ Ps (x, dy)Pt (x, dz)λ(dx) = Ps+t (y, dz)λ(dy).

(3.2.13)

ℝd

Proof. For a shorthand, we denote the left-hand side of (3.2.13) by m, and the righthand side by n. By λ(A + a) = λ(A) for a ∈ A , we have m(A × B) = ∫ Ps (x, A)Pt (x, B)λ(dx) = ∫ λ(dx) ∫ 1A (y + x)1B (z + x)μs (dy)μt (dz) ℝd

=

ℝd

∫ ρ(−y + z)μs (dy)μt (dz), ℝd ×ℝd

ℝd ×ℝd

3.2 Sample path properties of Lévy processes |

173

where ρ(X) = ∫ℝd 1A (x)1B (x + X)λ(dx). We may think of y, z as elements of A . By the assumption μt (A) = μt (−A), we can replace −y by y. Hence by the definition of the convolution of measures we have m(A × B) =

∫ ρ(y + z)μs (dy)μt (dz) = ∫ ρ(p)(μs ∗ μt )(dp). ℝd ×ℝd

ℝd

By the Markov property μs ∗ μt = μs+t , we also have m(A × B) = ∫ ρ(p)μs+t (dp) = ℝd

∫ 1A (x)1B (x + p)μs+t (dp)λ(dx) ℝd ×ℝd

= ∫ 1A (x)Ps+t (x, B)λ(dx) = n(A × B), ℝd

which implies (3.2.13). Lemma 3.42. Let the assumptions of Proposition 3.40 hold. Then (3.2.12) also holds. Proof. We prove the proposition by induction. Take n = 2. For simplicity, write Pj−1 = Ptj−1 −tj , and for the distribution of Ytj write μj . Let A = A1 × A2 . We see that ∫ P1 (x1 , dx2 )λ(dx1 ) = ∫ P1 (x1 , A2 )λ(dx1 ) = ∫ λ(dx1 ) ∫ μ1 (dy)1A1 (x1 )1A2 (x1 + y). A

A1

ℝd

ℝd

Since μj (−A) = μj (A) and λ(A + y) = λ(A) for y ∈ A , we see that ∫ P1 (x1 , dx2 )λ(dx1 ) = ∫ λ(dx1 ) ∫ μ1 (dy)1A1 (x1 )1A2 (x1 − y) A

ℝd

ℝd

= ∫ λ(dx1 ) ∫ μ1 (dy)1A1 (x1 + y)1A2 (x1 ) = ∫ P1 (x2 , dx1 )λ(dx2 ). ℝd

A

ℝd

Hence λ(dx1 )P1 (x1 , dx2 ) = P1 (x2 , dx1 )λ(dx2 ) follows. From this equality formally we can see that λ(dx1 )P1 (x1 , dx2 ) ⋅ ⋅ ⋅ Pn−1 (xn−1 , dxn )

= P1 (x2 , dx1 )λ(dx2 )P1 (x2 , dx3 ) ⋅ ⋅ ⋅ Pn−1 (xn−1 , dxn )

= P1 (x2 , dx1 )P1 (x3 , dx2 )λ(dx3 ) ⋅ ⋅ ⋅ Pn−1 (xn−1 , dxn )

= . . . = P1 (x2 , dx1 )P1 (x3 , dx2 ) ⋅ ⋅ ⋅ Pn−1 (xn , dxn−1 )λ(dxn ). To verify this rigorously, assume that (3.2.12) holds for n = 2, . . . , k. Let A = A1 ×⋅ ⋅ ⋅×Ak+1 . We have k

∫ (∏ Pi (xi , dxi+1 )) λ(dx1 ) = A

i=1

k−1

∫ A1 ×⋅⋅⋅×Ak

(∏ Pi (xi , dxi+1 )) Pk (xk , Ak+1 )λ(dx1 ) i=1

174 | 3 Lévy processes k−1

=

(∏ Pi (xi+1 , dxi )) Pk (xk , Ak+1 )λ(dxk ) = ∫ ξ (xk )Pk (xk , Ak+1 )λ(dxk ).

∫ A1 ×⋅⋅⋅×Ak

i=1

Ak

The second equality follows from the induction hypothesis, and ξ is given by ξ (xk ) =

k−1

∫

∏ Pi (xi+1 , dxi ).

A1 ×⋅⋅⋅×Ak−1 i=1

Since λ(dxk )Pk (xk , dxk+1 ) = Pk (xk+1 , dxk )λ(dxk+1 ) as measures, we see that k

∫ (∏ Pi (xi , dxi+1 )) λ(dx1 ) = A

i=1

∫ Ak ×Ak+1

ξ (xk )1Ak+1 (xk+1 )Pk (xk , dxk+1 )λ(dxk ) k

=

∫ Ak ×Ak+1

ξ (xk )1Ak+1 (xk+1 )Pk (xk+1 , dxk )λ(dxk+1 ) = ∫ (∏ Pi (xi+1 , dxi )) λ(dxk+1 ) A

i=1

follows. This implies that (3.2.12) also holds for n = k + 1. Example 3.43 (α-stable process). Let (Xt )t≥0 be a d-dimensional rotationally invariant α-stable process and consider its extension to (Xt )t∈ℝ defined on the whole real line ℝ. The distribution μt of Xt , t ≥ 0, is rotationally invariant, which implies that μt (A) = μt (−A) for A ∈ ℬ(ℝd ). Thus (Xt )t≥0 is time-shift invariant. Next we further consider time-shift invariance of two-sided Lévy processes, without assuming the reflection symmetry μt (A) = μt (−A). A crucial example is the socalled spin (in probability theory better known as the telegraph) process (σt )t∈ℝ defined by σt = (−1)Nt , where (Nt )t∈ℝ is a Poisson process on ℝ. The Poisson process is a non-negative integer-valued random process, not having reflection symmetry, and the generator A of (Nt )t≥0 given by Af (z) = f (z + 1) − f (z), z ∈ ℤ, is not symmetric. Thus (f , Ag)L2 (ℤ) = ∑ f ̄(z)(g(z + 1) − g(z)) = ∑ (f ̄(z − 1) − f ̄(z))g(z) ≠ (Af , g)L2 (ℤ) . z∈ℤ

z∈ℤ

However, σt is shift invariant. We consider this situation more generally next. Proposition 3.44 (Time-shift invariance). Let (Yt )t∈ℝ be an ℝd -valued two-sided Lévy process with Y0 = 0 almost surely, and ρ a function on ℝd such that ρ(−x) = ρ(x) for x ∈ ℝd . Define Zt = ρ(Yt ),

t ∈ ℝ.

Let λ be a shift invariant measure on ℝd , i. e., λ(A + a) = λ(A) for every A ∈ ℬ(ℝd ) and a ∈ A . Let t1 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . Then for all bounded functions fi , i = 2, . . . , n + m, and f1 , fn+m ∈ L2 (ℝd , λ(dx)), it follows that

3.2 Sample path properties of Lévy processes | n+m

n+m

∫ 𝔼x [ ∏ fi (Zti )] λ(dx) = ∫ 𝔼x [ ∏ fi (Zti +s )] λ(dx) i=1

ℝd

i=1

ℝd

175

(3.2.14)

for all −tn+m ≤ s ≤ −t0 . Here n

n

∫ 𝔼x [∏ fi (Zti )] λ(dx) = ∫ 𝔼 [∏ fi (ρ(Yti + x))] λ(dx). i=1

ℝd

i=1

ℝd

Proof. Let μt be the distribution of Yt on ℝd . The distribution of −Yt is given by λt = μt ∘ r, where r : ℝd → ℝd is defined by r(a) = −a. Let Pt (x, A) = 𝔼[1A (Yt + x)] and Qt (x, A) = 𝔼[1A (−Yt + x)] for A ∈ ℬ(ℝd ). Let gj = fj ∘ ρ. Thus fj (Zt ) = gj (Yt ) = gj (−Yt ) follows. We have for 0 ≤ s1 ≤ . . . ≤ sn , n

∫ 𝔼x [∏ gj (Ysj )] λ(dx) = j=1

ℝd

∫ ℝd ×ℝdn

=

∫

n

n

j=1

j=1

n

n

j=1

j=1

(∏ gj (xj )) ∏ Psj −sj−1 (xj−1 , dxj )λ(dx) (∏ gj (xj )) ∏ Qsj −sj−1 (xj−1 , dxj )λ(dx),

ℝd ×ℝdn

where s0 = 0 and x0 = x. Although t1 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m , for the reader’s convenience we assume that tj ≥ 0 for j = 1, . . . , n + m and −t1 ≤ . . . ≤ −tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m in the proof below, and show that n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∫ 𝔼x [∏ gi (Y−ti ) ∏ gj (Ytj )] λ(dx) = ∫ 𝔼x [∏ gi (Yt1 −ti ) ∏ gj (Yt1 +tj )] λ(dx). ℝd

ℝd

(3.2.15)

Thus (3.2.14) follows from (3.2.15). We have n

n+m

i=1

j=n+1

n

n+m

i=1

j=n+1

∫ 𝔼x [∏ gi (Y−ti ) ∏ gj (Ytj )] λ(dx) = ∫ 𝔼x [∏ gi (Yti )] 𝔼x [ ∏ gj (Ytj )] λ(dx). ℝd

ℝd

Hence we compute the left-hand side above as n+m

2

i=1

i=n

l. h. s. (3.2.15) = ∫ λ(dx) ∫ ( ∏ gi (xi )) Qtn (x, dxn ) (∏ Qti−1 −ti (xi , dxi−1 )) ℝd

ℝd(n+m)

n+m

× Ptn+1 (x, dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

176 | 3 Lévy processes By Lemma 3.45 below we have the equality of measures on ℝd × ℝd , i. e., ∫ Qtn (x, dxn )Ptn+1 (x, dxn+1 )λ(dx) = Ptn +tn+1 (xn , dxn+1 )λ(dxn ). ℝd

Thus we obtain l. h. s. (3.2.15) =

n+m

n

i=1

i=2

∫ ( ∏ gi (xi )) (∏ Qti−1 −ti (xi , dxi−1 )) λ(dxn ) ℝd(n+m)

n+m

× Ptn +tn+1 (xn , dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

(3.2.16)

On the other hand, r. h. s. (3.2.15) =

n+m

n

i=1

i=2

∫ ( ∏ gi (xi )) λ(dx1 ) (∏ Pti−1 −ti (xi−1 , dxi )) ℝd(n+m)

n+m

× Ptn +tn+1 (xn , dxn+1 ) ∏ Ptj −tj−1 (xj−1 , dxj ). j=n+2

(3.2.17)

By Lemma 3.45 again we also see that n

n

i=2

i=2

(∏ Qti−1 −ti (xi , dxi−1 )) λ(dxn ) = λ(dx1 ) (∏ Pti−1 −ti (xi−1 , dxi )) .

(3.2.18)

Comparing (3.2.16) and (3.2.17) with (3.2.18) we can conclude (3.2.15) and hence (3.2.14) follows for any s ∈ ℝ. To complete the proof of Proposition 3.44 we show Lemma 3.45. Lemma 3.45. Let the assumptions of Proposition 3.44 hold. (1) Let Pt (x, A) = 𝔼[1A (Yt + x)] and Qt (x, A) = 𝔼[1A (−Yt + x)]. Then ∫ Qs (x, dy)Pt (x, dz)λ(dx) = Ps+t (y, dz)λ(dy).

(3.2.19)

ℝd

(2) Equality (3.2.18) holds. Proof. (1) We denote the left-hand side of (3.2.19) by m and the right-hand side by n. We have m(A × B) = ∫ Qs (x, A)Pt (x, B)λ(dx) = ℝd

∫ ρ(−y + z)λs (dy)μt (dz), ℝd ×ℝd

where ρ(X) = ∫ℝd 1A (x)1B (x + X)λ(dx). By λt (A) = μt (−A) we see that

3.2 Sample path properties of Lévy processes |

m(A × B) =

177

∫ ρ(y + z)μs (dy)μt (dz) = ∫ ρ(p)(μs ∗ μt )(dp) ℝd ×ℝd

ℝd

= ∫ ρ(p)μs+t (dp) = ℝd

∫ 1A (x)1B (x + p)μs+t (dp)λ(dx) = n(A × B), ℝd ×ℝd

which implies (3.2.19). (2) We proceed by induction. For easing the notation we write Pj−1 = Ptj−1 −tj , Qj−1 = Qtj−1 −tj , and denote the distributions of Ytj and −Ytj by λj and μj , respectively. Assume n = 2, Let A = A1 × A2 . Since λj (−A) = μj (A) and λ(A + a) = λ(A) for a ∈ A , we see that ∫ P1 (x1 , dx2 )λ(dx1 ) = ∫ λ(dx1 ) ∫ 1A1 (x1 )1A2 (x1 + y)λ1 (dy) A

ℝd

ℝd

= ∫ λ(dx1 ) ∫ 1A1 (x1 + y)1A2 (x1 )μ1 (dy) ℝd

ℝd

= ∫ λ(dx2 ) ∫ 1A1 (x2 + y)1A2 (x2 )μ1 (dy) = ∫ Q1 (x2 , dx1 )λ(dx2 ). ℝd

A

ℝd

Hence λ(dx1 )P1 (x1 , dx2 ) = Q1 (x2 , dx1 )λ(dx2 ) follows. Suppose that (3.2.18) holds for n = 2, . . . , k. Let A = A1 × ⋅ ⋅ ⋅ × Ak+1 . We have k

k−1

∫ ∏ Pi (xi , dxi+1 )λ(dx1 ) = A i=1

(∏ Pi (xi , dxi+1 )) Pk (xk , Ak+1 )λ(dx1 )

∫

i=1

A1 ×⋅⋅⋅×Ak k−1

=

(∏ Qi (xi+1 , dxi )) Pk (xk , Ak+1 )λ(dxk ) = ∫ ξ (xk )Pk (xk , Ak+1 )λ(dxk ).

∫ A1 ×⋅⋅⋅×Ak

i=1

Ak

The second equality follows by the induction hypothesis, and ξ is given by ξ (xk ) =

k−1

∫

∏ Qi (xi+1 , dxi ).

A1 ×⋅⋅⋅×Ak−1 i=1

Since λ(dxk )Pk (xk , dxk+1 ) = Qk (xk+1 , dxk )λ(dxk+1 ) as measures, we see that k

∫ λ(dx1 ) (∏ Pi (xi , dxi+1 )) = i=1

A

∫ Ak ×Ak+1

ξ (xk )1Ak+1 (xk+1 )λ(dxk )Pk (xk , dxk+1 ) k

=

∫ Ak ×Ak+1

ξ (xk )1Ak+1 (xk+1 )Qk (xk+1 , dxk )λ(dxk+1 ) = ∫ (∏ Qi (xi+1 , dxi )) λ(dxk+1 ) A

follows. This implies that (3.2.18) also holds for n = k + 1.

i=1

178 | 3 Lévy processes Finally, we consider bridge measures for Lévy processes. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P x ), and recall that pXt denotes the probability transition kernel of (Xt )t≥0 . Definition 3.46 (Lévy bridge measure). Given a division 0 ≤ T1 ≤ t1 < . . . < tn ≤ T2 and x, y ∈ ℝd , the probability measure on (XD , ℬ(XD )) defined by x,y

P[T ,T ] (Xt1 ∈ A1 , . . . , Xtn ∈ An ) 1

2

= ∫

pXt1 −T1 (x − x1 )pXt2 −t1 (x1 − x2 ) ⋅ ⋅ ⋅ pXT2 −tn (xn − y) pXT −T (x − y) 2

ℝnd

1

n

n

i=1

i=1

(∏ 1Ai (xi )) ∏ dxi

(3.2.20)

is called Lévy bridge measure starting in x at t = T1 and ending in y at t = T2 . Furtherx,y more, the non-normalized measure given by pXT2 −T1 (x − y)P[T ,T ] is called conditional 1 2 Lévy measure. x,y

x,y

Write 𝔼[T ,T ] for expectation with respect to P[T ,T ] . From the definition it is directly 1

2

x,y

seen that P x and P[T ,T ] are related through 1

1

2

2

n

n

x,y

𝔼x [∏ fj (Xtj )] = ∫ pXT2 −T1 (x − y)𝔼[T ,T ] [∏ fj (Xtj )] dy j=1

1

ℝd

2

j=1

with T1 ≤ t1 ≤ . . . ≤ tn ≤ T2 and for every Borel measurable bounded function fj . In particular, n

n

x,y

𝔼x [f (XT1 ) (∏ fj (Xtj )) g(XT2 )] = ∫ pXT2 −T1 (x − y)f (x)g(y)𝔼[T ,T ] [∏ fj (Xtj )] dy. j=1

ℝd

1

2

j=1

3.3 Random measures and Lévy–Itô decomposition 3.3.1 Poisson random measures Definition 3.47 (Random measure). Let (S, S ) be a measurable space and (Ω, F , P) a probability space. Furthermore, let M = (M(A))A∈S be a collection of random variables M(A)(⋅) : Ω → ℝ indexed by A ∈ S ; M is a random measure whenever (1) M(0) = 0; ∞ (2) M(⋃∞ n=1 An ) = ∑n=1 M(An ) almost surely, provided that An ∩ Am = 0 for n ≠ m; (3) M(A) and M(B) are independent whenever A ∩ B = 0. The following is a random measure of fundamental interest. Definition 3.48 (Poisson random measure). If in addition to (1)–(3) of Definition 3.47 every M(A) has a Poisson distribution with intensity λ(A), i. e., if λ(A) < ∞,

3.3 Random measures and Lévy–Itô decomposition

P(M(A) = n) = e−λ(A)

λ(A)n n!

| 179

(3.3.1)

holds, with λ(A) = ∞ giving M(A) = ∞, then M is called a Poisson random measure. From (3.3.1) it follows that λ(A) = 𝔼[M(A)], and it is a measure on S since M is a random measure. Remarkably, a converse property also holds. Theorem 3.49. Given a σ-finite measure λ on a measurable space (S, S ), there exist a probability space (Ω, F , P) and a Poisson random measure M = (M(A))A∈S such that 𝔼[M(A)] = λ(A) for A ∈ S . Proof. Let Un ∈ S , n ∈ ℕ, such that 0 < λ(Un ) < ∞ and ⋃∞ m=1 Um = S. We take random variables {Xn (m)}n∈ℕ , m ∈ ℕ, and {Nn }n∈ℕ on (Ω, F , P) such that (1) for all n, m, Xn (m) is a Un -valued random variable with P(Xn (m) ∈ A) = λ(A)/λ(Un ), for A ∈ S ; (2) for all n we have that Nn is an integer-valued random variable such that P(Nn = k) =

1 λ(Un )k e−λ(Un ) , k!

k = 0, 1, 2, . . .

(3) Xn (m) and Nn , n, m ∈ ℕ, are mutually independent. N

m Write M(A) = ∑∞ m=1 Mm (A), where Mm (A) = ∑n=1 1A∩Um (Xn (m))1{Nm ≥1} , for A ∈ S . Let

A1 , . . . , Ak be pairwise disjoint sets and ⋃kj=1 Aj

= Ω, such that n1 + ⋅ ⋅ ⋅ + nk = n. We have

P(Mm (A1 ) = n1 , . . . , Mm (Ak ) = nk )

= P(Mm (A1 ) = n1 , . . . , Mm (Ak ) = nk |Mm (Ω) = n)P(Mm (Ω) = n).

Note that Mm (Ω) = Nm , and we arrive at the multinomial distribution P(Mm (A1 ) = n1 , . . . , Mm (Ak ) = nk |Mm (Ω) = n) n

n

j=1

j=1

= P (∑ 1A1 ∩Um (Xj (m)) = n1 , . . . , ∑ 1Ak ∩Um (Xj (m)) = nk ) λ(A1 ∩ Um ) n1 λ(Ak ∩ Um ) nk n! = ( ) ⋅⋅⋅( ) . n1 ! ⋅ ⋅ ⋅ nk ! λ(Um ) λ(Um ) We furthermore obtain P(Mm (A1 ) = n1 , . . . , Mm (Ak ) = nk ) =

λ(A1 ∩ Um ) n1 λ(Ak ∩ Um ) nk λ(Um )n −λ(Um ) n! ( ) ⋅⋅⋅( ) e n1 ! ⋅ ⋅ ⋅ nk ! λ(Um ) λ(Um ) n! k

= ∏ e−λ(Aj ∩Um ) j=1

λ(Aj ∩ Um )nj nj !

.

180 | 3 Lévy processes Summing over n1 , . . . , nk except for nj , we have P(Mm (Aj ) = nj ) = e−λ(Aj ∩Um )

λ(Aj ∩ Um )nj nj !

.

Thus Mm (A) is a Poisson random variable with intensity λ(A ∩ Um ) for every A. Since M(A) = ∑∞ m=1 Mm (A) and (Mm (A))m≥1 are independent of each other, and ∞

∞

m=1

m=1

𝔼[M(A)] = ∑ 𝔼[Mm (A)] = ∑ λ(A ∩ Um ) = λ(A), it follows that M(A) is a Poisson process whenever λ(A) < ∞. In case λ(A) = ∞, we have ∞

∞

m=1

m=1

1 λ(A ∩ Um ) ∧ a = ∞ 2 m=1 ∞

∑ P(Mm (A) ≥ 1) = ∑ (1 − e−λ(A∩Um ) ) ≥ ∑

with a > 0. Hence P(Mm (A) ≥ 1 for infinitely many m) = 1 by the Borel–Cantelli lemma. Thus M(A) = ∞ almost surely. This proves that (M(A))A∈S is a Poisson random measure with 𝔼[M(A)] = λ(A), for all A ∈ S . Next we define the counting measure associated with a Lévy process (Xt )t≥0 on a probability space (Ω, F , P). Every càdlàg function is bounded on compact intervals and attains its extrema there; moreover, every such function is Borel measurable. Denote Xt− = lims↑t Xs , t > 0, and define the jump at t by ΔXt = Xt − Xt− .

(3.3.2)

A càdlàg path can only have jump discontinuities, and the set 𝒥 = {t ∈ [0, ∞) | ΔXt ≠ 0}

(3.3.3)

of time points where a jump occurs is at most countable but does not have accumulation points. The Lévy processes with symbol (A, b, ν) can be classified by their jump properties: (1) type I: A = 0 and ν(ℝd \ {0}) < ∞; (2) type II: A = 0, ν(ℝd \ {0}) = ∞ and ∫0 t} = {Xt − Xτn−1 = 0}. Thus {τn − τn−1 > t} is independent of Fτn−1 by the strong Markov property of (Xt )t≥0 , and so τn − τn−1 is independent of Fτn−1 . On the other hand, τn − τn−1 is Fτn -measurable. In particular, τn − τn−1 and τm − τm−1 are independent for n ≠ m. By the strong Markov property of (Xt )t≥0 , we see that d

τn − τn−1 = inf {t > 0 | Xt+τn−1 − Xτn−1 ≠ 0} = inf {t > 0 | Xt − X0 ≠ 0} = τ1 . Thus τ1 , τ2 − τ1 , . . . , τn − τn−1 , . . . are independent and identically distributed. The following result gives a sufficient condition for a Lévy process to be a Poisson process. Lemma 3.51. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) such that it is almost surely increasing and ΔXt ∈ {0, 1}, for all t ≥ 0. Then (Xt )t≥0 is a Poisson process. Proof. Define a sequence of stopping times (τn )n∈ℕ by (3.3.6). Note that if Xt+s = 0, then Xs = 0. We have P(τ1 > t + s) = P(Xt+s = 0) = P(Xs = 0, Xt+s − Xs = 0)

= P(Xs = 0)P(Xt+s − Xs = 0) = P(Xs = 0)P(Xt = 0) = P(τ1 > s)P(τ1 > t). (3.3.7)

182 | 3 Lévy processes Note that t 󳨃→ P(τ1 > t) is decreasing, since t 󳨃→ Nt is increasing, and t 󳨃→ P(τ1 > t) is continuous at t = 0 since t 󳨃→ Nt is stochastically continuous. Thus a solution to the functional equation (3.3.7) is of the form P(τ1 > t) = e−λt with a suitable λ > 0. Thus P(Nt = 0) = P(τ1 > t) = e−λt . Hence e−λt = ∫t P(s)ds for the distribution function P(s) of τ1 , and then P(s)ds = ∞

λe−λs 1[0,∞) (s)ds. By Lemma 3.50, τ1 , τ2 − τ1 , . . . , τn − τn−1 , . . . are independent and identically distributed. Noting that τn = (τn − τn−1 ) + ⋅ ⋅ ⋅ + (τ1 − τ0 ), the distribution of τn is n n−1 s given by e−λs λ(n−1)! 1[0,∞) (s)ds. Then for k ≥ 0, P(Xt = n) = P(τn+1 > t, τn ≤ t) = P(τn+1 > t) − P(τn > t) ∞

= ∫e t

−λs λ

n+1 n

n!

s

∞

ds − ∫ e−λs t

λn sn−1 (λt)n −λt ds = e , (n − 1)! n!

showing that (Xt )t≥0 is a Poisson process. Lemma 3.52. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P). Define N(t, A) by (3.3.4). Then N(t, A) < ∞ almost surely, for all A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A.̄ Proof. Define a sequence of stopping times (τn )n∈ℕ by τ1 = inf{t > 0 | ΔXt ∈ A} and τn+1 = inf{t > τn | ΔXt ∈ A}, for n ≥ 1. Since the set of jumps in A has no accumulation point due to 0 ∈ ̸ A,̄ we have τ1 > 0 and τn → ∞ as n → ∞. Hence N(t, A) = ∑∞ n=1 1{τn ≤t} < ∞. Theorem 3.53. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P). Define N(t, A) by (3.3.4). Then we have the following properties: (1) Let A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A.̄ Then (N(t, A))t≥0 is a Poisson process with intensity μ(A). (2) The random processes N(t, A1 ), . . . , N(t, An ) are independent for all pairwise disjoint sets Aj ∈ ℬ(ℝd \ {0}) such that 0 ∉ Ā j , j = 1, . . . , n. Proof. Note that ΔN(t, A) = N(t, A) − N(t−, A) takes values in {0, 1} and t 󳨃→ N(t, A) is increasing. By Lemma 3.51 it is sufficient to show that (N(t, A))t≥0 is a Lévy process. We have N(t, A) − N(s, A) = n if and only if there exists s < t1 < . . . < tn ≤ t such that ΔXtj ∈ A, j = 1, . . . , n. Furthermore, ΔXu ∈ A if and only if there exists a ∈ A for which, given ε > 0, there exists δ > 0 such that |u − v| < δ implies that |Xv − Xu − a| < ε. Thus we deduce that {N(t, A) − N(s, A) = n} ∈ σ(Xv − Xu , s ≤ u < v ≤ t),

(3.3.8)

giving σ(N(t, A) − N(s, A)) ∈ σ(Xv − Xu | s ≤ u < v ≤ t). This implies that the increments of (N(t, A))t≥0 are independent. Since

3.3 Random measures and Lévy–Itô decomposition

| 183

N(t, A) − N(s, A) = |{s < r ≤ t | ΔXr ∈ A}| = |{0 < r ≤ t − s | ΔXr ∈ A}| = N(t − s, A) − N(0, A),

(N(t, A))t≥0 is stationary. Next we show stochastic continuity. Using the independence of increments, we see that P(N(t, A) = 0)

t 2t t (n − 1)t = P (N ( , A) = 0, N ( , A) − N ( , A) = 0, . . . , N(t, A) − N ( , A) = 0) n n n n n t = P (N ( , A) = 0) . n

This gives lim sup P(N(t, A) = 0) = lim lim sup P(N(t/n, A) = 0)n = lim lim sup P(N(s, A) = 0)n , t→0

n→∞

t→0

n→∞

n→∞

t→0

n

s→0

lim inf P(N(t, A) = 0) = lim lim inf P(N(t/n, A) = 0) = lim lim inf P(N(s, A) = 0)n . t→0

n→∞

s→0

We consider the following two cases: (1) limt→0 P(N(t, A) = 0) exists, i. e., limt→0 P(N(t, A) = 0) = 1 or = 0. (2) limt→0 P(N(t, A) = 0) does not exist, i. e., lim supt→0 P(N(t, A) = 0) = 1 and lim inft→0 P(N(t, A) = 0) = 0. Suppose case (2) holds. Since t 󳨃→ P(N(t, A) = 0) is decreasing, if P(N(t, A) = 0) > ε for some t, then lim inft→0 P(N(t, A) = 0) ≥ ε. Hence if lim inft→0 P(N(t, A) = 0) = 0, P(N(t, A) = 0) = 0 for all t and so lim supt→0 P(N(t, A) = 0) = 0, which yields a contradiction. Thus (2) cannot hold. Assume case (1) and limt→0 P(N(t, A) = 0) = 0. Thus limt→0 P(N(t, A) ≠ 0) = 1. Let A and B be disjoint. We have lim P(N(t, A ∪ B) ≠ 0) = lim(P(N(t, A) ≠ 0) + P(N(t, B) ≠ 0)).

t→0

t→0

Hence we deduce that limt→0 P(N(t, A) = 0) = 1 and limt→0 P(N(t, A) ≠ 0) = 0, which implies that limt→0 P(N(t, A) > ε) = 0 for arbitrary ε > 0. Thus part (1) of the proposition is proven. For part (2), in a similar way as (3.3.8) was obtained we see that the processes {N(t, A1 ) = n1 }, . . . , {N(t, Am ) = nm } are elements of an independent σ-field. The difference between the counting measure N(t, ⋅) and the intensity measure μ will be denoted by ̃ A) = N(t, A) − tμ(A), N(t,

A ∈ ℬ(ℝd \ {0}).

(3.3.9)

̃ A))t≥0 is called the associated If 0 ∉ A,̄ (N(t, A))t≥0 is a Poisson process and then (N(t, ̃ A). compensated Poisson process. Here are some basic properties of N(t, A) and N(t, (1) For every t > 0 and ω ∈ Ω, N(t, ⋅)(ω) is a counting measure on the measurable space (ℝd \ {0}, ℬ(ℝd \ {0})).

184 | 3 Lévy processes (2) For every A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A,̄ (N(t, A))t≥0 is a Poisson process with ̃ A))t≥0 is a martingale. intensity μ(A) = 𝔼[N(1, A)], and (N(t, Next consider the integral ∫A f (x)N(t, dx)(ω), which can be defined for every ω ∈ Ω, t ≥ 0, A ∈ ℬ(ℝd \ {0}), and suitable f . Suppose that 0 ∉ A.̄ For a step function f (x) = ∑nj=1 aj 1Aj (x), we have

n

n

∫ f (x)N(t, dx)(ω) = ∑ aj N(t, A ∩ Aj )(ω) = ∑ ∑ aj 1Aj (x)N(t, {x})(ω), x∈A j=1

j=1

A

where ∑x∈A is the sum over jumps x ∈ A such that x = ΔXs with some 0 ≤ s ≤ t. In particular, ∑x∈A is a finite sum for each path ω ∈ Ω. For a non-negative ℬ(ℝd \ {0})-measurable function f , there exists a sequence of step functions such that fn (x) ↑ f (x) for every x. It follows that ∫ f (x)N(t, dx) = ∑ f (x)N(t, {x}) a. s.,

(3.3.10)

x∈A

A

and (3.3.10) holds for general ℬ(ℝd \ {0})-measurable functions. We have the following on related characteristic functions. Theorem 3.54. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) and N(t, ⋅) the Poisson random measure associated with (Xt )t≥0 . Let μ be the intensity measure of N(t, ⋅). Suppose that A ∈ ℬ(ℝd \ {0}) and 0 ∉ A.̄ (1) Let f : ℝd → ℝd be Borel measurable and u ∈ ℝd . We have 𝔼 [ei ∫A u⋅f (x)N(t,dx) ] = et ∫A (e

iu⋅f (x)

−1)μ(dx)

.

(2) If f ∈ L1 (A, μ), then 𝔼[∫A f (x)N(t, dx)] = t ∫A f (x)μ(dx). (3) If f ∈ L2 (A, μ), then 𝔼[| ∫A f (x)N(t, dx)|2 ] = t ∫A |f (x)|2 μ(dx). Proof. Let f : ℝd → ℝd be Borel measurable. For y = (y1 , . . . , yd ) ∈ ℤd define Cyn = {x = (x1 , . . . , xd ) ∈ ℝd | 2−n (yj − 1) < xj ≤ 2−n yj , 1 ≤ j ≤ d}. We have ℝd = ⋃y∈ℤd Cyn , where Cyn ∩ Czn = 0 for y ≠ z. Choose a point xyn in each Cyn

and define a function fn : A → ℝd by fn (x) = xyn for x ∈ f −1 (Cyn ). Thus |f (x) − fn (x)| ≤ √d/2n . Let Yn = ∫ fn (x)N(t, dx) and Y = ∫ f (x)N(t, dx). We have |Yn (ω) − Y(ω)| ≤ A A N(t, A)(ω)√d/2n . Hence |Yn (ω) − Y(ω)| → 0 as n → ∞, for every ω ∈ Ω. Since Yn = ∑y∈ℤd xyn N(t, f −1 (Cyn )), we compute n

𝔼[eiu⋅∫A fn (x)N(t,dx) ] = ∏ 𝔼[ei(u⋅xy )N(t,f y∈ℤd

−1

(Cyn ))

] = ∏ et(e y∈ℤd

i(u⋅xyn )

−1)μ(f −1 (Cyn ))

= et ∫A (e

iu⋅fn (x)

−1)μ(dx)

.

3.3 Random measures and Lévy–Itô decomposition

|

185

We have ∫A fn (x)N(t, dx) → ∫A f (x)N(t, dx) as n → ∞ by (3.3.10). By the dominated convergence theorem, (1) holds. On replacing f by sf with s ∈ ℝ in (1) and differentiating both sides above at s = 0, we obtain (2) and (3). We can extend Theorem 3.54 to Poisson random measures. Let (S, S ) be a measurable space and M = (M(A))A∈S a Poisson random measure with intensity λ(⋅). Similarly to Theorem 3.54, we have the following result. Theorem 3.55. Let A ∈ S . (1) Suppose that f : ℝd → ℝd is Borel measurable and u ∈ ℝd . Then 𝔼 [ei ∫A u⋅f (x)M(dx) ] = et ∫A (e

iu⋅f (x)

−1)λ(dx)

.

(2) If f ∈ L1 (A, λ), then 𝔼[∫A f (x)M(dx)] = t ∫A f (x)λ(dx). (3) If f ∈ L2 (A, λ), then 𝔼[| ∫A f (x)M(dx)|2 ] = ∫A |f (x)|2 λ(dx). ̃ A). A similar result holds also for the compensated Poisson random measure N(t, Let f ∈ L1 (A, μ) and define ̃ dx) = ∫ f (x)N(t, dx) − t ∫ f (x)μ(dx). ∫ f (x)N(t, A

A

(3.3.11)

A

Theorem 3.56. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) and N(t, ⋅) the Poisson random measure associated with (Xt )t≥0 . Let A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A,̄ and f ∈ L1 (A, μ). Then we have the following: if (x) ̃ (1) 𝔼[ei ∫A f (x)N(t,dx) ] = et ∫A (e −1−if (x))μ(dx) . (2) 𝔼[∫A f (x)N(t, dx)] = 0. (3) In addition, suppose that f ∈ L2 (A, μ), g ∈ L2 (B, μ), and 0 ∉ B̄ ∈ ℬ(ℝd \ {0}). Then ̃ dx) ∫ g(x)N(t, ̃ dx)] = t ∫ f (x)g(x)μ(dx). 𝔼 [∫ f (x)N(t, B A∩B [A ]

(3.3.12)

̃ dx) and ∫ g(x)N(t, ̃ dx) are indepenIn particular, the random variables ∫A f (x)N(t, B dent whenever A ∩ B = 0. ̃ dx)) is a martingale with respect to the natural (4) The random process (∫A f (x)N(t, t≥0 filtration (Ft )t≥0 of (Xt )t≥0 . Proof. The proofs of (1)–(3) are similar to those of Theorem 3.54. Let f (x) = 1B (x) for B ∈ ℬ(ℝd \ {0}). We have ̃ dx) = N(t, A ∩ B) − tμ(A ∩ B). Yt = ∫ f (x)N(t, A

Since the increments of (N(t, A ∩ B))t≥0 are independent and we have 𝔼[N(t, A ∩ B)] = tμ(A ∩ B), it is direct that 𝔼[Yt |Fs ] = Ys . We obtain (4) by a limiting argument.

186 | 3 Lévy processes We have seen above that (N(t, A))t≥0 is a Poisson process for all A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A.̄ Now we will see also that (∫A f (x)N(t, dx))t≥0 is a compound Poisson process. Proposition 3.57 (Compound Poisson process). Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) and N(t, ⋅) the Poisson random measure associated with (Xt )t≥0 . Let f : ℝd → ℝ be measurable, and A ∈ ℬ(ℝd \ {0}) such that 0 ∉ A.̄ Define τn and Yn by τn = τnA = inf{t > 0 | ΔXt+τn−1 ∈ A} and Yn = ∫ f (x)N(τn , dx) − ∫ f (x)N(τn−1 , dx). A

A

Then we have the following: (1) (ΔXτn )n∈ℕ and {τn }n∈ℕ are independent; (2) Yn , n ∈ ℕ, are independent; (3) the random process (∫A f (x)N(t, dx))t≥0 is a compound Poisson process, in particular, a Lévy process. 3.3.2 Lévy–Itô decomposition Lévy processes are a class large enough to contain random processes with jumps. A natural question is if there is a canonical way of separating the continuous part from the part with discontinuous paths. As we will see now, any Lévy process can be decomposed into a sum of four independent random processes accounting for these components. To obtain this for a Lévy process (Xt )t≥0 , we define the total sum of jumps pathwise. For A ∈ ℬ(ℝd \ {0}), t

∫ ∫ zN(dsdz) = ∫ zN(tdz) = ∑ ΔXu 1A (ΔXu ) 0 A

0≤u≤t

A

is the sum of all the jumps taking values in set A up to time t. Since the paths of (Xt )t≥0 are càdlàg, this is a finite sum of random variables. ̃ dz))t≥0 and (∫ z N(t, ̃ dz))t≥0 are indepenNote that the random processes (∫A z N(t, B ̃ dz))t≥0 and (∫ zN(t, dz))t≥0 are independent dent as soon as A ∩ B = 0, and (∫A z N(t, B whenever A ∩ B = 0 and ∫|z|≤1 |z|μ(dz) < ∞. Lemma 3.54 allows to compute the characteristic functions of Zt(1) = ∫ zN(t, dz)

and

Zt(2) (ε) =

|z|≥1

giving 𝔼[eiuZt ] = etη (1)

(1)

(u)

and 𝔼[eiuZt

(2)

(ε)

] = etη

(2)

(u,ε)

η(1) (u) = ∫ (eiu⋅y − 1)μ(dy), |λ|≥1

̃ dz), ∫ z N(t, ε 0 ∫ ℜ( Ba (0)

1 ) du < ∞, η(u)

(3.5.9)

where η(u) is the Lévy symbol of (Xt )t≥0 . Example 3.85. Using the characterization given by the Chung–Fuchs criterion we have the following situations in some specific cases. (1) Since by using the Lévy–Khintchine representation, |η(u)| ≤ u ⋅ Au follows for all u ∈ ℝd , where A is the diffusion matrix in the Lévy triplet, every Lévy process with non-zero A is transient if d ≥ 3. (2) Brownian motion is recurrent if d ≤ 2 and transient if d ≥ 3. (3) A rotationally symmetric non-Gaussian α-stable process in d = 1 is recurrent if 1 ≤ α < 2 and transient if 0 < α < 1, and if d = 2 it is transient for all 0 < α < 2. Example 3.86. Let (Xt )t≥0 be a d-dimensional rotationally symmetric α-stable process, α ∈ (0, 2). We will investigate Riesz potentials G0α in Section 4.9.4, and use them here to characterize recurrence properties in terms of the related Green functions. (1) Let α < d. Then the process (Xt )t≥0 is transient with Riesz potential G0α (y

∞

− x) = ∫ pXt (y − x)dt = 𝒜d,α 0

1 , |y − x|d−α

x, y ∈ ℝd ,

where 𝒜d,α =

Γ((d − α)/2) . 2α π d/2 Γ(α/2)

(2) Let α ≥ d. Then the process is recurrent (pointwise recurrent when α > d = 1). In this case we can consider the compensated kernel, i. e., for α ≥ d we put ∞

G̃ 0α (y − x) = ∫ (pXt (y − x) − pXt (x0 )) dt, 0

where x0 = 0 for α > d = 1 and x0 = 1 for α = d = 1. In this case G̃ 0α (x) = {

− π1 log |x|

1 1 2Γ(α) cos(πα/2) |x|1−α

if α = d = 1, if α > d = 1.

206 | 3 Lévy processes It is known that in one-dimension the occupation time of Brownian motion is absolutely continuous with respect to Lebesgue measure and its density is the random process called Brownian local time (Lxt )t≥0 , i. e., we have a

y

Ut (a) = ∫ Lt dy. −a

This makes it possible to define the local time process of Brownian motion as the mesure de voisinage, introduced by P. Lévy and given by Lxt

t

1 = lim ∫ 1{Bs ∈(x−ε,x+ε)} ds, ε↓0 2ε

(3.5.10)

0

which then measures the amount of time over the interval [0, t] spent by Brownian motion near the point x and also gives an intuitively clear definition of the local time process. Local times can be defined also to more general Lévy processes but we will not discuss them here.

3.6 Subordinators and Bernstein functions 3.6.1 Subordinators and subordinate Brownian motion We conclude this chapter by discussing a special class of Lévy processes. Definition 3.87 (Subordinator). A one-dimensional Lévy process (Tt )t≥0 is called a subordinator whenever s ≤ t implies Ts ≤ Tt almost surely. d

Since Tt+s − Ts = Tt , for all s, t ≥ 0, the definition readily implies that Tt ≥ 0 almost surely, for every t > 0. A subordinator can be regarded as a random time since it is non-decreasing and non-negative. Theorem 3.88 (Characterization of subordinators). The Lévy symbol of a subordinator is ∞

η(u) = ibu + ∫ (eiuy − 1)λ(dy),

(3.6.1)

0

where b ≥ 0 and the Lévy measure satisfies λ((−∞, 0)) = 0

and

∞

∫ (y ∧ 1)λ(dy) < ∞.

(3.6.2)

0

Conversely, for every η(u) given by (3.6.1), there exists a subordinator (Tt )t≥0 whose Lévy symbol is of the form η(u).

3.6 Subordinators and Bernstein functions |

207

Proof. Suppose that (Tt )t≥0 is a subordinator with Lévy triplet (b, A, λ). Then (Tt )t≥0 has no negative jumps and is positive, thus λ((−∞, 0)) = 0 and A = 0 by the Lévy–Itô decomposition. Hence Tt = tb + lim ε↓0

xN(dsdx).

∫ (0,t]×(ε,∞)

Let Tt,ε = ∫(0,t]×(ε,∞) xN(dsdx). We see that Yt = limε↓0 Tt,ε = ∫(0,t]×(0,∞) xN(dsdx) exists in L2 sense and is bounded by Tt − bt. We have 𝔼[e−uTt,ε ] = exp [t ∫ (e−ux − 1)λ(dx)] [ (ε,∞) ] = exp [t ∫ (e−ux − 1 + ux1{|x|≤1} (x))λ(dx) − tu ∫ xλ(dx)] . [ (ε,∞) ] (ε,1]

(3.6.3)

As ε ↓ 0 we have 𝔼[e−uTt,ε ] → 𝔼[e−uYt ] > 0, and ∫(ε,∞) (e−ux − 1 − ux1{|x|≤1} (x))λ(dx)

tends to the integral over (0, ∞), which is finite. Hence ∫(0,1] xλ(dx) < ∞ by (3.6.3). Thus b ≥ 0, and (3.6.2)–(3.6.1) follow. Conversely, assume that (3.6.1)–(3.6.2) hold. Let (Tt )t≥0 be a Lévy process with Lévy symbol (3.6.1). We see that N((0, t] × (−∞, 0)) = tλ((−∞, 0)) = 0 almost surely, i. e., (Xt )t≥0 has no negative jumps. Hence by the Lévy–Itô decomposition we have Tt = tb +

∫

xN(dsdx).

(0,t]×(0,∞)

This shows that (Xt )t≥0 is increasing. By Theorem 3.88 it is seen that the map u 󳨃→ 𝔼[eiuTt ] = etη(u) can be analytically continued into the region {iu ∈ ℂ : u > 0}. This implies that the Laplace transform of Tt is well defined and given by 𝔼[e−uTt ] = e−tΨ(u) ,

(3.6.4)

where ∞

Ψ(u) = bu + ∫ (1 − e−uy )λ(dy)

(3.6.5)

0

for every u > 0. Example 3.89 (α/2-stable subordinator). Let b = 0, 0 < α < 2 and using (3.1.24) take ν(dx) =

1(0,∞) (x) α dx, 2Γ(1 − α/2) x 1+α/2

(3.6.6)

208 | 3 Lévy processes where Γ denotes the Gamma function. This is called α2 -stable subordinator and α/2

𝔼0 [e−uTt ] = e−tu holds.

Example 3.90 (Inverse Gaussian subordinator). Let (Ω, F , P) be a probability space, m ≥ 0 and δ > 0 given constants, and consider the first hitting time of one-dimensional standard Brownian motion with constant drift m Tt (m, δ) = inf{s > 0 | Bs + ms = δt}.

(3.6.7)

The process (Tt (m, δ))t≥0 is called inverse Gaussian subordinator for m > 0, and 𝔼[e−uTt (m,δ) ] = exp (−tδ (√2u + m2 − m))

(3.6.8)

holds. The distribution of Tt (m, δ) can be explicitly determined to be ρ(s) =

δt δmt 1 1 1 e exp (− (t 2 δ2 + m2 s)) . 3/2 √2π 2 s s 1

2

To see this, recall the exponential martingale of Brownian motion (eαBt − 2 α t )t≥0 , see Corollary 2.113. Consider the stopped process, (e tingale. Thus 1

αBTt ∧n − 21 α2 Tt ∧n

2

𝔼[eαBTt ∧n − 2 α Tt ∧n ] = 𝔼[eαB0 ] = 1,

)t≥0 , which is also a mar-

t ≥ 0, n ∈ ℕ.

Let An = {Tt ≤ n} and 1

1

2

1

2

2

𝔼[eαBTt ∧n − 2 α Tt ∧n ] = 𝔼[eαBTt ∧n − 2 α Tt ∧n 1An ] + 𝔼[eαBTt ∧n − 2 α Tt ∧n 1Acn ]. It is immediate to see that 1

1

2

2

𝔼 [eαBTt ∧n − 2 α Tt ∧n 1Acn ] ≤ e− 2 α n 𝔼 [1Acn eαBn ] 1

2

and since Bn < δt − mn on Acn , it follows that limn→∞ 𝔼[eαBTt ∧n − 2 α Tt ∧n 1Acn ] = 0. Furthermore, 1

1

2

2

lim 𝔼 [eαBTt ∧n − 2 α Tt ∧n 1An ] = 𝔼 [eα(δt−mTt ) e− 2 α Tt ] .

n→∞

1

2

This implies that 1 = 𝔼[eα(δt−mTt ) e− 2 α Tt ], and hence 1

e−αδt = 𝔼 [e− 2 α(α+2m)Tt ] follows. Setting α = √2u + m2 − m proves the claim.

3.6 Subordinators and Bernstein functions |

209

Definition 3.91 (Subordinate Brownian motion). Let (Bt )t≥0 be Brownian motion and (TtΨ )t≥0 a subordinator with Laplace exponent Ψ given by (3.6.4). The random process defined by (XtΨ )t≥0 = (BT Ψ )t≥0

(3.6.9)

t

is called subordinate Brownian motion with subordinator (TtΨ )t≥0 . Let (XtΨ )t≥0 be a subordinate Brownian motion as defined above. It follows that 𝔼[e

iξ ⋅XtΨ

∞

2

] = ∫ e−|ξ | y/2 pΨ t (y)dy, 0

Ψ where pΨ t denotes the distribution of Tt on [0, ∞). A subordinate Brownian motion is by definition the composition of Brownian motion and a subordinator, and it is a Lévy process. Hence the generator of (XtΨ )t≥0 is given by −Ψ(− 21 Δ). For instance, when Ψ(u) = (2u)α/2 , 0 < α < 2, we obtain that (XtΨ )t≥0 is a rotationally symmetric α-stable process with generator given by the fractional Laplacian −(−Δ)α/2 .

3.6.2 Bernstein functions Subordinators have a deep relationship with a specific class of functions. We will make use of this in the path integral representations of operators semigroups below. Definition 3.92 (Bernstein function). Let f ∈ C ∞ ((0, ∞)) with f ≥ 0. A function f is dn f called a Bernstein function whenever (−1)n dx n (x) ≤ 0, for all n ∈ ℕ. We denote the set of Bernstein functions by ℬ. It is possible to characterize Bernstein functions through completely monotone functions. Definition 3.93 (Completely monotone function). Let f ∈ C ∞ ((0, ∞)). The function f dn f ̇ is called a completely monotone function whenever (−1)n dx n (x) ≥ 0, for all n ∈ ℕ. By the definition, any completely monotone function f is positive and not increasing, and hence limx→∞ f (x) ≥ 0 exists. Proposition 3.94 (Characterization of completely monotone functions). Let f be a completely monotone function. Then there exists a unique measure μ on [0, ∞) such that f (u) = ∫ e−uy μ(dy).

(3.6.10)

[0,∞)

Conversely, whenever the right-hand side of (3.6.10) is finite, f (u) is a completely monotone function.

210 | 3 Lévy processes Proof. Suppose that f is completely monotone. We prove (3.6.10) in four steps. Step 1: Suppose that f (0+) = 1 and f (∞) = 0. By Taylor expansion we have a

n−1

(−1)k f (k) (a) (−1)n f (n) (s) f (u) = ∑ (a − u)k + ∫ (s − u)n−1 ds. k! (n − 1)! k=0

(3.6.11)

u

Let a → ∞. Since the first term at the right-hand side of (3.6.11) is positive, we have a

lim ∫

a→∞

u

(−1)n f (n) (s) (−1)n f (n) (s) (s − u)n−1 ds = ∫ (s − u)n−1 ds ≤ f (u). (n − 1)! (n − 1)! ∞

u

Hence (3.6.11) converges for every n as a → ∞. Thus every term converges to a nonnegative limit as a → ∞. For n ≥ 0, let (−1)n f (n) (a) (a − u)n . a→∞ n!

ρn (u) = lim

We note that this limit is independent of u > 0. Let cn = ∑n−1 k=0 ρk (u). We have f (u) = cn + ∫ [u,∞)

(−1)n f (n) (s) (s − u)n−1 ds. (n − 1)!

Clearly, f (u) ≥ cn ≥ 0. Since f (∞) = 0, it follows that cn = 0 for every n. Thus we have f (u) = ∫ [u,∞)

(−1)n f (n) (s) (s − u)n−1 ds. (n − 1)!

(3.6.12)

By the monotone convergence theorem we obtain 1 = lim f (u) = ∫ u→0

[0,∞)

(−1)n f (n) (s) n−1 s ds. (n − 1)!

(3.6.13)

ut n−1 ) f (t)dt, n + n

(3.6.14)

Substitution in (3.6.12) gives f (u) = ∫ (1 − [0,∞)

where fn (s) = {

(−1)n (n) f (n/s)(n/s)n+1 n!

0

s>0 s = 0.

Note that ∫[0,∞) fn (t)dt = 1 by (3.6.13). By Helly’s second theorem there exist a subsequence nk and a measure μ on (0, ∞) such that fnk (t)dt vaguely converges to μ(dt).

3.6 Subordinators and Bernstein functions |

211

Since fnk (t)dt, k ∈ ℕ, are probability measures, μ is a finite Radon measure. Let n−1

μ((0, ∞)) = m. We reset nk by n. Note that (1 − utn )+ → e−ut as n → ∞ uniformly in t ∈ [0, ∞). For every ε > 0 there exist K0 > 0 and n0 ∈ ℕ such that μ((K, ∞)) < ε 󵄨󵄨 󵄨󵄨 n−1 for every K > K0 , and supt>0 󵄨󵄨󵄨(1 − utn )+ − e−ut 󵄨󵄨󵄨 < ϵ for every n > n0 . Let K > K0 and 󵄨 󵄨 n > n0 . We have ∫ (1 − [0,∞)

ut n−1 ) f (t)dt − ∫ e−ut μ(dt) n + n [0,∞)

ut n−1 ut n−1 = ∫ (1 − ) fn (t)dt − ∫ e−ut μ(dt) + ∫ (1 − ) fn (t)dt − ∫ e−ut μ(dt). n + n + [0,K]

[0,K]

(K,∞)

(K,∞)

We estimate as 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ut n−1 󵄨󵄨 󵄨 −ut 󵄨󵄨 ∫ (1 − ) μ(dt) − ∫ e μ(dt)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n + 󵄨󵄨[0,K] 󵄨󵄨 [0,K] 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ut n−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −ut −ut −ut ≤ 󵄨󵄨󵄨 ∫ (1 − ) fn (t)dt − ∫ e fn (t)dt 󵄨󵄨󵄨 + 󵄨󵄨󵄨 ∫ e fn (t)dt − ∫ e μ(dt)󵄨󵄨󵄨󵄨 n + 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨[0,K] 󵄨󵄨 [0,K] [0,K] 󵄨󵄨[0,K] 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 ≤ ϵ ∫ fn (t)dt + 󵄨󵄨󵄨󵄨 ∫ e−ut fn (t)dt − ∫ e−ut μ(dt)󵄨󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨󵄨[0,K] 󵄨󵄨 [0,K] [0,K] On the other hand, we see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ut n−1 󵄨󵄨 󵄨 󵄨󵄨 ∫ (1 − ) μ(dt) − ∫ e−ut μ(dt)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n + 󵄨󵄨(K,∞) 󵄨󵄨 (K,∞) 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ut n−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −ut −ut −ut ≤ 󵄨󵄨󵄨 ∫ (1 − ) fn (t)dt − ∫ e fn (t)dt 󵄨󵄨󵄨 + 󵄨󵄨󵄨 ∫ e fn (t)dt − ∫ e μ(dt)󵄨󵄨󵄨󵄨 n + 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨(K,∞) 󵄨󵄨 (K,∞) (K,∞) 󵄨󵄨(K,∞) ≤ ε + e−uK (1 + m).

Since fn (t)dt vaguely converges to μ(dt), we have 󵄨󵄨 ∞ 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 ut n−1 󵄨󵄨 󵄨 −ut lim 󵄨󵄨 ∫ (1 − ) fn (t)dt − ∫ e μ(dt)󵄨󵄨󵄨 n→∞ 󵄨󵄨 󵄨󵄨 n + 󵄨󵄨 0 󵄨󵄨 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −uK −ut −ut ≤ 2ε + e (1 + m) + lim 󵄨󵄨󵄨 ∫ e fn (t)dt − ∫ e μ(dt)󵄨󵄨󵄨󵄨 = 2ε + e−uK (1 + m). n→∞ 󵄨 󵄨󵄨 󵄨󵄨[0,K] 󵄨󵄨 [0,K] 󵄨

212 | 3 Lévy processes Here we can take an arbitrary large K. Taking the limit of (3.6.14) along a subsequence nk , it follows that f (u) = ∫ e−ut μ(dt).

(3.6.15)

[0,∞)

Set u = 0. We have 1 = f (0) = μ([0, ∞)). Then μ is a probability measure. Uniqueness is further proven by the expression ∑ (−1)k

0≤k≤ux

uk dk ∫ e−ut μ(dt) = ∫ k! duk

∑

[0,∞) 0≤k≤ux

[0,∞)

(ut)k −ut e μ(dt). k!

From this we have lim

u→∞

∑ (−1)k

0≤k≤ux

uk d k ∫ e−ut μ(dt) = μ([0, ∞)). k! duk [0,∞)

This shows that μ can be recovered from its Laplace transform, and thus uniqueness follows. Step 2: Suppose that f (0+) < ∞ and f (∞) = 0. By looking at g(u) = f (u)/f (0+), we have g(u) = ∫[0,∞) e−ut μ(dt) with a probability measure μ. We see that f is represented by the measure f (0+)μ, i. e., f (u) = ∫ e−ut f (0+)μ(dt). [0,∞)

Step 3: Suppose that f (∞) = 0. Let fa (u) = f (u + a). The function fa is also completely monotone and fa (0+) = f (a) < ∞ and fa (∞) = 0. Thus there exists a unique finite measure μa on [0, ∞) such that fa (u) = ∫[0,∞) e−ut μa (dt). Since eat μa (dt) = ebt μb (dt), we can consistently define μ(dt) = eat μa (dt), for all a > 0. We have

f (u) = fu/2 (u/2) = ∫ e(−u/2)t μu/2 (dt) = ∫ e−ut e(u/2)t μu/2 (dt) = ∫ e−ut μ(dt). [0,∞)

[0,∞)

[0,∞)

Step 4: In this step we prove (3.6.10) for arbitrary completely monotone functions. Suppose that f (∞) = c > 0. Let g(u) = f (u) − c. The function g is completely monotone and g(∞) = 0. By Step 3 we have the respresentation g(u) = ∫[0,∞) e−ut μ(dt) with a finite measure μ. Thus f (u) = ∫ e−ut μ(dt) + c ∫ e−ut dδ0 , [0,∞)

[0,∞)

where δ0 is the Dirac measure with mass at 0. Thus μ + cδ0 represents f .

3.6 Subordinators and Bernstein functions |

213

Next we prove the converse. Write f (u) = ∫[0,∞) e−ut μ(dt). Fix u > 0 and pick ε ∈

(0, u). Since t n ≤ n!ε−n eεt for all t > 0, we find ∫ t n e−ut μ(dt) ≤ [0,∞)

n! n! ∫ t n e−(u−ε)t μ(dt) = n f (u − ε) εn ε [0,∞)

and this shows that we may use the differentiation lemma for parameter-dependent integrals to get (−1)n f (n) (u) = (−1)n ∫ t n [0,∞)

dn −ut e μ(dt) ≥ 0. dun

It follows from the definition that Bernstein functions are positive, increasing, and concave. Define a subset of ℬ by 󵄨󵄨 󵄨

ℬ0 = {Ψ ∈ ℬ 󵄨󵄨󵄨 lim Ψ(u) = 0} . u→0+

(3.6.16)

Examples of functions in ℬ0 include Ψ(u) = cuα/2 with c > 0 and 0 < α ≤ 2, and Ψ(u) = 1 − e−au with a ≥ 0. Furthermore, it can be seen that sums ∑nj=1 (1 − e−aj u ) of Bernstein functions are also Bernstein functions, and uα can be expressed as uα = ∞ ∫0 (1 − e−yu )ν(dy) by using the measure ν in (3.6.6). The right-hand side is derived from the analytic continuation of the Lévy symbol of subordinator (3.6.1). We will see next that ℬ0 can be characterized by subordinators. We define the following subset of Lévy measures. Definition 3.95 (Lévy measures and Bernstein functions). Let L be the set of Borel ∞ measures λ on ℝ \ {0} such that λ((−∞, 0)) = 0 and ∫0 (y ∧ 1)λ(dy) < ∞. Note that λ ∈ L satisfies ∫0 (y2 ∧1)λ(dy) < ∞, thus it is a Lévy measure. The important fact connecting L and ℬ0 is given by the next proposition. ∞

Proposition 3.96 (Characterization of Bernstein functions). For every Ψ ∈ ℬ0 there exists (b, λ) ∈ ℝ+ × L such that Ψ(u) = bu + ∫ (1 − e−uy )λ(dy).

(3.6.17)

(0,∞)

Conversely, the right-hand side of (3.6.17) is in ℬ0 for every (b, λ) ∈ ℝ+ × L . Proof. Let Ψ be a Bernstein function. Then Ψ󸀠 is a completely monotone function. According to Proposition 3.94 there exists a measure μ on [0, ∞) such that Ψ󸀠 (u) = ∫ e−uy μ(dy). [0,∞)

214 | 3 Lévy processes Let λ({0}) = b. We have u

u

Ψ(u) − Ψ(ε) = ∫ Ψ󸀠 (s)ds = (u − ε)b + ∫ ds ∫ e−ys μ(dy) ε

ε

(0,∞)

and 1 Ψ(u) − Ψ(0+) = ub + ∫ ds ∫ e−ys μ(dy) = ub + ∫ (1 − e−ys ) μ(dy). y (0,u)

(0,∞)

(0,∞)

Set a = Ψ(0+) and λ = y1 μ(dy)⌈(0,∞) . Then (3.6.17) follows. Furthermore, since we have (1 − e−1 )(1 ∧ t) ≤ 1 − e−t for t ≥ 0, we infer ∫ (1 ∧ y)λ(dy) ≤ (0,∞)

e e f (1) < ∞. ∫ (1 − e−y )λ(dy) = e−1 e−1 (0,∞)

Conversely, suppose that Ψ is given by (3.6.17). Since ye−uy ≤ y ∧ (eu)−1 and y ∧ (eu)−1 is integrable with respect to λ(dy), Ψ󸀠 (u) = ∫[0,∞) e−uy ν(dy), where ν(dy) = yλ(dy) + bδ0 (dy). This shows that Ψ󸀠 is completely monotone.

In the examples below we give some important cases of Bernstein functions, from which one can read off the basic parameters directly. Example 3.97. (1) Let Ψ(u) = uα/2 for 0 < α < 2. Then uα/2 =

α 1 ∫ (1 − e−uy ) 1+α/2 dy. 2Γ(1 − α/2) y (0,∞)

(2) Let Ψ(u) = (u + m2/α )α/2 − m for 0 < α < 2. Then (u + m2/α )α/2 − m =

2/α

α e−m y ∫ (1 − e−uy ) 1+α/2 dy. 2Γ(1 − α/2) y (0,∞)

(3) Let Ψ(u) = log(1 + u). Then log(1 + u) = ∫ (1 − e−uy ) (0,∞)

e−y dy. y

Due to Proposition 3.96, for Ψ ∈ ℬ0 there exists (b, λ) ∈ (ℝ+ , L ) such that (3.6.17) is satisfied. Proposition 3.98. The map ℬ0 → (ℝ+ , L ), Ψ 󳨃→ (b, λ), is a bijection.

3.6 Subordinators and Bernstein functions |

215

Proof. By Proposition 3.96 it is clear that the map is surjective. For a given Ψ ∈ ℬ0 the constant b is uniquely determined by b = lim

u→∞

Ψ(u) . u

Since f = dΨ/du is a completely monotone function, there exists a unique measure μ ∞ on [0, ∞) such that f (u) = ∫0 e−us dμ(s), for u > 0. On the other hand, we have ∞

∞

0

0

dΨ ̃ = b + ∫ e−uy yλ(dy) = ∫ e−uy λ(dy), du ̃ where λ(dy) = bδ0 + yλ(dy). Thus bδ0 + yλ(dy) = μ(dy). Since b is unique, the measure λ ∈ L for the Bernstein function Ψ is also uniquely determined. Denote the set of subordinators by S . By the one-to-one correspondence between ℬ0 and (ℝ+ , L ) we also have a bijection between S and ℬ0 . Proposition 3.99. If Ψ ∈ ℬ0 , then there exists a unique subordinator (Tt )t≥0 ∈ S such that 𝔼[e−uTt ] = e−tΨ(u) .

(3.6.18)

Conversely, if (Tt )t≥0 ∈ S , then there exists Ψ ∈ ℬ0 such that (3.6.18) holds. Proof. Let Ψ ∈ ℬ0 be given. There exists (b, λ) ∈ ℝ+ × L such that (3.6.17) holds. This means that there exists a subordinator (Tt )t≥0 with Lévy triplet (b, 0, λ). Moreover, (3.6.18) follows by analytic continuation of 𝔼[eiuTt ] = etη(u) with exponent η(u) = ibu + ∞ ∫0 (eiuy − 1)λ(dy). Conversely, let (Tt )t≥0 ∈ S be given. By (b, λ) ∈ ℝ+ × L associated with (Tt )t≥0 we define the Bernstein function by (3.6.17), and the result follows. By Propositions 3.6.17–3.6.18 it can be seen that there is a pairwise bijective relation among subordinators S , Bernstein functions ℬ0 , and (ℝ+ , L ). This will be useful in constructing path integral representations of semigroups generated by Bernstein functions of the Laplacian, discussed in the next chapter.

4 Feynman–Kac formulae 4.1 Schrödinger semigroups 4.1.1 Schrödinger equation and path integral solutions As we have explained in Chapter 1, a basic reason behind the interest in Feynman–Kac formulae is the analysis of the solutions of the Schrödinger equation

1 i𝜕t φ = − Δφ + Vφ (4.1.1) 2 with potential V, or its imaginary-time counterpart, the heat equation with spatially nonhomogeneous dissipation

1 −𝜕t φ = − Δφ + Vφ. (4.1.2) 2 In fact, the scope of Feynman–Kac formulae in the context of partial differential equations goes far beyond these two cases; however, in this book we have no space to discuss the PDE aspect separately. Let f : ℝ × ℝd → ℝ be a solution of the heat equation, i. e.,

Recall the formula

1 𝜕t f (t, x) = Δf (t, x), 2

f (0, x) = g(x),

(4.1.3)

f (t, x) = ∫ Πt (x − y)g(y) dy.

(4.1.4)

ℝd

known from the theory of partial differential equations. A comparison with 𝔼x [g(Bt )] = ∫ Πt (x − y)g(y) dy

(4.1.5)

ℝd

shows that, as we have said above, the solution of (4.1.3) can be obtained by running a Brownian motion, and f (t, x) = 𝔼x [g(Bt )]

(4.1.6)

is obtained. Using operator semigroup notation, (4.1.3) implies f = e(t/2)Δ g. In what follows we ask whether a similar representation is possible when − 21 Δ is replaced by − 21 Δ + V, with suitable V. Let now f : ℝ × ℝd → ℝ be a solution of 1 𝜕t f (t, x) = Δf (t, x) − V(x)f (t, x). 2

(4.1.7) 1

In semigroup notation, this problem translates then to studying f = e−t(− 2 Δ+V) g. In what follows, instead of equations in our considerations the basic input will be linear operators and one-parameter semigroups generated by them as above, although we will remember the deep relations with partial differential equations. https://doi.org/10.1515/9783110330397-004

218 | 4 Feynman–Kac formulae 4.1.2 Linear operators and their spectra In what follows we will be interested in various spectral properties of sums of selfadjoint operators. In the present section our main object will be Schrödinger operators 1 H =− Δ+V 2

(4.1.8)

defined on a suitable space as a sum of the Laplacian and the multiplication operator V viewed in a specific sense as a perturbation of the Laplacian. As we progress to take magnetic fields and spin into account, further operator terms will appear, and when we discuss relativistic quantum mechanics, the Laplacian will be replaced by its square root. In Volume 2 of the book we then turn to quantum field theory, where further types of operators will be added to Schrödinger operators. We start by briefly reviewing key concepts and facts of general linear operator theory used throughout this book, and next we will focus first on Laplacians and then on their perturbations. Let 𝒦 be a Hilbert space over the complex field ℂ with inner product (⋅, ⋅) and norm ‖ ⋅ ‖. When inner products or norms of several distinct Hilbert spaces are compared, we indicate the specific space in the subscript whenever necessary; for Lp spaces we use the standard notation ‖ ⋅ ‖p . Let ℋ be also a Hilbert space. The domain of a linear operator A : 𝒦 → ℋ will be denoted by D(A), its range by Ran(A), and its kernel by Ker(A). Recall that for two linear operators A and B we have A = B if and only if D(A) = D(B) and Aφ = Bφ for all φ ∈ D(A), and A ⊂ B if and only if D(A) ⊂ D(B) and Aφ = Bφ for all φ ∈ D(A); in the latter case B is said to be an extension of A, or from the opposite point of view, A is the restriction of B to D(A). An operator A on 𝒦 is densely defined if D(A) is a dense subset of 𝒦. Definition 4.1 (Bounded/unbounded operators). Let A : 𝒦 → ℋ be a linear operator. Whenever there exists c ≥ 0 such that ‖Aφ‖ℋ ≤ c‖φ‖𝒦 for all φ ∈ D(A), the operator A is called bounded, and otherwise it is called unbounded. A linear operator A is called strongly continuous at φ ∈ D(A) when there is a sequence (φn )n∈ℕ ⊂ D(A) such that φn → φ in strong convergence sense implies Aφn → Aφ also in strong convergence sense. If A is continuous for all φ ∈ D(A), A is called strongly continuous. A linear operator is bounded if and only if it is strongly continuous. The norm of a bounded operator A is defined as ‖A‖ = ‖A‖𝒦,ℋ =

‖Aφ‖ℋ . φ∈D(A),φ=0 ̸ ‖φ‖𝒦 sup

(4.1.9)

A bounded operator is a contraction if ‖A‖ ≤ 1. When 𝒦 is finite-dimensional, any linear operator on it is bounded. Next we give the notions of bounded operator topology we will often use.

4.1 Schrödinger semigroups | 219

Definition 4.2 (Topologies of bounded operators). Let (An )n∈ℕ be a sequence of bounded operators from 𝒦 to ℋ such that D(An ) = D, n ∈ ℕ, and A another bounded operator with D(A) = D. Then (An )n∈ℕ is (1) weakly convergent to A, denoted A = w − limn→∞ An , if limn→∞ (ψ, An φ − Aφ) = 0, for all φ ∈ D and for all ψ ∈ ℋ; (2) strongly convergent to A, denoted A = s-limn→∞ An , if limn→∞ ‖An φ − Aφ‖ℋ = 0, for all φ ∈ D; (3) uniformly convergent to A if limn→∞ ‖An − A‖ = 0. If A is a densely defined bounded operator with ‖Aφ‖ℋ ≤ c‖φ‖𝒦 , then it can be uniquely extended to an operator B such that A ⊂ B, D(B) = 𝒦 and ‖Bφ‖ℋ ≤ c‖φ‖𝒦 hold. In what follows we assume that the domain of a bounded operator is the whole space. Recall that the graph of a linear operator A : 𝒦 → ℋ is the subset G(A) = {(φ, Aφ) | φ ∈ D(A)} ⊂ 𝒦 ⊕ ℋ.

(4.1.10)

Definition 4.3 (Closed/closable operator). Let A : 𝒦 → ℋ be a linear operator. (1) A is a closed operator if G(A) is closed in 𝒦 ⊕ ℋ, in other words, if the sequence (φn )n∈ℕ satisfies Aφn → ϕ and φn → φ as n → ∞, then φ ∈ D(A) and moreover ϕ = Aφ holds. (2) A is a closable operator if there exists a closed operator B : 𝒦 → ℋ such that A ⊂ B. (3) The closure of A, denoted by A,̄ is the smallest closed extension of A, i. e., Ā is a closed extension of A and if B is another closed extension, then Ā ⊂ B. (4) A subspace 𝒟 ⊂ D(A) is a core of A if A⌈𝒟 = A, in other words, if for every φ ∈ D(A) there is a sequence (φn )n∈ℕ ⊂ 𝒟 such that φn → φ and Aφn → Aφ strongly as n → ∞. The bounded operators are closed. The next proposition ensures the existence of the closure of closable operators. Proposition 4.4. Let A be closable. Then the closure of A exists and G(A) = G(A)̄ holds. A linear operator A is injective if and only if Aφ = 0 implies φ = 0. In this case the inverse A−1 of A is defined by D(A−1 ) = Ran(A),

A−1 φ = ϕ

(4.1.11)

for Aϕ = φ. From this definition it follows that for φ ∈ Ran(A) and ϕ ∈ D(A), AA−1 φ = φ,

A−1 Aϕ = ϕ.

(4.1.12)

Definition 4.5 (Invertibility). A linear operator A is called invertible if it is injective and has a bounded inverse on Ran A.

220 | 4 Feynman–Kac formulae Note that invertibility of an operator and the existence of an inverse are properties that do not necessarily coincide. Definition 4.6 (Addition and multiplication of operators). The sum A + B of two linear operators A and B is defined by D(A + B) = D(A) ∩ D(B),

(A + B)φ = Aφ + Bφ.

(4.1.13)

The product AB is defined by D(AB) = {φ ∈ D(B) | Bφ ∈ D(A)},

(AB)φ = A(Bφ).

(4.1.14)

Note that in general it may happen that D(A) ∩ D(B) is not a dense subset in D(A) or D(B). Let B(𝒦, ℋ) denote the set of bounded operators from 𝒦 to ℋ with the whole space 𝒦 as their domain. For A, B ∈ B(𝒦, ℋ), A + B ∈ B(𝒦, ℋ) and ‖A + B‖ ≤ ‖A‖ + ‖B‖ hold, and for B ∈ B(𝒦, ℋ) and A ∈ B(ℋ, ℒ), AB ∈ B(𝒦, ℒ) and ‖AB‖ ≤ ‖A‖‖B‖ hold. In the following a central issue will be the spectral properties of some linear operators. Definition 4.7 (Resolvent and spectrum). Let A : 𝒦 → ℋ be closed. The resolvent set of A is ρ(A) = {λ ∈ ℂ | Ran(λ − A) = ℋ, λ − A is injective and (λ − A)−1 is bounded} (4.1.15) and its spectrum is defined as Spec A = ℂ \ ρ(A).

(4.1.16)

For λ ∈ ρ(A) the bounded operator RA (λ) = (λ − A)−1 is the resolvent of A. In the definition of the resolvent A must be assumed to be closed. The reason is as follows. If λ ∈ ρ(A), then RA (λ) = (λ −A)−1 is bounded and hence closed. Let j : 𝒦 ⊕ ℋ → ℋ ⊕ 𝒦 be defined by j(φ, ϕ) = (ϕ, φ). This is an isomorphism between 𝒦 ⊕ ℋ and ℋ ⊕ 𝒦, and G(B−1 ) = jG(B) follows. Let B be closed. Then G(B) is a closed set and hence G(B−1 ) is also closed. We conclude that B−1 is closed. Thus we proved that in general if B is closed and injective, then B−1 is also closed. In particular, λ − A is closed since so is (λ − A)−1 , from which follows that A must be closed. Furthermore, it can be also seen that if A is closable and injective, then (λ − A)−1 is also closable and (λ − A)−1 = (λ − A)−1 follows. For a closable operator A : 𝒦 → ℋ its resolvent set is defined by ρ(A) = {λ ∈ ℂ | Ran(λ − A) is dense, λ − A is injective, (λ − A)−1 is bounded}. For a closed operator A the conditions (1) Ran(λ − A) is dense, (2) λ − A is injective, and (3) (λ − A)−1 is bounded automatically imply that Ran(λ − A) = ℋ, and we can ̄ consequently see that ρ(A) = ρ(A).

4.1 Schrödinger semigroups | 221

The spectrum of A can be decomposed in three parts: (1) The point spectrum Specp (A) of A consists of the elements λ ∈ Spec(A) such that λ − A is not injective; the elements of the point spectrum are called eigenvalues, i. e., for every such λ there exists φλ ∈ D(A) \ {0}, such that Aφλ = λφλ , and φλ is an eigenvector (or eigenfunction when the vector space is a set of functions), while the number dim Ker(λ − A) is the multiplicity of eigenvalue λ. (2) The continuous spectrum Specc (A) of A consists of all λ ∈ Spec A for which λ − A has an inverse with a dense domain but it is not a bounded operator. (3) The residual spectrum Specr (A) of A is given by λ ∈ Spec(A) such that λ − A has an inverse, i. e., injective, but its domain is not dense. Thus the complex field ℂ is decomposed into the disjoint sets ℂ = ρ(A) ∪ Specp (A) ∪ Specc (A) ∪ Specr (A).

(4.1.17)

The resolvent set of a bounded operator A contains {z ∈ ℂ | |z| > ‖A‖} since (λ − A)−1 =

1 ∞ −n n ∑λ A λ n=0

holds in the uniform topology and the right-hand side of the equality is bounded for λ > ‖A‖. Thus the spectrum of bounded operators is a closed nonempty subset of {z ∈ ℂ | |z| ≤ ‖A‖}; however, in the case of unbounded operators both the spectrum and the resolvent set may be empty. The operators we will consider below are self-adjoint, therefore we do not need to address spectral theory in its full generality. Definition 4.8 (Adjoint operator). Let A : 𝒦 → ℋ be a densely defined linear operator. The adjoint operator A∗ of A is a linear operator from ℋ to 𝒦 defined by D(A∗ ) = {φ ∈ ℋ | ∃ψ ∈ ℋ such that (φ, Aϕ) = (ψ, ϕ), ∀ϕ ∈ D(A)} and A∗ φ = ψ. Since A is densely defined, A∗ is well defined. The adjoint operator A∗ is always closed, and whenever A is closed, moreover A = (A∗ )∗ holds. If A is closable, then Ā = (A∗ )∗ holds. It can be also checked that if D(A∗ ) is dense, then A is closable. For a closable operator A it follows that (A∗ ) = (A)̄ ∗ . If A, B ∈ B(𝒦, 𝒦), then A∗ , B∗ ∈ B(𝒦, 𝒦) and the ̄ ∗ , and (AB)∗ = B∗ A∗ hold. ̄ ∗ + bB properties ‖A∗ ‖ = ‖A‖, (A∗ )∗ = A, (aA + bB)∗ = aA Definition 4.9 (Symmetric operator and self-adjoint operator). Let A : 𝒦 → 𝒦 be a densely defined linear operator. A is said to be a symmetric operator if A ⊂ A∗ , a selfadjoint operator if A∗ = A, and an essentially self-adjoint operator if Ā is self-adjoint. Whenever D(A∗ ) is dense, A is closable; moreover, (A∗ )∗ = Ā for a closable operator. If A is symmetric, D(A∗ ) is dense by definition. Hence A is closable and Ā is a closed symmetric operator.

222 | 4 Feynman–Kac formulae In general, infinitely many self-adjoint extensions of a given symmetric operator exist. Note that if A ⊂ B, then B∗ ⊂ A∗ . Let A be symmetric and B a self-adjoint extension of A. We have A ⊂ B = B∗ ⊂ A∗ . Thus B = A∗ ⌈D with some domain D. Next let A be essentially self-adjoint and B a self-adjoint extension of A. Since A ⊂ B, it follows that (A∗ )∗ ⊂ B. Thus B = B∗ ⊂ ((A∗ )∗ )∗ = (A∗ )∗ and therefore B = (A∗ )∗ = A.̄ We summarize this in the following proposition. Proposition 4.10 (Self-adjoint extensions). (1) Let A be symmetric and B a self-adjoint extension of A. Then B is a restriction of A∗ . (2) Let A be essentially self-adjoint. Then Ā is the only self-adjoint extension of A, i. e., if B is self-adjoint and A ⊂ B, then Ā = B. (3) Let A be a symmetric operator and suppose that it has only one self-adjoint extension. Then A is essentially self-adjoint. A bounded operator A is symmetric if and only if it is self-adjoint. However, for an unbounded operator these two properties are in general different; indeed, a symmetric operator need not be closed and a closed symmetric operator need not be self-adjoint. Equivalent conditions to self-adjointness and essential self-adjointness of a symmetric operator are as follows. Proposition 4.11 (Equivalent conditions to self-adjointness). Let A be a symmetric operator in 𝒦. Then the following properties are equivalent: (1) A is self-adjoint; (2) A is closed and Ker(A∗ ± i) = {0}; (3) Ran(A ± i) = 𝒦; (4) A is closed and Ran(A ± i) = 𝒦. Proposition 4.12 (Equivalent conditions to essential self-adjointness). Let A be a symmetric operator in 𝒦. Then the following properties are equivalent: (1) A is essentially self-adjoint; (2) Ker(A∗ ± i) = {0}; (3) Ran(A ± i) = 𝒦. Self-adjointness removes some of the complications of the structure of the spectrum. The following facts are fundamental on the spectrum and resolvent of selfadjoint operators. Proposition 4.13 (Spectral properties of self-adjoint operators). Let A be a self-adjoint operator. Then (1) Spec(A) ⊂ ℝ; (2) the residual spectrum Specr (A) is empty; (3) the eigenvectors for distinct eigenvalues are orthogonal;

4.1 Schrödinger semigroups | 223

(4) for λ ∈ Spec(A) there exists a sequence (φn )n∈ℕ ⊂ D(A) such that ‖φn ‖ = 1 and limn→∞ ‖(λ − A)φn ‖ = 0; (5) if there exists c > 0 such that ‖(λ − A)φ‖ ≥ c‖φ‖ for all φ ∈ D(A), then λ ∈ ρ(A) and {z ∈ ℂ | |z − λ| < c} ⊂ ρ(A); (6) ‖RA (λ)‖ ≤ 1/|ℑλ| holds. A self-adjoint operator A with the property that (φ, Aφ) ≥ 0, for all φ ∈ D(A), is called a positive operator and denoted by A ≥ 0. Positivity of A is equivalent with Spec(A) ⊂ [0, ∞). Definition 4.14 (Unitary operator). A bounded operator U : 𝒦 → ℋ is called a unitary operator if it is an isometry, i. e., ‖Uφ‖ℋ = ‖φ‖𝒦 for all φ ∈ 𝒦, and Ran(U) = ℋ. The isometry condition can be equivalently formulated as U ∗ U = 1. Unitary operators can be used to define a conjugate operator to any given operator by keeping its spectrum unchanged. More precisely, if A is a closed operator and U is a unitary operator, then UAU −1 is well defined on UD(A) and closed; moreover, Spec(UAU −1 ) = Spec(A). For instance, Fourier transform is a unitary operator on L2 (ℝd ) and this equality can be used to compute the spectrum of the Laplacian; see Example 4.21. Furthermore, if A is self-adjoint, then UAU −1 is also self-adjoint on UD(A). Definition 4.15. Let (An )n∈ℕ be a sequence of unbounded operators. (1) An → A is said to converge in the sense of strong resolvent convergence if RAn (λ) strongly converges to RA (λ), for all λ ∈ ρ(A) ∩ ⋂∞ n=1 ρ(An ). (2) An → A is said to converge in the sense of uniform resolvent convergence if RAn (λ) uniformly converges to RA (λ), for all λ ∈ ρ(A) ∩ ⋂∞ n=1 ρ(An ). Convergence in strong and uniform resolvent sense are natural generalizations of strong and uniform convergence of bounded operators, respectively, to unbounded operators. 4.1.3 Spectral resolution The spectral theorem establishes a fundamental relationship between a linear operator and a measure on its spectrum. Let K be a Hilbert space and the set of projections on K be denoted by P(K ). Consider the map ℬ(ℝ) ∋ B 󳨃→ E(B) ∈ P(K ) with the properties (1) E(0) = 0 and E(ℝ) = 1; (2) if B = ⋃∞ n=1 Bn with Bm ∩ Bn = 0, m ≠ n, then n

E(B) = s-lim ∑ E(Bk ); n→∞

(3) E(B1 ∩ B2 ) = E(B1 )E(B2 ).

k=1

224 | 4 Feynman–Kac formulae A family of projections indexed by ℬ(ℝ) satisfying properties (1)–(3) above is called a projection-valued measure. For φ ∈ 𝒦 the Borel measure (φ, E(B)φ) on ℝ can be defined, and by the polarization identity it gives rise to the complex measure (φ, E(B)ψ), φ, ψ ∈ 𝒦 through (φ, E(B)ψ) =

1 4 1 ∑ ((φ + in ψ), E(B)(φ + in ψ)). 4 n=1 in

(4.1.18)

The spectral theorem says that there is a one-to-one correspondence between selfadjoint operators and projection-valued measures. For a self-adjoint operator A and the corresponding EA (λ) we define f (A) for a Borel measurable function f by (φ, f (A)ψ)𝒦 = ∫ f (λ)d(EA (λ)φ, ψ),

(4.1.19)

ℝ

with domain 󵄨󵄨 { } 󵄨 D(f (A)) = {φ ∈ 𝒦 󵄨󵄨󵄨 ∫ |f (λ)|2 d‖EA (λ)φ‖2 < ∞} . 󵄨󵄨 ℝ { } Note that the support of the measure EA coincides with Spec(A). Equality (4.1.19) is formally written as f (A) = ∫ℝ f (λ)dEA (λ) and is called the spectral resolution of f (A). The following algebraic relations hold. Proposition 4.16. Let A be a self-adjoint operator. Then for Borel measurable functions f and g, (1) D(f (A) + g(A)) = D((f + g)(A)) ∩ D(g(A)) = D((f + g)(A)) ∩ D(f (A)) and f (A) + g(A) ⊂ (f + g)(A); (2) D(f (A)g(A)) = D((fg)(A)) ∩ D(g(A)) and f (A)g(A) ⊂ (fg)(A). We will use the terminology of also another classification of the spectrum. Definition 4.17 (Discrete spectrum and essential spectrum). Let A be a self-adjoint operator. If λ ∈ Spec(A) is such that there is ε > 0 for which dim Ran EA (λ−ε, λ+ε) < ∞, then λ is said to be an element of the discrete spectrum of A, denoted Specd (A). If λ ∈ Spec(A) is such that dim Ran EA (λ − ε, λ + ε) = ∞ for all ε > 0, then λ is in the essential spectrum of A, denoted Specess (A). Note that the discrete and the essential spectra form a partition of the spectrum, i. e., Spec(A) = Specess (A) ⊔ Specd (A), where this is a disjoint union. Moreover, while Specess (A) is a closed set, Specd (A) need not be so. Furthermore, the following facts hold.

4.1 Schrödinger semigroups | 225

Proposition 4.18. A point λ ∈ Specd (A) if and only if both of the following properties hold: (1) λ is an isolated point of Spec(A), i. e., there exists ε > 0 such that Spec(A) ∩ (λ − ε, λ + ε) = {λ}; (2) λ is an eigenvalue of finite multiplicity, i. e., dim{φ ∈ 𝒦 | Aφ = λφ} < ∞. An eigenvalue of arbitrary multiplicity which is not an isolated point in the spectrum is called an embedded eigenvalue. There is a second possible breakup of the spectrum of a self-adjoint operator, which follows the canonical decomposition of a measure. Let A be a self-adjoint operator on a Hilbert space ℋ, and EA be the spectral resolution as above. By definition, a vector ψ ∈ ℋac if and only if the measure (ψ, EA (⋅)ψ) is absolutely continuous with respect to Lebesgue measure on ℝ. Similarly, ψ ∈ ℋsing if and only if the measure (ψ, EA (⋅)ψ) is singular with respect to Lebesgue measure. We then have ℋ = ℋac ⊕ ℋsing . Every eigenvector ψ belongs to ℋsing , and by definition ℋp is the closed linear hull of the set {ψ | ψ is eigenvector} and ℋsc = ℋsing ∩ ℋp⊥ . With these notations ℋ = ℋac ⊕ ℋp ⊕ ℋsc

holds. Consider the restrictions Aac = A⌈ℋac , Ap = A⌈ℋp and Asc = A⌈ℋsc . Furthermore, consider Specac (A) = Spec(Aac ) and Specsc (A) = Spec(Asc ). Definition 4.19 (Absolutely continuous spectrum and singular continuous spectrum). Specac (A) is called the absolutely continuous spectrum of A, and Specsc (A) the singular continuous spectrum of A . The set Specsing (A) = Spec(A⌈ℋsing ) is called the singular spectrum of A. Recall

that Specp (A) = {λ | λ is an eigenvalue} is the point spectrum. Note that Specp (A) = Spec(Ap ). It is possible to show that Spec(A) = Specac (A) ∪ Specp (A) ∪ Specsc (A), where the above sets need not be disjoint.

Example 4.20 (Multiplication operator). Let V be a Lebesgue measurable real-valued function called potential, and define a multiplication operator by (Vφ)(x) = V(x)φ(x) for a.e. x ∈ ℝd , with domain D(V) = {φ ∈ L2 (ℝd ) | Vf ∈ L2 (ℝd )}. Note that this operator is densely defined. Since V is real-valued, it is symmetric, and in fact it is a self-adjoint operator. Example 4.21 (Laplace operator). Recall that the weak partial derivative 𝜕j φ of φ ∈ L1loc (ℝd ) is a function h ∈ L1loc (ℝd ) such that ∫ h(x)f (x)dx = − ∫ φ(x)𝜕j f (x)dx, ℝd

ℝd

f ∈ C0∞ (ℝd ),

226 | 4 Feynman–Kac formulae and is the jth component of the gradient operator in weak sense. This notion reduces to the partial derivative (gradient) known from elementary calculus whenever φ ∈ C 1 (ℝd ). The Laplacian Δ in L2 (ℝd ) is the symmetric operator defined by d

Δ : φ 󳨃→ ∑ 𝜕j2 φ

(4.1.20)

j=1

for φ ∈ C0∞ (ℝd ). It is seen that T = Δ⌈C∞ (ℝd ) is essentially self-adjoint. We denote the 0

closure of T by the same symbol, i. e., Δ = T. The domain of Δ is the Sobolev space D(−(1/2)Δ) = H 2 (ℝd ) = {f ∈ L2 (ℝd ) | |k|2 f ̂ ∈ L2 (ℝd )},

where f ̂ and f ̌ denote the Fourier transform and inverse Fourier transform of f , respeĉ(k) = |k|2 f ̂(k) holds for every f ∈ H 2 (ℝd ). Since the range of the tively. Moreover, −Δf multiplication operator |k|2 is the positive semiaxis, it follows that −Δ has continuous spectrum and Spec(−Δ) = Specess (−Δ) = Specac (−Δ) = [0, ∞); in particular, −Δ is a positive operator. Example 4.22 (Fractional Laplacian). The operator (−Δ)α/2 in L2 (ℝd ) with domain ̂ α/2 f (k) = |k|α f ̂(k), is H α (ℝd ) = {f ∈ L2 (ℝd ) | |k|α f ̂ ∈ L2 (ℝd )}, 0 < α < 2, defined by (−Δ) called fractional Laplacian with exponent α/2. It is essentially self-adjoint on C0∞ (ℝd ), and its spectrum is Spec((−Δ)α/2 ) = Specess ((−Δ)α/2 ) = Specac ((−Δ)α/2 ) = [0, ∞). This is a case of a pseudo-differential operator. A related operator is the relativistic fractional Laplacian (−Δ + m2/α )α/2 − m, with m > 0. Finally, we discuss the so-called min-max principle, which is used to determine the eigenvalues of a self-adjoint operator bounded from below. Proposition 4.23 (Min-max principle). Let A be a self-adjoint operator in a Hilbert space K , and suppose it is bounded from below. Let n ∈ ℕ be arbitrary but fixed. Define μn (A) =

sup

ψ1 ,...,ψn−1 ∈K

inf

ψ∈D(A);‖ψ‖=1 ψ∈[ψ1 ,...,ψn−1 ]⊥

(ψ, Aψ),

(4.1.21)

where [ψ1 , . . . , ψn−1 ]⊥ = {ϕ ∈ K | (ψj , ϕ) = 0, j = 1, . . . , n−1}. Exactly one of the following properties holds: (1) there are n eigenvalues (counting multiplicity) below inf Specess (A) and μn (A) is the nth eigenvalue, counting multiplicity; (2) μn (A) = inf Specess (A), μn (A) = μn+1 (A) = μn+2 (A) = . . ., and there are at most n − 1 eigenvalues (counting multiplicity) below μn (A). The following useful tool is called the Rayleigh–Ritz method, applied in practice in order to get actual numerical estimates of eigenvalues.

4.1 Schrödinger semigroups | 227

Proposition 4.24 (Rayleigh–Ritz method). Let A be a self-adjoint operator bounded from below in a Hilbert space K . Let D ⊂ D(A) be an n-dimensional subspace, and P : K → D the projection to this subspace. We identify AD = PAP as a self-adjoint operator from D to itself, and let e1 ≤ . . . ≤ en be the ordered eigenvalues of AD . Let μj (A) be as given in (4.1.21). Then μj (A) ≤ ej for j = 1, . . . , n. In particular, if A has eigenvalues E1 , . . . , En at the bottom of its spectrum with E1 ≤ . . . ≤ En , then Ej ≤ ej for j = 1, . . . , n. An application of the min-max principle and the Rayleigh–Ritz method is in obtaining upper and lower bounds of the nth eigenvalue En of a self-adjoint operator A. By the min-max principle we have the lower bound En ≥

inf

ψ∈D(A);‖ψ‖=1 ψ∈[ψ1 ,...,ψn−1 ]⊥

(ψ, Aψ),

and the Rayleigh–Ritz method gives the upper bound En ≤ en . 4.1.4 Compact operators and trace ideals Many problems in classical mathematical physics can be reformulated in terms of integral equations. Consider the integral operator (Kf )(x) = ∫ K(x, y)f (y)dy ℝd

with integral kernel K(x, y). If this kernel is in L2 (ℝd × ℝd ), then K belongs to the following class of operators. Definition 4.25 (Compact operator). A bounded operator A : 𝒦 → ℋ is a compact operator whenever A takes an arbitrary bounded set in 𝒦 into a precompact set in ℋ. Equivalently, A is compact if and only if for every bounded sequence (φn )n∈ℕ , (Aφn )n∈ℕ has a subsequence convergent in ℋ. Some important properties of compact operators are summarized below. Proposition 4.26 (Riesz–Schauder theorem). Let A be a compact operator. (1) Spec(A) is a discrete set having no limit points except perhaps zero. (2) Every λ ∈ Spec(A) is an eigenvalue of finite multiplicity. Proposition 4.27. (1) Let C(𝒦) be the set of compact operators from 𝒦 to itself. Then C(𝒦) is an ideal of B(𝒦, 𝒦), i. e., AB, BA ∈ C(𝒦) for A ∈ C(𝒦) and B ∈ B(𝒦, 𝒦). Furthermore, K(𝒦) = B(𝒦, 𝒦)/C(𝒦) is simple, i. e., K(𝒦) has no nontrivial ideal. (2) C(𝒦) is closed in the uniform operator topology.

228 | 4 Feynman–Kac formulae The space K(𝒦) is called Calkin algebra. Example 4.28 (Integral operators). Let k ∈ L2 (ℝd × ℝd ). Then the integral operator A : f 󳨃→ ∫ℝd k(x, y)f (y)dy is a compact operator on L2 (ℝd ). We present some useful inequalities and criteria on compact operators without proof. Proposition 4.29 (Young’s inequality). Let 1 ≤ p, q, r ≤ ∞ such that

1 p

+ q1 + 1r = 2. Also,

let f ∈ Lp (ℝd ), g ∈ Lq (ℝd ), and h ∈ Lr (ℝd ). Then there exists a constant C(p, q, r, d) such that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ f (x)g(x − y)h(y)dxdy󵄨󵄨󵄨 ≤ C(p, q, r, d)‖f ‖p ‖g‖q ‖h‖r . 󵄨󵄨 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 󵄨ℝ ×ℝ 󵄨

(4.1.22)

The next is called Hardy–Littlewood–Sobolev inequality (or weak Young inequality). This relates with the situation above in which g is replaced by |x|−λ ∈ ̸ Lp (ℝd ), and is thus not covered by the Young inequality. Proposition 4.30 (Hardy–Littlewood–Sobolev inequality). Let p, r > 1 and 0 < λ < d, with p1 + dλ + 1r = 2. Let f ∈ Lp (ℝd ) and h ∈ Lr (ℝd ). Then there exists a constant C(λ, p, r, d) such that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −λ 󵄨󵄨 ∫ f (x)|x − y| h(y)dxdy󵄨󵄨󵄨 ≤ C(λ, p, r, d)‖f ‖p ‖h‖r . 󵄨󵄨 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 󵄨ℝ ×ℝ 󵄨

(4.1.23)

Example 4.31. Let k(x, y) = V 1/2 (x)|x − y|2−d V 1/2 (y) for d ≥ 2 with 0 ≤ V ∈ Ld/2 (ℝd ). Then ∫ℝd ×ℝd k(x, y)2 dxdy < ∞ by the Hardy–Littlewood–Sobolev inequality, and the integral operator A : f 󳨃→ ∫ℝd k(x, y)f (y)dy is compact. Further below we will see that it is actually of Hilbert–Schmidt class, see Definition 4.33. Next we introduce the weak Lp space Lpw (ℝd ) = {f : ℝd → ℂ | sup α|{x ∈ ℝd | |f (x)| > α}|1/p < ∞}. α>0

(4.1.24)

Here |E| denotes the Lebesgue measure of the measurable set E ⊂ ℝd . We note that Lp (ℝd ) ⊂ Lpw (ℝd ), since sup αp |{x ∈ ℝd | |f (x)| > α}| ≤ α>0

∫

|f (x)|p dx ≤ ‖f ‖pp .

|f (x)|>α

The expression supα>0 α|{x ∈ ℝd | |f (x)| > α}|1/p does not define a norm on the space Lpw (ℝd ), since the triangle inequality is not satisfied. For p > 1, however, there is an

4.1 Schrödinger semigroups | 229

alternative expression which is equivalent to supα>0 α|{x | |f (x)| > α}|1/p and is a norm. It is given by ‖f ‖p,w =

sup

∗

A⊂ℝd ,|A| p. Proof. Since there exists a constant C such that μn (A) ≤ Cn−1/p , it is immediate that q q ∞ −q/p ‖A‖q ≤ ∑∞ < ∞. n=1 μn (A) ≤ C ∑n=1 n Proposition 4.37 (Hölder inequality for trace ideals). Let 1/p = 1/q + 1/r, p ≥ 1, and A ∈ Iq , B ∈ Ir . Then AB ∈ Ip and ‖AB‖p ≤ ‖A‖q ‖B‖r holds. If p > 1, A ∈ Iq,w , B ∈ Ir,w , p then AB ∈ Ip,w and ‖AB‖p,w ≤ p−1 ‖A‖q,w ‖B‖r,w . From Proposition 4.37 we see that for 1 < p < ∞ and q = p/(p − 1), A ∈ Ip and B ∈ Iq satisfy AB ∈ I1 and |Tr(AB)| ≤ ‖AB‖1 ≤ ‖A‖p ‖B‖q . Thus the map Ip ∋ A 󳨃→ Tr(AB) ∈ ℂ defines the dual of Ip . It is, however, known that any ℓ ∈ Ip∗ is of the form ℓ(⋅) = Tr (⋅B) with some B ∈ Iq . We state this in the following proposition. Proposition 4.38 (Duality of Ip ). Let 1 < p < ∞ and q = p/(p − 1). Then Ip∗ = Iq in the sense that every ℓ ∈ Ip∗ has the form ℓ(A) = Tr(AB) with some B ∈ Iq and ‖ℓ‖Ip∗ = ‖B‖q . By this result we can identify Ip∗ with Iq for all 1 < p < ∞.

Next we consider trace ideals on the Hilbert space L2 (ℝd ). The most fundamental and useful examples of trace ideals on L2 (ℝd ) are obtained in the form f (x)g(−i∇). When f , g ∈ L2 (ℝd ), A = f (x)g(−i∇) is an integral operator with integral kernel A(x, y) = ̌ −y), satisfying A(⋅, ⋅) ∈ L2 (ℝd ×ℝd ) by Young’s inequality. Hence A ∈ I2 (2π)−d/2 f (x)g(x 2 d on L (ℝ ). A more general fact is described by the following result. Proposition 4.39. Let f ∈ Lp (ℝd ) and g ∈ Lp (ℝd ) with 2 ≤ p < ∞. Then it follows that A = f (x)g(−i∇) ∈ Ip on L2 (ℝd ) and ‖A‖p ≤ (2π)−d/p ‖f ‖p ‖g‖p .

4.1 Schrödinger semigroups | 231

In the following we construct a compact operator by a combination of elements of Lp (ℝd ) and Lpw (ℝd ). Proposition 4.40. Let 2 < p < ∞, g ∈ Lp (ℝd ) and u ∈ Lpw (ℝd ). Define the operator Bu,g h(k) =

1 u(k) ∫ eik⋅x g(x)h(x)dx (2π)d/2

(4.1.28)

ℝd

on L2 (ℝd ), with D(Bu,g ) = C0 (ℝd ). Then Bu,g can be extended to a compact operator on L2 (ℝd ), and Bu,g ∈ Ip,w ; moreover, there exists a constant Kp such that ‖Bu,g ‖∗p,w ≤ Kp ‖u‖p,w ‖g‖p .

(4.1.29)

Let F be Fourier transform on L2 (ℝd ). We denote FBu,g formally by u(−i∇x )g(x); see Proposition 4.73. Corollary 4.41. Let 2 < p < ∞, g ∈ Lp (ℝd ) and u ∈ Lpw (ℝd ). Define the operator Bu,g on L2 (ℝd ) by (4.1.28). Then Bu,g ∈ Iq for all q > p. Proof. By Proposition 4.40 it follows that Bu,g ∈ Ip,w . The corollary results then from Proposition 4.36. Example 4.42. Let d ≥ 3 and g ∈ Ld (ℝd ). Since u(k) = 1/|k| ∈ Ldw (ℝd ), by Proposition 4.40 and Corollary 4.41 we have Bu,g ∈ Id,w , and Bu,g ∈ Ip for p > d. Furthermore, B∗u,g Bu,g ∈ Id/2,w and, in particular, B∗u,g Bu,g ∈ Ip for any p > d/2. Finally, we discuss a convergence theorem. An N × N matrix (αnm )1≤n,m≤N (possibly N = ∞) is called doubly substochastic if ∑Nn=1 |αnm | < 1 for 1 ≤ m ≤ N, and ∑Nm=1 |αnm | < 1 for 1 ≤ n ≤ N. For example, if {fn }Nn=1 , {gm }Nm=1 are orthonormal sets on a Hilbert space 𝒦, the matrix (αnm )1≤n,m≤N = (|(fn , gm )𝒦 |2 )1≤n,m≤N is doubly substochastic. Proposition 4.43. Let A and B be compact operators. Then there exists a doubly substochastic matrix α = (αnm )1≤n,m≤∞ such that ∞

μn (A) − μn (B) = ∑ αnm μm (A − B), m=1

for all n ≥ 1, where μn is as given in Definition 4.33. In particular, |μn (A)−μn (B)| ≤ ‖A−B‖ for all n ≥ 1, with uniform norm. The next corollary can be immediately derived from Proposition 4.43. Corollary 4.44 (Convergence of μn (Am )). Suppose that (An )n∈ℕ and A are compact operators such that limm→∞ ‖Am − A‖ = 0. Then μn (Am ) → μn (A) as m → ∞, for all n ≥ 1. Under an extra condition combined with limm→∞ ‖Am − A‖ = 0, also Ip -convergence is obtained.

232 | 4 Feynman–Kac formulae Proposition 4.45 (Ip -convergence). Let 1 ≤ p < ∞. (1) Suppose that (Am )m∈ℕ , A, and B are compact operators such that for all n ≥ 1, μn (Am ) ≤ μn (B) and B ∈ Ip . Suppose, moreover, that Am → A in the operator norm as m → ∞. Then limm→∞ ‖Am − A‖p = 0. (2) Suppose that (Am )m∈ℕ , A ∈ Ip , and Am → A and A∗m → A∗ strongly as m → ∞ and ‖Am ‖p → ‖A‖p . Then limm→∞ ‖Am − A‖p = 0. (3) Suppose that (Am )m∈ℕ , A ∈ Ip , and Am → A, |Am | → |A|, and |A∗m | → |A∗ | weakly as m → ∞. Then limm→∞ ‖Am − A‖p = 0. The following result gives useful criteria guaranteeing that the spectrum of a selfadjoint operator is purely discrete. Proposition 4.46. Let A be a self-adjoint operator and bounded below. Then the following properties are equivalent: (1) (λ − A)−1 is a compact operator for some λ ∈ ρ(A); (2) (λ − A)−1 is a compact operator for all λ ∈ ρ(A); (3) the set {φ ∈ D(A) | ‖φ‖ < 1, ‖Aφ‖ ≤ b} is compact for all b; (4) the set {φ ∈ D(|A|1/2 ) | ‖φ‖ < 1, ‖|A|1/2 φ‖ ≤ b} is compact for all b. Below we will consider perturbations of given operators and address the question whether and how the essential spectrum of a self-adjoint operator changes under a perturbation. Perturbations by operators satisfying the condition below leave the essential spectrum invariant. Definition 4.47 (Relative compactness). Let A be self-adjoint. An operator B such that D(A) ⊂ D(B) is called relatively compact with respect to A or A-compact if and only if B(i + A)−1 is compact. Then we have Proposition 4.48. Let A be self-adjoint and B be relatively compact with respect to A. Then A + B with domain D(A) is closed and Specess (A + B) = Specess (A).

4.1.5 Schrödinger operators Now we turn to discussing the specific class of operators announced at the start of this chapter, and which will be a main object throughout this book. Definition 4.49 (Schrödinger operator). Let V be a real-valued multiplication operator in L2 (ℝd ), called potential. The linear operator 1 H =− Δ+V 2

(4.1.30)

4.1 Schrödinger semigroups | 233

with the dense domain 1 D(H) = D(− Δ) ∩ D(V) 2

(4.1.31)

is called a Schrödinger operator. By the definition of Δ it is readily seen that every φ ∈ D(−(1/2)Δ) is a bounded continuous function whenever d ≤ 3. Indeed, by the estimate ̂ 1 = ‖(1 + |k|2 )−1 (1 + |k|2 )φ‖ ̂ 1 ≤ ‖(1 + |k|2 )−1 ‖2 ‖(1 + |k|2 )φ‖ ̂ 2 < ∞, ‖φ‖ the Fourier transform of φ lies in L1 (ℝd ). Thus φ is a continuous bounded function ̂ by the Riemann–Lebesgue lemma. Moreover, by the scaling φ̂ r (k) = r d φ(rk) we have d−4 2 d/2 2 ̂ 1 ≤ cr ‖|k| φ‖ ̂ 2 + cr ‖φ‖ ̂ 2 , with a constant c independent of r. Since d ≤ 3, for ‖φ‖ arbitrary a > 0 there exists b such that ̂ 1 ≤ a‖Δφ‖2 + b‖φ‖2 . ‖φ‖∞ ≤ ‖φ‖

(4.1.32)

This bound is useful in estimating the relative bound of a given potential V with respect to the Laplacian. We are interested in defining Schrödinger operators as self-adjoint operators. One reason is that then the spectrum is guaranteed to be real, in match with the uses of quantum mechanics. Secondly, since the solution of the Schrödinger equation (4.1.1) is given by e−itH φ0 for φ(x, 0) = φ0 (x), in order to make sure that e−itH is a unitary map the question has to be addressed whether the symmetric operator H is self-adjoint or essentially self-adjoint. From the point of view of quantum mechanics, distinct selfadjoint extensions lead to different time evolutions; however, since several self-adjoint extensions of H may exist, a particular one must be chosen to describe the evolution of the system. Since for any self-adjoint extension K of H it is true that K ⊂ H ∗ , an important step is to find a core of H as then T = H⌈𝒟 is the unique self-adjoint extension of H⌈𝒟 . In general it is difficult to show self-adjointness or essential self-adjointness of a Schrödinger operator. Here we review some basic criteria; however, as we are interested in defining and studying the properties of perturbations of the Laplacian, first we give a notion of perturbation. Definition 4.50 (Relative boundedness). Given an operator A, an operator B is called A-bounded whenever D(A) ⊆ D(B) and ‖Bf ‖ ≤ a‖Af ‖ + b‖f ‖

(4.1.33)

with some numbers a, b ≥ 0, for all f ∈ D(A). In this case a is called a relative bound. Moreover, if the relative bound a can be chosen arbitrarily small, B is said to be infinitesimally small with respect to A.

234 | 4 Feynman–Kac formulae A classic result of self-adjoint perturbations of a given self-adjoint operator is offered by the following theorem. Theorem 4.51 (Kato–Rellich theorem). Suppose that A is a self-adjoint operator on ℋ, and B is a symmetric A-bounded operator with relative bound strictly less than 1. Then A + B is self-adjoint on D(A) and essentially self-adjoint on any core of A with (A + B)⌈D = A⌈D + B⌈D , where D is any core of A. Furthermore, if A is bounded from below by M, then the selfb adjoint operator A + B is also bounded from below by M − max { 1−a , a|M| + b}, where a and b are given by (4.1.33). Proof. It is sufficient to show that Ran(A + B ± ix) = ℋ for some x > 0. Since A is self-adjoint, Ran(A ± ix) = ℋ. Since A + B + ix = (B(A + ix)−1 + 1)(A + ix), it is sufficient to show that Ran(B(A + ix)−1 + 1) = ℋ. By the assumption we have ‖B(A + ix)−1 f ‖ ≤ a‖A(A + ix)−1 f ‖ + b‖(A + ix)−1 f ‖. From the identity ‖(A + ix)f ‖2 = ‖Af ‖2 + x 2 ‖f ‖2 it follows that ‖A(A + ix)−1 f ‖ ≤ ‖f ‖ and ‖(A + ix)−1 f ‖ ≤ ‖f ‖/x. Thus we have ‖B(A + ix)−1 f ‖ ≤ (a + b/x)‖f ‖ and ‖B(A + ix)−1 ‖ < 1 for sufficiently large x. Hence B(A + ix)−1 + 1 is invertible and Ran(A + B + ix) = ℋ follows for sufficiently large x. Similarly we can show that Ran(A + B − ix) = ℋ. We next show the second statement. Let D be a core of A. Setting A󸀠 = A⌈D and B󸀠 = B⌈D , we obtain ‖B󸀠 f ‖ ≤ a‖A󸀠 f ‖ + b‖f ‖

(4.1.34)

󸀠 for f ∈ D(A󸀠 ). For f ∈ D(A󸀠 ) there exists a sequence (fn )∞ n=1 ⊂ D(A ) such that fn → f 󸀠 󸀠 󸀠 and A fn → A f as n → ∞. By (4.1.34) we can also see that B fn is a Cauchy sequence, hence B󸀠 fn → B󸀠 f as n → ∞. Thus from the inequality ‖B󸀠 fn ‖ ≤ a‖A󸀠 fn ‖ + b‖fn ‖ and a limiting argument it follows that

‖B󸀠 f ‖ ≤ a‖A󸀠 f ‖ + b‖f ‖

(4.1.35)

for f ∈ D(A󸀠 ). Thus A󸀠 + B󸀠 is self-adjoint on D(A󸀠 ). By the above limiting argument we also see that (A󸀠 + B󸀠 )fn → A󸀠 f + B󸀠 f and fn → f , which implies that A󸀠 + B󸀠 f = A󸀠 f + B󸀠 f for f ∈ D(A󸀠 ). Hence A󸀠 + B󸀠 ⊃ A󸀠 + B󸀠 follows. Since A󸀠 + B󸀠 is a self-adjoint extension of A󸀠 + B󸀠 , it is trivial to see that A󸀠 + B󸀠 ⊃ A󸀠 + B󸀠 . Thus A󸀠 + B󸀠 = A󸀠 + B󸀠 . We see that A󸀠 + B󸀠 is self-adjoint and hence A + B is essentially self-adjoint on D. Finally, we estimate of A + B from below. Let −t < M. We can show that ‖B(A + t)−1 ‖ < 1 if b −t < M − max { 1−a , a|M| + b}. Thus for such t, Ran(A + B + t) = ℋ and (A + B + t)−1 = −1 −1 (A + t) (B(A + t) + 1)−1 is bounded. This implies that −t ∈ ̸ Spec(A + B).

4.1 Schrödinger semigroups | 235

Applied to Schrödinger operators, the natural class of V for the Kato–Rellich theorem is Lp (ℝd ) + L∞ (ℝd ). Example 4.52. Let d = 3 and V = V1 +V2 ∈ L2 (ℝ3 )+L∞ (ℝ3 ). Since ‖Vφ‖2 ≤ ‖V1 ‖2 ‖φ‖∞ + ‖V2 ‖∞ ‖φ‖2 and ‖φ‖∞ ≤ a‖Δφ‖2 + b‖φ‖2 , for arbitrary a > 0 and some b > 0 by (4.1.32), we have ‖Vφ‖2 ≤ a‖V1 ‖2 ‖Δφ‖2 + (b + ‖V2 ‖∞ )‖φ‖2 . Thus by Theorem 4.51 the Schrödinger operator −(1/2)Δ+V is self-adjoint on D(−(1/2)Δ) and bounded from below. Moreover, it is essentially self-adjoint on any core of −Δ. In particular, −(1/2)Δ − |x|−1 is self-adjoint on D(−(1/2)Δ) for d = 3. Example 4.53. The previous example can be extended to arbitrary d-dimension. Let V ∈ Lp (ℝd ) + L∞ (ℝd ) {

p = 2,

p > d/2,

d≤3

d ≥ 4.

(4.1.36)

In the case d ≥ 4 consider V ∈ Lp (ℝd ). Choose q = 2p/(p − 2) and r = 2p/(p + 2). From the Hölder and the Hausdorff–Young inequalities it follows that ‖Vf ‖2 ≤ ‖V‖p ‖f ‖q ≤ C‖V‖p ‖f ̂‖r . Thus ‖f ̂‖r = ‖(1 + ε|k|2 )−1 (1 + ε|k|2 )f ̂‖r ≤ ‖(1 + ε|k|2 )−1 ‖p ‖(1 + ε|k|2 )f ̂‖2 ≤ C 󸀠 ε−d/(2p) (ε‖Δf ‖2 + ‖f ‖2 ). For a given δ > 0 and a sufficiently small ε we see that ‖Vf ‖2 ≤ δ‖(−1/2)Δf ‖ + Cδ ‖f ‖2 . Thus V is infinitesimally small with respect to −Δ. In particular, H is self-adjoint on D(−(1/2)Δ) and essentially self-adjoint on any core of −Δ. From the discussion in Example 4.53 we conclude the following result. Proposition 4.54. Let V ∈ Lp (ℝd ), with p = 2 for d ≤ 3, and p > d/2 for d ≥ 4. Then there exists a constant C such that ‖Vf ‖2 ≤ C‖V‖p ‖(−(1/2)Δ + 1)f ‖

(4.1.37)

for f ∈ D(−(1/2)Δ). The Kato–Rellich theorem fails to be useful in investigating the self-adjointness of Schrödinger operators with a polynomial potential such as V(x) = |x|2 . In this case the following can be applied.

236 | 4 Feynman–Kac formulae Theorem 4.55 (Kato’s inequality). Let u ∈ L1loc (ℝd ) and suppose the distributional Laplacian is Δu ∈ L1loc (ℝd ). Then the inequality Δ|u| ≥ ℜ[(sgn u)Δu]

(4.1.38)

holds in distributional sense. Above we have the sign function sgn u = u(x)/|u(x)| whenever u(x) ≠ 0, and sgn u = 0 for u(x) = 0, defined as usual. The natural class for Kato’s inequality is L2loc (ℝd ). Corollary 4.56. Let V ∈ L2loc (ℝd ) be bounded below. Then H = (−1/2)Δ + V is essentially self-adjoint on C0∞ (ℝd ). Proof. By (3) of Proposition 4.11 it suffices to show that Ran(H⌈C∞ (ℝd ) +1) = L2 (ℝd ), 0

or equivalently, (u, ((−1/2)Δ + V + 1)f ) = 0 for ∀f ∈ C0∞ (ℝd ) implies that u = 0. Suppose that ((−1/2)Δ + V + 1)u = 0 in distributional sense. It follows that (1/2)Δu = Vu + u. By Kato’s inequality, Δ|u| ≥ ℜ([sgn u]Δu) = 2(V + 1)|u| ≥ 0. In particular, Δ|u| ≥ 0. Let vδ (x) = v(x/δ)δ−d , where v ≥ 0, v ∈ C0∞ (ℝd ) and ∫ℝd v(x)dx = 1. Consider wδ = |u| ∗ vδ .

We see that wδ ∈ D(−(1/2)Δ), Δwδ = |u| ∗ Δvδ = Δ|u| ∗ vδ ≥ 0, and (wδ , Δwδ ) ≤ 0 is trivial. Thus wδ = 0. Since wδ → |u| as δ → 0 in L2 (ℝd ), we have |u| = 0. Hence the statement follows. Corollary 4.56 can be extended to more general cases. Corollary 4.57. Let V = V1 + V2 be such that V1 ≥ 0, V1 ∈ L2loc (ℝd ), and V2 is −(1/2)Δ-bounded with a relative bound less than 1. Then H = (−1/2)Δ + V is essentially self-adjoint on C0∞ (ℝd ). Example 4.58. By Corollary 4.56 the operator −(1/2)Δ + P(x) with a polynomial P(x) bounded from below is essentially self-adjoint on C0∞ (ℝd ) and bounded from below. Example 4.59. By combining the Kato–Rellich theorem with Kato’s inequality it is seen that H = (−1/2)Δ+V with potential V = V1 +V2 such that V1 ∈ L2loc (ℝd ), V1 ≥ 0, and (−(1/2)Δ)1/2 -bounded V2 , is self-adjoint on D(−(1/2)Δ + V1 ) and essentially self-adjoint on C0∞ (ℝd ). To see this first note that −(1/2)Δ + V1 is essentially self-adjoint by Kato’s inequality. Secondly, V2 is (−(1/2)Δ + V1 )-bounded with infinitesimally small relative bound. Hence H is self-adjoint on D(−(1/2)Δ + V1 ) by the Kato–Rellich theorem. 4.1.6 Schrödinger operators through quadratic forms When the potential is too singular and D(−Δ) ∩ D(V) is too small (in the extreme case, it may reduce to {0}), then a Schrödinger operator cannot be defined as an operator

4.1 Schrödinger semigroups | 237

sum, however, it still can be defined in terms of a form sum. Recall that a quadratic form q on a Hilbert space 𝒦 is a map q : Q(q) × Q(q) → ℂ such that q(f , g) is linear in g and antilinear in f for f , g ∈ Q(q), where Q(q) denotes the form domain of q. The form is symmetric if q(f , g) = q(g, f ). If q(f , f ) ≥ −M‖f ‖2 for some M, then q is said to be semibounded. Note that a semibounded quadratic form is automatically symmetric if 𝒦 is defined over the complex field ℂ. A semibounded quadratic form q is closed if Q(q) is complete in the norm ‖f ‖+1 = (q(f , f ) + (M + 1)‖f ‖2 )1/2 with M as above. Moreover, if q is closed and D ⊂ Q(q) is dense in Q(q) in the norm ‖ ⋅ ‖+1 , then D is called a form core for q. By Riesz’s theorem there is a one-to-one correspondence between semibounded quadratic forms and self-adjoint operators bounded below. Let A be a self-adjoint operator bounded below. It is clear that A defines the semibounded quadratic form qA (f , g) = ∫ λd(EA (λ)f , g) ℝ

with form domain Q(qA ) = D(|A|1/2 ). Proposition 4.60. If q is a closed semibounded quadratic form, then there exists a unique self-adjoint operator A such that q = qA . Using this fact we can define the Schrödinger operator with a singular potential such as V(x) = δ(x). Also, the concept of relative boundedness of operators can be extended to forms as well. Definition 4.61 (Relative form-boundedness). Let A be a positive self-adjoint operator and B a self-adjoint operator such that Q(qB ) ⊃ Q(qA ) and |qB (f , f )| ≤ aqA (f , f ) + b(f , f )

(4.1.39)

for some a, b ≥ 0. Then B is said to be relatively form-bounded with respect to A or A-form-bounded. If B is A-bounded, then B is relatively form-bounded with respect to A with the same relative bound as the A-bound. Next we present a form analogue of the Kato–Rellich theorem. Theorem 4.62 (KLMN theorem). Let A be a positive self-adjoint operator and suppose that β is a symmetric quadratic form on Q(qA ) such that |β(f , f )| ≤ aqA (f , f ) + b(f , f ) holds with some a < 1 and b ≥ 0. Then there exists a unique self-adjoint operator C such that Q(qC ) = Q(qA ) and qC (f , g) = qA (f , g) + β(f , g) for f , g ∈ Q(qC ). Furthermore, C is bounded from below by −b and any domain of essential self-adjointness for A is a form core for C.

238 | 4 Feynman–Kac formulae Let now A and B be self-adjoint operators bounded from below such that Q(qA ) ∩ Q(qB ) is dense. The quadratic form q(f , g) = qA (f , g) + qB (f , g) with Q(q) = Q(qA ) ∩ Q(qB ) defines a closed semibounded quadratic form. Its associated self-adjoint operator is then denoted by C = A +̇ B. If a nonnegative self-adjoint operator B is A-form-bounded with relative bound strictly smaller than 1, then the KLMN theorem implies that a unique self-adjoint operator associated with the quadratic form qA − qB exists, and this is denoted by A −̇ B. Example 4.63. Let V(x) = |x|−α , 0 ≤ α < 2, and d = 3. In quantum mechanics the relation ‖|x|−1 f ‖ ≤ 4‖Δf ‖ is known as the uncertainty principle. It can be seen that V is infinitesimally small form-bounded with respect to −Δ, i. e., for every ε there exists bε > 0 such that ‖V 1/2 f ‖ ≤ ε‖(−Δ)1/2 f ‖ + bε ‖f ‖ for f ∈ Q(−Δ). Therefore there exists a self-adjoint operator K such that (f , Kg) = (f , (−Δ + V)g) for f , g ∈ C0∞ (ℝ3 ). Example 4.64. Let q(f , g) = (∇f , ∇g) + ∫ℝd V(x)f ̄(x)g(x)dx. The natural class of V such that inf{q(f , f )| ‖f ‖2 = 1, f ∈ H 1 (ℝd )} > −∞ is

Ld/2 (ℝd ) + L∞ (ℝd ), { { { p 2 V ∈ {L (ℝ ) + L∞ (ℝ2 ), p > 1, { { 1 1 ∞ 1 {L (ℝ ) + L (ℝ ),

d ≥ 3, d = 2,

(4.1.40)

d = 1.

Note that for such V the fact that f ∈ H 1 (ℝd ) implies by the Sobolev inequality that ∫ℝd |V(x)|f (x)|2 dx ≤ ‖V‖d/2 ‖f ‖22d/(d−2) ≤ C‖V‖d/2 ‖∇f ‖22 < ∞. Since q is semibounded on the form domain H 1 (ℝd ), there exists a self-adjoint operator associated with q.

The relationship between the convergence of sequences of quadratic forms and associated self-adjoint operators is the following. Let q1 and q2 be symmetric quadratic forms bounded from below; q1 ≥ q2 means that Q(q1 ) ⊂ Q(q2 ) and q1 (f , f ) ≥ q2 (f , f ), for all f ∈ Q(q1 ). A sequence (qn )n∈ℕ of symmetric quadratic forms bounded from below is nonincreasing (resp. nondecreasing) if qn ≥ qn+1 (resp. qn ≤ qn+1 ), for all n. It is well known that for any positive quadratic form q there exists a largest closable quadratic form qr that is smaller than q. Theorem 4.65 (Monotone convergence for forms). (1) Case of nonincreasing sequence: Let (qn )n∈ℕ be a nonincreasing sequence of densely defined, closed symmetric quadratic forms uniformly bounded from below, i. e., qn ≥ γ with a constant γ, and let Hn be the self-adjoint operator associated with qn . Consider the symmetric quadratic form q∞ defined by q∞ (f , f ) = lim qn (f , f ), n→∞

∞

Q(q∞ ) = ⋃ Q(qn ). n=1

(4.1.41)

Let H be the self-adjoint operator associated with (q∞ )r , the closure of (q∞ )r . Then s-limn→∞ (Hn − ζ )−1 = (H − ζ )−1 for ℜζ < γ.

4.1 Schrödinger semigroups | 239

(2) Case of nondecreasing sequence: Let (qn )n∈ℕ be a nondecreasing sequence of densely defined, closed symmetric quadratic forms, and let Hn be the self-adjoint operator associated with qn . Consider the symmetric quadratic form q∞ defined by q∞ (f , f ) = lim qn (f , f ), n→∞

Q(q∞ ) = {f | lim qn (f , f ) < ∞}. n→∞

(4.1.42)

Then q∞ is a closed symmetric quadratic form. Suppose, furthermore, that Q(q∞ ) is dense, and let H be the self-adjoint operator associated with q∞ . Then we have s-limn→∞ (Hn − ζ )−1 = (H − ζ )−1 , for Im ζ ≠ 0. Next we discuss a counterpart of relative compactness for forms. Definition 4.66 (Relative form-compactness). Let q be a closed semibounded quadratic form on 𝒦, and β a symmetric quadratic form such that Q(β) ⊃ Q(q) with the property that there exists a compact operator C such that β((A − γ)−1/2 f , (A − γ)−1/2 g) = (f , Cg). Here A is the unique self-adjoint operator such that q = qA and γ ∈ ℝ is such that A − γ ≥ 0. Then β is said to be relatively form-compact with respect to A or A-formcompact. If β = qB with a self-adjoint operator B, then B is said to be relatively formcompact with respect to A or A-form-compact. We have already seen in Proposition 4.48 that perturbations by relatively compact operators leave the essential spectrum invariant. A counterpart for relatively formcompact operators is the following result. Proposition 4.67. Let β be relatively form-compact with respect to a self-adjoint operator A bounded from below, and C be a self-adjoint operator such that qA + β = qC . Then Specess (C) = Spec(A). Example 4.68. Let 𝒦 = L2 (ℝd ) and A = (−1/2)Δ. Thus qA (f , g) = (A1/2 f , A1/2 g) with form domain Q(qA ) = D(A1/2 ). Consider the perturbation by a real-valued nonpositive potential V = −W. Define the quadratic form β(f , g) = −(W 1/2 f , W 1/2 g), and write R = W 1/2 (A + E)−1/2 for E ≥ 0. Suppose that β is relatively form-compact with respect to A. Since β((A + E)−1/2 f , (A + E)−1/2 g) = −(Rf , Rg), we see that R∗ R is compact, and thus W 1/2 (A + E)−1 W 1/2 = RR∗ is also compact. Conversely, if W 1/2 (A+E)−1 W 1/2 is compact, then the operator R∗ R is also compact, and the quadratic form β is relatively form-compact with respect to A. Thus we conclude that if W 1/2 (A + E)−1 W 1/2 is compact, then Specess ((−1/2)Δ −̇ W) = Specess ((−1/2)Δ) = [0, ∞). 4.1.7 Confining potentials and decaying potentials A natural question is how the spectrum of the Laplacian changes under a perturbation by a potential V. It is not our aim to discuss general spectral theory in this book; in-

240 | 4 Feynman–Kac formulae stead, we discuss some situations which will be relevant in our considerations further below. As a general feature, the structure of the spectrum of a Schrödinger operator H = 1 − 2 Δ + V is strongly influenced by the asymptotic properties of V at infinity, and there is a split in the behaviour according to the two main classes as defined below. Definition 4.69 (Confining / decaying potentials). Let V : ℝd → ℝ be a potential. We say that V is a confining potential if V(x) → ∞ as |x| → ∞, and a decaying potential if V(x) → 0 as |x| → ∞. We single out some basic examples often used in mathematical physics. Example 4.70. Some typical cases of confining potentials are given by the following choices: (1) Harmonic and anharmonic oscillators: Let V(x) = |x|2n , n ∈ ℕ. The case n = 1 describes the potential of the harmonic oscillator, and n ≥ 2 give anharmonic oscillators. (2) Double and multiple well potentials: The potential V(x) = |x|4 − b|x|2 , b > 0, is a symmetric double well potential. Multiple well potentials can be obtained by higher order polynomials. For decaying potentials we single out the following specific cases: (1) Potential wells: Let V(x) = −υ(x) with a compactly supported, non-negative bounded Borel function υ ≢ 0. Specifically, we can choose V(x) = −a1B(0,1) (bx), for a, b > 0. (2) Coulomb-type potentials: V(x) = −c|x|−β , with c, β > 0. (3) Yukawa-type potentials: V(x) = −c|x|−β e−b|x| , with b, c, β > 0. (4) Pöschl–Teller potential: V(x) = −a/ cosh2 (b|x|), with a, b > 0. (5) Morse potential: V(x) = a ((1 − e−b(|x|−r0 ) )2 − 1), with a, b, r0 > 0. Next we discuss some results which can be applied to these types of potentials to determine the structure of Spec(H). Theorem 4.71 (Rellich’s criterion). Let F and G be functions on ℝd such that lim|x|→∞ F(x) = ∞ and lim|x|→∞ G(x) = ∞. Then the set 󵄨󵄨 {f ∈ L2 (ℝd ) 󵄨󵄨󵄨 ∫ |f (x)|2 dx ≤ 1, ∫ F(x)|f (x)|2 dx ≤ 1, ∫ G(p)|f ̂(p)|2 dp ≤ 1} 󵄨 ℝd

ℝd

ℝd

is compact. Using this criterion the following interesting result is obtained. Theorem 4.72 (Confining potential). Let V ∈ L1loc (ℝd ) be bounded from below such that V(x) → ∞ as |x| → ∞. Then H = (−1/2)Δ +̇ V has compact resolvent. In particular, e−tH

4.1 Schrödinger semigroups |

241

is also compact for all t > 0 and H has a purely discrete spectrum, i. e., Spec(H) = Specd (H). Proof. By Proposition 4.46 it suffices to show that the set S = {f ∈ D(|H|1/2 ) | ‖f ‖ ≤ 1, ‖|H|1/2 f ‖2 ≤ b} is compact for all b. Since S is closed, we need only prove that S is contained in a compact set. Indeed, 󵄨󵄨 1 S ⊂ {f ∈ L2 (ℝd ) 󵄨󵄨󵄨 ∫ |f (x)|2 dx ≤ 1, ∫ V(x)|f (x)|2 dx ≤ b, ∫ |p|2 |f ̂(p)|2 dp ≤ b} 󵄨 2 ℝd

ℝd

ℝd

and the right-hand side is compact by Rellich’s criterion. From the identity e−tH = e−tH (H − z)(H − z)−1 and the boundedness of e−tH (H − z), it follows that e−tH is also compact. From this theorem we see that Schrödinger operators with confining potentials have a purely discrete spectrum. Next we consider Schrödinger operators with decaying potentials. The spectral properties of this class are markedly different from those with confining potentials. Let L∞,0 (ℝd ) = {f ∈ L∞ (ℝd ) | f (x) → 0 as |x| → ∞}. Define the operator P(−i∇) by P(−i∇)f = (2π)−d/2 ∫ℝd P(k)eik⋅x f ̂(k)dk. Proposition 4.73 (Compactness). Let P, Q ∈ L∞,0 (ℝd ). Then Q(x)P(−i∇) is compact. Proof. Let 1Λ ∈ C0∞ (ℝd ) be given by 1Λ (x) = {

1, 0,

|x| < Λ,

|x| > Λ + 1.

The product 1Λ (x)1Λ (−i∇) is an integral operator with kernel k(x, y) = (2π)−d/2 1Λ (x)1Λ̂ (y − x) ∈ L2 (ℝd × ℝd ). Thus 1Λ (x)1Λ (−i∇) is compact by Example 4.28. Since Q(x)1Λ (x) → Q(x) and 1Λ (ξ )P(ξ ) → P(ξ ) as Λ → ∞ in L∞ (ℝd ), we have Q(x)P(−i∇) = lim Q(x)1Λ (x)1Λ (−i∇)P(−i∇) Λ→∞

uniformly. Since Q(x)P(−i∇) is a uniform limit of a sequence of compact operators, it is compact by Proposition 4.26.

242 | 4 Feynman–Kac formulae Theorem 4.74 (Decaying potential). Let V be such that V ∈ Lp (ℝd ) + L∞,0 (ℝd ) {

p = 2,

p > d/2,

d ≤ 3,

d ≥ 4.

Then H = − 21 Δ + V is essentially self-adjoint on C0∞ (ℝd ) and Specess (H) = Specess (−Δ) = [0, ∞). Proof. Denote H0 = − 21 Δ. Essential self-adjointness follows by the Kato–Rellich theorem. By Proposition 4.73, V(H0 + 1)−1 is compact for V ∈ L∞,0 (ℝd ). Thus it suffices to show that V ∈ Lp (ℝd ) is relatively compact with respect to H0 . Let V ∈ Lp (ℝd ). For each ε > 0 we decompose V = Vε +Wε , where ‖Vε ‖p < ε and Wε ∈ L∞,0 (ℝd ). Indeed, let

1Λ (x) = {

1, 0,

|x| ≤ Λ,

|x| > Λ

and g

(N)

g, |g| < N, { { { = {N, g ≥ N, { { {−N, g ≤ −N.

V is decomposed as (N) V = (V1 +V − (V1Λ )(N) . ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Λ) =VΛN

(4.1.43)

=WΛN

Thus VΛN ∈ L∞,0 (ℝd ), WΛN ∈ Lp (ℝd ), and ‖WΛN ‖p < ε for sufficiently large Λ and N. Furthermore, VΛN → V in Lp as Λ, N → ∞. We have by (4.1.37) ‖(V(1 + H0 )−1 − VΛN (1 + H0 )−1 )f ‖ ≤ C‖V − VΛN ‖p ‖f ‖2 . Thus VΛN (1 + H0 )−1 → V(1 + H0 )−1 in the uniform topology and VΛN (1 + H0 )−1 is compact since VΛN ∈ L∞,0 (ℝd ). Hence V(1 + H0 )−1 is compact, and this completes the proof. Example 4.75. The following are two notable examples of Schrödinger operators with explicitly known spectra. (1) Harmonic oscillator: Choose, for simplicity, d = 1, and let ω > 0. The Schrödinger 2 operator H with V(x) = ω2 x 2 − ω2 describes the harmonic oscillator. In this case the spectrum is purely discrete. We have Spec(H) = Specd (H) = {ωn | n ∈ ℕ ∪ {0}}. (2) Hydrogen atom: Choose d = 3, and let γ > 0. The Schrödinger operator H with γ V(x) = − |x| describes the hydrogen atom. In this case Specess (H) = [0, ∞),

Specp (H) = Specd (H) = {−

γ2 | n ∈ ℕ}. 2n2

4.1 Schrödinger semigroups |

243

4.1.8 Strongly continuous operator semigroups We conclude this section by a summary of basic facts on operator semigroups. Recall from Definition 2.85 that a one-parameter family of bounded operators {St : t ≥ 0} on a Banach space ℬ is said to be a C0 -semigroup whenever (1) S0 = 1, (2) Ss St = Ss+t for all s, t, (3) St is strongly continuous in t, i. e., s-limt→0 St = S0 = 1. Every C0 -semigroup is generated by a uniquely associated operator. Definition 4.76 (Generator). Let {St : t ≥ 0} be a C0 -semigroup on a Banach space ℬ. Its generator is defined by the linear operator 1 Tf = s-lim (St f − f ) t↓0 t with domain D(T) = {f ∈ ℬ | s-limt↓0 1t (St f − f ) exists}. Note that T is densely defined, is closed, and determines the semigroup uniquely by the expression St = e−tT , for every t ≥ 0. Proposition 4.77. Let {St : t ≥ 0} be a C0 -semigroup on Banach space ℬ with generator T. The following properties hold for all t ≥ 0: d (1) St D(T) ⊂ D(T) and dt St φ = TSt φ; (2) there exist M ≥ 1 and a > 0 such that ‖St ‖ ≤ Meat ; t (3) ∫0 Sr φdr ∈ D(T) for all φ ∈ ℬ; t

t

(4) St φ − φ = T ∫0 Sr φdr; moreover St φ − φ = ∫0 Sr Tφdr if φ ∈ D(T); (5) if T is a bounded operator, then D(T) = ℬ and ‖St −S0 ‖ ≤ the map t 󳨃→ St is uniformly continuous.

M at (e −1)‖T‖. In particular a

With a given self-adjoint operator A, the semigroups e−itA and e−tA can be defined through the spectral resolution (4.1.19). In particular, {e−itA : t ∈ ℝ} is a strongly continuous one-parameter unitary group, and {e−tA : t ≥ 0} is a strongly continuous oneparameter semigroup whenever A is bounded below. Example 4.78 (Heat kernels). Some specific choices of interest of self-adoint generators T in L2 (ℝd ) giving rise to C0 -semigroups are of the form T = T(−i∇). By the Fourier transform F on L2 (ℝd ), it gives e−tT = Fe−tT(k) F −1 and represented by an integral operator with an integral kernel e−tT (x, y) = (Fe−tT(k) )(x − y), i. e., (e−tT f )(x) = ∫ e−tT (x, y)f (y)dy. ℝd

(1) T = − 21 Δ, has the integral kernel e(1/2)tΔ (x, y) =

1 exp(−|x − y|2 /2t); (2πt)d/2

244 | 4 Feynman–Kac formulae (2) T = (−Δ)α/2 , 0 < α < 2, has the integral kernel α/2

e−t(−Δ) (x, y) =

α 1 ∫ e−t|k| +ik⋅(x−y) dk; d (2π)

ℝd

moreover, for α = 1 the explicit formula 1/2

e−t(−Δ) (x, y) = Γ (

d+1 1 t ) (d+1)/2 2 2 π (t + |x − y|2 )(d+1)/2

holds; (3) T = (−Δ + m2 )1/2 − m, m > 0, has the integral kernel 2 1/2

e−t((−Δ+m )

−m)

(x, y) =

2 2 2 1 t √ 2 ∫ emt− (t +|x−y| )(|k| +m ) dk. d (2π) √t 2 + |x − y|2 ℝd

The cases above are examples of Feller semigroups associated with Brownian motion, α-stable processes, and relativistic stable processes, respectively. Conversely, given a semigroup here we present some basic results on their generators, of which we will often make use. Proposition 4.79 (Stone’s theorem). Let {Ut : t ∈ ℝ} be a strongly continuous oneparameter unitary group on a Hilbert space. Then there exists a unique self-adjoint operator A such that Ut = eitA , t ∈ ℝ. Proposition 4.80 (Hille–Yosida theorem). A linear operator T is the generator of a C0 -semigroup {St : t ≥ 0} on a Banach space if and only if (1) T is a densely defined closed operator; (2) there exists a ∈ ℝ such that (a, ∞) ⊂ ρ(T) and ‖(a − T)−m ‖ ≤

M , (λ − a)m

λ > a,

m = 1, 2, . . . ,

where ρ(T) denotes the resolvent set of T. Proposition 4.81 (Semigroup version of Stone’s theorem). Let {St : t ≥ 0} be a symmetric C0 -semigroup on a Hilbert space. Then there exists a self-adjoint operator A bounded below such that St = e−tA , t ≥ 0. Proof. Define the operator A by 1 −Af = s-lim (St − 1)f , t→0 t

󵄨󵄨 1 D(A) = {f 󵄨󵄨󵄨 s-lim (St − 1)f exists} . 󵄨 t→0 t

By the Hille–Yosida theorem and the fact that St is symmetric, A is a closed symmetric operator and for some a ∈ ℝ the resolvent ρ(A) contains (−∞, a). It is a standard

4.1 Schrödinger semigroups | 245

fact that the spectrum of a closed symmetric operator is one of the following cases: (a) the closed upper half-plane, (b) the closed lower-half plane, (c) the entire plane, (d) a subset of ℝ. Hence A must be (d), which implies that A is self-adjoint and bounded below. Since a given generator gives rise to a unique C0 -semigroup, we have St = e−tA . In case {St : t ≥ 0} is not a semigroup while s-limt↓0 t −1 (St − 1) does exist, the following result can be used. Proposition 4.82. Let A be a positive self-adjoint operator and ϱ(t) a family of selfadjoint operators with 0 ≤ ϱ(t) ≤ 1. Suppose that St = t −1 (1 − ϱ(t)) converges to A as t ↓ 0 in strong resolvent sense. Then s-lim ϱ(t/n)n = e−tA . n→∞

Proof. By the strong resolvent convergence of St/n to A, see Definition 4.15, we have that s-limn→∞ exp (−tSt/n ) = e−tA . It suffices to show that s-lim(exp(−tSt/n ) − ϱ(t/n)n ) = s-lim Gn (tSt/n ) = 0, n→∞

n→∞

where Gn (x) = {

e−x − (1 − x/n)n , e , −x

0 ≤ x ≤ n, x ≥ n.

It is easily seen that limn→∞ ‖Gn ‖∞ = 0 and the proposition follows. Next we consider semigroups generated by Schrödinger operators. Definition 4.83 (Schrödinger semigroup). The one-parameter operator semigroup {e−tH : t ≥ 0} defined by a Schrödinger operator H is a Schrödinger semigroup. Due to the additive structure of Schrödinger operators the following result will also be useful. Proposition 4.84 (Trotter product formula). Let A and B be positive self-adjoint operators such that A + B is essentially self-adjoint on D(A) ∩ D(B). Then s-lim(e−(t/n)A e−(t/n)B )n = e−t(A+B) , n→∞

where A + B denotes the closure of (A + B)⌈D(A)∩D(B) . Proof. Let ϱ(t) = e−(t/2)A e−tB e−(t/2)A . We have (e−(t/n)A e−(t/n)B )n = e−(t/2n)A ϱ(t/n)n−1 e−(t/2n)A e−(t/n)B .

(4.1.44)

246 | 4 Feynman–Kac formulae Since e−tA and e−tB are uniformly bounded in t ≥ 0 and strongly converge to the identity operator as t ↓ 0, it suffices to show that s-lim ϱ(t/n)n−1 = e−tC , n→∞

where C = A + B. We know that t −1 (1 − ϱ(t)) is a bounded self-adjoint operator and t −1 (1 − ϱ(t))ϕ = e−(t/2)A e−tB t −1 (1 − e−(t/2)A )ϕ + e−(t/2)A t −1 (1 − e−tB )ϕ + t −1 (1 − e−(t/2)A )ϕ → Cϕ as t → ∞, for ϕ ∈ D(A) ∩ D(B), which implies that t −1 (1 − ϱ(t)) → C in strong resolvent sense. Thus s-limn→∞ ϱ(t/n)n = e−tC follows by Proposition 4.82. Furthermore, s-lim(ϱ(t/n)n−1 − e−tC ) = s-lim ϱ(t/n)n−1 (1 − ϱ(t/n)) + s-lim(ϱ(t/n)n − e−tC ) = 0. n→∞

n→∞

n→∞

Hence the proposition follows. The Trotter product formula can further be extended to more general cases. Proposition 4.85. Let A and B be positive self-adjoint operators in a Hilbert space H with form domains Q(qA ) and Q(qB ). Denote by PAB the projection to Q(qA ) ∩ Q(qB ). Then s-lim(e−(t/n)A e−(t/n)B )n = e−t(A + B) PAB , ̇

n→∞

(4.1.45)

where A +̇ B is the quadratic form sum of A and B, i. e., the self-adjoint operator in PAB H associated with the densely defined closed quadratic form f 󳨃→ ‖A1/2 f ‖2 + ‖B1/2 f ‖2 .

4.2 Feynman–Kac formula for Schrödinger operators 4.2.1 Bounded smooth external potentials Now we turn to discussing probabilistic representations of Schrödinger semigroups given by Feynman–Kac formulae, which are the main object of this book. We begin by assuming that V ∈ C0∞ (ℝd ), which is the simplest case. For such a choice of potential the Schrödinger operator H is self-adjoint on D(H) = H 2 (ℝd ) and bounded from below by the Kato–Rellich theorem. Therefore the semigroup {e−tH : t ≥ 0} can be defined by the spectral resolution and the solution of (4.1.7) with initial condition g is given by f (x, t) = (e−tH g)(x). We will denote Brownian motion throughout by (Bt )t≥0 as discussed in Chapter 2, and its augmented filtration (FtBM )t≥0 as given by Definition 2.109 by simply (Ft )t≥0 , unless any confusion may arise. Theorem 4.86 (Feynman–Kac formula for Schrödinger operator). If V ∈ C0∞ (ℝd ), then for f , g ∈ L2 (ℝd ), t

(f , e−tH g) = ∫ 𝔼x [f (B0 )e− ∫0 V(Bs ) ds g(Bt )]dx ℝd

(4.2.1)

4.2 Feynman–Kac formula for Schrödinger operators |

247

holds. In particular, t

(e−tH g)(x) = 𝔼x [e− ∫0 V(Bs ) ds g(Bt )].

(4.2.2)

Proof. Define the map Kt on L2 (ℝd ) by t

(Kt f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )]. Since ‖Kt f ‖2 ≤ e−2 inf V ‖f ‖2 , the operator Kt is bounded on L2 (ℝd ) for every t ≥ 0. First we show that {Kt : t ≥ 0} is a symmetric C0 -semigroup on L2 (ℝd ). Define B̃ s = Bt−s − Bt , d s < t, for a fixed t > 0. By Proposition 2.100 we have B̃ s = Bs . Thus t

(f , Kt g) = ∫ 𝔼[f (x)e− ∫0 V(Bs +x)ds g(Bt + x)]dx ℝd t

= 𝔼[ ∫ f (x)e− ∫0 V(Bs +x)ds g(B̃ t + x)dx]. ̃

ℝd

Changing the variable x to y = B̃ t + x, we obtain t

̃ ̃ (f , Kt g) = 𝔼 [ ∫ f (y − B̃ t )e− ∫0 V(Bs −Bt +y)ds g(y)dy] [ℝd ] t

= ∫ 𝔼[f (y + Bt )e− ∫0 V(Bt−s +y)ds g(y)]dy = (Kt f , g),

(4.2.3)

ℝd t

i. e., Kt is symmetric. Write now Zt = e− ∫0 V(Bs )ds . The semigroup property follows directly from the Markov property of Brownian motion, i. e., for all s, t ≥ 0, (Ks Kt f )(x) = 𝔼x [Zs 𝔼Bs (Zt f (Bt ))] t

= 𝔼x [𝔼x [Zs e− ∫0 V(Bs+u )du f (Bs+t )|Fs ]] = 𝔼x [Zs+t f (Bs+t )] = Ks+t f (x). Strong continuity is implied by t

‖Kt f − f ‖ ≤ 𝔼[‖e− ∫0 V(⋅+Bs )ds f (⋅ + Bt ) − f ‖] → 0 as t ↓ 0, by using the dominated convergence theorem. We now know that on both sides of (4.2.1) there is a C0 -semigroup. What remains to show is that their generators are equal, i. e., 1t (Kt −1) converges to −H strongly in L2 on D(H). We will use Itô calculus for this. Suppose that f ∈ C0∞ (ℝd ) and put Yt = f (Bt ); then the Itô formula gives dZt = −VZt dt and dYt = ∇f ⋅ dBt + 21 Δfdt. By the product rule (2.4.44), 1 d(Zt Yt ) = Zt (−Vf (Bt ) + Δf (Bt )) dt + Zt ∇f (Bt ) ⋅ dBt 2 = −Zt (Hf )(Bt )dt + Zt ∇f (Bt ) ⋅ dBt .

(4.2.4)

248 | 4 Feynman–Kac formulae Taking expectation with respect to Wiener measure on both sides of the integrated t form of (4.2.4) and using 𝔼x [∫0 Zs ∇f ⋅ dBs ] = 0, we obtain t

t

x

(Kt f )(x) = f (x) − ∫ 𝔼 [Zs (Hf )(Bs )]ds = f (x) − ∫ Ks Hf (x)ds. 0

(4.2.5)

0

For f ∈ D(H) and ε > 0 there exists s0 > 0 with ‖Ks Hf − Hf ‖ ≤ ε, for all s < s0 . Thus by (4.2.5), with t < s0 t

󵄩󵄩 1 󵄩 󵄩󵄩 (K f − f ) + Hf 󵄩󵄩󵄩 ≤ 1 ∫ ‖−K Hf + Hf ‖ds ≤ ε 󵄩󵄩 󵄩󵄩 t s 󵄩t 󵄩 t 0

is obtained. Sine C0∞ (ℝd ) is a core of H by the Kato–Rellich theorem, for f ∈ D(H) there exists a sequence (fn )n∈ℕ ⊂ C0∞ (ℝd ) such that fn → f and Hfn → Hf strongly as n → ∞. Let Mt = 1t (Kt − I). We have 1t (Kt − I)f = 1t (Kt − I)fn + Mt (f − fn ). It can be seen that t 󵄩󵄩 󵄩2 󵄩󵄩𝔼[(e− ∫0 V(Bs +x)ds − 1)((f − f )(B + x))]󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n t 󵄩 󵄩 1 − ∫0t V(Bs +x)ds 2 ≤ ∫ 𝔼 [ 2 |e − 1| ] 𝔼[|(f − fn )(Bt + x)|2 ]dx t

‖Mt (f − fn )‖2 =

1 t2

ℝd 2 2tδ

≤ δ e ‖f − fn ‖2 , where δ = ‖V‖∞ . Let η > 0, and suppose that ‖f − fn ‖ ≤ η and ‖Hf − Hfn ‖ ≤ η for all n > N with some N. We conclude that for n > N, 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 (Kt − I)f + Hf 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 (Kt − I)fn + Hfn 󵄩󵄩󵄩 + ‖Mt (f − fn )‖ + ‖Hf − Hfn ‖ 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩󵄩󵄩 1 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩 (Kt − I)fn + Hfn 󵄩󵄩󵄩 + δetδ ‖f − fn ‖ + ‖Hf − Hfn ‖ ≤ 󵄩󵄩󵄩 (Kt − I)fn + Hfn 󵄩󵄩󵄩 + (δetδ + 1)η 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 uniformly in t. Thus we have limt→∞ ‖ 1t (Kt − I)f + Hf ‖ ≤ (δetδ + 1)η for every η > 0. This shows strong convergence of 1t (Kt − 1)f to −Hf for f ∈ D(H) as t ↓ 0.

Definition 4.87 (Feynman–Kac semigroup). The symmetric C0 -semigroup defined by t

L2 (ℝd ) ∋ f 󳨃→ Kt f = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )], t ≥ 0, is called the Feynman–Kac semigroup for the given V. It is seen that the solution of (4.1.7), like the solution of the heat equation, can be obtained by running a Brownian motion with an additional exponential weight applied. This gives a probabilistic representation of the solution or, equivalently, of the kernel of the semigroup e−tH . Using this picture we see that Brownian paths spending a long time in a region where V is large are exponentially penalized and thus their contribution to the expected value will be accordingly small. The dominant contribution will therefore come from paths that spend most of their time near the lowest values of V.

4.2 Feynman–Kac formula for Schrödinger operators |

249

4.2.2 Derivation through the Trotter product formula By using the Trotter product formula given in Proposition 4.84 we can give another proof of Theorem 4.86. Alternate proof of Theorem 4.86. Write H0 = −(1/2)Δ, let f1 , . . . , fn−1 ∈ L∞ (ℝd ), f0 , fn ∈ L2 (ℝd ), and write e−(t1 −t0 )H0 f1 e−(t2 −t1 )H0 f2 ⋅ ⋅ ⋅ e−(tn −tn−1 )H0 fn as ∏nj=1 e−(tj −tj−1 )H0 fj for notational convenience. By the finite-dimensional distributions of Brownian motion, see (2.3.4)–(2.3.5), for 0 = t0 ≤ t1 ≤ . . . ≤ tn we have n

n

j=1

j=1

n

n

( ∏ e−(tj −tj−1 )H0 fj )(x) = 𝔼x [ ∏ fj (Btj )]

(4.2.6)

and (f0̄ , ∏ e−(tj −tj−1 )H0 fj ) = ∫ 𝔼x [ ∏ fj (Btj )]dx. j=1

(4.2.7)

j=0

ℝd

The Trotter product formula combined with (4.2.7) gives n

(f , e−tH g) = lim (f , (e−(t/n)H0 e−(t/n)V )n g) = lim ∫ 𝔼x [f (B0 )g(Bt )e−(t/n) ∑j=1 V(Btj/n ) ]dx. n→∞

n→∞

ℝd

Since V is continuous and Bs is almost surely continuous in s, t

t n ∑ V(Btj/n ) → ∫ V(Bs )ds n j=1 0

almost surely as n → ∞. Thus we have t

(f , e−tH g) = ∫ 𝔼x [f (B0 )g(Bt )e− ∫0 V(Bs )ds ]dx. ℝd

Having the Feynman–Kac formula for smooth bounded potentials at hand, we are now interested in extending it to wider classes of V. We present here a first extension covering potentials as singular as V(x) = −1/|x|3 (see Section 4.3.3). Theorem 4.88 (Feynman–Kac formula for Schrödinger operator with singular external potential). Suppose that for V = V+ − V− we have V+ ∈ L1loc (ℝd \ S), where S is a closed set of measure zero, and V− is relatively form-bounded with respect to −(1/2)Δ with a relative bound strictly less than 1. Define H = −(1/2)Δ +̇ V+ −̇ V− . Then (4.2.1) holds. Proof. Suppose that V ∈ L∞ and Vn = ϕ(x/n)(V ∗ jn ), where jn = nd ϕ(xn) with ϕ ∈ C0∞ (ℝd ) such that 0 ≤ ϕ ≤ 1, ∫ℝd ϕ(x)dx = 1 and ϕ(0) = 1. We see that Vn (x) → V(x)

250 | 4 Feynman–Kac formulae almost everywhere. There is a set N ⊂ ℝd of Lebesgue measure zero such that Vn (x) → V(x) for x ∉ N , and 𝔼x [1{Bs ∈N } ] = ∫ 1N (y)Πt (y − x)dy = 0 ℝd

for every x ∈ N . We see that for every x ∈ N , t

x

x

t

0 = ∫ 𝔼 [1{Bs ∈N } ]ds = 𝔼 [∫ 1{Bs ∈N } ] ds 0 [0 ] by Fubini’s Theorem. Thus for every x ∈ N it can be seen that the Lebesgue measure of the set {t ∈ [0, ∞) | Bt (ω) ∈ N } is zero for almost every ω ∈ X . Hence it follows t t that ∫0 Vn (Bs )ds → ∫0 V(Bs )ds almost surely under 𝒲 x for x ∉ N . Therefore, by the dominated convergence theorem, t

t

∫ 𝔼x [f (B0 )g(Bt )e− ∫0 Vn (Bs )ds ]dx → ∫ 𝔼x [f (B0 )g(Bt )e− ∫0 V(Bs )ds ]dx ℝd

ℝd

as n → ∞. On the other hand, e−t(−(1/2)Δ+Vn ) → e−t(−(1/2)Δ+V) strongly, since −(1/2)Δ + Vn converges to −(1/2)Δ + V on a common core. Thus (4.2.1) holds for V ∈ L∞ (ℝd ). Now assume that V+ ∈ L1loc (ℝd \S) and V− is relatively form-bounded with respect to −(1/2)Δ. Let V+n (x) = {

V+ (x), n,

V+ (x) < n,

and

V+ (x) ≥ n,

V−m (x) = {

V− (x), m,

V− (x) < m, V− (x) ≥ m.

Write Vn,m = V+n − V−m . We have t

(f , e−t(−(1/2)Δ+Vn,m ) g) = ∫ 𝔼x [f (B0 )e− ∫0 Vn,m (Bs ) ds g(Bt )]dx.

(4.2.8)

ℝd

Denote h = −(1/2)Δ, and define the closed quadratic forms 1/2 1/2 1/2 1/2 qn,m (f , f ) = (h1/2 f , h1/2 f ) + (V+n f , V+n f ) − (V−m f , V−m f ),

1/2 1/2 qn,∞ (f , f ) = (h1/2 f , h1/2 f ) + (V+n f , V+n f ) − (V−1/2 f , V−1/2 f ),

q∞,∞ (f , f ) = (h1/2 f , h1/2 f ) + (V+1/2 f , V+1/2 f ) − (V−1/2 f , V−1/2 f ), whose form domains are respectively Q(qn,m ) = Q(h), Q(qn,∞ ) = Q(h), and Q(q∞,∞ ) = Q(h)∩Q(V+ ). Clearly, qn,m ≥ qn,m+1 ≥ qn,m+2 ≥ . . . ≥ qn,∞ and qn,m → qn,∞ in the sense of quadratic forms on ⋃∞ m=1 Q(qn,m ) = Q(h). Since qn,∞ is closed on Q(h), by Theorem 4.65 the associated positive self-adjoint operators satisfy h + V+n − V−m → h + V+n −̇ V− in strong resolvent sense, which implies that for all t ≥ 0, exp (−t (h + V+n − V−m )) → exp (−t (h + V+n −̇ V− ))

(4.2.9)

4.2 Feynman–Kac formula for Schrödinger operators | 251

strongly as m → ∞. Similarly, we have qn,∞ ≤ qn+1,∞ ≤ qn+2,∞ ≤ . . . ≤ q∞ and qn,∞ → q∞,∞ in form sense on {f ∈ ⋂∞ n=1 Q(qn,∞ ) | supn qn,∞ (f , f ) < ∞} = Q(h) ∩ Q(V+ ). Hence by Theorem 4.65 again we obtain exp(−t(h +̇ V+n −̇ V− )) → exp(−t(h +̇ V+ −̇ V− )), t ≥ 0,

(4.2.10)

in strong sense as n → ∞. By taking first m → ∞ and then n → ∞, it can be proven that both sides of (4.2.8) converge, i. e., the left-hand side of (4.2.8) converges by monotone convergence for forms by (4.2.9) and (4.2.10). On the other hand, let f , g ∈ L2 (ℝd ) and f , g ≥ 0. We have t

t

lim lim ∫ 𝔼x [f (B0 )e− ∫0 Vn,m (Bs )ds g(Bt )]dx = ∫ 𝔼x [f (B0 )e− ∫0 V(Bs )ds g(Bt )]dx

n→∞ m→∞

ℝd

ℝd

follows from the monotone convergence theorem for integrals for the limit m → ∞ and the dominated convergence theorem for the limit n → ∞, and hence it can be t

seen that the right-hand side of (4.2.8) converges to ∫ℝd 𝔼x [f (B0 )e− ∫0 V(Bs )ds g(Bt )]dx for

any f , g ∈ L2 (ℝd ) by the decomposition f = ℜf + iℑf , ℜf = ℜf+ − ℜf− , ℑf = ℑf+ − ℑf− , g = ℜg + iℑg, ℜg = ℜg+ − ℜg− , and ℑg = ℑg+ − ℑg− . 4.2.3 Kato-class potentials The Feynman–Kac formula we have derived so far covers only a limited range of interesting models of mathematical physics. First, the potential was not allowed to grow at infinity. This excludes cases when the particle is trapped in some region of ℝd , like under the harmonic potential V(x) = |x|2 . Secondly, it is desirable to allow local singularities, like for instance in the case of the Coulomb potential V(x) = −1/|x| in three dimensions. To include these and further cases of interest, we introduce here a large class of potentials and use it as a space of reference as far as we can. This space is Kato-class which on several counts is a natural function space, as will be made clear below. However, occasionally we will have to leave this space in order to draw stronger conclusions or because the conditions of Kato-class are too strong themselves for a particular statement we want to prove. Our objective in this section is to introduce Kato-class and discuss its basic properties, then define Schrödinger operators with Kato-class potentials as self-adjoint operators, and finally derive a corresponding Feynman–Kac formula. Definition 4.89 (Kato-class). (1) A potential V : ℝd → ℝ is called a Kato-class potential whenever lim sup ∫ |g(x − y)V(y)| dy = 0 r↓0 x∈ℝd

Br (x)

(4.2.11)

252 | 4 Feynman–Kac formulae holds, where Br (x) is the closed ball of radius r centered at x, and |x|, d = 1, { { { g(x) = {− log |x|, d = 2, { { 2−d d ≥ 3. {|x| ,

(4.2.12)

We denote this linear space by 𝒦(ℝd ). (2) V is local Kato-class if and only if 1K V ∈ 𝒦(ℝd ) for any compact set K ⊂ ℝd . The set of local Kato-class potentials is denoted by 𝒦loc (ℝd ). (3) V is Kato-decomposable whenever V = V+ − V− with V− ∈ 𝒦(ℝd ) and V+ ∈ 𝒦loc (ℝd ). For d = 1 we have an equivalent definition of Kato-class which we will use below. Proposition 4.90. Let d = 1. The potential V is of Kato-class if and only if sup ∫ |V(y)|dy < ∞. x∈ℝ

(4.2.13)

B1 (x)

Proof. Suppose (4.2.13) holds. We have sup ∫ |x − y||V(y)|dy ≤ r sup ∫ |V(y)|dy x∈ℝ

x∈ℝ

Br (x)

B1 (x)

for 0 < r < 1, thus V is of Kato-class. Next suppose that V is of Kato-class. For every ε > 0 there exists r > 0 such that supx∈ℝ ∫B (x) |x − y||V(y)|dy < ε. It follows that r

r sup 2 x∈ℝ

∫

|V(y)|dy ≤ sup x∈ℝ

r/2≤|x−y|≤r

∫

|x − y||V(y)|dy ≤ ε

r/2≤|x−y|≤r

and therefore sup x∈ℝ

∫

|V(y)|dy ≤

r/2≤|x−y|≤r

2ε . r

Since B1 (0) can be covered by a finite number N of rings, we have B1 (0) ⊂ ⋃Nj=1 Aj and Aj = {y | r/2 ≤ |xj − y| ≤ r} with suitable points xj . This then gives N

sup ∫ |V(y)|dy ≤ ∑ sup ∫ |V(y)|dy ≤ x∈ℝ

B1 (x)

j=1 x∈ℝ A +x j

2Nε < ∞. r

Before deriving a Feynman–Kac formula for such potentials it is useful to discuss some basic properties of Kato-class. We begin by explaining the definition. Let Πt be

4.2 Feynman–Kac formula for Schrödinger operators | 253

the probability transition kernel of Brownian motion given by (2.3.3), and f a nonnegative function. By Fubini’s Theorem we have ∞

𝔼x [ ∫ f (Bt )dt ] = ∫ H(x, y)f (y)dy, [0 ] ℝd

(4.2.14)

with integral kernel H(x, y) = ∫0 Πt (x, y)dt. In the limit t → ∞, Πt (x) behaves like ∞

(2πt)−d/2 , thus H diverges for d = 1 and 2. For d ≥ 3, on making the change of variable |x − y|2 /(2t) 󳨃→ t, Γ(d/2 − 1) 1 2π d/2 |x − y|d−2

H(x, y) =

(4.2.15)

is obtained, where Γ is the Gamma function. For lower dimensions we proceed by com̃ y) = ∫∞ (Πt (x, y) − h(t)) dt. For d = 1 pensation with a suitable additive term. Let H(x, 0 we choose h(t) = Πt (0, 0) = (2πt)−1/2 , yielding ̃ y) = H(x,

2

∞

1 e−|x−y| /(2t) − 1 dt = −|x − y| ∫ √2π √t 0

by making the same change of variable as above. For d = 2 write h(t) = Πt (0, v), with unit vector v = (01 ). We have ∞

2

−|x−y| /(2t) − e−1/(2t) 1 ̃ y) = 1 ∫ e H(x, dt = − log |x − y|. 2π t π 0

This shows that g in Definition 4.89 is actually given by the potential kernels of the Laplacian. The conditions in (4.2.12) limit the growth at infinity (𝒦(ℝd )) and the severity of local singularities (L1loc (ℝd )). These are illustrated by the following examples. Example 4.91. Let d ≥ 2. Kato-class allows singularities of the type |x|−α with any 0 ≤ α < 2, including in particular Coulomb potential V(x) = 1/|x|. In contrast, the function V(x) = |x| is not contained in 𝒦(ℝd ). Example 4.92. An application of the Hölder inequality gives an idea about the growth at infinity allowed. Let d = 3; then 1/p 1/q 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 1 q 󵄨󵄨 ∫ 󵄨 |V(y)| dy󵄨󵄨 ≤ ( ∫ dy) ( ∫ |V(y)| dy) , 󵄨󵄨 󵄨󵄨 |x − y| |x − y|p 󵄨󵄨 󵄨 Br (x) Br (x) Br (x)

with 1/p + 1/q = 1. The first factor is independent of x and when p < 3, it goes to zero as r → 0. Thus V ∈ 𝒦(ℝ3 ) if supx∈ℝ3 ∫B (x) |V(y)|q dy < ∞ for some r > 0 and some r q > 3/2. A direct calculation shows, for instance, that V(x) = |x1 |α e−|x1 |

β

(|x2 |2 +|x3 |2 )

,

x = (x1 , x2 , x3 )

254 | 4 Feynman–Kac formulae is in 𝒦(ℝ3 ) if β > 3α/2. This shows that Kato-class is essentially an integrability condition; pointwise divergence is allowed as long as the given integrals can be controlled. Example 4.93. Let V ∈ Lp (ℝd ) + L∞ (ℝd ) with p > d/2 and 1 ≤ p < ∞. Then V ∈ 𝒦(ℝd ); see Lemma 4.105. From the definiton of 𝒦(ℝd ) we furthermore obtain the properties below. Proposition 4.94 (Properties of 𝒦(ℝd )). (1) Let V, W ∈ 𝒦(ℝd ) and f , g be real-valued continuous functions on ℝd . Then fV + gW ∈ 𝒦(ℝd ). (2) 𝒦(ℝd ) ⊂ L1loc (ℝd ). (3) If V ∈ 𝒦(ℝd ), then supx∈ℝd ∫B (x) |V(y)| dy < ∞. 1

Proof. (1) follows directly from the definition of 𝒦(ℝd ). Consider statment (2). For d = 1 the statement in trivial by (4.2.13). Let d ≥ 2, K ⊂ ℝd be a compact set, and take ε > 0. There exists r > 0 such that supx∈ℝd ∫B (x) |g(x − y)V(y)| dy < ε. Let {a1 , . . . , an } ⊂ K be such that ∪nj=1 Bε (aj ) ⊃ K. We see that

r

n

n

∫ |V(y)|dy ≤ ∑ ∫ |V(y)|dy = C ∑ ∫ g(aj − y)|V(y)|dx ≤ nCε < ∞ K

j=1 B (a ) ε j

j=1 B (a ) ε j

with a constant C indepedent of {a1 , . . . , an }. Hence (2) follows. To show statment (3) set K = B1 (x) in the proof of (2). There exists a natural number n independent of x and {b1 , . . . , bn } ⊂ B1 (x) such that ∪nj=1 Bε (bj ) ⊃ B1 (x). In the same way as for (2) we have ∫B (x) |V(y)|dy ≤ nCε < ∞. 1

If V is Kato-decomposable, the positive part of V is in L1loc (ℝd ). We also have the following property. t

Lemma 4.95. Suppose that 0 ≤ V ∈ L1loc (ℝd ). Then 𝒲 x (∫0 V(Bs )ds = ∞) = 0 for a. e. x ∈ ℝd . In particular, 𝔼x [e

t − ∫0

V(Bs )ds

] > 0 for a. e. x ∈ ℝd .

Proof. Let N ∈ ℕ and 1N (x) = {

1, 0,

|x| < N, |x| ≥ N.

Since 1N V ∈ L1 (ℝd ), we have x

t

∫ 𝔼 [∫ 1N V(Bs )ds] dx ≤ t‖1N V‖L1 . [0 ] ℝ3 t

Hence for almost every fixed x ∈ ℝd , 𝒲 x (∫0 1N V(Bs )ds < ∞) = 1, and then there t

exists NN , dependent on x, such that 𝒲 x (NN ) = 0 and ∫0 1N V(Bs (ω))ds < ∞, for every

4.2 Feynman–Kac formula for Schrödinger operators | 255

t

ω ∈ X \ NN . Let N = ⋃∞ N=1 NN . Note that N depends on x. Thus ∫0 1N V(Bs (ω))ds < ∞, for every N ∈ ℕ and ω ∈ X \ N . Since Bs (ω) is continuous in s almost surely, for every ω ∈ X \ N there exists N = N(ω) ∈ ℕ such that V(Bs (ω)) = 1N V(Bs (ω)), for all 0 ≤ s ≤ t. Thus t

t

∫ V(Bs (ω))ds = ∫ 1N V(Bs (ω))ds < ∞, 0

ω∈X \N .

0

Next we present a first equivalent characterization of Kato-class. Proposition 4.96. A nonnegative function V is in 𝒦(ℝd ) if and only if t

x

lim sup 𝔼 [∫ V(Bs ) ds] = 0. t↓0 x∈ℝd [0 ]

(4.2.16)

Proof. As in (4.2.14), write for t ∈ ℝ t

x

t

𝔼 [∫ V(Bs ) ds] = ∫ ds ∫ Πs (x − y)V(y)dy. [0 ] 0 ℝd By the same change of variable as above, t

∫ Πs (x − y)ds = 0

1 1 (2π)d/2 |x − y|d−2

∞

∫

e−u ud/2−2 du = H(x, y; t)

|x−y|2 /(2t)

readily follows. The key observation now is that H(x, y; t) behaves like g(x − y) in Definition 4.89 whenever |x − y|2 /2t is small and vanishes rapidly when |x − y|2 /2t goes to infinity. This is obvious in the cases d ≥ 3 and is seen through integration by parts if d = 1 and 2. We give the details only in the simplest case d = 4; the others go by the same arguments, though require further estimates. The convenient feature of the case d = 4 is that we have an explicit formula for H(x, y; t) and t

|x−y|2 1 sup 𝔼x [∫ V(Bs ) ds] = sup ∫ e− 2t V(y)dy 2 4 (2π|x − y|) x∈ℝ4 [0 ] x∈ℝ ℝ4

(4.2.17)

follows. We split up the domain of integration at the right-hand side above into Aq = {y | |x − y| < t 1/q } and Bq = ℝ4 \ Aq for q > 0. Note that Aq shrinks to {x} when t → 0. If (4.2.16) holds, then the integral over A2 on the right-hand side of (4.2.17) vanishes with t → 0, uniformly in x. On A2 , exp(−|x − y|2 /2t) is bounded away from zero, hence V ∈ 𝒦(ℝ4 ) follows. Conversely, assume V ∈ 𝒦(ℝ4 ). Then by

256 | 4 Feynman–Kac formulae definition and the fact that the negative exponential is bounded from above, we obtain limt↓0 supx∈ℝ4 ∫A 2

3

2

|x−y| 1 e− 2t (2π|x−y|)2 1/6

V(y)dy = 0. On the other hand, on B3 we have

exp(−|x − y| /2t) ≤ exp(−|x − y|/t ), thus the right-hand side of (4.2.17) goes to zero on this set by dominated convergence. We present a second equivalent characterization of Kato-class recalling the asymptotic properties of the resolvent of the Laplacian. For d = 1 the kernel of the resolvent operator (−Δ + λ)−1 is explicitly given by (−Δ + λ)−1 (x, y) =

e−

√λ|x−y|

.

(4.2.18)

(−Δ + λ)−1 (x, y) = Cλ (x − y),

(4.2.19)

√λ

In what follows we assume d ≥ 2. We have

where Cλ (x) = (2π)−d/2 (

√λ |x|

(d−2)/2

)

K(d−2)/2 (√λ|x|),

(4.2.20)

with Kν (z) being the modified Bessel function of the third kind defined by Kν (z) =

1 z −ν z2 ( ) ∫ exp (− − t) t ν−1 dt. 2 2 4t ∞

(4.2.21)

0

We note that Kν (z) also has the integral representation Kν (z) =

∞

1 1 1 ∫ exp (− ( + t) z) t ν−1 dt. 2 2 t 0

We recall some further properties of Kν . It is well defined on the domain D = {z ∈ ℂ | ℜz > 0, ℜ(z 2 ) > 0} = {z ∈ ℂ | | arg z| < π/4} for any ν ∈ ℂ. The function Kν : D → ℂ is holomorphic and Kν = K−ν follows. Furthermore, it can be shown that for ℜν > −1/2 and z ∈ D, ∞

1

1 e−z s ν− 2 Kν (z) = ds. ∫ e−s sν− 2 (1 + ) 2z √2Γ(ν + 1 ) √z 2

√π

0

Inserting this identity into (4.2.20), for d ≥ 2, x ≠ 0 and λ > 0, we have (d−3)/2 e− λ|x| s −s (d−3)/2 Cλ (x) = e s (1 + ) ds, ∫ 2√λ|x| 2(d+1)/2 π (d−1)/2 Γ( d−1 ) |x|(d−1)/2 2

(√λ)(d−3)/2

√

∞

0

(4.2.22)

4.2 Feynman–Kac formula for Schrödinger operators | 257

where z α = exp(α(log |z| + i arg z)). In particular, for d = 3 we have Cλ (x) =

e− λ|x| . 4π|x| √

Take λ = 1. The kernel of the resolvent operator C1 (x), is called the Bessel potential. For λ = 0 we have C0 (x), which is the Riesz potential, that will be discussed in Section 4.9. The following estimates hold for Cλ (x) when λ > 0. Lemma 4.97. Let λ > 0. Then for every x ∈ ℝd \ {0} we have 1 1 e− λ|x| , { 2√π |λ|1/4 |x|1/2 { { { { { 3(d−3)/2 Γ(d−2) 1 |Cλ (x)| ≤ { 2d−1 π (d−1)/2 (1 + Γ( d−1 ) ) |x|d−2 , { 2 { { { { 3(d−3)/2 (1 + Γ(d−2) ) |λ|(d−3)/4 e−√λ|x| , Γ( d−1 ) |x|(d−1)/2 { 2d−1 π (d−1)/2 2 √

d = 2, d ≥ 3, √λ|x| ≤ 1, d ≥ 3, √λ|x| ≥ 1.

Proof. Let a > 0. By direct calculation ∞

∫ e−s s(d−3)/2 (1 + 0

s (d−3)/2 (3/2)(d−3)/2 (Γ( d−1 ) + Γ(d − 2)a(3−d)/2 ) 2 ) ds ≤ { √π 2a

follows. Combining this with (4.2.22) we obtain the desired results. For d = 2 and √λ|x| ≤ 1 there is a more explicit estimate. Lemma 4.98. Let d = 2, λ > 0 and √λ|x| ≤ 1. Then for x ∈ ℝ2 \ {0} we have |Cλ (x)| ≤

1 2 log ( ). √λ|x| 2π

Proof. We have directly ∞

1/λ|x|2

0

0

∞

4t 2 1 1 λ|x|2 1 − 1t 1 − λ|x| e dt + e−λ|x| ∫ e 2 dt. ∫ e− 4t −t dt ≤ ∫ 4π t t t

1/λ|x|2

By the assumption √λ|x| ≤ 1, we furthermore have 1

1/λ|x|2

∞

2 1 1 1 − 1t e−4t ≤ ∫ e− t dt + ∫ e dt + e−λ|x| ∫ dt t t t

0

1

1

∞

1/λ|x|2

∞

1

1

1

2 1 dt 1 1 ≤ ∫ e−t dt + ∫ + e−λ|x| ∫ e−4t dt ≤ 1 + log ( )+ . t t 4 λ|x|2

d ≥ 3, d=2

258 | 4 Feynman–Kac formulae Hence |Cλ (x)| ≤

1 5 1 1 e5/4 1 2 ( + log ( )) = (log ( )) ≤ (log ( )) , 2 √λ|x| 4π 4 4π 2π λ|x| λ|x|2

and the lemma follows. As |x − y| → 0, by Lemmas 4.97–4.98 we have |x − y|2−d , d ≥ 3, { { { Cλ (x − y) ∼ {− log |x − y|, d = 2, { { d = 1; {C, moreover, for every δ > 0, lim sup e|x−y| Cλ (x − y) = 0.

λ→∞ |x−y|>δ

Note that the kernel of the resolvent operator (−(1/2)Δ + λ)−1 is 2C2λ (x − y). We readily have supx∈ℝd ∫ℝd CE (x − y)|V(y)|dy → 0 as E → ∞ if and only if V ∈ 𝒦(ℝd ), and by comparison with Definition 4.89 the proposition below follows. We see that (−Δ + E)−1 |V|(x) = ∫ℝd CE (x − y)|V(y)|dy, and we use 󵄨 󵄨 ‖(−Δ + E)−1 |V|‖∞ = sup 󵄨󵄨󵄨󵄨(−Δ + E)−1 |V|(x)󵄨󵄨󵄨󵄨 . d

(4.2.23)

x∈ℝ

Proposition 4.99. We have V ∈ 𝒦(ℝd ) if and only if lim ‖(−Δ + E)−1 |V|‖∞ = 0.

(4.2.24)

E→∞

Proof. For the sake of completeness, we present a proof of equivalence between (4.2.24) and (4.2.16). Take τ in a suitable neighborhood of 0 and denote H0 = − 21 Δ. Clearly, 𝔼x [|V(Bs )|] = (e−sH0 |V|)(x). By Laplace transform, ∞

((H0 + E)−1 |V|)(x) = ∑

n=0

(n+1)T

∫ e−sE (e−sH0 |V|)(x)ds.

nT

A change of variable furthermore gives ∞

T

((H0 + E)−1 |V|)(x) = ∑ e−nTE ∫ e−sE (e−(s+nT)H0 |V|)(x)ds. n=0

0

By the semigroup property and using the integral kernel e−tH0 (x, y), we obtain T

∞

((H0 + E)−1 |V|)(x) = ∑ e−nTE ∫ (e−nTH0 )(x, y)dy ∫ e−sE (e−sH0 |V|)(y)ds. n=0

ℝd

0

4.2 Feynman–Kac formula for Schrödinger operators | 259

Take the supremum with respect y, which gives T

∞

((H0 + E) |V|)(x) ≤ ( sup ∫ e−sE (e−sH0 |V|)(y)ds) ∑ e−nTE ∫ (e−nTH0 )(x, y)dy. −1

y∈ℝd

n=0

0

ℝd

Since ∫ℝd (e−nTH0 )(x, y)dy = 1, we have T

∫ (e−nTH0 )(x, y)dy ≤

1 sup ∫ e−sE (e−sH0 |V|)(y)ds 1 − e−TE y∈ℝd 0

ℝd

≤

1 ‖(H0 + E)−1 |V|‖∞ . 1 − e−TE

Taking the supremum, this yields T

(1 − e

−TE

)‖(H0 + E) |V|‖∞ ≤ sup ∫ e−sE (e−sH0 |V|)(y)ds ≤ ‖(H0 + E)−1 |V|‖∞ . −1

y∈ℝd

0

Furthermore, trivially we have T

e

−TE

∫(e

T −sH0

|V|)(y)ds ≤ ∫ e

0

T −sE

(e

−sH0

|V|)(y)ds ≤ ∫(e−sH0 |V|)(y)ds.

0

0

T

Thus supy∈ℝd ∫0 (e−sH0 |V|)(y)ds is bounded by eTE ‖(H0 + E)−1 |V|‖∞ , for all T ≥ 0, and hence T

lim sup ∫(e−sH0 |V|)(x)ds ≤ ‖(H0 + E)−1 |V|‖∞ , T↓0 x∈ℝd

0

for all E. Therefore, if limE→∞ ‖(H0 +E)−1 |V|‖∞ = 0, then the right-hand side decreases with E, and optimizing over E implies (4.2.16). Conversely, if we assume (4.2.16), then by the above T

lim ‖(H0 + E) |V|‖∞

E→∞

−1

1 ≤ lim lim sup ∫(e−sH0 |V|)(x)ds = 0, T↓0 E→∞ 1 − e−TE x∈ℝd 0

i. e., (4.2.24) holds. The latter characterization is particularly interesting in view of the following property, which through the KLMN theorem implies that if V is Kato-class, then H is selfadjoint at least in form sense.

260 | 4 Feynman–Kac formulae Proposition 4.100. If there exist a, b > 0 and 0 < δ < 1 such that for all 0 < ε < 1 (f , |V|f ) ≤ ε‖√−Δf ‖2 + a exp(bϵ−δ )‖f ‖2 ,

f ∈ D(√−Δ),

(4.2.25)

then V ∈ 𝒦(ℝd ). Conversely, if V ∈ 𝒦(ℝd ), then D(√−Δ) ⊂ D(V 1/2 ) and V is −Δ-formbounded with infinitesimally small relative bound. Proof. Suppose that (4.2.25) holds. Since ∞

(−Δ + E)−1 |V|(x) = ∫ e−tE dt ∫ Πt (x − y)|V|(y)dy, 0

ℝd

we estimate Πt (x − y)|V(y)| for a fixed x. Let ϕ(⋅) = √Πt (x − ⋅). By the assumption (ϕ, |V|ϕ) ≤ ε‖√−Δϕ‖2 + a exp(bε−δ )‖ϕ‖2 = (ϕ, −Δϕ) + a exp(bε−δ ) ≤ ct −2 + a exp(bε−δ ) with a constant c independent of x. Hence 1

∞

0

1

󵄨󵄨 󵄨 󵄨󵄨(−Δ + E)−1 |V|(x)󵄨󵄨󵄨 ≤ (∫ + ∫ ) e−tE (ct −2 + a exp(bε−δ ))dt. 󵄨 󵄨

(4.2.26)

Taking ε = (1 + log |t|)2/(1+δ) , we have a exp(bε−δ ) ≤ t b aeb , thus (4.2.24) follows. Conversely, for V ∈ 𝒦(ℝd ) by duality it follows that ‖(−Δ + E)−1 |V|‖∞,∞ = ‖|V|(−Δ + E)−1 ‖1,1 , where ‖ ⋅ ‖p,p denotes bounded operator norm on Lp . Note that ‖(−Δ + E)−1 |V|‖∞,∞ = ‖(−Δ + E)−1 |V|‖∞ . By applying the Stein interpolation theorem to the operator-valued function F(z) = |V|z (−Δ + E)−1 |V|1−z on {z ∈ ℂ | ℜz ∈ [0, 1]}, it is seen that ‖|V|1/2 (−Δ + E)−1 |V|1/2 ‖2,2 ≤ ‖(−Δ + E)−1 |V|‖1,1 = ‖(−Δ + E)−1 |V|‖∞,∞ . Hence ‖|V|1/2 (−Δ + E)−1/2 ‖22,2 ≤ ‖(−Δ + E)−1 |V|‖∞,∞ → 0 as E → ∞. Since 1/2 ‖|V|1/2 f ‖ ≤ ‖(−Δ + E)−1 |V|‖1/2 ∞,∞ ‖(−Δ + E) f ‖,

V is −Δ-form-bounded with an infinitesimally small relative bound. Remark 4.101 (Stummel-class). Kato-class 𝒦(ℝd ) defined above can be regarded as the quadratic form analogue of Stummel-class. A potential V is in Stummel class whenever lim sup ∫ h(x − y)|V(y)|2 dy = 0

r→0 x∈ℝd

Br (x)

(4.2.27)

4.2 Feynman–Kac formula for Schrödinger operators |

261

holds, where 1 if d ≤ 3, { { { h(x) = {− log |x| if d = 4, { { 4−d if d ≥ 5. {|x|

(4.2.28)

The counterparts of Propositions 4.99–4.100 hold similarly. An equivalent condition to (4.101) is lim ‖(−Δ + E)−2 |V|2 ‖∞ = 0.

E→∞

Suppose that there are a, b > 0 and 0 < δ < 1 such that for all ε < 1 and all f ∈ D(−Δ), ‖Vf ‖2 ≤ ε‖ − Δf ‖2 + a exp(bε−δ )‖f ‖2 . Then V ∈ S(ℝd ). Remark 4.102. From the perspective of self-adjointness Kato-class is not a natural space, and certainly not the largest possible. A well-known example is the potential V(x) = |x|−2 | log |x||−γ , which is −Δ-form-bounded with relative bound 0 exactly when γ = 0, while it is of Kato-class if and only if γ > 1. t

Proposition 4.96 says that given V ∈ 𝒦(ℝd ) one can make 𝔼x [∫0 V(Bs )ds] arbitrarily small by taking t small, uniformly in x. We will now see that this implies exponent tial integrability of the random variable ∫0 V(Bs ) ds. The proof of this fact is based on Khasminskii’s lemma, which follows directly from the Markov property of Brownian motion but is so useful that we state it as a separate result. First we show a preliminary result. Lemma 4.103. Let t ≥ 0. Let f , g : X → ℝ be functions such that f is FtBM -measurable, 𝔼x [|f |] < ∞, and supy∈ℝ 𝔼y [|g|] < ∞. Let θt : X → X be a time shift given by (θt ω)(⋅) = ω(t + ⋅) and define θt∗ g(ω) = g ∘ θt (ω) = g(ω(⋅ + t)). Then 󵄨󵄨 x ∗ 󵄨󵄨 x y 󵄨󵄨𝔼 [fθt g]󵄨󵄨 ≤ 𝔼 [|f |] sup 𝔼 [|g|]. y∈ℝ

Proof. Note that |𝔼x [g(B⋅+t )|FtBM ]| = |𝔼Bt [g(B⋅ )]| ≤ supy∈ℝ 𝔼y [|g|]. Since f is FtBM measurable, |𝔼x [fθt∗ g]| = |𝔼x [f 𝔼x [θt∗ g|FtBM ]]| ≤ 𝔼x [|f | sup 𝔼y [|g|]] = sup 𝔼y [|g|]𝔼x [|f |], y∈ℝ

y∈ℝ

where we used the Markov property of Brownian motion. Lemma 4.104 (Khasminskii’s lemma). Let V ≥ 0 be a measurable function on ℝd with the property that for some t > 0 and α < 1 x

t

sup 𝔼 [∫ V(Bs ) ds] = α. x∈ℝd [0 ]

(4.2.29)

262 | 4 Feynman–Kac formulae Then

t

sup 𝔼x [e∫0 V(Bs ) ds ] ≤

x∈ℝd

1 . 1−α

(4.2.30)

Proof. By expanding the exponential in (4.2.30), it suffices to show that t

n

t

t

0

0

1 1 α ≥ sup 𝔼 [ [ ∫ V(Bs ) ds] ] = sup 𝔼x [ ∫ ds1 ⋅ ⋅ ⋅ ∫ dsn V(Bs1 ) ⋅ ⋅ ⋅ V(Bsn )]. n! n! d x∈ℝ x∈ℝd n

x

0

Since there are n! ways of ordering {s1 , . . . , sn } it suffices to prove [ ] αn ≥ sup 𝔼x [ ∫ V(Bs1 ) ⋅ ⋅ ⋅ V(Bsn )ds1 ⋅ ⋅ ⋅ dsn ] , x∈ℝd n

[Δn

(4.2.31)

]

where Δn = {(s1 , . . . , sn ) ∈ ℝ | 0 < s1 < . . . < sn }. For fixed s1 < . . . < sn−1 , Lemma 4.103 gives t

𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsn−1 ) ∫ V(Bsn ) dsn ] sn−1 [ ] x

x

t

≤ 𝔼 [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsn−1 )] sup 𝔼 [ ∫ V(Bsn ) dsn ] x∈ℝd [sn−1 ] ≤ α𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsn−1 )] .

The result follows by induction and an application of Fubini’s theorem. Now we are in the position to prove exponential integrability of the integral over the potential. As will be seen below, this is essentially all we need to prove the Feynman–Kac formula for Kato-decomposable potentials. When V− ∈ 𝒦(ℝd ), it can t

be seen that the exponent e∫0 V(Bs )ds is integrable with respect to Wiener measure so that 𝔼x [e

t ∫0

V(Bs )ds

] is bounded for all x.

Lemma 4.105. Let 0 ≤ V ∈ 𝒦(ℝd ). Then there exist β, γ > 0 such that t

sup 𝔼x [e∫0 V(Bs )ds ] < γeβt .

x∈ℝd

(4.2.32)

Furthermore, if V ∈ Lp (ℝd ) with p > d/2 and 1 ≤ p < ∞, then β ≤ c(p)1/ε Γ(ε)1/ε ‖V‖1/ε p , where ε = 1 −

d 2p

and c(p) = {

with

1 p

+

1 q

(4.2.33)

(2π)−d/2 , (2π)−d/2p qd/(2q) ,

p = 1, p>1

= 1. In particular, Lp (ℝd ) ⊂ 𝒦(ℝd ) for p > d/2 and 1 ≤ p < ∞.

(4.2.34)

4.2 Feynman–Kac formula for Schrödinger operators |

263

t

Proof. There exists t ∗ > 0 such that αt = supx∈ℝd 𝔼x [∫0 V(Bs )ds] < 1, for all t ≤ t ∗ , and αt → 0 as t → 0. By Khasminskii’s lemma we have t

sup 𝔼x [e∫0 V(Bs )ds ]
0, where [z] = max{w ∈ ℤ | w ≤ z}. Choosing γ = log{( 1−α1 ∗ )1/t t

∗

}, we arrive at (4.2.32).

(4.2.36) 1 1−αt ∗

and β =

Next we prove (4.2.33). Suppose V ∈ Lp (ℝd ) with p > d/2 and 1 ≤ p < ∞. Take p > 1. By the Hölder inequality we have 2 1 𝔼 [V(Bt )] ≤ ( ∫ e−|x−y| q/(2t) ) (2πt)d/2

x

1/q

‖V‖p .

ℝd

By direct computation 2 1 ( ∫ e−q|x−y| /(2t) dy) d/2 (2πt)

1/q

=

ℝd

qd/(2q) ; (2πt)d/(2p)

in particular, we have ‖𝔼x [V(Bt )]‖∞ ≤ c(p)t −d/(2p) ‖V‖p ,

p > 1.

Recall the Mittag-Leffler function defined by xk , Γ(1 + kb) k=0 ∞

mb (x) = ∑

where Γ denotes the Gamma function, x ∈ ℝ, and b > 0. It is known mb (x) satisfies 1/b

limx→∞ (mb (x) − b1 ex ) = 0 and there exists kb > 0 such that mb (x) ≤ kb ex

1/b

(4.2.37)

264 | 4 Feynman–Kac formulae for all x > 0. Let 0 ≤ s1 ≤ s2 ≤ . . . ≤ sk . By the Markov property of Brownian motion, 𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsk )]

= 𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsk−1 )𝔼x [V(Bsk )|ℱsBM ]] k−1

= 𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsk−1 )𝔼Bsk−1 [V(Bsk −sk−1 )]]

≤ 𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsk−1 )]c(p)‖V‖p (sk − sk−1 )−d/(2p)

≤ . . . ≤ c(p)k ‖V‖kp s−d/(2p) (s2 − s1 )−d/(2p) ⋅ ⋅ ⋅ (sk − sk−1 )−d/(2p) . 1 Hence it follows that k

t

t

t

t

[1 ] 𝔼 [ [∫ V(Bs )ds] ] = ∫ ds1 ∫ ds2 ⋅ ⋅ ⋅ ∫ 𝔼x [V(Bs1 ) ⋅ ⋅ ⋅ V(Bsk )]dsk k! s1 sk−1 ] ] 0 [ [0 x

t

t

≤ c(p)k ‖V‖kp ∫ ds1 ⋅ ⋅ ⋅ ∫ s−d/(2p) (s2 − s1 )−d/(2p) ⋅ ⋅ ⋅ (sk − sk−1 )−d/(2p) dsk 1 0

ε

= where ε = 1 −

(c(p)‖V‖p t Γ(ε)) Γ(1 + kε)

d 2p

k

sk−1

,

> 0 and we used the formula 1

∫ z 1+a (1 − z)a dz = 0

Γ(a)Γ(1 + a) , Γ(1 + 2a)

giving the relationship between the Gamma and Beta functions. By (4.2.37) it further follows that t

∞

sup 𝔼x [e∫0 V(Bs )ds ] ≤ ∑

x∈ℝd

k=0

(c(p)‖V‖p t ε Γ(ε))k Γ(1 + kε)

= mε (c(p)‖V‖p Γ(ε)t ε ) ≤ kε ec(p)

1/ε

1/ε ‖V‖1/ε p Γ(ε) t

.

Hence (4.2.33) is proven. Let p = 1. Then d = 1, and it follows directly that 𝔼x [V(Bt )] ≤ (2πt)−1/2 ‖V‖1 = c(1)t −1/2 ‖V‖1 .

(4.2.38)

Thus (4.2.33) is obtained in a similar way as for p > 1. 4.2.4 Feynman–Kac formula for Kato-decomposable potentials When V is Kato-decomposable, the operator − 21 Δ + V is not always an easy object to define. We have seen in Sections 4.1.5–4.1.6 that much analytic machinery is needed

4.2 Feynman–Kac formula for Schrödinger operators |

265

to decide whether it is essentially self-adjoint and on which domain. The strategy we will adopt here is to define the Feynman–Kac semigroup as given in Definition 4.87 for Kato-decomposable potentials, and identify Schrödinger operators for such potentials as generators of this semigroup. Let t

(Kt f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )]. By Lemma 4.105 the map Kt : L∞ → L∞ is bounded and linear whenever V is Katodecomposable. In fact, an even stronger property holds. Definition 4.106. Let (M, dm) be a measurable space. An operator T is said to be Lp -Lq bounded if there exists D ⊂ Lp (M) such that it is dense for all 1 ≤ p ≤ ∞, and T⌈D : Lp (M) → Lq (M) is bounded for all 1 ≤ p ≤ q ≤ ∞. In other words, Tp = T⌈D bounded from Lp (M) to Lq (M) with domain Lp (M).

Lp

is

Theorem 4.107 (Lp -Lq boundedness). Let V be Kato-decomposable. Then for every 1 ≤ p ≤ q ≤ ∞, the semigroup {Kt : t ≥ 0} exists and its elements act from Lp (ℝd ) to Lq (ℝd ) as bounded operators. Proof. By making use of the Riesz–Thorin theorem quoted below in Lemma 4.108 it suffices to prove that Kt is bounded from L1 (ℝd ) to L1 (ℝd ), from L1 (ℝd ) to L∞ (ℝd ), and from L∞ (ℝd ) to L∞ (ℝd ). Boundedness from L∞ (ℝd ) to L∞ (ℝd ) has already been shown in Lemma 4.105. We prove boundedness from L1 (ℝd ) to L1 (ℝd ) by using duality. For f ∈ L1 (ℝd ) and g ∈ L∞ (ℝd ) we have t

∫ f (x)Kt g(x) dx = ∫ 𝔼x [g(B0 )e− ∫0 V(Bs ) ds f (Bt )]dx. ℝd

(4.2.39)

ℝd

This follows from (4.2.3) in the proof of Theorem 4.86. By taking g = 1 and reading (4.2.39) from right to left we find ‖Kt f ‖1 ≤ ‖f ‖1 ‖Kt 1‖∞ ; hence Kt is bounded from L1 (ℝd ) to L1 (ℝd ). Since Kt is bounded from L∞ (ℝd ) to L∞ (ℝd ) and from L1 (ℝd ) to L1 (ℝd ), the Riesz–Thorin theorem implies that Kt is bounded from Lp (ℝd ) to Lp (ℝd ) for all p ≥ 1, and by the Markov property of Brownian motion it is a semigroup on all of these spaces. It remains to show that Kt is bounded from L1 (ℝd ) to L∞ (ℝd ). Consider the diagram Kt

Kt

L1 (ℝd ) 󳨀→ L2 (ℝd ) 󳨀→ L∞ (ℝd ). Assume first that f ∈ L2 (ℝd ). By the Schwarz inequality t

‖Kt f ‖2∞ ≤ sup (𝔼x [e−2 ∫0 V(Bs ) ds ]𝔼x [f (Bt )2 ]). x∈ℝd

(4.2.40)

266 | 4 Feynman–Kac formulae The first factor on the right-hand side above is bounded by Khasminskii’s lemma, 2 while the second is bounded due to 𝔼x [f (Bt )2 ] = (2πt)−d/2 ∫ℝd f (y)2 e−|x−y| /2t dy. Hence it follows that Kt : L2 (ℝd ) → L∞ (ℝd ) and ‖Kt f ‖∞ ≤ C‖f ‖2 . Now for 0 ≤ f ∈ L1 (ℝd ) choose g ∈ L2 (ℝd ), and by a similar reasoning as in (4.2.39) we find ∫ (Kt f ) (x)g(x) dx = ∫ f (x) (Kt g) (x) dx ≤ ‖Kt g‖∞ ‖f ‖1 . ℝd

ℝd

This can be extended to any f ∈ L2 (ℝd ) by separating f = (ℜf )+ −(ℜf )− +i(ℑf )+ −i(ℑf )− , where f+ = max{f (x), 0}, f− = − min{f (x), 0}, and 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 ∫ (Kt f ) (x)g(x) dx󵄨󵄨󵄨 ≤ 4‖Kt g‖∞ ‖f ‖1 . 󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝd 󵄨󵄨 Hence Kt f ∈ L2 (ℝd ) for f ∈ L1 (ℝd ), and for f ∈ L1 (ℝd ) we have ̃ ‖. ‖Kt f ‖∞ = ‖Kt/2 Kt/2 f ‖∞ ≤ C‖Kt/2 f ‖2 ≤ C‖f 1 Therefore Kt maps L1 (ℝd ) to L∞ (ℝd ) and the proof is complete. Lemma 4.108 (Riesz–Thorin theorem). Let T : Lpj (ℝd ) → Lqj (ℝd ) be a bounded operator, j = 1, 2, and 1 ≤ p1 , p2 , q1 , q2 ≤ ∞. Then T is bounded from Lr (ℝd ) to Ls (ℝd ) for every r, s such that 1−θ θ 1−θ θ 󵄨󵄨 1 1 {(r, s) 󵄨󵄨󵄨󵄨 ( , ) = ( + , + ) , 0 < θ < 1} . p1 p2 q1 q2 󵄨 r s By Theorem 4.107 we see that Kt f is a function in Lq (ℝd ) for f ∈ Lp (ℝd ). Hence as a function in Lq (ℝd ), Kt f (x) is defined for x ∈ ℝd \ Nt,f with a null set Nt,f . Using t

the representstion (Kt f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )] however, we can see in the theorem below that x 󳨃→ Kt f (x) is continous for all x ∈ ℝd in this version. To see the continuity of Kt f (x) in x we make use of the smoothing effect of e−t(−Δ/2) . We define e−t(−Δ/2) f (x) for every x ∈ ℝd by e−t(−Δ/2) f (x) = ∫ Πt (x − y)f (y)dy ℝd

for f ∈ Lp (ℝd ) with 1 ≤ p ≤ ∞. Proposition 4.109 (Smoothing effect). Let f ∈ Lp (ℝd ) with some 1 ≤ p ≤ ∞. Then e−t(−Δ/2) f is continuous. Proof. Suppose p = 1. Since e−t(−Δ/2) f (x) = ∫ℝd Πt (x − y)f (y)dy, the statement follows by the dominated convergence theorem. Let p = ∞ and define f (x) fn (x) = { 0

|x| ≤ n, |x| > n.

4.2 Feynman–Kac formula for Schrödinger operators |

267

Since fn ∈ L1 (ℝd ) for each n ∈ ℕ, the function e−t(−Δ/2) fn (x) is continuous in x. Choose n > |x|. We then have |e−t(−Δ/2) fn (x) − e−t(−Δ/2) f (x)| ≤ ∫ Πt (x − y)|f (y)|dy |y|≥n

≤ ‖f ‖∞

Πt (z)dz ≤ ‖f ‖∞ ∫ Πt (z)dz.

∫ |z|≥n+|x|

|z|≥n

The right-hand side converges to zero as n → ∞, uniformly in x. Hence the uniform limit e−t(−Δ/2) f (x) is continuous in x. Finally, take 1 < p < ∞ and let g ∈ Lp (ℝd ). Define gn (x) = {

g(x) n

|g(x)| ≤ n, |g(x)| > n.

We see that e−t(−Δ/2) gn (x) is continuous in x for each n. By the Hölder inequality we have 1/q

|e

−t(−Δ/2)

gn (x) − e

−t(−Δ/2)

q

g(x)| ≤ ( ∫ Πt (x − y) dy) ℝd

1/p p

( ∫ |gn (y) − g(y)| dy)

≤ ‖Πt ‖Lq (ℝd ) ‖gn − g‖Lp (ℝd )

ℝd

for 1/p + 1/q = 1. The right-hand side converges to zero as n → ∞, uniformly in x. Hence e−t(−Δ/2) g(x) is continuous in x. Lemma 4.110. Let V be a Kato-decomposable potential and f ∈ L∞ (ℝd ). Then t

e−r(−Δ/2) Kt−r f (x) = 𝔼x [e− ∫r V(Bs )ds f (Bt )],

t ≥ r > 0.

(4.2.41)

In particular, (4.2.41) is continuous in x for every r > 0. t−r

Proof. We have e−r(−Δ/2) Kt−r f (x) = 𝔼x [𝔼Br [e− ∫0 V(Bs )ds f (Bt−r )]]. The Markov property implies the lemma. Furthermore, continuity follows by Proposition 4.109. Theorem 4.111. Let V be Kato-decomposable and f ∈ Lp (ℝd ) for 1 ≤ p ≤ ∞. Then Kt f is a continuous function for every t > 0. Proof. First we make a preliminary observation. Suppose that V ∈ 𝒦(ℝd ). In the proof of Lemma 4.104 we have x

sup 𝔼 [|1 − e

x∈ℝd

t

− ∫0 V(Bs ) ds

t

k

1 |] ≤ ∑ sup 𝔼x [( ∫ |V(Bs )| ds) ] k! x∈ℝd k=1 ∞

≤

supx∈ℝd 𝔼 1−

x

t [∫0

0

|V(Bs )|ds]

t supx∈ℝd 𝔼x [∫0

|V(Bs )|ds]

t→0

󳨀→ 0

(4.2.42)

268 | 4 Feynman–Kac formulae by Proposition 4.96. Note that Khasminskii’s lemma implies that for small enough T, t

sup sup 𝔼x [e∫0 |V(Bs )| ds ] < ∞.

(4.2.43)

x∈ℝd 0≤t≤T

We can now proceed to prove the theorem. Since Kt f = Kt/2 Kt/2 f and Kt/2 f ∈ L∞ (ℝd ) whenever f ∈ Lp (ℝd ), 1 ≤ p ≤ ∞, it suffices to consider f ∈ L∞ (ℝd ). The essential idea is to apply the smoothing property seen in Proposition 4.109. We have Kt f (x) − Kt f (xn ) = Kt f (x) − e−r(−Δ/2) Kt−r f (x) + e−r(−Δ/2) Kt−r f (x) − e−r(−Δ/2) Kt−r f (xn ) + e−r(−Δ/2) Kt−r f (xn ) − Kt f (xn ). By the smoothing effect we see that e−r(−Δ/2) Kt−r f (x) − e−r(−Δ/2) Kt−r f (xn ) → 0 as n → ∞ for every r > 0. It is sufficient to show that Kt f (x) − e−r(−Δ/2) Kt−r f (x) → 0 as r → 0, uniformly in x. First assume V ∈ 𝒦(ℝd ). For every r > 0, let gr (x) = e−r(−Δ/2) Kt−r f (x). By Lemma 4.110 we have τ

t

‖gr − Kt f ‖∞ = sup 𝔼x [(1 − e− ∫0 V(Bs ) ds )e− ∫r V(Bs ) ds f (Bt )] x∈ℝd

r

r

r→0

≤ ‖f ‖∞ ( sup sup 𝔼x [e− ∫0 V(Bs ) ds ])( sup 𝔼x [1 − e− ∫0 V(Bs ) ds ]) 󳨀→ 0. r≤t

x∈ℝd

x∈ℝd

In the last line we have used Lemma 4.103 and (4.2.42). Thus Kt f is continuous. Next let V = V+ − V− be Kato-decomposable, and consider for R > 0 the function VR = V+R − V− with V+R (x) = V+ (x) if |x| ≤ R and V+R (x) = 0 if |x| > R. We have VR ∈ 𝒦(ℝd ) by the definiton of the Kato-decomposable property, and the above reasoning applies. Let M ⊂ BR (0) be compact, where BR (0) denotes the ball of radius R centered t t at the origin. We now claim that ∫0 VR (Bs ) ds = ∫0 V(Bs ) ds for each path ω ∈ X starting in M and not leaving the ball up to time t. Hence t

t

sup |𝔼x [e− ∫0 V(Bs ) ds f (Bt )] − 𝔼x [e− ∫0 VR (Bs ) ds f (Bt )]| x∈M

t 󵄨󵄨 t 󵄨󵄨 ≤ sup 𝔼x [e∫0 V− (Bs ) ds 󵄨󵄨󵄨󵄨e− ∫0 V+ (Bs )ds − 1󵄨󵄨󵄨󵄨 |f (Bt )|1{sup0≤s≤t |Bs |≥R} ] 󵄨 󵄨 x∈M t

≤ 2‖f ‖∞ sup 𝔼x [e∫0 V− (Bs ) ds 1{sup0≤s≤t |Bs |≥R} ] x∈M

t

≤ 2‖f ‖∞ sup (𝔼x [e−2 ∫0 V− (Bs ) ds ]) x∈M

1/2

1/2

(𝔼x [1{sup0≤s≤t |Bs |≥R} ])

.

The first factor above is bounded since V− is Kato-class, and the second goes to zero as R → ∞ by Lévy’s maximal inequality 𝔼x [1{sup0≤s≤t |Bs |≥R} ] ≤ 2𝔼x [1{|Bt |>R} ].

4.2 Feynman–Kac formula for Schrödinger operators |

269

Hence Kt f is a locally uniform limit of continuous functions, and Kt f (x) is continuous in x. The last part of the above argument is of some independent interest, and we present it separately. Corollary 4.112. Let V be Kato-decomposable and f ∈ Lp (ℝd ) with 1 ≤ p ≤ ∞. Then e−tH f is the local uniform limit of the sequence (e−tHn f )n∈ℕ with Hn = − 21 Δ + Vn and Vn with compact support. One possible choice in the corollary above is Vn = 1{|x|≤n} V. In particular, each Vn is in 𝒦(ℝd ) as opposed to being only Kato-decomposable. We are now ready to define H with Kato-class potential as a self-adjoint operator and give the Feynman–Kac formula for e−tH . Theorem 4.113 (Feynman–Kac formula for Schrödinger operator with Kato-decomposable potential). Assume that V is Kato-decomposable. Then {Kt : t ≥ 0} defined by t

(Kt f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )] is a symmetric C0 -semigroup on L2 (ℝd ). Moreover, there exists a unique self-adjoint operator K bounded from below such that Kt = e−tK , for every t ≥ 0. Proof. We already know that {Kt : t ≥ 0} is a semigroup of bounded operators on L2 (ℝd ), and (g, Kt f ) = (Kt g, f ) can be proven in the same way as (4.2.3). To see strong continuity, by denseness it suffices to prove ‖Kt f − f ‖ → 0 as t ↓ 0 for all bounded L2 functions f with compact support. Let Qt = etΔ/2 . We see that ‖Qt f − f ‖ → 0 as t ↓ 0. We only need to show that t

x 󳨃→ Kt f (x) − Qt f (x) = 𝔼x [(e− ∫0 V(Bs ) ds − 1)f (Bt )] converges to 0 in L2 (ℝd ) as t ↓ 0. Using the proof of Theorem 4.111 and Corollary 4.112 it follows that Kt f − Qt f → 0 as t ↓ 0 pointwise in ℝd . Moreover, t

(Kt f (x) − Qt f (x))2 ≤ 𝔼x [(e− ∫0 V(Bs ) ds − 1)2 ]𝔼x [f (Bt )2 ]. Here the first factor is bounded uniformly in t and x by Lemma 4.105, while the t-supremum over the second one is bounded and exponentially decaying at infinity, thus integrable. Thus Kt f − Qt f → 0 in L2 (ℝd ) by the dominated convergence theorem. Hence by the triangle inequality ‖Kt f − f ‖ ≤ ‖Kt f − Qt f ‖ + ‖Qt f − f ‖ the statement follows. Furthermore, by the Hille–Yosida theorem there exists a unique self-adjoint operator K bounded below such that Kt = e−tK , for every t ≥ 0. Definition 4.114 (Schrödinger operator with Kato-decomposable potential). Let V be a Kato-decomposable potential. We define the Schrödinger operator for the Kato-decomposable potential V by K given in Theorem 4.113.

270 | 4 Feynman–Kac formulae By Proposition 4.100 a Kato-decomposable potential is the sum of an L1loc (ℝd )-potential and a −Δ-form-bounded potential with infinitesimally small relative bound. This implies that K with a Kato-decomposable potential V generally coincides with the Schrödinger operator − 21 Δ + V defined by the quadratic form sum. Remark 4.115. Suppose that V = V+ − V− such that V+ ∈ L1loc (ℝd ) and V− ∈ 𝒦(ℝd ). t

We can also prove that (Kt f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )] is a symmetric C0 -semigroup on L2 (ℝd ), and for every 1 ≤ p ≤ q ≤ ∞ the semigroup {Kt : t ≥ 0} exists and its elements act from Lp (ℝd ) to Lq (ℝd ) as bounded operators. Kt f (x) is, however, not necessarily continuous in x.

4.3 Properties of Schrödinger operators and semigroups 4.3.1 Kernel of the Schrödinger semigroup From the perspective of functional analysis and the theory of partial differential equations it is a crucial point that for every t > 0 the operator e−tH has an integral kernel given by the Feynman–Kac formula. This is addressed in the next theorem. Theorem 4.116. Let V be a Kato-decomposable potential. Then e−tK is an integral operator in Lp (ℝd ) for every 1 ≤ p ≤ ∞. Moreover, the integral kernel ℝd × ℝd ∋ (x, y) 󳨃→ Kt (x, y) is jointly continuous in x and y, and is given by t

x,y

Kt (x, y) = Πt (x − y)𝔼[0,t] [e− ∫0 V(Bs ) ds ].

(4.3.1)

x,y

Here 𝒲[0,t] is the Brownian bridge measure defined in (2.3.18). t

Proof. By Theorem 4.113 we have (e−tK f )(x) = 𝔼x [e− ∫0 V(Bs ) ds f (Bt )], for every f ∈ Lp (ℝd ), and by Theorem 4.111 the above equality holds pointwise as the right-hand side is a continuous function. Thus x,y

t

(e−tK f )(x) = ∫ Πt (x − y)f (y)𝔼[0,t] [e− ∫0 V(Bs ) ds ]dy, ℝd

which shows that (4.3.1) holds for all x and almost all y. Next we show continuity. Let s = t/3. We have Kt (x, y) = ∫ Ks (x, v)Ks (v, w)Ks (w, y) dw dv = ∫ Ks (x, v)Ks (y, w)Ks (v, w) dw dv. ℝ2d

ℝ2d

The product Ks (x, v)Ks (y, w) = Πs (x − v)Πs (y − w)𝔼[0,s]

(x,v),(y,w)

s

(1)

[e− ∫0 (V(Br

)+V(B(2) r )) dr

]

4.3 Properties of Schrödinger operators and semigroups | 271

is the kernel of the Schrödinger operator with Kato-class potential 1 1 K̃ = − Δx − Δy + V(x) + V(y), 2 2 and (x, y) 󳨃→ V(x) + V(y) is Kato-decomposable in ℝ2d since V is in ℝd , where B(i) t ,i = 1, 2, denote two independent d-dimensional Brownian motions. On the other hand, the map (v, w) 󳨃→ Ks (v, w) is bounded. One way to see this is to recall that e−sK is bounded from L1 (ℝd ) to L∞ (ℝd ) and sup(v,w)∈ℝ2d |Ks (v, w)| = supf ∈L1 (ℝd ),‖f ‖1 =1 ‖e−tK f ‖∞ < ∞. Hence Kt (x, y) = (e−t K Ks )(x, y) ̃

with Ks ∈ L∞ (ℝd × ℝd ) and is thus jointly continuous in x and y by Theorem 4.111. The above proof has the virtue of simplicity; using more sophisticated arguments, it can be shown that Kt (x, y) is even jointly continuous in x, y and t for t > 0. 4.3.2 Positivity improving and uniqueness of ground state In this section we start our discussion of eigenfunctions of Schrödinger operators by using Feynman–Kac formulae. Our first goal is to discuss uniqueness of the ground state, and then we turn briefly to their possible higher multiplicities, and furthermore to fundamental questions of existence and absence of ground states in the sections to follow. Definition 4.117 (Ground state). Let H be a self-adjoint operator bounded from below. An eigenfunction φ0 satisfying Hφ0 = E(H)φ0 ,

E(H) = E0 = inf Spec(H)

(4.3.2)

is called a ground state of H, and the bottom of the spectrum E0 is called ground state energy. The nonnegative integer m(H) = dim Ker(H − E(H))

(4.3.3)

is called the multiplicity of the ground state. If m(H) = 1, the ground state is said to be unique. A simple but useful fact is that the kernel e−tH (x, y) is strictly positive. This is readt

ily implied by (4.3.1) whenever e− ∫0 V(Bs )ds is strictly positive for almost every ω ∈ X . In particular, e−tH is positivity improving in the sense of the following definition.

Definition 4.118 (Positivity preserving/improving operator). Let (M, μ) be a σ-finite measure space.

272 | 4 Feynman–Kac formulae (1) A nonzero function f ∈ L2 (M, dμ) is called positive if f ≥ 0 holds μ-almost everywhere. Moreover, f is called strictly positive if f > 0 holds μ-almost everywhere. (2) A bounded operator A on L2 (M, dμ) is called a positivity preserving operator if (f , Ag) ≥ 0, for all positive f , g ∈ L2 (M, dμ). A is called a positivity improving operator if (f , Ag) > 0, for all positive f , g ∈ L2 (M, dμ). Example 4.119. Since etΔ f (x) = ∫ℝd Πt (x)f (x)dx, the operator etΔ is positivity improving. Moreover, e−ia⋅(−i∇) is also a positivity preserving operator, as it produces the shift e−ia(−i∇) f (x) = f (x − a).

Positivity preserving operators of the form e−tE(−i∇) have a deep connection with Lévy processes. As seen in Example 3.29, the square root of the Laplacian generates a stable Lévy process. The Lévy–Khintchine formula (3.1.8) has the following important relationship with positivity preserving operators. Proposition 4.120. Let E : ℝd → ℂ. (1) Suppose that the real part of E is bounded from below. Then the following statements are equivalent: (a) e−tE(−i∇) is a positivity preserving semigroup; (b) e−tE(u) is a positive definite distribution, i. e., for all ϕ ∈ C0∞ (ℝd ) we have ∫ℝd ∫ℝd e−tE(u) ϕ(x − u)ϕ(x)dudx ≥ 0; (c) E(−u) = E(u), and it is conditionally negative definite, i. e., n

∑ E(pi − pj )zī zj ≤ 0,

i,j=1

for every n ∈ ℕ and z = (z1 , . . . , zn ) ∈ ℂn with ∑ni=1 zi = 0; (d) there exists a triplet (b, A, ν), where b ∈ ℝd , A is a positive definite symmetric matrix, and ν is a Lévy measure such that 1 −E(u) = a + iu ⋅ b − u ⋅ Au + ∫ (eiu⋅y − 1 − iu ⋅ y1{|y|≤1 })ν(dy) 2

(4.3.4)

ℝd \{0}

with some a ∈ ℝ. (2) Suppose that E is a spherically symmetric and polynomially bounded continuous function with E(0) = 0, with its real part bounded from below. Suppose, moreover, that ΔE in distributional sense is positive definite. Then E is conditionally negative definite. Write E(u) = √|u|2 + m2 − m,

m ≥ 0;

then by a straightforward calculation Δu E(u) = (d − 1)(|u|2 + m2 )−1/2 + m2 (|u|2 + m2 )−3/2 .

4.3 Properties of Schrödinger operators and semigroups | 273

2

2

2

Since (|u|2 + m2 )−β = dβ ∫0 t β−1 e−t(|u| +m ) dt and e−t|u| is positive definite, we conclude by Proposition 4.120 that E(u) is conditionally negative definite, which implies that −E(u) can be represented as in (4.3.4). Since the left-hand side is real and E(u)/|u|2 → 0 as |u| → 0, its Lévy triplet is (b, A, ν) = (0, 0, ν). Thus we have ∞

−√|u|2 + m2 + m = ∫ (eiu⋅y − 1 − iu ⋅ y1{|y|0

The following result gives the Feynman–Kac formula for e−E(−i∇) . Proposition 4.121. Let (Xt )t≥0 be a Lévy process on a probability space (Ω, F , P) with characteristic triplet (0, 0, ν) in (4.3.5). Then (g, e−tE(−i∇) f ) = ∫ 𝔼P [g(x)f (x + Xt )]dx. ℝd

The Lévy measure ν in (4.3.5) can be determined exactly. The integral kernel of the operator e−tE(−i∇) is given by the Fourier transform Fe−tE(u) F −1 and thus e−tE(−i∇) (x, y) = 2 (

2 2 m (d+1)/2 tK(d+1)/2 (m√|x − y| + t ) ) , 2π (t 2 + |x − y|2 )(d+1)/4

(4.3.6)

where Kν is the modified Bessel function of the third kind (4.2.21). Using that Kν (z) ∼ 1 Γ(ν)( 21 z)−ν as |z| ↓ 0, we obtain 2 1 (g, E(−i∇)f ) = lim [(g, f ) − (g, e−tE(−i∇) f )] t→0 t =(

m ) 2π

(d+1)/2

∫ ∫ ℝd ℝd

(g(x) − g(y))(f (x) − f (y)) K(d+1)/2 (m√|x − y|2 )dxdy. |x − y|(d+1)/2

For m = 0 the integral form of (g, E(−i∇)f ) is given by (g, E(−i∇)f ) =

Γ((d + 1)/2) (g(x) − g(y))(f (x) − f (y)) dxdy + m(g, f ). ∫ ∫ 2π (d+1)/2 |x − y|d+1 ℝd ℝd

Hence it is seen that the Lévy measure associated with E(u) for m = 0 is ν−1 (dy) =

1 d+1 1 Γ( ) d+1 dy 2 π (d+1)/2 |y|

and for m > 0 m ν−1 (dy) = 2 (

m (d+1)/2 1 ) K(d+1)/2 (m|y|)dy. (d+1)/2 2π |y|

(4.3.7)

274 | 4 Feynman–Kac formulae Remark 4.122. Let ν−α (dy) =

d+α 2α Γ( 2 ) 1 dy, π d/2 |Γ(− α2 )| |y|d+α

0 < α < 2,

where Γ(−x) = − x1 Γ(1 − x) for x > 0. The Lévy process (Xt )t≥0 with triplet (0, 0, ν−α ) is a rotationally symmetric stable process with index α and its generator is −(−Δ)α/2 . A useful application of the positivity properties of operators is in showing uniqueness of the ground state of self-adjoint operators. Theorem 4.123 (Perron–Frobenius theorem). Let (M, μ) be a σ-finite measure space and K be a self-adjoint operator bounded from below in L2 (M, dμ). Suppose that inf Spec(K) is an eigenvalue and e−K is positivity improving. Then inf Spec(K) is nondegenerate and the eigenfunction corresponding to the eigenvalue inf Spec(K) is strictly positive. Proof. Put λ = inf Spec(K). We have ‖e−K ‖ = e−λ . Let f be an eigenfunction corresponding to the eigenvalue λ. Since e−K maps real-valued functions to real-valued functions, ℜf is also an eigenfunction and therefore we may assume f to be real-valued from the outset. Since e−K is positivity improving and linear, we find (g, e−K f ) = (g, e−K f + − e−K f − ) ≤ (g, e−K f + + e−K f − ) = (g, e−K |f |) for all g ≥ 0 and all f = f + − f − ∈ L2 with f + > 0 and f − > 0. Thus e−λ ‖f ‖2 = (e−K f , f ) ≤ (e−K |f |, |f |) ≤ ‖e−K ‖‖f ‖2 = e−λ ‖f ‖2 , implying (e−K f , f ) = (e−K |f |, |f |). This further implies (e−K f + , f − ) = −(e−K f − , f + ).

(4.3.8)

Since f + , f − , e−K f + , and e−K f − are all nonnegative, both sides of (4.3.8) must vanish, and this can only happen if either f + or f − vanishes almost everywhere as e−K improves positivity. Hence f has a definite sign and, for instance, f ≥ 0 can be assumed. However, since f = e−K e+λ f , it follows that f is in fact strictly positive. The above reasoning applies to any eigenfunction corresponding to the eigenvalue λ. This shows that there cannot be two linearly independent such eigenfunctions since then they could be chosen orthogonal which is impossible as they both would have to be strictly positive. Hence f is unique. Theorem 4.124 (Positivity improving operators). Suppose that V+ ∈ L1loc (ℝd ) and V− is relatively form bounded with respect to − 21 Δ with a relative bound strictly smaller than one. Let H = 21 Δ +̇ V+ −̇ V− . Then {e−tH : t ≥ 0} is positivity improving. In particular, for H = − 21 Δ + V with a Kato-decomposable V, the semigroup {e−tH : t ≥ 0} is positivity improving.

4.3 Properties of Schrödinger operators and semigroups | 275

Proof. Since {e(t/2)Δ : t ≥ 0} is positivity improving, ∫ℝd 𝔼x [f (B0 )g(Bt )]dx > 0 for every non-negative f and g. The measure of M = {(x, ω) ∈ ℝd × X | f (x)g(Bt (ω) + x) > 0} is t

positive. For V+ ∈ L1loc (ℝd ), it is seen in Lemma 4.95 that e− ∫0 V(Bs +x)ds > 0 for almost every (x, ω) ∈ ℝd × X . We have t

(f , e−tH g) = ∫ 𝔼[f (x)g(Bt + x)e− ∫0 V(Bs +x)ds ]dx ℝd

t

≥ ∫ f (x)g(Bt + x)e− ∫0 V(Bs +x)ds dxd𝒲 > 0, M

and e−tH is positivity improving for all t ≥ 0. Theorem 4.125 (Uniqueness of ground state). Suppose that V+ ∈ L1loc (ℝd ) and V− is relatively form bounded with respect to − 21 Δ with a relative bound strictly smaller than one. Let H = 21 Δ +̇ V+ −̇ V− . Then the ground state of H is unique whenever it exists. In particular, for H = − 21 Δ+V with a Kato-decomposable V, the ground state of H is unique whenever it exists. Proof. By Theorem 4.124, e−tH is positivity improving. Uniqueness follows by the Perron–Frobenius theorem.

4.3.3 Degenerate ground state and Klauder phenomenon Now we turn to presenting an example where e−tH is not positivity improving. To break positivity down, we construct a potential V such that t

∫ 𝒲 x (e− ∫0 V(Bs )ds = 0) dx > 0. ℝd

This requires the potential V to be singular in a suitable form. Let 3

M1 = {x = (x1 , x2 , x3 ) ∈ ℝ | x3 > 0}, 3

M2 = {x = (x1 , x2 , x3 ) ∈ ℝ | x3 < 0}, M = M1 ∪ M2 .

Define

1 1 ν H(ν) = − Δ + |x|2 + . 2 2 |x − 𝜕M |3

(4.3.9)

Here 𝜕M is the boundary of M and |x − 𝜕M | denotes the distance between x and 𝜕M . Since the potential in (4.3.9) is singular, the self-adjointness of H(ν), ν ≠ 0, is unclear. To see this, we use the following general result.

276 | 4 Feynman–Kac formulae Proposition 4.126. Let D be an open set in ℝd and V ≥ 0 on D. Suppose that there is a uniform Lipschitz continuous function h on every compact subset of D such that (1) ∑di=1 (𝜕i h)2 ≤ e2h on D; (2) limx→𝜕D h(x) = ∞ (if D is not bounded, ∞ is regarded as a point of 𝜕D); (3) there exists δ > 0 such that (f , (−Δ + V)f ) ≥ (1 + δ)(f , e2h f ) for f ∈ C0∞ (D). Then −Δ + V is essentially self-adjoint on C0∞ (D). Lemma 4.127. H is essentially self-adjoint on C0∞ (M ), for all ν ∈ ℝ. Proof. Let ν + 21 |x|2 , |3 Vε (x) = { |x−𝜕M ν + 21 |x|2 , ε3

|x − 𝜕M | < ε, otherwise.

Since V = Vε + T, where T is a bounded operator, it suffices to show that −Δ + 2Vε is essentially self-adjoint on C0∞ (M ). Define h = log Vε1/2 ; h can be directly shown to satisfy (2) and (3) in Proposition 4.126. We check (1), which is equivalent with having 0 ≤ 4Vε3 − ∑3i=1 (𝜕i Vε )2 . For |x − 𝜕M | ≥ ε, it is immediate to see that 4Vε3 − ∑3i=1 (𝜕i Vε )2 > 0 for sufficiently small ε. We have for |x − 𝜕M | < ε, 3

4Vε3 − ∑(𝜕i Vε )2 ≥ 4 ( i=1

3 ν 1 1 9ν2 + |x|2 ) − ( + |x|2 ) . 3 2 2 |x − 𝜕M |8 |x − 𝜕M |

Take sufficiently small ε. The right-hand side is positive, and the lemma follows. Since H(0) ≤ H(ν), we have μn (H(0)) ≤ μn (H(ν)), where μn (K) =

sup

inf

ϕ1 ,...,ϕn−1 ψ∈D(K),‖ψ‖=1 ψ∈[ϕ1 ,...,ϕn−1 ]⊥

(ψ, Kψ).

By the fact that μn (H(0)) = n + 21 it is seen that μn (H(ν)) → ∞ as n → ∞. In particular, by the min-max principle H(ν) has a purely discrete spectrum. Further, by Theorem 4.88 we have (f , e

−tH(ν)

t

x

g) = ∫ 𝔼 [f (x)g(Bt ) exp(− ∫ 0

ℝ3

ν + |Bs |2 ds)]dx. |Bs − 𝜕M |3

(4.3.10)

Lemma 4.127 gives H(ν) = HD (ν), where HD (ν) denotes H(ν) with Dirichlet boundary condition. We see that t

(f , e−tHD (ν) g) =

∫ f (x)g(Bt ) exp(− ∫ M1 ∪M2

0

ν + |Bs |2 ds)d𝒲 x dx, |Bs − 𝜕M |3

(4.3.11)

4.3 Properties of Schrödinger operators and semigroups | 277

with Mj = {(x, ω) ∈ ℝ3 × X | x + Bs ∈ Mj , s ∈ [0, t]} denoting the set of paths starting at x and confined to Mj . Comparing (4.3.10) with (4.3.11), we see that t

∫ 0

ds =∞ |Bs (ω) − 𝜕M |3

for ω ∈ ℝ3 × X \ (M1 ∪ M2 ). Thus such paths do not contribute to (4.3.10) at all. Hence (f , e−tH(ν) g) = 0 for f and g for which supp f ⊂ M1 and supp g ⊂ M2 . Thus e−tH(ν) is not positivity improving. Moreover, we have lim (f , e−tH(ν) g) =

ν→0

∫ f (x)g(Bt )d𝒲 x dx ≠ (f , e−tH(0) g). M1 ∪M2

This vestigial effect of singular potentials is called Klauder phenomenon and shows that once a singular potential is turned on, its effect continues to be felt after the potential is turned off. We have seen that for e−tH(ν) positivity improving breaks down. This opens the possibility to construct a non-unique ground state. Corollary 4.128. The ground state of H(ν) is two-fold degenerate. Proof. It can be seen that H(ν) can be reduced by L2 (Mj ). With Hj (ν) = H(ν)⌈L2 (Mj ) we have H(ν) = H1 (ν) ⊕ H2 (ν). Let Ej = inf Spec(Hj (ν)); then E1 = E2 by symmetry. Thus the corollary follows. By replacing the potential ν|x −𝜕M |−3 with ν|x −𝜕M |−2−ε , ε > 0, we can show the same results as in Lemma 4.127 and Corollary 4.128.

4.3.4 Existence and non-existence of ground states In this section we present criteria of existence and non-existence of ground states of Schrödinger operators with Kato-decomposable potentials by using the Feynman–Kac formula. Also, we show a representation formula for ground states in case they exist. Let V be a Kato-decomposable potential, consider 1 H = − Δ + V, 2 and denote inf Spec(H) = E0 . Lemma 4.129. Let ρ ∈ L2 (ℝd ) such that ess infx∈K ρ(x) > 0 for every compact set K ⊂ ℝd , and E(⋅) be the spectral measure of H associated with ρ. Then inf supp E = E0 .

278 | 4 Feynman–Kac formulae Proof. Let 𝒪 = {f ∈ L2 (ℝd ) | supp f ⊂ ⋃N>0 BN }, where BN = {x ∈ ℝd | |x| ≤ N}. Since e−tH is positivity improving, we have for f ∈ 𝒪 that (f , e−tH f ) ≤ C 2 (ρ, e−tH ρ), where C =

ess supx∈ℝd |f (x)| . ess infx∈supp|f | |ρ(x)|

Hence we have

1 1 (f , e−tH f ) log(ρ, e−tH ρ) ≤ − lim log = inf supp Ef , t→∞ t t→∞ t C2

inf supp E = − lim

where Ef is the spectral measure of H associated with f . Thus we obtain inf supp E ≤ inf supp Ef ,

f ∈ 𝒪.

Since E0 = inff ∈𝒪 inf supp Ef , we obtain inf supp E ≥ E0 ≥ inf supp E. For ρ ∈ L2 (ℝd ) define γρ (t) =

(ρ, e−tH ρ)2 . ‖e−tH ρ‖2

Formally γρ (t) → |(ρ, φ0 )|2 > 0 as t → ∞ if ρ is a strictly positive function and a ground state φ0 exists. Using this quantity we obtain criteria on the existence and absence of a ground state. Theorem 4.130 (Criterion for existence of ground state). Let ρ ∈ L2 (ℝd ) with the property that ess infx∈K ρ(x) > 0 for every compact set K ⊂ ℝd , and denote limt→∞ γρ (t) = a. If a > 0, then H has a ground state. Proof. Since a > 0, for sufficiently large t it follows that √γρ (t) > b with some b > 0. Let E(⋅) be the spectral measure of H associated with ρ, and write δ = inf supp E. Then by Lemma 4.129 it follows that δ = inf Spec(H) = E0 . By the definition of δ we see that E([δ, δ + ε)) ≠ 0 for every ε > 0, and E([δ, δ + ε)) is strongly convergent to E({δ}) as ε ↓ 0. We have √γρ (t) =

∫[δ,δ+ε) e−(λ−δ)t dE + ∫[δ+ε,∞) e−(λ−δ)t dE 1/2

(∫[δ,∞) e−2(λ−δ)t dE)

.

By the Schwarz inequality we furthermore have ∫[δ,δ+ε) e−(λ−δ)t dE (∫[δ,∞)

1/2 e−2(λ−δ)t dE)

≤ E([δ, δ + ε))

1/2

(∫[δ,δ+ε) e−2(λ−δ)t dE) (∫[δ,∞)

e−2(λ−δ)t dE)

1/2

1/2

≤ E([δ, δ + ε))1/2 .

Next consider the ratio at the left hand side above. Since the denominator and the numerator can be estimated as

4.3 Properties of Schrödinger operators and semigroups | 279

∫

e−(λ−δ)t dE ≤ e−εt ,

[δ+ε,∞)

∫ e−2(λ−δ)t dE ≥ [δ,∞)

∫

e−2(λ−δ)t dE ≥ e−εt E([δ, δ + ε/2)),

[δ,δ+ε/2)

we see that ∫[δ+ε,∞) e−(λ−δ)t dE

1/2

(∫[δ,∞) e−2(λ−δ)t dE)

≤

e−εt e−εt/2 ≤ . e−εt/2 E([δ, δ + ε/2))1/2 E([δ, δ + ε/2))1/2

It then follows that √γρ (t) ≤ E([δ, δ + ε))1/2 +

e−εt/2 . E([δ, δ + ε/2))1/2

Taking t → ∞ on both sides above, we obtain b ≤ E([δ, δ + ε)) for every ε > 0, which tends to b ≤ E({δ}) in the limit ε ↓ 0. Hence 0 ≠ E({δ}) = E({E0 }) and thus a ground state of H exists. Theorem 4.130 gives a sufficient condition for H to have a ground state. A similar argument can also be used to show absence of ground states of H. Theorem 4.131 (Criterion for absence of ground state). Let ρ ∈ L2 (ℝd ) be a non-negative function, not identically zero. If a = limt→∞ γρ (t) = 0, then H has no ground state. Proof. We may suppose that inf Spec(H) = 0, so that limt→∞ e−tH = 1{0} (H) in strong sense. If 0 is an eigenvalue of H, i. e., H has a ground state φ0 , then a = (ρ, φ0 ) > 0 since φ0 is strictly positive by the fact that e−tH is positivity improving. Thus a = 0 implies that H has no ground state. Now we assume that a ground state φ0 of H exists. Since it is strictly positive, (f , φ0 ) ≠ 0 for 0 < f ∈ L2 (ℝd ). From this it follows that e−tH f = φ0 . t→∞ ‖e−tH f ‖

s − lim

This expression can be rewritten in terms of Feynman–Kac formula like t

φ0 (x) = s − lim

t→∞

𝔼x [e− ∫0 V(Bs )ds f (Bt )] t 󵄨 󵄨2 ( ∫ℝd 󵄨󵄨󵄨𝔼x [e− ∫0 V(Bs )ds f (Bt )]󵄨󵄨󵄨 dx)

1/2

.

(4.3.12)

Next we show that under an appropriate condition formula (4.3.12) can be extended also to f ∈ ̸ L2 (ℝd ) and we can choose, for instance, f = 1.

280 | 4 Feynman–Kac formulae Theorem 4.132 (Ground state representation formula). Suppose that a ground state t

φ0 of H exists and assume that the function ℝd ∋ x 󳨃→ 𝔼x [e− ∫0 V(Bs )ds ] ∈ L2 (ℝd ) and t

lim lim 𝔼x [e− ∫0 (V(Bs )−E0 )ds 1|Bt |≥R ] = 0

(4.3.13)

R→∞ t→∞

strongly in L2 (ℝd ). Then t

φ0 (x) = s − lim

𝔼x [e− ∫0 V(Bs )ds ]

.

(4.3.14)

󵄩󵄩 a 󵄩󵄩 󵄩󵄩 a 󵄩󵄩 at,N 󵄩󵄩󵄩 󵄩󵄩󵄩 at,N 󵄩󵄩 t 󵄩 󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩 − φ0 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 t − − φ 󵄩󵄩 󵄩󵄩 . 0 󵄩󵄩 ‖at ‖ 󵄩󵄩 󵄩󵄩 ‖at ‖ ‖at,N ‖ 󵄩󵄩󵄩 󵄩󵄩󵄩 ‖at,N ‖ 󵄩󵄩

(4.3.15)

t→∞

t 󵄨 󵄨2 ( ∫ℝd 󵄨󵄨󵄨𝔼x [e− ∫0 V(Bs )ds ]󵄨󵄨󵄨 dx)

1/2

Proof. Let 1N be the indicator function on [−N, N]d and denote t

at,N (x) = 𝔼x [e− ∫0 V(Bs )ds 1N ] = (e−tHp 1N )(x), t

at (x) = 𝔼x [e− ∫0 V(Bs )ds ]. Thus

Since 1N ∈ L2 (ℝd ) and (φ0 , 1N ) ≠ 0, it is seen that 󵄩󵄩 a 󵄩󵄩 󵄩 t,N 󵄩 lim 󵄩󵄩󵄩 − φ0 󵄩󵄩󵄩 = 0. 󵄩󵄩 t→∞ 󵄩 󵄩 ‖at,N ‖ We also have 󵄩󵄩 a at,N 󵄩󵄩󵄩 ‖a − at,N ‖ 󵄩󵄩 t 󵄩󵄩 ≤ 2 t − . 󵄩󵄩 󵄩 ‖at,N ‖ 󵄩󵄩 ‖at ‖ ‖at,N ‖ 󵄩󵄩 Note that ‖at,N ‖2 = ∫[E

p ,∞)

e−2λt d‖E(λ)1N ‖2 ≥ e−2E0 t ‖E({E0 })1N ‖2 = e−2E0 t |(1N , φ0 )|2 ,

where E(⋅) denotes the spectral measure for H. By using (4.3.13) it follows that 󵄩󵄩 x − ∫t (V(B )−E ) 󵄩󵄩 s 0 󵄩󵄩𝔼 [e 0 1|Bt |≥N ]󵄩󵄩󵄩 ‖at − at,N ‖ 󵄩󵄩 󵄩 = 0. lim lim ≤ lim lim N→∞ t→∞ N→∞ t→∞ ‖at,N ‖ (1N , φ0 ) By (4.3.15) we obtain for every ε > 0 that 󵄩󵄩 a 󵄩󵄩 󵄩󵄩 a at,N 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 < ε lim 󵄩󵄩󵄩 t − φ0 󵄩󵄩󵄩 ≤ lim 󵄩󵄩󵄩 t − 󵄩󵄩 t→∞ 󵄩󵄩 ‖at ‖ ‖at,N ‖ 󵄩󵄩󵄩 t→∞ 󵄩 󵄩 ‖at ‖ for every N > N0 with some N0 , as required.

(4.3.16)

4.3 Properties of Schrödinger operators and semigroups | 281

Remark 4.133. The representation formula in Theorem 4.132 gives a probabilistic interpretation to ground states. Recall the local time process (Lxt )t≥0 at position x ∈ ℝd as defined in (3.5.10) for Brownian motion (Bt )t≥0 . (We note that when local time does not exist, this observation can be applied to the occupation measure.) Then we have t

y

∫ V(Bs )ds = ∫ V(y)Lt dy. 0

ℝd

This shows that one can actually think of V as being a test function for local time and y

𝔼x [e− ∫ℝd V(y)Lt dy ] y 󵄩 t→∞ 󵄩 x − ∫ℝd V(y)Lt dy 󵄩 ]󵄩󵄩󵄩2 󵄩󵄩𝔼 [e

φ0 (x) = s − lim

holds, i. e., V is a weight function giving the proportion of time paths will spend in specific locations in space under the “potential landscape” determined by V. A ground state results then as a long-time phenomenon, during which paths would “scan” the potential landscape, and it forms on the condition that this landscape allows paths to spend sufficient times in local regions. This aspect will be further discussed in the next section. For illustration we show how to use the above formula to determine the ground state of the harmonic oscillator explicitly. The following lemma is a special case of Lemmas 5.26–5.27 below, where also the proofs are presented. Lemma 4.134. For every x, y ∈ ℝd and t > 0 we have 1

t

2

𝔼x [e− 2 ∫0 |Bs | ds ] = (

d/2 1 |x|2 ) exp (− ) cosh t 2 coth t

and 1

t

2

𝔼x [e− 2 ∫0 |Bs | ds |Bt = y] = ( Example 4.135. Choose V(x) =

d/2 t |y − xe−t |2 |y|2 − |x|2 |y|2 ) exp (− + + ). sinh t 2 2t 1 − e−2t |x|2 . 2

Then (4.3.14) holds and gives φ0 (x) =

|x| 1 e− 2 π d/4

2

.

Proof. Let Hosc = − 21 Δ + 21 |x|2 corresponding to the d-dimensional harmonic oscillator. A standard calculation shows that inf Spec(Hosc ) = E0 =

d 2

and φ0 (x) =

satisfy Hosc φ0 =

d φ . 2 0

π

1

e− d/4

|x|2 2

282 | 4 Feynman–Kac formulae Then by applying Lemma 4.134 we obtain 1

t

2

𝔼x [e− 2 ∫0 |Bs | ds ] 1/2 󵄨󵄨 1 t 2 󵄨󵄨2 ( ∫ℝd 󵄨󵄨󵄨𝔼x [e− 2 ∫0 |Bs | ds ]󵄨󵄨󵄨 dx) 󵄨 󵄨

=

e−

|x|2 1 2 coth t

( ∫ℝd e−|x|

2

1 coth t

dx)

→

1/2

π

1

e− d/4

|x|2 2

as t → ∞. Remark 4.136. We note that while in Example 4.135 we could do without, condition (4.3.13) holds also for this case. Since E0 = d/2, by making use of Lemma 4.134 again we see that 1

t

1

2

t

2

eE0 t 𝔼x [e− 2 ∫0 |Bs | ds 1|Bt |>R ] = edt/2 ∫ Πt (y)𝔼x [e− 2 ∫0 |Bs | ds |Bt = y]dy d/2

t

=

1 e ) ∫ ( sinh t (2π)d/2 |y|>R

|y|>R

exp (−

|y − xe−t |2 |y|2 − |x|2 + ) dy. 2 1 − e−2t

Thus 󵄩󵄩 󵄩󵄩2 |y|2 1 t 2 󵄨󵄨 󵄨󵄨2 lim lim 󵄩󵄩󵄩󵄩eE0 t 𝔼x [e− 2 ∫0 |Bs | ds 1|Bt |>R ]󵄩󵄩󵄩󵄩 = lim 󵄨󵄨󵄨 ∫ e− 2 dy󵄨󵄨󵄨 = 0 󵄨 R→∞ t→∞ 󵄩 R→∞ 󵄨 󵄩 |y|>R

and (4.3.13) is satisfied. 4.3.5 Sojourn times and existence of bound states We know from Theorem 4.72 that confining potentials have discrete spectra, and thus the related Schrödinger operators have a countable set of bound states. The existence of bound states for Schrödinger operators with decaying potentials is, however, not similarly guaranteed and this depends on further properties of the potential. In this section we will see that bound states exist also for decaying potentials as soon as they allow the related processes to spend sufficiently long times in given regions of space. For the discussion it will be sufficient to consider compactly supported potentials, and we start by the Schrödinger operator with a one-dimensional potential well 1 Hv = − Δ − v1{|x|≤a} (x) 2

(4.3.17)

on L2 (ℝ), with the well having boundaries at ±a, a > 0, and depth v > 0. The Feynman–Kac formula gives (f , e−tHv g) = ∫ f (x)𝔼x [evUt (a) g(Bt )]dx ℝ

(4.3.18)

4.3 Properties of Schrödinger operators and semigroups | 283

for f , g ∈ L2 (ℝ), where t

Ut (a) = ∫ 1{|Bs |≤a} ds 0

is the occupation time of a one-dimensional Brownian motion in the potential well (see also (3.5.8)). From (4.3.18) it follows that x,y

(f , e−tHv g) = ∫ ∫ f (x)g(y)Πt (x − y)𝔼[0,t] [evUt (a) ]dxdy,

(4.3.19)

ℝℝ x,y

where 𝔼[0,t] denotes expectation with respect a Brownian bridge measure. Thus the integral kernel of e−tHv is given by

x,y

e−tHv (x, y) = Πt (x − y)𝔼[0,t] [evUt (a) ] .

(4.3.20)

We recall without proof the following strong ergodic property of one and twodimensional Brownian motion, known as the Kallianpur–Robbins laws. Proposition 4.137 (Kallianpur–Robbins laws). Let W ∈ L1 (ℝd ) and suppose that W̄ = ∫ℝd W(x)dx > 0. Then for every u > 0 we have (d = 1)

t

u

∫ W(Bs )ds 2 2 lim 𝒲 ( 0 ≤ u) = √ ∫ e−z /2 dz; ̄ t→∞ π √ W t x

(4.3.21)

0

(d = 2) lim 𝒲 x (

t→∞

t

∫0 W(Bs )ds ≤ log u) = 1 − e−u . W̄ log t

(4.3.22)

Here the limits do not depend on x ∈ ℝd . tain

Consider d = 1 and W(x) = 1{|x|≤a} (x). From the Kallianpur–Robbins laws we oblim 𝒲 x (

t→∞

u

2 Ut (a) 2 ≤ u) = √ ∫ e−z /2 dz. π 2a√t

0

In particular, we have ∞

2 lim 𝒲 (Ut (a) ≥ 2au√t) ≥ √ ∫ e−z t→∞ π x

u

2

/2

dz > 0.

Using (4.3.23) we can give a path integration proof of the following theorem.

(4.3.23)

284 | 4 Feynman–Kac formulae Theorem 4.138. Let d = 1. For every v > 0 there exists a bound state of Hv . Proof. Consider the set G = {|Bt | > bt α }, where b > 0 and α > 0. Note that (bt α −x)2

∞

2 2 2 e− 2t t→∞ 𝒲 (G) = √ 󳨀→ 0, ∫ e−y /2t dy ≤ √ πt πt bt α − x

x

(4.3.24)

bt α −x

whenever α > 21 . Hence we have with such α lim 𝒲 x (Ut (a) ≥ 2au√t)

t→∞

= lim 𝒲 x (Ut (a) ≥ 2au√t |Gc )𝒲 x (Gc ) + lim 𝒲 x (Ut (a) ≥ 2au√t |G)𝒲 x (G) t→∞

t→∞

= lim 𝒲 (Ut (a) ≥ 2au√t |Gc ) t→∞

x

(4.3.25)

due to (4.3.24). Consider the set E = {B0 = x} ∪ {Ut (a) ≥ 2au√t}, and choose f (x) = 1{|x|≤b} and g(y) = 1{|y|≤bt α } with some b > 0. In particular, on E paths satisfy t

e−2auv t−∫0 V(Bs )ds ≥ 1. Since on x, we furthermore have √

1 ∫ f (x)dx 2b ℝ

= 1 and the probabilities above do not depend

lim 𝒲 x (Ut (a) ≥ 2au√t)

t→∞

=

1 󵄨 ∫ f (x) lim ∫ g(y)𝒲 x (Ut (a) ≥ 2au√t 󵄨󵄨󵄨 Bt ∈ dy) dx t→∞ 2b ℝ

ℝ

1 x,y ≤ ∫ lim ∫ f (x)g(y)𝔼[0,t] [1]dxdy 2b t→∞ ℝ

ℝ

t 1 √ x,y ≤ ∫ lim e−2auv t ∫ f (x)g(y)𝔼[0,t] [e− ∫0 V(Bs )ds ]dxdy 2b t→∞

ℝ

ℝ

1 √ = lim ∫ e−2auv t f (x)dx ∫ e−tHv (x, y)g(y)dy, 2b t→∞ ℝ

(4.3.26)

ℝ

where the last step is obtained by using (4.3.20). Next suppose, to the contrary, that Hv is a nonnegative operator. Then e−tHv is contraction and hence |(f , e−tHv g)| ≤ ‖f ‖‖g‖ follows. We note that Specess (Hv ) = [0, ∞) by Theorem 4.74. The dominated convergence theorem implies by the expression above and (4.3.23) that 0 < lim 𝒲 x (Ut (a) ≥ 2au√t) ≤ t→∞

≤

1 √ lim e−2auv t (f , e−tHv g) 2b t→∞

1 √ √ lim e−2auv t ‖f ‖‖g‖ = lim e−2auv t t α/2 = 0 t→∞ 2b t→∞

for every v > 0. This is a contradiction showing that for every v > 0 there is a negative discrete spectrum of Hv and a bound state exists.

4.3 Properties of Schrödinger operators and semigroups |

285

This argument can be extended to the two-dimensional case using the second statement in the Kallianpur–Robbins laws. Consider now the Schrödinger operator with a two-dimensional potential well 1 Hv = − Δ − v1{|x|≤a} , 2

(4.3.27)

where we assume a, v > 0 again. Theorem 4.139. Let d = 2. For every v > 0 there exists a bound state of Hv . Proof. Since the potential is now rotationally symmetric, we use polar coordinates. The operator (4.3.27) is unitary equivalent with 1 𝜕2 1 𝜕 1 𝜕2 H̃ v = (− − − v1 ) ⊗ 1 − ⊗ {r≤a} 2 𝜕r 2 2r 𝜕r 2r 2 𝜕θ2 acting in L2 (ℝ2 ) ≅ L2 ([0, ∞)) ⊗ L2 (S1 ). Here 2

L (S1 ). It is known that value, and e

±iθl

𝜕2 Spec( 𝜕θ 2)

=

{−l2 }∞ l=0 ,

𝜕2 𝜕θ2

is the Laplace–Beltrami operator on

each −l2 is a two-fold degenerate eigen-

are eigenfunctions associated with −l2 . We have ∞ l2 H̃ v = ⨁ (Hr,v + 2 ) 2r l=0

2 under L2 (ℝ2 ) ≅ ⨁∞ l=0 L ([0, ∞)), where

Hr,v = −

1 𝜕2 1 𝜕 − − v1{r≤a} . 2 𝜕r 2 2r 𝜕r

Thus to prove the theorem it is sufficient to show that the Hamiltonian Hr,v has a bound state in L2 ([0, ∞)). Note that 0 Hr,v =

1 𝜕2 1 𝜕 + 2 2 𝜕r 2r 𝜕r

is the generator of a two-dimensional Bessel process (Rt )t≥0 on a probability space (Ω, F , P y ), starting from y ≥ 0. It follows that for f , g ∈ L2 ([0, ∞)) y

(f , e−tHr,v g) = ∫ f (y)𝔼P [evUt (a) g(Rt )]dy. [0,∞)

From the two-dimensional case it follows from the Kallianpur–Robbins laws that lim P y (

t→∞

where

Ut (a) ≤ u) = 1 − e−u , πa2 log t

u ≥ 0, y ∈ ℝ2 ,

t

Ut (a) = ∫ 1{Rs ≤a} (Rs )ds. 0

286 | 4 Feynman–Kac formulae Consider the set G = {Rt > bt α }. Using the probability transition kernel of (Rt )t≥0 , i. e., r − y22t+r2 yr e I0 ( ) , t t where I0 (x) = ∑∞ k=0 all α >

1 2

(x/2)2k k!Γ(k+1)

is the modified Bessel function of the first kind, we have for

∞

y2 +r 2 1 yr ∫ re− 2t I0 ( ) dr = 0. t→∞ t t

lim P y (Rt ≥ bt α ) = lim

t→∞

bt α −y

Hence by taking conditional expectation on G and f (y) = 1{y≤b} , g(r) = 1{r≤bt α } , we have similarly as in (4.3.26) that 󵄨 lim P y (Ut (a) ≥ πa2 u log t) = lim P y (Ut (a) ≥ πa2 u log t 󵄨󵄨󵄨 Gc )

t→∞

t→∞

∞

≤

∞

t 2 1 y,r ∫ lim ∫ f (y)g(r)e−πa uv log t ∫ e− ∫0 V(Rs )ds dP[0,t] dydr b t→∞

0

=

0

XD

∞

∞

0

0

2 1 lim ∫ t −πa uv f (y)dy ∫ e−tHr,v (y, r)g(r)dr. b t→∞

y,r

Here P[0,t] denotes bridge measure. We take α and u such that 2πa2 v > α/u. Assuming to the contrary that Hr,v is nonnegative and hence a contraction, we obtain like in the one-dimensional case 0 < e−u ≤ lim P y (Ut (a) ≥ πa2 u log t) ≤ t→∞

2 1 lim e−πa uv log t (f , e−tHr,v g) b t→∞

2 2 1 ≤ lim t −πa uv ‖f ‖‖g‖ = lim t −πa uv+α/2 = 0. t→∞ b t→∞

This is a contradiction, implying the statement. Remark 4.140. We note that by a calculation one can show that c √t, 𝔼[Ut (a)] = { 1 c2 log t,

d = 1, d = 2,

where c1 = c1 (a) > 0 and c2 = c2 (a) > 0, which explains the time-scales appaearing in the Kallianpur–Robbins laws. Thus the above theorems show that the existence of a bound state can be understood to be a consequence of paths spending sufficiently long times in particular subsets of space and it gives bounds on how long these average total times must be. The proofs use the fact that, since in one and two dimensions Brownian motion is recurrent, specific ergodic properties hold on its additive functionals (given here by the Kallianpur–Robbins laws). On the other hand, three- or

4.3 Properties of Schrödinger operators and semigroups | 287

higher-dimensional Brownian motion is transient and such a result cannot hold. In this case indeed we know that the potential well needs to be deep enough for a bound state to occur. Next we discuss the relationship of sojourn times and bound states from a different perspective, and show that actually the eigenfunctions for a potential well can even be reconstructed by using such random times. Consider Hv defined by (4.3.17). The spectrum of this operator is Spec(Hv ) = Specess (Hv )∪Specd (Hv ) with Specess (Hv ) = [0, ∞). By general theory we know that all eigenvalues E0 < E1 < E2 < . . . are simple. Since the potential is even, it follows that φ(−x) is an eigenfunction whenever φ(x) is too. Since all the eigenvalues are simple, we have φ(−x) = ±φ(x), i. e., they are either even or odd. Furthermore, the eigenfunction φn corresponding to eigenvalue En has exactly n zeroes. To determine the discrete component Specd (Hv ) of the spectrum, we need to solve the system of differential equations −φ󸀠󸀠 (x) − vφ(x) = Eφ(x), { 󸀠󸀠 −φ (x) = Eφ(x),

|x| ≤ a, |x| > a,

with boundary conditions given by the requirement that φ and φ󸀠 are continuous functions at ±a. A straightforward calculation shows that the eigenvalues E = −|E| corresponding to even and odd eigenfunctions satisfy respectively the transcendental equations tan (a√2(v − |E|)) = √

|E| v − |E|

and

cot (a√2(v − |E|)) = −√

|E| . v − |E|

(4.3.28)

It can be verified by elementary analysis that these equations have a unique solution En ∈ (−v, 0) for each n ∈ ℕ and for every a, v > 0. Moreover, the total number of eigenvalues is N + 1, N = 0, 1, 2, . . ., determined by the inequality Nπ √ (N + 1)π ≤ 2va < . 2 2

The eigenfunctions are φn (x) = {

An e−√2|En ||x| 1{|x|>a} + Bn cos(√2(v − |En |) x)1{|x|≤a} , An e

−√2|En ||x|

1{|x|>a} + Bn sin(√2(v − |En |) x)1{|x|≤a} ,

n = 2m,

n = 2m + 1

with m = 0, 1, 2, . . ., and normalization constants An and Bn . In particular, for the ground state we have φ0 (x) = A0 e−√2|E||x| 1{|x|>a} + B0 cos(√2(v − |E|) x)1{|x|≤a}

(4.3.29)

with A0 = √

√2|E|

1 + a√2|E|

ea√2|E| cos(a√2(v − |E|)),

where for simplicity we write E0 = E.

B0 = √

√2|E|

1 + a√2|E|

,

288 | 4 Feynman–Kac formulae Now we show that the eigenfunctions can be determined by (4.3.18) in terms of occupation times, i. e., φn (x) = 𝔼x [e−|En |t+vUt (a) φn (Bt )],

t > 0, n ∈ ℕ.̇

(4.3.30)

For simplicity we only discuss the ground state in detail. Consider the first hitting times defined by τa = inf{t > 0 | Bt = a} for Brownian motion starting at zero, and τa,b = τa ∧ τb for the levels a < 0 < b. Recall from Lemma 2.60 that these are almost surely finite stopping times with respect to the natural filtration of Brownian motion. From (4.3.30) we have x

φ0 (x) = 𝔼[e−|E0 |t+vUt (a) φ0 (Bt + x)], where we denote t

t

0

0

Utx (a) = ∫ 1{|Bs +x|≤a} ds = ∫ 1{−a−x≤Bs ≤a−x} ds. Note that Uτx−a−x,a−x (a) = τ−a−x,a−x whenever |x| < a and is zero otherwise. Since the stopping times are almost surely finite, we furthermore have φ0 (x) = 𝔼[e

x −|E0 |t∧τ−a−x,a−x +vUt∧τ (a) −a−x,a−x

→ 𝔼[e

φ0 (Bt∧τ−a−x,a−x + x)]

−|E0 |τ−a−x,a−x +vUτx−a−x,a−x (a)

φ0 (Bτ−a−x,a−x + x)] as t → ∞.

Recall the Laplace transforms 𝔼[e−uτa ] = e−

√2u|a|

and

𝔼[e−uτa,b ] =

cosh (√2u b+a ) 2 cosh (√2u b−a ) 2

,

u ≥ 0.

(4.3.31)

Let x > a. By path continuity τ−a−x,a−x = τa−x , and we have φ0 (x) = 𝔼[e−|E0 |τa−x φ0 (Bτa−x + x)] = φ0 (a)𝔼[e−|E0 |τa−x ] = φ0 (a)e−√2|E0 |(x−a) . We obtain similarly for x < −a that τ−a−x,a−x = τ−a−x and φ0 (x) = 𝔼[e−|E0 |τ−a−x φ0 (Bτ−a−x + x)] = φ0 (−a)e−√2|E0 |(−x−a) = φ0 (a)e

√2|E0 |(x+a)

using φ0 (−a) = φ0 (a). When −a < x < a we apply the two-barrier formula in (4.3.31) and obtain φ0 (x) = 𝔼[e(v−|E0 |)τ−a−x,a−x φ0 (Bτ−a−x,a−x + x)] = 𝔼[e(v−|E0 |)τ−a−x,a−x φ0 (Bτ−a−x,a−x + x)1{τ−a−x τa−x } ]

4.3 Properties of Schrödinger operators and semigroups | 289

= φ0 (a)

cos(√2(v − |E0 |)x)

cos(√2(v − |E0 |)a)

.

The constant φ0 (a) can be determined by the normalization condition, which then yields the expression of the ground state given by (4.3.29). Note that for odd eigenfunctions φ(−a) = −φ(a) and therefore the cos(√2(v − |E0 |)x) factors are replaced by sin(√2(v − |E0 |)x). Remark 4.141. The argument can also be extended to higher dimensions. Denote by Br (a) a ball of radius r centered in a, and consider the Schrödinger operator H = − 21 Δ − v1Br (a) . Define the stopping times TBa = inf{t > 0 : Xt ∈ Ba } and

τBa = inf{t > 0 : Xt ∉ Ba }.

Using the Ciesielski–Taylor formulae 𝔼x [e−uτBr (a) ] = ( 𝔼x [e−uTBr (a) ] = (

r ) |x − a|

d−2 2

r ) |x − a|

d−2 2

I d−2 (|x|√2u) 2

I d−2 (r √2u) 2

K d−2 (|x|√2u) 2

K d−2 (r √2u)

,

2

where I d−2 and K d−2 are the modified Bessel functions of the first and third kind, re2

spectively, i. e.,

2

Iν (z) = Kν (z) =

z ν 21−ν

Γ(ν +

1

1

∫ e−zy (1 − y2 )ν− 2 dy

1 )√π 2 0 ∞

z ν √π

2ν Γ(ν +

1

∫ e−zy (y2 − 1)ν− 2 dy.

1 ) 2 1

Then by a similar argument as above we obtain φ0 (x) = A0 (

a ) |x|

d−2 2

K d−2 (|x|√2|E0 |)1{|x|>a} + B0 ( 2

a ) |x|

d−2 2

I d−2 (|x|√2(v − |E0 |))1{|x|≤a} , 2

where the constants can be determined from L2 -normalization as before. 4.3.6 The number of eigenfunctions with negative eigenvalues In this section we prove the celebrated Lieb–Thirring inequality, giving an estimate on the number of eigenfunctions in the level sets of nonpositive eigenvalues of a Schrödinger operator. From the point of view of quantum mechanics these eigenfunctions describe bound states. Let {Ej }Nj=1 be nonpositive eigenvalues of a Schrödinger

290 | 4 Feynman–Kac formulae operator − 21 Δ−W with a negative potential −W. A heuristic motivation of Lieb–Thirring inequality is the following semiclassical approximation: N

∑ |Ej |β ∼ j=1

1 (2π)d

β 1 ∫ [( |p|2 − W(x)) ] dpdx 2 −

ℝd ×ℝd

1 1 = ∫ dx (2π)d 2β ℝd

(2W(x) − r 2 )β r d−1 |Sd−1 |dr

∫ 0≤r≤√2W(x) π/2

=

1 2β+(d/2) |Sd−1 | ∫ sin2β+1 θ cosd−1 θdθ ∫ W(x)β+(d/2) dx. (2π)d 2β 0

ℝd

Inserting the (d − 1)-dimensional surface area of unit sphere |Sd−1 | = 2π d/2 /Γ(d/2) and π/2

using the formula ∫0

sin2β+1 θ cosd−1 θdθ = 21 Γ(β + 1)Γ(d/2)/Γ(β + 1 + d/2), we have N

∑ |Ej |β ∼ acd,β ∫ |W(x)|β+(d/2) dx, j=1

where acd,β =

ℝd

Γ(β + 1) 1 . (2π)d/2 Γ(β + 1 + d/2)

Throughout this section we denote 1 H0 = − Δ, 2 so that H = H0 + V. Let ET (⋅) be the spectral resolution of the self-adjoint operator T and write 1(⋅) (T) = ET (⋅)L2 (ℝd ). Define NE (V) = dim 1(−∞,−E] (H) = #{eigenvalues of H ≤ −E}, N0 (V) = lim NE (V).

E > 0,

E↓0

Remark 4.142. In this section we discuss H such that V is relatively form-compact with respect to H0 . Then the essential spectrum of H is identical with that of H0 and it follows that Specess (H) = [0, ∞) and (−∞, 0)∩Spec(H) is purely discrete. When 0 is an accumulation point of the discrete spectrum, N0 (V) = ∞ = dim 1(−∞,0) (H) follows. On the other hand, when 0 is not an accumulation point, we have N0 (V) = dim 1(−∞,0) (H). Thus in all cases for a relatively form-compact V it follows that N0 (V) = dim 1(−∞,0) (H) = #{eigenvalues of H < 0}. Note that N0 (V) does not include a count of zero eigenvalues.

4.3 Properties of Schrödinger operators and semigroups | 291

Before estimating N0 (V) we briefly discuss the Birman–Schwinger principle. Recall the kernel of −Δ + m2 introduced in (4.2.20). Let Cm (x, y) denote its integral kernel, which is the Green function defined by (−Δ + m2 )Cm (x, y) = δ(x − y). Alternatively, by Fourier transform Cm (x, y) = Cm (x − y) =

1 e−ip⋅(x−y) dp. ∫ |p|2 + m2 (2π)d ℝd

As was seen in (4.2.20) this can be computed explicitly, i. e., Cm (x) =

1 m (d−2)/2 ( ) K(d−2)/2 (m|x|), (2π)d/2 |x|

(4.3.32)

where Kν (x) > 0 is the modified Bessel function of the third kind (4.2.21). The fact e−m|x|

2m Cm (x) = { e−m|x|

,

, 4π|x|

d = 1, d=3

was shown in Lemmas 4.97–4.98. Many properties of Cm (x, y) can be deduced from (4.3.32) and the known properties of Bessel functions. Here we give a summary of the properties we need for our purposes. Proposition 4.143. The Green function Cm (x, y) = Cm (x−y) has the following properties: (1) Cm (x, y) > 0; e (2) for m|x − y| > 0 it behaves like Cm (x, y) ≤ const m(d−3)/2 |x−y| (d−1)/2 , for all d ≥ 3; −m|x−y|

|x − y|−d+2 , (3) for m|x − y| ∼ 0 it behaves like Cm (x, y) ∼ { − log(m|x − y|), (4) limm→0 Cm (x, y) = C0 (x, y), for all d ≥ 3; 1 (5) C0 (x, y) = Γ(d/2−1) , for all d ≥ 3. 4π d/2 |x−y|d−2

d ≥ 3, d = 2;

Now we look at potentials V for which |V|1/2 (−Δ+m2 )−1 |V|1/2 is a compact operator, i. e., when V is relatively form-compact with respect to −Δ. Lemma 4.144 (Compactness of Birman–Schwinger kernel). Let K = |V|1/2 (−Δ + m2 )−1 |V|1/2 . (d = 1) Let V ∈ L1 (ℝ) and m > 0. Then K ∈ I2 and ‖K‖2 ≤ C‖V‖1 . (d = 2) Let V ∈ Lp (ℝ2 ) for p > 1 and m > 0. Then K ∈ Ip and ‖K‖p ≤ C‖V‖p . (d ≥ 3) Let V ∈ Ld/2 (ℝd ) and m ≥ 0. Then K ∈ I2 and ‖K‖2 ≤ C‖V‖d/2 .

292 | 4 Feynman–Kac formulae Proof. For d = 1 it is trivial to see that ∫ |V(x)|Cm (x − y)2 |V(y)|dxdy ≤ C‖V‖21 . ℝ×ℝ

Thus it is immediate that K ∈ I2 and ‖K‖22 ≤ C‖V‖21 . Consider d = 2. By Proposition 4.39 we see that (−Δ + m2 )−1/2 |V|1/2 ∈ I2p for all p > 1 since (|k|2 + m2 )−1/2 ∈ L2p (ℝ2 ) and |V|1/2 ∈ L2p (ℝ2 ). Thus by Proposition 4.37, K ∈ Ip and ‖K‖p ≤ C‖V 1/2 ‖22p = C‖V‖p . Finally, let d ≥ 3. It suffices to discuss the case of m = 0 only, since (f , |V|1/2 (−Δ + m2 )−1 |V|1/2 f ) ≤ (f , |V|1/2 (−Δ)−1 |V|1/2 f ). By the Hardy–Littlewood–Sobolev inequality we have ∫ |V(x)|C0 (x − y)2 |V(y)|dxdy ≤ C‖V‖2d/2 . ℝd ×ℝd

This gives K ∈ I2 and ‖K‖2 ≤ C‖V‖d/2 . For d ≥ 3 a similar estimate to Lemma 4.144 can be obtained due to Proposition 4.40, studying the map Lpw (ℝd ) × Lp (ℝd ) ∋ (u, g) → Bu,g ∈ Ip,w . Lemma 4.145. Let d ≥ 3 and V ∈ Ld/2 (ℝd ). Then for p > d/2, |V|1/2 (−Δ)−1 |V|1/2 ∈ Ip ,

(4.3.33)

‖|V|1/2 (−Δ)−1 |V|1/2 ‖p ≤ Dp ‖V‖d/2

(4.3.34)

and

follow with suitable Dp > 0. Proof. Note that u(k) = 2|k|−1 ∈ Ldw (ℝd ). We have Bu,|V|1/2 ∈ Id,w , hence (−Δ)−1/2 |V|1/2 = FBu,|V|1/2 ∈ Id,w . It follows that |V|1/2 (−Δ)−1 |V|1/2 ∈ Id/2,w by Proposition 4.37, and then (4.3.33). Furthermore, by (4.1.29) we have ‖(−Δ)−1/2 |V|1/2 ‖d,w ≤ C‖|V|1/2 ‖d and by Proposition 4.37 we see again that ‖|V|1/2 (−Δ)−1 |V|1/2 ‖d/2,w ≤ C 2 ‖|V|1/2 ‖2d = C 2 ‖V‖d/2 . Using (4.36), we then arrive at (4.3.34). Let V = V+ − V− be such that V− ∈ L∞ (ℝd ). Since H0 − V− ≤ H0 + V in the sense of quadratic forms, the number of negative eigenvalues of H0 +V is smaller than that of H0 − V− . Therefore, instead of H0 + V we may consider H0 − V− . We assume in what follows that V(x) = −W(x) ≤ 0 such that V is relatively formcompact with respect to H0 . Let H(α) = H0 − αW.

4.3 Properties of Schrödinger operators and semigroups | 293

Since −W is H0 -form-compact, the essential spectrum of H(α) stays unchanged, i. e., Specess (H(α)) = Specess (H0 ) = [0, ∞). Furthermore, H(α) is bounded from below since −W is also relatively form-bounded with an infinitesimally small relative bound. Notice that for λ ≥ 0 we have H(α)ψ = −λψ exactly when (H0 + λ)ψ = αWψ. From this we get that Wψ ∈ D((H0 + λ)−1 ) and ψ = α(H0 +λ)−1 Wψ. Since ψ ∈ D(W 1/2 ), the identity W −1/2 W 1/2 ψ = α(H0 +λ)−1 W 1/2 W 1/2 ψ gives W 1/2 ψ = αW 1/2 (H0 + λ)−1 W 1/2 W 1/2 ψ. We then conclude that for λ ≥ 0 we have H(α)ψ = −λψ if and only if W 1/2 (H0 + λ)−1 W 1/2 φ = where φ = W 1/2 ψ. Put

1 φ, α

Kλ = W 1/2 (H0 + λ)−1 W 1/2 .

This is also called Birman–Schwinger kernel, for which −λ ∈ Spec(H(α))

⇐⇒

1 ∈ Spec(Kλ ) α

holds. Lemma 4.146. Let ej (λ) be the jth eigenvalue of Kλ , i. e., ‖Kλ ‖ = e1 (λ) ≥ e2 (λ) ≥ . . . ≥ 0. Then the map (0, ∞) ∋ λ 󳨃→ ej (λ) is continuous, nonincreasing, i. e., ej (λ + a) ≤ ej (λ) for any a > 0, and limλ→∞ ej (λ) = 0. Proof. Let λ ∈ (0, ∞). We show that limλ󸀠 →λ ‖Kλ − Kλ󸀠 ‖ = 0, i. e., λ 󳨃→ Kλ is uniformly continuous on (0, ∞). It is direct to see that ‖Kλ − Kλ󸀠 ‖ = |λ − λ󸀠 |‖W 1/2 (H0 + λ󸀠 )−1 (H0 + λ)−1 W 1/2 ‖ ≤

|λ − λ󸀠 | ‖W 1/2 (H0 + λ󸀠 )−1/2 ‖‖(H0 + λ)−1/2 W 1/2 ‖. √λλ󸀠

Note that ‖W 1/2 (H0 + λ󸀠 )−1/2 ‖ ≤ ‖(H0 + λ󸀠 )−1/2 (H0 + λ)1/2 ‖‖(H0 + λ)−1/2 W 1/2 ‖ ≤ C‖(H0 + λ)−1/2 W 1/2 ‖,

where C is independent of λ󸀠 . Hence we have 1 ‖Kλ − Kλ󸀠 ‖ ≤ C|λ − λ󸀠 | ‖(H0 + λ)−1/2 W 1/2 ‖2 . λ This yields that λ 󳨃→ Kλ is uniformly continuous and for ε > 0 there exists δ > 0 such that |(ϕ, (Kλ − Kλ󸀠 )ϕ)| ≤ ε‖ϕ‖2 for |λ − λ󸀠 | ≤ δ and for arbitrary ϕ. We have (ϕ, Kλ󸀠 ϕ) − ε‖ϕ‖2 ≤ (ϕ, Kλ ϕ) ≤ (ϕ, Kλ󸀠 ϕ) + ε‖ϕ‖2

294 | 4 Feynman–Kac formulae for all ϕ. From this and the min-max principle we see that ej (λ󸀠 ) − ε ≤ ej (λ) ≤ ej (λ󸀠 ) + ε and thus |ej (λ) − ej (λ󸀠 )| ≤ ε for |λ − λ󸀠 | < δ, and the continuity of ej (⋅) follows. It is immediate to see that (ϕ, Kλ ϕ) ≥ (ϕ, Kλ󸀠 ϕ) for λ ≤ λ󸀠 . Consequently ej (⋅) is decreasing by the min-max principle again. Finally, let ej (λ) ∈ Spec(Kλ ) if and only if −λ ∈ Spec(H(1/ej (λ))). Since H(1/ej (λ)) = H0 −̇ 1/ej (λ)W is bounded below by −1/ej (λ) × c with some c by the KLMN theorem, −λ ≥ −1/ej (λ) × c and from this it follows that c/λ ≥ ej (λ). Thus limλ→∞ ej (λ) = 0. Proposition 4.147 (Birman–Schwinger principle). Suppose that V ≤ 0 and it is H0 -formcompact. Then NE (V) = dim 1[1,∞) (KE ), N0 (V) ≤ dim 1[1,∞) (K0 ).

−E < 0,

(4.3.35) (4.3.36)

Proof. For −E < 0, NE (V) = #{e ∈ Spec(H(1)) | e ≤ −E} = #{λ ∈ [E, ∞) | Spec(Kλ ) ∋ 1} = #{e ∈ Spec(KE ) | e ≥ 1} = dim 1[1,∞) (KE ).

(4.3.37)

The second equality of (4.3.37) can be derived from the fact that −λ ∈ Spec(H(1)) ⇐⇒ 1 ∈ Spec(Kλ ),

(4.3.38)

and the third is derived from Lemma 4.146. The relation (4.3.35) follows from (4.3.37). To obtain (4.3.36) note that N0 (E) = limE↓0 NE (V) and KE ≤ K0 . Next we extend the Birman–Schwinger principle to a version including zero eigenvalues. We proceed by more abstract arguments. Let 𝒦 be a separable Hilbert space over ℂ and T a self-adjoint operator in 𝒦 such that inf Spec(T) = 0 and 0 ∈ ̸ Specp (T). Let S be a self-adjoint operator such that S ≤ 0 and LE = |S|1/2 (T + E)−1 |S|1/2 is compact for E ≥ 0. The Birman–Schwinger principle says that dim 1(−∞,0) (T + S) ≤ dim 1[1,∞) (L0 ). We want to extend this to dim 1(−∞,0] (T + S) ≤ dim 1[1,∞) (L0 ),

(4.3.39)

so that the left-hand side in (4.3.39) includes the number of nonpositive eigenvalues of T + S. In particular, the bound dim 1[1,∞) (L0 ) < 1 yields that T + S has no nonpositive eigenvalues. To prove (4.3.39) we use the following lemma.

4.3 Properties of Schrödinger operators and semigroups | 295

Lemma 4.148. Let α > 1. Then dim 1(−∞,0] (T + S) ≤ dim 1(−∞,0) (T + αS). Proof. We proceed by contradiction. Suppose that dim 1(−∞,0) (T + αS) < dim 1(−∞,0] (T + S). Since the left-hand side above is finite, we write dim 1(−∞,0) (T + αS) = n ≥ 0. Then dim 1(−∞,0] (T + S) ≥ n + 1. Let 𝒦 ⊃ {ϕ1 , . . . , ϕn+1 } be an arbitrary orthonormal system of 1(−∞,0] (T + S), and L = linear hull of {ϕ1 , . . . , ϕn+1 } an (n + 1)-dimensional subspace of 𝒦. Consider a projection P : 𝒦 → L . The restriction A = P(T + αS)P⌈L can be regarded as a self-adjoint operator from L to itself. Denote the eigenvalues of A by μ̃ j , j = 1, . . . , n + 1, ordered as μ̃ 1 ≤ μ̃ 2 ≤ . . . ≤ μ̃ n+1 . We can compare the eigenvalues of T + αS and A. Define μj =

sup

ψ1 ,...,ψj−1 ∈K

inf

ψ∈D(A);‖ψ‖=1 ψ∈[ψ1 ,...,ψj−1 ]⊥

(ψ, Aψ).

Inequality μj ≤ μ̃ j for j = 1, . . . , n + 1 can be derived by the Rayleigh–Ritz technique in Proposition 4.24. We have (f , (T + αS)f ) = (f , (T + S)f ) + (α − 1)(f , Sf ) ≤ 0 for all f ∈ L . Suppose that 0 = (g, (T + αS)g) for some g ∈ L . Then (g, (T + S)g) = (α − 1)(g, Sg) = 0 and (g, Tg) = 0 follow. However Specp (T) ∋ ̸ 0 implies that g = 0. Consequently, A is strictly negative and μ̃ j < 0 for j = 1, . . . , n + 1 follows. In particular, μn+1 ≤ μ̃ n+1 < 0. This yields dim 1(−∞,0) (T + αS) ≥ n + 1, contradicting dim 1(−∞,0) (T + αS) = n, and thus the lemma follows. Theorem 4.149. Suppose that L0 is compact. Then dim 1(−∞,0] (T + S) ≤ dim 1[1,∞) (L0 ). Proof. By Lemma 4.148 we have dim 1(−∞,0] (T + S) ≤ lim dim 1(−∞,0) (T + αS). α↑1

(4.3.40)

Note that dim 1(−∞,0) (T + αS) = limE↑0 dim 1(−∞,E] (T + αS), and by the Birman– Schwinger principle we have dim 1T+αS ((−∞, E]) ≤ dim 1αLE ([1, ∞)) for all E < 0. Since LE ≤ L0 , dim 1αLE ([1, ∞)) ≤ dim 1[1,∞) (αL0 ) follows. Using these results and (4.3.40) we have dim 1(−∞,0] (T + S) ≤ lim dim 1[1,∞) (αL0 ) ≤ dim 1[1,∞) (L0 ), α↑1

and the theorem follows.

296 | 4 Feynman–Kac formulae An immediate consequence leads to an extension of Proposition 4.147. Define N̄ E (V) = dim 1(−∞,−E] (H) = #{eigenvalues of H ≤ −E},

E ≥ 0.

Note that N̄ E (V) = NE (V) for −E < 0 and N̄ 0 (V) ≥ N0 (V). Corollary 4.150 (Extended Birman–Schwinger principle). Suppose that V ≤ 0 and it is H0 -form-compact. Then N̄ E (V) = dim 1[1,∞) (KE ), N̄ 0 (V) ≤ dim 1[1,∞) (K0 ).

−E < 0,

Proof. We replace T, S, and L0 by H0 , V, and K0 , respectively, in Theorem 4.149. Corollary 4.150 has the following immediate implication. Corollary 4.151 (Absence of ground state). Let d ≥ 3. Suppose that V ≤ 0 is H0 -formcompact and the operator norm of K0 satisfies ‖K0 ‖ < 1. Then H has no nonpositive eigenvalues. In particular, the bottom of the spectrum of H is not an eigenvalue, i. e., H has no ground state. Proof. By the extended Birman–Schwinger principle N̄ 0 (V) ≤ dim 1[1,∞) (K0 ) = 0 follows, and Specess (H) = [0, ∞). Note that in general it is difficult to verify whether zero is an eigenvalue of H. We give a sufficient condition for H to have no zero eigenvalue in Proposition 4.168 below. However, as shown in Corollary 4.151, ‖K0 ‖ < 1 implies that zero is no eigenvalue of H. From now on we will estimate N̄ 0 (V) from above. It can be seen directly that W 1/2 (H0 + αW + λ)−1 W 1/2 = Kλ (1 + αKλ )−1 .

(4.3.41)

On the other hand, the integral kernel of W

1/2

(H0 + αW + λ) W −1

1/2

=W

1/2

∞

∫ e−λt e−t(H0 +αW) W 1/2 dt 0

can be expressed by using the conditional Wiener measure, (W 1/2 (H0 + αW + λ)−1 W 1/2 )(x, y) ∞

t

x,y

= W 1/2 (x)W 1/2 (y) ∫ e−λt 𝔼[0,t] [e−α ∫0 W(Bs )ds ] Πt (x − y)dt.

(4.3.42)

0

Let g(y) = e

−αy

and F(x) = x(1 + αx)−1 , i. e., ∞

F(x) = x ∫ e−y g(xy)dy. 0

(4.3.43)

4.3 Properties of Schrödinger operators and semigroups | 297

Together with (4.3.41), (4.3.42) can be reformulated as ∞

t

x,y

F(Kλ )(x, y) = W 1/2 (x)W 1/2 (y) ∫ e−λt 𝔼[0,t] [g (∫ W(Bs )ds)] Πt (x − y)dt. 0 0 [ ]

(4.3.44)

Moreover, when W is continuous, the right-hand side of (4.3.42) is also continuous in x and y. By a general result, Tr F(Kλ ) can be evaluated by setting x = y. Hence t

∞

[ ] Tr F(Kλ ) = ∫ e−λt Πt (0)dt ∫ W(x)𝔼x,x [0,t] g (∫ W(Bs )ds) dx. 0 0 [ ] ℝd

(4.3.45)

Lemma 4.152. Suppose that W is H0 -form-compact and continuous, and let f (y) = yg(y) = ye−αy and F(x) = x(1 + αx)−1 . Then it follows that t

∞

dt [ ] Tr F(Kλ ) = ∫ e−λt Πt (0) ∫ 𝔼x,x [0,t] f (∫ W(Bs )ds) dx. t 0 [ 0 ] ℝd

(4.3.46)

Proof. It suffices to show that t

t 1 [f (∫ W(Bs )ds) e−α ∫0 W(Bs )ds ] dx ∫ 𝔼x,x [0,t] t [ 0 ] ℝd

t

t

−α ∫0 W(Bs )ds ] [ = ∫ W(x)𝔼x,x dx. [0,t] g (∫ W(Bs )ds) e 0 [ ] ℝd

(4.3.47)

Let Ur = e−rH(−α) We−(t−r)H(−α) . Note that Ur is compact, and its integral kernel can be computed as r

t−r

(Ur f )(x) = 𝔼x [e−α ∫0 W(Bs )ds W(Br )𝔼Br [e−α ∫0

W(Bs )ds

f (Bt−r )]]

t

= 𝔼x [e−α ∫0 W(Bs )ds W(Br )f (Bt )] t

x,y

= ∫ Πt (x − y)𝔼[0,t] [e−α ∫0 W(Bs )ds W(Br )] f (y)dy. ℝd

Hence the kernel is given by t

x,y

Ur (x, y) = Πt (x − y)𝔼[0,t] [e−α ∫0 W(Bs )ds W(Br )] and it is continuous in x and y. Thus it follows that t

−α ∫0 W(Bs )ds Tr Ur = ∫ Ur (x, x)dx = Πt (0) ∫ 𝔼x,x W(Br )] dx. [0,t] [e ℝd

(4.3.48)

ℝd

t

From the identity Tr Ur = Tr U0 it follows that ∫0 1t Tr Ur dr = Tr U0 . Inserting (4.3.48) into this yields thus (4.3.47).

298 | 4 Feynman–Kac formulae Furthermore, a more general formula holds. Lemma 4.153. Let 0 ≤ W ∈ Lp (ℝd ) with d/2, { { p = {> 1, { {1,

d ≥ 3, d = 2, d = 1.

Fix λ > 0 for d = 1, 2 and λ ≥ 0 for d ≥ 3. Let f be a nonnegative lower semicontinuous function on [0, ∞) with f (0) = 0 and let f , g, and F be related by ∞

F(x) = x ∫ e−y g(xy)dy,

f (y) = g(y)y,

(4.3.49)

0

and F(‖Kλ ‖) < ∞. We have t

∞

1 [ ] Tr F(Kλ ) = ∫ e−λt Πt (0)dt ∫ 𝔼x,x [0,t] f (∫ W(Bs )ds) dx t 0 [ 0 ] ℝd ∞

= ∫e 0

−λt

t x,x [ Πt (0)dt ∫ W(x)𝔼[0,t] g (∫ W(Bs )ds)] dx, 0 [ ] ℝd

(4.3.50)

(4.3.51)

where both sides above may be infinite. We prove this lemma after Lemmas 4.154–4.157. In Lemmas 4.154–4.157 we assume that d ≥ 3. In the cases d = 1 and 2 the proof is similar. We also write Kλ (W) for Kλ when we emphasize the dependence of W, and (a, b) ⊂ (0, ∞) in the proofs of Lemmas 4.154–4.157. Note that W ∈ Lp (ℝd ) implies that (−1/2)Δ +̇ W is bounded from below; see Example 4.64. Furthermore, we see that F(x) < ∞ for every 0 ≤ x ≤ ‖Kλ ‖. Lemma 4.154. Suppose that λ > 0, W ∈ C0∞ (ℝd ), and g ∈ C0 ((a, b)). Let f , g, and F be related by (4.3.49). Then (4.3.50) and (4.3.51) hold. Proof. Denote the finite linear sum of functions of the form g(y) = e−αy by C = linear hull of {e−αy | α ≥ 0}, which by the Stone–Weierstrass theorem is dense in the space of continuous functions on [0, ∞) vanishing at infinity. Hence there exists a sequence (gn )n∈ℕ ⊂ C such that ‖gn − g‖∞ → 0 as n → ∞. We write F(x) = Fg (x) = ∞ ∞ x ∫0 e−y g(xy)dy to emphasize the dependence of g. Since Fg (x) = ∫0 e−z/x g(z)dz, g 󳨃→ Fg is linear and Fg (x) is monotone increasing. Hence ∞

x,y

t

Fgn (Kλ )(x, y) = W 1/2 (x)W 1/2 (y) ∫ e−λt 𝔼[0,t] [gn (∫ W(Bs )ds)] Πt (x − y)dt 0 0 [ ]

(4.3.52)

4.3 Properties of Schrödinger operators and semigroups | 299

for gn ∈ C by (4.3.44). The integral operator defined by the integral kernel given by the left-hand side (resp. the right-hand side) of (4.3.52) is denoted by An (resp. Bn ). We have for Φ, Ψ ∈ L2 (ℝd ), ∞

(Φ, An Ψ)L2 (ℝd ) = ∫ e−y (Φ, Kλ gn (yKλ )Ψ)L2 (ℝd ) dy 0 ∞

→ ∫ e−y (Φ, Kλ g(yKλ )Ψ)L2 (ℝd ) dy = (Φ, Fg (Kλ )Ψ)L2 (ℝd ) 0

as n → ∞. On the other hand, from ‖gn ‖∞ ≤ ‖gn − g‖∞ + ‖g‖∞ it follows that for all n ≥ 1, ‖gn ‖∞ < C with some C, and together with it we see that 1/2

(Φ, Bn Ψ) =

∫ Φ(x)Ψ(y)W(x) W(y)

∞

1/2

∫e 0

ℝd ×ℝd

→

−λt

1/2

∫ Φ(x)Ψ(y)W(x) W(y)

1/2

∞

∫e 0

ℝd ×ℝd

−λt

t x,y 𝔼[0,t] [gn (∫ W(Bs )ds)]Πt (x 0

t x,y 𝔼[0,t] [g(∫ W(Bs )ds)]Πt (x 0

− y)dxdy

− y)dxdy

as n → ∞ by the dominated convergence theorem. We see that Fg (Kλ ) is an integral operator with integral kernel Fg (Kλ )(x, y) = W

1/2

(x)W

1/2

∞

(y) ∫ e

−λt

0

t x,y [ 𝔼[0,t] g (∫ W(Bs )ds)] Πt (x 0 [ ]

− y)dt

(4.3.53)

for g ∈ C0 ((a, b)). It can be also checked that Fg (Kλ ) ∈ I1 for g ∈ C0 ((a, b)) by the fact Fg (x) ≤ Cm x m for any m ≥ 1. To see this, let h(x) = g(x)/x m . We have ‖h‖∞ < ∞, since supp g ⊂ (a, b) and a > 0. Note that ∞

∞

Fg (x) = x ∫ e (xy) h(xy)dy ≤ (‖h‖∞ ∫ e−y ym dy) x m+1 . −y

m

0

Thus for m ≥ 1, Tr Fg (Kλ ) ≤

0

Cm+1 Tr Kλm+1 Fg (Kλ ) ∈ I1

=

Cm+1 ‖Kλ ‖m+1 m+1

< ∞, since Kλ ∈ I2 . We have

for g ∈ C0 ((a, b)).

(4.3.54)

Since the right-hand side of (4.3.53) is jointly continuous in x and y which can be seen by the expression in terms of Brownian bridge, Fg (Kλ )(x, y) ∞

t

s s s = W 1/2 (x)W 1/2 (y) ∫ e−λt 𝔼0𝒲 [g (∫ W((1 − )x + y − Bt + Bs )ds)] Πt (x − y)dt, t t t 0 0 [ ] it follows that Tr(F(Kλ )) = ∫ℝd F(Kλ )(x, x)dx < ∞ and (4.3.51). Equality (4.3.50) can be proven in the same way as (4.3.51).

300 | 4 Feynman–Kac formulae Lemma 4.155. Suppose that λ > 0, W ∈ L∞ (ℝd ) with compact support, and g ∈ C0 ((a, b)). Let f , g, and F be related by (4.3.49). Then (4.3.50) and (4.3.51) hold. Proof. There exists a sequence Wn ∈ C0∞ (ℝd ) such that Wn ↑ W almost everywhere and Wn ↑ W in Ld/2 (ℝd ) as n → ∞. Since Kλ (W) ∈ I2 , in the same way as (4.3.54) we can show that F(Kλ (W)) ∈ I1 .

(4.3.55)

Since the operator norm ‖Kλ (Wn ) − Kλ (W)‖ is estimated as ‖Kλ (Wn ) − Kλ (W)‖ ≤ C‖W − Wn ‖d/2 by Lemma 4.144 with a constant C > 0, Kλ (Wn ) → Kλ (W) as n → ∞ in the operator norm, and then it also follows that F(Kλ (Wn )) → F(Kλ (W))

(4.3.56)

as n → ∞ in the operator norm. Since Wn ≤ W, we also see that μm (F(Kλ (Wn ))) ≤ μm (F(Kλ (W)))

(4.3.57)

for each m, where μm (A) denotes the mth eigenvalue of √A∗ A arranged in nonincreasing order with each eigenvalue occurring by their multiplicity; (4.3.55), (4.3.56), and (4.3.57) imply that Tr F(Kλ (Wn )) → Tr F(Kλ (W))

(4.3.58)

as n → ∞ by Proposition 4.45. On the other hand, there exist a compact set S ⊂ ℝd and A > 0 such that Wn (x) ≤ A1S (x) for x ∈ ℝd . Let g(x) = h(x)x m as in Lemma 4.154. Thus we have t

t

0

0

m

e−λt Wn (x)g (∫ Wn (Bs )ds) ≤ ‖h‖∞ e−λt Wn (x) (∫ Wn (Bs )ds) ≤ ‖h‖∞ e−λt 1S (x)Am+1 t m and e−λt 1S (x)Am+1 t m ∈ L1 ([0, ∞) × ℝd , dt ⊗ dx). Using this we can show that t

∞

[ ] ∫ e−λt Πt (0)dt ∫ Wn (x)𝔼x,x [0,t] g (∫ Wn (Bs )ds) dx 0 0 [ ] ℝd ∞

t

[ ] → ∫ e−λt Πt (0)dt ∫ W(x)𝔼x,x [0,t] g (∫ W(Bs )ds) dx 0 0 [ ] ℝd by the dominated convergence theorem, and hence (4.3.51) is proven. Equality (4.3.50) can be shown similarly.

4.3 Properties of Schrödinger operators and semigroups |

301

Lemma 4.156. Suppose that λ > 0, W ∈ Ld/2 (ℝd ), and g ∈ C0 ((a, b)). Let f , g, and F be related by (4.3.49). Then (4.3.50) and (4.3.51) hold. Proof. Let Wn ∈ L∞ (ℝd ) with compact support be such that Wn ↑ W almost everywhere and in Ld/2 (ℝd ). In a similar manner to (4.3.54) we can see that F(Kλ (W)) ∈ I1 . Since Tr F(Kλ (Wn )) is monotone increasing, Tr F(Kλ (Wn )) → Tr F(Kλ (W))

(4.3.59)

is shown in a similar way to (4.3.58). On the other hand, the identity ∞

Tr F(Kλ (Wn )) = ∫ e 0

−λt

t x,x [ Πt (0)dt ∫ Wn (x)𝔼[0,t] g (∫ Wn (Bs )ds)] dx 0 [ ] ℝd

(4.3.60)

implies that the right-hand side of (4.3.60) is monotone increasing with respect to n. Then the monotone convergence theorem yields ∞

lim ∫ e

n→∞

−λt

0 ∞

= ∫e

−λt

0

t x,x [ Πt (0)dt ∫ Wn (x)𝔼[0,t] g (∫ Wn (Bs )ds)] dx 0 [ ] ℝd

t x,x [ Πt (0)dt ∫ W(x)𝔼[0,t] g (∫ W(Bs )ds)] dx 0 [ ] ℝd

< ∞.

Together with (4.3.59), (4.3.51) follows and (4.3.50) can be similarly proven. Lemma 4.157. Suppose that λ > 0, W ∈ Ld/2 (ℝd ), and g is a lower semicontinuous positive function. Let f , g, and F be related by (4.3.49), and Fg (‖Kλ ‖) < ∞. Then (4.3.50) and (4.3.51) hold. Proof. Since any lower semicontinuous function is a monotone limit of continuous functions, there exist positive continuous functions gn , n ∈ ℕ, such that gn (x) ↑ g(x) for each x. Let χn ∈ C0∞ ((1/n, n)) be such that 0 ≤ χn (x) ≤ 1, χn (x) ↑ 1 for x ∈ (0, ∞) as n → ∞. Let ḡn (x) = χn (x)gn (x). We have ḡn (x) ↑ g(x) and supp ḡn ⊂ (1/n, n). Hence Fḡn (Kλ ) ∈ I1 and ∞

t

[ ̄ ] Tr Fḡn (Kλ ) = ∫ e−λt Πt (0)dt ∫ W(x)𝔼x,x [0,t] gn (∫ W(Bs )ds) dx. d 0 0 [ ] ℝ

(4.3.61)

Note that Tr Fḡn (Kλ ) ∈ I1 is monotone increasing with respect to n, and condition Fg (‖Kλ ‖) < ∞ implies that Fg (Kλ ) is compact and Fḡn (Kλ ) → Fg (Kλ ) as n → ∞ in the operator norm. We have μm (Fḡn (Kλ )) → μm (Fg (Kλ )) as n → ∞ for each m by Proposition 4.44. It is sufficient to consider two cases.

302 | 4 Feynman–Kac formulae (1) Case limn→∞ Tr Fḡn (Kλ ) < ∞: By the monotone convergence theorem Fg (Kλ ) ∈ I1 t

x,x and e−λt W(x)g (∫0 W(Bs )ds) ∈ L1 ([0, ∞) × ℝd × X , dt ⊗ dx ⊗ 𝒲[0,t] ), and then t

∞

[ ] Tr Fg (Kλ ) = ∫ e−λt Πt (0)dt ∫ W(x)𝔼x,x [0,t] g (∫ W(Bs )ds) dx 0 0 [ ] ℝd

(4.3.62)

follows and (4.3.51) is obtained. (2) Case limn→∞ Tr Fḡn (Kλ ) = ∞: Equality Tr Fg (Kλ ) = ∞ follows and the right-hand side of (4.3.62) is also infinity, showing (4.3.51). Equality (4.3.50) is similarly proven. Proof of Lemma 4.153. We see that Kδ = |Kδ | = |Kδ∗ | and (Φ, Kδ Ψ) = ((−Δ)−1/2 W 1/2 Φ, (−Δ)1/2 (−Δ + δ)−1 (−Δ)1/2 (−Δ)−1/2 W 1/2 Ψ). Both (−Δ)−1/2 W 1/2 Φ and (−Δ)−1/2 W 1/2 Ψ are well defined since d ≥ 3, and then we see that lim(Φ, Kδ Ψ) = ((−Δ)−1/2 W 1/2 Φ, (−Δ)−1/2 W 1/2 Ψ) = (Φ, W 1/2 (−Δ)−1 W 1/2 Ψ), δ↓0

which implies that Kδ → K0 weakly as δ ↓ 0. Furthermore, Kδ ∈ I2 and ‖Kδ ‖2 can be expressed as ‖Kδ ‖22 =

∞

∫ W(x)Cδ (x − y)2 W(y)dxdy = ∑ (W 1/2 ϕn , (−Δ + δ)−1 W 1/2 ϕn )2 n=1

ℝd ×ℝd

for any complete orthonormal system {ϕn }n∈ℕ in L2 (ℝd ). From this we see that the integral ∫ℝd ×ℝd W(x)Cδ (x − y)2 W(y)dxdy is monotone increasing as δ ↓ 0, and ∞

∫ W(x)Cδ (x − y)2 W(y)dxdy ≤ ∑ (W 1/2 ϕn , (−Δ)−1 W 1/2 ϕn )2 = ‖K0 ‖22 < ∞. n=1

ℝd ×ℝd

We see that lim ∫ W(x)Cδ (x − y)2 W(y)dxdy = δ↓0

ℝd ×ℝd

∫ W(x)C0 (x − y)2 W(y)dxdy ℝd ×ℝd

by the monotone convergence theorem. This identically implies that lim ‖Kδ ‖2 = ‖K0 ‖2 . δ↓0

(4.3.63)

Hence ‖Kδ − K0 ‖2 → 0 as δ ↓ 0 by Proposition 4.45. In particular, Kδ converges to K0 in the operator norm, and then F(Kδ ) → F(K0 ) in the operator norm. Hence we can conclude that μn (F(Kδ )) ↑ μn (F(K0 )) as δ ↓ 0 for each n.

4.3 Properties of Schrödinger operators and semigroups |

303

(1) Case Tr F(Kδ ) < ∞ and limδ↓0 Tr F(Kδ ) < ∞: By the monotone convergence theorem t

x,x we have F(K0 ) ∈ I1 and W(x)g (∫0 W(Bs )ds) ∈ L1 ([0, ∞) × ℝd × X , dt ⊗ dx ⊗ 𝒲[0,t] ), and then

Tr Fg (K0 ) =

t x,x [ ∫ Πt (0)dt ∫ W(x)𝔼[0,t] g (∫ W(Bs )ds)] dx. 0 0 [ ] ℝd ∞

(4.3.64)

(2) Case Tr F(Kδ ) < ∞ and limδ↓0 Tr F(Kδ ) = ∞, or Tr F(Kδ ) = ∞ for some δ > 0: Then Tr Fg (K0 ) = ∞ follows and the right-hand side of (4.3.64) is also infinity, which shows (4.3.51). Corollary 4.158. Let the conditions of Lemma 4.153 hold, and suppose that there exists ∞ ℓ such that g(x) ≤ ℓ(x)x m and ∫0 e−y ym ℓ(xy)dy < ∞ for x ∈ [0, ∞) with m = 2 for d ≥ 3, m > 1 for d = 2, and m = 1 for d = 1. Then Tr F(Kλ ) < ∞ and (4.3.50) and (4.3.51) are satisfied. Proof. In Lemmas 4.154–4.157, ∫0 e−y ym ℓ(xy)dy < ∞ for arbitrary x ∈ [0, ∞) implies that Fg (Kλ ) ∈ I1 since Fg (x) ≤ Cm x m . Then the corollary follows. ∞

We can now state the Lieb–Thirring inequality. Theorem 4.159 (Lieb–Thirring inequality). Let V = V+ − V− . Suppose that d ≥ 3 and V− ∈ Ld/2 (ℝd ). Then there exists a constant ad , independent of V, such that N̄ 0 (V) ≤ ad ∫ |V− (x)|d/2 dx. ℝd

Proof. Let F, f , and g be functions related by (4.3.49), and suppose that f is convex ∞ and ∫0 f (x)x −1 x −d/2 dx < ∞. Hence Tr F(Kλ ) < ∞ and (4.3.50) and (4.3.51) are satisfied. Note that F is monotone increasing, thus F(x)/F(1) ≥ 1, for x ≥ 1. By Lemma 4.153 we have N̄ 0 (V) ≤ F(1)−1 Tr F(K0 ) ∞

= F(1)−1 ∫ Πt (0) 0 ∞

t

≤ F(1)−1 ∫ Πt (0) 0

t

dt ds ] [ ) dx ∫ 𝔼x,x [0,t] f (∫ tW(Bs ) t t [ 0 ] ℝd dt 1 ∫ dx ∫ 𝔼x,x [f (tW(Bs ))] ds, t t [0,t] ℝd

0

where we applied Jensen’s inequality to the convex function f . By using the definition of Brownian bridge 𝔼x,x [0,t] [h(Bs )] =

1 ∫ h(y)Πs (x − y)Πt−s (y − x)dy Πt (0) ℝd

304 | 4 Feynman–Kac formulae the right-hand side above can be computed as F(1)

t

∞

−1

dt ds ∫ ∫ dx ∫ ∫ f (tW(y))Πs (x − y)Πt−s (y − x)dy t t 0

ℝd

= F(1)

0

ℝd t

∞

dt ds ∫ Πt (0) ∫ ∫ f (tW(y))dy t t

−1

0

0

ℝd

∞

=

1 ∫ t −1 t −d/2 dt ∫ f (tW(y))dy. (2π)d/2 F(1) 0

(4.3.65)

ℝd

Changing the variable t to s/W(y), we furthermore have =

∞

1

(2π)d/2 F(1)

∫ f (s)s−1 s−d/2 ds ∫ W(y)d/2 dy = ad ∫ W(y)d/2 dy, 0

ℝd

ℝd

where ad =

1

∞

(2π)d/2 F(1)

∫ f (s)s s

−1 −d/2

ds =

(2π)−d/2 ∫0 f (s)s−1 s−d/2 ds ∞

∫0 f (s)s−1 e−s ds ∞

0

.

(4.3.66)

For d = 1, 2 we have the following corollary. Corollary 4.160. Let V ∈ L1 (ℝ) for d = 1, and V ∈ Lp (ℝ2 ) with p > 1 for d = 2. Suppose that V is negative and not identically zero. Then H0 + αV has a negative eigenvalue for every α > 0. Proof. Let f = |V|1/2 ϕ be such that ϕ ∈ L2 (ℝd ) and f ∈ L2 (ℝd ) ∩ L1 (ℝd ). We have (f , (H0 + λ)−1 f ) = ∫ ℝd

|f ̂(p)|2 dp → ∞ |p|2 /2 + λ

as λ ↓ 0 for d = 1, 2, since f ̂ is continuous and f ̂(p) ≠ 0 for p near zero. Thus (|V|1/2 ϕ, (H0 + λ)−1 |V|1/2 ϕ)/‖ϕ‖2 ≤ ‖Kλ ‖ → ∞ as λ ↓ 0, which means together with the fact −λ ∈ Spec(H(α)) ⇐⇒ 1/α ∈ Spec(Kλ ) that for every α > 0 there exists λ > 0 such that 1/α = ‖Kλ ‖, since λ 󳨃→ ‖Kλ ‖ is continuous. Then 1/α ∈ Spec(Kλ ) which implies that −λ ∈ Spec(H(α)). Hence for every α > 0 there exists −λ < 0 such that −λ ∈ Spec(H(α)). Remark 4.161. For dimensions d = 1 or 2, the Schrödinger operator H = H0 + V with V ≤ 0 (not identically zero) has at least one negative eigenvalue by Corollary 4.160. This may be compared with the case of d ≥ 3 where by the Lieb–Thirring inequality there are no nonpositive eigenvalues if ‖V‖d/2 is sufficiently small.

4.3 Properties of Schrödinger operators and semigroups |

305

The above ideas also allow to study other sums of negative eigenvalues of H, such as the Riesz means m

N β (V) = ∑ |Ej |β ,

m ≤ ∞,

j=1

for β > 0. This is a byproduct of the estimate of NE (V) which can be used to define the counting measure dNE (V) on [0, ∞). The map [0, ∞) ∋ E 󳨃→ NE (V) ∈ ℕ is decreasing and we have ∞

β

N (V) = − ∫ E β dNE (V). 0

Integration by parts gives ∞

N β (V) = − ∫ E β 0

∞

dNE (V) dE = β ∫ E β−1 NE (V)dE, dE

β > 0.

(4.3.67)

0

First we make the following observation. Let E > 0. Since (|k|2 + E)−1/2 ∈ Lp (ℝd ) for all p > d, Proposition 4.39 yields (−Δ + E)−1/2 V−1/2 ∈ I2β+d for V− ∈ Lβ+(d/2) (ℝd ). Thus we have V−1/2 (−Δ + E)−1 V−1/2 ∈ Iβ+(d/2) . In particular, NE (V) ≤ NE (V− ) < ∞ for every E > 0, and by the Birman–Schwinger principle NE (V− ) ≤ dim 1[1,∞) (V−1/2 (−Δ + E)−1 V−1/2 ). Theorem 4.162 (Lieb–Thirring inequality). Suppose d ≥ 3, β > 0, and let V = V+ − V− , with V− ∈ Lβ+(d/2) (ℝd ). Then ∞

N β (V) ≤ ad,β ∫ |V− (x)|β+(d/2) dx, 0

where ad,β =

1 βad ∫0 z β−1 (1

− z)

d/2

dz, with ad in (4.3.66).

Proof. We denote the negative part of V by −W. Step 1: First we suppose that supp W is compact and let E > 0. We write the eigenvalue equation Hf = −Ef as (H + E)f = 0. Since V + E ≥ −W + E ≥ −(−W + E)− , we have NE (V) ≤ N̄ 0 (−(−W + E)− ). d d We also see that W 1/2 ∈ Ld (ℝd ) ∩ Ld+2β (ℝd ). In particular, (−W + E)1/2 − ∈ L (ℝ ) and 1/2 −1 1/2 it follows that (−W + E)− (−Δ) (−W + E)− is compact. Consequently, in a similar manner to (4.3.65) in the proof of Theorem 4.159 we see that ∞

NE (V) ≤ N̄ 0 (−(−W + E)− ) = F(1)−1 ∫ 0

Πt (0) dt ∫ f (t(−(−W(y) + E)− ))dy. t ℝd

(4.3.68)

306 | 4 Feynman–Kac formulae Here f is a nonnegative lower semicontinuous function on [0, ∞) with f (0) = 0, and ∞ f , F are related by F(x) = x ∫0 e−y g(xy)dy and f (y) = g(y)y. Combining (4.3.67) and (4.3.68), we obtain ∞

∞

0

0

N β (V) = β ∫ E β−1 NE (V)dE ≤ β ∫ E β−1 N̄ 0 (−(−W(y) + E)− )dE. Inserting (4.3.68) in the right-hand side above, we have ∞

∞

0

0

N β (V) ≤ βF(1)−1 ∫ E β−1 dE ∫

Πt (0) dt ∫ f (t(−(−W(y) + E)− ))dy. t ℝd

Since the integration domain with respect to E is actually [0, W(y)], we have β

N (V) ≤ βF(1)

W(y)

∞

−1

Π (0) ∫ t dt ∫ dy ∫ E β−1 f (t(W(y) − E))dE t 0

= βF(1)

0

ℝd

1

∞

−1

Π (0) ∫ t dt ∫ dy ∫ W(y)β z β−1 f (tW(y)(1 − z))dz. t 0

ℝd

0

Changing the variable t to s = t/(1 − z) and using Πt (0) = (2πt)−d/2 , we see that ∞

N β (V) ≤ βF(1)−1 ∫ 0

1

Πs (0) ds ∫ W(y)β f (sW(y))dy ∫ z β−1 (1 − z)d/2 dz. s 0

ℝd

Thus as in the proof of Theorem 4.159, we obtain N β (V) ≤ ad,β ∫ W(y)β+(d/2) dy. ℝd

This then shows the statement for W with compact support. Step 2: Next assume that W 1/2 ∈ Ld+2β (ℝd ), and let Wn = W1[−n,n] . We show that NE (−Wn ) → NE (−W)

(4.3.69)

as n → ∞ for every E > 0. Let KE (n) = Wn1/2 (−Δ + E)−1 Wn1/2 and KE = W 1/2 (−Δ + E)−1 W 1/2 for E > 0, and {μj (KE (n))}nj=1 (resp. {μj (KE )}nj=1 ) denote the eigenvalues of KE (n) (resp. KE ). From Proposition 4.39 it directly follows that ‖(−Δ + E)−1/2 Wn1/2 − (−Δ + E)−1/2 W 1/2 ‖d+2β

≤ (2π)−d/(d+2β) ‖W 1/2 − Wn1/2 ‖Ld+2β (ℝd ) ‖(|k|2 + E)−1/2 ‖Ld+2β (ℝd ) → 0

4.3 Properties of Schrödinger operators and semigroups |

307

as n → ∞. This implies that ‖(−Δ + E)−1/2 Wn1/2 − (−Δ + E)−1/2 W 1/2 ‖ → 0 as n → ∞, and hence KE (n) → KE uniformly as n → ∞. Thus by Corollary 4.44 we see that μj (KE (n)) → μj (KE ) as n → ∞ for j ≥ 1 and E > 0. Recall the identity NE (−Wn ) = dim 1[1,∞) (KE (n)). We conclude that dim 1[1,∞) (KE (n)) → dim 1[1,∞) (KE ) < ∞ as n → ∞, and (4.3.69) follows. Finally, together with (4.3.69) and taking the limit n → ∞ on both sides of ∞

N β (−Wn ) = β ∫ E β−1 NE (−Wn )dE ≤ ad,β ∫ |Wn (x)|β+(d/2) dx, 0

ℝd

we have N β (−W) ≤ ad,β ∫ |W(x)|β+(d/2) dx. ℝd

4.3.7 Application to canonical commutation relations The Lieb–Thirring inequality discussed in Theorem 4.162 can be applied to construct pairs of operators satisfying canonical commutation relations (CCRs). CCRs are a fundamental tool in quantum physics. In one-dimensional quantum mechanics the momentum operator P = −id/dx and the position operator Q = the multiplication by x satisfy CCR: [P, Q] = −i1 on D(PQ)∩D(QP). There is a physical folklore such that the pair of time–energy is a counterpart of that of position–momentum. So we want to define a time operator as a symmetric operator satisfying CCR with respect to the Hamiltonian of a quantum system. More precisely, we are interested in finding an operator T associated with a given self-adjoint operator S such that [S, T] = −i1

(4.3.70)

on D(ST) ∩ D(TS), and we call T a time operator associated with S. When the pair (S, T) satisfies CCR, it is known that either S or T is unbounded. Hence it may occur that D(ST) ∩ D(TS) is not dense or empty. The so-called weak CCR is defined by replacing CCR (4.3.70) with a bilinear form: (Sϕ, Tψ) − (Tϕ, Sψ) = −i(ϕ, ψ),

ϕ, ψ ∈ D(S) ∩ D(T).

(4.3.71)

A weak time operator T associated with S is a symmetric operator satisfying (4.3.71). A weak time operator can be furthermore extended to the so-called ultraweak time operator. A densely defined symmetric quadratic form t(⋅, ⋅) : L2 (ℝd ) × L2 (ℝd ) → ℂ is an ultraweak time operator associated with a self-adjoint operator S if and only if t(Sϕ, ψ) − t(ϕ, Sψ) = −i(ϕ, ψ)

(4.3.72)

308 | 4 Feynman–Kac formulae for all ψ, ϕ ∈ D with some domain D , where D is called an ultraweak CCR domain for (S, t). Note that D is not necessarily dense. If T is a weak time operator, then T ̂ ψ) = (ϕ, Tψ) with defines the ultraweak time operator t ̂ : L2 (ℝd ) × L2 (ℝd ) → ℂ by t(ϕ, ultraweak CCR domain D(S) ∩ D(T). CCR is closely related to the pair of unitary groups. We say that the pair of selfadjoint operators (A, B) satisfies the Weyl relation if and only if e−isA e−itB = eist e−itB e−isA holds for all s, t ∈ ℝ. A Weyl relation implies CCR, and the pair (P, Q) satisfies the Weyl relation; in general, however, CCR does not imply the Weyl relation. If (A, B) satisfies the Weyl relation, B is called the ultrastrong time operator associated with A. It is known as the von Neumann uniqueness theorem that if pair (A, B) satisfies the Weyl relation and there is no invariant domain with respect to e−isA and e−itB , then A ≅ P and B ≅ Q. Here ≅ describes a unitary equivalence. When S is bounded from below, this theorem tells us that there exists no symmetric operator T such that (S, T) satisfies the Weyl relation, since S ≇ P. Thus instead of the Weyl relation the so-called weak Weyl relation is introduced. The pair (A, B) satisfies the weak Weyl relation if and only if (1) A is self-adjoint; (2) B is symmetric; (3) e−itA D(B) ⊂ D(B) and Be−itA ψ = e−itA (B + t)ψ,

∀ψ ∈ D(B), t ∈ ℝ.

It is clear that Weyl relations imply weak Weyl relations, and weak Weyl relations do CCR. A symmetric operator T is a strong time operator associated with a self-adjoint operator S if and only if the pair (S, T) satisfies the weak Weyl relation. Lemma 4.163. Suppose that a strong time operator T associated with a self-adjoint operator S exists. Then the following assertions follow. (1) The closure T̄ is also a strong time operator. (2) If S is bounded below, then T has no self-adjoint extension. (3) Spec(S) = Specac (S). By this lemma we may assume that the strong time operator is a closed symmetric operator in what follows, and there is no strong time operator if S has point spectrum. Example 4.164 (Aharonov–Bohm operator). Let Pj = −id/dxj and Qj be the multiplication by xj for j = 1, . . . , d in L2 (ℝd ). A strong time operator associated with Pj is Qj . An important example includes the Aharonov–Bohm operator TAB , which is a strong time operator associated with 21 ∑dj=1 Pj2 and defined by TAB =

1 d ∑ (Q P −1 + Pj−1 Qj )⌈Dj , 2 j=1 j j

(4.3.73)

4.3 Properties of Schrödinger operators and semigroups |

309

with Dj = {ρ(Pj2 )D(Qj ) | ρ ∈ C0∞ (ℝd \ {0})}. We note that TAB has no self-adjoint extensions. In summary, we saw the hierarchy for relations such that Weyl relation → weak Weyl relation → CCR → weak CCR → CCR in the sense of quadratic form (4.3.72); furthermore, we defined several classes of time operators: ultrastrong time operator ⊂ strong time operator ⊂ time operator ⊂ weak time operator ⊂ ultraweak time operator.

Next we review a time operator associated with a self-adjoint operator S such that Spec(S) = {Ej }∞ j=1 with E1 < E2 < . . . and limn→∞ En = ∞. In this case there exists no strong time operator by (3) of Lemma 4.163. Let Senα = En enα , α = 1, . . . , Mn , and Mn 1 (enα , emβ ) = δnm δαβ , where Mn denotes the multiplicity of En . Let ēn = √M enα . ∑α=1 Note that (ēn , ēm ) = δnm . Suppose that ∑∞ j=J we can see that

1 Ej2

n

< ∞ for some J ≥ 1. Then for a fixed n

2

Em 1 1 1 = ∑ ≤M ∑ 2 0, and a positive differentiable function h(r) defined on [R, ∞) it follows that (a) V1 and V2 are locally bounded on SR = {x ∈ ℝd | |x| > R}, and V1 is strictly negative on SR ; (b) d s+1 sup (r V1 (rw)) ≤ −r s h(r) dr w∈Sd−1 (c)

for r > R, where w is a coordinate on the (d − 1)-dimensional unit sphere Sd−1 ; lim

r→∞

r −1 + r supw∈Sd−1 |V2 (rw)| h(r)

= 0;

(d) dh(r)/dr ≤ Ch2 (r) on SR ; (e) ∫S h(|x|)2 |f (x)|2 dx < ∞ and ∫S |V1 (x)||f (x)|2 dx < ∞ for any f ∈ D(H). R

R

Then 0 is no eigenvalue of H. We give the following examples. (1) Suppose that V1 (x) = v(w)r −μ + R(x) and V2 (x) = o(r −1−μ/2 ), where μ ∈ [0, 2), supw∈Sd−1 v(w) < 0, R(x) = o(r −μ ), and supw∈Sd−1 R(rw) = o(r −μ−1 ). Then V = V1 + V2 satisfies assumptions stated above and 0 ∈ ̸ Specp (H) follows. (2) Let a > 0 be arbitrary. Putting μ = 0 and v(w) = −a < 0 in (1), we can see that 0 is not an eigenvalue of − 21 Δ + V2 − a, i. e., − 21 Δ + V2 has no positive eigenvalues. Suppose (3) and (4) of Assumption 4.166. Then ∞

Tp ϕ = i ∑ ( ∑ n=1

(ēm , ϕ)

1 m=n ̸ E n

−

1 Em

) ēn

is a time operator associated with Hp−1 by Lemma 4.165. Since Hp = f (Hp−1 ), where f (x) = x −1 , formally the time operator T associated with Hp is given by T=

1 1 1 1 (Tp 󸀠 −1 + 󸀠 −1 Tp ) = − (Tp Hp−2 + Hp−2 Tp ). 2 2 f (Hp ) f (Hp )

To verify this formula rigorously, instead of considering operators we define the symmetric quadratic form tp : D(Tp ) × D(Tp ) → ℂ on ℋp by − 1 ((Tp ϕ, Hp−2 ψ) + (Hp−2 ϕ, Tp ψ)) , tp (ϕ, ψ) = { 2 0, otherwise.

312 | 4 Feynman–Kac formulae We formally write tp (ϕ, ψ) = (ϕ, Tψ) and T = − 21 (Tp Hp−2 + Hp−2 Tp ). Note that it is however not clear whether D(Hp−2 ) ⊃ Tp D(Tp ) or not. Hence we may not define tp as a symmetric operator. Now we define the densely defined symmetric quadratic form tH (⋅, ⋅) : L2 (ℝd ) × L2 (ℝd ) → ℂ under the decomposition L2 (ℝd ) = ℋac ⊕ ℋp by tH (ϕ1 ⊕ ϕ2 , ψ1 ⊕ ψ2 ) = (ϕ1 , Tac ψ1 ) + tp (ϕ2 , ψ2 )

(4.3.75)

for ϕ1 , ψ1 ∈ D(Tac ) and ϕ2 , ψ2 ∈ D(Tp ). Theorem 4.169 (Ultraweak time operator). Suppose Assumption 4.166. Then tH is an ultraweak time operator associated with H with ultraweak CCR domain D(Tac ) ⊕ H −1 E , where H −1 E = linear hull of { E1 ēn − E1 ēm | n, m ∈ ℕ}. n

m

Note that H E is not necessarily dense. −1

Proof. It is sufficient to show that tp is an ultraweak time operator associated with Hp with ultraweak CCR domain Hp−1 E . We shall prove this. Let t = −2tp . Let ϕ󸀠 = Hp−1 ϕ, ψ󸀠 = Hp−1 ψ ∈ Hp−1 E . We see that t(Hp ϕ󸀠 , ψ󸀠 ) − t(ϕ󸀠 , Hp ψ󸀠 ) = t(ϕ, Hp−1 ψ) − t(Hp−1 ϕ, ψ). By the definition of t we have t(Hp ϕ󸀠 , ψ󸀠 ) − t(ϕ󸀠 , Hp ψ󸀠 ) = (Tp ϕ, Hp−3 ψ) + (Hp−2 ϕ, Tp Hp−1 ψ) − (Hp−3 ϕ, Tp ψ) − (Tp Hp−1 ϕ, Hp−2 ψ) = (Hp−1 Tp ϕ, Hp−2 ψ) − (Hp−2 ϕ, Hp−1 Tp ψ) + (Hp−2 ϕ, Tp Hp−1 ψ) − (Tp Hp−1 ϕ, Hp−2 ψ). The first two terms of the most right-hand side above can be computed by using [Hp−1 , Tp ] = −i1 on E as (Hp−1 Tp ϕ, Hp−2 ψ) − (Hp−2 ϕ, Hp−1 Tp ψ) = 2i(Hp−1 ϕ, Hp−1 ψ) + (Tp Hp−1 ϕ, Hp−2 ψ) − (Hp−2 ϕ, Tp Hp−1 ψ). Hence we conclude that t(Hp ϕ󸀠 , ψ󸀠 ) − t(ϕ󸀠 , Hp ψ󸀠 ) = 2i(ϕ󸀠 , ψ󸀠 ). Then the theorem follows. Now we give several examples of V such that H has an ultraweak time operator. It is sufficient to check Assumption 4.166. Suppose that V is of the form V(x) =

W(x) (|x|2 + 1)1/2+ε

(4.3.76)

for some ε > 0, where W : ℝd → ℝ and W(−Δ + i)−1 is compact. If V is of the form (4.3.76), then V is called the Agmon potential. Agmon potentials form a linear space of −Δ-bounded perturbations of relative bound zero. In particular, H is self-adjoint on D(H0 ). The perturbation by Agmon potential V leaves the essential spectrum of H0 invariant, i. e., Specess (H) = Specess (H0 ) = [0, ∞). Following facts are known.

4.3 Properties of Schrödinger operators and semigroups | 313

Lemma 4.170. Let V be an Agmon potential. Then we have the following: (1) Specsc (H) = 0; (2) the wave operator Ω(H, H0 ) = s − limt→∞ e−itH eitH0 exists and is complete; in particular, Specac (H) = [0, ∞); (3) the set of positive eigenvalues of H is a discrete subset in (0, ∞). Example 4.171 (Agmon potential). (1) It is known that any G ∈ Lp (ℝd ) for d/2 < p < ∞ and p ≥ 2 is relatively compact G(x) with respect to −Δ. Then V(x) = (1+|x| 2 )1/2+ε , ε > 0, is an Agmon potential. (2) V(x) =

U(x) , (1+|x|2 )1/2+ε

ε > 0, with U ∈ L∞ (ℝd ) with d ≥ 3 is an Agmon potential.

Theorem 4.172. Let d = 3 and V be a nonpositive continuous spherically symmetric Agmon potential, i. e., V(x) = W(r), where r = |x|. Suppose that (1) V(x) ≤ − |x|a2−δ for |x| > R with some R > 0, a > 0 and δ > 0; (2) ∫b |W(r)|dr < ∞ for some b > 0 and V ∈ L2loc (ℝd \ {0}); ∞

(3) ∫ℝ3 |V(x)|7/2 dx < ∞.

Then an ultraweak time operator associated with H exists. Proof. By Proposition 4.170, Specsc (H) = 0 and then H = Hac ⊕ Hp . Under (1) and (2) ∞ we can see that Spec(H) = {Ej }∞ j=1 ∪ [0, ∞), Specp (H) = {0} ∪ {Ej }j=1 , 0 ∈ ̸ Specp (H), and Specac (H) = [0, ∞). Since the wave operator Ω(H, H0 ) exists, Tac = ΩTAB Ω∗ is a strong time operator associated with Hac by Proposition 4.167. Furthermore, (3) im2 plies ∑∞ j=1 Ej < ∞ by the Lieb–Thirring inequality. Then the ultraweak time operator tp associated with Hp exists with the ultraweak CCR domain Hp−1 E . We set t(ϕ, ψ) = (ϕ1 , Tac ψ1 ) + tp (ϕ2 , ψ2 ) for ϕ = ϕ1 ⊕ ϕ2 and ψ = ψ1 ⊕ ψ2 under the identification L2 (ℝ3 ) = ℋac ⊕ ℋp . Hence symmetric quadratic form t is the ultraweak time operator associated with H with ultraweak CCR domain D(Tac ) ∩ H −1 E . U(x) Example 4.173. Let d = 3. Suppose that U ∈ L∞ (ℝ3 ). Then V(x) = (1+|x| 2 )1/2+ε is an Agmon potential for all ε > 0. Suppose that U is negative, continuous, and spherically symmetric and satisfies U(x) = 1/|x|α for |x| > R with 0 < α < 1 and some R > 0. For each α, we can choose ε > 0 such that 2ε + α < 1. Hence V satisfies (1), (2), and (3) of Theorem 4.172. Hence an ultraweak time operator tH associated with Hp exists, and an ultraweak time operator associated with H exists.

Example 4.174. Let d = 3. The hydrogen atom Schrödinger operator is defined by 1 Hhyd = H0 − |x| . It is known that Specsc (Hhyd ) = 0, Specp (Hhyd ) = {− 21 j−2 }∞ j=1 , and

−4 Specac (Hhyd ) = [0, ∞). Thus ∑∞ < ∞ and 0 ∈ ̸ Specp (Hhyd ). Then an ultraweak j=1 j time operator associated with Hhyd,p exists. Moreover, the modified wave operator

314 | 4 Feynman–Kac formulae ΩD (Hhyd , H0 ) is defined by ΩD (Hhyd , H0 ) = s − limt→∞ eitHhyd UD (t) with some unitary operator UD (t). Ω = ΩD (H, H0 ) plays the roll of Ω in Proposition 4.167. Hence there exists a strong time operator associated with Hhyd,ac . Together with them we can conclude that an ultraweak time operator associated with Hhyd exists. 4.3.8 Exponential decay of eigenfunctions A sufficient condition for the existence of a ground state of a Schrödinger operator is that lim inf|x|→∞ V(x) = ∞. However, ground states exist also for many other choices of V, such as Coulomb potential in three dimensions. In many cases these ground states decay exponentially at infinity, while for potentials growing at infinity they may decay even faster. An intuitive derivation of the order of decay of the eigenfunctions is obtained as follows. Suppose the potential is of the form V(x) ∼ |x|2m and the ground state of the n form ψ(x) ∼ e−|x| . By the eigenvalue equation − 21 Δψ + |x|2m ψ = Eψ we have 1 − (n(n − 1)|x|n−2 + n2 |x|2n−2 )ψ + |x|2m ψ ∼ Eψ. 2 Comparing the leading order terms for large |x| at the two sides we expect that n = m+1. This can in fact be proven by using the Feynman–Kac formula. Suppose ψ ∈ L2 (ℝd ) with Hψ = λψ for some λ ∈ ℝ. By adding a constant to V we may assume λ = 0. Thus t

ψ(x) = e−tH ψ(x) = 𝔼x [e− ∫0 V(Bs ) ds ψ(Bt )]. Therefore the value of ψ(x) is obtained by running a Brownian motion from x under the potential V. This suggests that starting in regions where V is large, the effect will be quite small as most of the Brownian paths take a long time to leave a finite region. Formally, t

t

|ψ(x)|2 ≤ ‖ψ‖2∞ 𝔼x [e−2 ∫0 V+ (Bs ) ds ]𝔼x [e−2 ∫0 V− (Bs ) ds ] by the Schwarz inequality. By Khasminskii’s lemma the expectation involving V− grows at most exponentially in t with a constant that can be estimated conveniently. Now we present Carmona’s argument in detail. This will work for any eigenfunction of H and not just the ground state; therefore we assume φ ∈ L2 (ℝd ) with Hφ = Eφ. For the present purposes we introduce two further classes of potentials V. Definition 4.175. The potential classes 𝕍upper and 𝕍lower are defined by the following properties.

4.3 Properties of Schrödinger operators and semigroups | 315

𝕍upper : (1) (2) lower 𝕍 : (1) (2)

V ∈ 𝕍upper if and only if V = W − U such that U ≥ 0 and U ∈ Lp (ℝd ) for p > d/2 and 1 ≤ p < ∞; W ∈ L1loc (ℝd ) and W∞ = infx∈ℝd W(x) > −∞. V ∈ 𝕍lower if and only if V = W − U such that U ≥ 0 and U ∈ Lp (ℝd ) for p > d/2 and 1 ≤ p < ∞; W ≥ 0 and W ∈ 𝒦loc (ℝd ).

Let V = W − U ∈ 𝕍upper . We have W ∈ L1loc (ℝd ) and 0 ≥ −U ∈ Lp (ℝd ) implies −U ∈ 𝒦(ℝd ). Hence e−tH : L2 (ℝd ) → L∞ (ℝd ) by Theorem 4.107 and Remark 4.115; in particular, φ ∈ L∞ (ℝd ). A fundamental estimate of which we will make use in showing the spatial decay of eigenfunctions is given in the following lemma. Lemma 4.176 (Carmona’s estimate). Let V ∈ 𝕍upper . Then for every t, a > 0 and every 0 < α < 1/2, there exist constants D1 , D2 , D3 > 0 such that α a2 t

m

|φ(x)| ≤ t −d/2 D1 eD2 ‖U‖p t eEt (D3 e− 4 where m = (1 −

d −1 ) 2p

e−tW∞ + e−tWa (x) )‖φ‖,

(4.3.77)

and Wa (x) = inf{W(y) | |x − y| < a}.

Proof. By the Schwarz inequality t

t

|φ(x)| ≤ etE (𝔼x [e−4 ∫0 W(Bs )ds ])1/4 (𝔼x [e+4 ∫0 U(Bs )ds ])1/4 𝔼[|φ(x + Bt )|2 ]1/2 .

(4.3.78)

Note that 𝔼[|φ(x + Bt )|2 ] = ∫ Πt (y)|φ(x + y)|2 dy ≤ (2πt)−d/2 ‖φ‖2 . ℝd

Let A = {ω ∈ X | sup0≤s≤t |Bs (ω)| > a}. It follows from Lévy’s maximal inequality that ∞

2 2 2 𝔼[1A ] ≤ 2𝔼[1{|Bt |≥a} ] = |Sd−1 | ∫ e−r /2 r d−1 dx ≤ ξα e−αa /t (2π)d/2

a/√t

with some ξα , for 0 < α < 1/2, where |Sd−1 | is the surface area of the (d − 1)dimensional unit sphere. The first factor in (4.3.78) is estimated as t

t

t

𝔼x [e−4 ∫0 W(Bs )ds ] = 𝔼0 [1A e−4 ∫0 W(Bs +x)ds ] + 𝔼x [1Ac e−4 ∫0 W(Bs )ds ] 2

≤ e−4tW∞ 𝔼0 [1A ] + e−4tWa (x) ≤ ξα e−αa /t e−4tW∞ + e−4tWa (x) .

(4.3.79)

Next we estimate the second factor. Since U is in Kato-class, there exist constants D1 , D2 > 0 such that t

m

𝔼x [e−4 ∫0 U(Bs )ds ] ≤ D1 eD2 ‖U‖p t

(4.3.80)

316 | 4 Feynman–Kac formulae by Lemma 4.105. Setting D3 = ξα1/4 , we obtain the lemma by using the inequality (a + b)1/4 ≤ a1/4 + b1/4 for a, b ≥ 0. Corollary 4.177 (Confining potential). Let V = W − U ∈ 𝕍upper . Suppose that W(x) ≥ γ|x|2n outside a compact set K, for some n > 0 and γ > 0. Take 0 < α < 1/2. Then there exists a constant C1 > 0 such that αc

n+1

|φ(x)| ≤ C1 ‖φ‖e− 16 |x| ,

(4.3.81)

where c = infx∈ℝd \K W|x|/2 (x)/|x|2n . Proof. Since supx∈ℝd |φ(x)| < ∞, it suffices to show all the statements for sufficiently large |x|. Note that W|x|/2 (x) ≥ c|x|2n for x ∈ ℝd \ K. We have the bounds |x|W|x|/2 (x)1/2 ≥ c|x|n+1

|x|W|x|/2 (x)−1/2 ≤ c|x|1−n

and

for x ∈ ℝd \ K. Inserting t = t(x) = |x|W|x|/2 (x)−1/2 and a = a(x) = α

n+1

m

1−n

|φ(x)| ≤ ‖φ‖(c|x|n+1 )−d/2 e− 16 c|x| D1 e(D2 ‖U‖p +Ep )c|x| (D3 ec|x| for x ∈ ℝd \ K, where m = (1 −

d −1 ) . 2p

1−n

|x| 2

in (4.3.77), we have

|W∞ |

α

n+1

+ e−(1− 16 )c|x| )

Hence (4.3.81) follows.

For V = W − U ∈ 𝕍upper define Σ = lim inf|x|→∞ V(x). Since −U ∈ Lp (ℝd ), lim inf|x|→∞ (−U(x)) = 0 and hence Σ = lim inf|x|→∞ W(x). Moreover, Σ ≥ W∞ holds. Corollary 4.178 (Decay property: upper bound). Let V = W − U ∈ 𝕍upper . (1) Decaying potential: Suppose that Σ > E, Σ > W∞ , and let 0 < β < 1. Then there exists a constant C2 > 0 such that |φ(x)| ≤ C2 ‖φ‖ e

β

− 8√ 2

Σ−E √Σ−W∞

|x|

.

(4.3.82)

(2) Confining potential: Suppose that lim|x|→∞ W(x) = ∞. Then there exist constants C, δ > 0 such that |φ(x)| ≤ C‖φ‖ e−δ|x| .

(4.3.83)

Proof. Since supx∈ℝd |φ(x)| < ∞, it is again sufficient to show the statements for large enough |x|. Decaying case: Rewrite formula (4.3.77) as m

α a2 t

|φ(x)| ≤ t −d/2 D1 eD2 ‖U‖p t (D3 e− 4

e−t(W∞ −E) + e−t(Wa (x)−E) )‖φ‖,

(4.3.84)

d −1 where m = (1− 2p ) . Upon flipping signs, with Σ = lim inf|x|→∞ (−W− (x)) and Σ > W∞ , it is possible to choose a decomposition V = W −U ∈ 𝕍upper such that D2 ‖U‖m p ≤ (Σ−Ep )/2

4.3 Properties of Schrödinger operators and semigroups | 317

since lim inf|x|→∞ (−U(x)) = 0. Inserting t = t(x) = ε|x| and a = a(x) = we have

|x| 2

in (4.3.84),

m

α D1 eD2 ‖U‖p ε|x| (D3 e− 16ε |x| e−ε|x|(W∞ −E) + e−ε|x|(W|x|/2 (x)−E) )‖φ‖ d/2 (ε|x|) α 1 1 D1 ≤ (D3 e−( 16ε +ε(W∞ −E)− 2 ε(Σ−E))|x| + e−ε((W|x|/2 (x)−E)− 2 (Σ−E))|x| )‖φ‖. d/2 (ε|x|)

|φ(x)| ≤

Choosing ε = √α/16/√Σ − W∞ , the exponent in the first term above becomes α 1 1 + ε(W∞ − E − ε(Σ − E) = ε(Σ − E). 16ε 2 2 Moreover, we see that lim inf|x|→∞ W|x|/2 (x) = Σ, and we obtain ε

|φ(x)| ≤ C2 ‖φ‖ e− 2 (Σ−E)|x| for sufficiently large |x|. Thus (4.3.82) follows. Confining case: In this case for any c > 0 there exists large N > 0 such that W|x|/2 (x) ≥ c, for all |x| > N. Inserting t = t(x) = ε|x| and a = a(x) = |x| in (4.3.77), we obtain 2 m

α D1 eD2 ‖U‖p ε|x| (D3 e− 16ε |x| e−ε|x|(W∞ −E) + e−ε|x|(W|x|/2 (x)−E) )‖φ‖ d/2 (ε|x|) α m m D1 ≤ (D3 e−( 16ε −εD2 ‖U‖p +ε(W∞ −E))|x| + e−ε(c−E−D2 ‖U‖p )|x| )‖φ‖ (ε|x|)d/2

|φ(x)| ≤

for |x| > N. Choosing sufficiently large c and sufficiently small ε such that α − εD2 ‖U‖m p + ε(W∞ − E) > 0 16ε

and

c − E − D2 ‖U‖m p > 0,

we obtain |φ(x)| ≤ C 󸀠 e−δ |x| for large enough |x|. Thus (4.3.83) follows. 󸀠

By using the Feynman–Kac formula we can also establish a lower bound on positive eigenfunctions. Lemma 4.179. Let V = W − U ∈ 𝕍lower . Then the ground state φ is continuous and strictly positive. t

Proof. We have φ(x) = e−t(H−E(H)) φ(x) = 𝔼x [e− ∫0 (V(Bs )−E(H))ds φ(Bt )], for all t ≥ 0. Note that V is Kato-decomposable. Hence φ(x) is continuous in x by Theorem 4.111. Since e−tH is positivity improving, φ is strictly positive a.e. Let N = {x ∈ ℝd | φ(x) = 0}, which is a set of Lebesgue measure zero. Suppose that x ∈ N and then 0 = φ(x) = t

t

𝔼x [e− ∫0 (V(Bs )−E(H))ds φ(Bt )]. Since e− ∫0 (V(Bs )−E(H))ds > 0 a.e., we have 𝔼x [φ(Bt )] = 0. This implies 0 = ∫ℝd φ(y)Πt (x − y)dy and we see that φ(y) = 0 a.e. This contradicts φ ≠ 0. Hence N = 0 and the statement follows.

318 | 4 Feynman–Kac formulae Next we derive a preliminary estimate to the lower decay bound below. For a, t ≥ 0 and G ⊂ ℝ define P(a, G, t) = { sup |Bs | ≤ a, Bt ∈ G}. 0≤s≤t

It is a known result that 𝒲 (P(a, [c, d], t)) =

1 ∫ √2πt

∞

∑ (−1)k exp (−

[c,d] k=−∞

(u − 2ka)2 ) du. 2t

(4.3.85)

2 Note that ∑∞ k=−∞ exp (−(u − 2ka) /2t) < ∞ for every u, and ∞

∫

∑ exp (−

[c,d] k=−∞

(u − 2ka)2 ) du < ∞. 2t

Lemma 4.180. Suppose that a, α, t > 0 are such that a > 2α and a2 /t > β. Here β is the unique positive solution of F(ξ ) = 1 − (e−21β/8 + e−5β/8 + e−165β/8 ) = 0. Then for every interval [x − α, x + α] ⊂ [−a, a] it follows that 𝒲 (P(a, [x − α, x + α], t)) ≥

a2 α a2 F ( ) e− 2t . √2πt t

(4.3.86)

Proof. Let x+α

∞ 1 (u − 2ka)2 g(x) = 𝒲 (P(a, [x − α, x + α], t) = ) du. ∫ ∑ (−1)k exp (− √2πt 2t k=−∞ x−α

It can be seen that g(x) = g(−x). Choose x ∈ [0, a − α] and let x+α

f (x) = 𝒲 (P(a, [x, x + α], t)) =

∞ 1 (u − 2ka)2 ) du. ∫ ∑ (−1)k exp (− √2πt 2t k=−∞ x

We have g(x) ≥ f (x). Since df (x)/dx ≤ 0, f is monotonously decreasing and we get a

f (x) ≥ f (a − α) ≥

∞ 1 (u − 2ka)2 ) du. ∫ ∑ (−1)k exp (− √2πt 2t k=−∞ a−α

Estimating the right hand side we obtain a

∞

∫ ∑ (−1)k exp (−

a−α k=−∞

a2 (u − 2ka)2 ) du ≥ α (e− 2t + I(a, t) − J(a, t)) , 2t

4.3 Properties of Schrödinger operators and semigroups | 319

where I(a, t) =

∑

m∈ℤ\{0}

I(m) =

∑

m∈ℤ\{0}

exp (−

J(a, t) = ∑ J(m) = ∑ exp (− m∈ℤ

m∈ℤ

|(2 − 8m)a|2 ), 8t

|(5 − 8m)a|2 ). 8t

We see that I(−m) − J(m + 1) > 0 for m ≥ 1, and I(m) − J(−m − 1) > 0 for m ≥ 1. Hence we have a

∞

∫ ∑ (−1)k exp (−

a−α k=−∞

a2 (u − 2ka)2 ) du ≥ α (e− 2t − J(0) − J(1) − J(−1)) . 2t

a2

Write ξ = a2 /t; so e− 2t − J(0) − J(1) − J(−1) = e−ξ /2 F(ξ ), and the result follows. Recall that eigenfunctions corresponding to E(H) are strictly positive. Lemma 4.181. Let V = W −U ∈ 𝕍lower . Suppose that φ is a strictly positive eigenfunction of H at eigenvalue E. Also, let x ∈ ℝd \ {0}, and α1 , . . . , αd , a1 , . . . , ad , b1 , . . . , bd and t > 0 satisfy, for j = 1, . . . , d, a2j /t > β,

aj > 2αj ,

|[−aj , aj ] ∩ [−xj − bj , −xj + bj ]| > 2αj ,

(4.3.87)

where |A| denotes the Lebesgue measure of A. Then d

φ(x) ≥ ε(b)etE e−t Wa (x) ∏ ̃

j=1

αj

√2πt

F(

a2j t

a2j

) e− 2t .

(4.3.88)

where W̃ a (x) = sup {W(y) | |yj − xj | < aj , j = 1, . . . , d} ε(b) = inf {φ(y) | |yj | ≤ bj , j = 1, . . . , d}. Proof. Note that φ is continuous and φ(x) > 0 for all x ∈ ℝd . This can be shown as in Lemma 4.179, and implies that 0 < ε(b) < ∞. Hence t

t

t

φ(x) = etE 𝔼x [φ(Bt )e− ∫0 W(Bs )ds e∫0 U(Bs )ds ] ≥ etE 𝔼x [φ(Bt )e− ∫0 W(Bs )ds 1A ] ≥ ε(b)etE e−t Wa (x) 𝔼x [1A ].

(4.3.89)

̃

Here j

A = {|Bt + xj | ≤ bj , sup |Bjs | ≤ aj , j = 1, . . . , d} 0≤s≤t

=

j {Bt

∈ [−bj − xj , bj − xj ], sup |Bjs | ≤ aj , j = 1, . . . , d}. 0≤s≤t

320 | 4 Feynman–Kac formulae There exists kj such that [kj − αj , kj + αj ] ⊂ [−aj , aj ] ∩ [−xj − bj , xj + bj ], and hence ⋂dj=1 P(aj , [kj −αj , kj +αj ], t) ⊂ A. The independence of P(aj , [kj −αj , kj +αj ], t), j = 1, . . . , d, and the estimate (4.3.87) yield d

d

j=1

j=1

𝔼x [1A ] ≥ ∏ 𝔼x [1P(aj ,[kj −αj ,kj +αj ],t) ] ≥ ∏

αj

√2πt

F(

a2j t

a2j

) e− 2t .

(4.3.90)

Combining this with (4.3.89) and (4.3.90), we obtain (4.3.88). Theorem 4.182 (Lower bound on ground state). Let V = W − U ∈ 𝕍lower . Suppose that W(x) ≤ γ|x|2m outside a compact set with γ > 0 and m > 1, and φ(x) ≥ 0. Then there m+1 exist constants δ, D > 0 such that φ(x) ≥ De−δ|x| . Proof. Since ‖φ‖∞ < ∞, it suffices to prove the corollary for large enough |x|. Set t = |x|−(m−1) , aj = 1 + |xj |, αj = 1/2, and bj = 1 for j = 1, . . . , d. It can be directly checked that with these choices (4.3.87) is satisfied. Note that W̃ a (x) ≤ γ24m |x|2m . We see that F(ξ ) < 1 and limξ →∞ F(ξ ) = 1. Hence F(a2j /t) = F((1 + |xj |)2 |x|m−1 ) ≥ 1/2 for sufficiently large |x|. On inserting to (4.3.88) we obtain φ(x) ≥ ε(b) (

d

−(m−1) |x|(m−1)/2 E −γ24m |x|m−1 − 21 ∑dj=1 (1+|xj |)2 |x|m−1 ) e|x| e e 4√2π

d

≥

4m m−1 ε(b) |x|(m−1)/2 ( ) e−(γ2 +2)|x| . 2 4√2π

Here we assumed that e|x| 4m

m+1

−(m−1)

E

> 1/2 for sufficiently large |x|. We have

ε(b) 2

(m−1)/2

d

( |x|4√2π ) >

De−(δ−γ2 −2)|x| with some D for sufficiently large |x| and δ > γ24m + 2, and thus conm−1 clude that φ(x) ≥ De−δ|x| .

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials 4.4.1 Feynman–Kac–Itô formula The Schrödinger operator (4.1.30) gives the total energy of a quantum particle in a force field −∇V. In some cases of electrodynamics there is an interaction with a magnetic field that has to be taken into account. This is done by adding a vector potential a = (a1 , . . . , ad ) to the Schrödinger operator. A formal definition of the Schrödinger operator with vector potentials is then given by 1 H(a) = (−i∇ − a)2 + V. 2

(4.4.1)

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 321

Lemma 4.183. Suppose that a ∈ (L2p (ℝd )+L∞ (ℝd ))d , V ∈ Lp (ℝd )+L∞ (ℝd ), and (∇⋅a) ∈ Lp (ℝd ) + L∞ (ℝd ), where p = 2 for d ≤ 3 and p > d/2 for d ≥ 4. Then H(a) is self-adjoint on D(−(1/2)Δ) and essentially self-adjoint on any core of − 21 Δ. Proof. Under the assumption 1 1 1 H(a) = − Δ + (i∇ ⋅ a) + a ⋅ (i∇) + a ⋅ a + V 2 2 2

(4.4.2)

follows. It is easy to see that a ⋅ a, ∇ ⋅ a, and V are infinitesimally small multiplication operators with respect to − 21 Δ. Furthermore, a ⋅ ∇ is also infinitesimally small. Indeed, we have ‖aμ 𝜕μ f ‖ ≤ ‖aμ ‖2p ‖𝜕μ f ‖q ≤ ‖aμ ‖2p (ε‖Δf ‖2 + bε ‖f ‖) for every ε > 0, where q = 2p/(p−1). The second inequality follows from the Hausdorff– Young inequality as ‖𝜕μ f ‖q ≤ ‖kμ f ̂‖q󸀠 ≤ ‖(1 + |k|)−α ‖2p ‖(1 + |k|)α kμ f ̂‖2 , d where q󸀠 = 2p/(p + 1) and α ∈ ( 2p , 1). By the Kato–Rellich theorem H(a) is self-adjoint

on D(−(1/2)Δ), and essentially self-adjoint on any core of − 21 Δ.

In this section our goal is to construct a Feynman–Kac formula for e−tH(a) . A drift transformation applied to Brownian motion gives t

t

1

1

2

𝔼x [f (Bt )e∫0 a(Bs )⋅dBs − 2 ∫0 a(Bs ) ds ] = e−t(− 2 Δ−a⋅∇) f (x). On replacing a by −ia, formally t

t

1

1

2

𝔼x [f (Bt )e−i ∫0 a(Bs )⋅dBs + 2 ∫0 a(Bs ) ds ] = e−t(− 2 Δ+ia⋅∇) f (x) is obtained. Let W = V + 21 a⋅a+ 21 (i∇⋅a). By (4.4.2) the “imaginary” drift transformation yields t

1

t

t

𝔼x [f (Bt )e−i(∫0 a(Bs )⋅dBs + 2 ∫0 ∇⋅a(Bs )ds)−∫0 V(Bs ds ] t

1

t

2

t

= 𝔼x [f (Bt )e−i ∫0 a(Bs )⋅dBs + 2 ∫0 a(Bs ) ds−∫0 W(Bs )ds ] = (e−tH(a) f )(x). We turn this rigorous in the next theorem. Theorem 4.184 (Feynman–Kac formula for Schrödinger operator with vector potential, Feynman–Kac–Itô formula). Suppose that a ∈ (Cb2 (ℝd ))d and V ∈ L∞ (ℝd ). Then for f , g ∈ L2 (ℝd ), t

t

(f , e−tH(a) g) = ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds g(Bt )]dx. ℝd

(4.4.3)

322 | 4 Feynman–Kac formulae In particular, t

t

(e−tH(a) f )(x) = 𝔼x [e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds f (Bt )].

(4.4.4)

Recall that the notation ∘dBs in the stochastic integral above stands for Stratonovich integral. t

t

Proof. Write Zt = eXt , Xt = −i ∫0 a(Bs ) ∘ dBs − ∫0 V(Bs )ds, and Yt = f (Bt ). Define (Qt f )(x) = 𝔼x [Zt f (Bt )]. Note that V(Bt ) and (∇ ⋅ a)(Bt ) are L1loc (ℝ, dt) almost surely, and aj (Bt ) ∈ M 2 (0, t). Thus, since i dXt = −ia ⋅ dBt − (∇ ⋅ a)dt − Vdt, 2 we arrive at 1 i 1 dZt = Zt dXt + Zt (dXt )2 = Zt ( − (∇ ⋅ a) − a ⋅ a − V)dt + Zt (−ia) ⋅ dBt 2 2 2 by the Itô formula. Suppose that f ∈ C0∞ (ℝd ). By dYt = 21 Δfdt + ∇f ⋅ dBt and the product formula, i 1 d(Zt Yt ) = Zt (− (∇ ⋅ a)dt − (a ⋅ a)dt + (−ia) ⋅ dBt − Vdt) Yt 2 2 1 + Zt ( Δfdt + ∇f ⋅ dBt ) + Zt (−ia) ⋅ ∇fdt 2 − H(a)fZt dt + Zt (∇f + (−ia)f ) ⋅ dBt is obtained. This gives t

(Qt f )(x) − f (x) = ∫(Qs H(a)f )(x)ds.

(4.4.5)

0

Now suppose that f ∈ D(H(a)). Since C0∞ (ℝd ) is a core of H(a), there exists a sequence (fn )n∈ℕ ⊂ C0∞ (ℝd ) such that fn → f and H(a)fn → H(a)f as n → ∞ strongly. Qt is bounded with ‖Qt ‖ ≤ etδ (2πt)−d/2 and δ = supx∈ℝd |V(x)|. We have Qt fn − fn → Qt f − f t

t

and ∫0 Qs H(a)fn ds → ∫0 Qs H(a)fds as n → ∞ in L2 (ℝd ). Taking a subsequence n󸀠 , for t

almost everywhere x ∈ ℝd , (Qt fn󸀠 )(x)−fn󸀠 (x) → (Qt f )(x)−f (x) and ∫0 (Qs H(a)fn󸀠 )(x)ds → t

∫0 (Qs H(a)f )(x)ds. Hence we have (4.4.5) for f ∈ L2 (ℝd ). In a similar manner as in Theorem 4.86 we see that 1 s-lim (Qt f − f ) = −H(a)f . t→0 t

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 323

Then it is sufficient to show that Qt is a symmetric C0 -semigroup. We directly see that s

t

Qs Qt f (x) = 𝔼x [e−i ∫0 a(Br )∘dBr 𝔼Bs [f (Bt )e−i ∫0 a(Bs )∘dBs ]] s

s+t

= 𝔼x [e−i ∫0 a(Br )∘dBr 𝔼x [f (Bs+t )e−i ∫s

a(Br )∘dBr

= Qs+t f (x).

|Fs ]]

Recall that (Ft )t≥0 is the natural filtration of Brownian motion. Hence the semigroup property follows. Let B̃ s = Bt−s − Bt , 0 ≤ s ≤ t. We have t

t

̃ ̃ ̃ (f , Qt g) = ∫ f (x)𝔼x [e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds g(B̃ t )]dx ℝd t

t

̃ ̃ ̃ = 𝔼0 [ ∫ f (x)e−i ∫0 a(x+Bs )∘dBs e− ∫0 V(x+Bs )ds g(x + B̃ t )] dx. [ℝd ]

Let 1 Ij = (a(x + B̃ tj/n ) + a(x + B̃ t(j−1)/n ))(B̃ tj/n − B̃ t(j−1)/n ). 2 t

̃ s in L2 (X ) as n → ∞, and a subsequence (∑n Ij )n󸀠 We have ∑nj=1 Ij → ∫0 a(x + B̃ s ) ∘ dB j=1 t ̃ s almost surely. We reset n󸀠 as n. Changing the variable x converges to ∫0 a(x + B̃ s ) ∘ dB to y − B̃ t , we have 󸀠

n

t

̃ ̃ ̃ (f , Qt g) = lim 𝔼0 [ ∫ f (y − B̃ t )e−i ∑j=1 Ij e− ∫0 V(y−Bt −Bs )ds g(y)]dy, n→∞

ℝd

where 1 Ij̃ = (a(y − B̃ t + B̃ jt/n ) + a(y − B̃ t + B̃ t(j−1)/n ))(B̃ tj/n − B̃ t(j−1)/n ). 2 Recall that a ∈ (Cb2 (ℝd ))d . In L2 (X ) sense it follows that n

t

j=1

0

lim ∑ Ij̃ = − ∫ a(y + Bs ) ∘ dBs .

n→∞

This gives t

t

(f , Qt g) = ∫ 𝔼x [f (Bt )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds ]g(x)dx = (Qt f , g).

(4.4.6)

ℝd

Hence Qt is symmetric. Strong continuity of Qt with respect to t is obtained in the same way as in the proof of Theorem 4.86.

324 | 4 Feynman–Kac formulae 4.4.2 Alternative proof of the Feynman–Kac–Itô formula As before, the Trotter product formula can once again be employed to prove the Feynman–Kac–Itô formula. Here we give this proof of Theorem 4.184, which will be useful in constructing the functional integral representation of the Pauli–Fierz model in Volume 2. Proof. Suppose a ∈ (Cb2 (ℝd ))d and V = 0. Write Ks (x, y) = Πs (x − y)eih , where

(4.4.7)

1 h = h(x, y) = (a(x) + a(y)) ⋅ (x − y). 2

Define the family of integral operators ϱs : L2 (ℝd ) → L2 (ℝd ), ϱs f (x) = ∫ Ks (x, y)f (y)dy,

s ≥ 0,

(4.4.8)

ℝd

where ϱ0 f (x) = f (x). Note that ϱs is symmetric and ‖ϱs ‖ ≤ 1. Let f , g ∈ C0∞ (ℝd ). It is directly seen that d 1 (g, ϱs f ) = ∫ g(x)dx ∫ Πs (x − y) ( Δy eih f (y)) dy. ds 2 ℝd

Here we used that

dΠs (x ds

ℝd

− y) = 21 Δy Πs (x − y). Since

𝜕j eih f = (i𝜕j h ⋅ f + 𝜕j f )eih ,

𝜕j2 eih f = {𝜕j2 f + 2i𝜕j h ⋅ 𝜕j f + (i𝜕j2 h + (i𝜕j h)2 )f }eih , 1 𝜕j h = {𝜕j a(y) ⋅ (x − y) − (aj (x) + aj (y))}, 2 1 𝜕j2 h = 𝜕j2 a(y) ⋅ (x − y) − 𝜕j aj (y), 2

where 𝜕j = 𝜕yj , j = 1, . . . , d, we have lim ∫ Πs (x − y)(Δy eih f (y))dy = −(−Δf − 2a ⋅ (−i∇f ) − (−i∇a − a ⋅ a)f ) = −2H(a)f .

s→0

ℝd

Thus lims→0

d (g, ϱs f ) ds

= (g, −H(a)f ) and hence lim(g, t −1 (1 − ϱt )f ) = (g, H(a)f ).

t→0

(4.4.9)

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 325

n

We show next that ϱ2t/2n converges to a symmetric semigroup. For f , g ∈ L2 (ℝd ), n (g, ϱ2t/2n f )

= ∫ g(x)e

n

i ∑2j=1 h(xj−1 ,xj )

2n

2n

j=1

j=0

f (xn )( ∏ Πt/2n (xj − xj−1 )) ∏ dxj

ℝd

with x0 = x. Therefore, n

lim (g, ϱ2t/2n f )

n→∞

2n

1 = lim ∫ 𝔼 [g(B0 )f (Bt ) exp (−i ∑ (a(Btj/2n ) + a(Bt(j−1)/2n ))(Btj/2n − Bt(j−1)/2n ))]dx n→∞ 2 j=1 x

ℝd

t

= ∫ 𝔼x [g(B0 )f (Bt ) exp (−i ∫ a(Bs ) ∘ dBs )] dx 0 [ ] ℝd

(4.4.10)

n

by Corollary 2.134. Hence | limn→∞ (g, ϱ2t/2n f )| ≤ ‖g‖ ‖f ‖. By the Riesz theorem there is a n

bounded operator St such that limn→∞ (g, ϱ2t/2n f ) = (g, St f ). It is symmetric since ϱs is symmetric. Equation (4.4.10) implies that t

St f (x) = 𝔼x [f (Bs )e−i ∫0 a(Bs )∘dBs ]. Hence S0 = 1, s-limt→0 St = 1, and Ss St = Ss+t follow. Thus St is a symmetric C0 -semigroup, so there exists a self-adjoint operator A such that e−tA = St by Proposition 4.81. It suffices to show that A = H(a). This can be checked without using the Itô formula. We have 1 1 n ( (e−tA − 1)g, f ) = lim ( (ϱ2t/2n − 1)g, f ) n→∞ t t 2n −1

= lim ∑ n→∞

j=0

1

n 1 2n 2n 2n (j/2n ) s] n ( (ϱ − 1)g, ϱ f ) = lim ( (ϱt/2n − 1)g, ϱ[2 ∫ n f ) ds, n t/2 t/2 t/2 n n→∞ 2 t t

0

where the brackets denote the integer part in this formula. Since for g ∈ D(−(1/2)Δ), 2n (ϱt/2n − 1)g = −H(a)g, n→∞ t

w − lim

the norm ‖(2n /t)(ϱt/2n − 1)g‖ is uniformly bounded in n. Moreover, we see that n

s] s-limn→∞ ϱ[2 = e−stA , s ∈ [0, 1], by Corollary 4.186. Thus we conclude that t/2n 1

1 ( (e−tA − 1)g, f ) = ∫(−H(a)g, e−tsA f )ds. t

(4.4.11)

0

As t → 0 on both sides above, we get (g, Af ) = (H(a)g, f ), implying that Ag = H(a)g. Hence A = H(a) and the proposition follows for V = 0.

326 | 4 Feynman–Kac formulae Let now V be continuous and bounded. Note that for f0 , fn ∈ L2 (ℝd ) and fj ∈ L (ℝd ) for j = 1, . . . , n − 1, it follows that for 0 = t0 < t1 < . . . < tn , ∞

n

n

j=1

j=1

tn

(f0 , ∏ e−(tj tj−1 )H(a) fj ) = ∫ 𝔼x [f0 (B0 )( ∏ fj (Btj ))e−i ∫0 ℝd

a(Bs )∘dBs

]dx.

(4.4.12)

This can be proven by using the Markov property of Brownian motion as n

(f0 , ∏ e−(tj −tj−1 )H(a) fj ) j=1

t1

= ∫ 𝔼x [f0 (B0 )f1 (Bt1 )e−i ∫0 ℝd

t2

= ∫ 𝔼x [f0 (B0 )f1 (Bt1 )e−i ∫0

a(Bs )∘dBs

a(Bs )∘dBs

t2 −t1

𝔼Bt1 [f1 (Bt1 )e−i ∫0

a(Bs )∘dBs

G2 (Bt2 −t1 )]]dx

f1 (Bt1 )G2 (Bt2 −t1 )]dx,

ℝd

where Gj (x) = fj (x)(∏ni=j+1 e−(ti −ti−1 )H(a) fi )(x). Repeating this procedure we arrive at (4.4.12). From this and the Trotter product formula we have (f , e−tH(a) g) = lim (f , (e−(t/n)H(a) e−(t/n)V )n g) n→∞

t

n

= lim ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∑j=1 V(Bjt/n ) ]dx n→∞

ℝd

t

t

= ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds ]dx. ℝd

Finally, we can extend the result for smooth V in a similar way as done in Theorem 4.88. Now we prove the statements used in the proof above. Lemma 4.185. Let An be a positive and contracting self-adjoint operator such that s-limn→∞ An = A. Then for all s ≥ 0 we have that s-limn→∞ Asn = As . Proof. Suppose that 0 < s ≤ 1/2. Since An is positive, we have A2s n f

∞

sin(2πs) = ∫ λ2s−1 (An + λ)−1 An fdλ. π 0

2s s s From this follows that A2s n → A weakly as n → ∞, thus An → A strongly. For 0 ≤ s s/2 2 s s ≤ 1, s-limn→∞ An = s-limn→∞ (An ) = A , since An is a contraction. Repeating this procedure the result follows.

Corollary 4.186. We have s-limn→∞ ϱ[ns] = e−stA for s ≥ 0. t/n

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 327

Proof. Write An = ϱ2n t/n . The operator An is positive, contracting, and self-adjoint such that s-limn→∞ An = e−2tA holds. We have A(ns+1)/n ≤ ϱ2[ns] ≤ Asn in form sense. From this n t/n

2[ns] estimate, Lemma 4.185, and the fact that s-limn→∞ A1/n n = 1, we obtain (f , ϱt/n f ) →

(f , e−2tsA f ) as n → ∞. Hence s-limn→∞ ϱ[ns] = e−tsA follows. t/n

4.4.3 Extension to singular external and vector potentials The Feynman–Kac–Itô formula derived in Sections 4.4.1 and 4.4.2 can further be extended to singular vector potentials a = (a1 , . . . , ad ). By using the form technique, we are able to define H(a) as a self-adjoint operator. Let Dμ = −i𝜕μ − aμ : L2 (ℝd ) → D 󸀠 (ℝd ),

(4.4.13)

with Schwartz distribution space D 󸀠 (ℝd ) over ℝd . Define the quadratic form d

qa (f , g) = ∑ (Dμ f , Dμ g) + (V 1/2 f , V 1/2 g) μ=1

(4.4.14)

with domain Q(qa ) = ⋂dμ=1 {f ∈ L2 (ℝd ) | Dμ f ∈ L2 (ℝd )} ∩ D(V 1/2 ). Lemma 4.187. Suppose that a ∈ (L2loc (ℝd ))d and V ∈ L1loc (ℝd ) with V ≥ 0. Then qa is a closed symmetric form. Proof. Let (fn )n ⊂ Q(qa ) be a Cauchy sequence with respect to the norm ‖ ⋅ ‖0 = (qa (⋅, ⋅) + ‖ ⋅ ‖2 )1/2 ; then V 1/2 fn , Dμ fn are Cauchy sequences in L2 (ℝd ). Thus there exist g, f ∈ L2 (ℝd ) such that Dμ fn → g and V 1/2 fn → V 1/2 f as n → ∞ in L2 (ℝd ). This gives (f , Dμ ϕ) = (g, ϕ), for all ϕ ∈ C0∞ (ℝd ). Therefore g = Dμ f in D 󸀠 (ℝd ) distribution sense, and hence Q(qa ) is complete with respect to ‖ ⋅ ‖0 . By Lemma 4.187 there exists a self-adjoint operator h such that (f , hg) = qa (f , g),

f ∈ Q(qa ), g ∈ D(h).

(4.4.15)

Definition 4.188 (Schrödinger operator with singular external and singular vector potential (form version)). Suppose that a ∈ (L2loc (ℝd ))d and V ∈ L1loc (ℝd ). We denote the self-adjoint operator h in (4.4.15) by H(a). A sufficient condition for C0∞ (ℝd ) to be a core of H(a) is the following.

328 | 4 Feynman–Kac formulae Proposition 4.189. (1) Let V ∈ L1loc (ℝd ) with V ≥ 0. If a ∈ (L2loc (ℝd ))d , then C0∞ (ℝd ) is a form core of H(a). (2) Let V ∈ L2loc (ℝd ) with V ≥ 0. If a ∈ (L4loc (ℝd ))d and ∇ ⋅ a ∈ L2loc (ℝd ), then C0∞ (ℝd ) is an operator core. We can also construct a Feynman–Kac–Itô formula for the form-defined H(a). Lemma 4.190. Let 0 ≤ V and V ∈ L1loc (ℝd ). Suppose that a(n) , a ∈ (L2loc (ℝd ))d and 2 d (n) a(n) μ → aμ in Lloc (ℝ ) as n → ∞. Then H(a ) converges to H(a) in strong resolvent sense. (n) ∞ d Proof. Put Hn = H(a(n) ), H = H(a), and D(n) μ = −i𝜕μ − aμ . Note that C0 (ℝ ) is a form

core of H(a) by Proposition 4.189. Write un = (Hn + i)−1 f for f ∈ L2 (ℝd ). Thus ‖un ‖ ≤ ‖f ‖ and d

2 1/2 2 2 ∑ ‖D(n) μ un ‖ + ‖V un ‖ ≤ ‖f ‖ .

μ=1

1/2 Since un , D(n) μ un , and V un are bounded, there exist a subsequence un󸀠 and some vec) tors u, v, wμ ∈ L2 (ℝd ) such that un󸀠 → u, V 1/2 un󸀠 → v and D(n μ un󸀠 → wμ weakly. Since, 󸀠

) 󸀠 d 1/2 furthermore, V 1/2 un󸀠 → V 1/2 u and D(n μ un󸀠 → Dμ u in D (ℝ ), we have v = V u and 󸀠

Dμ u = wμ , and hence u ∈ Q(qa ). For ϕ ∈ C0∞ (ℝd )

qa (u, ϕ) = lim (un , Hn ϕ) = (f − iu, ϕ). n→∞

(4.4.16)

As C0∞ (ℝd ) is a form core of H, (4.4.16) implies that u ∈ D(H) and f = (H + i)u. Hence u = (H + i)−1 f follows. Therefore, (Hn + i)−1 → (H + i)−1 in weak sense, and similarly, (Hn − i)−1 → (H − i)−1 weakly. Since i ‖(Hn + i)−1 f ‖2 = (f , (Hn + i)−1 f − (Hn − i)−1 f ) → ‖(H + i)−1 f ‖2 , 2 strong resolvent convergence of Hn follows. Lemma 4.191. Let V ≥ 0 in L1loc (ℝd ), a ∈ (L2loc (ℝd ))d , and ∇ ⋅ a ∈ L1loc (ℝd ). Then (f , e−tH(a) g) has the same functional integral representation as in (4.4.3). Proof. First suppose that V ∈ L∞ (ℝd ). Note that since aμ (Bs ) ∈ ⋃T>0 𝕄2T by the ast

sumption that a ∈ (L2loc (ℝd ))d we have 𝒲 x (| ∫0 a(Bs ) ⋅ dBs | < ∞) = 1; moreover, by t

Lemma 4.95, 𝒲 x (| ∫0 ∇ ⋅ a(Bs )ds| < ∞) = 1 also holds. By using a mollifier we can

2 d choose a sequence a(n) ∈ C0∞ (ℝd ), n = 1, 2, . . . , such that a(n) μ → aμ in Lloc (ℝ ) and

∇ ⋅ a(n) → ∇ ⋅ a in L1loc (ℝd ) as n → ∞. Let 1R = χ(x1 /R) ⋅ ⋅ ⋅ χ(xd /R), R ∈ ℕ, where χ ∈ C0∞ (ℝ) such that 0 ≤ χ ≤ 1, χ(x) = 1 for |x| < 1 and χ(x) = 0 for |x| ≥ 2. Since 1R a(n) → a(n) as R → ∞ in (L2loc (ℝd ))d and a(n) → a as n → ∞ in (L2loc (ℝd ))d , it follows

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 329

from Lemma 4.190 that e−tH(1R a ) → e−tH(1R a) , as n → ∞ and e−tH(1R a) → e−tH(a) as R → ∞ in strong sense. Furthermore, (4.4.3) remains true for a replaced by 1R a(n) ∈ C0∞ (ℝd ). Since 1R a(n) ∈ (C0∞ (ℝd ))d and 1R a(n) → 1R a in (L2loc (ℝd ))d as n → ∞, it follows that (n)

t

t

∫ 1R (Bs )a(n) (Bs ) ⋅ dBs → ∫ 1R (Bs )a(Bs ) ⋅ dBs 0

(4.4.17)

0

and since ∇ ⋅ (1R a(n) ) = (∇1R ) ⋅ a(n) + 1R (∇ ⋅ a(n) ) → (∇1R ) ⋅ a + 1R (∇ ⋅ a) in L1 (ℝd ), it follows that t

t

∫ ∇ ⋅ (1R (Bs )a(n) (Bs ))ds → ∫ {(∇1R (Bs )) ⋅ a(Bs ) + 1R (Bs )(∇ ⋅ a(Bs ))} ds 0

(4.4.18)

0

strongly in L1 (X , d𝒲 x ). Thus there is a subsequence n󸀠 such that (4.4.17) and (4.4.18) hold almost surely for n replaced by n󸀠 , and therefore (4.4.3) follows by a limiting argument for a replaced by 1R a. Let Ω+ (R) = {ω ∈ X | Ω− (R) = {ω ∈ X | and

max

Bμs (ω) ≤ R},

min

Bμs (ω) ≥ −R},

0≤s≤t,1≤μ≤d 0≤s≤t,1≤μ≤d

󵄨󵄨 t 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 I(R) = 󵄨󵄨∫ 1R (Bs )a(Bs ) ⋅ dBs − ∫ a(Bs ) ⋅ dBs 󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨󵄨0 󵄨󵄨 0

Since x

d

x

𝒲 (Ω− (R)) = 𝒲 (Ω+ (R)) =

μ ∏ 𝒲 x (|Bt | μ=1

d

R

2 2 ≤ R) = ( ∫ e−y /(2t) dy) , √2πt

0

and I(R) = 0 on Ω+ ∩ Ω− , we have d

∞

2 2 𝒲 (I(R) ≥ ε) = 𝒲 (I(R) ≥ ε, Ω+ (R) ∪ Ω− (R) ) ≤ 2 ( ∫ e−y /(2t) dy) . √2πt

x

x

c

c

R

Hence limR→∞ 𝒲 x (I(R) ≥ ε) = 0. Thus there exists a subsequence R󸀠 such that t t ∫0 1R󸀠 (Bs )a(Bs ) ⋅ dBs → ∫0 a(Bs ) ⋅ dBs as R󸀠 → ∞, almost surely. It can be seen in a t

similar way that with a subsequence (4.4.18) converges to ∫0 ∇ ⋅ a(Bs )ds almost surely. t

t

Thus ∫0 1R󸀠󸀠 (Bs )a(Bs ) ∘ dBs → ∫0 a(Bs ) ∘ dBs , almost surely. Since t

t

t

|f (B0 )g(Bt )e− ∫0 V(Bs )ds e−i ∫0 1R󸀠󸀠 (Bs )a(Bs )∘dBs | = |f (B0 )g(Bt )|e− ∫0 V(Bs )ds

330 | 4 Feynman–Kac formulae and the right-hand side is integrable, (4.4.3) holds for a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ) by the dominated convergence theorem. Finally, by a similar argument as in the proof of Theorem 4.88, we can extend (4.4.3) to V ∈ L1loc (ℝd ). Define H 0 (a) by H(a) with V = 0. Lemma 4.192. Let a ∈ (L2loc (ℝd ))d , ∇⋅a ∈ L1loc (ℝd ), and V be a real-valued multiplication operator. (1) Suppose V is relatively form-bounded with respect to − 21 Δ with relative bound b. Then V is also relatively form-bounded with respect to H 0 (a) with a relative bound not larger than b. (2) Suppose V is relatively bounded with respect to − 21 Δ with relative bound b. Then V is also relatively bounded with respect to H 0 (a) with a relative bound not larger than b. Proof. In virtue of Lemma 4.191 we have |(f , e−tH Since (H 0 (a) + E)−1/2 =

1 √π

0

(a)

g)| ≤ (|f |, et(1/2)Δ |g|).

∫0 t −1/2 e−t(H ∞

0

(a)+E)

(4.4.19)

dt, E > 0, (4.4.19) implies that

|(H 0 (a) + E)−1/2 f |(x) ≤ (−(1/2)Δ + E)−1/2 |f |(x)

(4.4.20)

for almost every x ∈ ℝd . Hence we have |V(x)|1/2 |(H 0 (a) + E)−1/2 f |(x) ≤ |V(x)|1/2 (−(1/2)Δ + E)−1/2 |f |(x) and ‖ |V|1/2 (H 0 (a) + E)−1/2 f ‖ ‖ |V|1/2 (−(1/2)Δ + E)−1/2 |f |‖ ≤ . ‖f ‖ ‖f ‖ Similarly, by using (H 0 (a) + E)−1 = ∫0 e−t(H ∞

0

(a)+E)

(4.4.21)

dt, E > 0, we have

‖ |V|(H 0 (a) + E)−1 f ‖ ‖ |V|(−(1/2)Δ + E)−1 |f |‖ ≤ . ‖f ‖ ‖f ‖

(4.4.22)

On taking the limit E → ∞, the right-hand sides of (4.4.21) and (4.4.22) converge to b. Hence (1) follows by (4.4.21) and (2) by (4.4.22). Let V be such that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to − 21 Δ with a relative bound strictly smaller than one. The KLMN theorem and Lemma 4.192 allow to define H 0 (a) +̇ V+ −̇ V− as a self-adjoint operator.

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 331

Definition 4.193 (Schrödinger operator with singular external and singular vector potential). Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Let V be such that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to − 21 Δ with a relative bound strictly smaller than one. Then the Schrödinger operator with external potential V and vector potential a is defined by H(a) = H 0 (a) +̇ V+ −̇ V− .

(4.4.23)

Theorem 4.194 (Feynman–Kac formula for Schrödinger operator with singular external potential and singular vector potential). Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Suppose that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to − 21 Δ with a relative bound strictly smaller than one. Then the Feynman–Kac formula for (f , e−tH(a) g) is given by (4.4.3). Proof. The proof is a minor modification of the proof of Theorem 4.88. Let V+n (x) = {

V+ (x), n,

V+ (x) < n,

V−m (x) = {

V+ (x) ≥ n,

V− (x), m,

V− (x) < m, V− (x) ≥ m.

Write Vn,m = V+n − V−m . We have (f , e−t(H

0

(a)+Vn,m )

t

t

g) = ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,m (Bs ) ds g(Bt )]dx.

(4.4.24)

ℝd t

The only difference with Theorem 4.88 is that e−i ∫0 a(Bs )∘dBs appears in the integrand. Write h = H 0 (a) and define the closed quadratic forms 1/2 1/2 1/2 1/2 qn,m (f , f ) = (h1/2 f , h1/2 f ) + (V+n f , V+n f ) − (V−m f , V−m f ),

1/2 1/2 qn,∞ (f , f ) = (h1/2 f , h1/2 f ) + (V+n f , V+n f ) − (V−1/2 f , V−1/2 f ),

q∞,∞ (f , f ) = (h1/2 f , h1/2 f ) + (V+1/2 f , V+1/2 f ) − (V−1/2 f , V−1/2 f ), whose form domains are respectively Q(qn,m ) = Q(h), Q(qn,∞ ) = Q(h), and Q(q∞,∞ ) = Q(h) ∩ Q(V+ ). By the same arguments of monotone convergence of quadratic forms as in the proof of Theorem 4.88 we conclude that for all t ≥ 0, exp (−t (h + V+n − V−m )) → exp (−t (h + V+n −̇ V− ))

(4.4.25)

exp(−t(h +̇ V+n −̇ V− )) → exp(−t(h +̇ V+ −̇ V− ))

(4.4.26)

as m → ∞, and

as n → ∞ strongly. By taking first n → ∞ and then m → ∞ it can be proven that both sides of (4.4.24) converge, i. e., the left-hand side of (4.4.24) converges by (4.4.25) and (4.4.26). On the other hand, we have the estimate t

t

∫ 𝔼x [|f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,m (Bs ) ds g(Bt )|]dx ℝd

332 | 4 Feynman–Kac formulae t

≤ ∫ 𝔼x [|f (B0 )|e− ∫0 Vn,m (Bs ) ds |g(Bt )|]dx ℝd

t

≤ ∫ 𝔼x [|f (B0 )|e− ∫0 Vn,∞ (Bs )ds |g(Bt )|]dx < ∞, ℝd

where boundednes of the third term can be obtained from the proof Theorem 4.88. Moreover, t t t 󵄨󵄨 󵄨 󵄨󵄨f (B )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,m (Bs ) ds g(B )󵄨󵄨󵄨 ≤ |f (B )| e− ∫0 Vn,∞ (Bs ) ds |g(B )|, 󵄨󵄨 0 t 󵄨󵄨 0 t 󵄨 󵄨

and the dominated convergence theorem yields t

t

lim ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,m (Bs ) ds g(Bt )]dx

m→∞

ℝd

t

t

= ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,∞ (Bs ) ds g(Bt )]dx. ℝd

Furthermore, since t

t

∫ 𝔼x [|f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,∞ (Bs ) ds g(Bt )|]dx ℝd

t

≤ ∫ 𝔼x [|f (B0 )|e− ∫0 Vn,∞ (Bs ) ds |g(Bt )|]dx < ∞ ℝd

and t t t 󵄨󵄨 󵄨 󵄨󵄨f (B )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn󸀠 ,∞ (Bs ) ds g(B )󵄨󵄨󵄨 ≤ |f (B )| e− ∫0 Vn,∞ (Bs ) ds |g(B )| 󵄨󵄨 0 t 󵄨󵄨 0 t 󵄨 󵄨

for n ≤ n󸀠 , the dominated convergence theorem again yields t

t

t

t

lim ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 Vn,∞ (Bs ) ds g(Bt )]dx

n→∞

ℝd

= ∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs ) ds g(Bt )]dx. ℝd

Together with these, the right-hand side of (4.4.24) converges to t

t

∫ 𝔼x [f (B0 )e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds g(Bt )]dx ℝd

for all f , g ∈ L2 (ℝd ) by taking first m → ∞ and then n → ∞.

4.4 Feynman–Kac formula for Schrödinger operators with vector potentials | 333

We used inequality (4.4.19) in the proof of Lemma 4.192. This can be extended to operators with singular external potentials. Corollary 4.195 (Diamagnetic inequality). Under the assumptions of Theorem 4.194 we have |(f , e−tH(a) g)| ≤ (|f |, e−tH(0) |g|),

t ≥ 0.

(4.4.27)

4.4.4 Kato-class potentials and Lp -Lq boundedness We can also add Kato-class potentials to Schrödinger operators with vector potentials a. Let V be Kato-decomposable, a ∈ (L2loc (ℝd ))d , and ∇ ⋅ a ∈ L1loc (ℝd ). Define t

t

Kt f (x) = 𝔼x [e−i ∫0 a(Bs )∘dBs e− ∫0 V(Bs )ds f (Bt )].

(4.4.28)

Since the absolute value of the first exponential is 1, the operator Kt is well defined and bounded on L2 (ℝd ), for every t ≥ 0. Theorem 4.196 (Feynman–Kac formula for Schrödinger operator with Kato-decomposable potential and vector potential). Suppose that V is Kato-decomposable, a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Then {Kt : t ≥ 0} is a symmetric C0 -semigroup and there exists the unique self-adjoint operator K(a) bounded from below such that Kt = e−tK(a) ,

t ≥ 0.

Proof. The proof is similar to Theorem 4.184. Definition 4.197 (Schrödinger operator with Kato-decomposable potential and vector potential). Let a ∈ (L2loc (ℝd ))d , ∇ ⋅ a ∈ L1loc (ℝd ) and V be Kato-decomposable. Then we define the Schrödinger operator with Kato-decomposable potential V and vector potential a by the self-adjoint operator K(a) given in Theorem 4.196. Proposition 4.198 (Lp -Lq boundedness and continuity of integral kernel). Suppose that V is Kato-decomposable and a ∈ (L2loc (ℝd ))d with ∇⋅a ∈ L1loc (ℝd ). Let 1 ≤ p ≤ q ≤ ∞. Then the following hold: (1) e−tK(a) is Lp -Lq bounded, for every t ≥ 0; (2) if |a|2 ∈ 𝒦(ℝd ) and ∇ ⋅ a ∈ 𝒦(ℝd ), then Kt f (x) is continuous in x for f ∈ Lp (ℝd ) with 1 ≤ p ≤ ∞; (3) if |a|2 ∈ 𝒦(ℝd ) and ∇ ⋅ a ∈ 𝒦(ℝd ), the integral kernel ℝd × ℝd ∋ (x, y) 󳨃→ Kt (x, y) is jointly continuous in x and y, and is given by x,y

t

t

Kt (x, y) = Πt (x − y)𝔼[0,t] [e− ∫0 V(Bs ) ds e−i ∫0 a(Bs )∘dBs ]. x,y

Here 𝒲[0,t] is the Brownian bridge measure defined in (2.3.18).

(4.4.29)

334 | 4 Feynman–Kac formulae Proof. By the diamagnetic inequality we have |e−tK(a) f (x)| ≤ e−tK(0) |f |(x). In Theorem 4.107 we have shown that e−tK(0) is hypercontractive. Using the above estimate, it follows that e−tK(a) is bounded from L1 (ℝd ) to L1 (ℝd ), from L∞ (ℝd ) to L∞ (ℝd ), and from L1 (ℝd ) to L∞ (ℝd ). This proves (1). Part (2) can be seen by a minor modification of the proof of Theorem 4.111. It suffices to consider f ∈ L∞ (ℝd ). Let xn → x as n → ∞. We have Kt f (x) − Kt f (xn ) = Kt f (x) − e−r(−Δ/2) Kt−r f (x) + e−r(−Δ/2) Kt−r f (x) − e−r(−Δ/2) Kt−r f (xn ) + e−r(−Δ/2) Kt−r f (xn ) − Kt f (xn ) and t

t

e−r(−Δ/2) Kt−r f (x) = 𝔼x [e− ∫r V(Bs )ds e−i ∫r a(Bs )∘dBs f (Bt )].

(4.4.30)

By the smoothing effect, see Proposition 4.109, e−r(−Δ/2) Kt−r f (x)−e−r(−Δ/2) Kt−r f (xn ) → 0 as n → ∞ for every r > 0. It is sufficient to show that Kt f (x) − e−r(−Δ/2) Kt−r f (x) → 0 as r → 0, uniformly in x. First assume V ∈ 𝒦(ℝd ). For every r > 0, let gr (x) = r

r

e−r(−Δ/2) Kt−r f (x). By (4.4.30) and the independence of 1 − e− ∫0 V(Bs ) ds e−i ∫0 a(Bs )∘dBs and t

t

e− ∫r V(Bs ) ds e−i ∫r a(Bs )∘dBs f (Bt ), we have

r r t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 t ‖gr − Kt f ‖∞ ≤ sup 𝔼x [󵄨󵄨󵄨1 − e− ∫0 V(Bs ) ds e−i ∫0 a(Bs )∘dBs 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨e− ∫r V(Bs ) ds e−i ∫r a(Bs )∘dBs f (Bt )󵄨󵄨󵄨󵄨] 󵄨 󵄨 󵄨 󵄨 d x∈ℝ r r 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ A sup (𝔼x [󵄨󵄨󵄨1 − e−i ∫0 a(Bs )∘dBs 󵄨󵄨󵄨] + 𝔼x [󵄨󵄨󵄨1 − e− ∫0 V(Bs ) ds 󵄨󵄨󵄨] ). 󵄨 󵄨 󵄨 󵄨 x∈ℝd t

Here A = ‖f ‖∞ supx∈ℝd supr≤t 𝔼x [e− ∫r V(Bs ) ds ]. In the same way as in the proof of Ther

orem 4.111 we see that supx∈ℝd 𝔼x [1 − e− ∫0 V(Bs ) ds ] → 0 as r → 0. For the second term we have 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 r 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 r 󵄨󵄨 󵄨󵄨] 󵄨 −i ∫0 a(Bs )∘dBs 󵄨󵄨󵄨 x [󵄨󵄨󵄨 x [ 1 󵄨󵄨󵄨 󵄨 𝔼 [󵄨󵄨1 − e 󵄨󵄨] ≤ 𝔼 󵄨󵄨∫ a(Bs )dBs 󵄨󵄨 + 𝔼 󵄨󵄨∫ ∇ ⋅ a(Bs )ds󵄨󵄨󵄨] . 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 󵄨󵄨] 󵄨󵄨] [󵄨󵄨0 [ 󵄨󵄨0 x

By Schwarz inequality and the Itô-isometry we have 1/2 1/2 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 r 󵄨󵄨2 r 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 x x 2 𝔼 [󵄨󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨] ≤ (𝔼 [󵄨󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨 ]) = (𝔼 [∫ |a(Bs )| ds]) . 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨] 󵄨󵄨 ] [󵄨󵄨0 [󵄨󵄨0 [0 ] x

Since |a|2 ∈ 𝒦(ℝd ), it follows that 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 [ lim sup 𝔼 󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨] = 0. 󵄨󵄨 󵄨󵄨 r→0 x∈ℝd 󵄨󵄨] [󵄨󵄨0 x

4.5 Feynman–Kac formula for unbounded semigroups and Stark effect |

335

We can also see that 󵄨󵄨 r 󵄨󵄨 r 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨] x[ [ lim sup 𝔼 󵄨󵄨∫ ∇ ⋅ a(Bs )ds󵄨󵄨 ≤ lim sup 𝔼 ∫ |∇ ⋅ a(Bs )|ds] = 0 󵄨󵄨 󵄨󵄨 r→0 x∈ℝd 󵄨󵄨] r→0 x∈ℝd [󵄨󵄨0 [0 ] x

since ∇ ⋅ a ∈ 𝒦(ℝd ). Thus we find that Kt f (x) is continuous in x. In the case when V is Kato-decomposable, (2) can be proven in the same way as Theorem 4.111. Finally we prove (3). The proof is similar to Theorem 4.116. We have t

x,y

t

(e−tK f )(x) = ∫ Πt (x − y)f (y)𝔼[0,t] [e− ∫0 V(Bs ) ds e−i ∫0 a(Bs )∘dBs ]dy, ℝd

which shows that (4.4.29) holds for all x and almost all y. Next we show the continuity of integral kernel by a similar trick to the proof of Theorem 4.116. Let s = t/3. We have Kt (x, y) = ∫ Ks (x, v)Ks (v, w)Ks (w, y) dw dv = ∫ Ks (x, v)Ks (y, w)Ks (v, w) dw dv. ℝ2d

ℝ2d

The product Ks (x, v)Ks (y, w) = Πs (x − v)Πs (y − w)𝔼[0,s]

(x,v),(y,w)

s

(1)

[e− ∫0 (V(Br

s

s

(1) (1) (2) (2) )+V(B(2) r )) dr −i ∫0 a(Br )∘dBr −i ∫0 a(Br )∘dBr

e

]

is the kernel of the Schrödinger operator in L2 (ℝ2d ): 1 1 K̃ = (−i∇x − a(x))2 + (−i∇y − a(y))2 + V(x) + V(y), 2 2 where B(i) t , i = 1, 2, denote two independent d-dimensional Brownian motions. We see that V(x) + V(y) is Kato-decomposable, and ∇x ⋅ a(x) + ∇y ⋅ a(y) and |a(x)|2 + |a(y)|2 are Kato-class in ℝ2d . On the other hand, the map (v, w) 󳨃→ Ks (v, w) is bounded. Hence Kt (x, y) = (e−t K Ks )(x, y) ̃

with Ks ∈ L∞ (ℝd × ℝd ) and is thus jointly continuous in x and y by statment (2).

4.5 Feynman–Kac formula for unbounded semigroups and Stark effect Throughout Sections 4.2–4.4 the Schrödinger operators H were required to be bounded from below, so that e−tH is bounded for all t ≥ 0. In what follows we derive a Feynman– Kac formula for an unbounded semigroup e−tH with interesting applications.

336 | 4 Feynman–Kac formulae Definition 4.199 (Stark Hamiltonian). Let E ∈ ℝ3 . The operator 1 H(E) = − Δ + E ⋅ x + V(x) 2 on L2 (ℝ3 ) is called Stark Hamiltonian. The Stark Hamiltonian describes the interaction between a charged quantum particle and an external electrostatic field E, leading to a shift and split-up of the particle’s 1 spectral lines. When V(x) = − |x| ,the operator H(0) has a point spectrum. In contrast, the spectrum of H(E) is the whole of ℝ when E ≠ 0; in particular, it has no point spectrum at all. Lemma 4.200 (Faris–Levine). Let V and W be real-valued measurable functions on ℝd such that V(x) ≥ −c|x|2 − d, with c, d ≥ 0, and V ∈ L2loc (ℝd ). Suppose that (1) there exists a dense subset D such that D ⊂ D(−(1/2)Δ) ∩ D(V) ∩ D(W), xj D ⊂ D, and 𝜕j D ⊂ D so that − 21 Δ + V + W + 2c|x|2 is essentially self-adjoint on D; (2) − a2 Δ + W is bounded from below on D for some a < 1. Then − 21 Δ + V + W is essentially self-adjoint on D.

Two immediate consequences of Lemma 4.200 are as follows. Corollary 4.201. Let V1 and V2 be such that (1) V2 < − 21 Δ with a relative bound less than 1; (2) V1 ≥ −c|x|2 − d for some c and d, and V1 ∈ L2loc (ℝd ).

Then − 21 Δ + V1 + V2 is essentially self-adjoint on C0∞ (ℝd ). Proof. We apply Lemma 4.200. Let V = −c|x|2 − d and W = V1 + V2 + c|x|2 + d. It trivially holds that V ≥ −c|x|2 − d and C0∞ ⊂ D(−(1/2)Δ) ∩ D(V) ∩ D(W). The operator − 21 Δ+V +W +2c|x|2 = − 21 Δ+(V1 +2c|x|2 )+V2 is essentially self-adjoint on C0∞ (ℝd ), since by assumption (1) above V1 +2c|x|2 +d > 0 and V1 +2c|x|2 ∈ L2loc (ℝd ); see Corollary 4.57. Moreover, −(a/2)Δ + W is bounded from below. Hence by Lemma 4.200 the statement follows. The second corollary covers the Stark Hamiltonian. Corollary 4.202. Let V < − 21 Δ with a relative bound less than 1. Then the Stark Hamiltonian H(E) is essentially self-adjoint on C0∞ (ℝ3 ). Lemma 4.203. Consider the Coulomb potential V(x) = −1/|x|. Then for all E ∈ ℝ3 , inf Spec(H(E)) = −∞. In particular, e−tH(E) is unbounded. Proof. Let H0 (E) = − 21 Δ + E ⋅ x. Since V < 0, it suffices to show that Spec(H0 (E)) = ℝ. By rotation symmetry this can be reduced to considering the case E = (ε, 0, 0). Let x = (x0 , x⊥ ) ∈ ℝ × ℝ2 and p = (p0 , p⊥ ) be its conjugate momentum. The Hamiltonian 3 becomes H0 (E) = 21 (p20 + p2⊥ ) + εx0 . Let U‖ = ei(1/6ε)p0 and F⊥ denote Fourier transform

4.5 Feynman–Kac formula for unbounded semigroups and Stark effect |

337

with respect to x⊥ . We see that U = F⊥ U‖ : L2 (ℝx × ℝ2k⊥ ) → L2 (ℝ3 ) maps S (ℝ3 ) onto itself. Furthermore, we have 1 U −1 H0 (E)U = k⊥2 + εx0 2

(4.5.1)

on S (ℝ3 ). Since S (ℝ3 ) is a core of both H0 (E) and the multiplication operator on the right-hand side of (4.5.1), H0 (E) and T = 21 k⊥2 +ϵx0 are unitary equivalent as self-adjoint operators. Since Spec(T) = ℝ, the lemma follows. Next we establish a Feynman–Kac formula for a class of unbounded semigroups including in particular the Stark Hamiltonian. Recall that H = − 21 Δ + V. Our assumption on V is that for every ε > 0 there exists Cε such that V(x) ≥ −ε|x|2 − Cε . It is useful to rewrite the Feynman–Kac formula by using Brownian bridge starting from a at t = 0 and ending in b at t = T, i. e., the solution of the stochastic differential equation dXt =

b − Xt dt + dBt , T −t

0 ≤ t < T,

X0 = a.

(4.5.2)

This equation can be solved exactly to obtain t

Xt = a (1 −

t t 1 ) + b + (T − t) ∫ dB , T T T −s s

t < T,

(4.5.3)

0

see Example 2.161. Note that Xt → b almost surely as t → T. Hence, for continuous V and bounded from below, (f , e−tH g) =

∫ f (x)g(y)Q(x, y; V, t)Πt (x − y)dxdy,

(4.5.4)

ℝd ×ℝd

where by (4.5.3) we have t

s s Q(x, y; V, t) = 𝔼 [exp (− ∫ V ((1 − ) x + y + αs ) ds)] , t t 0 [ ]

(4.5.5)

and αs is a Gaussian random variable defined by αs = {

s 1 dBr , t−r

(t − s) ∫0 0,

Note that 𝔼[αs ] = 0 and 𝔼[αs αs󸀠 ] = s(1 − section we prove the following theorem.

s󸀠 ) t

s < t, s = t.

for s ≤ s󸀠 ≤ t. In the remainder of this

Theorem 4.204 (Feynman–Kac formula for unbounded semigroup). Assume that V is continuous and for every ε > 0 there exists Cε such that V(x) ≥ −ε|x|2 − Cϵ . Let f , g ∈ L2 (ℝd ) have compact support. Then f , g ∈ D(e−tH ) and (4.5.4) holds.

338 | 4 Feynman–Kac formulae In order to prove this theorem we first need the following lemma. Lemma 4.205. Let An and A be self-adjoint operators such that An → A in strong resolvent sense. Also, let f be a continuous function and ψ ∈ D(f (An )) for all n ∈ ℕ. Then we have the following. (1) If supn∈ℕ ‖f (An )ψ‖ < ∞, then ψ ∈ D(f (A)). (2) If supn∈ℕ ‖f (An )2 ψ‖ < ∞, then f (An )ψ → f (A)ψ in the strong sense as n → ∞. Proof. (1) Let m, { { { fm (x) = {f (x), { { {−m,

f (x) ≥ m,

|f (x)| ≤ m,

f (x) ≤ −m.

As fm is bounded, fm (An ) → fm (A) strongly as n → ∞. Since ‖fm (A)ψ‖ = lim ‖fm (An )ψ‖ ≤ sup ‖fm (An )‖ ≤ sup ‖f (An )‖, n→∞

n∈ℕ

n∈ℕ

we have supm∈ℕ ‖fm (A)ψ‖ < ∞. This implies ψ ∈ D(f (A)). (2) Since ‖(f (An ) − fm (An ))ψ‖ ≤ m−1 ‖f (An )2 ψ‖, it follows that fm (An )ψ → f (An )ψ, uniformly in n. Then f (An )ψ → f (A)ψ since fm (An )ψ → fm (A)ψ. Proof of Theorem 4.204. Let Vn = max{V(x), −n} and Hn = − 21 Δ + Vn . Since C0∞ (ℝd ) is a common core of Hn , it follows that Hn → H as n → ∞ in strong resolvent sense. For f ∈ L2 (ℝd ) with compact support we prove sup ‖e−tHn f ‖ < ∞.

(4.5.6)

n∈ℕ

Note that t

s

t

s

s

2

s

e− ∫0 Vn ((1− t )x+ t y+αs ) ≤ e∫0 ε((1− t )x+ t y+αs ) +Cε ds ≤ eCε t+2εt(x

2

By Jensen’s inequality e

t

2εt ∫0 αs2 dst

t

0

and therefore t

𝔼[e

2εt ∫0 αs2 dst

2

≤ ∫ e2εtαs

t

ds , t

2

] ≤ ∫ 𝔼[e2εtαs ] 0

ds . t

A direct computation gives 2

𝔼[e2εtαs ] =

2 2 1 1 , ∫ e2εty e−y /2κs dy = √2πκs √π(1 − 4εtκs )

ℝ

t

+y2 )+2εt ∫0 αs2 dst

.

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 339

where κs = s(1 − st ) is the covariance of αs , and we assume ε < 1. Set ε = δ/t 2 with some δ < 1. Since κs ≤ κt/2 = t/4, we have Q(x, y; V, t) ≤

e Cε t

√π(1 − δ)

e2δ(x

2

+y2 )/t

(4.5.7)

.

Hence ‖e−tHn f ‖2 ≤

e2Cε t

δ

√π(1 − δ)

∫ |f (x)f (y)|Π2t (x − y)e t (x

2

+y2 )

dxdy < ∞,

ℝd ×ℝd

uniformly in n. Thus (4.5.6) follows. By Lemma 4.204 (1) we have f ∈ D(e−tH ) and e−tHn f → e−tH f strongly as n → ∞. Thus the left-hand side of (4.5.4) converges, and so does the right-hand side by (4.5.7) and the dominated convergence theorem.

4.6 Feynman–Kac formula for relativistic Schrödinger operators 4.6.1 Relativistic Schrödinger operator Relativity theory says that in the high-energy regime the kinetic energy of a body in motion should be proportional to the absolute value of its momentum p rather than its square. In fact, the energy of a classical relativistic particle of mass m is √|p|2 c2 + m2 c4 , where c is the speed of light, and this justifies the definition of the relativistic kinetic operator √−ℏ2 c2 Δ + m2 c4 − mc2 .

(4.6.1)

Since the constant mc2 is subtracted, the bottom of its spectrum is zero. Using a basic convention in mathematical physics, from now on we choose units in which c = 1 = ℏ. In this section we consider Feynman–Kac formulae for relativistic Schrödinger operators of a more general form. Definition 4.206 (Relativistic Schrödinger operator with vector potential). Suppose that a ∈ (L2loc (ℝd ))d and V ∈ L∞ (ℝd ). Then relativistic Schrödinger operator with vector potential a is defined by HR (a) = (2H 0 (a) + m2 )1/2 − m + V,

(4.6.2)

where H 0 (a) is the self-adjoint operator given in Definition 4.188 with V = 0. Note that the square-root term in the definition above is given through the spectral projection of the self-adjoint operator H 0 (a). Denote by HR0 (a) the operator HR (a) with V = 0.

340 | 4 Feynman–Kac formulae Proposition 4.207. (1) Suppose that a ∈ (L4loc (ℝd ))d and ∇ ⋅ a ∈ L2loc (ℝd ). Then C0∞ (ℝd ) is an operator core of HR0 (a). (2) Suppose that a ∈ (L2loc (ℝd ))d . Then C0∞ (ℝd ) is a form core of HR0 (a). Proof. For simplicity, in this proof we write HR0 (a) = HR and H 0 (a) = H. (1) First note that there exist nonnegative constants c1 and c2 such that ‖HR f ‖ ≤ c1 ‖Hf ‖ + c2 ‖f ‖

(4.6.3)

for all f ∈ D(H). Hence C0∞ (ℝd ) is contained in D(HR ). Since HR is a nonnegative selfadjoint operator, HR + 1 has a bounded inverse, and we use the fact that C0∞ (ℝd ) is a core of HR if and only if HR C0∞ (ℝd ) is dense in L2 (ℝd ). Let g ∈ L2 (ℝd ) and suppose that (g, (HR + 1)f ) = 0, for all f ∈ C0∞ (ℝd ). Then C0∞ (ℝd ) ∋ f 󳨃→ (g, HR f ) = −(g, f ) defines a continuous functional which can be extended to L2 (ℝd ). Thus g ∈ D(HR ) and 0 = ((HR + 1)g, f ). Since C0∞ (ℝd ) is dense, we have (HR + 1)g = 0, and hence g = 0 since HR + 1 is one-to-one, proving the assertion. (2) Note that ‖HR1/2 f ‖2 ≤ c1 ‖H 1/2 f ‖2 + c2 ‖f ‖2 for f ∈ Q(h) = D(H 1/2 ), and C0∞ (ℝd ) is contained in Q(HR ) = D(HR1/2 ). Since HR1/2 + 1 has also bounded inverse, it is seen by the same argument as above that C0∞ (ℝd ) is a core of HR1/2 or a form core of HR . A key element in deriving the Feynman–Kac formula is a use of the Lévy subordinator studied in Example 3.90. In the notation of the example, we choose δ = 1, γ = m so that we have the subordinator Tt = Tt (1, m) on a suitable probability space (𝒯 , ℬ𝒯 , P) such that 𝔼0P [e−uTt ] = e−t(

√2u+m2 −m)

(4.6.4)

.

For simplicity, we write 𝔼x,0 for 𝔼x,0 𝒲×P . Theorem 4.208 (Feynman–Kac formula for relativistic Schrödinger operator with vector potential). Suppose that V ∈ L∞ (ℝd ), a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Then Tt

(f , e−tHR (a) g) = ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0

t

a(Bs )∘dBs − ∫0 V(BTs )ds

e

]dx.

(4.6.5)

ℝd

Proof. We divide the proof in four steps. Step 1: Suppose V = 0. Then Tt

0

(f , e−tHR (a) g) = ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0 ℝd

a(Bs )∘dBs

]dx.

(4.6.6)

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 341

Denote by Eλ the spectral projection of the self-adjoint operator H 0 (a). We have 0

(f , e−tHR (a) g) = ∫ e−t(

√2λ+m2 −m)

d(f , Eλ g).

(4.6.7)

[0,∞)

By (4.6.4) we obtain (f , e

−tHR0 (a)

∞

g) = ∫ 𝔼0P [e−λTt ]d(f , Eλ g) = 𝔼0P [(f , e−Tt H

0

(a)

g)].

0

Equality (4.6.6) follows from inserting the Feynman–Kac–Itô formula for e−Tt H Tt

0

(f , e−Tt HR (a) g) = ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0

a(Bs )∘dBs

0

(a)

, i. e.,

(4.6.8)

]dx,

ℝd

discussed in Section 4.4.3. Step 2: Let 0 = t0 < t1 < . . . < tn and f0 , fn ∈ L2 (ℝd ) and fj ∈ L∞ (ℝd ) for j = 1, . . . , n − 1. We prove next that n

n

0

Tt

(f0 , ∏ e−(tj tj−1 )HR (a) fj ) = ∫ 𝔼x,0 [f (B0 )( ∏ fj (BTt ))e−i ∫0 j=1

j=1

ℝd

a(Bs )∘dBs

j

(4.6.9)

]dx.

0

We use the shorthand notation Gj (x) = fj (x)(∏ni=j+1 e−(ti −ti−1 )HR (a) fi )(x). The left-hand side of (4.6.9) can be written as Tt −t 1 0

∫ 𝔼x,0 [f (B0 )e−i ∫0

a(Bs )∘dBs

G1 (BTt −t )]dx. 1

ℝd

0

By the Markov property of (Bt )t≥0 we have n

0

(f0 , ∏ e−(tj −tj−1 )HR (a) fj ) j=1

Tt

= ∫ 𝔼x,0 [f (B0 )e−i ∫0

1

a(Bs )∘dBs

B Tt

Tt −t 2 1

𝔼0P 𝔼𝒲 1 [f1 (B0 )e−i ∫0

a(Bs )∘dBs

G2 (BTt −t )]]dx 2

ℝd Tt

= ∫ 𝔼x,0 [f (B0 )e−i ∫0

1

a(Bs )∘dBs

𝔼0P [f1 (BTt )e 1

ℝd

Tt −t +Tt 1 2 1 t1

−i ∫T

a(Bs )∘dBs

G2 (BTt

1

1

+Tt2 −t1 )]]dx.

The right-hand side above can be rewritten as Tt

∫ 𝔼x,0 [f (B0 )e−i ∫0 ℝd

1

a(Bs )∘dBs

Tt

Tt −t 2 1

f1 (BTt )𝔼P 1 [e−i ∫0 1

a(Bs )∘dBs

G2 (BTt −t )]]dx. 2

1

342 | 4 Feynman–Kac formulae By the Markov property of (Tt )t≥0 again we can see that Tt

= ∫ 𝔼x,0 [f (B0 )e−i ∫0 ℝd

Tt

= ∫ 𝔼x,0 [f (B0 )e−i ∫0

1

a(Bs )∘dBs

f1 (BTt )e

Tt

−i ∫T

1

2

a(Bs )∘dBs

2 t1

a(Bs )∘dBs

G2 (BTt )]dx 2

f1 (BTt )G2 (BTt )]dx. 1

ℝd

2

Continuing by the above procedure, (4.6.9) is obtained. Step 3: Suppose now that V ∈ L∞ and it is continuous. We show (4.6.5) under this condition. By the Trotter product formula, 0

(f , e−tHR (a) g) = lim (f , (e−(t/n)HR (a) e−(t/n)V )n g) n→∞

Tt

= lim ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0 n→∞

n a(Bs )∘dBs − ∑j=1 (t/n)V(BTtj/n )

e

]dx

ℝd

= r. h. s. (4.6.9). Here we used the fact that for each path (ω, τ) the function V(BTs (τ) (ω)) is continuous in s except for at most finitely many points. Note that Ts almost surely has no accumulation points in any compact interval. Thus t

n

t ∑ V(BTtj/n ) → ∫ V(BTs )ds n j=1 0

as n → ∞, for each path as a Riemann integral. This completes the proof of the claim. Step 4: Finally, to complete the proof of the theorem, define Vn in the same manner as in the proof of Theorem 4.235. Note that Vn is bounded and continuous, and moreover, Vn (x) → V(x) as n → ∞ for x ∉ N , where N is a set of Lebesgue measure zero. Note that 𝔼x,0 [1N (BTs )] = 𝔼0 [1{Ts >0} ∫ 1N (x + y)ks (y)dy] + 1N (x)𝔼0 [1{Ts =0} ] = 0 ℝd

for x ∈ N , where ks (x) is the distribution of the random variable BTs given by (4.3.6). Hence t

0 = ∫𝔼 0

x,0

[1N (BTs )]ds = 𝔼

x,0 [

t

∫ 1N (BTs )ds] . [0 ]

For a.e. (ω, τ) the Lebesgue measure of {t ∈ [0, ∞) | BTt (τ) (ω) ∈ N } is zero. Therefore t

t

∫0 Vn (BTs )ds → ∫0 V(BTs )ds as n → ∞ for a.e. (ω, τ). Moreover, Tt

∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0 ℝd

t

a(Bs )∘dBs − ∫0 Vn (Bs )ds

e

]dx

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 343 Tt

→ ∫ 𝔼x,0 [f (B0 )g(BTt )e−i ∫0

t

a(Bs )∘dBs − ∫0 V(Bs )ds

e

]dx

ℝd 0

0

as n → ∞. On the other hand, e−t(HR (a)+Vn ) → e−t(HR (a)+V) strongly as n → ∞, since HR0 (a) + Vn converges to HR0 (a) + V on the common domain D(HR0 (a)). The Feynman–Kac formula of e−tHR (a) can be extended to more singular external potentials. The key idea is similar to that applied in the case of e−tH(a) discussed in Theorem 4.88. Lemma 4.209. Let a ∈ (L2loc (ℝd ))d , ∇ ⋅ a ∈ L1loc (ℝd ), and let V be a real-valued multiplication operator. (1) Suppose V is relatively form-bounded with respect to √−Δ + m2 − m with relative bound b. Then V is also relatively form-bounded with respect to HR0 (a) with a relative bound not larger than b. (2) Suppose V is relatively bounded with respect to √−Δ + m2 −m with relative bound b. Then V is also relatively bounded with respect to HR0 (a) with a relative bound not larger than b. Proof. In virtue of Theorem 4.208 we have 0

|(f , e−tHR (a) g)| ≤ (|f |, e−t(

√−Δ+m2 −m)

|g|).

(4.6.10)

The proof is similar to Lemma 4.192. Let V be such that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to √−Δ + m2 − m with a relative bound strictly smaller than one. The KLMN theorem and Lemma 4.209 can be used to define HR0 (a) +̇ V+ −̇ V− as a self-adjoint operator. Definition 4.210 (Relativistic Schrödinger operator with singular external potential and singular vector potential). Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Let V be such that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to √−Δ + m2 −m with a relative bound strictly smaller than one. Then the relativistic Schrödinger operator with external potential V and vector potential a is defined by HR (a) = HR0 (a) +̇ V+ −̇ V− .

(4.6.11)

Theorem 4.211 (Feynman–Kac formula for relativistic Schrödinger operator with singular external potential and singular vector potential). Suppose a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Let V be such that V+ ∈ L1loc (ℝd ) and V− is relatively form-bounded with respect to √−Δ + m2 − m with a relative bound strictly smaller than one. Then the Feynman–Kac representation of (f , e−tHR (a) g) is given by (4.4.3). Proof. The proof is similar to the limiting argument in Theorem 4.194.

344 | 4 Feynman–Kac formulae Corollary 4.212 (Diamagnetic inequality). Under the same assumptions as in Theorem 4.211 we have |(f , e−tHR (a) g)| ≤ (|f |, e−tHR (0) |g|).

(4.6.12)

We can also show by using the Feynman–Kac formula that e−tHR (0) , with a = 0, is positivity improving. Theorem 4.213 (Positivity improving). Suppose that V+ ∈ L1loc (ℝd ) and V− is relatively form bounded with respect to √−Δ + m2 − m with a relative bound strictly smaller than one. Then {e−tHR (0) : t ≥ 0} is positivity improving. Proof. Since {e−t(−Δ/2) : t ≥ 0} is positivity improving and e−t( {e

−t √−Δ+m2

√−Δ+m2 −m)

: t ≥ 0} is also positivity improving and we have ∫ℝd 𝔼

for every non-negative f , g. For V+ ∈ t

x,0

= 𝔼[e−Tt (−Δ/2) ],

[f (B0 )g(BTt )]dx > 0 1 d Lloc (ℝ ), in a similar way to Lemma 4.95 it can be d

seen that e− ∫0 V(BTs +x)ds > 0 for almost every (x, ω, ϖ) ∈ ℝ × X × T . The measure of M = {(x, ω, ϖ) ∈ ℝd × X × T | f (x)g(BTt (ϖ) (ω) + x) > 0} is positive and we have t

(f , e−tHR (0) g) ≥ ∫ f (x)g(BTt + x)e− ∫0 V(BTs +x)ds dxd𝒲 dP > 0. M

Thus e−tHR (0) is positivity improving for all t ≥ 0. 4.6.2 Relativistic Kato-class potentials In Section 4.6.1 we constructed the Feynman–Kac formula of e−tHR (a) . Similarly to K(a), we can define the relativistic Schrödinger operator KR (a) as the generator of a C0 -semigroup defined through Feynman–Kac formula with general external potentials. Let pRt (x) =

1 √ 2 2 ∫ e−ix⋅ξ e−t( |ξ | +m −m) dξ d (2π) ℝd

and define ∞

gR (x) = ∫ e−t pRt (x)dt. 0

gR (x) gives the kernel of (√−Δ + m2 − m + 1)−1 , i. e., (f , (√−Δ + m2 − m + 1)−1 h) =

∫ f ̄(x)gR (x − y)h(y)dxdy ℝd ×ℝd

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 345

holds. Formally gR (x) can be represented as gR (x) =

1 e−ik⋅x dk, ∫ (2π)d √|k|2 + m2 − m + 1

(4.6.13)

ℝd

but it is not clear that the right-hand side of (4.6.13) is finite. In order to give a clear meaning to this, we need to define the right-hand side of (4.6.13) in the sense of L2 . We introduce three classes of external potentials given by {

d

t

󵄨󵄨 󵄨 󵄨

x,0

}

𝒱1 = {V : ℝ → ℝ 󵄨󵄨󵄨 lim sup ∫ 𝔼𝒲×P [V(BTs )]ds = 0} , 󵄨 t↓0 d

x∈ℝ 0 } 󵄨󵄨 󵄨󵄨 d −1 𝒱2 = {V : ℝ → ℝ 󵄨󵄨 lim sup ((√−Δ + m2 − m + λ) V) (x) = 0} , 󵄨󵄨 λ→∞ x∈ℝd

{

{ { { {

d

󵄨󵄨 󵄨 󵄨󵄨

} } gR (x − y)V(y)dy = 0} ; } δ↓0 x∈ℝd |x−y| 0.

(4.6.19)

In particular, (4.6.19) is continous in x for every r > 0. Proof. We have e−r(

√−Δ+m2 −m)

t−r

R Kt−r f (x) = 𝔼x,0 [𝔼BTr ,0 [e− ∫0 t−r

T

V(BTs )ds −i ∫0 t−r a(Bs )∘dBs

e

Tt−r (ϖ)

f (BTt−r )]].

a(Bs )∘dBs For a fixed ϖ consider 𝔼BTr (ϖ) [e− ∫0 V(BTs (ϖ) )ds e−i ∫0 f (BTt−r (ϖ) )]. For simplicity write Tt for Tt (ϖ). The Markov property of Brownian motion yields t−r

𝔼BTr [e− ∫0

T

V(BTs )ds −i ∫0 t−r a(Bs )∘dBs

e

f (BTt−r )]

348 | 4 Feynman–Kac formulae t−r

= 𝔼x [e− ∫0

T

t−r V(BTs +Tr )ds −i ∫T

e

r

+Tr

a(Bs )∘dBs

f (BTt−r +Tr )|ℱTt ] ,

which implies that t−r

𝔼x [𝔼BTr [e− ∫0 = 𝔼x [e

t−r − ∫0

T

V(BTs )ds −i ∫0 t−r a(Bs )∘dBs

e

V(BTs +Tr )ds

e

t

= 𝔼x [e− ∫r V(BTs−r +Tr )ds e

T +T −i ∫T t−r r r T

−i ∫T t−r r

+Tr

f (BTt−r )]]

a(Bs )∘dBs

f (BTt−r +Tr )]

a(Bs )∘dBs

f (BTt−r +Tr )].

For a fixed r we notice that Tt = Tt −Tr +Tr =T ̇ t−r +Tr for all t ≥ r. Taking the expectation with respect to the subordinator, we have t−r

𝔼x,0 [𝔼BTr ,0 [e− ∫0 = 𝔼x,0 [e

t − ∫r

T

V(BTs )ds −i ∫0 t−r a(Bs )∘dBs

e

V(BTs )ds

e

T −i ∫T t r

a(Bs )∘dBs

f (BTt−r )]]

f (BTt )]

which shows the lemma. Next we introduce the following condition. We say that a measurable function f on ℝd is in ℒ(ℝd ) if Tr

lim sup 𝔼x,0 [∫ |f (Bs )|ds] = 0. r→0 x∈ℝd [0 ] Proposition 4.222 (Lp -Lq boundedness and continuity of integral kernel). Suppose that V is relativistic Kato-decomposable and a ∈ (L2loc (ℝd ))d with ∇ ⋅ a ∈ L1loc (ℝd ). Let 1 ≤ p ≤ q ≤ ∞. Then the following hold: (1) e−tKR (a) is Lp -Lq bounded, for every t ≥ 0; (2) If |a|2 , ∇ ⋅ a ∈ ℒ(ℝd ) and V is uniformly locally integrable, then KtR f (x) is continuous in x for f ∈ Lp (ℝd ) with 1 ≤ p ≤ ∞; (3) If |a|2 , ∇ ⋅ a ∈ ℒ(ℝd ) and V is uniformly locally integrable, then the integral kernel ℝd × ℝd ∋ (x, y) 󳨃→ KtR (x, y) is jointly continuous in x and y, and is given by x,y

t

Tt

KtR (x, y) = pXt (x − y)𝔼[0,t] [e− ∫0 V(BTs ) ds e−i ∫0

a(Bs )∘dBs

].

(4.6.20)

x,y

Here P[0,t] is the Lévy bridge measure for (BTt )t≥0 .

(4) Let a = 0 and V be uniformly locally integrable. Then e−tKR (0) is positivity improving. Proof. By the diamagnetic inequality we have |e−tKR (a) f (x)| ≤ e−tKR (0) |f |(x). Part (1) can be proven similarly to Theorem 4.107 with Brownian motion (Bt )t≥0 replaced by the subordinate Brownian motion (BTt )t≥0 . Claim (2) is also proven by a minor modification of the proof of Theorem 4.111. It suffices to consider f ∈ L∞ (ℝd ). Let xn → x as n → ∞. We have

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 349

KtR f (x) − KtR f (xn ) = KtR f (x) − e−r( + e−r(

√−Δ+m2 −m)

R Kt−r f (x) − e−r(

√−Δ+m2 −m)

√−Δ+m2 −m)

R Kt−r f (x)

R Kt−r f (xn ) + e−r(

√−Δ+m2 −m)

R Kt−r f (xn ) − KtR f (xn ).

By the smoothing effect in Proposition 4.220 we see that e−r(

√−Δ+m2 −m)

R Kt−r f (x) − e−r(

√−Δ+m2 −m)

R Kt−r f (xn ) → 0

as n → ∞, for every r > 0. It is sufficient to show that KtR f (x) − e−r(

√−Δ+m2 −m)

R Kt−r f (x) → 0

as r → 0, uniformly in x. Let V ∈ 𝒦R (ℝd ) and let gr (x) = e−r( r − ∫0

T (ϖ) −i ∫0 r

a(Bs )∘dBs independence of 1−e V(BTs (ϖ) ) ds e and e for fixed ϖ, and by Lemma 4.221 we have

t − ∫r

√−Δ+m2 −m)

V(BTs (ϖ) ) ds

e

R Kt−r f (x). By the

T (ϖ) −i ∫T t(ϖ) r

a(Bs )∘dBs

f (Bt )

T r Tr 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 t −i ∫ t a(B )∘dB ‖gr − Kt f ‖∞ ≤ sup 𝔼x,0 [󵄨󵄨󵄨󵄨1 − e− ∫0 V(BTs ) ds e−i ∫0 a(Bs )∘dBs 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨e− ∫r V(BTs ) ds e Tr s s f (BTt )󵄨󵄨󵄨󵄨] 󵄨 󵄨󵄨 󵄨 x∈ℝd Tr r 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ B sup (𝔼x,0 [󵄨󵄨󵄨󵄨1 − e−i ∫0 a(Bs )∘dBs 󵄨󵄨󵄨󵄨] + 𝔼x,0 [󵄨󵄨󵄨1 − e− ∫0 V(BTs ) ds 󵄨󵄨󵄨] ). 󵄨 󵄨 󵄨 󵄨 d

x∈ℝ

t

Here B = ‖f ‖∞ supx∈ℝd supr≤t 𝔼x,0 [e− ∫r V(BTs ) ds ]. Since V is uniformly locally integrable, t

lim sup ∫ 𝔼x,0 [V(BTs )]ds = 0 t↓0 x∈ℝd

(4.6.21)

0

r

follows. We see then that supx∈ℝd 𝔼x,0 [1 − e− ∫0 V(BTs ) ds ] → 0 as r → 0, in the same way as in the proof of Theorem 4.111. For the second term we have 𝔼

x,0

󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 1 󵄨 󵄨󵄨 −i ∫ a(B )∘dB x,0 x,0 󵄨 󵄨 󵄨 s s 󵄨󵄨 [󵄨󵄨󵄨󵄨1 − e 0 󵄨󵄨] ≤ 𝔼 [󵄨󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨] + 𝔼 [ 󵄨󵄨󵄨∫ ∇ ⋅ a(Bs )ds󵄨󵄨󵄨] . 2 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨] 󵄨󵄨] [󵄨󵄨 0 [ 󵄨󵄨 0

For each Tr = Tr (ϖ), by Schwarz inequality and the Itô-isometry we have 1/2 1/2 󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 Tr 󵄨󵄨2 Tr 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨] 󵄨󵄨 ] x [󵄨󵄨 x [󵄨󵄨 x[ 2 ] 𝔼 󵄨󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨 ≤ (𝔼 [󵄨󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨 ]) = (𝔼 ∫ |a(Bs )| ds ) . 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨] 󵄨󵄨 ] [󵄨󵄨󵄨 0 [0 ] [󵄨󵄨 0

Since |a|2 ∈ ℒ(ℝd ), it follows that lim sup 𝔼

r→0 x∈ℝd

󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ a(Bs )dBs 󵄨󵄨󵄨] = 0. 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨] [󵄨󵄨 0 󵄨

x,0 [󵄨󵄨

350 | 4 Feynman–Kac formulae We can also see that lim sup 𝔼

r→0 x∈ℝd

󵄨󵄨 Tr 󵄨󵄨 Tr 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ ∇ ⋅ a(Bs )ds󵄨󵄨󵄨] ≤ lim sup 𝔼x,0 [∫ |∇ ⋅ a(Bs )|ds] = 0 󵄨󵄨 󵄨󵄨 󵄨󵄨] r→0 x∈ℝd [󵄨󵄨󵄨 0 [0 ] 󵄨

x,0 [󵄨󵄨

since ∇ ⋅ a ∈ ℒ(ℝd ). Thus KtR f (x) is continuous in x. Next, let V = V+ − V− be relativistic Kato-decomposable, and consider for R > 0 the function VR = V+R − V− with V+R (x) = V+ (x) if |x| ≤ R and V+R (x) = 0 if |x| > R. We have VR ∈ 𝒦R (ℝd ). Let M ⊂ BR (0) be compact, where BR (0) denotes the ball of radius R centered at the origin. Hence t Tt t Tt 󵄨󵄨 󵄨󵄨 sup 󵄨󵄨󵄨𝔼x,0 [e− ∫0 V(Bs ) ds e−i ∫0 a(Bs )∘dBs f (Bt )] − 𝔼x,0 [e− ∫0 VR (Bs ) ds e−i ∫0 a(Bs )∘dBs f (Bt )]󵄨󵄨󵄨 󵄨 󵄨 x∈M t

≤ 2‖f ‖∞ sup 𝔼x,0 [e∫0 V− (BTs ) ds 1{sup0≤s≤t |BT x∈M

t

≤ 2‖f ‖∞ sup (𝔼x,0 [e−2 ∫0 V− (Bs ) ds ])

1/2

x∈M

s

|≥R} ]

(𝔼x,0 [1{sup0≤s≤t |BT

s

1/2 . |≥R} ])

The first factor above is bounded since V− is relativistic Kato-class, and the second goes to zero as R → ∞ by Lévy’s maximal inequality 𝔼x,0 [1{sup0≤s≤t |BT

s

|≥R} ]

= 𝔼0 𝔼x [1{sup0≤s≤t |BT

s

|≥R} ]

≤ 2𝔼0 𝔼x [1{|BT |>R} ]. t

Hence KtR f (x) is continuous in x. Finally, to prove (3) we again proceed similarly to Theorem 4.116. We have x,y

t

Tt

(e−tKR f )(x) = ∫ pXt (x − y)f (y)𝔼[0,t] [e− ∫0 V(BTs ) ds e−i ∫0

a(Bs )∘dBs

]dy,

ℝd

which shows that (4.4.29) holds for all x and almost all y. Next we show the continuity of integral kernel. Let s = t/3. We have KtR (x, y) = ∫ KsR (x, v)KsR (v, w)KsR (w, y) dw dv = ∫ KsR (x, v)KsR (y, w)KsR (v, w) dw dv. ℝ2d

ℝ2d

The product KsR (x, v)KsR (y, w) = pXs (x − v)pXs (y − w)𝔼[0,s]

(x,v),(y,w)

s

Ts

[e− ∫0 (V(BTr )+V(BTr )) dr e−i ∫0 (1)

(2)

T

s (1) (2) (2) a(B(1) r )∘dBr −i ∫0 a(Br )∘dBr

is the kernel of the relativistic Schrödinger operator in L2 (ℝ2d ): K̃ R = √(−i∇x − a(x))2 + m2 − m + √(−i∇y − a(y))2 + m2 − m + V(x) + V(y),

]

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 351

where B(i) t , i = 1, 2, denote two independent d-dimensional Brownian motions. We see that V(x) + V(y) is relativistic Kato-decomposable, and ∇x ⋅ a(x) + ∇y ⋅ a(y) and |a(x)|2 +|a(y)|2 are in ℒ(ℝ2d ). On the other hand, the map (v, w) 󳨃→ KsR (v, w) is bounded. Hence KtR (x, y) = (e−t KR KsR )(x, y) with KsR ∈ L∞ (ℝ2d ) and is jointly continuous in x and ̃

t

y by statement (2). Finally, to show (4) note that (4.6.21) we have e− ∫0 V(BTs )ds > 0 a.e. for sufficiently small t. Hence (f , e−tHR (0) g) > 0 for every non-negative f and g for all t > 0 by the semigroup property. 4.6.3 Decay of eigenfunctions In this subsection we study the decay of eigenfunctions of relativistic Schrödinger operators. In Section 4.3.8 we have discussed the exponential decay of eigenfunctions of Schrödinger operators via Carmona’s estimate. For relativistic Schrödinger operators the study of the decay of eigenfunctions is more complicated than for classical Schrödinger operators, and our basic tool will be a martingale related to the Feynman– Kac semigroup. For simplicity, we denote Xt = BTt ,

t ≥ 0,

with the subordinator (Tt )t≥0 used above. Recall that (Xt )t≥0 is a Lévy process such that its generator is the self-adjoint operator √−Δ + m2 − m. Let φ be an eigenfunction of the relativistic Schrödinger operator HR (0) such that HR (0)φ = Eφ.

(4.6.22)

Also, as above, we have t

e−tHR (0) f (x) = 𝔼0 [f (Xt )e− ∫0 V(Xs +x)ds ]. Consider the random process (Mt (x))t≥0 with t

Mt (x) = eEt e− ∫0 V(Xs +x)ds φ(Xt + x),

t ≥ 0.

(4.6.23)

Then it is seen that φ(x) can be represented in terms of (Mt (x))t≥0 as φ(x) = (e−t(HR (0)−E) φ)(x) = 𝔼0 [Mt (x)] = 𝔼x [Mt (0)]

(4.6.24)

for a. e. x ∈ ℝd and for all t ≥ 0. Lemma 4.223 (Martingale properties). (Mt (x))t≥0 is a martingale with respect to the natural filtration of (Xt )t≥0 .

352 | 4 Feynman–Kac formulae Proof. Let Fs = σ(Xr , 0 ≤ r ≤ s). By the Markov property of (Xt )t≥0 we readily see that for t ≥ s s

t

𝔼0 [Mt (x)|Fs ] = eEt e− ∫0 V(Xr +x)dr 𝔼0 [e− ∫s V(Xr )dr φ(Xt + x)|Fs ] s

t−s

= eEt e− ∫0 V(Xr +x)dr 𝔼Xs [e− ∫0 = eEs e

s − ∫0

V(Xr +x)dr (t−s)E

= eEs e

s − ∫0

V(Xr +x)dr

e

V(Xr )dr

𝔼Xs [e

t−s − ∫0

φ(Xt−s + x)] V(Xr )dr

φ(Xt−s + x)]

(e−(t−s)(HR (0)−E) φ) (Xs ) = Ms .

Let τ be a stopping time with respect to the natural filtration (Ft )t≥0 of (Xt )t≥0 . Then (Mt∧τ )t≥0 is also a martingale by Theorem 2.73, which implies by (4.6.24) that φ(x) = 𝔼0 [Mt∧τ (x)].

(4.6.25)

The distribution function pXt of Xt can be expressed in several ways. Let t > 0. (1) It is given by pXt (x)

2

m t 2 2 {2 ( 2π ) (t 2 +|x|2 )(d+1)/4 K(d+1)/2 (m√|x| + t ), ={ d+1 −(d+1)/2 t , {Γ( 2 )π (t 2 +|x|2 )(d+1)/2

m > 0, m = 0,

(4.6.26)

where Kν denotes the modified Bessel function of the third kind (4.2.21). (2) We alternatively have the representation: 2 2 1 t √ 2 2 ∫ emt e− (|x| +t )(|p| +m ) dp. d (2π) √|x|2 + t 2

pXt (x) =

(4.6.27)

ℝd

(3) By subordination, explicitly using the distribution functions of Brownian motion (Bt )t≥0 and of the subordinator (Tt )t≥0 , we also have ∞

2

pXt (x) = emt ∫ e−m u Π2u (x)θt (du),

(4.6.28)

0

where θt (du) = π −1/2 tu−3/2 e−t

2

/(4u)

du.

Lemma 4.224. If V is relativistic Kato-decomposable and uniformly locally integrable, then φ ∈ L∞ (ℝd ). Proof. Since supy∈ℝd pXt (y) < ∞ for t > 0 and V is relativistic Kato-decomposable, we have that 0

|φ(x)| ≤ (𝔼 [e

1/2

1/2

t

−2 ∫0 V(Xr +x)dr

])

( ∫ |φ(y + ℝd

with a constant C.

x)|2 pXt (y)dy)

≤ C‖φ‖

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 353

Now we study the decay properties of the eigenfunction φ. Theorem 4.225 (Confining potential: upper bound). Let V be relativistic Kato-decomposable and uniformly locally integrable. Suppose that lim|x|→∞ V(x) = ∞. Let φ be an eigenfunction corresponding to an eigenvalue E. Then for every positive constant c > 0 there exists a positive constant D(c) such that |φ(x)| ≤ D(c)e−c|x| . Proof. For every R > 0, let τR = inf{t > 0 | |Xt | > R}. τR is a stopping time and we have t∧τR

φ(x) = 𝔼0 [e(t∧τR )E e− ∫0

V(Xs +x)ds

φ(Xt∧τR + x)].

Hence we have |φ(x)| ≤ ‖φ‖∞ 𝔼0 [e−(WR (x)−E)t∧τR ], where WR (x) = inf|y−x|≤R V(y). Note that lim|x|→∞ WR (x) − E = ∞ and WR (x) − E > 0 for large |x|. Thus |φ(x)| ≤ ‖φ‖∞ (𝔼0 [1{τR 0 there exists a constant Dε > 0 such that |φ(x)| ≤ Dε e−mε |x| ,

(4.6.29)

where m mε = { √2m|E| − E 2 − ε

if |E| > m, if |E| ≤ m.

(2) If m = 0, then there exists a constant c > 0 such that |φ(x)| ≤

c . 1 + |x|d+1

(4.6.30)

Proof. Let V = V+ − V− and τR (x) = inf{t > 0 | |Xt + x| ≤ R}, which is a stopping time. For ε > 0 choose R large enough such that sup|y|>R V− (y) < ε, and ε small enough such that q = −(E + ε) > 0. Thus t∧τR (x) 󵄨󵄨 󵄨󵄨 V(Xs +x)ds |φ(x)| = 󵄨󵄨󵄨󵄨𝔼0 [e(t∧τR (x))E e− ∫0 φ(Xt∧τR (x) + x)]󵄨󵄨󵄨󵄨 ≤ ‖φ‖∞ 𝔼0 [e−q(t∧τR (x)) ]. 󵄨 󵄨

Let μqR be the q-capacitary measure of {y ∈ ℝd | |y| ≤ R}. By (3.5.6) we have lim 𝔼0 [e−q(t∧τR (x)) ] = 𝔼0 [e−qτR (x) ] = ∫ uq (x − y)μqR (dy),

t→∞

(4.6.31)

|y|≤R

where q

∞

u (x) = ∫ pXt (x)e−qt dt. 0

Let m > 0. By (4.6.28) we have the upper bounds −√2mq−q2 |x|

Ce { { { |x| d2 −1 q u (x) ≤ { { { Ce−m|x| d { |x| 2 −1

if q ≤ m,

(4.6.32)

otherwise

with a constant C, for all |x| ≥ 1. For |E| ≤ m, from (4.6.32) it follows that 0

𝔼 [e

−qτR (x)

e− {C ∫ ] ≤ { |y|≤R {CR ,

√2mq−q2 |x−y| q μR (dy)

≤ C 󸀠 e−(

√2m|E|−E 2 +ε)|x|

,

|x| ≥ R + 1, |x| < R + 1.

Here C, C 󸀠 , and CR are positive constants. Hence (4.6.29) follows for the case |E| ≤ m. Similarly, we can show (4.6.29) for |E| > m by (4.6.32).

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 355

Let m = 0. It is seen by (4.6.26) that

c1 c2 ≤ uq (x) ≤ d+1 |x|d+1 |x|

(4.6.33)

for |x| > 1 with positive constants c1 and c2 . Together with (4.6.31), this implies 0

𝔼 [e

−qτR (x)

{∫ ] ≤ { |y|≤R { CR ,

c2 μq (dy) |x−y|d+1 R

≤

c2󸀠 , |x|d+1

|x| ≥ R + 1, |x| < R + 1.

Here CR is a constant. Thus (4.6.30) follows. Finally, we show a lower bound on the ground state of HR (0). Theorem 4.227 (Lower bound). Let φ0 be the ground state of HR (0) and continuous. Suppose that V is continuous and lim|x|→∞ V(x) = 0. (1) Let m = 0. Then there exists a constant c > 0 such that φ0 (x) ≥

c . 1 + |x|d+1

(4.6.34)

(2) Let m > 0. Then there exists a constant c > 0 such that φ0 (x) ≥ ce−mε |x| .

(4.6.35)

Proof. Note that φ0 is strictly positive. In a similar way to the proof of Theorem 4.226, with τR (x) = inf{t > 0||Xt + x| ≤ R}, and ε > 0, we choose R is large enough such that sup|y|>R V− (y) < ε and ε small enough such that q = −(E + ε) > 0. We see that t∧τR (x)

V(Xs +x)ds

t∧τR (x)

V+ (Xs +x)ds

φ0 (x) = 𝔼0 [e(t∧τR (x))E e− ∫0 ≥ 𝔼0 [e(t∧τR (x))E e− ∫0 τR (x)

= 𝔼0 [eτR (x)E e− ∫0

V+ (Xs +x)ds

φ0 (Xt∧τR (x) + x)] φ0 (Xt∧τR (x) + x)]

φ0 (XτR (x) + x)]

≥ inf φ0 (y)𝔼0 [e−qτR (x) 1{τR (x) 0. By (4.6.27) we have pXt (x) ≥

1 t √ 2 2 e−m|x| ∫ emt e− |x| +t |p| dp ≥ cd (|x|2 + t 2 )−(d+1)/2 te−m|x| (2π)d √|x|2 + t 2 d ℝ

356 | 4 Feynman–Kac formulae for a positive constant cd > 0. From this we have ∞

∞

uq (x) = ∫ e−qt pXt (x)dt ≥ cd e−m|x| ∫ 0

0

(|x|2

te−qt dt = cn ϕ(|x|)e−m|x| . + t 2 )(d+1)/2

(4.6.37)

Here ϕ(x) is comparable to |x|−d when |x| → ∞. This lower bound improves for q ≤ m. From (4.6.28) we have √2mq−q2 |x|

uq (x) ≥ ed ψ(|x|)e−

(4.6.38)

with a constant ed and a function ψ(x) which is comparable to |x|−d when |x| → ∞. The lower bound for m > 0 follows from (4.6.38) and (4.6.37), and for m = 0 it follows from (4.6.33).

4.6.4 Non-relativistic limit To close this section, we consider a limit in which relativistic Schrödinger operators turn into classical Schrödinger operators studied previously. For the passage to this so-called non-relativistic limit we go back to (4.6.1) and re-introduce the speed of light c (while keep ℏ = 1) as a parameter, and define HR (0) = √−c2 Δ + m2 c4 − mc2 + V. We write Hc for HR (0), and by using the Feynman–Kac formula next we show that Hc → H∞ as c → ∞ in a specific sense, and the limit operator is the Schrödinger operator H∞ = −

1 Δ + V. 2m

We call this non-relativistic limit. For every c > 0 consider the subordinator (Ttc )t≥0 with parameter c such that c

𝔼P [e−uTt ] = e−t(

√2c2 u+m2 c4 −mc2 )

.

Then the related Feynman–Kac formulae become t

− ∫ V(B c )ds (f , e−tHc g) = ∫ 𝔼x𝒲 [f ̄(x)[𝔼P [g(BTtc )e 0 Ts ]]dx, ℝd

t

(f , e−tH∞ g) = ∫ 𝔼x𝒲 [f ̄(x)g(Bt/m )e− ∫0 V(Bs/m )ds ]dx. ℝd

(4.6.39)

4.6 Feynman–Kac formula for relativistic Schrödinger operators | 357

On comparison, one can expect that Ttc converges to the constant (non-random) subordinator mt as c → ∞, so that c

lim 𝔼P [e−uTt ] = e−tu/m .

(4.6.40)

c→∞

We show this convergence in the lemma below extended to a class of functions of the subordinator. Lemma 4.228. If f is a bounded continuous function on ℝ, then lim 𝔼P [f (Ttc )] = f (t/m).

c→∞

Proof. Let f ∈ S (ℝ). We have c 1 ∫ f ̌(k)𝔼P [e−ikTt ]dk. √2π

𝔼P [f (Ttc )] =

ℝ

The distribution of Ttc is given by ρct (s) =

ct mc2 t −3/2 1 c2 t 2 e s exp (− ( + m2 c2 s)) 1[0,∞) (s). √2π 2 s c

We can compute the expectation 𝔼P [e−ikTt ] in terms of the Bessel function 1 z −β z2 Kβ (z) = ( ) ∫ exp (− − s) sβ−1 ds, 2 2 4s ∞

β ∈ ℂ,

0

for z ∈ {z ∈ ℂ | ℜz > 0, ℜ(z 2 ) > 0} = {z ∈ ℂ | | arg z| < π/4}. We obtain c

∞

𝔼P [e−ikTt ] = ∫ 0

ct mc2 t −3/2 1 c2 t 2 e s exp (− ( + (2ki + m2 c2 )s)) ds. √2π 2 s

Put z = (2ki + m2 c2 )1/2 ct, taking the principal value of the square root so that (2ki + m2 c2 )1/2 = (4k 2 + m4 c4 )1/4 eiθ/2 , with tan θ = 2k/m2 c2 and |θ| < π/2. Using Kβ , we have c 2 2 𝔼P [e−ikTt ] = √ √zemc t K−1/2 (z). π

Since K−1/2 (z) = K1/2 (z) = √ π2 e√z , we see that −z

c

𝔼P [e−ikTt ] = e−(

√i2k+m2 c2 −mc)ct

.

358 | 4 Feynman–Kac formulae Furthermore, we have c(√2ki + m2 c2 − mc) = c((4k 2 + m4 c4 )1/4 − mc)eiθ/2 + mc2 (eiθ/2 − 1). The first term converges to zero as c → ∞. For the second term it can be seen that mc2 (eiθ/2 − 1) = mc2 (cos θ/2 − 1 + i sin θ/2). Since mc2 (cos θ/2 − 1) ∼ mc2 (θ/2)2 /2 ∼ mc2 (k/m2 c2 )2 /2 ∼ 0 and imc2 sin θ/2 ∼ imc2 θ/2 ∼ ik/m as c → ∞, we get c

lim 𝔼P [e−ikTt ] = e−ik(t/m)

c→∞

and lim 𝔼P [f (Ttc )] =

c→∞

1 ∫ f ̌(k)e−ik(t/m) = f (t/m). √2π ℝ

When f is chosen to be a bounded continuous function, it can be uniformly approximated by functions in S (ℝ) and the proof of the lemma is completed by a simple limiting argument. Remark 4.229. In Lemma 4.228 we actually show that ρct (s) → δ(s − mt ) as c → ∞. This can also formally be seen through the following argument. We have the estimate 1 c2 t 2 ( s + m2 c2 s) ≥ mtc2 , in which equality holds only for s = t/m. Substituting s = t/m 2 2 2

2

in ρct (s), we have emc t exp (− 21 ( c st + m2 c2 s)) = 1 and ρct (t/m) = (t/m)−3/2 √ct2π → ∞ as c → ∞. On the other hand, ρct (s) → 0 for s ≠ t/m as c → ∞ due to having then 1 2

2 2

( c st + m2 c2 s) > mtc2 .

Next we derive the non-relativistic limit of e−tHc . We put a strong restriction on the potential V just for the sake of simplicity. Proposition 4.230. If V is a bounded continuous function, then s − lim e−tHc = e−tH∞ .

(4.6.41)

c→∞

Proof. We suppose that V is non-negative without loss of generality. It is enough to show the weak limit lim (f , e−tHc g) = (f , e−tH∞ g).

(4.6.42)

c→∞

Since Hc ≥ 0 for every c > 0, ‖e−tHc ‖ ≤ 1 for all c > 0. It is also sufficient to show (4.6.42) for arbitrary f , g in S (ℝ) by a limiting argument. Note that by Lemma 4.228 it can be seen that 𝔼ν [|f (Ttc ) − f (t/m)|] → 0 as c → ∞. It follows that (f , e−tHc g) − (f , e−tH∞ g) = ∫ 𝔼x𝒲 [f ̄(x)𝔼P [(g(BTtc ) − g(Bt/m ))e ℝd

t

− ∫0 V(BT c )ds s

]]dx

4.7 Feynman–Kac formula for Schrödinger operators with spin t

| 359

t

− ∫ V(B c )ds + ∫ 𝔼x𝒲 [f ̄(x)g(Bt/m )𝔼P [(e 0 Ts − e− ∫0 V(Bs/m )ds )]]dx. ℝd

It is straightforward to see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 󵄨 − ∫ V(B )ds c x lim 󵄨󵄨󵄨 ∫ 𝔼𝒲 [f ̄(x)𝔼P [(g(BTtc ) − g(Bt/m ))e 0 Ts ]]dx 󵄨󵄨󵄨󵄨 c→∞ 󵄨 󵄨󵄨 󵄨󵄨 d 󵄨󵄨 󵄨ℝ

≤ eδt lim ∫ 𝔼x𝒲 [|f ̄(x)|𝔼P [|g(BTtc ) − g(Bt/m )|]]dx = 0, c→∞

ℝd

where δ = ‖V‖∞ . We also see that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 t t 󵄨󵄨 󵄨 − ∫ V(B )ds c x − ∫ V(B )ds lim 󵄨󵄨󵄨 ∫ 𝔼𝒲 [f ̄(x)g(Bt/m )𝔼P [(e 0 Ts − e 0 s/m )]]󵄨󵄨󵄨󵄨 c→∞ 󵄨 󵄨󵄨 󵄨󵄨󵄨ℝd 󵄨󵄨 t

≤ lim ∫ 𝔼x𝒲 [|f ̄(x)g(Bt/m )| ∫ 𝔼P |V(BTsc ) − V(Bs/m )|ds] = 0. c→∞ 0 [ ] ℝd This yields the result. We note that Proposition 4.230 can also be shown by operator theory. Since C0∞ (ℝd ) is a core of Hc , c > 0, and H∞ , it follows that limc→∞ Hc f = H∞ f for all f ∈ C0∞ (ℝd ), implying Hc → H∞ as c → ∞ in strong resolvent sense by the general result below. Proposition 4.231. Let (An )n∈ℕ and A be self-adjoint operators on a Hilbert space ℋ, and suppose that D is a common core to all An and A. If An f → Af as n → ∞ for every f ∈ D, then An → A as n → ∞ in strong resolvent sense. Proof. Let z ∈ ℂ with ℑz ≠ 0. Take f ∈ D and g = (A + z)f . Then (An + z)−1 g − (A + z)−1 g = (An + z)−1 (A − An )f converges to 0 as n → ∞, since Af − An f → 0 and (An + z)−1 is uniformly bounded with ‖(An +z)−1 ‖ ≤ 1/|ℑz|. Since D is a core for A, the image (A+z)D is dense, and we obtain that (An + z)−1 g → (A + z)−1 g as n → ∞ for all g ∈ ℋ.

4.7 Feynman–Kac formula for Schrödinger operators with spin 4.7.1 Schrödinger operators with spin

1 2

In this section we choose d = 3 and consider the path integral representation of the Pauli operator, which includes spin 21 . The effect of spin will be described in terms of a ±1-valued jump process, and the Schrödinger operator will be defined on L2 (ℝ3 ×ℤ2 ; ℂ)

360 | 4 Feynman–Kac formulae instead of L2 (ℝ3 ; ℂ2 ). This construction has the advantage of using scalar-valued functions rather than ℂ2 -valued functions, thus in the Feynman–Kac formula a scalarvalued integral kernel will occur. Let σ1 , σ2 , σ3 be the 2 × 2 Pauli matrices given by σ1 = (

0 1

1 ), 0

σ2 = (

0 i

−i ), 0

σ3 = (

1 0

0 ). −1

(4.7.1)

Each σμ is symmetric and traceless. They satisfy the anticommutation relations {σμ , σν } = 2δμν

(4.7.2)

and thereby 3

σμ σν = i ∑ εμνλ σλ , λ=1

(4.7.3)

where εαβγ denotes the Levi-Cività tensor given by

ε

αβγ

1, { { { = {−1, { { {0,

αβγ is an even permutation of 123, αβγ is an odd permutation of 123, otherwise.

Specifically, we have σ1 σ2 = iσ3 , σ2 σ3 = iσ1 , and σ3 σ1 = iσ2 and therefore, alternatively, iσ1 , iσ2 , iσ3 can be seen as the infinitesimal generators of the Lie group SU(2). Definition 4.232 (Schrödinger operator with vector potential and spin 1/2). Consider the potentials V ∈ L∞ (ℝ3 ) and a ∈ (Cb2 (ℝ3 ))3 . The Schrödinger operator with vector potential a and spin 21 is defined by HS (a) =

1 (σ ⋅ (−i∇ − a))2 + V 2

(4.7.4)

acting in L2 (ℝ3 ; ℂ2 ). On expanding (4.7.4) by using the anticommutation relations (4.7.3), we obtain 1 HS (a) = H(a) − σ ⋅ b, 2

(4.7.5)

where b = (b1 , b2 , b3 ) = ∇ × a is a magnetic field. Let a ∈ (Cb2 (ℝ3 ))3 , b ∈ (L∞ (ℝ3 ))3 , and V ∈ L∞ (ℝ3 ). Then HS (a) is self-adjoint on D(−(1/2)Δ) and bounded from below; moreover, it is essentially selfadjoint on any core of − 21 Δ as a consequence of the Kato–Rellich theorem.

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 361

Next we transform HS (a) into an operator on the set of ℂ-valued functions on a suitable space. Let ℤ2 be the set of the square roots of identity, i. e., ℤ2 = {θ1 , θ2 },

(4.7.6)

where θα = (−1)α = {

−1, 1,

α = 1,

α = 2.

(4.7.7)

By the identification L2 (ℝ3 ; ℂ2 ) ∋ (

f (x, +1) ) 󳨃→ f (x, θ) ∈ L2 (ℝ3 × ℤ2 ; ℂ) f (x, −1)

(4.7.8)

and (4.7.5), HS (a) can be reduced to the self-adjoint operator Hℤ2 (a) defined in the Hilbert space L2 (ℝ3 × ℤ2 ) giving 1 1 Hℤ2 (a)f (x, θ) = (H(a) − θb3 (x)) f (x, θ) − (b1 (x) − iθb2 (x))f (x, −θ), 2 2

(4.7.9)

where x ∈ ℝ3 and θ ∈ ℤ2 . Thus HS (a) can be regarded as the operator Hℤ2 (a) on the space of ℂ-valued functions with configuration space ℝ3 × ℤ2 . 4.7.2 A jump process In order to derive a Feynman–Kac formula for e−tHℤ2 (a) , in addition to Brownian motion we need a Lévy process accounting for the spin. We give this three-dimensional Lévy process (Zt )t≥0 on a probability space (𝒮 , ℬ𝒮 , P), with characteristics (b, A, ν) such that ν(ℝ3 \ {0}) = 1. For I ∈ ℬ(ℝ3 ) let N(t, I) = |{0 ≤ s ≤ t | ΔZs ∈ I}|

(4.7.10)

be the counting measure associated with (Zt )t≥0 . Define the random process (Nt )t≥0 on (𝒮 , ℬ𝒮 , P) by Nt = N(t, ℝ3 \ {0}).

(4.7.11)

Note that (Nt )t≥0 is a Poisson process with intensity 1. Consider the measure dNt =

∫ N(dtdz) ℝ3 \{0}

(4.7.12)

362 | 4 Feynman–Kac formulae on ℝ+ . The compensator of Nt is given by t and 𝔼P [e−αNt ] = et(e t+

−α

−1)

. Since

t

[ ] [ ] 𝔼P [∫ ∫ f (s, τ, z)N(dsdz)] = 𝔼P [∫ ∫ f (s, τ, z)dsν(dz)] , [0

ℝ3 \{0}

[0 ℝ3 \{0}

]

]

we have for f = f (s, τ), (s, τ) ∈ ℝ+ × 𝒮 , t+

t

𝔼P [∫ f (s, τ)dNs ] = 𝔼P [∫ f (s, τ)ds] , [0

[0

]

(4.7.13)

]

independent of z ∈ ℝ3 \ {0}. For each τ ∈ 𝒮 there exist n = n(τ) ∈ ℕ and points of discontinuity of t 󳨃→ Nt , 0 < s1 = s1 (τ), . . . , sn = sn (τ) ≤ t, such that t+

n

n

j=1

j=1

∫ f (s, Ns )dNs = ∑ f (sj , Nsj ) = ∑ f (sj , j). 0

(4.7.14)

Since 𝔼P [Nt ] = t and P(Nt = N) = t N e−t /N!, the expectation of (4.7.14) reduces to a Lebesgue integral, i. e., t+

t

t ∞

n

s 𝔼P [∫ f (s, Ns )dNs ] = 𝔼P [∫ f (s, Ns )ds] = ∫ ∑ f (s, n) e−s ds. n! [0 ] [0 ] 0 n=0 Furthermore, Proposition 3.74 yields the following Itô type formula. Proposition 4.233. Suppose that hi (t) = hi (t, τ) ∈ 𝔽, i = 1, . . . , d, are independent of z ∈ ℝd \ {0}. Let X i = (Xti )t≥0 be the semimartingale given by dXti = f i ⋅ dBt + g i dt + hi dNt ,

i = 1, . . . , d,

and F ∈ C 2 (ℝd ), where f i = f i (t, ω) and g i = g i (t, ω). Then n

t

n

t

F(Xt ) = F(X0 ) + ∑ ∫ Fi (Xs )(f i ⋅ dBs ) + ∑ ∫ Fi (Xs )g i ds i=1 0

+

t

t+

0

0

i=1 0

1 n ∑ ∫ Fij (Xs )(f i ⋅ f j )ds + ∫(F(Xs− + h(s−)) − F(Xs− ))dNs . 2 i,j=1

Here Fi = 𝜕i F and Fij = 𝜕i 𝜕j F. t+

t+

Proof. Note that ∫0 hi (s, ω)dNs = ∫0 ∫ℝd \{0} hi (s, ω)N(dsdz) by the definition of dNs . The result is obtained by a direct application of the Itô formula.

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 363

Let N + α = (Nt + α)t≥0 and 𝔼αP [f (N)] = 𝔼P [f (N + α)]. It will be convenient to use the joint particle and spin process as a single ℝ3 × ℤ2 -valued random process as given below. Definition 4.234. Define the ℝ3 × ℤ2 -valued stochastic process (qt )t≥0 on the probability space (X × 𝒮 , ℬ(X ) × ℬ𝒮 , 𝒲 x × P) by qt = (Bt , θNt )

(4.7.15)

where θNt = (−1)Nt . x,α For simplicity, we write 𝔼x,α 𝒲×P = 𝔼 . To determine the generator of the process (qt )t≥0 , let σF be the fermionic harmonic oscillator defined by

1 σF = (σ3 + iσ2 )(σ3 − iσ2 ). 2

(4.7.16)

Note that, in fact, σF = −σ1 + 12×2 , where 12×2 is the 2 × 2 identity matrix. A direct computation yields (f , e−t(−(1/2)Δ+εσF ) g) = ∑ ∫ 𝔼x,α [f (q0 )g(qt )εNt ]dx. α=1,2

(4.7.17)

ℝ3

Thus setting ε = 1, we see that the generator of qt is −(− 21 Δ+σF ) under the identification L2 (ℝ3 × ℤ2 ) ≅ L2 (ℝ3 ; ℂ2 ). In particular, (f , e−t(−(1/2)Δ−σ1 ) g) = et ∑ ∫ 𝔼x,α [f (q0 )g(qt )]dx. α=1,2

ℝ3

4.7.3 Feynman–Kac formula for the jump process In this section we derive the Feynman–Kac formula of e−tHℤ2 (a) by making use of the random process (qt )t≥0 defined in (4.7.15). Theorem 4.235 (Feynman–Kac formula for Schrödinger operator with vector potential and spin 1/2). Let a ∈ (Cb2 (ℝ3 ))3 , b ∈ (L∞ (ℝ3 ))3 , and V ∈ L∞ (ℝ3 ). If t

󵄨󵄨 󵄨󵄨 1 ∫ ds ∫ 󵄨󵄨󵄨󵄨log √b1 (y)2 + b2 (y)2 󵄨󵄨󵄨󵄨 Πs (y − x)dy < ∞ 2 󵄨 󵄨 0

(4.7.18)

ℝ3

for all (x, t) ∈ ℝ3 × ℝ+ , then the Feynman–Kac-type formula (f , e−tHℤ2 (a) g) = et ∑ ∫ 𝔼x,α [f (q0 )g(qt )eZt ]dx α=1,2

ℝ3

(4.7.19)

364 | 4 Feynman–Kac formulae holds. Here t

t

t

0

0

0

1 Zt = −i ∫ a(Bs ) ∘ dBs − ∫ V(Bs )ds + ∫ θNs b3 (Bs )ds 2 t+

1 + ∫ log ( (b1 (Bs ) − iθNs b2 (Bs ))) dNs . 2 0

Before turning to the proof of Theorem 4.235 here is a heuristic argument. Since the diagonal part − 21 θb3 (x) of Hℤ2 (a) acts as an external potential up to the sign θ = t

±, we have formally the integral ∫0 (−1/2)θNs b3 (Bs )ds in Zt . This explains why the offt

diagonal part ∫0 log ((1/2)(b1 (Bs ) − iθNs b2 (Bs ))) dNs appears in Zt . Let 1 W(x, −θ) = log ( (b1 (x) − iθb2 (x))) 2

(4.7.20)

and consider t+

(KtS f )(x, σ) = et 𝔼x,α [f (Bt , θNt )e∫0

W(Bs ,−θNs− )dNs

].

First assume, for simplicity, that W has no zeroes. The generator −K S of KtS can be computed by Itô’s formula for semimartingales yielding t+

d(e∫0

W(Bs ,−θNs )dNs

t+

) = e∫0

W(Bs ,−θNs− )dNs

(eW(Bt ,−θNt ) − 1)dNt .

(4.7.21)

On the other hand, t

t

d(e− ∫0 V(Bs )ds ) = e− ∫0 V(Bs )ds (−V(Bt ))dt,

(4.7.22)

t

implying that (e−t(−(1/2)Δ+V) f )(x) = 𝔼x [e− ∫0 V(Bs )ds f (Bt )]. Comparing (4.7.21) and (4.7.22) it is clear that the Itô formula gives the differential for continuous processes and the difference for discontinuous ones. From (4.7.21) it follows that the generator K S of KtS is 1 −K S f (x, θ) = (− Δ − eW(x,−θ) + 1) f (x, −θ). 2 S

t

Thus e−tK f (x, θα ) = et 𝔼Bt ,α [f (Bs , θNt )e∫0 W(x,−θNs− )dNs ], giving rise to the special form of the off-diagonal part. We rewrite this term as 1 − (b1 (x) − iθb2 (x)) = −eW(x,−θ) + 1 − 1. 2 Hence (4.7.20) follows.

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 365

Remark 4.236. We prove Proposition 4.235 by making use of the Itô formula. In ort+ der for the Itô formula to apply, however, the integrand in ∫0 ⋅ ⋅ ⋅ dNs must be predictable with respect to the given filtration. Note that θNs is right-continuous in s for each (ω, τ) ∈ X × 𝒮 , so we define θNs− = limε↑0 θNs−ε . Thus θNs− is left-continuous and W(Bs , −θNs− ) is predictable, i. e., W(Bs , −θNs− ) is σ(Br , Nr ; 0 ≤ r ≤ s) measurable and left-continuous in s, for every (ω, τ) ∈ X × 𝒮 . This allows then an application of the t+ Itô formula to ∫0 W(Bs , −θNs− )dNs . Proof of Theorem 4.235. Write U(Bs , θNs ) = − 21 θNs b3 (Bs ) + V(Bs ) for the diagonal part. The off-diagonal part W(Bs , −θNs− ) is predictable and, in order to apply Itô’s formula, t+

we need to check that | ∫0 W(Bs , −θNs− )dNs | is bounded almost surely. Indeed, 󵄨󵄨 󵄨󵄨 t+ t 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 1 󵄨󵄨 x,α [ 󵄨󵄨 x,α [ 󵄨󵄨 ] ∫ W(Bs , −θNs− )dNs 󵄨󵄨 ≤ 𝔼 ∫ 󵄨󵄨󵄨log ( √b1 (Bs )2 + b2 (Bs )2 )󵄨󵄨󵄨󵄨 dNs ] 󵄨󵄨𝔼 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 󵄨󵄨 [0 ]󵄨󵄨 [0 ] t

󵄨󵄨 󵄨󵄨 1 = ∫ ds ∫ Πs (y − x) 󵄨󵄨󵄨󵄨log ( √b1 (y)2 + b2 (y)2 )󵄨󵄨󵄨󵄨 dy 2 󵄨 󵄨 0

ℝ3

t+

is bounded by the assumption; hence | ∫0 W(Bs , −θNs− )dNs | < ∞, almost surely. Define KtS : L2 (ℝ3 × ℤ2 ) → L2 (ℝ3 × ℤ2 ) by KtS g(x, θα ) = 𝔼x,α [eZt g(Bt , θNt )]. 1/2 M t (M−1)t/2 It can be seen that ‖KtS g‖ ≤ VM e e ‖g‖, where 󸀠

M󸀠 =

1 sup |b (x)|, 2 x∈ℝ3 3

M=

1 sup (b2 (x) + b22 (x)), 4 x∈ℝ3 1

t

VM = sup 𝔼x [e−2 ∫0 V(Bs )ds ], x∈ℝ3

which are all finite. Thus KtS is bounded. Since Zt is continuous at t = 0 for each (ω, τ) ∈ X × 𝒮 , dominated convergence yields ‖KtS g − g‖ ≤ ∑ ∫ 𝔼x,α [|g(x, θ) − g(Bt , θNt )eZt |]dx → 0 α=1,2

ℝ3

as t → 0. Since (Bt )t≥0 and (Nt )t≥0 are independent Markov processes, it follows similarly as in the proof of Theorem 4.184 that S (KsS KtS g)(x, θα ) = Ks+t g(x, θα ),

i. e., {KtS : t ≥ 0} is a C0 -semigroup. Denote the generator of KtS by the closed operator −h. We will see below that KtS = e−th = e−t(Hℤ2 (a)+1) . By Proposition 4.233 it follows that t+

θNt − θN0 = −2 ∫ θNs− dNs , 0

366 | 4 Feynman–Kac formulae and g(Bt , θNt ) − g(x, θN0 ) t

t

t+

0

0

0

1 = ∫ ∇g(Bs , θNs ) ⋅ dBs + ∫ Δg(Bs , θNs )ds + ∫ (g(Bs , −θNs− ) − g(Bs , θNs− )) dNs , 2 as well as

t

t

1 e − 1 = ∫ e (−ia(Bs )) ∘ dBs + ∫ eZs (−V(Bs ) + (−ia(Bs ))2 − U(Bs , θNs )) ds 2 Zt

Zs

0

0

t+

+ ∫ eZs− (eW(Bs ,−θNs− ) − 1)dNs . 0

By the product rule, d(eZt g) = deZt ⋅ g + eZt ⋅ dg + deZt ⋅ dg, and two identities above we have eZt g(Bt , θNt ) − g(x, θN0 ) t

1 1 = ∫ eZs ( Δ − ia(Bs ) ⋅ ∇ − a(Bs )2 − V(Bs ) − U(Bs , θNs )) g(Bs , θNs )ds 2 2 0

t

+ ∫ eZs (∇g(Bs , θNs ) − ia(Bs ) ⋅ g(Bs , θNs )) ⋅ dBs 0

t+

+ ∫ eZs− (g(Bs , −θNs− )eW(Bs ,−θNs− ) − g(Bs , θNs− ))dNs . 0

Take expectation on both sides above. The martingale part vanishes and by (4.7.13) we obtain 𝔼

x,α

Zt

t

[e g(Bt , θNt ) − g(x, θα )] = ∫ 𝔼x,α [G(s)]ds, 0

where

1 i 1 G(s) = eZs ( Δ − ia(Bs ) ⋅ ∇ − ∇ ⋅ a(Bs ) − a(Bs )2 − V(Bs ) − U(Bs , θNs ))g(Bs , θNs ) 2 2 2 + eZs− (g(Bs , −θNs− )eW(Bs ,−θNs− ) − g(Bs , θNs− )),

with s > 0, and 1 i 1 G(0) = ( Δ − ia(x) ⋅ ∇ − ∇ ⋅ a(x) − a(x)2 − V(x) − U(x, θα ) − 1)g(x, θα ) 2 2 2 + eW(x,−θα ) g(x, −θα )

= −(HS (a) + 1)g(x, θα ).

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 367

At s = 0, G(s) is continuous for each (ω, τ) ∈ X × 𝒮 , hence t

1 1 lim (f , (KtS − 1)g) = lim ∫ ds ∑ ∫ f (x, θα )𝔼x,α [G(s)]dx = (f , −(Hℤ2 (a) + 1)g). t→0 t t→0 t α=1,2 0

ℝ3

Since C0∞ (ℝ3 × ℤ2 ) is a core of Hℤ2 (a), equality (4.7.19) follows. 4.7.4 Extension to singular external potentials and singular vector potentials Next we extend Theorem 4.235 to singular external potentials V and singular vector potentials a as in the spinless case, following a similar strategy as in Section 4.4.3. Define the quadratic form 3

qaS (f , g) = ∑ (σμ Dμ f , σμ Dμ g) + (V 1/2 f , V 1/2 g) μ=1

(4.7.23)

with domain Q(qaS ) = ⋂3μ=1 {f ∈ L2 (ℝ3 ) | Dμ f ∈ L2 (ℝ3 )} ∩ D(V 1/2 ). Lemma 4.237. Let V ≥ 0 be L1loc (ℝ3 ) and a ∈ (L2loc (ℝ3 ))3 . Then qaS is a closed symmetric form. The proof is very similar to Lemma 4.187. From here it follows that there exists a selfadjoint operator hS such that (f , hS g) = qaS (f , g),

f ∈ Q(qaS ), g ∈ D(hS ).

(4.7.24)

A sufficient condition for C0∞ (ℝ3 ) to be a core of hS can also be obtained in a similar way to the spinless case; see Proposition 4.189. Proposition 4.238. The following properties hold: (1) Let V ∈ L1loc (ℝ3 ) with V ≥ 0. If a ∈ (L2loc (ℝ3 ))3 , then C0∞ (ℝ3 ) is a form core of hS . (2) Let V ∈ L2loc (ℝ3 ) with V ≥ 0. If a ∈ (L4loc (ℝ3 ))3 , ∇ ⋅ a ∈ L2loc (ℝ3 ), and b ∈ (L2loc (ℝ3 ))3 , then C0∞ (ℝ3 ) is an operator core of hS . In case (2) above the operator hS can be realized as 1 1 1 hS f = − Δf − a ⋅ (−i∇)f + (− a ⋅ a − (−i∇ ⋅ a) − σ ⋅ b) f + Vf . 2 2 2

(4.7.25)

Instead of hS we define the Schrödinger operator with spin by the sum of the kinetic term and spin. Assume that a ∈ (L2loc (ℝ3 ))3 , 0 ≤ V ∈ L1loc (ℝ3 ), and the magnetic field b satisfies b ∈ (L∞ (ℝ3 ))3 . Let H 0 (a) be given by Definition 4.188 with V = 0. It can

368 | 4 Feynman–Kac formulae be seen that H 0 (a) − 21 σ ⋅ b is self-adjoint on D(H 0 (a)). We regard b to be independent of a, and define 1 HS (a, b) = H 0 (a) +̇ V − σ ⋅ b. 2 Let Hℤ2 (a, b) be the unitary transformed operator to L2 (ℝ3 × ℤ2 ) of HS (a, b), and Hℤ0 2 (a, b) denote Hℤ2 (a, b) with V = 0. From now on we discuss more general external potentials V and vector potentials a. Consider general vector potentials possibly having zeroes. Note that (4.7.18) is a sufficient condition making sure that t+

∫ |W(Bs , −θNs− )|dNs < ∞,

a. e. (ω, τ) ∈ X × 𝒮 .

(4.7.26)

0

When, however, b1 (x)−iθb2 (x) vanishes for some (x, θ), it is not clear that (4.7.26) holds. This case is relevant and Theorem 4.235 must be improved since we have to construct the path integral representation of e−tHℤ2 (a,b) in which the off-diagonal part b1 − iθb2 of Hℤ2 (a, b) has zeroes or a compact support. The off-diagonal part of HS (a, b), however, in general may have zeroes. For instance, aμ for all μ = 1, 2, 3 have a compact support, and so does the off-diagonal part of b = ∇ × a. Therefore, in order to avoid that the off-diagonal part vanishes, we introduce 1 1 Hℤε 2 (a, b)f (x, θ) = (H(a, b) − θb3 (x)) f (x, θ) − Ψε ( (b1 (x) − iθb2 (x))) f (x, −θ), 2 2 (4.7.27) where Ψε (z) = z + εψε (z)

(4.7.28)

for z ∈ ℂ and ε > 0, with the indicator function 1, ψε (z) = { 0, Thus

|z| < ε/2,

|z| ≥ ε/2.

(4.7.29)

󵄨󵄨 󵄨 󵄨󵄨Ψ ( 1 (b − iθb ))󵄨󵄨󵄨 > ε > 0. 󵄨󵄨 ε 2 󵄨󵄨 2 1 󵄨 󵄨 2

Let t > 0 be fixed and consider the set W = Wt defined by t+

{ } 1 3 W = {∫ log ( √b1 (x + Bs )2 + b2 (x + Bs )2 ) dNs > −∞} ⊂ ℝ × X × 𝒮 . 2 {0 } Lemma 4.239. Let W be given by (4.7.30). Then it follows that 󵄨󵄨 t+ 󵄨󵄨 1 lim 󵄨󵄨󵄨󵄨e∫0 log(Ψε ( 2 (b1 (Bs +x)−iθs− b2 (Bs +x))))dNs 1W c 󵄨󵄨󵄨󵄨 = 0. ε→0 󵄨 󵄨

(4.7.30)

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 369

Proof. Let W(X) = 21 √b1 (X)2 + b2 (X)2 . We have t+ 󵄨󵄨 ∫t+ log(Ψ ( 1 (b (x+B )−iθ b (x+B ))))dN 󵄨󵄨 ∫ log(W(x+Bs )+ε)dNs 󵄨󵄨e 0 ε 2 1 s Ns 2 s s 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 ≤ e 0 󵄨 󵄨

Note that for every (x, ω, τ) ∈ ℝ3 × X × 𝒮 there exist a number n = n(τ) and random jump times r1 (τ), . . . , rn (τ) such that t+

n

∫ log(W(x + Bs (ω)) + ε)dNs = ∑ log(W(x + Brj (τ) (ω)) + ε). j=1

0

We see that (x, ω, τ) ∈ W c if and only if there exists r such that 0 < r < t, s 󳨃→ Ns is discontinuous at s = r, and b1 (x + Br (ω))2 + b2 (x + Br (ω))2 = 0. Hence for (x, ω, τ) ∈ 𝒲 c , there exists ri such that t+

n

∫ log(W(x + Bs ) + ε)dNs = ∑ log(W(x + Brj ) + ε) + log ε, j=i̸

0

and t+

e∫0 t+

Thus limε→0 |e∫0

log(W(x+Bs )+ε)dNs

log(W(x+Bs )+ε)dNs

≤e

∑nj=i̸ log(W(x+Brj )+ε) log ε

e

.

1W c | = 0, and the lemma follows.

Theorem 4.240 (Feynman–Kac formula for Schrödinger operator with singular vector potential and spin 21 ). Suppose that V ∈ L∞ (ℝ3 ), a ∈ (L2loc (ℝ3 ))3 , ∇ ⋅ a ∈ L1loc (ℝ3 ) and b ∈ (L∞ (ℝ3 ))3 . Let W be given by (4.7.30). Then (f , e−tHℤ2 (a,b) g) = et ∑ ∫ 𝔼x,α [f (q0 )g(qt )eZt 1W ]dx. α=1,2

(4.7.31)

ℝ3

Proof. We show this theorem through a similar limiting argument to Lemma 4.191. 2 3 Suppose first that a(n) ∈ (C0∞ (ℝ3 ))3 , n ∈ ℕ, such that a(n) μ → aμ in Lloc (ℝ ) as n → ∞. Take 1R (x) = χ(x1 /R)χ(x2 /R)χ(x3 /R), R > 0, where χ ∈ C0∞ (ℝ) such that 0 ≤ χ ≤ 1, χ(x) = 1 for |x| < 1, and χ(x) = 0 for |x| ≥ 2. In the same way as in Lemma 4.190 we can show that ε

e−tHℤ2 (1R a e

(n)

,b)

−tHℤε (1R a,b) 2

ε

→ e−tHℤ2 (1R a,b) →e

−tHℤε (a,b) 2

as n → ∞, as R → ∞

(4.7.32) (4.7.33)

in strong sense. In the proof of Lemma 4.191 we have already shown that there exists 󸀠 t t a subsequence n󸀠 such that ∫0 1R a(n ) (Bs ) ∘ dBs → ∫0 1R a(Bs ) ∘ dBs almost surely as

370 | 4 Feynman–Kac formulae t

t

n󸀠 → ∞, and R󸀠 such that ∫0 1R󸀠 a(Bs ) ∘ dBs → ∫0 a(Bs ) ∘ dBs . We reset n󸀠 and R󸀠 by n and R, respectively. Let t

t

0

0

Ztε = −i ∫ a(Bs ) ∘ dBs − ∫ V(Bs )ds t

t+

1 1 + ∫ θNs b3 (Bs )ds + ∫ log(Ψε ( (b1 (Bs ) − iθNs b2 (Bs ))))dNs 2 2 0

0

and Ztε (n, R) be Ztε with a replaced by 1R a(n) . Hence we conclude that ε

ε

lim lim ∫ 𝔼x,α [f (q0 )g(qt )eZt (n,R) ]dx = ∫ 𝔼x,α [f (q0 )g(qt )eZt ]dx.

R→∞ n→∞

ℝ3

ℝ3

Combining this with (4.7.32) and (4.7.33), we obtain ε

ε

(f , e−tHℤ2 (a,b) g) = et ∑ ∫ 𝔼x,α [f (q0 )g(qt )eZt ]dx. α=1,2

(4.7.34)

ℝ3

We also see that ε

ε

lim ∫ 𝔼x,α [f (q0 )g(qt )eZt ]dx = ∫ 𝔼x,α [lim f (q0 )g(qt )eZt ]dx

ε→0

ℝ3

ℝ3

ε→0

and ε

ε

ε

lim f (q0 )g(qt )eZt = lim f (q0 )g(qt )eZt 1W + lim f (q0 )g(qt )eZt 1W c = f (q0 )g(qt )eZt 1W .

ε→0

Since e

ε→0

−tHℤε (a,b) 2

ε→0

is strongly convergent to e−tHℤ2 (a,b) as ε → 0, (4.7.31) also follows.

By Lemma 4.191 we have a diamagnetic inequality for Hℤ2 (a, b). Corollary 4.241 (Diamagnetic inequality). Under the assumptions of Lemma 4.238, |(f , e−tHℤ2 (a,b) g)| ≤ (|f |, e−tHℤ2 (0,b0 ) |g|),

(4.7.35)

where b0 = (√b21 + b22 , 0, b3 ). Proof. Let W(X) = 21 √b1 (X)2 + b2 (X)2 . The corollary follows from the fact that t+

|e∫0

log( 21 (b1 (Bs )−iθNs b2 (Bs ))dNs )

t+

| ≤ e∫0

log W(Bs )dNs

and |(f , e−tHℤ2 (a,b) g)| ≤ ∑ ∫ 𝔼x,α [|f (q0 )|g(qt )|eZt ]dx, ̂

α=1,2

ℝ3

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 371

where the exponent Ẑ t is defined by t

t

t+

0

0

0

1 Ẑ t = − ∫ V(Bs )ds + ∫ θNs b3 (Bs )ds + ∫ log W(Bs )dNs . 2 Theorem 4.242 (Feynman–Kac formula for Schrödinger operator with singular external potential, singular vector potential, and spin 21 ). Suppose that a ∈ (L2loc (ℝ3 ))3 , ∇ ⋅ a ∈ L1loc (ℝ3 ), and b satisfies b ∈ (L∞ (ℝ3 ))3 . (1) Let V be a real-valued multiplication operator relatively bounded (resp. formbounded) with respect to − 21 Δ with a relative bound c. Then V is also relatively bounded (resp. form-bounded) with respect to Hℤ0 2 (a, b) with a relative bound not exceeding c. (2) Let V be such that V+ ∈ L1loc (ℝ3 ) and V− is relatively form-bounded with respect to − 21 Δ with a relative bound less than 1. Then Hℤ0 2 (a, b) +̇ V+ −̇ V−

(4.7.36)

can be defined as a self-adjoint operator. This also will be denoted by Hℤ2 (a, b). (3) Let V satisfy the same assumptions as in (2). Then the Feynman–Kac formula of (f , e−tHℤ2 (a,b) g) is given by (4.7.34) and (4.7.31). Proof. By the diamagnetic inequality ‖|W|α (Hℤ0 2 (a, b) + E)−α f ‖ ‖f ‖

≤

‖|W|α (Hℤ0 2 (0, b0 ) + E)−α f ‖ ‖f ‖

(4.7.37)

for α = 1 and 21 , (1) follows by the same argument as Lemma 4.192 and the boundedness of the magnetic field bj . Statement (2) is implied by the KLMN theorem. Thus by Theorem 4.235 and a minor modification of the proof of Theorem 4.194, also (3) follows. Definition 4.243 (Schrödinger operator with singular external potential, singular vector potential, and spin 21 ). Assume that a ∈ (L2loc (ℝ3 ))3 and that the magnetic field b satisfies b ∈ (L∞ (ℝ3 ))3 . Let V be such that V+ ∈ L1loc (ℝ3 ) and V− is relatively formbounded with respect to − 21 Δ with a relative bound less than 1. Then Hℤ2 (a, b) is defined by (4.7.36).

4.7.5 Decay of eigenfunctions and martingale properties In a similar way to the proof of the decay of eigenfunctions of Schrödinger operators we can also show the decay of eigenfunctions of Schrödinger operators with spin. As before, we consider potentials in the classes 𝕍upper and 𝕍lower , given in Definition 4.175.

372 | 4 Feynman–Kac formulae We note that, under the assumption that b ∈ (L∞ (ℝ3 ))3 , we have 1 2 √b + b22 ∈ L∞ (ℝ3 ). 2 1

(4.7.38)

1 1 E ∗ = (‖b3 ‖∞ + ‖√b21 + b22 ‖2∞ + 1) 2 4

(4.7.39)

b3 ∈ L∞ (ℝ3 ), Write and consider

Hℤ2 (a, b)φ = Eφ. Let V = W − U ∈ 𝕍upper . We have W ∈ L1loc (ℝd ), and 0 ≥ −U ∈ Lp (ℝd ) implies that −U ∈ 𝒦(ℝd ), thus V is Kato-decomposable. Lemma 4.244. Suppose that b ∈ (L∞ (ℝ3 ))3 . Let V = W − U ∈ 𝕍upper . Then for every t, a > 0 and 0 < α < 1/2 there exist constants D1 , D2 , D3 > 0 such that m

|φ(x)| ≤ D1 ‖φ‖eD2 ‖U‖p t e(E+E where m = (1 −

d −1 ) 2p

∗

)t

α a2 t

(D3 e− 4

e−tW∞ + e−tWa (x) ),

(4.7.40)

and Wa (x) = inf{W(y) | |x − y| < a}.

Proof. Since φ(x, θα ) = e−t(Hℤ2 (a,b)−E) φ(x, θα ) = e(E+1)t 𝔼x,α [φ(Bt , θNt )eZt 1W ], by the Schwarz inequality it follows that 1/2

|φ(x, θα )| ≤ etE et (𝔼x,α [|φ(Bt , θNt )|2 ]) t

1/2

1

× (𝔼x [e−2 ∫0 V(Bs )ds ]) (𝔼α [et‖b3 ‖∞ +2Nt log ‖ 2 ≤ etE et et(E Here we used 𝔼α [eNt C ] = et(e

C

∗

−1)

−1)

1/2

t

])

1/2

1/2

(𝔼x,α [|φ(Bt , θNt )|2 ]) (𝔼x [e−2 ∫0 V(Bs )ds ]) .

. It can be seen that ∞

𝔼x,α [|φ(Bt , θNt )|2 ] = ∫ Πt (y) ∑ |φ(x + y, θα+n )|2 n=0

ℝ3

√b21 +b22 ‖∞

t n −t e dy n!

≤ ∫ Πt (y)(|φ(x + y, +1)|2 + |φ(x + y, −1)|2 )dy ≤ ‖φ‖2 . ℝ3

Hence we obtain t

|φ(x, θα )| ≤ et(E+E ) (𝔼x [e−2 ∫0 V(Bs )ds ]) ∗

1/2

‖φ‖

and then t

|φ(x)| ≤ et(E+E ) (𝔼x [e−4 ∫0 W(Bs )ds ]) ∗

1/4

t

1/4

(𝔼x [e+4 ∫0 U(Bs )ds ])

1/2

𝔼[|φ(x + Bt )|2 ] .

In the same way as with the estimate (4.3.78), with E replaced by E + E ∗ the lemma follows.

4.7 Feynman–Kac formula for Schrödinger operators with spin

| 373

The above Carmona-type estimate yields the required decay properties; the proof is similar to Corollaries 4.177–4.178. Corollary 4.245 (Decay property). Suppose (4.7.38). Let V = W − U ∈ 𝕍upper . (1) Suppose that W(x) ≥ γ|x|2n outside a compact set K, for some n > 0 and γ > 0. Take 0 < a < 1/2. Then there exists a constant C1 > 0 such that |φ(x, θα )| ≤ C1 ‖φ‖ exp (−

ac n+1 |x| ) , 16

(4.7.41)

where c = infx∈ℝd \K W|x|/2 (x)/|x|2n . (2) Let Σ = lim inf|x|→∞ V(x). Suppose that Σ > E + E ∗ , Σ > W∞ , and let 0 < β < 1. Then there exists a constant C2 > 0 such that |φ(x, θα )| ≤ C2 ‖φ‖ exp (−

β Σ − E − E∗ |x|) . 8√2 √Σ − W∞

(4.7.42)

(3) Suppose that lim|x|→∞ W(x) = ∞. Then there exist constants C, δ > 0 such that |φ(x, θα )| ≤ C‖φ‖ exp (−δ|x|) .

(4.7.43)

Finally, we show a martingale property of the process associated with Schrödinger operators with spin, as a counterpart of Lemma 4.223. By using this martingale together with a suitable choice of stopping times one can then estimate the decay of eigenfunctions of Hℤ2 (a, b) by an alternative approach than above. For simplicity we assume throughout that 𝔼xP

t

[∫ | log 1 √b1 (Bs )2 + b2 (Bs )2 |ds] < ∞, 2 [0 ]

a. e. x ∈ ℝ3 ,

i. e., the measure of W c is zero. We recall that the exponent of the Feynman–Kac formula of e−tHℤ2 (a,b) is given by Zt = ZtV + ZtA + ZtS with t

ZtV

= − ∫ V(Bs )ds, 0

ZtA

t

t+

0

0

t

= −i ∫ a(Bs ) ∘ dBs , 0

ZtS = − ∫ Ud (Bs , θNs )ds + ∫ log (−Uo (Bs , −θNs− )) dNs . − 21 Yb3 (x)

We write Ud (x, Y) = and Uo (x, −Y) = − 21 (b1 (x) − iYb2 (x)) for notational simplicity. Let φ be an eigenfunction such that Hℤ2 (a, b)φ = Eφ. Also, let Zt (x, α) = ZtV (x)+ ZtA (x)+ZtS (x, α) be given by Zt with Bs and Ns replaced by Bs +x and Ns +α, respectively, i. e., ZtV (x)

t

= − ∫ V(Bs + x)ds, 0

ZtA (x)

t

= −i ∫ a(Bs + x) ∘ dBs , 0

374 | 4 Feynman–Kac formulae

ZtS (x, α)

t

t+

= − ∫ Ud (Bs + x, θNs +α )ds + ∫ log (−Uo (Bs + x, −θNs− +α )) dNs . 0

0

Define the random process (Mt (x, α))t≥0 by Mt (x, α) = et(E+1) eZt (x,α) φ(Bt + x, θNt +α ),

t ≥ 0,

and the filtration Mt = σ(qr , 0 ≤ r ≤ t),

t ≥ 0.

Note that e−t(Hℤ2 (a,b)−E) Ψp = φ for all t ≥ 0, and thus 𝔼x,α [Mt (0, 0)] = 𝔼0,0 [Mt (x, α)] = φ(x, θα ).

(4.7.44)

Theorem 4.246 (Martingale property). The random process (Mt (x, α))t≥0 is a martingale with respect to (Mt )t≥0 , i. e., 𝔼0,0 [Mt (x, α)|Ms ] = Ms (x, α) for t ≥ s. Proof. We prove the case when (x, α) = (0, 0) for simplicity; the proof for (x, α) ≠ (0, 0) is similar. Let Zt ([u, v]) be defined by Zt with the integration domain [0, t] replaced by [u, v], and write Mt = Mt (0, 0). We see that 𝔼0,0 [Mt |Ms ] = et(E+1) eZt ([0,s]) 𝔼0,0 [eZt ([s,t]) φ(qt )|Ms ] . By the Markov property of the ℝd × ℤ2 -valued random process (Bt , Nt )t≥0 we have t−s

𝔼0,0 [eZt ([s,t]) φ(qt )|Ms ] = 𝔼Bs ,Ns [e− ∫0

t−s

V(Br )dr −i ∫0

e

t−s

a(Br )∘dBr ∫0

e

Ud (Br ,θr )dr K

e φ(qt−s )] .

(4.7.45)

The off-diagonal part K in (4.7.45) is K=

∑

s −∞} ⊂ ℝ × X × 𝒮 × 𝒯 . 2

{0

}

Lemma 4.251. Let R be given by (4.8.9). It follows that 󵄨󵄨 Tt + 󵄨󵄨 1 lim 󵄨󵄨󵄨󵄨e∫0 log(Ψε ( 2 (b1 (Bs +x)−iθs− b2 (Bs +x))))dNs 1Rc 󵄨󵄨󵄨󵄨 = 0. ε→0 󵄨 󵄨 Proof. The proof is a minor modification of that of Lemma 4.239. Let 1 W(X) = √b1 (X)2 + b2 (X)2 . 2

(4.8.9)

378 | 4 Feynman–Kac formulae We have T + 󵄨󵄨 ∫Tt + log(Ψ ( 1 (b (x+B )−iθ b (x+B ))))dN 󵄨󵄨 ∫ t log(W(x+Bs )+ε)dNs 󵄨󵄨e 0 ε 2 1 s Ns 2 s s 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 ≤ e 0 󵄨 󵄨

We note that for every (x, ω, τ, ϖ) ∈ ℝ3 × X × 𝒮 × 𝒯 there exist a number n = n(τ, ϖ) and random jump times r1 (τ), . . . , rn (τ) such that Tt (ϖ)+

n

0

j=1

∫ log(W(x + Bs (ω)) + ε)dNs = ∑ log(W(x + Brj (τ) (ω)) + ε).

We see that (x, ω, τ, ϖ) ∈ R c if and only if there exists r such that for 0 < r < Tt (ϖ) the map s 󳨃→ Ns is discontinuous at s = r, and b1 (x + Br (ω))2 + b2 (x + Br (ω))2 = 0. Hence for (x, ω, τ, ϖ) ∈ R c , there exists ri such that Tt +

n

0

j=i̸

∫ log(W(x + Bs ) + ε)dNs = ∑ log(W(x + Brj ) + ε) + log ε,

Tt +

and e∫0

log(W(x+Bs )+ε)dNs

≤e

∑nj=i̸ log(W(x+Brj )+ε) log ε

e

. Thus it follows that

󵄨󵄨 Tt + 󵄨󵄨 lim 󵄨󵄨󵄨󵄨e∫0 log(W(x+Bs )+ε)dNs 1Rc 󵄨󵄨󵄨󵄨 = 0. ε→0 󵄨 󵄨 Theorem 4.252 (Feynman–Kac formula for relativistic Schrödinger operator with singular vector potential and spin 1/2). Let V ∈ L∞ (ℝ3 ). Suppose that a ∈ (L2loc (ℝ3 ))3 , ∇ ⋅ a ∈ L1loc (ℝ3 ) and b ∈ (L∞ (ℝ3 ))3 . Then (f , e−tHRℤ2 (a,b) g) = ∑ ∫ 𝔼x,α,0 [eTt f (q̃ 0 )g(q̃ t )eZt 1Rc ]dx, ̃

α=1,2

(4.8.10)

ℝ3

where R is given by (4.8.9). Proof. The proof is a simple modification of Theorem 4.240, i. e., qt , Zt , and Z̃ tε in the proof of Theorem 4.240 are replaced by q̃ t , Z̃ t , and Tt

t

0

0

Z̃ tε = −i ∫ a(Bs ) ∘ dBs − ∫ V(BTs )ds Tt

Tt +

1 1 + ∫ θNs b3 (Bs )ds + ∫ log(Ψε ( (b1 (Bs ) − iθNs b2 (Bs ))))dNs , 2 2 0

respectively.

0

4.8 Feynman–Kac formula for relativistic Schrödinger operators with spin

| 379

From the Feynman–Kac formula we can derive a diamagnetic inequality for the Hamiltonian with spin HRℤ2 (a, b). However, this expression is not as simple as that for Hℤ2 (a, b) derived in Corollary 4.241, where we obtained the diamagnetic inequality 0

0

|(f , e−tHℤ2 (a,b) g)| ≤ (|f |, e−tHℤ2 (0,b0 ) |g|) with b0 = (√b21 + b22 , 0, b3 ). Note that Hℤ0 2 (0, b0 ) can be negative since inf Spec(Hℤ0 2 (0, b0 )) ≤ inf Spec(Hℤ0 2 (a, b)).

(4.8.11)

The difficulty for the relativistic case is that the formula (f , e−t

√2H+m2 −m

g) = 𝔼P [(f , e−Tt H g)]

m≥0

holds only for positive self-adjoint operators H. To derive the diamagnetic inequality for HRℤ2 (a, b) we define H̃ Rℤ2 (a, b) = √2(Hℤ0 (a, b) − E) + m2 − m + V, 2

(4.8.12)

where E = inf Spec (Hℤ0 2 (0, b0 )), and Hℤ0 2 (a, b) − E ≥ 0 follows from (4.8.11). Thus H̃ Rℤ2 (0, b0 ) = √2(Hℤ2 (0, b0 ) − E) + m2 − m + V

(4.8.13)

is well-defined. From the Feynman–Kac formula in Theorem 4.252 we immediately obtain Corollary 4.253 (Diamagnetic inequality). Under the assumptions of Theorem 4.252, |(f , e−t HRℤ2 (a,b) g)| ≤ (|f |, e−t HRℤ2 (0,b0 ) |g|) ̃

̃

(4.8.14)

holds. Using this inequality we can now give a similar result to Theorem 4.242. Theorem 4.254 (Feynman–Kac formula for relativistic Schrödinger operator with singular external potential, singular vector potential, and spin 1/2). Suppose that a ∈ (L2loc (ℝ3 ))3 , ∇ ⋅ a ∈ L1loc (ℝ3 ), and b ∈ (L∞ (ℝ3 ))3 . (1) Let V be a relatively bounded (resp. form-bounded) potential with respect to √−Δ + m2 with a relative bound c. Then V is also relatively bounded (resp. form0 bounded) with respect to HRℤ 2 (a, b) with a relative bound not exceeding c. (2) Let V = V+ − V− be such that V+ ∈ L1loc (ℝ3 ) and V− is relatively form-bounded with respect to √−Δ + m2 with a relative bound less than 1. Then 0 HRℤ (a, b) +̇ V+ −̇ V− 2

(4.8.15)

can be defined as a self-adjoint operator. This is also denoted by HRℤ2 (a, b).

380 | 4 Feynman–Kac formulae (3) Let V satisfy the same assumptions as in (2). Then the Feynman–Kac formula of (f , e−tHRℤ2 (a,b) g) is given by (4.8.10). Proof. Denote S = − 21 σ ⋅ b0 and E = inf Spec(Hℤ0 2 (0, b0 )). Note that ‖√−Δ + m2 f ‖2 = ‖√2Hℤ0 (0, b0 ) − 2E + m2 f ‖2 + (f , −2(S − E)f ). 2

Since |(f , 2Sf )| ≤ κ‖f ‖2 with a constant κ, we have ‖√−Δ + m2 f ‖2 ≤ ‖√2Hℤ0 (0, b0 ) − 2E + m2 f ‖2 + (|2E| + κ)‖f ‖2 . 2

Together with ‖Vf ‖ ≤ c‖√−Δ + m2 f ‖ + ε‖f ‖2 with a constant ε, we have ‖Vf ‖ ≤ c‖√2Hℤ0 (0, b0 ) − 2E + m2 f ‖ + C‖f ‖ 2

(4.8.16)

with a constant C. By (4.8.16) and the diamagnetic inequality in Corollary 4.253, V is also relatively bounded with respect to √2Hℤ0 (a, b) − 2E + m2 with the same relative 2 bound c. Moreover, since the difference T = √2Hℤ0 (a, b) − 2E + m2 − √2Hℤ0 (a, b) + m2 2

2

is a bounded operator with ‖T‖ ≤ √|2E|, V is also relatively bounded with respect to √2Hℤ0 2 (a, b) + m2 with relative bound c. Thus (1) follows. The form version is similarly proven. The proofs of (2) and (3) are the same as those of (2) and (3) of Theorem 4.242. Essential self-adjointness of √(σ ⋅ (−i∇ − a))2 + m2 −m+V on ℂ2 ⊗C0∞ (ℝ3 ) can also be obtained from Proposition 4.238 and (1) of Theorem 4.254. Theorem 4.255 (Essential self-adjointness of relativistic Schrödinger operator with spin). Let V be relatively bounded with respect to √−Δ + m2 with a relative bound less than one. Suppose that a ∈ (L4loc (ℝ3 ))3 , ∇ ⋅ a ∈ L2loc (ℝ3 ), and ∇ × a = b satisfies b ∈ (L∞ (ℝ3 ))3 . Then √(σ ⋅ (−i∇ − a))2 + m2 − m + V is essentially self-adjoint on ℂ2 ⊗ C0∞ (ℝ3 ). Proof. By Proposition 4.247 we can see that √(σ ⋅ (−i∇ − a))2 + m2 is essentially self0 adjoint on ℂ2 ⊗ C0∞ (ℝ3 ). Note that √(σ ⋅ (−i∇ − a))2 + m2 − m = HRS (a, b), since the 2 0 operator (σ ⋅ (−i∇ − a)) can be expanded and equals HS (a, b). (1) of Theorem 4.254 yields that V is relatively bounded with respect to √(σ ⋅ (−i∇ − a))2 + m2 with a relative bound less than one. Hence the theorem follows by the Kato–Rellich theorem. This leads us to the following definition. Definition 4.256 (Relativistic Schrödinger operator with singular external potential, singular vector potential, and spin 1/2). Suppose that a ∈ (L2loc (ℝ3 ))3 and the magnetic field b satisfies b ∈ (L∞ (ℝ3 ))3 . Let V = V+ − V− be such that V+ ∈ L1loc (ℝ3 ) and V− is relatively form-bounded with respect to √−Δ + m2 with a relative bound less than 1. Then HRℤ2 (a, b) is defined by (4.8.15).

4.8 Feynman–Kac formula for relativistic Schrödinger operators with spin

| 381

4.8.2 Martingale properties In this section we consider martingale properties for the processes associated with HRℤ2 , similarly to relativistic Schrödinger operators and Schrödinger operators with spin discussed before. Let φ be an eigenfunction of HRℤ2 (a, b) such that HRℤ2 (a, b)φ = Eφ with eigenvalue E ∈ ℝ. Consider the stochastic process (Yt )t≥0 defined by Yt = etE eTt eZt φ(q̃ t ), ̃

t ≥ 0,

on (X × 𝒮 × 𝒯 , ℬ(X ) × ℬ𝒮 × ℬ𝒯 , 𝒲 × PN × PT ). Furthermore, for (x, α) ∈ ℝ3 × {1, 2} we define Yt (x, α) = etE eTt eZt (x,α) φ(q̃ t (x, α)), ̃

t ≥ 0,

where q̃ t (x, α) = (BTt + x, θα+NT ) and Z̃ t (x, α) = Z̃ tV (x) + Z̃ tA (x) + Z̃ tS (x, α) is given by t

t

Z̃ tV (x) = − ∫ V(BTs + x)ds, 0

Tt

Z̃ tA (x) = −i ∫ a(Bs + x) ∘ dBs ,

Tt

Tt +

0

0

0

Z̃ tS (x, α) = − ∫ Ud (Bs + x, θα+Ns )ds + ∫ log (−Uo (Bs + x, −θα+Ns− )) dNs . Here the functions Uo and Ud mapping ℝ3 × ℤ2 to ℝ are given by Ud (x, Y) = − 21 Yb3 (x) and Uo (x, −Y) = − 21 (b1 (x) − iYb2 (x)). We have 𝔼x,α,0 [Yt ] = 𝔼0,0,0 [Yt (x, α)] = φ(x, θα ). We introduce a filtration under which (Yt )t≥0 is a martingale. Define Yt (ϖ) and Yt (x, α, ϖ) for every ϖ ∈ 𝒯 by Yt and Yt (x, α), respectively, with the subordinator Tt replaced by the random number Tt (ϖ) ≥ 0. For each ϖ ∈ 𝒯 let 𝒢t (ϖ) = σ((Br , Nr ), 0 ≤ r ≤ Tt (ϖ)) ⊂ ℬ(X ) × ℬ𝒮 (1)

and define (1)

𝒢t

= {E ∈ ℬ(X ) × ℬ𝒮 × ℬ𝒯 | Eϖ1 ∈ 𝒢t(1) (ϖ) ∀ϖ ∈ 𝒯 },

where Eϖ1 = {(ω, η) ∈ X × ℬ𝒮 | (ω, η, ϖ) ∈ E}.

382 | 4 Feynman–Kac formulae We also define (2)

𝒢t

2 = {E ∈ ℬ(X ) × ℬ𝒮 × ℬ𝒯 | Eω,η ∈ σ(Tr : 0 ≤ r ≤ t) ∀(ω, η) ∈ X × 𝒮 },

2 where Eω,η denotes the section of E at (ω, η) ∈ X × 𝒮 , i. e., 2 Eω,η = {ϖ ∈ 𝒯 | (ω, η, ϖ) ∈ E}.

We see that 𝒢t(1) and 𝒢t(2) are sub-σ-fields of ℬ(X )× ℬ𝒮 × ℬ𝒯 . Define the filtration (𝒢t )t≥0 by (1)

(2)

𝒢t = 𝒢t ∩ 𝒢t ,

t ≥ 0.

The conditional expectation 𝔼0,0,0 [Yt (x, α)|𝒢t(1) ] = 𝔼0,0,0 [Yt (x, α)|𝒢t(1) ](⋅, ⋅, ⋅) is a random process on X × 𝒮 × 𝒯 and 𝔼0,0 [Yt (x, α, ϖ)|𝒢t(1) (ϖ)](⋅, ⋅) on X × 𝒮 . The relationship is given in the following lemma. Lemma 4.257. It follows that 𝔼0,0,0 [Yt (x, α)|𝒢t(1) ](⋅, ⋅, ϖ) = 𝔼0,0 [Yt (x, α, ϖ)|𝒢t(1) (ϖ)](⋅, ⋅)

(4.8.17)

for a.e. ϖ ∈ 𝒯 . Proof. Let A ∈ 𝒢t(1) with section Aϖ ∈ 𝒢t(1) (ϖ), and C ∈ ℬ𝒯 . We have 𝔼0,0,0 [1C 1A Yt (x, α)] = ∫ 1C 𝔼0,0 [1Aϖ Yt (x, α, ϖ)]dPT (ϖ) 𝒯

= ∫ 1C 𝔼0,0 [1Aϖ (⋅, ⋅)𝔼0,0 [Yt (x, α, ϖ)|𝒢t(1) (ϖ)] (⋅, ⋅)] dPT (ϖ). 𝒯

On the other hand, we have 𝔼0,0,0 [1C 1A Yt (x, α)] = 𝔼0,0,0 [1C 1A 𝔼0,0,0 [Yt (x, α)|𝒢t(1) ]] = ∫ 1C 𝔼0,0 [1Aϖ (⋅, ⋅)𝔼0,0,0 [Yt (x, α)|𝒢t(1) ] (⋅, ⋅, ϖ)] dPT (ϖ). 𝒯

Since C ∈ ℬ𝒯 is arbitrary, a comparison of the two sides above gives 𝔼0,0 [1Aϖ (⋅, ⋅)𝔼0,0 [Yt (x, α, ϖ)|𝒢t(1) (ϖ)] (⋅, ⋅)] = 𝔼0,0 [1Aϖ (⋅, ⋅)𝔼0,0,0 [Yt (x, α)|𝒢t(1) ] (⋅, ⋅, ϖ)] .

(4.8.18)

Let A = A󸀠 × 𝒯 be a rectangle. We have Aϖ = A󸀠 ∈ ℬ(X ) × ℬ𝒮 for all ϖ ∈ 𝒯 . Substituting A into (4.8.18), we have 𝔼0,0 [1A󸀠 (⋅, ⋅)𝔼0,0 [Yt (x, α, ϖ)|𝒢t(1) (ϖ)] (⋅, ⋅)] = 𝔼0,0 [1A󸀠 (⋅, ⋅)𝔼0,0,0 [Yt (x, α)|𝒢t(1) ] (⋅, ⋅, ϖ)] . Since A󸀠 ∈ ℬ(X ) × ℬ𝒮 is arbitrary, (4.8.17) follows.

| 383

4.8 Feynman–Kac formula for relativistic Schrödinger operators with spin

Lemma 4.258 (Martingale properties). The random process (Yt (x, α))t≥0 is a martingale with respect to (𝒢t )t≥0 , i. e., 𝔼0,0,0 [Yt (x, α)|𝒢s ] = Ys (x, α) for t ≥ s. Proof. We show the case when (x, α) = (0, 0) to keep the notation simple; the proof for (x, α) ≠ (0, 0) is similar. Note that 𝔼0,0,0 [Yt |𝒢s ] = 𝔼0,0,0 [Yt |𝒢s(1) ∩ 𝒢s(2) ] = 𝔼0,0,0 [𝔼0,0,0 [Yt |𝒢s(1) ]|𝒢s(2) ]. We first compute 𝔼0,0 [Yt (ϖ)|𝒢s(1) (ϖ)]. Write Tv

v

Tv

Tv +

Z̃ t ([u, v]) = − ∫ V(BTr )dr − i ∫ a(Br ) ∘ dBr − ∫ Ud (Br , θNr )dr + ∫ log(−Uo (Br , −θNr− ))dNr u

Tu

Tu

Tu

and, for every ϖ ∈ 𝒯 , Tv (ϖ)

v

Z̃ t ([u, v], ϖ) = − ∫ V(BTr (ϖ) )dr − i ∫ a(Br ) ∘ dBr u

Tu (ϖ)

Tv (ϖ)

Tv (ϖ)+

Tu (ϖ)

Tu (ϖ)

− ∫ Ud (Br , θNr )dr + ∫ log(−Uo (Br , −θNr− ))dNr and q̃ t (ϖ) = (BTt (ϖ) , θNT (ϖ) ), t ≥ 0. Since Tt (ϖ) is nonrandom, we see in a similar way t to the nonrelativistic case that 𝔼0,0 [Yt (ϖ)|𝒢s(1) (ϖ)] t

= etE eTt (ϖ) eZt ([0,s],ϖ) 𝔼BTs (ϖ) ,NTs (ϖ) [e− ∫s V(BTr (ϖ)−Ts (ϖ) )dr e ̃

×e

T (ϖ)

− ∫T t(ϖ) Ud (Br−Ts (ϖ) ,θNr−T s

s (ϖ)

)dr

e

T (ϖ)

−i ∫T t(ϖ) a(Br−Ts (ϖ) )∘dBr s

T (ϖ)+

∫T t(ϖ) log(−Uo (Br−Ts (ϖ) ,−θN(r−T s

s (ϖ))−

))dNr

× φ(BTt (ϖ)−Ts (ϖ) , θNT (ϖ)−T (ϖ) ) ] , s

t

where we used the Markov property of ((Bt , Nt ))t≥0 . Hence by Lemma 4.257 we have 𝔼0,0,0 [Yt |𝒢s(1) ] = etEp eTs eZt ([0,s]) Zt,s , ̃

where t

Zt,s = eTt −Ts 𝔼BTs ,NTs [e− ∫s V(BTr −Ts )dr e ×e

T

T

−i ∫T t a(Br−Ts )∘dBr s

T +

− ∫T t Ud (Br−Ts ,θNr−T )dr ∫T t log(−Uo (Br−Ts ,−θN(r−T s

Here Zt,s is given by

s

e

s

s )−

))dNr

φ(BTt −Ts , θNT −T )] . t

s

384 | 4 Feynman–Kac formulae t

u

eu−v 𝔼Bu ,Nv [e− ∫s V(BTr −u )dr e−i ∫v u

× e− ∫v

a(Br−v )∘dBr

u+

Ud (Br−v ,θNr−v )dr ∫v log(−Uo (Br−v ,−θN(r−v)− ))dNr

e

φ(Bu−v , θNu−v )]

evaluated at u = Tt and v = Ts . We note that 𝔼0,0,0 [f |𝒢s(2) ](ω1 , ω2 , ⋅) = 𝔼0P [f (ω1 , ω2 , ⋅)|Ns ](⋅),

(4.8.19)

where Ns = σ(Tr , 0 ≤ r ≤ s). Since etE eTs eZt ([0,s]) is measurable with respect to 𝒢s(2) , by (4.8.19) we consider the conditional expectation of Zt,s giving ̃

t

T −Ts

t 󵄨 𝔼0,0,0 [Zt,s 󵄨󵄨󵄨󵄨𝒢s(2) ] = 𝔼0P [eTt −Ts 𝔼BTs ,NTs [e− ∫s V(BTr −Ts )dr e−i ∫0 Tt −Ts

× e− ∫0

(Tt −Ts )+

Ud (Br ,θNr )dr ∫0

e

a(Br )∘dBr

log(−Uo (Br ,−θNr− ))dNr

󵄨󵄨 φ(BTt −Ts , θNT −T )]󵄨󵄨󵄨󵄨 Ns ] . t s 󵄨

By the Markov property of (Tt )t≥0 we have t

T

Tt−s

= 𝔼Ps [eTt−s 𝔼BT0 ,N0 [e− ∫s V(BTr−s )dr e−i ∫0 Tt−s

× e− ∫0

T

a(Br )∘dBr

Ud (Br ,θNr )dr ∫0 t−s log(−Uo (Br ,−θNr− ))dNr

e

+

φ(BTt−s , θNT )]] . t−s

Since 𝔼uP [f (T⋅ )] = 𝔼0P [f (T⋅ + u)], we see that t

Tt−s −T0

= 𝔼0P [eTt−s −T0 𝔼BT0 +u ,NT0 +u [e− ∫s V(BTr−s −T0 )dr e−i ∫0 Tt−s −T0

× e− ∫0

= 𝔼BTs ,NTs ,0 [eTt−s e Tt−s

× e− ∫0

(Tt−s −T0 )+

Ud (Br ,θNr )dr ∫0

e

t−s − ∫0

V(BTr )dr T

e

log(−Uo (Br ,−θNr− ))dNr

T −i ∫0 t−s

+

φ(BTt−s −T0 , θNT

t−s −T0

)]]⌈

u=Ts

a(Br )∘dBr

Ud (Br ,θNr )dr ∫0 t−s log(−Uo (Br ,−θNr− ))dNr

e

a(Br )∘dBr

φ(q̃ t−s )]

= (e−(t−s)HRℤ2 (a,b) φ) (q̃ s ). Hence we conclude that 𝔼0,0,0 [Yt |𝒢s ] = esE eTs eZt ([0,s]) (e−(t−s)(HRℤ2 (a,b)−E) φ)(q̃ s ) = Ys ̃

and the result follows.

4.8.3 Decay of eigenfunctions In this section we estimate the decay of eigenfunctions of HRℤ2 by using the martingale properties established in Lemma 4.258. We will use the following assumptions.

4.8 Feynman–Kac formula for relativistic Schrödinger operators with spin

|

385

Assumption 4.259. Let the following properties hold: (1) b ∈ (L∞ (ℝ3 ))3 ; (2) m∗ < m2 /2, where

1 ‖√b21 + b22 ‖2∞ + 1; 4

m∗ = ‖b3 ‖∞ +

(3) V is uniformly locally integrable and is in relativistic Kato-class, i. e., t

lim sup 𝔼x,0 [∫ V(BTr )dr ] = 0. t↓0 x∈ℝ3 [0 ] In (4.7.39) we define E ∗ = m∗ /2. Lemma 4.260. If Assumption 4.259 holds, then φ ∈ L∞ (ℝd ) and t∧τ

|φ(x, θα )|2 ≤ C‖φ‖2 𝔼0,0 [e2(t∧τ)E e−2 ∫0

V(BTr +x)dr

] 𝔼0,0 [em

∗

Tt∧τ

(4.8.20)

],

for every stopping time τ with respect to (𝒢s )s≥0 and t ≥ 0. Here C > 0 is a constant. Proof. Let W(X) = 21 √b1 (X)2 + b2 (X)2 . Recall that Yt = etE eTt eZt φ(q̃ t ), and note that ̃

φ(x, θα ) = 𝔼x,α,0 [Yt ] for every t. The Schwarz inequality yields t

Tt

|φ(x, θα )|2 ≤ e2tE 𝔼x,α,0 [e2Tt e−2 ∫0 V(BTr )dr e∫0 t

T +

|b3 (Br )|dr 2 ∫0 t log W(Br )dNr

e

] 𝔼x,α,0 [|φ(q̃ t )|2 ]

≤ e2tE 𝔼0,0 [e−2 ∫0 V(BTr +x)dr eTt m ]𝔼x,α,0 [|φ(q̃ t )|2 ].

(4.8.21)

∗

Note that ∞

∞

𝔼x,α,0 [|φ(q̃ t )|2 ] = ∫ ρ(s, t)ds ∫ Πs (y)dy ∑ |φ(x + y, θα+n )|2 0

n=0

ℝ3 2

2

sn −s e n!

∞

≤ ∫ (|φ(x + y, 1)| + |φ(x + y, −1)| )dy ∫ ρ(s, t)Πs (y)ds 0

ℝd

2

≤ C‖φ‖

with a constant C, and where ρ(⋅, t) is the distribution of Tt . Furthermore, take q and p m2 1 1 such that 2m ∗ > q > 1 and p + q = 1. By the Hölder inequality we get t

t

𝔼x,0 [e−2 ∫0 V(BTr )dr eTt m ] ≤ (𝔼x,0 [e−2p ∫0 V(BTr )dr ]) ∗

1/p

1/q

(𝔼x,0 [eqTt m ]) ∗

.

386 | 4 Feynman–Kac formulae The first term at the right-hand side above satisfies 1/p

t

sup (𝔼x,0 [e−2p ∫0 V(BTr )dr ])

< ∞,

x∈ℝd

since V is of relativistic Kato-class, and 𝔼x,0 [eqTt m ] = 𝔼0ν [eqTt m ] = e+t(m− ∗

∗

√m2 −2qm∗ )

< ∞.

Hence φ ∈ L∞ (ℝd ). Note that by the martingale property of Yt (x, α), φ(x, θα ) = 𝔼0,0,0 [Yt∧τ (x, α)] for every stopping time τ and t ≥ 0. The estimate (4.8.20) follows from (4.8.21) with t replaced by t ∧ τ. Choosing a suitable stopping time we can estimate the decay of eigenfunctions of HRℤ2 . Lemma 4.261. Let τR = inf{t | |BTt | > R}. Then τR is a stopping time with respect to the filtration (𝒢t )t≥0 . Proof. It suffices to show that {τR ≤ t} ∈ 𝒢t . Note that {τR ≤ t} = ⋃ (A(ϖ), ϖ), ϖ∈𝒯

where A(ϖ) = {ω ∈ X | sup0≤s≤t |BTs (ϖ) (ω)| > R} × 𝒮 ∈ 𝒢t(1) (ϖ). Thus it follows that {τR ≤ t} ∈ 𝒢t(1) . Moreover,

{τR ≤ t} =

⋃

(ω, η, B(ω)),

(ω,η)∈X ×𝒮

where B(ω) = {ϖ ∈ 𝒯 | sup0≤s≤t |BTs (ϖ) (ω)| > R}. Therefore {τR ≤ t} ∈ 𝒢t(2) and hence {τR ≤ t} ∈ 𝒢t . Theorem 4.262 (Confining potentials). Suppose that lim|x|→∞ V(x) = ∞ and let Assumption 4.259 hold. Then for every a > 0 there exists b > 0 such that |φ(x, θα )| ≤ be−a|x| . Proof. We have by Lemma 4.260 t∧τR

|φ(x, θα )| ≤ √C‖φ‖ (𝔼0,0 [e2E(t∧τR ) e−2 ∫0

V(BTr +x)dr

1/2

])

(𝔼0,0 [em

∗

Tt∧τR

])

1/2

,

t ≥ 0.

Let W(x) = WR (x) = inf{V(y) | |x − y| < R}, and note that lim|x|→∞ W(x) − E = ∞. In particular, we may assume that W(x) − E > 0. This gives t∧τR

(𝔼0,0 [e2E(t∧τR ) e−2 ∫0

V(BTr +x)dr

1/2

])

4.8 Feynman–Kac formula for relativistic Schrödinger operators with spin

≤ (𝔼0,0 [1{τR τR } e2(E+ε)τR ] ≤ e2(E+ε)t + C1 e−mε |x| ,

t ≥ 0,

where mε = {

m 2√m|E| − |E|2

if 2|E| > m, if 2|E| ≤ m.

Also, note that 𝔼x,0 [em

∗

Tt∧τR

] ≤ 2e(m−

√m2 −2m∗ )t

.

Therefore, √m2 −2m∗ )t/2

|φ(x, θα )| ≤ (e(E+ε)t + C1 e−mϵ |x|/2 )√2e(m−

.

Upon inserting t = δ|x| with sufficiently small δ, the theorem follows from (4.8.25).

4.9 Feynman–Kac formula for nonlocal Schrödinger operators 4.9.1 Nonlocal Schrödinger operators Schrödinger operators with other kinetic terms than (−Δ)1/2 , which were discussed in Section 4.6, are interesting for further applications in mathematical physics and the theory of stochastic processes. In this section we consider more general cases which we call nonlocal Schrödinger operators. Recall that an operator A is called local whenever supp Af ⊂ supp f for all f ∈ D(A), and nonlocal otherwise. For definiteness, we will consider only a specific class of nonlocal Schrödinger operators. Recall from from (3.6.16) that ℬ0 describes the set of Bernstein functions Ψ such that limu→0+ Ψ(u) = 0.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 389

Definition 4.264 (Nonlocal Schrödinger operator). Let Ψ ∈ ℬ0 and V ∈ L∞ (ℝd ). We call the operator 1 H Ψ = Ψ (− Δ) + V 2

(4.9.1)

nonlocal Schrödinger operator with Bernstein function Ψ of the Laplacian. Note that Ψ (− 21 Δ) is defined through the spectral projection of the Laplacian. Furthermore, since V is bounded, H Ψ is self-adjoint on D(Ψ(− 21 Δ)). Let (TtΨ )t≥0 be the subordinator defined on a probability space (𝒯 , ℬ𝒯 , P) and uniquely associated with Ψ, see Proposition 3.99. Recall subordinate Brownian motion (XtΨ )t≥0 = (BT Ψ )t≥0 ,

(4.9.2)

t

as given in Definition 3.91. We have the following probabilistic representation of nonlocal Schrödinger operators with Bernstein functions of the Laplacian. Theorem 4.265 (Feynman–Kac formula for nonlocal Schrödinger operator). Suppose that V ∈ L∞ (ℝd ). Then we have for f , g ∈ L2 (ℝd ), t

Ψ

Ψ

Ψ − ∫0 V(Xs )ds Ψ (f , e−tH g) = ∫ 𝔼x,0 ] dx. 𝒲×P [f (X0 )g(Xt )e

(4.9.3)

ℝd

Proof. Replacing the subordinator (Tt )t≥0 in Theorem 4.208 with (TtΨ )t≥0 , the proof is similar. 4.9.2 Vector potentials In this section we consider nonlocal Schrödinger operators with vector potential a and their Feynman–Kac formulae. We assume throughout that a ∈ (L2loc (ℝd ))d . By Definition 4.188 we denote the self-adjoint operator h in (4.4.15) by h(a). Definition 4.266 (Nonlocal Schrödinger operator with vector potential). Let Ψ ∈ ℬ0 and assume a ∈ (L2loc (ℝd ))d . Whenever V is a real-valued bounded potential, we define the nonlocal Schrödinger operator with vector potential a by H Ψ (a) = Ψ(h(a)) + V.

(4.9.4)

Lemma 4.267. Take Ψ ∈ ℬ0 . (1) If a ∈ (L4loc (ℝd ))d and ∇ ⋅ a ∈ L2loc (ℝd ), then C0∞ (ℝd ) is a core of Ψ(h(a)). (2) If a ∈ (L2loc (ℝd ))d , then C0∞ (ℝd ) is a form core of Ψ(h(a)). Proof. (1) There exist nonnegative constants c1 and c2 such that Ψ(u) ≤ c1 u + c2 for all u ≥ 0. This gives the bound ‖Ψ(h(a))f ‖ ≤ c1 ‖h(a)f ‖ + c2 ‖f ‖

(4.9.5)

390 | 4 Feynman–Kac formulae for all f ∈ D(h(a)). Hence it can be proven that C0∞ (ℝd ) is contained in D(Ψ(h(a))) and (Ψ(h(a)) + 1)C0∞ (ℝd ) is dense. Thus (1) follows. (2) Note that ‖Ψ(h(a))1/2 f ‖2 ≤ c1 ‖h(a)1/2 f ‖2 + c2 ‖f ‖2 for f ∈ Q(h(a)) = D(h(a)1/2 ), and C0∞ (ℝd ) is contained in Q(Ψ(h(a))) = D(Ψ(h(a))1/2 ). Since Ψ(h(a))1/2 + 1 has also a bounded inverse, it is seen by the same argument as above that C0∞ (ℝd ) is a core of Ψ(h(a))1/2 or a form core of Ψ(h(a)). Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). We have t

(f , e−th(a) g) = ∫ 𝔼x [f (B0 )g(Bt )e−i ∫0 a(Bs )∘dBs ] dx.

(4.9.6)

ℝd

We turn to constructing a functional integral representation for nonlocal Schrödinger operators defined by (4.9.4). Theorem 4.268 (Feynman–Kac formula for nonlocal Schrödinger operator with singular vector potential). Suppose that Ψ ∈ ℬ0 and V ∈ L∞ (ℝd ). Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Then we have (f , e−tH

Ψ

(a)

TtΨ

−i ∫0 g) = ∫ 𝔼x,0 𝒲×P [f (B0 )g(BT Ψ )e t

ℝd

t

a(Bs )∘dBs − ∫0 V(BTsΨ )ds

e

] dx.

(4.9.7)

Proof. The proof is similar to Theorem 4.208 applying TtΨ instead of Tt . Corollary 4.269 (Diamagnetic inequality). Let Ψ ∈ ℬ0 and V ∈ L∞ (ℝd ), and suppose that a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Then |(f , e−tH

Ψ

(a)

g)| ≤ (|f |, e−tH

Ψ

(0)

(4.9.8)

|g|)

and the energy comparison inequality inf Spec(H Ψ (0)) ≤ inf Spec(H Ψ (a)) holds. Proof. By Theorem 4.268 we have |(f , e−tH

Ψ

(a)

g)| ≤ ∫ 𝔼x,0 𝒲×P [|f (B0 )||g(BT Ψ )|e ℝd

t

t

− ∫0 V(BT Ψ )ds s

] dx.

The right-hand side above coincides with that of (4.9.8), and the second statement follows directly from (4.9.8). By making use of the functional integral representation we can now also consider more singular external potentials. Lemma 4.270. Let a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). (1) Suppose that V is relatively form-bounded with respect to Ψ(− 21 Δ) with relative bound b. Then V is also relatively form-bounded with respect to Ψ(h(a)) with a relative bound not larger than b.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 391

(2) Suppose that V is relatively bounded with respect to Ψ(− 21 Δ) with relative bound b. Then V is also relatively bounded with respect to Ψ(h(a)) with a relative bound not larger than b. Proof. In virtue of Corollary 4.269 we have 1

|(f , e−tΨ(h(a)) g)| ≤ (|f |, e−tΨ(− 2 Δ) |g|). Then the proof is similar to Theorem 4.192. Theorem 4.271 (Self-adjointness for H Ψ (a)). (1) Suppose that a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ), and let V be relatively bounded with respect to Ψ(− 21 Δ) with relative bound strictly smaller than one. Then H Ψ (a) is self-adjoint on D(Ψ(h(a))) and bounded from below. Moreover, it is essentially self-adjoint on any core of Ψ(h(a)). (2) Suppose, furthermore, that a ∈ (L4loc (ℝd ))d and ∇ ⋅ a ∈ L2loc (ℝd ). Then C0∞ (ℝd ) is an operator core of H Ψ (a). Proof. By (2) of Lemma 4.270, V is also relatively bounded with respect to Ψ(h(a)) with a relative bound strictly smaller than one. Then (1) follows from the Kato–Rellich theorem. Statement (2) follows from Lemma 4.267. Lemma 4.270 also allows Ψ(h(a)) + V be defined in form sense. Let V = V+ − V− . Lemma 4.270 implies that whenever V− is form-bounded with respect to Ψ(− 21 Δ) with a relative bound strictly smaller than one, it is also form-bounded with respect to Ψ(h(a)) with a relative bound strictly smaller than one. Moreover, assume that V+ ∈ L1loc (ℝd ). We see that for a ∈ (L2loc (ℝd ))d , Q(Ψ(h(a)))∩Q(V+ ) ⊃ C0∞ (ℝd ) by Lemma 4.267. In particular, Q(Ψ(h(a))) ∩ Q(V+ ) is dense. Define the quadratic form q(f , f ) = (Ψ(h(a))1/2 f , Ψ(h(a))1/2 f ) + (V+1/2 f , V+1/2 f ) − (V−1/2 f , V−1/2 f )

(4.9.9)

on Q(Ψ(h(a))) ∩ Q(V+ ). Definition 4.272 (Nonlocal Schrödinger operator with singular external potential and singular vector potential). Let a ∈ (L2loc (ℝd ))d , ∇ ⋅ a ∈ L1loc (ℝd ), and V = V+ − V− be such that V+ ∈ L1loc (ℝd ) and V− is form-bounded with respect to Ψ(− 21 Δ) with a relative bound strictly smaller than 1. We denote the self-adjoint operator associated with the quadratic form q in (4.9.9) by Ψ(h(a)) +̇ V+ −̇ V− . Since we need the conditions a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ) to show the relative form-boundedness of V− with respect to Ψ(h(a)), a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ), they are assumed in Definition 4.272. Now we extend Theorem 4.268 to potentials expressed in terms of form sums.

392 | 4 Feynman–Kac formulae Theorem 4.273 (Feynman–Kac formula for nonlocal Schrödinger operator with singular external potential and singular vector potential). Suppose that a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Let V = V+ − V− be such that V+ ∈ L1loc (ℝd ) and V− is infinitesimally small with respect to Ψ(− 21 Δ) in form sense. Then the functional integral representation given by Theorem 4.268 also holds for Ψ(h(a)) +̇ V+ −̇ V− . Proof. The proof is similar to Theorem 4.194. 4.9.3 Ψ-Kato-class potentials Next we consider an appropriate extension of Kato-class to subordinate Brownian motion. Since our input is an operator and not directly a process, we need the following conditions. Assumption 4.274. Let Ψ ∈ ℬ0 be such that 2

∫ e−tΨ(|ξ | /2) dξ < ∞,

t > 0.

(4.9.10)

ℝd

Let Ψ ∈ ℬ0 and (b, λ) ∈ ℝ+ × L be its corresponding nonnegative drift coeffi∞ cient and Lévy measure, i. e., Ψ(u) = bu + ∫0 (1 − e−uy ) λ(dy). It is clear that if b > 0, 1

then (4.9.10) is satisfied. If b = 0 but ∫0 λ(dy) < ∞, (4.9.10) is not satisfied, since 1

supu≥0 Ψ(u) < ∞. Thus Ψ obeying (4.9.10) at least satisfies ∫0 λ(dy) = ∞ when b = 0. In this case we have 1

1

0

0

1

2 |u|2 |u|2 y Ψ( ) ≥ ∫(1 − e−|u| y/2 )λ(dy) ≥ (1 − e−1 ) ∫ ( ∧ 1) λ(dy) ≥ (1 − e−1 ) ∫ λ(dy). 2 2

1

2/|u|2 1

Thus for b = 0 and ∫0 λ(dy) = ∞, assuming that there exists ρ such that ∫2/|u|2 λ(dy) ≥ ρ(u) and ∫ℝd e−tρ(ξ ) dξ < ∞, Assumption 4.274 holds. Under this assumption we define pΨ t (x) =

2 1 ∫ e−ix⋅ξ e−tΨ(|ξ | /2) dξ d (2π)

(4.9.11)

ℝd

and ∞

GλΨ (x) = ∫ e−λt pΨ t (x)dt.

(4.9.12)

0

Ψ The function pΨ t (x) denotes the probability transition kernel of (Xt )t≥0 and the function GλΨ is the integral kernel of the resolvent (Ψ(− 21 Δ) + λ)−1 with λ > 0, i. e., −1 1 (f , (Ψ(− Δ) + λ) g) = 2

∫ f (x)g(y)GλΨ (x − y)dxdy. ℝd ×ℝd

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 393

Ψ ∞ d Clearly, pΨ t (x) and Gλ (x) are spherically symmetric. For f ∈ C0 (ℝ ) it follows that Ψ 𝔼0,0 𝒲×P [f (Xt )] = ∫ f (x)pt (x)dx.

(4.9.13)

ℝd

Hence for nonnegative f ∈ C0∞ (ℝd ), the right-hand side of (4.9.13) is nonnegative and d thus pΨ t (x) ≥ 0 for almost every x ∈ ℝ . By a limiting argument with f → 1 we can also Ψ 1 d Ψ see that pt ∈ L (ℝ ) and ‖pt ‖L1 (ℝd ) = 1 by (4.9.13). We moreover compute GλΨ as GλΨ (x)

∞

1 1 r (d−1)/2 √r|x|J(d−2)/2 (r|x|)dr = ∫ (2π)d |x|(d−1)/2 λ + Ψ(r 2 /2) 0

with the Bessel function of the first kind π

Jν (s) =

∞ (s/2)ν (−1)n s 2n+ν ( ) . ∫ eis cos θ (sin θ)2ν dθ = ∑ 1 n!Γ(n + ν + 1) 2 √πΓ(ν + 2 ) n=0 0

Note that supu≥0 √uJν (u) < ∞. Let ‖f ‖l1 (L∞ ) = ∑ sup |f (x)|, z∈ℤd x∈Cz

where Cz denotes the unit cube centered at z ∈ ℤd . We further introduce a second assumption on the probability transition kernel pΨ t . Assumption 4.275. The probability transition kernel pΨ t satisfies for every δ > 0 sup ‖1{|x|>δ} pΨ t ‖l1 (L∞ ) < ∞. t>0

(4.9.14)

Let f be a real-valued function on ℝd . When r 󳨃→ f (rx) is nonincreasing on [0, ∞) for all x ∈ ℝd , we say that f is radially nonincreasing. In d = 1 for a radially nonincreasing L1 function f it can be seen, by the definition of l1 (L∞ ), that there exists a constant Cδ = Cδ (f ) such that for each δ > 0, ‖1{|x|>δ} f ‖l1 (L∞ ) ≤ Cδ ‖f ‖L1 .

(4.9.15)

In the general case d ≥ 2 it can be also seen that (4.9.15) holds for all radially nonincreasing f . In particular, Assumption 4.275 is satisfied whenever pΨ t is radially noninΨ creasing, since ‖pt ‖L1 = 1. Example 4.276 (α/2-stable subordinator). For Ψ(u) = uα/2 , 0 < α < 2, it is clear that Assumption 4.274 is satisfied. It is also known that the distribution density of BT Ψ t (which in this case is a symmetric α-stable process) is radially nonincreasing. This is obtained by a unimodality argument of spherically symmetric distribution functions.

394 | 4 Feynman–Kac formulae Example 4.277 (Relativistic α/2-stable subordinator). Let Ψ(u) = √2u + m2 −m, m ≥ 0. It is clear that Assumption 4.274 is satisfied. The distribution function pΨ t of BT Ψ is t

expressed as (4.6.27). Then pΨ t is indeed radially nonincreasing. We introduce three classes of potentials, given by {

t

󵄨󵄨 󵄨 󵄨

d

x,0

}

Ψ

𝒱1 = {V : ℝ → ℝ 󵄨󵄨󵄨 lim sup ∫ 𝔼𝒲×ν [V(Xs )]ds = 0} , 󵄨 t↓0 d x∈ℝ

0 } 󵄨󵄨 󵄨 󵄨󵄨 1 󵄨 󵄨 d −1 𝒱2 = {V : ℝ → ℝ 󵄨󵄨󵄨 lim sup 󵄨󵄨󵄨󵄨((Ψ(− Δ) + λ) V) (x)󵄨󵄨󵄨󵄨 = 0} , 󵄨󵄨 λ→∞ x∈ℝd 󵄨 2 󵄨

{

{ {

󵄨󵄨 󵄨 󵄨

d

𝒱3 = {V : ℝ → ℝ 󵄨󵄨󵄨 lim sup 󵄨 δ↓0 { d

{

x∈ℝ

} } G1Ψ (x − y)V(y)dy = 0} . } |x−y| 0 is given. (1) Let V ∈ 𝒱2 and λ > 0. We have t

Ψ ∫ 𝔼x,0 𝒲×P [V(Xs )]ds 0

t

≤ ∫e

λ(t−s)

0

Ψ 𝔼x,0 𝒲×P [V(Xs )]ds

λt

∞

Ψ ≤ e ∫ e−λs 𝔼x,0 𝒲×P [V(Xs )]ds. 0

Also, ∞

1 ε Ψ −1 sup ∫ e−λs 𝔼x,0 𝒲×P [V(Xs )]ds ≤ sup ((Ψ(− Δ) + λ) V) (x) < 2 2 x∈ℝd x∈ℝd 0

for sufficiently large λ. Let eλt < 2 with small enough t. Then we get t

Ψ ∫ 𝔼x,0 𝒲×P [V(Xs )]ds ≤ ε 0

for sufficiently small t, and V ∈ 𝒱1 follows. (2) Let α(V) = supx∈ℝd ∫x+C V(y)dy. We have 0

Ψ (pΨ s ∗ V)(x) = ∑ ∫ ps (y)V(x − y)dy z∈ℤd C

z

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 395

Ψ ≤ ∑ (sup pΨ s (x)) ∫ V(x − y)dy ≤ α(V)‖ps ‖l1 (L∞ ) . z∈ℤd

−sΨ(ξ Since |pΨ s (x)| ≤ ∫ℝd e

2

/2)

x∈Cz

Cz

dξ < ∞ by (4.9.10), we obtain Ψ ‖1|y|≤δ pΨ s ‖l1 (L∞ ) ≤ C‖ps ‖∞ < ∞

with C = C(δ). We furthermore have Ψ Ψ Ψ Ψ ‖pΨ s ‖l1 (L∞ ) ≤ ‖1|y|>δ ps ‖l1 (L∞ ) + ‖1|y|≤δ ps ‖l1 (L∞ ) ≤ ‖1|y|>δ ps ‖l1 (L∞ ) + C‖ps ‖∞ < ∞

by (4.9.14). In particular, we get Ψ Ψ Ψ sup 𝔼x,0 𝒲×P [V(Xs )] ≤ sup (ps ∗ V)(x) ≤ α(V)‖ps ‖l1 (L∞ ) .

x∈ℝd

x∈ℝd

(4.9.19)

t

Ψ Let V ∈ 𝒱1 and supx∈ℝd ∫0 𝔼x,0 𝒲×P [V(Xs )]ds ≤ ε/2 for sufficiently small t. We see that by (4.9.19) ∞

1 Ψ sup ((Ψ(− Δ) + λ)−1 V) (x) = sup ∫ e−λs 𝔼x,0 𝒲×P [V(Xs )]ds 2 x∈ℝd x∈ℝd 0

t

∞

Ψ −λs x,0 ≤ sup ∫ e−λs 𝔼x,0 𝔼𝒲×P [V(XsΨ )]ds 𝒲×P [V(Xs )]ds + sup ∫ e x∈ℝd

x∈ℝd

0

t

∞

≤

ε −λs + α(V) sup ‖pΨ ds ≤ ε s ‖l1 (L∞ ) ∫ e 2 s≥t t

for sufficiently large λ. Thus V ∈ 𝒱2 follows. Next let V ∈ 𝒱3 . We split the integral as ∫ G1Ψ (x − y)V(y)dy = ℝd

∫

G1Ψ (x − y)V(y)dy +

|x−y|≤δ

∫

G1Ψ (x − y)V(y)dy.

|x−y|>δ

Since λ → GλΨ (z) is a nonincreasing function, it follows that ε > sup 2 x∈ℝd

∫

G1Ψ (x − y)V(y)dy ≥ sup

x∈ℝd

|x−y| 0. On the other hand, we have by (4.9.19) that ∞

∫ |x−y|>δ

GλΨ (x − y)V(y)dy = ∫ e−λs ds ∫ pΨ s (y)V(x − y)dy 0

|y|>δ

396 | 4 Feynman–Kac formulae ∞

−λs ≤ α(V) (sup ‖1|y|>δ pΨ ds < s ‖l1 (L∞ ) ) ∫ e s>0

0

ε 2

for sufficiently large λ. Thus we obtain sup ∫ GλΨ (x − y)V(y)dy ≤ ε

x∈ℝd

ℝd

for large enough λ and V ∈ 𝒱2 follows. (3) Let V ∈ 𝒱2 . Choose λ such that supx∈ℝd ((Ψ(− 21 Δ) + λ)−1 V)(x) < ε. Note that |z| < δ(λ) with sufficiently small δ(λ) implies that G1Ψ (z) ≤ cGλΨ (z). Thus we see that sup

x∈ℝd

G1Ψ (x − y)V(y)dy ≤ sup c

∫

x∈ℝd

|x−y| 0, we have

󵄨󵄨 ∞ 󵄨󵄨 ∞ ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 −λs Ψ 󵄨󵄨 −λs Ψ Ψ e p (x)ds ≤ e p (0)ds ≤ (sup p (0)) e−λs ds < ∞. ∫ ∫ ∫ 󵄨󵄨 󵄨󵄨 s s s 󵄨󵄨 󵄨󵄨 s≥t 󵄨󵄨 t 󵄨󵄨 t t t

GλΨ0 (x) → ∞ implies that ∫0 e−λ0 s pΨ s (x)ds → ∞ as |x| → 0 for every t > 0. Hence GλΨ (x) → ∞ as |x| → 0 for all λ > 0. Moreover, from lim

|x|→0

G1Ψ (x) GλΨ (x)

t

= lim

|x|→0

−s Ψ ∫0 e−s pΨ s (x)ds + ∫t e ps (x)ds ∞

t

−λs pΨ (x)ds ∫0 e−λs pΨ s (x)ds + ∫t e s ∞

it follows that lim

|x|→0

G1Ψ (x) GλΨ (x)

t

= lim

|x|→0

∫0 e−s pΨ s (x)ds t

∫0 e−λs pΨ s (x)ds

.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 397

Let λ ≥ 1. We have lim|x|→0 G1Ψ (x)/GλΨ (x) ≥ 1. On the other hand, t

∫0 e−s pΨ s (x)ds

lim

t

∫0 e−λs pΨ s (x)ds

|x|→0

≤ e(λ−1)t .

Hence 1 ≤ lim

G1Ψ (x)

|x|→0

GλΨ (x)

≤ e(λ−1)t

and t is arbitrary. This means that (4.9.20) is satisfied. The case λ < 1 can be discussed similarly. Now we are able to define Kato-class potentials with respect to the operator Ψ(− 21 Δ). Definition 4.280 (Ψ-Kato-class and Ψ-Kato-decomposable potential). Let tions 4.274–4.275 hold. A potential V is Ψ-Kato-class if lim sup

δ↓0 x∈ℝd

∫

Assump-

G1Ψ (x − y)V(y)dy = 0.

|x−y| 0. We have GλΨ (x)

∞

∞

0

0

te−λt e−m|x| Ce−m|x| ue−λu|x| ≥C∫ dt = du. ∫ |x|d (1 + u2 )(d+1)/2 (√|x|2 + t 2 )d+1

Hence it follows that lim|x|→0 GλΨ (x) = ∞ for every λ > 0, and Proposition 4.279 yields that lim|x|→0 G1Ψ (x)/GλΨ (x) = 1. This implies that 𝒱2 ⊂ 𝒱3 , where 𝒱2 is given by (4.9.17) and 𝒱3 by (4.9.18). Next we show a Feynman–Kac formula for nonlocal Schrödinger operators with Bernstein functions of the Laplacian and Ψ-Kato-decomposable potentials. Lemma 4.284. Let V be a uniformly locally integrable and positive Ψ-Kato-class potential with Ψ ∈ ℬ0 , and let Assumptions 4.274–4.275 hold. Then for every t ≥ 0, t

Ψ

∫0 V(Xs )ds sup 𝔼x,0 ] < ∞. 𝒲×P [e

x∈ℝd

(4.9.23)

t

Ψ Proof. Since V satisfies limt↓0 supx∈ℝd ∫0 𝔼x,0 𝒲×P [V(Xs )]ds = 0, the lemma is proven in a similar way to Lemma 4.105 by an application of Khasminskii’s lemma.

The following expression then results in a similar way to Theorem 4.113. Theorem 4.285 (Feynman–Kac formula for nonlocal Schrödinger operator with Ψ-Katodecomposable potential). Let Ψ ∈ ℬ0 and Assumptions 4.274–4.275 hold. Let V be a uniformly locally integrable and Ψ-Kato-decomposable potential. Consider t

Ψ

− ∫0 V(Xs )ds KtΨ f (x) = 𝔼x,0 f (XtΨ )] 𝒲×P [e

(4.9.24)

for f ∈ L2 (ℝd ). Then {KtΨ : t ≥ 0} is a symmetric C0 -semigroup on L2 (ℝd ). Moreover, there exists a unique self-adjoint operator K Ψ bounded from below such that Ψ

KtΨ = e−tK ,

t ≥ 0.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 399

Definition 4.286 (Nonlocal Schrödinger operator with Ψ-Kato-decomposable potential). Let Ψ ∈ ℬ0 and V be a Ψ-Kato-decomposable potential. We define the nonlocal Schrödinger operator with Ψ-Kato-decomposable potential V by the self-adjoint operator K Ψ given in Theorem 4.285. The next result says that we can define a Feynman–Kac semigroup for Ψ-Katodecomposable potentials and singular external potentials. Theorem 4.287 (Feynman–Kac formula for nonlocal Schrödinger operator with Ψ-Katodecomposable potential and vector potential). Choose Ψ ∈ ℬ0 and let Assumptions 4.274–4.275 hold. Also, let V be Ψ-Kato-class and uniformly locally integrable potential, a ∈ (L2loc (ℝd ))d , and ∇ ⋅ a ∈ L1loc (ℝd ). Consider TtΨ

−i ∫0 KtΨ,a f (x) = 𝔼x,0 𝒲×P [e

t

a(Bs )∘dBs − ∫0 V(BTsΨ )ds

e

f (BT Ψ )] . t

Then {KtΨ,a : t ≥ 0} is a symmetric C0 -semigroup. In particular, there exists a self-adjoint operator K Ψ (a) bounded from below such that KtΨ,a = e−tK

Ψ

(a)

,

t ≥ 0.

Proof. The proof is a modification of the proofs of Theorems 4.196 and 4.184. Definition 4.288 (Nonlocal Schrödinger operator with Ψ-Kato-decomposable potential and vector potential). Let Assumptions 4.274–4.275 hold, and V be a Ψ-Kato-class and uniformly locally integrable potential. Suppose that a ∈ (L2loc (ℝd ))d and ∇ ⋅ a ∈ L1loc (ℝd ). Then the nonlocal Schrödinger operator with Ψ-Kato-decomposable potential V and vector potential a is defined by K Ψ (a) given in Theorem 4.287. We can show also the continuity of KtΨ f (x) in x. To see this we similarly make use Ψ

of the smoothing effect of e−tK . Define Ψ

x,0 e−tK f (x) = ∫ pΨ t (x − y)f (y)dy = 𝔼 [f (BTt )],

x ∈ ℝd , t ≥ 0,

ℝd

for f ∈ Lp (ℝd ) with 1 ≤ p ≤ ∞. Proposition 4.289 (Smoothing effect). Let Assumptions 4.274–4.275 hold. Ψ (1) Let f ∈ Lp (ℝd ) with p = 1 or p = ∞. Then e−tK f is continuous. q d (2) Let f ∈ Lp (ℝd ) with some 1 < p < ∞. Suppose that pΨ t ∈ L (ℝ ) for 1/p + 1/q = 1. Ψ

Then e−tK f is continuous.

Proof. The proof is similar to the relativistic case in Proposition 4.220. Note that pΨ t ∈ Ψ

−tK L∞ (ℝd ) ∩ L1 (ℝd ). Replacing pXt by pΨ f (x) is continuous in x for t , it is seen that e p = 1 and p = ∞. Consider then case of 1 < p < ∞. Let g ∈ Lp (ℝd ) and define

400 | 4 Feynman–Kac formulae

gn (x) = {

g(x) n

Ψ |g(x)| ≤ n, We see that e−tK gn (x) is continuous in x for every n ∈ ℕ. |g(x)| > n. Ψ

Ψ

By the Hölder inequality we have |e−tK gn (x) − e−tK g(x)| ≤ ‖pΨ t ‖Lq (ℝd ) ‖gn − g‖Lp (ℝd ) . Ψ

The right-hand side converges to zero as n → ∞ uniformly in x, hence x 󳨃→ e−tK g(x) is continuous.

Lemma 4.290. Let Assumptions 4.274–4.275 hold. Suppose that V is a Ψ-Kato-decomposable potential and let f ∈ L∞ (ℝd ). Then Ψ

Ψ e−rK Kt−r f (x) = 𝔼x,0 [e

TΨ

t t − ∫r V(BT Ψ )ds −i ∫T Ψ a(Bs )∘dBs

e

s

r

f (BT Ψ )], t

t ≥ r > 0.

(4.9.25)

In particular, (4.9.25) is continous in x for every r > 0. Proof. The proof is similar to that of Lemma 4.221. We introduce the following condition. We say that a measurable function f on ℝd is in ℒΨ (ℝd ) if TrΨ

[ ] lim sup 𝔼x,0 [ ∫ |f (Bs )|ds] = 0.

r→0 x∈ℝd

[0

]

Proposition 4.291 (Lp -Lq boundedness and continuity of integral kernel). Let Assumptions 4.274–4.275 hold. Suppose that V is Ψ-Kato-decomposable and a ∈ (L2loc (ℝd ))d with ∇ ⋅ a ∈ L1loc (ℝd ). Let 1 ≤ p ≤ q ≤ ∞. Then the following hold: Ψ

(1) e−tK (a) is Lp -Lq bounded, for every t ≥ 0. q d Ψ (2) If |a|2 , ∇ ⋅ a ∈ ℒ(ℝd ) and pΨ t ∈ L (ℝ ) for 1/p + 1/q = 1, then Kt f (x) is continuous in p d x for f ∈ L (ℝ ) with 1 ≤ p ≤ ∞. (3) If |a|2 , ∇ ⋅ a ∈ ℒ(ℝd ), then the integral kernel ℝd × ℝd ∋ (x, y) 󳨃→ KtΨ (x, y) is jointly continuous in x and y, and is given by x,y

KtΨ (x, y) = pΨ t (x − y)𝔼[0,t] [e

t

Ψ

− ∫0 V(BT Ψ ) ds −i ∫Tt a(Bs )∘dBs 0 s

e

].

(4.9.26)

x,y

Here P[0,t] is the Lévy bridge measure for (BT Ψ )t≥0 .

(4) If a = 0, then e−tK

Ψ

(0)

is positivity improving.

t

Proof. Part (1) can be proven similarly to Theorem 4.107 with Brownian motion (Bt )t≥0 replaced by the subordinate Brownian motion (BT Ψ )t≥0 . By the Riesz–Thorin theorem Ψ

t

it suffices to show that e−tK (a) is bounded as an operator of (1) L∞ (ℝd ) → L∞ (ℝd ), (2) L1 (ℝd ) → L1 (ℝd ), and (3) L1 (ℝd ) → L∞ (ℝd ). By the diamagnetic inequality we have Ψ |e−tKR (a) f (x)| ≤ e−tKR (0) |f |(x). Hence we can prove (1)–(3) for e−tK (0) in a similar way to Theorem 4.107. Claims (2), (3) and (4) also follow by a minor modification of the proof of Theorem 4.111.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 401

4.9.4 Fractional Kato-class potentials In this section we consider a special class of Ψ-Kato class, which is related to stable processes and fractional Schrödinger operators. Furthermore, it includes an extension of classical potential theory. Definition 4.292 (Fractional Kato-class potential). Let α ∈ (0, 2) and Ψ(u) = uα/2 . We say that V is in fractional Kato-class 𝒦α = 𝒦α (ℝd ) in the sense of Definition 4.280. We also say that V = V+ − V− is fractional Kato-decomposable when V+ ∈ 𝒦α and α V− ∈ L1loc (ℝd ). When 1K V ∈ 𝒦α for every compact subset K ⊂ ℝd , we write V ∈ 𝒦loc . Let Ψ(u) = uα/2 . Then (XtΨ )t≥0 is a rotationally symmetric α-stable process generated by −(−Δ)α/2 , and we write Gλα for GλΨ . We compute the kernel function G0α for every dimension d. Formally it is given by G0α (x) =

1 e−ik⋅x dk. ∫ |k|α (2π)d ℝd

We want to vary α from 0 to 2, but now we consider G0α also for α ∈ (0, d] with d ≥ 1. To compute the Fourier transform hα,d (x) of |k|−α , we use the equality ∫ ℝd

f ̂(k) dk = ∫ f ̄(x)hα,d (x)dx, |k|α

f ∈ S (ℝd ).

(4.9.27)

ℝd

Note that the integrand at the left-hand side above has a singularity at k = 0 which is integrable whenever α < d. In case when α = d, we need to choose f ∈ S (ℝd ) such that at least f ̂(0) = 0 is satisfied. It can be seen that the domain of the self-adjoint operator |k|−α is 󵄨󵄨 󵄨 { } 󵄨2 󵄨 D(|k|−α ) = {f ∈ L2 (ℝd ) 󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨󵄨|k|−α f ̂(k)󵄨󵄨󵄨󵄨 dk < ∞} . 󵄨󵄨 { } ℝd Thus f ∈ S (ℝd ) with f ̂(0) ≠ 0 does not belong to D(|k|−α ). Let 0 < α < d. For the computation of the Fourier transform it is useful to apply the equality ∞

2

∫ e−t|k| t β dt = 0

Γ(1 + β) , |k|2β+2

k ≠ 0,

for β > −1. Let f ∈ S (ℝd ). Putting β = (α − 2)/2, we have ∞

∫ ℝd

2 f ̂(k) 1 dk = ∫ t (α−2)/2 ( ∫ f ̂(k)e−t|k| dk) dt. α |k| Γ(α/2)

0

ℝd

(4.9.28)

402 | 4 Feynman–Kac formulae The requirement 0 < α < d yields (α − 2)/2 > −1 and thus the left-hand side of (4.9.28) is finite. Note also that −|x|2 /4t

e dx. ∫ f ̂(k)e−t|k| dk = ∫ f ̄(x) (2πt)d/2 2

ℝd

ℝd

Hence 2

∞

∫ ℝd

c f ̂(k) 1 e−|x| /4t 1 dk = dt) dx = d−α ∫ f ̄(x) d−α dx. (4.9.29) ∫ f ̄(x) ( ∫ t (α−2)/2 α |k| Γ(α/2) cα (2πt)d/2 |x| 0

ℝd

ℝd

Here cκ = Γ(κ/2)2κ/2 . We conclude that −(d−α) −α = c ̂ cα |k| d−α |x|

̂ −α/2 f = |k|−α f ̂, we have ̂ = (2π)−d/2 f ̂ ∗ ĝ and (−Δ) in the sense of (4.9.29). Since fg (f , (−Δ)−α/2 g) = 𝒜(d, α) ∫ f ̄(x) ℝd ×ℝd

1 g(y)dxdy |x − y|d−α

(4.9.30)

for f , g ∈ S (ℝd ), with 𝒜(d, α) =

1 cd−α Γ((d − α)/2) = . d/2 cα 2π Γ(α/2)π d/2 2α

Equality (4.9.30) can further be extended to f ∈ L2 (ℝd ) and g ∈ Lp (ℝd ) with p = and (−Δ)−α/2 g(x) = 𝒜(d, α) ∫ ℝd

2d , d+2α

1 g(y)dy ∈ L2 (ℝd ) |x − y|d−α

by a use of the Hardy–Littlewood–Sobolev inequality. We obtain G0α (x) = 𝒜(d, α)

1 , |x|d−α

0 < α < d.

Let α = d. The map α 󳨃→ 𝒜(d, α) has a singularity at α = d, and for f ∈ S (ℝd ), the

integral ∫ℝd

f ̂(k) dk |k|d

is not finite in general. Define { ̊ S (ℝ ) = {f ∈ S (ℝd ) { d

󵄨󵄨 󵄨󵄨 } 󵄨󵄨 f ̂(k) 󵄨󵄨 ∫ 󵄨󵄨 |k|d dk < ∞ } . 󵄨󵄨 d } 󵄨ℝ

̊ d ) implies that f ̂(0) = 0, i. e., ∫ d f (x)dx = 0. Let ε > 0 and In particular, f ∈ S (ℝ ℝ d ̊ f ∈ S (ℝ ). By the results above we have ∫ f ̂(k) ℝd

c 1 1 dk = ε ∫ f ̄(x) ε dx. cd−ε |x| |k|d−ε ℝd

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 403

Consider the right-hand side above. As ε → 0, ε

cε 2ε Γ(1 + ε/2) 1 = d/2−1 → d/2−1 . cd−ε 2 Γ(d/2 − ε/2) 2 Γ(d/2)

Here we used the identity εΓ(ε/2) = 2Γ(1 + ε/2). Hence by subtraction of this term lim ( ε↓0

cε c 1 1 1 1 |x|−ε − d/2−1 ) = lim (ε ε |x|−ε − d/2−1 ) . ε↓0 cd−ε cd−ε 2 Γ(d/2) ε 2 Γ(d/2) ε

The right-hand side above can be computed as =

d 2ε Γ(1 + ε/2) 1 ( d/2−1 |x|−ε )⌈ = D(d) + log |x|−1 , dε 2 Γ(d/2 − ε/2) Γ(d/2)2d/2−1 ε=0

where D(d) =

(log 2+Γ󸀠 (d/2)/2)Γ(d/2)+Γ󸀠 (d/2)/2 2d/2−1 Γ(d/2)2

and Γ󸀠 (x) = ∫0 e−u ux log udu is the derivative ∞

of the Gamma function. By using this fact and ∫ℝd f (x)dx = 0 we see that lim ε↓0

cε c 1 1 ) dx ∫ f ̂(x)|x|−ε dx = lim ∫ f ̄(x) ( ε |x|−ε − d/2−1 ε↓0 cd−ε cd−ε 2 Γ(d/2) ε ℝd

ℝd

= ∫ f ̄(x) (D(d) + ℝd

1 1 1 1 log ) dx = ∫ f ̄(x) log dx. |x| |x| Γ(d/2)2d/2−1 Γ(d/2)2d/2−1 ℝd

Thus 1 1 1 ∫ f ̂(k) d dk = ∫ f ̄(x) log dx d/2−1 |x| |k| Γ(d/2)2

ℝd

(4.9.31)

ℝd

̊ d ), and we obtain for f ∈ S (ℝ ̂ −d = − |k|

1 log |x| Γ(d/2)2d/2−1

in the sense of (4.9.31). This gives G0d (x) = −

1 log |x|. Γ(d/2)π d/2 2d−1

We summarize these results in the following proposition. Proposition 4.293 (Riesz potential). It follows that 𝒜(d, α)|x|α−d , 𝒜(d, d) log |x|−1 ,

G0α (x) = {

0 < α < d, α = d,

in the sense that ̂ = ∫ f ̂(k)|k|−α g(k)dk ℝd

∫ f ̄(x)G0α (x − y)g(y)dxdy, ℝd ×ℝd

(4.9.32)

404 | 4 Feynman–Kac formulae ̊ d ) for α = d. Here whenever f , g ∈ S (ℝd ) for 0 < α < d, and for f , g ∈ S (ℝ Γ((d−α)/2) d/2 2α ,

𝒜(d, α) = { Γ(α/2)π1

0 < α < d,

Γ(d/2)π d/2 2d−1

α = d.

,

The expression given by (4.9.32) is called Riesz potential. Example 4.294 (Newton potential for d ≥ 2). The limit case α = 2 yields the integral kernel of (−Δ)−1 , and G02 (x) is called the Newton potential, given by 1

G02 (x) = { 2π

Ad

log |x|−1 , 1 |x|2−d , (d−2)

d = 2, d ≥ 3,

where Ad denotes the (d − 1)-dimensional area of the unit sphere in ℝd , i. e., Ad = 2π d/2 /Γ(d/2). For d = 1 we can define the Newton potential in distributional sense by 1 G02 (x) = − |x|, 2

d = 1.

It is direct to see that −

d2 1 (− |x|) = δ(x) 2 2 dx

and

−

d2 1 (− |x| ∗ f ) = f (x) 2 2 dx

for every f ∈ S (ℝ), in distributional sense. Now we give a criterion for fractional Kato-class. Proposition 4.295. Let V be a nonnegative function and α ∈ (0, 2). (1) If 0 < α ≤ d and lim sup

δ↓0 x∈ℝd

G0α (x − y)V(y)dy = 0,

∫ |x−y|0} dx, s

(4.9.60)

ℝd

where pΨ s/|V(x)| (0) =

2 1 ∫ e−sΨ(|ξ | /2)/|V(x)| dξ . d (2π)

ℝd

We note that the right-hand side of (4.9.60) may not be finite; this depends on the choice of f .

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 417

Proof. Let F be a function as in (4.9.59). Since F is a monotone increasing function, we have N̄ 0 (V) ≤ F(1)−1 Tr(F(K0Ψ )) = F(1)

∞

−1

dt ∫ pΨ t (0) t

0

∫

𝔼0,0 𝒲×P

ℝd

󵄨󵄨 t 󵄨󵄨 [ f (∫ t|V(X Ψ + x)| ds )󵄨󵄨󵄨 X Ψ = 0] dx. s t 󵄨󵄨󵄨󵄨 t 󵄨 0 [ ]

By Jensen’s inequality, 󵄨󵄨 ∞ t dt ds 󵄨󵄨󵄨󵄨 Ψ −1 Ψ 0,0 [ Ψ ̄ N0 (V) ≤ F(1) ∫ pt (0) ∫ 𝔼𝒲×P ∫ f (t|V(Xs + x)|) 󵄨󵄨 Xt = 0] dx. t t 󵄨󵄨󵄨 󵄨 0 [0 ] ℝd t ds t

Using ∫0

= 1 and swapping dx and d𝒲 × dP, we obtain ∞

dt N̄ 0 (V) ≤ F(1)−1 ∫ pΨ ∫ f (t|V(x)|)dx. t (0) t 0

ℝd

When V(x) = 0, also f (tV(x)) = 0. This implies that the right-hand side above equals ∞

F(1)−1 ∫ pΨ t (0) 0

dt ∫ f (t|V(x)|)1{|V(x)|>0} dx. t ℝd

Changing the variable from t|V(x)| to s and integrating with respect to s, the bound ∞ (4.9.60) follows from F(1) = ∫0 e−s s−1 f (s)ds. Next we discuss how the Lieb–Thirring inequality (4.9.60) in fact depends on the Bernstein function Ψ. To make this expression more explicit we note that the diagonal part of the probability transition kernel has the representation pΨ t (0)

∞

1 = dξ ) dr. ∫ e−r ( ∫ 1 √ 2 { Ψ(ξ /2)≤√r/t} (2π)d 0

(4.9.61)

ℝd

Denote by BΨ r (x) a ball of radius r centered in x in the topology of the metric given by dΨ (ξ , η) = √Ψ(|η − ξ |2 /2),

(4.9.62)

d Ψ Ψ i. e., BΨ r (x) = {y ∈ ℝ |d (x, y) ≤ r}. Note that d (ξ , η) = 0 if and only if ξ = η, since Ψ is ∞ concave and a C -function. The integral ∫ℝd 1 √ 2 dξ is the volume of BΨ (0) √r/t Ψ

in this metric. If d satisfies the condition

{ Ψ(ξ /2)≤√r/t}

∫ 1BΨ (x) dy ≤ c ∫ 1BΨr (x) dy, ℝd

2r

ℝd

x ∈ ℝd , r > 0,

(4.9.63)

418 | 4 Feynman–Kac formulae with a constant c > 0 independent of x and r, then dΨ is said to have the volume doubling property. When dΨ has this property, then it furthermore follows that c1 ∫ 1 √ ℝd

{ Ψ(ξ 2 /2)≤√r/t}

dξ ≤ pΨ t (0) ≤ c2 ∫ 1 √ ℝd

{ Ψ(ξ 2 /2)≤√r/t}

dξ

(4.9.64)

with some constants c1 and c2 . We use the following lemma without proof. Lemma 4.315 (Volume doubling). A necessary and sufficient condition for Ψ ∈ ℬ0 to give rise to a volume doubling dΨ is lim inf u→0

Ψ(Cu) > 1, Ψ(u)

lim inf u→∞

Ψ(Cu) >1 Ψ(u)

for some C > 1. In particular, this lemma implies that Ψ increases at infinity as a (possibly fractional) power. Corollary 4.316. Suppose that Ψ ∈ ℬ0 is strictly monotone increasing. Then under the assumptions of Theorem 4.314 we have f (s) r|V(x)| d/2 −r N̄ 0 (V) ≤ A ∫ ds ∫ dx ∫ (Ψ−1 ( )) e dr, s s ∞

∞

0

ℝd

(4.9.65)

0

where A=

1

1

. ∞ (2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds

Furthermore, if dΨ has the volume doubling property, then f (s) |V(x)| d/2 N̄ 0 (V) ≤ c2 A ∫ ds ∫ (Ψ−1 ( )) dx, s s ∞

0

(4.9.66)

ℝd

where c2 is given in (4.9.64). Proof. Since under the assumption the function Ψ ∈ ℬ0 is invertible and its inverse is increasing, the proof is straightforward by using Ker Ψ = {0}, (4.9.61), (4.9.64), and ∫1√

ℝd

{

Ψ(ξ 2 /2)≤√x}

dξ = ∫ 1{ξ 2 ≤2Ψ−1 (x)} dξ = ℝd

π d/2

d Γ ( d2 ) 2

d/2

(2Ψ−1 (x))

.

In case when Ψ ∈ ℬ0 has a scaling property, we can derive a more explicit formula.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 419

Corollary 4.317. Suppose that Ψ ∈ ℬ0 is strictly monotone increasing and the assumptions of Theorem 4.314 hold. In addition, assume that there exists γ > 0 such that Ψ(au) = aγ Ψ(u) for all a, u ≥ 0, and ∫0 f (s)s ∞

d −1− 2γ

ds < ∞. Then

d/2 N̄ 0 (V) ≤ B ∫ (Ψ−1 (|V(x)|)) dx,

(4.9.67)

ℝd

where B=

∞ −1− d 2γ

d Γ( 2γ + 1)

∫0 s

f (s)ds

. ∞ (2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds

Proof. The inverse function Ψ−1 has the scaling property Ψ−1 (av) = a1/γ Ψ−1 (v). Thus the corollary follows. Instead of the scaling property suppose now that there exists λ > 0 such that Ψ(u) ≥ Cuλ with a constant C > 0. This inequality holds for at least large enough u if dΨ has the volume doubling property. We have a similar formula to Corollary 4.317. Corollary 4.318. Suppose that Ψ ∈ ℬ0 is strictly monotone increasing and the assumptions of Theorem 4.314 hold. If Ψ(u) ≥ Cuλ , then N̄ 0 (V) ≤ B󸀠 ∫ |V(x)|d/2λ dx,

(4.9.68)

ℝd

where B = 󸀠

C −d/2λ

d

∫0 f (s)s−1− 2λ ds ∞

(2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds ∞

.

Proof. The bound Ψ(u) ≥ Cuλ gives Ψ−1 (u) ≤ C −1/λ u1/λ , and the corollary follows. In some special cases of Bernstein functions Ψ we can derive more explicit forms of the Lieb–Thirring inequality. Recall the Sobolev space H s (ℝd ) = {f ∈ L2 (ℝd )|f ̂(k)(1 + |k|2 )s ∈ L2 (ℝd )}

(4.9.69)

for s ≥ 0. In what follows we discuss some key specific cases of the Lieb–Thirring inequality, where we assume for simplicity that V is nonpositive and continuous. Example 4.319 (Fractional Schrödinger operators). Let Ψ(u) = (2u)α/2 and H Ψ = (−Δ)α/2 + V, 0 < α ≤ 2. For f , g ∈ H α/2 (ℝd ) we define the quadratic form Q(f , g) = ((−Δ)α/4 f , (−Δ)α/4 g) − (|V|1/2 f , |V|1/2 g).

(4.9.70)

420 | 4 Feynman–Kac formulae Let V ∈ Ld/α (ℝd ) for α ∈ (0, 2). From the Sobolev inequality ‖f ‖q ≤ C‖(−Δ)α/2 f ‖p

(4.9.71)

for q = pd/(d − αp) and d > αp with some constant C, it follows that 1 1 ‖f ‖22d ≥ (|V|1/2 f , |V|1/2 f )‖V‖−1 d/α . C C d−α

‖(−Δ)α/4 f ‖22 ≥

(4.9.72)

The estimate gives Q(f , f ) ≥ 0 when ‖V‖d/α < 1/C. There is a bounded function λ(x) such that h = V − λ satisfies ‖h‖d/α < 1/C. Thus V = h + λ and λ ∈ L∞ (ℝd ), thus V is form-bounded with respect to (−Δ)α/2 with a relative bound strictly smaller than 1. In particular, inff ∈H α/2 (ℝd ) Q(f , f ) > −∞. Furthermore (−Δ)α/4 |V|1/2 ∈ I2d/α,w by Proposition 4.40 and |V|1/2 (−Δ)α/2 |V|1/2 ∈ Id/α,w follows from Proposition 4.37. Hence |V|1/2 (−Δ)α/2 |V|1/2 ∈ Ip ,

d . α

p>

(4.9.73)

Thus Assumption 4.310 is verified. It follows that pΨ t (0) =

α C (α, d) 1 ∫ e−t|ξ | dξ = 1 d/α , d (2π) t

(4.9.74)

ℝd

where C1 (α, d) =

2π d/2 Γ(d/α) . α(2π)d Γ(d/2)

With the constant L(1) α,d

= C1 (α, d)

∫0 s−1−d/α f (s)ds ∞

∫0 e−s s−1 f (s)ds ∞

,

we have by Theorem 4.314 that N̄ 0 (V) ≤ L(1) ∫ |V(x)|d/α dx. α,d

(4.9.75)

ℝd

Example 4.320 (Fractional relativistic Schrödinger operators). Let 0 < α < 2 and m > 0. Choose Ψ(u) = (2u + m2/α )α/2 − m and H Ψ = (−Δ + m2/α )α/2 − m + V. Using (4.9.72) we can also derive ‖(−Δ + m2/α )α/4 f ‖22 ≥ ‖(−Δ)α/4 f ‖22 ≥

1 (|V|1/2 f , |V|1/2 f )‖V‖−1 d/α . C

(4.9.76)

Hence V ∈ Ld/α (ℝd ) is relatively form-bounded with respect to (−Δ + m2/α )α/2 − m with relative bound strictly smaller than 1. Suppose that V ∈ Ld/α (ℝd ) ∩ Ld/2 (ℝd ). We have |V|1/2 (−Δ + m2/α )−α/2 |V|1/2 ∈ Ip ,

p > d/α

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 421

in a similar way to (4.9.73). Thus Assumption 4.310 is satisfied. For m > 0 we have pΨ t (0) =

2 2/α α/2 1 ∫ e−t((|ξ | +m ) −m) dξ d (2π)

ℝd

and the bound pΨ t (0) ≤

α αr |Sd−1 | C (α, d) C (α, d) − t − αr t ( ∫ e 2(2m2/α )1−α/2 r d−1 dr + ∫ e 22−α/2 r d−1 dr) ≤ 2 d/2 + 3 d/α , d (2π) t t 2

r≤m1/α

r>m1/α

with d/2 1 2 d C2 (α, d) = ( (2m2/α )1−α/2 ) Γ ( ) 2 α 2

and

1 22−α/2 C3 (α, d) = ( ) α α

d/α

d Γ( ). α

In case m = 0, see (4.9.74). With L(2) α,d

= C2 (α, d)

∫0 s−1−d/2 f (s)ds ∞

∫0 e−s s−1 f (s)ds ∞

and

L(3) α,d

= C3 (α, d)

∫0 s−1−d/α f (s)ds ∞

∫0 e−s s−1 f (s)ds ∞

which are independent of V, we have N̄ 0 (V) ≤ L(2) ∫ |V(x)|d/2 dx + L(3) ∫ |V(x)|d/α dx. α,d α,d ℝd

(4.9.77)

ℝd

Example 4.321 (Relativistic Schrödinger operators). Let Ψ(u) = √2u + m2 − m and H Ψ = (−Δ + m2 )1/2 − m + V for m > 0. This is the case of α = 1 in Example 4.320. Hence V ∈ Ld/2 (ℝd ) is relatively form-bounded with respect to (−Δ + m2 )1/2 − m with relative bound strictly smaller than 1. Let V ∈ Ld/2 (ℝd ) ∩ Ld (ℝd ). Then we have N̄ 0 (V) ≤ L(2) ∫ |V(x)|d/2 dx + L(3) ∫ |V(x)|d dx. 1,d 1,d ℝd

ℝd

Example 4.322 (Sums of different stable generators). Let Ψ(u) = (2u)α/2 + (2u)β/2 , 0 < α, β < 2, and H Ψ = (−Δ)α/2 + (−Δ)β/2 + V. Relative boundedness of V follows similarly as above, whenever V ∈ Ld/α (ℝd ) ∩ Ld/β (ℝd ). Let α ≥ β. We have |V|1/2 ((−Δ)α/2 + (−Δ)β/2 ) |V|1/2 −1

= |V|1/2 ((−Δ)(α−β)/2 + 1) (−Δ)−β/2 |V|1/2 ≤ |V|1/2 (−Δ)−β/2 |V|1/2 . −1

In a similar way to (4.9.73) it is obtained that |V|1/2 ((−Δ)α/2 + (−Δ)β/2 ) |V|1/2 ∈ Ip , −1

p > min{d/α, d/β}.

422 | 4 Feynman–Kac formulae Thus Assumption 4.310 is satisfied. By Theorem 4.314 and using d

−α pΨ ∧t t (0) ≤ c (t

− dβ

t>0

),

(4.9.78)

with some constant c > 0, we obtain N̄ 0 (V) ≤ Lα ∫ |V(x)|d/α dx + Lβ ∫ |V(x)|d/β dx, ℝd

ℝd

with Lγ =

∫0 s−1−d/γ f (s)ds ∞

c

(2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds ∞

,

γ = α, β.

Example 4.323 (Jump-diffusion operators). Let Ψ(u) = u + buα/2 , 0 < α < 2, and b ∈ (0, 1]. Then H Ψ = −Δ + b(−Δ)α/2 + V. By (4.9.72) whenever V ∈ Ld/2 (ℝd ) ∩ Ld/α (ℝd ), V is relatively form-bounded with respect to −Δ + b(−Δ)α/2 with relative bound strictly smaller than 1. We have |V|1/2 (−Δ + b(−Δ)α/2 )−1 |V|1/2 = |V|1/2 ((−Δ)1−α/2 + b) (−Δ)−α/2 |V|1/2 ≤ |V|1/2 (−Δ)−α/2 |V|1/2 . −1

Then |V|1/2 (−Δ + b(−Δ)α/2 )−1 |V|1/2 ∈ Ip ,

p > d/α

and Assumption 4.310 is satisfied. Using the estimate 2

−d/2 pΨ ∧ (bt)−d/α ) ∧ (t −d/2 e−|x−y| /ct + (bt)−d/α ∧ t (x − y) ≤ (t

bt ), |x − y|d+α

with some c > 0, we have −d/2 pΨ ∧ (bt)−d/α t (0) ≤ t

and by Theorem 4.314 we obtain N̄ 0 (V) ≤ D ∫ |V(x)|d/2 dx + Dα ∫ |V(x)|d/α dx, ℝd

ℝd

where D=

1

∞

(2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds ∞

and Dα =

∫0 s−1−d/2 f (s)ds

b−d/α

∫0 s−1−d/α f (s)ds ∞

(2π)d/2 d2 Γ ( d2 ) ∫0 e−s s−1 f (s)ds ∞

.

(4.9.79)

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 423

4.9.8 Decay of eigenfunctions In this section we briefly summarize results on the spatial decay of bound states for nonlocal Schrödinger operators. These have been obtained via Feynman–Kac type representations, however, by an extensive use of techniques of probabilistic potential theory which are beyond the scope of this book, therefore we refer the reader to the literature for details of proof. In our framework here we use directly a random process as a basic input, however, we also discuss the related operators. Let (Xt )t≥0 be a Lévy process with values in ℝd , d ≥ 1, with probability measure P x of the process starting from x ∈ ℝd , and denote by 𝔼x expectation with respect to this measure. Recall that (Xt )t≥0 is a Markov process with respect to its natural filtration satisfying the strong Markov property and having càdlàg paths. Denote by pt (x) its transition probability density, whenever it exists, let D ⊂ ℝd be an open bounded set, and consider τD = inf {t ≥ 0 | Xt ∉ D}, i. e., the first exit time of (Xt )t≥0 from D. Then the transition densities pD (t, x, y) of the process killed on exiting D also exist and are given by the Dynkin–Hunt formula pD (t, x, y) = pt (y − x) − 𝔼x [pt−τD (y − XτD )1{τD 0.

The Green function of the process (Xt )t≥0 on D is thus GD (x, y) = ∫0 pD (t, x, y)dt, for x, y ∈ D. We will use the following class of processes. Below we adopt the notation f ≍ g to mean that there exist C1 , C2 > 0 such that C1 f ≤ g ≤ C2 f , and C(X) for a constant dependent by a process (Xt )t≥0 . ∞

Definition 4.324 (Jump-paring Lévy processes). We call a Lévy process (Xt )t≥0 a symmetric jump-paring Lévy process if it satisfies the following properties: (1) Its Lévy triplet is (0, A, ν) such that the Lévy measure ν(dx) = ν(x)dx satisfies (i) ν(E) = ν(−E), for every Borel set E ⊂ ℝd \ {0}; (ii) ν(ℝd \ {0}) = ∞. (2) There exists a non-increasing function g : (0, ∞) → (0, ∞) such that (i) ν(x) ≍ g(|x|), for all x ∈ ℝd \ {0}; (ii) there is a constant C1 = C1 (X) > 0 such that g(r) ≤ C1 g(r + 1), for all r ≥ 1; (iii) there is a constant C2 = C2 (X) > 0 such that ∫ g(|x − y|)g(|y|)dy ≤ C2 g(|x|), |x−y|>1 |y|>1

(3) There exists tb = tb (X) > 0 such that supx∈ℝd ptb (x) < ∞. (4) For all 0 < p < q < r < ∞ we have sup

sup GB(0,r) (x, y) < ∞.

x∈B(0,p) y∈B(0,q)c

|x| ≥ 1.

(4.9.80)

424 | 4 Feynman–Kac formulae d

Condition 1 (i) above says that the process is symmetric, i. e., −Xt = Xt , for all t ≥ 0, and 1 (ii) implies that P x is absolutely continuous with respect to Lebesgue measure and thus P x (Xt ∈ dy) has a density, which we denote by pt (y − x). While the other conditions are technical and self-explanatory, condition 2 (iii) has a structural importance. It provides a control of the convolutions of ν with respect to large jumps, saying that double (and by iteration, any multiple) large jumps of the process are stochastically dominated by single large jumps. Heuristically this may be interpreted as a basic preference of the process to get to a location far away from the point of departure through a single large jump rather than via a sequence of large jumps, for which reason we refer to this as the jump-paring property. Condition 1 (i) in Definition 4.324 implies that the Lévy process (Xt )t≥0 has characteristic function 𝔼0 [eiξ ⋅Xt ] = e−tψ(ξ ) ,

ξ ∈ ℝd , t > 0,

whose exponent given by the Lévy–Khintchine formula simplifies to 1 ψ(ξ ) = Aξ ⋅ ξ + ∫ (1 − cos(ξ ⋅ z))ν(dz), 2

(4.9.81)

ℝd

where A is the diffusion matrix and ν(dz) = ν(z)dz is the Lévy measure with intensity ν. Thus the generator L of the process is determined by ̂ (ξ ) = −ψ(ξ )̂f (ξ ), Lf

ξ ∈ ℝd , f ∈ D(L),

(4.9.82)

with domain D(L) = {f ∈ L2 (ℝd ) | ψ̂f ∈ L2 (ℝd )}. The operator L is negative definite and self-adjoint with core C0∞ (ℝd ), and we have Lf (x) =

∫ (f (x + z) − f (x) − 1{|z|≤1} (z)z ⋅ ∇f (x)) ν(z)dz,

x ∈ ℝd , f ∈ C0∞ (ℝd ).

ℝd \{0}

The jump-paring class has a non-trivial overlap with subordinate Brownian motions in the sense that neither class contains the other. Next we give some examples and counterexamples to the jump-paring class. Example 4.325. The following Lévy processes satisfy Definition 4.324: (1) Specific cases of subordinate Brownian motions with characteristic exponents ψ such that e−tb ψ ∈ L1 (ℝd ) for some tb > 0. Examples include: (i) Rotationally symmetric α-stable process: Let ψ(ξ ) = |ξ |α , α ∈ (0, 2). In this case g(r) = r −d−α and L = −(−Δ)α/2 . (ii) Mixtures of independent rotationally symmetric stable processes: Let ψ(ξ ) = a|ξ |α + b|ξ |β , with 0 < β < α < 2 and a, b > 0. We have g(r) = aC(α)r −d−α + bC(β)r −d−β and L = −a(−Δ)α/2 − b(−Δ)β/2 .

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 425

(iii) Jump-diffusion process: Let ψ(ξ ) = a|ξ |α + b|ξ |2 , 0 < α < 2, a, b > 0, i. e., the process is a mixture of a rotationally symmetric α-stable process and an independent Brownian motion. In this case g(r) = r −d−α and L = −a(−Δ)α/2 + bΔ. (iv) Rotationally symmetric geometric α-stable process: Let ψ(ξ ) = log(1 + |ξ |α/2 ), 0 < α < 2. In this case g(r) = r −d (1 + r)−α and L = log(1 − Δα/2 ). Notice that ψ is now a slowly varying function at infinity. In contrast to the previous examples, in this case there is tb (α) > 0 for which the transition probability densities are unbounded for 0 < t ≤ tb (α), though they are bounded for t > tb (α). (v) Relativistic rotationally symmetric α-stable process Let ψ(ξ ) = (|ξ |2 + m2/α )α/2 − 1/α m, α ∈ (0, 2), m > 0. In this case g(r) = e−m r r −d−α (1 + r (d+α−1)/2 ) and L = −(−Δ + m2/α )α/2 + m. (2) Symmetric Lévy processes with subexponentially localized Lévy measures. Let β (Xt )t≥0 be a symmetric Lévy process with intensity g(r) = e−ar r −d−δ (1 + r)d+δ−γ , where a > 0, β ∈ (0, 1], δ ∈ [0, 2) and γ > (d + 1)/2. This class includes a large family of (exponentially) tempered symmetric stable processes (a > 0, β = 1, γ = δ + d, δ ∈ (0, 2)) and the rotationally symmetric relativistic stable processes above. In contrast, the following two examples do not belong to the jump-paring class: (3) Rotationally symmetric geometric 2-stable (gamma variance) process: Let ψ(ξ ) = log(1 + |ξ |2 ). In this case g(r) = r −d e−r (1 + r)(d−1)/2 and L = − log(1 − Δ). In this example the Lévy measure does not satisfy the jump-paring condition 2 (iii). (4) Iterated symmetric geometric α-stable process: Let ψ(ξ ) = log(1 + logα (1 + |ξ |α )), 0 < α < 2. It can be checked directly that the transition densities are unbounded for every t > 0 and condition (3) fails. Next we give the class of potentials which will be used in this context. Definition 4.326 (X-Kato class). Let (Xt )t≥0 be a Lévy process. We say that the potential V : ℝd → ℝ belongs to X-Kato class 𝒦X if it satisfies t

lim sup 𝔼x [∫ |V(Xs )|ds] = 0. t↓0 x∈ℝd [0 ] Also, we say that V is an X-Kato decomposable potential, denoted V ∈ 𝒦±X , whenever V = V+ − V− ,

with V− ∈ 𝒦X

and

X V+ ∈ 𝒦loc ,

where V+ , V− denote the positive and negative parts of V, respectively, and where V+ ∈ X 𝒦loc means that V+ 1K ∈ 𝒦X for all compact sets K ⊂ ℝd . d X It is straightforward to see that L∞ loc (ℝ ) ⊂ 𝒦loc . Moreover, by stochastic continuity of X 1 d (Xt )t≥0 also 𝒦loc ⊂ Lloc (ℝ ), and thus an X-Kato decomposable potential is always lo-

426 | 4 Feynman–Kac formulae cally absolutely integrable. Note that the condition defining X-Kato class allows local singularities of V. Define t

Tt f (x) = 𝔼x [e− ∫0 V(Xs )ds f (Xt )] ,

f ∈ L2 (ℝd ), t > 0.

(4.9.83)

Using the Markov property and stochastic continuity of the process it can be shown that {Tt : t ≥ 0} is a strongly continuous semigroup of symmetric operators on L2 (ℝd ), which we call the Feynman–Kac semigroup associated with the process (Xt )t≥0 and potential V. We summarize basic properties of the Feynman–Kac semigroup, which we only state for the class defined in Definition 4.324 though they are valid more generally. Proposition 4.327. Let (Xt )t≥0 be a symmetric jump-paring Lévy process and V ∈ 𝒦±X . Then the following properties hold: (1) For all t > 0, every Tt is a bounded operator on every Lp (ℝd ) space, 1 ≤ p ≤ ∞. The operators Tt : Lp (ℝd ) → Lp (ℝd ) for 1 ≤ p ≤ ∞, t > 0, and Tt : Lp (ℝd ) → L∞ (ℝd ) for 1 < p ≤ ∞, t ≥ tb , and Tt : L1 (ℝd ) → L∞ (ℝd ) for t ≥ 2tb are bounded, with some tb > 0. (2) For all t ≥ 2tb , Tt has a bounded measurable kernel u(t, x, y) symmetric in x and y, i. e., Tt f (x) = ∫ℝd u(t, x, y)f (y)dy, for all f ∈ Lp (ℝd ) and 1 ≤ p ≤ ∞. (3) For all t > 0 and f ∈ L∞ (ℝd ), Tt f is a bounded continuous function. (4) For all t ≥ 2tb the operators Tt are positivity improving, i. e., Tt f (x) > 0 for all x ∈ ℝd and f ∈ L2 (ℝd ) such that f ≥ 0 and f ≠ 0 a.e. Furthermore, by the Hille–Yosida theorem there exists a self-adjoint operator H bounded from below such that e−tH = Tt . We call the operator H = −L + V defined in form sense and with dense domain in L2 (ℝd ) a nonlocal Schrödinger operator with kinetic term −L, where L is the infinitesimal generator of the process (Xt )t≥0 , and potential V. Next we assume that H has a non-empty discrete spectrum, and consider the spatial decay of the related eigenfunctions. When V is a confining potential, then Tt are compact operators for all t > 0, and thus Spec(H) = Specd (H). When V is a decaying potential, then Tt are not compact operators and we need to assume that Specd (H) ≠ 0. Let φn be an eigenfunction at eigenvalue λn , n = 0, 1, 2, . . ., solving the equation Hφn = λn φn . As before, we denote by φ0 the unique and strictly positive ground state corresponding to λ0 = inf Spec(H). Note that since φn = eλn tb Ttb φn , by (1) and (3) of Proposition 4.327 we have that Tt (Lp (ℝd )) ⊂ L∞ (ℝd ) and Tt (L∞ (ℝd )) ⊂ Cb (ℝd ) for every t > 2tb and p ≥ 1,

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 427

it follows that each φn is a bounded continuous function and thus it makes sense to study their pointwise decay at infinity. For an open set D ⊂ ℝd and an X-Kato decomposable potential V we will use the function ΛVD (x)

x

τD

t

= 𝔼 [ ∫ e− ∫0 V(Xs )ds dt ] , [0 ]

x ∈ D,

where τD is the first exit time of (Xt )t≥0 from D. Intuitively, ΛVD (x) is the mean sojourn time in D under the potential of the process (Xt )t≥0 starting from an interior point x before its first exit from D. Also, like before, we denote a ball of radius r centered in z by Br (z). In the statements below we will not keep track of the constants, thus the same C may mean a different constant from line to line. We then have the following results. First we discuss the case of confining potentials. Theorem 4.328 (Case of confining potentials). If (Xt )t≥0 is a jump-paring Lévy process with Lévy intensity ν and V ∈ 𝒦±X is a confining potential, then for every δ ≥ 0 for which λ0 + δ > 0 there exist constants Cn = Cn (X, V, δ) and Rn = Rn (X, V, δ) > 0 such that |φn (x)| ≤ Cn ‖φn ‖∞ ΛV+δ B1 (x) (x) ν(x),

|x| ≥ Rn ,

n = 0, 1, 2, . . .

(4.9.84)

Moreover, for the ground state we have φ0 (x) ≍ ΛV+δ B1 (x) (x) ν(x),

|x| ≥ R0 .

(4.9.85)

The above result can be interpreted as saying that the decay of the ground state at far out points is driven by how soon on average paths leave small neighbourhoods. Since for confining potentials C1 sup1 V ≤ ΛVD ≤ C2 inf1 V , with appropriate constants C1 , C2 , D D for sufficiently temperately growing potentials we can get more explicit bounds. Corollary 4.329. If further to the assumptions of Theorem 4.328 also supy∈B1 (x) V(y) ≤ C infy∈B1 (x) V(y) holds with some C ≥ 1 for large enough |x|, then we have |φn (x)| ≤ Cn󸀠

ν(x) , V(x)

and φ0 (x) ≍

n = 0, 1, 2, . . . , ν(x) , V(x)

whenever |x| is large enough. The above expression shows a neat separation of the contributions of the jump intensity and of the potential into the decay, and points to a mechanism of balance between the killing effect of the potential and jump activity. Another consequence is the following surprising property.

428 | 4 Feynman–Kac formulae Corollary 4.330. If the assumptions of Theorem 4.328 hold, then there exists a constant Cn = Cn (X, V) such that |φn (x)| ≤ Cn ‖φn ‖∞ φ0 (x),

x ∈ ℝd , n ∈ ℕ.

This ground state domination holds in the case of classical Schrödinger operators H = − 21 Δ + V if the semigroup {e−tH : t ≥ 0} is intrinsically ultracontractive, while it is straightforward to see that it fails for the harmonic oscillator, when intrinsic ultracontractivity does not hold (see Definition 5.3 and Proposition 5.5 in Section 5.1.1 below). However, for nonlocal Schrödinger operators related to jump-paring Lévy processes and X-Kato decomposable potentials it holds also in lack of intrinsic ultracontractivity. Next we turn to decaying potentials. Now both V(x) and ν(x) tend to zero as |x| → ∞ so that far out the process under V behaves increasingly similar to the unperturbed process. Therefore an expression as in Corollary 4.329 resulting from the killing effect can no longer hold, and the contribution of the potential becomes more subtle. Theorem 4.331 (Case of decaying potentials: lower bound). Let (Xt )t≥0 be a jumpparing Lévy process with Lévy intensity ν, V ∈ 𝒦±X a decaying potential, and φ an eigenfunction of H at eigenvalue λ < 0. Then the following hold: (1) If property 2 (ii) in Definition 4.324 holds, then there exist C = C(λ) > 0 and R = R(λ) > 0 such that φ(x) ≥ Cν(x),

|x| ≥ R0 .

(2) If either property 2 (ii) in Definition 4.324 holds but 2 (iii) does not hold, or neither hold, then lim sup |x|→∞

φ(x) = ∞. ν(x)

In the event of the eigenvalue being sufficiently low-lying below zero (i. e., the bottom of the essential spectrum), the following result holds. Theorem 4.332 (Case of decaying potentials: upper bounds). Let (Xt )t≥0 be a jumpparing Lévy process with Lévy intensity ν, V ∈ 𝒦±X a decaying potential, and φ an eigenfunction of H at eigenvalue λ < 0. If there exists λ∗ = λ∗ (X) > 0 such that φ is an eigenfunction for an eigenvalue λ ∈ (−∞, −λ∗ ), then there exist C = C(X, λ) and R = R(X, λ) such that |φ(x)| ≤ C ‖φ‖∞ ν(x),

|x| ≥ R.

(4.9.86)

For arbitrary eigenvalues, when thus they can be any near to zero, the situation is more complicated and depends more closely on the nature of the jumps. If ν is heavytailed, we have the following.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 429

Theorem 4.333 (Upper bound for decaying potentials and heavy-tailed jump kernel). Let (Xt )t≥0 be a jump-paring Lévy process with Lévy intensity ν, V ∈ 𝒦±X a decaying potential, and φ an eigenfunction of H at eigenvalue λ < 0. Assume, furthermore, that there exists c > 0 such that for all r ≥ 1 we have ν(x − y) ≤ cν(x), whenever |y| ≤ r and |x| ≥ 2r. Then (4.9.86) holds with suitable constants C and R. For lighter-tailed ν the extra condition in Theorem 4.333 breaks down and we need a more detailed control in terms of suitable parameter functions relating to some intrinsic jump preferences of the processes involved. Define K1X (s) = sup |x|≥s

∫|x−y|>s, |y|>s ν(x − y)ν(y)dy ν(x)

,

s ≥ 1.

(4.9.87)

This is a non-increasing function and gives the optimal constant C in the bound ∫ ν(x − y)ν(y)dy ≤ Cν(x),

|x| ≥ s,

|x−y|>s |y|>s

for any fixed s ≥ 1. Thus 1/K1X (s) is the rate of preference of single jumps of size at least s over double jumps of size at least s each, i. e., when K1X (s) decreases with s → ∞, this preference improves. Next let 0 < s1 < s2 < s3 ≤ ∞ and define K2X (s1 , s2 , s3 ) = inf {C ≥ 1 | ν(x − y) ≤ C ν(x), |y| ≤ s1 , s2 ≤ |x| < s3 }.

(4.9.88)

This function is non-decreasing in s1 ∈ (0, s2 ), for every fixed 0 < s2 < s3 . Whenever s3 = ∞ and s2 ≫ s1 ≫ 1, the ratio 1/K2X (s1 , s2 , s3 ) measures the rate of preference of a direct large jump from x ∈ Bs2 (0)c to zero over a jump from a point z situated in the s1 -neighbourhood of x to zero. In our applications, the second case corresponds to the situation when the process moving from x to zero first fluctuates inside Bs1 (x) and then makes one large jump to the origin. Thus K2X is the inverse rate of preference of the scenario “no small steps but direct large jump” over the scenario “first small steps, then large jump”. Then we have the following result covering this case. Theorem 4.334 (Upper bound for decaying potentials and light-tailed jump kernel). Let (Xt )t≥0 be a jump-paring Lévy process with Lévy intensity ν and transition probability density pt (x), V ∈ 𝒦±X a decaying potential, and φ an eigenfunction of H at eigenvalue λ < 0. Denote by Φ(r) = sup|ξ |≤r ψ(ξ ), r > 0, the symmetrized Lévy exponent of (Xt )t≥0 . Furthermore, assume the following conditions: (1) ∫ℝd |x|2 ν(dx) < ∞; (2) there exists κ1 ≥ 2 such that lims→∞ K1X (κ1 s)K2X (s, κ1 s, ∞) = 0; (3) there exists κ2 < ∞ such that for all s1 ≥ 1 we have lim sups→∞ K2X (s1 , s, ∞) ≤ κ2 ; (4) there exists C1 > 0 such that sup|x|≥r pt (x) ≤ C1 t Φ(1/r) , t > 0, r ≥ 1; rd

430 | 4 Feynman–Kac formulae (5) there exist C2 ≥ 1 and R > 0 such that pt (x) ≤ C2 pt (y), for every t > 0 and |x| ≥ |y| ≥ R satisfying |x − y| ≤ 1. Then (4.9.86) holds with R and a suitable constant C. To understand what condition (2) in Theorem 4.334 means first notice that by the definition of K1X it follows that ∫ ν(x − z)ν(z)dz ≤ K1X (s2 )ν(x),

for all x such that |x| > s2 ,

|x−z|>s2 |z|>s2

and using the definition of K2X we have ν(x) ≤ K2X (s1 , s2 , ∞)ν(y),

for all y such that |y| > s2 , |x − y| < s1 .

Thus 1/(K1X (s2 )K2X (s1 , s2 , ∞)) measures the rate of preference of a single large jump from y to the origin (“direct single large jump” scenario) over two large jumps from x ∈ Bs1 (y) to the origin (“first small steps, then two large jumps” scenario). This means that condition (2) corresponds to the situation when the first scenario outdoes the second at a rate which improves when the length of the long jumps increases appropriately with the scale of the smaller fluctuations. Secondly, condition (3) says that the “no small steps but direct large jump” scenario above outdoes the “first small steps, then large jump” scenario at a rate 1/κ2 , no matter how large the smaller fluctuations are (i. e., there is always a suitably larger jump for which the former is preferred to occur). Conditions (4)–(5) are just technical and are met by a large selection of processes. Remark 4.335. Note that for decaying potentials ΛVD ≍ C(D), where C(D) is a constant which in (4.9.86) appears amalgamated in a constant prefactor. Theorems 4.332–4.334 show that, under appropriate extra conditions, the decay continues to be driven by the same mechanism as for confining potentials. However, as Theorem 4.331 shows, when the jump-paring condition does not hold, the decay of bound states is necessarily slower than the fall-off rate of the jump kernel. The decay rates for specific jump-paring Lévy processes and confining potentials are straightforward to derive from the above results. The most detailed results are known from explicit solutions of the eigenvalue problem, which will be summarized in Section 4.9.9 below. Here we discuss just one example for confining potentials, and a generic example for decaying potentials. Example 4.336 (Jump-diffusions). Consider the operator L = −(−Δ)α/2 + aΔ generating the jump-diffusion process (Zt )t≥0 , Zt = Xt + Bt , where (Xt )t≥0 is a rotationally symmetric α-stable process and (Bt )t≥0 is an independent Brownian motion with diffusion matrix A = aId (where Id is the identity matrix in ℝd ). For a confining potential V ∈ 𝒦±X the ground state of H = −L + V decays as predicted by Theorem 4.328 or Corollary 4.329, i. e., the decay rate is determined by the fractional Laplacian and the jump

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 431

component of the process. These results provide the leading order of decay only and it can be expected that a more refined asymptotics will contain a further term of smaller order coming from the Brownian component of the process. Example 4.337. Consider a decaying potential V ∈ 𝒦±X , and a symmetric jump-paring Lévy process (Xt )t≥0 with generator L and profile function of the Lévy intensity of the form β

g(r) = e−cr r −δ ,

(4.9.89)

with suitable parameters c > 0, β, δ ≥ 0. Assume that a ground state φ0 of the nonlocal Schrödinger operator H = −L + V exists. The following ground state decay behaviours occur: (1) Polynomially decaying Lévy intensities: If c = 0 and δ > d, then φ0 (x) ≍ |x|−δ for all λ0 < 0. (2) Stretched-exponentially decaying Lévy intensities: If c > 0, β ∈ (0, 1) and δ ≥ 0, β then φ0 (x) ≍ e−c|x| |x|−δ for all λ0 < 0. (3) Exponentially decaying Lévy intensities: If c > 0, β = 1, and δ > d+1 , and there 2 exists λX∗ such that |λ0 | > λX∗ , then we have φ0 (x) ≍ e−c|x| |x|−δ . When the eigenvalue λ0 is arbitrary, then for large enough |x| for every ε > 0 there is a Cε > 0 such that φ0 (x) ≥ Cε max {e−̃c

√|λ0 |+ε|x|

, e−c|x| |x|−δ } .

φ (x)

0 Moreover, if δ ≤ d+1 , then lim sup|x|→∞ ν(x) = ∞. 2 (4) Super-exponentially decaying Lévy intensities: If c > 0, β > 1, and δ ≥ 0, then for every ε > 0 there is a Cε > 0 such that

φ0 (x) ≥ Cε e−̃c

√min{|λ0 |+ε,1} |x|(log |x|)(β−1)/β

for large enough |x| and any λ0 < 0. Remark 4.338. It is seen that the above results improve on the decay estimates of eigenfunctions for the relativistic Schrödinger operator presented in Section 4.6.3. Remark 4.339 (Regime change in the decay rates). The example above shows that there is a split in the decay behaviour as the tails of the Lévy intensity are progressively made lighter from polynomial and sub-exponential through exponential to super-exponential decays of ν. It is seen that if |λ0 | is small with respect to the edge of the essential spectrum of the generator of the free process and ν decreases exponentially, then the decay rate of φ0 is slower than ν, with essential contribution of λ0 . However, when |λ0 | is large enough, the fall-off is again dominated by ν as long as the basic jump-paring condition holds. In particular, this includes the exponential case with δ > d+1 . On the other hand, when δ ≤ d+1 the jump-paring condition fails, the 2 2 decay of φ0 is slower than the decay of ν, no matter how large |λ0 | is.

432 | 4 Feynman–Kac formulae 4.9.9 Massless relativistic harmonic oscillator In this section our aim is to discuss the specific one-dimensional nonlocal Schrödinger operator Hrosc = √−Δ + x 2 .

(4.9.90)

We will derive its spectrum and various properties of its eigenfunctions. Note that √−Δ is essentially self-adjoint on C0∞ (ℝ) and Spec(√−Δ) = Specess (√−Δ) = [0, ∞). Since the potential is positive and growing at infinity, we know that Spec(H) ⊂ [0, ∞) is discrete. Consider the eigenvalue equation √−Δφn + x 2 φn = En φn ,

φn ∈ H 1 (ℝ), n ∈ ℕ.

(4.9.91)

In this section we use another numbering and denote the ground state by φ1 . Theorem 4.225 directly implies that x 2 φn ∈ L1 (ℝ) for n ∈ ℕ. This also means that φn ∈ L1 (ℝ) for every n ∈ ℕ. The Fourier transform of φn is denoted by φ̂ n , and the eigenvalue equation (4.9.91) transforms into −φ̂ 󸀠󸀠 n (y) + (|y| − En )φ̂ n (y) = 0, We have |φ̂ 󸀠󸀠 n (y)| ≤

1 ‖x 2 φn ‖L1 (ℝ) , √2π

y ∈ ℝ.

(4.9.92)

y ∈ ℝ.

(4.9.93)

Furthermore, by multiplying and dividing, and using the Schwarz inequality, 1/2

‖φ̂ n ‖L1 (ℝ)

1 ≤ (∫ dy) 1 + y2 ℝ

2

2

1/2

(∫(1 + y )|φ̂ n (y)| dy)

= √π‖φn ‖H 1 (ℝ) .

ℝ

Hence it follows that φ̂ n ∈ C 2 (ℝ) ∩ L1 (ℝ) for all n ∈ ℕ. Note, moreover, that if φ̂ n (y) is a solution of (4.9.92), then so is φ̂ n (−y). Thus it suffices to consider (4.9.92) only for y > 0, and construct odd and even solutions on the whole of ℝ. We then know that the odd and even solutions satisfy, respectively, φ̂ n (0+) = 0,

n = 2, 4, 6, . . . ,

and

φ̂ 󸀠n (0+) = 0,

n = 1, 3, 5, . . . .

(4.9.94)

Our goal in what follows is to study the spectrum and the eigenfunctions of Hrosc given by (4.9.91) through the L1 solutions of (4.9.92) satisfying the boundary conditions (4.9.94). Let y > 0. Setting ϕn (y) = φ̂ n (y − En ) in (4.9.92) and making the linear transformation y − En 󳨃→ y, the eigenvalue equation reduces to the Airy differential equation ϕ󸀠󸀠 n (y) − yϕn (y) = 0,

(4.9.95)

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 433

whose solutions are expressed in terms of the Airy functions as ϕn (y) = c1,n Ai(y) + c2,n Bi(y). Here Ai(x) = Bi(x) =

1 t3 ∫ cos ( + xt) dt, π 3 ∞

0 ∞

1 t3 t3 ∫ [sin ( + xt) + exp (− + xt)] dt. π 3 3 0

Since φ̂ n ∈ L1 (ℝ) and the function Bi tends to infinity as x → ∞, we have c2,n = 0 for all n. Without loss of generality we take c1,n = 1 for all n, and we normalize these functions in a next step. For even solutions on the full real line we thus obtain φ̂ n (y) = Ai(|y| − En ),

y ∈ ℝ.

(4.9.96)

The boundary conditions (4.9.94) for this case give Ai󸀠 (−En ) = 0. For the odd solutions we have φ̂ n (y) = sgn(y)Ai(|y| − En ),

y ∈ ℝ,

(4.9.97)

and the boundary conditions imply Ai(−En ) = 0. By the Parseval identity and the fact that (yAi(y)2 − (Ai󸀠 (y))2 )󸀠 = Ai(y)2 , which is a straightforward consequence of the Airy equation (4.9.95), we obtain 2

2

∞

‖φn ‖ = ‖φ̂ n ‖ = 2 ∫ Ai(y − En )2 dy = 2 ((Ai󸀠 (−En ))2 + En Ai(−En )2 ) . 0

We have thus proved the following result. Theorem 4.340. The spectrum of Hrosc is purely discrete given by Spec(Hrosc ) = {Ek }∞ k=0 with E2k−1 = −a󸀠k

and

E2k = −ak ,

for k ∈ ℕ, where ak and a󸀠k denote the zeroes of the functions Ai and Ai󸀠 in decreasing order. They are all simple, the eigenfunctions φ2k−1 (x) are even and φ2k (x) are odd. Furthermore, the Fourier transforms of the normalized eigenfunctions φ̂ n are given by Ai(|y|−E )

n { √2En Ai(−En ) , φ̂ n (y) = { sgn(y)Ai(|y|−En ) { √2Ai󸀠 (−En ) ,

n = 1, 3, 5, . . . , n = 2, 4, 6, . . . .

434 | 4 Feynman–Kac formulae We present some further results on the eigenvalues and eigenfunctions without proof, and refer the interested reader to the literature. To have a more explicit idea of the dependence of the eigenvalues on the index k, we make use of the asymptotic expansions and estimates for the zeroes of the Airy function and its derivative. Corollary 4.341 (Asymptotic behavior of eigenvalues). We have as k → ∞ E2k−1 ∼ g (

3π (4k − 1)) 8

and

E2k ∼ f (

3π (4k − 3)) , 8

where g(t) = t 2/3 (1 −

7 −2 35 −4 181223 −6 18683371 −8 t + t − t + t − ⋅ ⋅ ⋅) , 48 288 207360 1244160

f (t) = t 2/3 (1 +

5 −2 5 −4 77125 −6 108056875 −8 t − t + t − t + ⋅ ⋅ ⋅) . 48 36 82944 6967296

Moreover, for k ∈ ℕ E2k−1 ≤ (

2/3 2/3 3π 3π 3 5 (4k − 1)) ≤ E2k ≤ ( (4k − 1)) (1 + arctan ( )) . 8 8 2 18π(4k − 1)

A numerical calculation for the first few eigenvalues gives E1 ≅ 1.01879297164747,

E3 ≅ 3.24819758217983,

E5 ≅ 4.82009921117874,

E2 ≅ 2.33810741045976,

E4 ≅ 4.08794944413097, E6 ≅ 5.52055982809555.

In particular, these expressions give the following spectral gap estimate. 2/3

Corollary 4.342 (Spectral gap). It follows that E2 − E1 ≥ ( 3π ) 8

(32/3 − 1).

Another consequence is a result on the asymptotic behavior of the heat trace at zero and a Weyl-type formula on the asymptotic distribution of eigenvalues. Let N(E) = |{n ∈ ℕ | En ≤ E}| be the counting measure of the number of eigenvalues En of Hrosc up to level E > 0. Corollary 4.343 (Weyl-type law). It follows that ∞

lim t 3/2 ∑ e−En t =

t→0+

and lim

E→∞

n=1

1 , √π

N(E) 4 = . 3π E 3/2

Next we summarize some further information obtained on the eigenfunctions.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 435

Corollary 4.344 (Asymptotic behavior of eigenfunctions). For every k ∈ ℕ and N = 2, 3, . . ., we have as |x| → ∞ the asymptotic expansions φ2k−1 (x) = √

p (a󸀠 ) p (a󸀠 ) p (a󸀠 ) 2 1 ( 3 4 k − 5 6 k + ⋅ ⋅ ⋅ + (−1)N 2N−12N k ) + O ( 2N+2 ) , 󸀠 −ak x x x x

φ2k (x) = √2 (

q4 (ak ) q6 (ak ) q (a ) 1 − + ⋅ ⋅ ⋅ + (−1)N 2N2N+1k ) + O ( 2N+3 ) , x7 x5 x x

where pn and qn are polynomials of degree n defined by the recursive relations pn+1 (x) = p󸀠n (x) + xqn (x),

qn+1 (x) = pn (x) + qn󸀠 (x)

with p0 (x) ≡ 1 and q0 (x) ≡ 0. Furthermore, we have the following corollary. Corollary 4.345 (Properties of eigenfunctions). (1) The eigenfunctions φn are real analytic functions on ℝ with Maclaurin expansions φ2k−1 (x) = √ φ2k (x) =

m

∞ w (a )(−1) 2m 2 1 x , ∑ 2m k 󸀠 󸀠 (2m)! −ak Ai(ak ) m=0 󸀠

w2m+1 (ak )(−1)m 2m+1 x , (2m + 1)! Ai󸀠 (ak ) m=0 √2

∞

∑

where k ∈ ℕ and wn (x) = ∫0 Ai(u + x)un du. (2) The eigenfunctions φn have a finite number of zeroes. (3) The eigenfunctions φn are uniformly bounded. (4) The ground state φ1 is decreasing on (0, ∞), and there exist x1 > x0 > 0 such that φ1 is concave on [−x0 , x0 ] and is convex on (−∞, −x1 ] and [x1 , ∞). ∞

Finally, we have the following heat kernel estimates. Corollary 4.346 (Properties of semigroup). (1) (Long times) There exists a constant c > 1 such that 1 e a1 t e a1 t ≤ e−tHrosc (x, y) ≤ c , 4 4 4 c (1 + x )(1 + y ) (1 + x )(1 + y4 ) 󸀠

󸀠

for every t > 1 and x, y ∈ ℝ. (2) (Short times) For every t ∈ (0, 1] and x, y ∈ ℝ we have e−tHrosc (x, y) ≈

t t 2 + |x − y|2

whenever

t
0. Denote p = −i dx and define the functions

g(x) = 2x − sin(2x),

h(x) =

1 1 + g(x)2

f (x) = (√(p + 1)2 + m2 + √(p − 1)2 + m2 ) h(x) and φ(x) = f (x) sin x 1 V(x) = λ − (√p2 + m2 − m) φ(x), φ(x)

(4.9.101) (4.9.102)

where λ = √1 + m2 − m > 0. Theorem 4.348. Let H be given by (4.9.100), V by (4.9.102), and φ by (4.9.101). If m ≥ 146, then V is a real-valued smooth potential with the property that V(x) = O(1/|x|), and λ, φ satisfy the eigenvalue equation Hφ = λφ, φ ∈ D(H). Furthermore, there exists a constant C > 0 such that 2 C ≤ |φ(x)| ≤ , 1 + x2 1 + x2

x ∈ ℝ.

438 | 4 Feynman–Kac formulae We note that the restriction m ≥ 146 is inessential, and by a scaling a similar result applies for all m > 0. Indeed, with a > 0 and (Ua g)(x) = a1/2 g(ax), the operator H is unitary equivalent to Ha = Ua HUa−1 = a1 (√p2 + (am)2 − am + aV(ax)). Then by Theorem 4.348 we can construct a smooth decaying potential such that Ha has a positive eigenvalue for any a with am > 146. We can use this basic result to derive a result also in three dimensions.

Corollary 4.349. Let m ≥ 146, write W(x) = V(|x|), x ∈ ℝ3 , and consider the operator Hr = √−Δ + m2 − m + W(x) on L2 (ℝ3 ). Then ζ (x) =

φ(|x|) √4π|x|

is in D(Hr ) = D(√−Δ) and satisfies the eigenvalue equation Hr ζ = λζ with the same eigenvalue λ = √1 + m2 − m. Next we note that in the non-relativistic limit the potentials, the eigenvalues and eigenfunctions constructed in the previous section converge to the expressions obtained by von Neumann and Wigner. To obtain this, similarly to Section 4.6.4 we restore the speed of light c > 0 as a parameter in the operator. Let 1 √ ( (p + 1)2 + m2 c2 + √(p − 1)2 + m2 c2 ) h(x), 2mc φc (x) = fc (x) sin x, fc (x) =

λc = c (√1 + m2 c2 − mc) , Vc (x) = λc − c

(√p2 + m2 c2 − mc) φc (x) φc (x)

.

Then we define the relativistic Hamiltonian with c by Hc = √c2 p2 + m2 c4 − mc2 + Vc (x). By Theorem 4.348 we see that the eigenvalue equation Hc φc = λc φc holds for all c > 146/m. Theorem 4.350. For every fixed m > 0 we have the following non-relativistic limit: lim φc (x) = sin(x)h(x) = φ∞ (x),

c→∞

lim λ c→∞ c

=

1 , 2m

lim Vc (x) =

c→∞

p2 φ∞ (x) 1 (1 − ), 2m φ∞ (x)

uniformly in C 2 (ℝ),

x ∈ ℝ \ πℕ.

In the three-dimensional case we retrieve the expressions (4.9.98)–(4.9.99). With a similar notation as above we obtain

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 439

1 Corollary 4.351. For every fixed m > 0 we have limc→∞ λc = 2m , limc→∞ ζc (x) = 2 3 φNW (x), uniformly in C (ℝ ), and limc→∞ Wc (x) = VNW (x), for all x ∈ ℝ3 .

Finally, we consider the delicate situation of eigenvalues at the continuum edge. In Corollary 4.151 and Proposition 4.168 we have seen some conditions ruling out zero eigenvalues for classical Schrödinger operators, and discussed that in general it is difficult to determine whether zero is an eigenvalue or not, which may be important information for specific applications. First we show by an example that zero eigenvalues can occur for nonlocal Schrödinger operators as well and then we will study some intriguing behaviours of the related eigenfunctions. We note that there are many more examples of such potentials. Let κ > 0, α ∈ (0, 2), and P be a solid harmonic polynomial, homogeneous of degree l ≥ 0, i. e., satisfying P(cx) = cl P(x) for all c > 0, and ΔP = 0. Recall Gauss’ regularized hypergeometric function 2 F1 (

a

󵄨 ∞ (a)k (b)k k b 󵄨󵄨󵄨 1 󵄨󵄨 z) = z , ∑ 󵄨 c 󵄨󵄨 Γ(c + 1) k=0 k!

z ∈ ℂ,

for a, b, c ∈ ℂ such that c is not a negative integer, with the Pochhammer symbol (p)k = p(p + 1) ⋅ ⋅ ⋅ (p + k − 1). Theorem 4.352. Let P be a solid harmonic polynomial of degree l ∈ ℕ,̇ and write δ = d + 2l for d ∈ ℕ. Also, let θ > 0, and 0 < κ ≤ δ+α with α ∈ (0, 2). Consider 2 Vκ,α (x) = − φκ (x) =

δ+α 2 α √θ α δ+α α Γ( ) Γ ( + κ) (1 + θ|x|2 )κ 2 F1 ( 2 Γ(κ) 2 2

P(x) (1 + θ|x|2 )κ

δ 2

α 2

󵄨 + κ 󵄨󵄨󵄨 󵄨󵄨 −θ|x|2 ) , 󵄨󵄨 󵄨 (4.9.103)

for x ∈ ℝd . Then the following hold: (1) The equality Hκ(α) φκ = (−Δ)α/2 φκ + Vα,κ (x)φκ = 0

holds in distributional sense. For κ ∈ (0, 4δ ] the functions φκ are zero-resonances and for κ ∈ ( 4δ , δ+α ) they are zero-eigenfunctions of Hκ(α) . 2

(2) We have lim|x|→∞ Vκ,α (x) = 0 if and only if 0 < κ < (3) Asymptotic behaviour at infinity: O (|x|−α ) { { { {O (|x|−2α ) |Vκ,α (x)| = { {O (|x|−α log |x|) { { 2κ−δ−α ) {O (|x| as |x| → ∞.

if κ if κ if κ if κ

δ+α . 2

∈ (0, δ2 ) \ { δ−α }, 2 δ−α = 2 , = δ2 , ∈ ( δ2 , δ+α ), 2

440 | 4 Feynman–Kac formulae Furthermore,

δ α Γ( −κ)Γ( +κ) 2 2 , for κ ∈ (0, δ2 ) \ { δ−α } δ−α 2 Γ(κ)Γ( −κ) 2 δ+α Γ( ) 2 , (ii) lim|x|→∞ |x|2α Vκ,α (x) = −2α √θα δ−α for κ = δ−α 2 Γ( 2 ) δ+α α Γ( ) |x| α√ α 2 (iii) lim|x|→∞ log(1+|x| θ , for κ = δ2 2 ) Vκ,α (x) = −2 α Γ(− )Γ( δ2 ) 2 δ+α δ 2κ−δ−α α √ α Γ(κ− 2 )Γ( 2 ) (iv) lim|x|→∞ |x| Vκ,α (x) = −2 θ Γ(κ)Γ(− α ) , for κ ∈ ( δ2 , δ+α ). 2 2 δ−α δ+α For sufficiently large |x| we have Vκ,α (x) > 0 if κ ∈ ( 2 , 2 ) and Vκ,α (x) κ ∈ ( 4δ , δ−α ]. 2

(i) lim|x|→∞ |x|α Vκ,α (x) = −2α √θα

(4)

< 0 if

For the remaining part of this section we discuss the decay properties of zeroenergy eigenfunctions and resonances by using Feynman–Kac representations. From the results in Sections 4.3.8, 4.6.3 and 4.9.8 we have seen that the decay rates for an eigenfunction φ crucially depend on the gap between the eigenvalue λ < 0 corresponding to φ and the continuum edge Σ = 0. Now we are interested in decay properties in the conditions when this gap is zero. We will use the same notations and concepts as in Section 4.9.8, and only indicate here the differences. First we give the definition of the class of processes that we will use, which slightly differs from the jump-paring class given by Defnition 4.324. Consider the operator L with symbol ψ as defined by (4.9.81)–(4.9.82). We will use the symmetrization Φ(r) = sup|ξ |≤r ψ(ξ ), r > 0, of the symbol ψ as above. We consider the following class of Lévy processes and related operators L. Definition 4.353. Let (Xt )t≥0 be a Lévy process with Lévy–Khintchine exponent ψ as in (4.9.81) and Lévy triplet (0, A, ν), satisfying the following conditions. (1) There exist a non-increasing function g : (0, ∞) → (0, ∞) and constants C1 , C2 > 0 such that ν(x) ≍ C1 g(|x|),

x ∈ ℝd \ {0} ,

and

g(|x|) ≤ C2 g(2|x|),

|x| ≥ 1.

(2) There exists tb > 0 such that supx∈ℝd p(tb , x) = p(tb , 0) < ∞. (3) There exists a constant C3 = C3 (X) such that sup

x,y: |x−y|≥s/8

GB(0,s) (x, y) ≤ C3

Φ(1/s) , sd

s ≥ 1.

Note that the difference from Definition 4.324 is that condition (1) requires a doubling property of the profile function of ν instead of the more general jump-paring property. These conditions are satisfied by isotropic and anisotropic stable processes, layered stable processes etc, and the related operators include the fractional Laplacian and generators of Lévy processes comparable with the fractional Laplacian.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 441

Choosing X-Kato decomposable potentials from K±X as given in Definition 4.326, we define the Feynman–Kac semigroup as in (4.9.83). Also, define 1 Hf = − lim (Tt f − f ), t↓0 t

f ∈ Lp (ℝd ),

such that the strong limit exists in Lp (ℝd ). We denote the set of all such functions by DLp H and call it the Lp -domain on H. Then H is a closed unbounded operator such that DLp H is dense in Lp (ℝd ). For p = 2 this operator can be identified with the self-adjoint operator H = −L + V defined in form sense. Let V ∈ 𝒦±X be a decaying potential. We consider solutions φ ∈ Lp (ℝd ), p ≥ 1, φ ≢ 0, of the equation Hφ = 0.

(4.9.104)

In the L2 -framework Σ = inf Specess (H) = inf Specess (−L) = 0 is the edge of the essential spectrum of H and (4.9.104) can be understood as the eigenvalue equation at Σ. Whenever the solution φ to (4.9.104) is such that φ ∈ L2 (ℝd ), we call it a zero-energy eigenfunction (or zero-energy bound state) and then 0 is an eigenvalue. Otherwise, we call both (by a slight abuse of language) a zero-resonance. When the eigenvalue is zero, the decay no longer can benefit from the relative position of the eigenvalue from the continuum edge as in the situations discussed in Section 4.9.8. As it turns out, now it depends on vestigial effects of the potential like its sign at infinity. Thus we split the discussion according to this. First we consider potentials V positive at infinity in the sense that V(x) → 0 as |x| → ∞, and there exists r0 > 0 such that V(x) ≥ 0 for |x| ≥ r0 . (4.9.105) We will use the notation V∗ (x) =

inf

r0 ≤|y|≤ 32 |x|

V(y),

|x| ≥ r0 .

Notice that V∗ (x) is a radial and non-increasing function such that V∗ (x) ≥ 0, |x| ≥ r0 . Theorem 4.354 (Upper bound). Let (Xt )t≥0 be a Lévy process with Lévy–Khintchine exponent ψ as in (4.9.81) and Lévy intensity ν, such that the assumptions in Definition 4.353 hold. Let V ∈ 𝒦±X such that (4.9.105) holds with a suitable r0 > 0, and φ be a solution of (4.9.104). Then we have the following: (1) If lim

|x|→∞

1 Φ ( |x| )

V∗ (x)

= 0 and

∫ |x|>2r0

ν(x) dx < ∞, V∗ (x)

442 | 4 Feynman–Kac formulae then there exists C > 0 and R ≥ 2r0 such that |φ(x)| ≤ C ‖φ‖∞

ν(x) , V∗ (x)

|x| > R.

In particular, φ ∈ L1 (ℝd ). (2) If lim

1 Φ ( |x| )

|x|→∞

V∗ (x)

= 0 and

∫ |x|>2r0

ν(x) dx = ∞, V∗ (x)

and ν(z) [ ν(x) ] sup [ exp(η∗ dz)] < ∞, ∫ V (x) V |x|≥2r0 ∗ ∗ (z) 2r0 ≤|z|≤|x| [ ] with a constant η∗ > 0, then there exist C > 0 and R ≥ 2r0 such that |φ(x)| ≤ C ‖φ‖∞ Φ(

1

ν(x) exp(η∗ V ∗ (x)

∫ r0 ≤|z|≤|x|

ν(z) dz), V∗ (z)

|x| > R.

|x| (3) If lim inf|x|→∞ V (x) > 0, φ ≥ 0, and φ ∈ Lp (ℝd ) for some p ∈ (1, ∞), then there ∗ exists C > 0 such that

)

p

φ(x) ≤ C ‖φ‖p (

(∫|z|>|x| ν(z) p−1 dz)

p−1 p

1 Φ( |x| )

+

1 ), |x|d/p

|x| > 2r0 .

The next result deals with the case when φ has no definite sign, but it is antisymmetric with respect to a given (d − 1)-dimensional hyperplane p ⊂ ℝd with 0 ∈ p, and has a definite sign on both of the corresponding half-spaces. By rotating the coordinate system if necessary, we choose p = {x ∈ ℝd | x1 = 0} and assume that φ(−x1 , x2 , . . . , xd ) = −φ(x1 , x2 , . . . , xd ), x = (x1 , . . . , xd ) ∈ ℝd ,

and φ(x1 , . . . , xd ) ≥ 0 whenever x1 > 0.

(4.9.106)

Theorem 4.355. Let the assumptions of Theorem 4.354 hold, and furthermore suppose that there exist C0 > 0 and R0 > 0 such that |ν(z1 ) − ν(z2 )| ≤ C0

ν(z2 ) |z − z2 |, |z2 | 1

|z1 | ≥ |z2 | ≥ R0 .

(4.9.107)

Let φ be a solution of (4.9.104) such that (4.9.106) is satisfied. Then the following hold:

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 443 Φ(

1

)

|x| (1) If lim|x|→∞ V (x) = 0, ∫|x|>2r ∗ 0 C1 > 0 such that

ν(x) dx V∗ (x)

< ∞, φ ∈ L1 (ℝd ), and there exists a constant

ν(y) ν(x) ≤ C1 , V∗ (y) V∗ (x)

|y| ≥

|x| ≥ 2r0 , 2

(4.9.108)

then there exists C > 0 and R ≥ 2r0 ∨ 4R0 such that |φ(x)| ≤ C(‖φ‖∞ ∨ ‖φ‖1 ) for |x1 | > R. (2) If lim inf|x|→∞

1 Φ( |x| ) V∗ (x)

ν(x) Φ(1/|x|) 1 ( ∨( V∗ (x) V∗ (x) |x|

∫ 2r0 0 and φ ∈ Lp (ℝd ), p > 1, then there exists C > 0 such that p

|φ(x)| ≤ C ‖φ‖p (

(∫|z|>|x| ν(z) p−1 dz)

p−1 p

1 Φ( |x| )

+

1 ), |x|d/p

|x1 | > 2r0 .

Now we give a lower estimate on the eigenfunctions. We use the notation V ∗ (x) = sup V(y),

|x| ≥ 2r0 .

|y|≥ |x| 2

Theorem 4.356 (Lower bound). Let the assumptions of Theorem 4.354 hold and φ be a positive solution of (4.9.104). Then we have the following: (1) If lim|x|→∞ such that

1 Φ( |x| ) ∗ V (x)

= 0 and ∫|x|>2r

0

ν(x) dx V ∗ (x)

φ(x) ≥ C (2) If lim|x|→∞ such that

1 Φ( |x| ) ∗ V (x)

= 0 and ∫|x|>2r

φ(x) ≥ C

(3) If lim inf|x|→∞ such that

1 Φ( |x| ) ∗ V (x)

0

ν(x) , V ∗ (x)

ν(x) dx V ∗ (x)

ν(x) exp (η∗ V ∗ (x)

> 0 and ∫|x|>2r

0

< ∞, then there exist C > 0 and R ≥ 2r0

= ∞, then there exist η∗ , C > 0 and R ≥ 2r0

∫ 2r0 ≤|y|≤|x|

ν(x) 1 dx Ψ( |x| )

φ(x) ≥ C

|x| > R.

ν(x)

1 Φ ( |x| )

ν(y) dy) , V ∗ (y)

|x| > R.

< ∞, then there exist C > 0 and R ≥ 2r0

,

|x| > R.

444 | 4 Feynman–Kac formulae (4) If lim inf|x|→∞

1 Φ( |x| ) V ∗ (x)

R ≥ 2r0 such that

φ(x) ≥ C

> 0 and ∫|x|>2r ν(x)

1 Φ ( |x| )

0

ν(x) 1 dx Ψ( |x| )

exp (η∗

= ∞, then there exist η∗ , C > 0 and

∫ 2r0 ≤|y|≤|x|

ν(y)

1 Φ ( |y| )

dy) ,

|x| > R.

Next we consider potentials negative at infinity in the sense that there exists r0 > 0 and C > 0 such that 0 ≤ −V(x) ≤ CΦ(1/|x|), |x| ≥ r0 .

(4.9.109)

Note that indeed V(x) → 0 as |x| → ∞, and the condition above also covers potentials with compact support such as potential wells. We present a counterpart of Theorem 4.355 (3) in the case when the potential is negative at infinity and the negative nodal domain of φ is a subset of a given half-space. By rotating the coordinate system if necessary, without loss of generality we assume that there exists l ∈ ℝ such that supp φ− ⊂ {y ∈ ℝd | y1 < l}.

(4.9.110)

Theorem 4.357. Let (Xt )t≥0 be a Lévy process with Lévy–Khintchine exponent ψ as in (4.9.81) and Lévy intensity ν, such that the assumptions in Definition 4.353 hold. Consider V ∈ L∞ (ℝd ) satisfying (4.9.109). Moreover, suppose that for every ε ∈ (0, 1) there exists M ≥ 1 such that Φ(r) ≤ εΦ(Mr), with r ∈ (0, 1]. If φ is a solution of (4.9.104) such that φ ∈ Lp (ℝd ), for some p > 1, and (4.9.110) holds, then for every ε ∈ (0, 1) there exist C > 0 and R > 3r0 such that p

φ(x) ≤ C (‖φ‖p ∨ ‖φ‖∞ ) (

(∫|y|>|x| ν(y) p−(1+ε) dy) Φ(1/|x|)

1−ε

p−(1+ε) p

+

1

d

|x| p

)

,

x1 ≥ R.

Moreover, if φ(l + x1 , x2 , . . . , xd ) = −φ(l − x1 , x2 , . . . , xd ), x ∈ ℝd , with l given by (4.9.110), ̃ > 3r0 ∨ |l| such that the same upper bound is true for φ(x) replaced then there exists R ̃ by |φ(x)|, whenever |x1 | ≥ R. To highlight these results, we particularize to fractional Schrödinger operators. First consider potentials that are positive at infinity. Theorem 4.358. Let L = −(−Δ)α/2 , 0 < α < 2, and V ∈ 𝒦±X for which there exists r0 > 0 such that V(x) > 0 and V(x) ≍ |x|−β , for |x| ≥ r0 , with some β > 0. Suppose that there exists a positive function φ ∈ Cb (ℝd ) which is a solution of (4.9.104). Then the following hold: (1) If β < α, then there exist constants C1 , C2 > 0 such that C1 C2 ≤ φ(x) ≤ , d+α−β (1 + |x|) (1 + |x|)d+α−β

In particular, φ ∈ Lp (ℝd ), for every p ≥ 1.

x ∈ ℝd .

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 445

(2) If β ≥ α, then there exist ρ ∈ (0, 1) and a constant C3 > 0 such that C3 , (1 + |x|)d−ρ

φ(x) ≥

x ∈ ℝd .

d In particular, φ ∉ Lp (ℝd ), for every p ∈ [1, d−ρ ]. On the other hand, if φ ∈ Lp (ℝd ) for some p > 1, then there exists C4 > 0 such that

φ(x) ≤

C4 , (1 + |x|)d/p

In particular, if φ ∈ Lp (ℝd ) with p = C4 > 0 such that φ(x) ≤

d d−ρ−ε

x ∈ ℝd .

for some ε ∈ (0, 1), then there exists

C4 , (1 + |x|)d−ρ−ε

x ∈ ℝd .

Notice that under the assumptions of Theorem 4.358 it follows that φ ∈ L1 (ℝd ) if and only if α > β. It can already be seen from the above theorem that there is a transition in the localization properties of φ when α > β changes to α ≤ β. For a closer understanding of this transition around α ≈ β, we consider a more refined class of potentials. Theorem 4.359. Let L = −(−Δ)α/2 , 0 < α < 2, and V ∈ 𝒦±X for which there exists r0 > 0 such that V(x) > 0 and V(x) ≍ |x|−α (log |x|)δ , for |x| ≥ r0 , with some δ > 0. Suppose that there exists a positive function φ ∈ Cb (ℝd ) which is a solution of (4.9.104). Then the following hold. (1) If δ > 1, then there exist constants C1 , C2 > 0 such that C1 C2 ≤ φ(x) ≤ , (1 + |x|)d (log(1 + |x|))δ (1 + |x|)d (log(1 + |x|))δ

x ∈ ℝd .

In particular, φ ∈ Lp (ℝd ), for every p ≥ 1. (2) If δ = 1, then there exist 0 < ρ1 ≤ 1 ≤ ρ2 and constants C4 , C5 > 0 such that (1 +

C1 d |x|) (log(1

+

|x|))1−ρ1

≤ φ(x) ≤

(1 +

C2 d |x|) (log(1

+ |x|))1−ρ2

,

x ∈ ℝd .

In particular, φ ∈ Lp (ℝd ) for every p > 1, but φ ∉ L1 (ℝd ). (3) If δ ∈ (0, 1), then there exist 0 < ρ1 ≤ 1 ≤ ρ2 and constants C6 , C7 > 0 such that C6

ρ1

1−δ

ρ2

1−δ

e 1−δ (log |x|) e 1−δ (log |x|) ≤ φ(x) ≤ C7 , d δ (1 + |x|) (log(1 + |x|)) (1 + |x|)d (log(1 + |x|))δ

x ∈ ℝd .

In particular, φ ∈ Lp (ℝd ), for every p > 1, but φ ∉ L1 (ℝd ). (4) If δ ≤ 0, then exactly the same bounds and Lp -properties hold as in (2) of Theorem 4.358.

446 | 4 Feynman–Kac formulae Remark 4.360. From the results above it is seen that the possible localization properties of the positive zero-energy eigenfunctions or zero-resonances for decaying potentials positive at infinity splits naturally into disjoint regimes representing the following three different scenarios. For simplicity assume that V is a potential that is positive at infinity and temperately varying so that V(x) ≍ V ∗ (x) ≍ V∗ (x) for large enough |x|. Let r0 > 0 be large enough, and define h(r) =

∫ r0 0. Clearly, in case h tribute to the lower rate. By a direct inspection, one can see that these results compare well with the exact results in Theorem 4.352 (3). In particular, it can be checked that in the range d2 < κ < d+α 2 d Scenario 1 prevails, for κ = d2 we have Scenario 2, and for 0 ∨ d−α < κ < we have 2 2 Scenario 3. Finally, we present a result for the case of fractional Schrödinger operators with potentials that are negative at infinity.

4.9 Feynman–Kac formula for nonlocal Schrödinger operators | 447

Theorem 4.361. Let L = −(−Δ)α/2 , 0 < α < 2, and V ∈ 𝒦±X for which there exists r0 > 0 such that V(x) ≤ 0 and |V(x)| ≤ C|x|−α , for |x| ≥ r0 , with some C > 0. Suppose that there exists a function φ ∈ Lp (ℝd ), for some p > 1, which is a solution to (4.9.104) with the property that there exists i ∈ {1, 2, . . . , d} such that φ(x1 , . . . , −xi , . . . , xd ) = −φ(x1 , . . . , xi , . . . , xd ) and supp φ− ⊂ {x ∈ ℝd | xi ≤ 0}. Then for every q ∈ (0, dp ) there exist C = C(q) and R > 0 such that |φ(x)| ≤ C (‖φ‖p ∨ ‖φ‖∞ )

1 , |x|q

|xi | ≥ R.

Remark 4.362. From the above it is seen that the decay of ground states at zero eigenvalue depends essentially on two factors. On the one hand, the sign of the potential at infinity makes a qualitative difference. From the decay results above one can appreciate that a positive tail of the potential has a soft bouncing effect tending to contain paths in compact regions, while a negative potential allows the paths to spread more out to infinity. This difference makes the analysis of potentials negative at infinity much more difficult than of potentials positive at infinity. On the other hand, the decay depends on mean sojourn times spent in some regions by the paths. Since (by assuming V(x) ≍ V ∗ (x) ≍ V∗ (x), for simplicity) 1 𝔼 [τB|x| (0) ] ≍ Φ(1/|x|) 0

and

τB|x|/2 (x)

x

ΛB|x|/2 (x) (x) = 𝔼 [ ∫ 0

t

e− ∫0 V(Xs )ds dt] ≍

1 , V(x)

we see that now the decay is governed by the quantity 1 Φ ( |x| )

V(x)

≍

ΛB|x|/2 (x) (x)

𝔼0 [τB|x| (0) ]

,

so that the decays relate to global survival times (in large balls of radii proportional to |x| itself), rather than local survival times as in (4.9.85), which makes the occurrence of zero eigenvalues a very small and delicate effect. Like in the case of decaying potentials with strictly negative eigenvalues, see Remark 4.339, here we see further qualitative transitions between decay regimes. When ΛB|x|/2 (x) (x) = o(𝔼0 [τB|x| (0) ]) as in Scenarios (1)–(2) above, the potential has a relatively pronounced effect, making the paths have a slight but decisive preference for neighbourhoods around the origin than for far out neighbourhoods, even if these regions are now very large. This is especially reflected in the decay by V entering explicitly in Scenario (1). When, however, ΛB|x|/2 (x) (x) = O(𝔼0 [τB|x| (0) ]) as in Scenario (3), the effect of the potential is extremely weak also in relative terms, and the two lifetimes (one under the potential, the other free of the potential) evolve on the same scale, getting very near to, though still differing from, the situation of free fluctuations. This probabilistic picture also gives a hint about the very “birth” of bound states at zero energy level.

5 Gibbs measures associated with Feynman–Kac semigroups 5.1 Ground state transform and related processes 5.1.1 Ground state transform and the intrinsic semigroup As we have seen in Chapter 4, the Feynman–Kac representations of the semigroups generated by standard Schrödinger operators − 21 Δ + V and nonlocal Schrödinger operators Ψ(− 21 Δ) + V with Bernstein functions Ψ ∈ ℬ0 of the Laplacian, are formally the same but with the essential difference of featuring different processes. This allows us to unify the discussion so that we obtain (1) standard Schrödinger operators and Brownian motion for Ψ(u) = u; (2) fractional Schrödinger operators and rotationally symmetric α-stable processes for Ψ(u) = (2u)α/2 , 0 < α < 2; (3) relativistic Schrödinger operators and rotationally symmetric relativistic α-stable processes for Ψ(u) = (2u + m2/α )α/2 − m, 0 < α < 2, m > 0; (4) other nonlocal Schrödinger operators and subordinate Brownian motions with further possible choices of Ψ. In this section therefore H will be any of these Schrödinger operators, {Kt : t ≥ 0} the related Feynman–Kac semigroup, and (Xt )t≥0 the Lévy process (Brownian motion or a jump process) generated by −Ψ(− 21 Δ), unless Ψ is further specified. Also, we mean by Ψ-Kato decomposable potentials any of the cases as given by Definitions 4.89, 4.280 or 4.292. The Feynman–Kac semigroup {Kt : t ≥ 0} has the peculiarity that in general Kt 1ℝd ≠ 1ℝd . Now we introduce a semigroup related to the Feynman–Kac semigroup which, however, does keep unit mass and thus becomes a Markov semigroup. This is called the intrinsic Feynman–Kac semigroup. We assume that V is Ψ-Kato-decomposable, and the corresponding Schrödinger operator is defined through the Feynman–Kac formula, as in Definition 4.114 or (4.9.24). We suppose, moreover, that H has a normalized ground state φ0 > 0, i. e., Hφ0 = E0 φ0 ,

E0 = inf Spec(H),

‖φ0 ‖2 = 1.

(5.1.1)

̄ 0 = 0. We write H̄ = H − E0 so that Hφ Definition 5.1 (Ground state transform). Let H be a standard or nonlocal Schrödinger operator with Kato-decomposable potential V, and φ0 > 0 be its normalized unique ground state. Consider the probability measure dN0 = φ20 (x)dx https://doi.org/10.1515/9783110330397-005

(5.1.2)

450 | 5 Gibbs measures associated with Feynman–Kac semigroups on ℝd , and the Hilbert space L2 (ℝd , dN0 ). The ground state transform is the unitary map 2

d

2

d

U : L (ℝ , dN0 ) → L (ℝ , dx),

f 󳨃→ φ0 f .

In fact, the ground state transform is a case of Doob’s h-transform familiar from the theory of random processes. From now on we use the simpler notation L2 (dN0 ) = L2 (ℝd , dN0 ). Define the operator L = U −1 H̄ U

(5.1.3)

with domain D(L) = {f ∈ L2 (dN0 ) | U f ∈ D(H)}. Also, let E0 t ̃t (x, y) = e Kt (x, y) , K φ0 (x)φ0 (y)

(5.1.4)

where Kt (x, y) denotes the integral kernel of Kt . Definition 5.2 (Intrinsic Feynman–Kac semigroup). The one-parameter semigroup ̃t : t ≥ 0} acting on L2 (dN0 ) with integral kernel K ̃t (x, y), i. e., {K ̃t f (x) = ∫ K ̃t (x, y)f (y)dN0 (y), K

(5.1.5)

ℝd

is called intrinsic Feynman–Kac semigroup for the potential V. The generator of the intrinsic semigroup is the operator L given by (5.1.3). Note that ̃t are even though the Lp -norms of the operators Kt can be larger than 1, the operators K always contractions. Moreover, the intrinsic Feynman–Kac semigroup is more natural ̃t 1 d = 1 d , for every t > 0. than {Kt : t ≥ 0} since K ℝ ℝ In view of Examples 4.72 and 4.74 it can be expected that there are important differences between Schrödinger (and therefore Feynman–Kac) semigroups depending on the behavior of V at infinity. The following is a strong regularizing property. Definition 5.3 (Ultracontractivity properties). The semigroup {e−tH : t ≥ 0} is called (1) ultracontractive if e−tH is contractive from L2 (ℝd ) to L∞ (ℝd ) for all t > 0; (2) intrinsically ultracontractive (IUC) if e−tL is ultracontractive on L2 (dN0 ), t > 0; (3) asymptotically intrinsically ultracontractive (AIUC) if there exists t0 > 0 such that e−tL is ultracontractive on L2 (dN0 ), for all t > t0 . Theorem 5.4 (Intrinsic ultracontractivity). Let H be a Schrödinger operator with a Katodecomposable potential V and ground state φ0 . Consider its Feynman–Kac semigroup {Kt : t ≥ 0}. Then {e−tH : t ≥ 0} is IUC (resp. AIUC) if and only if for every t > 0 (resp. if

5.1 Ground state transform and related processes | 451

for every t > t0 , for some t0 > 0) there is a constant CV,t > 0 such that Kt (x, y) ≤ CV,t φ0 (x)φ0 (y),

x, y ∈ ℝd ,

(5.1.6)

or equivalently, x, y ∈ ℝd .

̃t (x, y) ≤ CV,t , K

(5.1.7)

A consequence of intrinsic ultracontractivity is that a similar lower bound on the (1) kernel also holds, i. e., for every t > 0 there is a constant CV,t > 0 such that (1) Kt (x, y) ≥ CV,t φ0 (x)φ0 (y),

x, y ∈ ℝd .

(5.1.8)

An immediate consequence of this is that if the semigroup is intrinsically ultracontractive, then φ0 ∈ L1 (ℝd ). A classic result for the Feynman–Kac semigroup generated by Schrödinger operators H = − 21 Δ + V identifying the borderline case for IUC in a specific growth-class, is the following fact. Proposition 5.5. If H = − 21 Δ + V with V(x) = |x|β , then the semigroup {Kt : t ≥ 0} is intrinsically ultracontractive if and only if β > 2. Moreover, if β > 2, then cϱ(x) ≤ φ0 (x) ≤ Cϱ(x), |x| > 1, holds with some C, c > 0 and β

ϱ(x) = |x|− 4 +

d−1 2

e

2 − 2+β |x|1+β/2

.

Remark 5.6. We note that there is no proper borderline potential for IUC to hold (therefore the proviso on the growth class in the theorem above). Indeed, a sufficient condition for IUC is ∞

∫

r0

dr < ∞, √V(r)

for some r0 > 0,

(5.1.9)

which for radial potentials is also necessary. For instance, for the potential V(x) = |x|2 (log |x|)2 (log log |x|)2 ⋅ ⋅ ⋅ (log ⋅ ⋅ ⋅ log |x|)2 (log ⋅ ⋅ ⋅ log |x|)2+δ , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (m − 1)-times

m-times

m ∈ ℕ, δ ≥ 0,

this condition is satisfied if and only if δ > 0. The study of intrinsic ultracontractivity can be extended also to nonlocal Schrödinger operators. Since this requires different techniques than discussed so far and go beyond the scope of this book, we give some remarks instead of detailed theorems and refer the interested reader to the literature. The following result holds for a large class of nonlocal operators but we only state it for fractional and relativistic Schrödinger operators.

452 | 5 Gibbs measures associated with Feynman–Kac semigroups Proposition 5.7. Consider Ψ(u) = uα/2 and Ψ(u) = (2u + m2/α )α/2 − m, 0 < α < 2, m > 0, denote by ν the corresponding Lévy intensity, and let V be a Ψ-Kato decomposable potential. The following hold: V(x) (1) If lim|x|→∞ |log = ∞, then the semigroup {Kt : t ≥ 0} is IUC. ν(x)| (2) If there exist R, C > 0 such that {Kt : t ≥ 0} is AIUC.

V(x) |log ν(x)|

≥ C for all |x| ≥ R, then the semigroup

The above result can be upgraded to necessary and sufficient conditions under a further restriction on the potential by requiring them to grow temperately in the sense that supy∈B V(y) ≤ C infy∈B V(y) with a constant C > 0, for every unit ball B in BR (0)c . Remark 5.8 (Borderline potentials). In contrast with standard Schrödinger operators, nonlocal Schrödinger operators do have borderline potentials of minimal growth for AIUC to hold. This is given by equivalence classes of potentials defined by the asymptotic comparability V ≍ |log ν|, where ν is the Lévy density of the process generated by −Ψ(− 21 Δ). Borderline classes can be classified by whether they are of logarithmic order (when − log ν(x) ≍ log |x| for large enough |x|), linear order (when − log ν(x) ≍ |x| for large enough |x|), or sublinear but faster than logarithmic order (when − log ν(x) ≍ |x|β , 0 < β < 1, for large enough |x|) interpolating between them, while there are no further cases. It follows from Proposition 5.7 that for the fractional Laplacian (−Δ)α/2 the borderline potential is of logarithmic order, and it is of linear order for the relativistic Laplacian (−Δ + m2/α )α/2 . This indicates that for nonlocal Schrödinger semigroups it is “easier” to achieve intrinsic ultracontractivity, and it can be explained by a more efficient mixing mechanism in path space due to the possibility of jumps in contrast with the continuous paths of Brownian motion. It can be seen that neither the Brownian component nor the small-jump component of the process has any effect on AIUC. For our purposes below an important consequence of AIUC is the following property. Proposition 5.9. The following two conditions are equivalent. (1) The semigroup {Kt : t ≥ 0} is AIUC. (2) The property ̃t (x, y) t→∞ K 󳨀→ 1

(5.1.10)

holds, uniformly in (x, y) ∈ ℝd × ℝd . Proof. The implication (2) ⇒ (1) is immediate, thus we only show the converse statement. We have by the semigroup property for every x, y ∈ ℝd and t > t0

5.1 Ground state transform and related processes | 453

󵄨󵄨̃ 󵄨 󵄨󵄨Kt (x, y) − 1󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 Kt0 (x, z)Kt−2t0 (z, w)Kt0 (w, y) e−E0 t φ0 (x)φ0 (y) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 ∫ dzdw − 󵄨󵄨 e−E0 t φ0 (x)φ0 (y) e−E0 t φ0 (x)φ0 (y) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝd ×ℝd 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Kt0 (x, z)φ(z)(Kt−2t0 (z, w) − e−E0 (t−2t0 ) φ0 (z)φ0 (w))Kt0 (w, y)φ0 (w) 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨 ∫ dzdw 󵄨󵄨 󵄨󵄨 e−E0 t φ0 (x)φ0 (z)φ0 (w)φ0 (y) 󵄨󵄨 󵄨󵄨 ℝd ×ℝd 󵄨 󵄩󵄩 Kt (x, y) 󵄩󵄩2 󵄨 󵄨 󵄩 󵄩󵄩 0 ≤ eE0 t 󵄩󵄩󵄩 ∫ 󵄨󵄨󵄨󵄨Kt−2t0 (z, w) − e−E0 (t−2t0 ) φ0 (z)φ0 (w)󵄨󵄨󵄨󵄨 φ0 (z)φ0 (w)dzdw 󵄩 󵄩󵄩 φ0 (x)φ0 (y) 󵄩󵄩󵄩∞ d d ℝ ×ℝ

1/2

≤ Ce

E0 t

( ∫ |Kt−2t0 (z, w) − e

−E0 (t−2t0 )

2

φ0 (z)φ0 (w)| dzdw)

.

ℝd ×ℝd

The last factor at the right-hand side is the Hilbert–Schmidt norm of the operator Kt−2t0 − e−E0 (t−2t0 ) 1φ0 , where 1φ0 : L2 (ℝd ) → L2 (ℝd ) is the projection onto the onedimensional subspace of L2 (ℝd ) spanned by φ0 . This gives ∞

̃t (x, y) − 1| ≤ CeE0 t ( ∑ e−2Ek (t−2t0 ) ) |K k=1

1/2

∞

1/2

= Ce2t0 E1 e−(E1 −E0 )t ( ∑ e−2(Ek −E1 )(t−2t0 ) ) , k=1

where E0 < E1 ≤ . . . are the eigenvalues of H. By the dominated convergence theorem the last sum converges to the multiplicity of E1 as t → ∞. Since E1 > E0 , (5.1.10) follows. The difference E1 − E0 > 0 is the (fundamental) spectral gap. Remark 5.10 (Other contractivity properties). Apart from intrinsic ultracontractivity, there are further smoothening semigroup properties of interest: (1) The semigroup {e−tL : t ≥ 0} is supercontractive if for every p ∈ (2, ∞) and t > 0 the operators e−tL are bounded from L2 (dN0 ) to Lp (dN0 ). The semigroup {e−tH : t ≥ 0} is intrinsically supercontractive (ISC) if {e−tL : t ≥ 0} is supercontractive. (2) The semigroup {e−tL : t ≥ 0} is hypercontractive if for every p ∈ (2, ∞) there exists tp > 0 such that for every t ≥ tp the operators e−tL are bounded from L2 (dN0 ) to Lp (dN0 ). The semigroup {e−tH : t ≥ 0} is called intrinsically hypercontractive (IHC) if {e−tL : t ≥ 0} is hypercontractive. (3) For p ∈ (2, ∞] we say that the operator e−tH is Lp -ground state-dominated if 1 −tH e 1ℝd ∈ Lp (dN0 ). φ 0

For standard Schrödinger operators there is a natural hierarchy of these properties. For instance, in the case of H = −Δ + |x|α (log(1 + |x|))β the related intrinsic semigroup is not hypercontractive if α < 2, it is hypercontractive but not supercontractive if α = 2,

454 | 5 Gibbs measures associated with Feynman–Kac semigroups β = 0, it is supercontractive but not ultracontractive if α = 2, 0 < β ≤ 2, and it is ultracontractive if α = 2, β > 2 or if α > 2. This shows that ultracontractivity is essentially stronger than supercontractivity, which is essentially stronger than hypercontractivity. For necessary, sufficient or joint conditions, and mutual implications we refer to the literature (see a discussion in Chapter 6). One interesting fact is that for a large subclass of nonlocal Schrödinger operators with Bernstein functions of the Laplacian and a large class of potentials ISC is equivalent with IUC, and IHS is equivalent with AIUC. On the other hand, for standard Schrödinger operators all these properties are different. 5.1.2 Ground state-transformed processes as solutions of SDE Since L is a Markov generator on L2 (dN0 ), we can associate a stochastic differential equation with it. The advantage of this is that it sheds light on the structure of the process generated by L, which we call ground state-transformed process, by identifying its drift, diffusion and jump components. Consider first the standard Schrödinger operator H = − 21 Δ + V, and assume that φ0 > 0 is its unique ground state. Since C0∞ (ℝd ) is a core also for L, a simple calculation shows that on this subspace we have 1 Lf = − Δf − ∇ log φ0 ⋅ ∇f . 2

(5.1.11)

This then leads to the stochastic differential equation dXt = ∇ log φ0 (Xt )dt + dBt ,

X0 = x,

(5.1.12)

whose solution (Xtx )t≥0 satisfies 𝔼[f (Xtx )] = (e−tL f )(x),

t ≥ 0.

(5.1.13)

However, for given V in general little information is available about the regularity of φ0 , and thus it is not clear if ∇ log φ0 is well-defined as a function at all and one can view (5.1.12) as an SDE in the classical sense discussed in Chapter 2 or more general setups are necessary. Also, even if the SDE has a proper meaning, due to the generally unbounded and non-linear drift coefficient, the study of existence and uniqueness of solutions is not straightforward as in many interesting cases the Lipschitz condition (2.4.53)–(2.4.54) in Theorem 2.152 will not hold. Instead of this general study we will discuss two specific cases at the end of this section, for which the SDE can be explicitly solved and which illuminate the mechanisms due to which solutions exist. Consider now nonlocal Schrödinger operators H = Ψ(− 21 Δ) + V such that the Lévy process (Xt )t≥0 has Lévy triplet (0, A, ν), and V is such that H has a unique ground state φ0 > 0. Then on C0∞ (ℝd ) we have

5.1 Ground state transform and related processes | 455

1 Lf = σ∇ ⋅ σ∇f + σ∇ log φ0 ⋅ σ∇f + 2

∫ 01 0

φ0 (Xs− )

(5.1.14)

where (Bt )t≥0 is d-dimensional Brownian motion with covariance matrix σ, N is a Poisson random measure on [0, ∞) × (ℝd \ {0}) × [0, ∞) with intensity dt ν(z)dz dw, and ̃ N(ds, dz, dw) is the related compensated Poisson measure. We have the following result, which we present without proof since it would require more preparation on SDE with jumps than presented in Section 2.4.5 above. Theorem 5.11. Suppose that the function x 󳨃→ ∇ log φ0 (x), x ∈ ℝd , is locally bounded. Then the stochastic differential equation (5.1.14) has a weak solution. While this leaves uniqueness open, in the sections below we construct processes which are weak solutions of the stochastic differential equations (5.1.12) or (5.1.14). Remark 5.12. Ground state-transformed processes are interesting objects on their own, even without the direct relevance of Feynman–Kac type representations. As it can be seen from (5.1.14), the related SDE has position-dependent coefficients which destroy stationarity of increments, and which are generally unbounded. These processes belong to the class of Lévy-type (Feller) processes, whose general theory is in the making. Also, from the equation one can appreciate a second interpretation of potentials on the level of processes after the sojourn and local times studied in Section 4.3.5, showing that the effect of the potential is a position-dependent drift and a bias in the jump intensity. For an analysis of global and local sample path properties we refer the reader to the literature, as discussed in Chapter 6 below. We conclude this section by a discussion of two specific cases of ground statetransformed processes via SDE for potentials of fundamental interest. Example 5.13 (Ground state-transformed process for the harmonic oscillator). Consider 1 d2 mω2 2 ω Hosc = − + x − (5.1.15) 2m dx2 2 2

456 | 5 Gibbs measures associated with Feynman–Kac semigroups on L2 (ℝ). The parameter ω > 0 is the frequency of the oscillator. The ground state of Hosc is explicitly given by φosc (x) = (ω/π)1/4 e−ωx

2

/2

.

Since log φosc ∈ C 1 (ℝ), formula (5.1.12) gives the SDE dXt = −ωXt dt + dBt ,

X0 = x,

which is a case of the Langevin equation studied in Example 2.159. A unique strong solution of this SDE is the Ornstein–Uhlenbeck process Xtx

t

= xe

−ωt

+ ∫ e−ω(s−t) dBs .

(5.1.16)

0

We take up this example for further discussion in Section 5.1.5 below. Example 5.14 (Ground state-transformed process for potential wells). Consider now the Schrödinger operator with one-dimensional potential well Hv = −

d2 − v1{|x|≤a} dx2

with v > 0. Using the explicit formula (4.3.29) and the fact that φ0 ∈ C 1 (ℝ), the SDE (5.1.12) becomes dXt = dBt − (√2|E| sgn(Xt )1{|Xt |>a} + √2(v − |E|) tan(√2(v − |E|) Xt )1{|Xt |≤a} ) dt. (5.1.17) Let λ be an eigenvalue of an even eigenfunction of Hν . Recall that |λ| satisfies that tan (a√2(v − |λ|)) = √

|λ| , v − |λ|

(5.1.18)

which implies that X tan X = Y, { 2 X + Y 2 = 2νa2 ,

(5.1.19)

where X = a√2(V − |λ|) and Y = a√|λ|. The ground state energy λ = E of Hν is the unique solution of (5.1.19) such that 0 < X < π/2. Hence √2(v − |E|)
0, with compact support 𝒟 ⊂ ℝd . Suppose that the Schrödinger operator H = Ψ(− 21 Δ) − v1𝒟 , Ψ ∈ ℬ0 , has a ground state φ0 at eigenvalue λ0 . Then v − |λ0 | ≤ λ1𝒟 ,

(5.1.21)

where λ1𝒟 is the principal eigenvalue of Ψ(− 21 Δ) in 𝒟 with Dirichlet exterior condition on 𝒟c . Denote by (Xt )t≥0 the Lévy process generated by −Ψ(− 21 Δ), by P x its probabilty measure starting at x ∈ ℝd , and by τ𝒟 = inf{t > 0 | Xt ∈ 𝒟c } the first exit time of the process from 𝒟. Then from the Feynman–Kac formula we get that t

φ0 (x) = 𝔼x [e∫0 (v1𝒟 (Xs )+λ0 )ds φ0 (Xt )]

(5.1.22)

for almost every x. The right-hand side is continuous in x and gives a continuous version of φ0 , which we choose from now on. Since the semigroup {e−tHν : t ≥ 0} is positivity improving, φ0 (x) > 0 for x ∈ ℝ \ N with a null set N. Suppose that φ0 (y) = 0 t

for some y. Then we have 𝔼y [e∫0 (v1𝒟 (Xs )+λ0 )ds ψ0 (Xt )] = 0, but the integrand is nont

negative and we see that e∫0 (v1𝒟 (Xs )+λ0 )ds φ0 (Xt ) = 0 almost surely under P y . Since t ∫0 (v1𝒟 (Xs )+λ0 )ds

e > 0 almost surely, φ0 (Xt ) = 0 almost surely with respect to P y . Thus y y we have 𝔼y [φ0 (Xt )] = ∫ℝ φ0 (x)Pt (x)dx = 0, where Pt (x) denotes the distribution function of Xt which is strictly positive. This implies that φ0 (x) = 0 for almost every x, and contradicts that φ0 (x) is strictly positive for almost every x. This means that the continuous version of φ0 is strictly positive, in particular, miny∈𝒟 φ0 (y) > 0. From (5.1.22) we have t

φ0 (x) ≥ 𝔼x [e∫0 (v1𝒟 (Xs )+λ0 )ds φ0 (Xt )1{t t), y∈𝒟

x ∈ 𝒟,

for every t ≥ 0. By taking logarithms on both sides and dividing by t > 0, we get v − |λ0 | ≤ − lim sup t→∞

1 log P x (τ𝒟 > t) = λ1𝒟 , t

giving (5.1.21). When Ψ(u) = u and 𝒟 = [−a, a], this principal Dirichlet eigenvalue is π2 λ1𝒟 = 8a 2 , which indeed gives (5.1.20). Condition (5.1.20) ensures that the drift is bounded and (5.1.17) has a unique strong solution by Theorem 2.152. Note that the everywhere continuity of the drift term is equivalent to condition (4.3.28), and the drift is piecewise differentiable except for x = ±a. Indeed, from the expression it is seen that at the boundaries of the potential well there is a regime change of the drift and as soon as a path exits [−a, a], then due to the sgn(Xt ) term it is pulled back to the origin at a constant speed √2|E|. This is the stabilizing mechanism in this case making sure that paths do not explode, i. e., do not go to infinity in finite time, and thus a solution exists. We consider now (5.1.17) and choose the interval I = [−a, a]. Let dXt = −√2(v − |E|) tan(√2(v − |E|) Xt )dt + dBt

458 | 5 Gibbs measures associated with Feynman–Kac semigroups and consider the stopping time τ±a = inf{t > 0 | |Xt | = ±a}. We construct a weak solution of this stochastic differential equation for times t ≤ τ±a . It readily follows that the scale function defined in (2.4.90) in this case is S(x) =

1

a√2(v − |E|)

tan(√2(v − |E|)x).

Thus an application of the Itô formula to the process (Yt )t≥0 = (S(Xt ))t≥0 gives 1 dYt = dBt . 1 + 2(v − |E|)Yt2 d

Using Lemma 2.170 we obtain Yt = B1h−1 (t) , where (B1t )t≥0 denotes a one-dimensional Brownian motion and h(t) =

t

1 + (√2(v − |E|)B1t )2

.

5.1.3 P(ϕ)1 -processes with continuous paths Now we discuss a second approach to ground state-transformed processes. In this section we consider standard Schrödinger operators H = − 21 Δ + V and use the related Markov operators L given by (5.1.3) to construct a diffusion directly via extension of its finite dimensional distributions. Consider two-sided Brownian motion and denote by X = C(ℝ, ℝd ) the space of continuous paths on the real line ℝ as before. In this section we construct a probability measure 𝒩0x , x ∈ ℝd , on (X, ℬ(X)) such that the coordinate process (Xt )t∈ℝ is a diffusion process generated by L. Theorem 5.15. Suppose that V is Kato-decomposable and the Schrödinger operator H = − 21 Δ + V has a strictly positive ground state φ0 . Let L = U −1 H̄ U be the ground state transform of H, and Xt (ω) = ω(t) the coordinate process on (X, ℬ(X)). Then there exists a probability measure 𝒩0x on (X, ℬ(X)) satisfying the following properties: (1) (Initial distribution) 𝒩0x (X0 = x) = 1; (2) (Reflection symmetry) (Xt )t≥0 and (Xs )s≤0 are independent, and d

X−t = Xt ,

t ∈ ℝ;

(3) (Diffusion property) let Ft = σ(Xs , 0 ≤ s ≤ t), +

Ft = σ(Xs , t ≤ s ≤ 0) −

be given filtrations; then (Xt )t≥0 and (Xs )s≤0 are diffusion processes with respect to (Ft+ )t≥0 and (Ft− )t≤0 , respectively, i. e., 𝔼𝒩0x [Xt+s |Fs+ ] = 𝔼𝒩0x [Xt+s |σ(Xs )] = 𝔼𝒩 Xs [Xt ], 0

5.1 Ground state transform and related processes | 459

− 𝔼𝒩0x [X−t−s |F−s ] = 𝔼𝒩0x [X−t−s |σ(X−s )] = 𝔼𝒩 X−s [X−t ] 0

for s, t ≥ 0, and ℝ ∋ t 󳨃→ Xt ∈ ℝd is almost surely continuous, where 𝔼𝒩 Xs means 0 𝔼𝒩 y evaluated at y = Xs ; 0 (4) (Shift invariance) let −∞ < t0 ≤ t1 ≤ . . . ≤ tn < ∞; then n

∫ 𝔼𝒩0x [∏ fj (Xtj )]dN0 = (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ e−(tn −tn−1 )L fn )L2 (dN0 ) j=0

ℝd

(5.1.23)

for f0 , fn ∈ L2 (dN0 ) and fj ∈ L∞ (ℝd ), j = 1, . . . , n − 1, and the finite-dimensional distributions of the process are shift invariant, i. e., n

n

∫ 𝔼𝒩0x [ ∏ fj (Xtj )]dN0 = ∫ 𝔼𝒩0x [ ∏ fj (Xtj +s )]dN0 ,

ℝd

j=1

j=1

ℝd

s ∈ ℝ.

Definition 5.16 (P(ϕ)1 -process). Let H = − 21 Δ+V and its ground state φ0 be given, and consider the probability space (X, ℬ(X), 𝒩0x ). The coordinate process (Xt )t∈ℝ , Xt (ω) = ω(t) on this space is called a P(ϕ)1 -process for H. We proceed now to prove existence and some properties of P(ϕ)1 -processes in several steps. For arbitrary time points 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn , n ∈ ℕ, define the set function 𝒫{t0 ,...,tn } : ×nj=0 ℬ(ℝd ) → ℝ by n

𝒫{t0 ,...,tn } (×i=0 Ai ) = (1A0 , e

−(t1 −t0 )L

1A1 ⋅ ⋅ ⋅ e−(tn −tn−1 )L 1An )L2 (dN0 )

(5.1.24)

and for t ≥ 0, 𝒫{t} : ℬ(ℝd ) → ℝ by 𝒫{t} (A) = (1, e

−tL

1A )L2 (dN0 ) = (1, 1A )L2 (dN0 ) .

(5.1.25)

Step 1: The family of set functions given by (5.1.24)–(5.1.25) satisfies the consistency condition n

n+m

d

n

𝒫{t0 ,...,tn+m } ((×i=0 Ai ) × (×i=n+1 ℝ )) = 𝒫{t0 ,...,tn } (×i=0 Ai ),

for all m, n ∈ ℕ. Hence by the Kolmogorov extension theorem there exists a probability measure 𝒫∞ on ((ℝd )[0,∞) , σ(𝒜)), where d

d

𝒜 = {ω : ℝ → ℝ | ω⌈Λ ∈ E, E ∈ (ℬ(ℝ )) , Λ ⊂ ℝ, |Λ| < ∞} |Λ|

(5.1.26)

such that (5.1.27)

𝒫{t} (A) = 𝔼𝒫∞ [1A (Yt )] , n

n

𝒫{t0 ,...,tn } (×i=0 Ai ) = 𝔼𝒫∞ [ ∏ 1Aj (Ytj )], j=0

n ≥ 1,

(5.1.28)

460 | 5 Gibbs measures associated with Feynman–Kac semigroups for A, A0 , . . . , An ∈ 𝒜, where Yt (ω) = ω(t), ω ∈ (ℝd )[0,∞) , is the coordinate process. The process (Yt )t≥0 on the probability space ((ℝd )[0,∞) , σ(𝒜), 𝒫∞ ) satisfies n

(f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ e−(tn −tn−1 )L fn )L2 (dN0 ) = 𝔼𝒫∞ [ ∏ fj (Ytj )],

(5.1.29)

(1, f )L2 (dN0 ) = (1, e−tL f )L2 (dN0 ) = 𝔼𝒫∞ [f (Yt )] = 𝔼𝒫∞ [f (Y0 )]

(5.1.30)

j=0

for fj ∈ L∞ (ℝd ), j = 0, 1, . . . , n, and 0 ≤ t0 < t1 < . . . < tn , for all n ∈ ℕ. Step 2: Next we show that this process has a continuous version. Lemma 5.17. The process (Yt )t≥0 has a continuous version. Proof. By the Kolmogorov–Čentsov theorem it suffices to show that the estimate 𝔼𝒫∞ [|Yt − Ys |2n ] ≤ |t − s|n , n ≥ 2, holds. Let s < t. We have d 2n 2n ̄ 𝔼𝒫∞ [|Yt − Ys |2n ] = ∑ ∑ ( ) (−1)k (φ, x μ e−(t−s)H x μ φ). k μ=1 k=0

The last term can be represented in terms of Brownian motion as μ

t−s

μ

(φ0 , xμ e−(t−s)H x μ φ0 ) = e(t−s)E ∫ 𝔼x [B0 φ0 (B0 )Bt−s φ0 (Bt−s )e− ∫0 ̄

V(Br )dr

]dx.

ℝd

By Schwarz inequality, t−s

𝔼𝒫∞ [|Yt − Ys |2n ] = e(t−s)E ∫ 𝔼x [|Bt−s − B0 |2n φ0 (B0 )φ0 (Bt−s )e− ∫0 ℝd

V(Br )dr

1/2

t

]dx 1/2

≤ e(t−s)E 𝔼[|Bt−s − B0 |2n ( ∫ e−2 ∫0 V(Br +x)dr φ0 (x)2 dx) ( ∫ φ0 (Bt−s + x)2 dx) ] ℝd

ℝd t

1/2

≤ e(t−s)E ‖φ0 ‖(𝔼[|Bt−s − B0 |4n ]) 𝔼 [( ∫ e−2 ∫0 V(Br +x)dr φ0 (x)2 dx)] . [ ℝd ] Since V is Kato-decomposable, we furthermore have t

≤ e−(t−s)E ‖φ0 ‖2 𝔼[|Bt−s − B0 |4n ]1/2 sup 𝔼x [e−2 ∫0 V(Bs )ds ] x∈ℝd

t

≤ √C2n |t − s|n e−(t−s)E ‖φ0 ‖2 sup 𝔼x [e−2 ∫0 V(Bs )ds ]. x∈ℝd

Here we used that 𝔼[|Bt − Bs |2n ] = Cn |t − s|n .

5.1 Ground state transform and related processes | 461

Let now Y = (Y t )t≥0 be the continuous version of (Yt )t≥0 on the probability space ((ℝd )[0,∞) , σ(𝒜), 𝒫∞ ). Denote the image measure of 𝒫∞ on (X , ℬ(X )) with respect to Y by ℳ = 𝒫∞ ∘ Y .

(5.1.31)

−1

̃t (ω) = ω(t), for ω ∈ X . We thus constructed a We identify the coordinate process by Y d ̃ = (Y ̃t )t≥0 on (X , ℬ(X ), ℳ) such that Y t = ̃t for t ≥ 0. (5.1.29) and random process Y Y ̃ (5.1.30) can be expressed in terms of (Yt )t≥0 as n

̃t )], (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )L2 (dN0 ) = 𝔼ℳ [ ∏ fj (Y j j=0

(1, f )L2 (dN0 ) = (1, e

−tL

f )L2 (dN0 )

̃t )] = 𝔼ℳ [f (Y ̃0 )]. = 𝔼ℳ [f (Y

Step 3: Define a probability measure on X by x

̃0 = x) ℳ (⋅) = ℳ( ⋅ |Y

(5.1.32)

̃0 is dN0 , note that for all x ∈ ℝd . Since the distribution of Y ℳ(A) = ∫ 𝔼ℳx [1A ]dN0 (x). ℝd

̃t )t≥0 on (X , ℬ(X ), ℳx ) satisfies The random process (Y n

̃t )]dN0 , (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )L2 (dN0 ) = ∫ 𝔼ℳx [ ∏ fj (Y j j=0

ℝd

̃0 )]dN0 = ∫ f (x)dN0 . (1, e−tL f )L2 (dN0 ) = (1, f )L2 (dN0 ) = ∫ 𝔼ℳx [f (Y ℝd

(5.1.33)

ℝd

̃t )t≥0 is a Markov process on (X , ℬ(X ), ℳx ) with respect Lemma 5.18. The process (Y ̃s , 0 ≤ s ≤ t), t ≥ 0. to the natural filtration σ(Y Proof. Let pt (x, A) = (e−tL 1A )(x),

A ∈ ℬ(ℝd ),

t ≥ 0.

(5.1.34)

Note that pt (x, A) = 𝔼[1A (Xtx )]. Thus by (5.1.33) the finite-dimensional distributions of ̃t )t≥0 are (Y n

n

n

j=1

j=1

̃t )] = ∫ (∏ 1A (xj )) ∏ pt −t (xj−1 , dxj ) 𝔼ℳx [ ∏ 1Aj (Y j j j j−1 j=1

ℝnd

(5.1.35)

462 | 5 Gibbs measures associated with Feynman–Kac semigroups with t0 = 0 and x0 = x. We show that pt (x, A) is a probability transition kernel. Note that e−tL is positivity improving. Thus 0 ≤ e−tL f ≤ 1, for all functions f such that 0 ≤ f ≤ 1, and e−tL 1 = 1 follows, so pt (x, ⋅) is a probability measure on ℝd with pt (x, ℝd ) = 1. Secondly, pt (⋅, A) is trivially Borel measurable with respect to x. Thirdly, by the semigroup property e−sL e−tL 1A = e−(s+t)L 1A the Chapman–Kolmogorov identity (2.2.53) follows directly, and hence pt (x, A) is a probability transition kernel. Thus ̃t )t≥0 is a Markov process by (5.1.35) and Proposition 2.82. (Y ̃t )t≥0 to a process on the whole real line ℝ. Consider X ̃ = X × X , Step 4: We extend (Y M ̃ = ℬ(X ) × ℬ(X ), and ℳ̃ x = ℳx × ℳx . Let (X̃ t )t∈ℝ be the random process defined by ̃t (ω1 ), Y X̃ t (ω) = { ̃−t (ω2 ), Y

t ≥ 0, t < 0,

(5.1.36)

for ω = (ω1 , ω2 ) ∈ X ̃ , on the product space (X ̃ , M ,̃ ℳ̃ x ). Note that X̃ 0 = x almost surely under ℳ̃ x , and X̃ t is continuous in t almost surely. It is trivial to see that X̃ t , d t ≥ 0, and X̃ s , s ≤ 0, are independent, and X̃ t = X̃ −t . Step 5: We can now complete the proof of the theorem. Proof of Theorem 5.15. Denote the image measure of ℳ̃ x on (X, ℬ(X)) with respect to (X̃ t )t≥0 by x

x

𝒩0 = ℳ̃ ∘ X̃ .

(5.1.37)

−1

Let Xt (ω) = ω(t), t ∈ ℝ, ω ∈ X, be the coordinate process. We have d

̃t Xt = Y

d

(t ≥ 0),

̃−t Xt = Y

(t ≤ 0).

(5.1.38)

̃t )t≥0 and (Y ̃−t )t≤0 are Markov processes with respect to the Since by Step 3 above (Y ̃s , 0 ≤ s ≤ t) and σ(Y ̃s , −t ≤ s ≤ 0), respectively, (Xt )t≥0 and (Xt )t≤0 natural filtrations σ(Y are also Markov processes with respect to (Ft+ )t≥0 resp. (Ft− )t≤0 , where Ft+ = σ(Xs , 0 ≤ s ≤ t) and Ft− = σ(Xs , −t ≤ s ≤ 0). Thus the diffusion property follows. We also see d

that (Xs )s≤0 and (Xt )t≥0 are independent and X−t = Xt by (5.1.38) and Step 4 above. Hence reflection symmetry follows. Shift invariance follows by the lemma below; for any s ∈ ℝ, n

n

∫ 𝔼𝒩0x [ ∏ fj (Xtj )]dN0 = ∫ 𝔼𝒩0x [ ∏ fj (Xtj +s )]dN0 .

ℝd

j=0

ℝd

j=0

Lemma 5.19. Let −∞ < t0 ≤ t1 ≤ . . . ≤ tn . Then n

∫ 𝔼𝒩0x [ ∏ fj (Xtj )]dN0 = (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )L2 (dN0 ) .

ℝd

j=0

(5.1.39)

5.1 Ground state transform and related processes | 463

Proof. Let t0 ≤ . . . ≤ tn ≤ 0 ≤ tn+1 ≤ . . . ≤ tn+m . By the independence of (Xs )s≤0 and (Xt )t≥0 , n+m

n

n+m

j=0

j=n+1

∫ 𝔼𝒩0x [ ∏ fj (Xtj )]dN0 = ∫ 𝔼𝒩0x [∏ fj (Xtj )]𝔼𝒩0x [ ∏ fj (Xtj )]dN0 .

ℝd

j=0

ℝd

We furthermore have n+m

𝔼𝒩0x [ ∏ fj (Xtj )] = (e−tn+1 L fn+1 e−(tn+2 −tn+1 )L fn+2 ⋅ ⋅ ⋅ fn+m−1 e−(tn+m −tn+m−1 )L fn+m )(x) (5.1.40) j=n+1

and

n+m

n+m

j=0

j=0

𝔼𝒩0x [ ∏ fj (Xtj )] = 𝔼𝒩0x [ ∏ fj (X−tj )] = (e+tn L fn e−(tn −tn−1 )L fn−1 ⋅ ⋅ ⋅ f2 e−(t1 −t0 )L f1 )(x). (5.1.41)

By (5.1.40) and (5.1.41) we obtain n+m

∫ 𝔼𝒩0x [ ∏ fj (Xtj )]dN0 = (e+tn L fn ⋅ ⋅ ⋅ f2 e−(t1 −t0 )L f1 , e−tn+1 L fn+1 ⋅ ⋅ ⋅ e−(tn+m −tn+m−1 )L fn+m )L2 (dN0 )

ℝd

j=0

= (f1 , e−(t1 −t0 )L f2 ⋅ ⋅ ⋅ fn+m−1 e−(tn+m −tn+m−1 )L fn+m )L2 (dN0 ) .

Hence (5.1.39) follows. Remark 5.20. By Theorem 5.15 we have (e−tL f )(x) = 𝔼𝒩0x [f (Xt )].

(5.1.42)

The right-hand side of (5.1.42) can be also represented in terms of Brownian motion resulting in 𝔼𝒩0x [f (Xt )] =

eE0 t x − ∫0t V(Bs )ds 𝔼 [e f (Bt )φ0 (Bt )] φ0 (x)

(5.1.43)

as a function in L2 (ℝd ). Define the probability measure 𝒩0 on X × ℝd by d𝒩0 = d𝒩0x ⊗ dN0 .

(5.1.44)

Using this measure we have the following change-of-measure formula whose proof immediately follows from Lemma 5.19. Theorem 5.21 (Feynman–Kac formula for P(ϕ)1 -processes). If f , g ∈ L2 (dN0 ), then (f , e−tL g)L2 (dN0 ) = (fφ0 , e−t(H−E0 ) gφ0 )L2 (ℝd ) = 𝔼𝒩0 [f ̄(X0 )g(Xt )].

(5.1.45)

An immediate corollary of this theorem is the following. Corollary 5.22 (Invariant measure). The probability measure φ20 (x)dx defined on (ℝd , ℬ(ℝd )) is an invariant measure of the P(ϕ)1 -process (Xt )t≥0 .

464 | 5 Gibbs measures associated with Feynman–Kac semigroups 5.1.4 Dirichlet principle In this subsection we consider a property of running maxima of P(ϕ)1 -processes (Xt )t≥0 . Proposition 5.23. Let T, Λ > 0 and suppose that f ∈ C(ℝd ) ∩ D(L1/2 ). Then it follows that 𝒩0 ( sup |f (Xs )| ≥ Λ) ≤ 0≤s≤T

e ((f , f )L2 (dN0 ) + T(L1/2 f , L1/2 f )L2 (dN0 ) )1/2 . Λ

(5.1.46)

(Here e is the base of the natural logarithm.) Proof. Denote Tj = Tj/2n , j = 0, 1, . . . , 2n , and fix T and n. Let G = {x ∈ ℝd | |f (x)| ≥ Λ},

(5.1.47)

τ = inf {Tj ≥ 0 | XTj ∈ G}.

(5.1.48)

and define the stopping time

The identity 𝒩0 (

sup

j∈{0,1,...,2n }

|f (XTj )| ≥ Λ) = 𝒩0 (τ ≤ T)

directly follows. We estimate the right-hand side above. Let 0 < ϱ < 1 be fixed and choose a suitable ϱ later. We see that by the Schwarz inequality with respect to dN0 , 1/2

𝒩0 (τ ≤ T) = 𝔼𝒩0 [1{τ≤T} ] ≤ 𝔼𝒩0 [ϱ

τ−T

τ

] ≤ ϱ 𝔼𝒩0 [ϱ ] ≤ ϱ −T

−T

( ∫ (𝔼x𝒩0 [ϱτ ])2 dN0 ) ℝd

,

(5.1.49)

where 𝔼x𝒩0 = 𝔼𝒩0x . Let ψ ≥ 0 be any function such that ψ(x) ≥ 1 on G. The Dirichlet principle (shown in the lemma below) implies that ∫ (𝔼x𝒩0 [ϱτ ])2 dN0 ≤ (ψ, ψ) + ℝd

n

ϱT/2 −(T/2n )L )ψ). n (ψ, (1 − e T/2 1−ϱ

Inserting |f (x)| ≥ 1, { |f (x)| Λ = , Λ

x ∈ G,

x ∈ ℝd \ G

(5.1.50)

5.1 Ground state transform and related processes | 465

into ψ in (5.1.50), we have n

n 1 ϱT/2 1 ≤ 2 (f , f ) + (|f |, (1 − e−(T/2 )L )|f |). n 2 T/2 Λ Λ 1−ϱ

∫ (𝔼x𝒩0 [ϱτ ])2 dN0 ℝd

Since e−(T/2

n

(5.1.51)

is positivity improving and thus

)L

(|f |, (1 − e−(T/2

n

)L

)|f |) ≤ (f , (1 − e−(T/2

n

)L

)f ),

we have, by (5.1.49), 1/2

n

𝒩0 (

sup

j∈{0,1,...,2n }

Set ϱ = e−1/T . By 𝒩0 (

|f (XTj )| ≥ Λ) ≤ n

ϱT/2 n 1−ϱT/2

sup

j∈{0,1,...,2n }

Since (f , (1 − e−(T/2

n

)L

𝒩0 (

ϱ−T ϱT/2 −(T/2n )L ((f , f ) + )f )) . n (f , (1 − e Λ 1 − ϱT/2

≤ 2n , we have

|f (XTj )| ≥ Λ) ≤

n e ((f , f ) + 2n (f , (1 − e−(T/2 )L )f ))1/2 . Λ

(5.1.52)

)f ) ≤ (T/2n )(L1/2 f , L1/2 f ), we obtain sup

j∈{0,1,...,2n }

|f (XTj )| ≥ Λ) ≤

e ((f , f ) + T(L1/2 f , L1/2 f ))1/2 . Λ

(5.1.53)

Take n → ∞ on both sides of (5.1.53). By the dominated convergence theorem lim 𝒩0 (

n→∞

sup

j∈{0,1,...,2n }

|f (XTj )| ≥ Λ) = 𝒩0 ( lim

sup

n→∞ j∈{0,1,...,2n }

|f (XTj )| ≥ Λ) .

Since f (Xt ) is continuous in t, it follows that lim

sup

n→∞ j∈{0,1,...,2n }

|f (XTj )| = sup |f (Xs )|. 0≤s≤T

It remains to show (5.1.50). Lemma 5.24 (Dirichlet principle). Let 0 < ϱ < 1. Fix n and set Tj = Tj/2n , j = 0, 1, . . . , 2n . Let G ⊂ ℝd be measurable and τ = inf{Tj ≥ 0 | XTj ∈ G}. Then for every function ψ ≥ 0 such that ψ(x) ≥ 1 on G, it follows that ∫ (𝔼x𝒩0 [ϱτ ])2 dN0 ≤ (ψ, ψ)L2 (dN0 ) + ℝd

n

ϱT/2 −(T/2n )L )ψ)L2 (dN0 ) . n (ψ, (1 − e T/2 1−ϱ

466 | 5 Gibbs measures associated with Feynman–Kac semigroups Proof. Define ψϱ (x) = 𝔼x𝒩0 [ϱτ ]. By the definition of τ we see that ψϱ (x) = 1,

x ∈ G,

(5.1.54)

since τ = 0 when Xs starts from the inside of G. Let Ft = σ(Xs , 0 ≤ s ≤ t) be the natural filtration of (Xt )t≥0 . By the Markov property of (Xt )t≥0 it is directly seen that e−(T/2

n

)L

X

ψϱ (x) = 𝔼x𝒩0 [𝔼𝒩T/2 [ϱτ ]] = 𝔼x𝒩0 [𝔼x𝒩0 [ϱτ∘θT/2n |FT/2n ]] = 𝔼x𝒩0 [ϱτ∘θT/2n ], n

0

(5.1.55)

where θt is the shift on X defined by (θt ω)(s) = ω(s + t) for ω ∈ X. Note that (τ ∘ θT/2n )(ω) = τ(ω) − T/2n ≥ 0,

(5.1.56)

when x = X0 (ω) ∈ ℝd \ G. Hence by (5.1.55) and (5.1.56) we have the identity n

ϱT/2 e−(T/2

n

)L

ψϱ (x) = ψϱ (x),

x ∈ ℝd \ G.

(5.1.57)

It is trivial to see that ∫ (𝔼x𝒩0 [ϱτ ])2 dN0 = (ψϱ , ψϱ ) ≤ (ψϱ , ψϱ ) + ℝd

n

ϱT/2 −(T/2n )L )ψϱ ). n (ψϱ , (1 − e T/2 1−ϱ

By using relation (5.1.57) we can compute the right-hand side above as n

(ψϱ 1G , ψϱ 1G ) +

ϱT/2 −(T/2n )L )ψϱ ). n (ψϱ 1G , (1 − e T/2 1−ϱ

(5.1.58)

Since n

(ψϱ 1G , (1 − e−(T/2

= (ψϱ 1G , (1 − e = (ψϱ 1G , (1 − e ≤ (ψϱ 1G , (1 − e

)L

)ψϱ )

−(T/2n )L −(T/2n )L −(T/2n )L

)ψϱ 1G ) + (ψϱ 1G , (1 − e−(T/2 )ψϱ 1G ) − (ψϱ 1G , e

−(T/2n )L

n

)L

)ψϱ 1ℝd \G )

ψϱ 1ℝd \G )

)ψϱ 1G ),

we have ∫ (𝔼x𝒩0 [ϱτ ])2 dN0 ℝd

n

ϱT/2 −(T/2n )L ≤ (ψϱ 1G , ψϱ 1G ) + )ψϱ 1G ). n (ψϱ 1G , (1 − e 1 − ϱT/2

(5.1.59)

Note that ψϱ 1G (x) ≤ ψ(x) for all x ∈ ℝd . We have n

(ψϱ 1G , ψϱ 1G ) +

ϱT/2 −(T/2n )L )ψϱ 1G ) n (ψϱ 1G , (1 − e T/2 1−ϱ n

ϱT/2 −(T/2n )L ≤ ‖ψ‖ + )ψ). n (ψ, (1 − e T/2 1−ϱ 2

A combination of (5.1.59) and (5.1.60) then proves the lemma.

(5.1.60)

5.1 Ground state transform and related processes | 467

5.1.5 Mehler’s formula In this section we consider the special case Hosc = −

1 Δ + Vosc 2m

with Vosc (x) =

mω2 2 ω |x| − 2 2

m, ω > 0, describing the harmonic oscillator, which is one of the few models for which an explicit formula is available for the kernel of the Schrödinger semigroup. This kernel is given by Mehler’s formula. We will give three proofs by using three different but related approaches. First we proceed by the ground state transform and obtain Mehler’s formula via the kernel of the generator of the intrinsic operator (generator of the P(ϕ)1 -process). Next we obtain it via a computation of the integral given by the Feynman–Kac formula. Finally, we also show a proof using operator theory. Derivation via ground state transform. Let d = 1 and consider the Schrödinger operator Hosc = −

1 d2 mω2 2 ω + x − 2 2m dx 2 2

(5.1.61)

on L2 (ℝ). Here m is the mass of the oscillator, and ω is the traditional notation for its frequency which should not be confused with other meaning before. Hosc is essentially self-adjoint on C0∞ (ℝ) by Corollary 4.56, and from Example 4.75 we know that Spec(Hosc ) = {ωn | n ∈ ℕ ∪ {0}}. The ground state is explicitly given by φosc (x) = (

mω 1/4 −mωx2 /2 ) e . π

(5.1.62)

Furthermore, the complete set of eigenfunctions is given by φn (x) = (

mω

4n (n!)2 π

1/4

)

e−mωx

2

/2

Hn (√mωx),

(5.1.63)

where Hn , n = 1, 2, . . . , denote the Hermite polynomials defined by Hn (x) = (−1)n ex

2

dn −x2 e . dxn

Theorem 5.25 (Mehler’s formula). It follows that e−tHosc (x, y) = (

1/2 mω mω (1 + e−2ωt )(x 2 + y2 ) − 4xye−ωt ) exp (− ). −2ωt 2 π(1 − e ) 1 − e−2ωt

(5.1.64)

468 | 5 Gibbs measures associated with Feynman–Kac semigroups Proof. Let m, ω, σ > 0 and consider the stochastic differential equation dXt = −ωXt dt + σdBt ,

B0 = x.

(5.1.65)

As seen in Example 2.159, its solution is the Ornstein–Uhlenbeck process Xtx

t

= xe

−ωt

+ σ ∫ e−ω(t−s) dBs .

(5.1.66)

0

By a direct calculation 𝔼[Xtx ] = xe−ωt ,

𝔼[Xtx Xsx ] = x 2 e−ω(t+s) +

σ2 (1 − e−2ω(s∧t) ) 2ω

(5.1.67)

follow from (5.1.66). The generator of (Xt )xt≥0 is L=

σ 2 d2 d − ωx . 2 dx2 dx

(5.1.68)

Now consider (5.1.61). Write Xt⋅ = Xt and 𝔼P [. . .] = ∫−∞ 𝔼[. . .]φ2osc (x)dx. Note that ∞

𝔼P [Xt ] = 0

and

𝔼P [Xs Xt ] =

σ 2 −ω|s−t| e . 2ω

(5.1.69)

Clearly, φosc ∈ C 2 (ℝ). The ground state transform Ug : L2 (ℝ, φ2osc dx) → L2 (ℝ), Ug f = φosc f , gives Losc = Ug−1 Hosc Ug = −

1 d2 d + ωx , 2m dx2 dx

which on putting 1/m = σ 2 coincides with (5.1.68) apart from sign. We have n

n

j=0

j=1

𝔼P [ ∏ fj (Xtj )] = (1, f0 ( ∏ e−(tj −tj−1 )Losc fj )1)

L2 (ℝ,φ2osc dx)

for f0 , . . . , fn ∈ L∞ (ℝ). Noting that Hosc xφosc = ωφosc , we can also check (5.1.69) by the formal calculation 𝔼P [Xt Xs ] = (φosc , xe

−(t−s)Hosc

xφosc ) = e

∞

−ω|t−s|

∫ x 2 φ2osc (x)dx = −∞

σ 2 −ω|t−s| e . 2ω

Next we view e−tHosc as an operator in L2 (ℝ) and derive its kernel. By (5.1.66), 𝔼[f (Xtx )]

t

= 𝔼 [f (xe−ωt + σ ∫ e−ω(t−s) dBs )] . 0 [ ]

5.1 Ground state transform and related processes | 469

For every fixed t > 0, t

rt = σ ∫ e

−ω(t−s)

dBs = σe

0

−ωt

t

1 (e Bt − ∫ Bs eωs ds) ω ωt

0

is a Gaussian random variable with zero mean and covariance 𝔼[rt2 ] =

σ2 (1 − e−2ωt ) = σ 2 κt , 2ω

with κt =

1 − e−2ωt , 2ω

t ≥ 0.

Thus e−tHosc f (x) = Ug e−tLosc Ug−1 f (x) = φosc (x)𝔼[(f /φosc )(Xtx )]. Since Xtx = xe−ωt + rt with the Gaussian random variable rt , the right-hand side above can be computed as 1 (2πσ 2 κt )

∞

φ (x) ∫ 1/2 osc

−∞

2

f (xe−ωt + y) − 2σ|y|2 κ t dy. e φosc (xe−ωt + y)

(5.1.70)

Changing the variable xe−ωt + y to z and setting σ 2 = 1/m, we further obtain ∞

= ∫ f (z) (2πκt /m)−1/2 −∞

φosc (x) |z − xe−ωt |2 exp (− ) dz. φosc (z) 2κt /m

(5.1.71)

From this e−tHosc (x, z) = √ =(

−2xze−ωt + (ωκt + e−2ωt )x 2 + (1 − ωκt )z 2 m exp (− ) 2πκt 2κt /m

1/2 mω (1 + e−2ωt )(x 2 + z 2 ) − 4xze−ωt ) exp (− ). π(1 − e−2ωt ) 2(1 − e−2ωt )/(mω)

Derivation via the Feynman–Kac formula. Here we consider the d-dimensional variant 1 1 d Hosc = − Δ + |x|2 − , 2 2 2 with parameters m = ω = 1. The Feynman–Kac formula gives dt

1

t

2

e−tHosc f (x) = e− 2 𝔼x [e− 2 ∫0 |Bs | ds f (Bt )], thus we are interested in computing expectations involving integrals as in the exponent at the right hand side.

470 | 5 Gibbs measures associated with Feynman–Kac semigroups Lemma 5.26. Let a ∈ ℝd , β ≥ 0 and b ∈ ℝ. Then 2

b2

𝔼a [e−β|Bt | − 2

t

∫0 |Bs |2 ds

]=(

d/2 b |a|2 b b + 2β coth(bt) ) exp (− ). b cosh(bt) + 2β sinh(bt) 2 b coth(bt) + 2β

Proof. If b = 0 the proof is straightforward, so we suppose that b ≠ 0. We have 2

b2

𝔼a [e−β|Bt | − 2

t

∫0 |Bs |2 ds

2

b2

] = 𝔼0 [e−β|Bt +a| − 2

t

∫0 |Bs +a|2 ds

].

We choose as to be −b(Bs + a) in the Girsanov formula. Define the d-dimensional random process (Zt )t≥0 by dZt = −b(Bt + a) ⋅ dBt −

b2 |B + a|2 dt. 2 t

A computation gives t

b b −b ∫(Bs + a) ⋅ dBs = − (|Bt |2 − dt + a ⋅ Bt ) = − (|Bt + a|2 − dt − |a|2 ) 2 2 0

and hence we obtain t

Zt = Z0 −

b b2 (|Bt + a|2 − dt − |a|2 ) − ∫ |Bs + a|2 ds. 2 2 0

Define the new probability measure P̂ = eZt 𝒲 0 . By the Girsanov formula, under P̂ t

B̂ t = (Bt + a) + b ∫(Bs + a)ds

(5.1.72)

0

is a Brownian motion starting at a, thus Wt = B̂ t − a is a standard Brownian motion. Putting Xt = Bt + a, we rewrite (5.1.72) as t

Xt = a + Wt − b ∫ Xs ds,

t ≥ 0.

0

This is a Langevin equation and we conclude that its unique solution (Xt )t≥0 is a d-dimensional Ornstein–Uhlenbeck process starting at a such that t

Xt = e

−bt

a + ∫ e−b(t−s) dWs . 0

5.1 Ground state transform and related processes |

471

By using P̂ we obtain b2

2

𝔼0 [e−β|Bt +a| − 2

t

∫0 |Bs +a|2 ds

2

b

2

2

] = 𝔼P̂ [e−β|Xt | + 2 (|Xt | −|a| −dt) ] b

b

2

2

= 𝔼P̂ [e−(β− 2 )|Xt | ]e− 2 |a| e− b

bdt 2

(5.1.73)

.

2

We determine 𝔼P̂ [e−(β− 2 )|Xt | ]. Note that by (5.1.71) the distribution of Xt on ℝd is given by ρt (z) =

1 |z − e−bt a| exp (− ), 2κt (2πκt )d/2

where κt =

1 − e−2bt e−bt = sinh(bt). 2b b

Hence we have 𝔼P̂ [f (Xt )] =

1 |z − e−bt a| f (z) exp (− ) dz. ∫ 2κt (2πκt )d/2 ℝd

2

Take f (z) = e−ξ |z| with ξ = β − b2 . With η = e−bt a ∈ ℝd we get 𝔼P̂ [e

−ξ |Xt |2

|η|2

1

1

η

2

( −1) −(ξ + 2κ )|z− | 2κt 1 t (2κt )(ξ + 2κ1 ) 2κt (ξ + 2κ1 ) t t ]= e e dz. ∫ (2πκt )d/2 ℝd

By integration we obtain η 1 1 󵄨󵄨󵄨 ) 󵄨z − ∫ exp (− (ξ + 2κt 󵄨󵄨 (2πκt )d/2 2κ (ξ + t d ℝ

=

󵄨󵄨2 󵄨 ) dz

1 󵄨󵄨󵄨 ) 2κt

−d/2 bdt β 1 1 −d/2 d/2 −d/2 (ξ + ) π = (2κ ξ + 1) = (cosh(bt) + 2 sinh(bt)) e2 . t 2κt b (2πκt )d/2 (5.1.74)

Furthermore, |η|2 1 ( 2κt (2κt )(ξ +

1 ) 2κt

− 1) =

2β

( − 1)e−2bt −ξ b|a|2 b|a|2 |η|2 = − (1 + b ) 2κt ξ + 1 2 2 2κt ξ + 1 2β

=

b|a|2 b|a|2 b coth(bt) + 1 − . 2 2 coth(bt) + 2β b

Combining (5.1.73)–(5.1.75), the lemma follows.

(5.1.75)

472 | 5 Gibbs measures associated with Feynman–Kac semigroups We can derive Mehler’s formula as an application of Lemma 5.26. By the Feynman– Kac formula we have 1

t

2

(f , e−tHosc g) = ∫ f ̄(x)𝔼x [e− 2 ∫0 (|Bs | −d)ds g(Bt )]dx. ℝd 1

t

2

Let Y = e− 2 ∫0 |Bs | ds and define 𝔼x [Y|Bt = y] = h(x, y). Then (f , e−tHosc g) = edt/2

∫ f ̄(x)g(y)Πt (y)h(x, y)dxdy ℝd ×ℝd

from which we obtain e−tHosc (x, y) = edt/2 Πt (y)h(x, y). Lemma 5.27. For x, y ∈ ℝd and b ∈ ℝ we have b2

𝔼x [e− 2

t

∫0 |Bs |2 ds

|Bt = y] = (

d/2 bt |y − e−bt x|2 b(|y|2 − |x|2 ) |y|2 ) exp (− + + ), sinh(bt) 2κt 2 2t

where κt = (e−bt /b) sinh(bt) = (1 − e−2bt )/(2b). b2

t

2

Proof. We write X = e− 2 ∫0 |Bs | ds and 𝔼x [X|Bt = y] = h(x, y). Note that h(x, Bt ) = 𝔼x [X|σ(Bt )]. We will make use of the formula 2 1 e−|x−y| /2ε → δ(x − y) d/2 (2πε)

(5.1.76)

as ε ↓ 0, where the convergence is in distributional sense. It follows that 1 1 2 2 1 1 𝔼x [e− 2ε |Bt −y| X] = lim ∫ Πt (z)e− 2ε |z−y| h(x, z)dz = Πt (y)h(x, y). d/2 d/2 ε→∞ (2πε) ε→∞ (2πε)

lim

ℝd

We see that by Proposition 5.26 1 2 1 1 lim 𝔼x [e− 2ε |Bt −y| X] d/2 Πt (y) ε↓0 (2πε) 1 2 b 2 2 1 1 = lim 𝔼 ̂ [e− 2ε |Bt −y| + 2 |Bt | ]e−b|x| /2 e−bdt/2 Πt (y) ε↓0 (2πε)d/2 P b|x|2 bdt 1 1 1 − 1 |z−y|2 + b2 |z|2 − 2κ1 |z−η|2 t = e− 2 e− 2 lim dz ∫ e 2ε d/2 d/2 ε↓0 (2πε) Πt (y) (2πκt )

h(x, y) =

ℝd

Recall that η = ae−bt . By (5.1.76) we have h(x, y) =

2 2 2 1 1 e−b|x| /2 e−bdt/2 eb|y| /2 e−|y−η| /(2κt ) d/2 Πt (y) (2πκt ) 2

2

2

2

= e|y| /2t (bt/ sinh(bt))d/2 e−b|x| /2 eb|y| /2−|y−η| /(2κt ) .

5.1 Ground state transform and related processes |

473

Example 5.28. Using Lemma 5.27 we can explicitly calculate various examples of h(x, y). d/2

bt (1) h(0, 0) = ( sinh(bt) )

.

2

(2) h(a, a) = h(0, 0) exp (− |a| (2bt tanh bt2 − 1)). 2t 2

(3) h(a, 0) = h(0, 0) exp (− |a| bt coth(bt)). 2t 2

(4) h(0, a) = h(0, 0) exp (− |a| (bt coth(bt) − 1)). 2t Mehler’s formula is now an easy consequence of Lemma 5.27. Theorem 5.29 (Mehler’s formula). We have the expression e−tHosc (x, y) = (

d/2 1 (|x|2 + |y|2 )(1 + e−2t ) − 4(x, y)e−t ) exp (− ). π(1 − e−2t ) 2(1 − e−2t )

Proof. Set b = 1 and write Πt (y) = (2πt)−d/2 exp(−|y|2 /(2t)) in Lemma 5.27. This gives e−tHosc (x, y) = Πt (y)etd/2 h(x, y) =( =(

d/2 1 |y − xe−t |2 |x|2 |y|2 ) exp (− − + ) 2κt 2 2 π(1 − e−2t )

d/2 1 (|x|2 + |y|2 )(1 + e−2t ) − 4(x, y)e−t ) exp (− ). π(1 − e−2t ) 2(1 − e−2t )

Derivation via operator theory. Mehler’s formula can also be obtained through analytic continuation t 󳨃→ −it as a corollary of the theorem below. Again we simply choose m = ω = 1. Let Kt be the integral kernel of e−itHosc , and denote T = {kπ | k ∈ ℤ}. Theorem 5.30. The kernel Kt : ℝ × ℝ → ℂ has the following expressions: (1) for every t ∈ ℝ \ T, Kt (x, y) =

x 2 +y2 1 e−i( 2 √2πi sin t

xy cot t− sin ) t

;

(5.1.77)

(2) for every t ∈ T, a. e. x ∈ ℝ, and f ∈ D(Hosc ) ∞

lim ∫ Ks (x, y)f (y)dy = f (x). s→t

−∞

In particular, e−tHosc (x, y) =

1

√π(1 − e−2t )

holds for all x, y ∈ ℝ and t > 0.

exp (

4xye−t − (x 2 + y2 )(1 + e−2t ) ) 2(1 − e−2t )

(5.1.78)

474 | 5 Gibbs measures associated with Feynman–Kac semigroups Proof. By eigenfunction expansion the solution of the initial value problem −i𝜕t ψ = Hosc ψ,

ψ(x, 0) = ϕ(x)

can be written as ∞

ψ(x, t) = ∫ Kt (x, y)ϕ(y)dy,

(5.1.79)

−∞

with integral kernel ∞

Kt (x, y) = ∑ φn (x)φn (y)e−iEn t . n=0

(5.1.80)

A combination of (5.1.63) and (5.1.80) leads to Kt (x, y) =

∞ 2 it 1 − x2 +y 1 e 2 − 2 ∑ n Hn (x)Hn (y)e−int . √π 2 n! n=0 2

n

(5.1.81)

2

d −y Next, the formula Hn (y) = (−1)n ey dy known for Hermite polynomials allows to ne replace the sum appearing at the right-hand side of (5.1.81) by 2

2 1 1 d n (− e−it ) Hn (x) e−y . n! 2 dy n=0

∞

ey ∑ 2

The generating function e2sz−z = ∑∞ n=0 for the above the formal expression 2

ey exp (−xe−it

1 H (s)z n n! n

of Hermite polynomials gives then

2 d 1 d2 − e−2it 2 ) e−y . dy 4 dy

(5.1.82)

In the following we will argue that this expression actually makes sense. 2 Clearly, the exponential operator acting on e−y factorizes. Both operators are unbounded; however, if the second can be defined, the first is well defined on the same d domain. The first operator is a shift by the prefactor of dy . We consider 𝒦z = exp (z

d2 ), dy2

z ∈ ℂ,

and make sense of it on a set including z = τ = −(1/4)e−2it . Take the open disc D = {z ∈ ℂ | |z| < 41 }. Note that for every z ∈ D the function 2

2

f (y) = e−y is an analytic vector of 𝒦z , i. e., 𝒦z e−y ∈ L2 (ℝ), and is analytic with respect to z, i. e., ∞ 󵄩 󵄩󵄩 z n dn −y2 󵄩󵄩󵄩 ∞ |z|n e 󵄩󵄩󵄩 ≤ ∑ ∑ 󵄩󵄩󵄩 󵄩 n! dyn 󵄩󵄩 n=0 n! n=0 󵄩

∞ 󵄩󵄩 dn 󵄩 1 ∞ |z|n 2n −k2 /4 󵄩󵄩 󵄩 −y2 󵄩 ‖k e ‖ = ∑ cn |z|n , ∑ 󵄩󵄩 n e 󵄩󵄩󵄩 = 󵄩󵄩 dy 󵄩󵄩 √2 n=0 n! n=0

5.1 Ground state transform and related processes |

475

where cn = (π/2)1/4 √(4n − 1)!!/n!. It is seen that cn /cn+1 → 1/4 as n → ∞ and this explains the way D was defined. For every z ∈ D define 𝒜z = {f ∈ D(𝒦z ) | f is analytic vector of 𝒦z }. This set 2 contains, for instance, e−y . For every z ∈ D and f ∈ 𝒜z the derivative with respect to z of 𝒦z exists. Denote g(y, z) = (𝒦z f )(y); then we have 𝜕z g = 𝜕y2 g,

g(y, 0) = f (y)

since z = 0 ∈ D, i. e., g satisfies the heat equation with the initial condition above, for every z ∈ D. This observation allows to express g(y, z). Define 󵄨󵄨 { 󵄨 Df = {z ∈ D̄ 󵄨󵄨󵄨 󵄨󵄨 {

∞ 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 1 󵄨󵄨 } − (y−u) 󵄨󵄨 4z du 󵄨󵄨 < ∞, y ∈ ℝ f (u)e ∫ 󵄨󵄨 󵄨󵄨 } 󵄨󵄨 √z 󵄨󵄨 󵄨 󵄨 −∞ } 2

as a subset of the closure of D; in particular, we write Dexp for f (y) = e−y . An easy calculation shows that Dexp lies in D and coincides with the complement in D of the closed disk of radius 1/8 centered in (−1/8, 0). It is checked directly that for every z ∈ Df ∞

(y−u)2 1 Ff (y, z) = ∫ f (u)e− 4z du, 2√πz

(5.1.83)

−∞

where the square root is the one with the smallest argument, is a solution of the heat equation above. Moreover, Ff is analytic since |𝜕Ff /𝜕z| < ∞. Take now z = ζ ∈ (0, 41 ). We have in L2 sense that Ff (y, ζ ) = g(y, ζ ), hence by uniqueness of the analytic continuation g(y, z) = (𝒦z f )(y),

z ∈ Df .

(5.1.84)

This representation can be extended to the circle 𝜕D. Pick a number ε such that |ℜτ| ≤ ε < 1/4 and define 𝒜εz = 𝒦ε L2 (ℝ) ⊂ 𝒜z . Take a sequence (zn )n∈ℕ ⊂ D such that ε zn → τ for some τ ∈ Dh and h ∈ ⋂∞ n=1 𝒜zn . It is easily seen that pointwise 𝒦zn h → 2

1/(2√πτ) ∫−∞ h(u) exp(− (⋅−u) )du as n → ∞. By the Lebesgue dominated convergence 4τ theorem it follows then for every φ ∈ C0∞ that ∞

1 (⋅ − u)2 (φ, 𝒦zn h) 󳨀→ (φ, ) du) ∫ h(u) exp (− 4τ 2√πτ ∞

as n → ∞.

−∞

ε On the other hand, 𝒦τ h is well defined for every h ∈ ⋂∞ n=1 𝒜zn since 𝒦τ h = 𝒦τ+ε fh , with ∞ some fh ∈ ⋂n=1 𝒜zn , and ε + ℜτ ≥ 0. Moreover, 𝒦zn h → 𝒦τ h in strong sense. This follows again through dominated convergence by using the continuity of the exponential function, the equalities 2

‖𝒦τ h − 𝒦zn h‖ = ‖𝒦τ+ 1 fh − 𝒦z 4

1 n+ 4

2

∞

1

1

fh ‖ = ∫ |e−(τ+ 4 )λ − e−(zn + 4 )λ |2 dEfh (λ), 0

476 | 5 Gibbs measures associated with Feynman–Kac semigroups 1

1

and the estimate |e−(τ+ 4 )λ −e−(zn + 4 )λ |2 ≤ 4, uniform in λ, and making use of the spectral theorem. Hence also (φ, 𝒦zn h) → (φ, 𝒦τ h) as n → ∞, ∀φ ∈ C0∞ , and by comparison ∞

(y−u)2 1 (𝒦τ h)(y) = ∫ h(u)e− 4τ du 2√πτ

(5.1.85)

−∞

ε follows, for every h ∈ ⋂∞ n=1 𝒜zn , τ ∈ Dh , and almost every y ∈ ℝ.

2

2 −y Take an fh ∈ ⋂∞ . n=1 𝒜zn and ε ≥ |ℜτ|, and look at the solution of exp(ε𝜕y )fh = e −εk 2 ̂ −k 2 /4 On taking Fourier transforms, e f (k) = e follows; f ̂ , and thus also f , will be h

2

h

h

ε ε L2 whenever ε < 1/4, and then e−y ∈ ⋂∞ n=1 𝒜zn . This also explains the way 𝒜z was 2

defined. By putting now h(y) = e−y in (5.1.85) and using (5.1.82), (5.1.77) results by easy manipulations for every τ with |ℜτ| ≤ ε < 1/4. This argument excludes the values of time corresponding to τ = (−1/4, 0), i. e., the set T. The reason is that this is the only value of τ that falls outside of Dexp , and for these values the kernel becomes singular described by a δ-distribution. Take t ∈ T and a sequence (tn )n∈ℕ ⊂ ℝ \ T such that tn → t. Since e−itHosc is a strongly con󸀠 tinuous unitary group, we have with a subsequence (tm )m∈ℕ for every L2 function φ e−itm Hosc φ(x) → e−itHosc φ(x), for almost every x ∈ ℝ. Moreover, for almost every x ∈ ℝ ∞ we have ∫−∞ Ktm󸀠 (x, y)φ(y)dy → φ(x), which is seen directly. 󸀠

5.1.6 P(ϕ)1 -processes with càdlàg paths Finally we consider nonlocal Schrödinger operators H = Ψ(− 21 Δ) + V and construct P(ϕ)1 -processes for them. We assume that the potential is in Ψ-Kato class, and is such that H has a unique ground state φ0 > 0. We denote the intrinsic operator LΨ = U −1 H̄ U ,

H̄ = H − E0 , E0 = inf Spec(H),

with the same notation of the ground state transform as in Section 5.1.1. A main difference from the case in Section 5.1.3 is that here we have to deal with jump processes. Recall that XD denotes the càdlàg path space on the full real line ℝ. Our main aim is to prove that there exists a probability measure M0x on the space (XD , ℬ(XD )) such that for f , g ∈ L2 (ℝd ) and Ψ-Kato class potential V Ψ

(f , e−|t−s|L g) = ∫ 𝔼M0x [f (Ys )g(Yt )] φ20 (x)dx,

t, s ∈ ℝ,

(5.1.86)

ℝd

holds, with coordinate process (Yt )t∈ℝ . Define dM0 = φ20 (x)dx

(5.1.87)

and as above write L2 (dM0 ) for L2 (ℝd , dM0 ) for simplicity. As before, we denote by XD and XD̄ the càdlàg and càglàd path spaces on [0, ∞), respectively.

5.1 Ground state transform and related processes |

477

Theorem 5.31. Let V be a Ψ-Kato decomposable potential such that H = Ψ(− 21 Δ) + V has a unique ground state φ0 > 0, and the ground state transform of H̄ is LΨ . Denote by (Yt )t∈ℝ the coordinate process on (XD , ℬ(XD )) and consider the filtrations (Ft+ )t≥0 = σ(Ys : 0 ≤ s ≤ t) and (Ft− )t≤0 = σ(Ys : t ≤ s ≤ 0). Then for all x ∈ ℝd there exists a probability measure M0x on (XD , ℬ(XD )) satisfying the following properties: (1) (Initial distribution) M0x (Yt = x) = 1; (2) (Reflection symmetry) (Yt )t≥0 and (Yt )t≤0 are independent and d

Y−t = Yt ,

t ∈ ℝ;

(3) (Markov property)(Yt )t≥0 is a Markov process with respect to (Ft+ )t≥0 , and (Yt )t≤0 with respect to (Ft− )t≤0 ; (4) (Shift invariance) let −∞ < t0 ≤ t1 ≤ . . . ≤ tn < ∞, n ∈ ℕ; then the finitedimensional distributions with respect to dM0 are given by n

Ψ

Ψ

∫ 𝔼M0x [∏ fj (Ytj )] dM0 = (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn ) j=0

ℝd

L2 (dM0 )

(5.1.88)

for f0 , fn ∈ L2 (dM0 ), and fj ∈ L∞ (ℝd ), j = 1, . . . , n − 1, and they are shift invariant, i. e., n

n

∫ 𝔼M0x [∏ fj (Ytj )] dM0 = ∫ 𝔼M0x [∏ fj (Ytj +s )] dM0 , j=0

ℝd

j=0

ℝd

s ∈ ℝ.

Proof. The proof runs is parallel with Theorem 5.15, here we present an outline. Let 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn with arbitrary n ∈ ℕ, and the set function 𝒫{t0 ,...,tn } : ×nj=0 ℬ(ℝd ) → ℝ be defined by n

𝒫{t0 ,...,tn } (×j=0 Aj ) = (1A0 , e 𝒫{t0 } (A) = (1, e

−t0 LΨ

1A )

−(t1 −t0 )LΨ

L2 (dM0 )

Ψ

1A1 ⋅ ⋅ ⋅ 1An−1 e−(tn −tn−1 )L 1An )

L2 (dM0 )

= (1, 1A )L2 (dM0 ) .

,

(5.1.89) (5.1.90)

Step 1: Denote Λ = {S ⊂ ℝ | |S| < ∞}. It can be verified directly that the family of set functions (𝒫S )S∈Λ given above satisfies the consistency condition n

n+m

d

n

𝒫{t0 ,...,tn+m } ((×j=0 Aj ) × (×j=n+1 ℝ )) = 𝒫{t0 ,...,tn } (×j=0 Aj ),

for all m, n ∈ ℕ. Hence by the Kolmogorov extension theorem there exists a probability measure 𝒫∞ on the space ((ℝd )[0,∞) , σ(𝒜)), where d

d

𝒜 = {ω : ℝ → ℝ | ω⌈Λ ∈ E, E ∈ (B (ℝ )) , Λ ⊂ ℝ, |Λ| < ∞}, |Λ|

(5.1.91)

478 | 5 Gibbs measures associated with Feynman–Kac semigroups such that n

n

𝒫{t0 ,...,tn } (×j=0 Aj ) = 𝔼𝒫∞ [∏ 1Aj (Ztj )] ,

n ∈ ℕ,

j=0

𝒫{t} (A) = 𝔼𝒫∞ [1A (Zt )].

Here (Zt )t≥0 is the coordinate process. Thus the stochastic process (Zt )t≥0 on the probability space ((ℝd )[0,∞) , σ(𝒜), 𝒫∞ ) satisfies n

Ψ

Ψ

𝔼𝒫∞ [∏ fj (Ztj )] = (f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn ) j=0

Ψ

𝔼𝒫∞ [f0 (Zt0 )] = (1, e−t0 L f0 )

L2 (dM0 )

L2 (dM0 )

,

= (1, f0 )L2 (dM0 )

(5.1.92) (5.1.93)

for fj ∈ L∞ (ℝd ), j = 1, . . . , n − 1, f0 , fn ∈ L2 (dM0 ), and 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn . Step 2: Next we prove the existence of both a càdlàg version of (Zt )t≥0 and a càglàd version of (Zt )t≥0 by checking the Dynkin–Kinney condition given in Theorem 3.34. By Lemma 5.32 below there exists a càdlàg version Z 󸀠 = (Zt󸀠 )t≥0 of (Zt )t≥0 on the probability space ((ℝd )[0,∞) , σ(𝒜), 𝒫∞ ). Denote the image measure of 𝒫∞ on (XD , ℬ(XD )) by 󸀠 −1

𝒬 = 𝒫∞ ∘ (Z ) . d Let (Ỹ t )t≥0 be the coordinate process on (XD , ℬ(XD ), 𝒬) such that Zt󸀠 = Yt̃ . (5.1.92) and (5.1.93) can be expressed in terms of (Ỹ t )t≥0 as Ψ

Ψ

(1, e−tL f0 )

L2 (dM0 )

n

Ψ

(f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )

L2 (dM0 )

= 𝔼𝒬 [∏ fj (Ỹ tn )] ,

= (1, f0 )L2 (dM0 ) = 𝔼𝒬 [f0 (Ỹ t )].

j=0

(5.1.94) (5.1.95)

Note that a càglàd version Z 󸀠󸀠 = (Zt󸀠󸀠 )t≥0 of (Zt )t≥0 can also be constructed on the space ((ℝd )[0,∞) , σ(𝒜), 𝒫∞ ). In a similar way to (Zt󸀠 )t≥0 it can be shown that there exists a probability measure 𝒬̄ on the càglàd measurable space (XD̄ , ℬ(XD̄ )) such that the coordinate process (Ỹ t )t≥0 satisfies both (5.1.94) and (5.1.95) with 𝒬 replaced by 𝒬̄ and 󸀠󸀠 −1

𝒬̄ = 𝒫∞ ∘ (Z )

holds. Step 3: Since XD and XD̄ are Polish spaces by Corollary 3.37, by Theorem 2.41 we can define regular conditional measures 𝒬x ( ⋅ ) = 𝒬( ⋅ |Ỹ 0 = x) and 𝒬̄ x ( ⋅ ) = 𝒬( ⋅ |Ỹ 0 = x) for x ∈ ℝd on (XD , ℬ(XD )) and (XD̄ , ℬ(XD̄ )), respectively. Since the distribution of Y0

5.1 Ground state transform and related processes |

479

is dM0 , we have 𝒬(A) = ∫ℝd 𝔼𝒬x [1A ]dM0 and 𝒬̄ (A) = ∫ℝd 𝔼𝒬̄ x [1A ]dM0 . The coordinate process (Ỹ t )t≥0 on the probability space (XD , ℬ(XD ), 𝒬x ) satisfies Ψ

Ψ

(1, e−tL f0 )

n

Ψ

(f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )

L2 (dM0 )

L2 (dM0 )

= ∫ 𝔼𝒬x [∏ fj (Ỹ tj )] dM0 , j=0

ℝd

= (1, f0 )L2 (dM0 ) = ∫ 𝔼𝒬x [f0 (Ỹ t )]dM0

(5.1.96) (5.1.97)

ℝd

and the coordinate process (Ỹ t )t≥0 on (XD̄ , ℬ(XD̄ ), 𝒬̄ x ) also satisfies Ψ

Ψ

(1, e−tL f0 )

n

Ψ

(f0 , e−(t1 −t0 )L f1 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )L fn )

L2 (dM0 )

L2 (dM0 )

= ∫ 𝔼𝒬̄ x [∏ fj (Ỹ tj )] dM0 , j=0

ℝd

= (1, f0 )L2 (dM0 ) = ∫ 𝔼𝒬̄ x [f0 (Ỹ t )]dM0 .

(5.1.98) (5.1.99)

ℝd

In Lemma 5.33 below we will show that (Ỹ t )t≥0 is a Markov process on (XD , ℬ(XD ), 𝒬x ) with respect to the natural filtration (Ft )t≥0 , and (Ỹ t )t≥0 is a Markov process on (XD̄ , ℬ(XD̄ ), 𝒬̄ x ) with respect to the natural filtration (Ft )t≥0 . Step 4: Now we construct a random process on the full real line ℝ. Consider the product sample space Ω̂ = XD × XD̄ with product σ-field F ̂ = ℬ(XD ) × ℬ(XD̄ ) and product measure 𝒬x̂ = 𝒬x × 𝒬̄ x , respectively. Let Ŷ t be the coordinate process given by ω1 (t), Ŷ t (ω) = { ω2 (−t),

t ≥ 0, t 0 be fixed. Then for every ε > 0 the probability 𝒫∞ (|Zt − Zs | > ε) tends to zero uniformly as |s − t| → 0 for s, t ∈ [0, T]. Proof. Note that the right-hand side of (5.1.92) can be expressed directly in terms of the subordinate Brownian motion (Xt )t≥0 on a probability space (Ω, F , P), i. e., n

tn

𝔼𝒫∞ [∏ fj (Ztj )] = ∫ φ0 (x)𝔼xP [e− ∫0 j=0

ℝd

(V(Xs )−E0 )ds

n

(∏ fj (Xtj )) φ0 (Xtn )] dx. j=0

(5.1.100)

480 | 5 Gibbs measures associated with Feynman–Kac semigroups Denote by Bε (x) the d-dimensional closed ball centered in x with radius ε, and let 0 ≤ s < t ≤ T. By (5.1.100) we have x

𝒫∞ (|Zt − Zs | > ε) = ∫ φ0 (x)𝔼P [e

t−s

− ∫0 (V(Xr )−E0 )dr

φ0 (Xt−s )1Bε (x)c (Xt−s )] dx.

ℝd

The Schwarz inequality yields 𝒫∞ (|Zt − Zs | > ε)

≤ ∫ φ0 (x) (𝔼xP [φ20 (Xt−s )])

1/2

t−s

(𝔼xP [1Bε (x)c (Xt−s )e−2 ∫0

(V(Xr )−E0 )dr

1/2

])

dx.

(5.1.101)

ℝd

Applying the Schwarz inequality again gives t−s

𝔼xP [1Bε (x)c (Xt−s )e−2 ∫0

(V(Xr )−E0 )dr

] ≤ (𝔼xP [1Bε (x)c (Xt−s )])

1/2

t−s

(𝔼xP [e−4 ∫0

(V(Xr )−E0 )dr

1/2

])

≤ CP 0 (|Xt−s | > ε)1/2 , t−s

where C = supx∈ℝd (𝔼xP [e−4 ∫0 Schwarz inequality that

(V(Xr )−E0 )dr

])1/2 . Thus we can conclude by (5.1.101) and the

𝒫∞ (|Zt − Zs | > ε) ≤ C

1/2

‖φ0 ‖2 P 0 (|Xt−s | > ε)1/4 ,

which goes to zero as |t − s| → 0 by stochastic continuity of (Xt )t≥0 . The uniform stochastic convergence is proven in the same way as in Lemma 3.35. Lemma 5.33. (1) The coordinate process (Ỹ t )t≥0 is a Markov process on (XD , ℬ(XD ), 𝒬x ) with respect to the natural filtration (Ft )t≥0 . (2) The coordinate process (Ỹ t )t≥0 is a Markov process on (XD̄ , ℬ(XD̄ ), 𝒬̄ x ) with respect to the natural filtration (Ft )t≥0 . Proof. We only show (1), part (2) can be proven similarly. Let Ψ

ut (x, A) = e−tL 1A (x) for every A ∈ ℬ(ℝd ), x ∈ ℝd , and t ≥ 0. Clearly, ut (x, A) = 𝔼𝒬x [1A (Ỹ t )] and, by (5.1.96) and (5.1.97) the finite-dimensional distributions of (Ỹ t )t≥0 are given by n

n

n

j=0

j=0

j=0

𝔼𝒬x [∏ 1Aj (Ỹ tj )] = ∫ (∏ 1Aj (xj )) ∏ utj −tj−1 (xj−1 , dxj ) ℝd

Ψ

(5.1.102)

with t0 = 0 and x0 = x. By using the properties of the semigroup {e−tL : t ≥ 0} it can be checked directly that ut (x, A) is a probability transition kernel, thus (Ỹ t )t≥0 is a Markov process with finite-dimensional distributions given by (5.1.102).

5.2 Gibbs measures on path space

| 481

Definition 5.34 (Càdlàg / fractional P(ϕ)1 -process). Consider the probability space (XD , ℬ(XD ), M0x ) and the coordinate process (Yt )t∈ℝ on this space satisfying (1)–(4) in Theorem 5.31. We call (Yt )t∈ℝ càdlàg P(ϕ)1 -process for the given V. When the process is associated with a fractional Schrödinger operator, we call it fractional P(ϕ)1 -process for V. Using this measure we have the following change-of-measure formula. Theorem 5.35 (Feynman–Kac formula for càdlàg P(ϕ)1 -process). For all f , g ∈ L2 (dM0 ) we have Ψ

(f , e−tL g)L2 (dM0 ) = (fφ0 , e−t(H−E0 ) gφ0 )L2 (ℝd ) = ∫ 𝔼M0x [f ̄(Y0 )g(Yt )]dM0 (x). ℝd

Here (Yt )t∈ℝ denotes the coordinate process. Proof. The proof is immediate from (5.1.88).

5.2 Gibbs measures on path space 5.2.1 From Feynman–Kac formulae to Gibbs measures In this section we take a different view of the Feynman–Kac formula. In its simplest form for a Schrödinger operator H = − 21 Δ + V with suitable V, i. e., t

e−tH f (x) = 𝔼x [e− ∫0 V(Bs )ds f (Bt )] =

t

∫

e− ∫0 V(Bs (ω))ds f (Bt (ω))d𝒲 x (ω)

C(ℝ+ ;ℝd )

it suggests that the effect of the potential exerted on the paths is a contribution by the t

exponential weight function e− ∫0 V(Bs )ds modifying the distribution of paths described by Wiener measure starting from x, corresponding to the case when V ≡ 0. Similarly to the Girsanov transform, this leads to a change of measure (after normalization) for every t > 0. Such probability measures are widely encountered in various contexts, and are known as Gibbs measures in the mathematical physics literature. Gibbs measures have been introduced first to serve for a rigorous study of thermodynamic equilibrium states of lattice spin systems, but gradually proved to be a more generally applicable and rather fertile concept. When we try to interpret them in the context of Brownian sample paths, a related particle model can similarly be found. To see this, recall the proof of the Feynman–Kac formula using the Trotter product formula in Section 4.2.2. There we have seen that for a Schrödinger operator H as above with a smooth bounded potential V, for f , g ∈ L2 (ℝd ) we have (f , e−tH g) = lim (f , (e−(t/n)H0 e−(t/n)V )n g). n→∞

482 | 5 Gibbs measures associated with Feynman–Kac semigroups By Theorem 4.116 the integral kernel Kt (x, y) of e−tH is continuous in x and y. Writing out the integral kernels of e−(t/n)H0 , and letting f and g approach δ-distributions yields Kt (x, y) = lim ∫ n→∞

ℝdn

n 1 e− ∑j=0 (2πt/n)d/2

1 |x −xj |2 −∑nj=1 2t/n j+1

V(xj )

dx1 ⋅ ⋅ ⋅ dxn ,

(5.2.1)

where x0 = x, xn+1 = y, and x1 , . . . , xn are intermediate positions. For any fixed finite n, the right-hand side above can be interpreted to describe the following situation. Consider a one-dimensional chain of length n+2, consisting of interacting particles placed in the positions x1 , . . . , xn ∈ ℝd , and pinned down at its two ends at x and y. Let the neighboring particles be coupled by a quadratic potential, while each particle being subjected to an external potential V. The total energy of the chain is given then by the exponent in (5.2.1). The energetically optimal position of the chain is a configuration that minimizes this energy. In this case it comes about as a compromise between a straight line through x and y, and a configuration in which all particles in the positions xj are in the global minimum of V (assuming there is one). On the other hand, entropy makes the possible configurations of the chain fluctuate at inverse temperature n/2t around this energetically optimal state. The expression in (5.2.1) defines a finite measure on (ℝd )n , which can be normalized to a probability measure. This gives then the probability distribution of finding the chain in a subset of positions, and it describes its equilibrium state in which energy and entropy balance each other in an optimal way. A peculiarity of the model (5.2.1) is that the limit n → ∞ is taken. Different from the infinite volume limit, which is standard in the classical theory of Gibbs measures and will be discussed below, this is a type of continuum limit for the particle system: while the number of particles grows like n, the pair interaction energy 2tn |xj+1 − xj |2 increases at the same rate. This couples nearby particles increasingly strongly, forcing them to occupy the same position with large probabilities, and eventually leads to a Brownian t

path perturbed by the density e− ∫0 V(Bs )ds . Thus one can view the right-hand side of the Feynman–Kac formula t

x,y

Kt (x, y) = Πt (x − y) ∫ e− ∫0 V(Bs )ds d𝒲[0,t]

(5.2.2)

X

appearing in Theorem 4.116 as the probabilistic description of an elastic continuous string fixed at both ends, under the effect of an external potential V. Our interest in this chapter is, informally, to study probability measures of the form x,y

dQI (ω|x, y) = e−UI (ω) dPI (ω),

(5.2.3)

on path spaces indexed by a collection of appropriate bounded intervals I ⊂ ℝ, and the extensions of these probability measures to full path space indexed by ℝ. This

5.2 Gibbs measures on path space

| 483

problem formulates for any of the two-sided Markov processes (Xt )t∈ℝ generated by the kinetic terms of the (standard and nonlocal) Schrödinger operators discussed so x,y far. Thus for every I the measure PI is the appropriate bridge measure for (Xt )t∈ℝ , and UI (ω) is the energy functional associated with path (Xs (ω))s∈I , which in the case of (5.2.2) is Ut (ω) = − ∫I V(Xs (ω))ds. The basic difficulty in extending (QI )I∈𝒫f (ℝ) , where 𝒫f (ℝ) denotes a family of bounded subintervals of the real line to be further specified below, is that attempting to take a direct limit proves typically to be unworkable as UI almost surely diverges in this limit. This will require a strategy which we borrow from mathematical physics, and we will follow the Dobrushin–Lanford–Ruelle (DLR) construction so that we look for probability measures Q on a path space indexed by ℝ whose family of probability measures conditional on paths outside I match with QI , for every I ∈ 𝒫f (ℝ). This problem is reminiscent of the strategy of the Kolmogorov extension theorem, however, with the essential difference that here the extension is made by starting from prescribed conditional measures rather than marginals. As a next step we then find that, similarly to the so-called thermodynamic limit of classical Gibbs measures, solutions to the DLR problem can be constructed by taking suitable limits of QIn ( ⋅ |x, y) as In → ℝ on letting n → ∞. However, unsurprisingly, the techniques that we need to develop will have to differ from the classical approaches as here we are in an essentially different context. Although it is possible to consider Gibbs measures for continuous time random processes in a great generality, our primary interest is to use them later in studying some spectral properties of specific operators related to models of quantum theory, and therefore we restrict attention to particular classes of functionals Ut . We give a sample of applications below involving Brownian motion or jump Lévy processes perturbed by one-body or pair interaction potentials, of progressively increasing complexity. Example 5.36. Various choices of the energy functional are possible as follows. Some will be considered in Volume 2 in applications to concrete models. For simplicity of notation we do not explicitly feature ω. (1) The case t

Ut = ∫ V(Bs )ds,

(5.2.4)

0

where (Bt )t≥0 is Brownian motion, relates to path measures of P(ϕ)1 -processes for Schrödinger operators with Kato-decomposable potential V. (2) Let t

Ut = ∫ V(Xs )ds, 0

(5.2.5)

484 | 5 Gibbs measures associated with Feynman–Kac semigroups where (Xt )t≥0 is a jump Lévy process, for instance, a rotationally symmetric α-stable process obtained for a fractional Schrödinger operator (−Δ)α/2 + V, α ∈ (0, 2). The case α = 1 describes a massless relativistic quantum particle. A similar expression can be obtained when (Xt )t≥0 is a relativistic α-stable process generated by the operator (−Δ − m2/α )α/2 + V, m > 0, and α = 1 describes a massive relativistic quantum particle. (3) Let t

t t

Ut = ∫ V(Xs )ds + ∫ ∫ W ϕ (Xs − Xr , s − r)dsdr, 0

(5.2.6)

0 0

where W ϕ (x, s) = −

̂ 2 −ik⋅x−|k||s| |ϕ(k)| 1 e dk, ∫ 4 |k| ℝd

is a pair interaction potential, with a sufficiently fast decaying, spherical symmetric real-valued function ϕ. In Volume 2 we will see how this arises, and how by using them it is possible to derive and prove the ground state properties of quantum field models, such as the Nelson model of an electrically charged spinless quantum particle linearly coupled to a scalar boson field. (4) Changing W ϕ (x, s) above to Wpol (x, s) = −

1 −|s| e 4|x|

(5.2.7)

describes the polaron model. A variant, called bipolaron model, differs by the fact that it consists of two dressed electrons coupled to the same phonon field, which repel each other by Coulomb interaction. In this case 2

t t

t

Ut = α ∫ ∫ ℰ (Bs , Br , B̃ s , B̃ r , s − r)dsdr − g ∫ 0 0

0

1 ds, |Bs − B̃ s |

where ℰ (Bs , Br , B̃ s , B̃ r , u) = Wpol (Bs − Br , u) + 2Wpol (Bs − B̃ r , u) + Wpol (B̃ s − B̃ r , u)

with α < 0 being the polaron–phonon coupling parameter, and g > 0 is the strength of the Coulomb repulsion between the two polarons. In this case the reference measure is a product of two Wiener measures of the independent Brownian motions (Bt )t∈ℝ and (B̃ t )t∈ℝ . (5) Let t t

Ut = ∫ ∫ δ(Bs − Br )dsdr. 0 0

(5.2.8)

5.2 Gibbs measures on path space

| 485

This gives the intersection local time of Brownian motion describing a polymer model with short-range soft-core interaction encouraging to avoid selfintersections. (6) Let t

t t

Ut = ∫ V(Bs )ds + ∫ ∫ W(Bs − Br , s − r)dBs ⋅ dBr 0

(5.2.9)

0 0

with some choices of W which we do not write down explicitly here. This describes the Pauli–Fierz model of a charged particle interacting with a vector boson field. The difference from the cases above consists in having double stochastic integrals here instead of double Riemann integrals. Gibbs measures of this type will be discussed when we study the Pauli–Fierz model in Volume 2. (7) Let t t

Ut = ∫ ∫ 0 0

1 dB ⋅ dBr . |Bs − Br | s

(5.2.10)

This is used in describing turbulent fluids. The above formal expression gives the total energy of a vorticity field concentrated along Brownian curves Xt ∈ ℝ3 obtained from a given divergence-free velocity field. Our main concern here will be to construct Gibbs measures covering Examples (1)–(3). A main benefit will be a study, through the appropriate Feynman–Kac-type formulae, of ground state properties of quantum models in terms of averages of suitable functions on path space with respect to these Gibbs measures, which will be carried out in Volume 2.

5.2.2 Gibbs measures on Brownian paths Let X = C(ℝ; ℝd ), F = ℬ(X), and consider two-sided ℝd -valued Brownian motion (Bt )t∈ℝ on this space, as defined in Section 2.3.2. We keep denoting the corresponding Wiener measure starting from x ∈ ℝd by 𝒲 x , as before. With I ⊂ ℝ we use the notation FI for the sub-σ-field generated by (Xt )t∈I , and TI = FI c , where I c = ℝ \ I, as well as FT and TT for I = [−T, T], T > 0. The tail field will be denoted by T = ⋂N∈ℕ TN , and the Lebesgue measure of I by |I|. Definition 5.37 (Index set and state space). The set {I ⊂ ℝ | |I| < ∞} of bounded intervals of the real line is the index set, and the state space is ℝd . In the context of Gibbs measures there is an underlying probability measure giving the distribution of paths without any potential, called reference measure, which

486 | 5 Gibbs measures associated with Feynman–Kac semigroups we will denote by P. In view of the applications discussed in Example 5.36 we also introduce pair interaction potentials apart from one-body external potentials. Definition 5.38 (Potentials). (1) A Borel measurable function V : ℝd → ℝ is called an external potential. We call V a P-admissible external potential whenever 0 < 𝔼P [e− ∫I V(Bs ) ds ] < ∞

(5.2.11)

for every bounded interval I ⊂ ℝ. (2) A function W : ℝd × ℝd × ℝ × ℝ → ℝ is called a pair interaction potential. We say that W is admissible whenever sup ∫ sup |W(x, y, s, t)| ds < ∞. t∈ℝ

ℝ

x,y∈ℝd

(5.2.12)

If W(x, y, s, t) = W(x, y, s − t, 0), we use the simpler notation W(x, y, t); similarly, we write W(x, t) in case W(x, y, s, t) = W(x − y, 0, s − t, 0). For brevity we will henceforth refer to the potentials jointly satisfying the conditions above as admissible. The condition on the pair potential is rather strong; however, as we move on we will introduce weaker regularity conditions. In the Feynman–Kac formula considered at the beginning of this chapter, (5.2.11) holds for Kato-decomposable potentials V by Lemma 4.105. While n-body potentials can be defined in a straightforward way, in what follows we will not need more general potentials than the one-body and pair interaction potentials V and W. Definition 5.39. For fixed ω̄ ∈ X denote by PTω̄ the unique measure on (X, F ) such that for all functions f bounded and measurable with respect to FT and all g bounded and measurable with respect to TT ̄ ω)̄ 𝔼Pω̄ [fg] = 𝔼P [f |TT ](ω)g( T

(5.2.13)

holds. For a general measurable f the integral ∫X f dPTω̄ can be obtained from (5.2.13) by

approximation of f . In other words, PTω̄ is the product measure on C([−T, T]; ℝd ) × ̄ C([−T, T]c ; ℝd ) of the regular conditional probability of P given ω(t) = ω(t), for all c d ̄ |t| > T, and the Dirac measure concentrated at ω on C([−T, T] ; ℝ ). Thus, intuitively, PTω̄ is the measure obtained from P by fixing the path to ω̄ outside [−T, T] and allowing it to fluctuate inside [−T, T]. It is clear that PTω̄ is indeed a version of P( ⋅ |TT ). We emphasize that there many choices of possible versions since conditional expectations are only defined up to sets of measure zero. By the law of the iterated logarithm, the measure 𝒲 x is concentrated on a set of paths that is characterized by the asymptotic behavior of Bt (ω) as |t| → ∞.

5.2 Gibbs measures on path space

| 487

Thus had we defined P( ⋅ |TT ) arbitrarily on the set N ⊂ X of measure zero consisting of all paths not having this precise asymptotic behavior, we would still have obtained a version of the conditional expectation. Our definition of PTω̄ reduces this element of arbitrariness considerably, and we will need this when defining Gibbs measures. Without restricting generality, we choose symmetric bounded intervals from the index set. Write Λ(S, T) = ([−S, S] × [−T, T]) ∪ ([−T, T] × [−S, S]), Λ(T) = (ℝ × [−T, T]) ∪ ([−T, T] × ℝ),

where S, T > 0. For S ≤ T and admissible potentials V, W we use the notation T

ℋΛ(S,T) (B) = ∫ V(Bs ) ds + ∫ W(Bs , Bt , |s − t|) ds dt,

(5.2.14)

Λ(S,T)

−T

and define ℋΛ(T) by replacing Λ(S, T) with Λ(T) in (5.2.14). We will also write ℋT instead of ℋΛ(T,T) . To ease the notation we do not explicitly indicate the ω-dependence of these functionals. Now we can define Gibbs measures on Brownian paths for the potentials V, W. Definition 5.40 (Gibbs measure). A probability measure Q on (X, F ) is called a Gibbs measure with respect to reference measure P and admissible potentials V, W if for every T > 0, (1) Q|FT ≪ P|FT for all T > 0; (2) for every bounded F -measurable function f , 𝔼Q [f |TT ](ω)̄ =

𝔼Pω̄ [fe−ℋΛ(T) ] T

𝔼Pω̄ [e−ℋΛ(T) ]

,

Q-a. s.

(5.2.15)

T

A probability measure QT on (X, F ) is called finite volume Gibbs measure for the interval [−T, T] if (1󸀠 ) QT |FS ≪ P|FS for all S < T; (2󸀠 ) for all 0 < S < T and every bounded F -measurable function f , 𝔼QT [f |TS ](ω)̄ =

𝔼Pω̄ [fe−ℋΛ(S,T) ] S

𝔼Pω̄ [e−ℋΛ(S,T) ]

,

QT -a. s.

(5.2.16)

S

In our terminology “Gibbs measure” corresponds to “infinite volume Gibbs measure” in the classical theory. We refer to the path ω̄ as a boundary condition or boundary path. The normalization factor ZT (ω)̄ = 𝔼Pω̄ [e−ℋΛ(T) ] T

is traditionally called partition function.

488 | 5 Gibbs measures associated with Feynman–Kac semigroups Remark 5.41 (DLR equations). Conditions (2) and (2󸀠 ) in the above definition are traditionally called Dobrushin–Lanford–Ruelle (DLR) equations. These are in fact equalities and can be viewed as counterparts for conditional measures of the Kolmogorov consistency relations for marginal measures. Remark 5.42 (Gibbs specification). To make it more visible that a Gibbs measure is a probability measure which has prescribed conditional probability kernels, we give the following equivalent definition of a Gibbs measure. We say that a probability measure Q on (X, F ) is a Gibbs measure for the admissible potentials V, W if for every A ∈ ̄ is a version of the conditional probability F and T > 0 the function ω̄ 󳨃→ QT (A, ω) Q(A|TT ), i. e., ̄ Q(A|TT )(ω)̄ = QT (A, ω), where QT (A, ω)̄ =

A ∈ F , T > 0, a. e. ω̄ ∈ X,

1 ̄ ∫ 1A e−ℋΛ(T) dPTω , ZT (ω)̄

A ∈ F , ω̄ ∈ X.

(5.2.17)

X

Remark 5.43. To understand why (1) in Definition 5.40 is necessary, consider the spē Writing the (on a first sight more natural) expression cial version PTω̄ of 𝔼P [ ⋅ |TT ](ω). 𝔼Q [f |TT ] =

𝔼P [fe−ℋΛ(T) |TT ] 𝔼P [e−ℋΛ(T) |TT ]

(5.2.18)

instead of (5.2.15) is in most cases meaningless. The problem is that the left-hand side of the above equality is only defined uniquely outside a set N of Q-measure zero, while the right-hand side is only defined uniquely outside a set M of P-measure zero. In many cases of interest, P and Q are mutually singular on X, and then (5.2.18) is no condition at all. Thus in our case sets of measure zero do matter and extra care is needed. The reason why we require (1) explicitly is the same. Without it the two sides of (2) may refer to functions that are uniquely defined only on disjoint subsets of the probability space. In the case of a finite-dimensional state space and the measure of a Feller Markov process as reference measure, this inconvenience can be avoided and then (1) is unnecessary. Constructing a Gibbs measure for given potentials V and W may be difficult. However, constructing a finite volume Gibbs measure under the given regularity conditions is straightforward. Proposition 5.44. Let P be the reference measure, T > 0, and V, W be admissible potentials. If 0 < 𝔼P [e−ℋT ] < ∞, then dQT =

e−ℋT dP 𝔼P [e−ℋT ]

is a finite volume Gibbs measure for [−T, T].

5.2 Gibbs measures on path space

| 489

Proof. By assumption e−ℋT is P-integrable. Thus QT ≪ P, and in particular QT |FT ≪ P|FT . To check DLR consistency, let f and g be bounded, and suppose that g is TS -measurable. Then for T > S 𝔼P [e−ℋT ]𝔼QT [g

𝔼P⋅ [fe−ℋΛ(S,T) ] S

𝔼P⋅ [e−ℋΛ(S,T) ]

] = 𝔼P [𝔼P [g

S

= 𝔼P [g

𝔼P⋅ [fe−ℋΛ(S,T) ] S

𝔼P⋅ [e−ℋΛ(S,T) ]

𝔼P⋅ [fe−ℋΛ(S,T) ]

󵄨 −ℋT 󵄨󵄨󵄨 e 󵄨󵄨TS ]] 󵄨󵄨 𝔼P⋅ [e−ℋΛ(S,T) ] S S

e−ℋ[−T,T]2 \Λ(S,T) 𝔼P [e−ℋΛ(S,T) |TS ]]

S

= 𝔼P [e

−ℋT

]𝔼QT [fg].

The last equality is due to the fact that 𝔼P⋅ [e−ℋΛ(S,T) ] is a version of the conditional S

expectation 𝔼P [e−ℋΛ(S,T) |TS ]. Dividing by 𝔼P [e−ℋT ] shows that condition (2) in Definition 5.40 is verified.

Remark 5.45 (Stochastic and sharp boundary conditions). The finite volume Gibbs measure that we constructed above corresponds to free boundary conditions. A similar proof as above shows that for every ω̄ ∈ X, the probability measure dQω T = ̄

e−ℋΛ(T) ̄ dP ω 𝔼Pω̄ [e−ℋΛ(T) ] T

(5.2.19)

T

is a finite volume Gibbs measure. We call it a finite volume Gibbs measure with sharp boundary condition ω.̄ Occasionally, instead of prescribing sharp boundary conditions by fixing path positions, it is more convenient to give only their probability distributions. Let h : X → ℝ be a TT -measurable nonnegative function such that 0 < 𝔼P [he−ℋT ] < ∞. Then as in Proposition 5.44 we can show that dQhT =

he−ℋΛ(T) dP 𝔼P [he−ℋΛ(T) ]

is a finite volume Gibbs measure for [−T, T]. The density h can then be regarded as a stochastic boundary condition. Next we show that limits of finite volume Gibbs measures yield Gibbs measures in the sense of Definition 5.40. The following notion of convergence will be used. Definition 5.46 (Local convergence). A sequence (mn )n∈ℕ of probability measures on X is said to be locally convergent to a probability measure m if for every 0 < T < ∞ and every A ∈ FT , mn (A) → m(A) as n → ∞.

490 | 5 Gibbs measures associated with Feynman–Kac semigroups Remark 5.47. (1) In many cases Gibbs measures are mutually singular with respect to their reference measures, while finite volume Gibbs measures are absolutely continuous with respect to them. Thus the topology of local convergence is the strongest reasonable for such Gibbs measures. In particular, in many cases it is possible to find a tail measurable function f such that ∫X f dPN = 0 for all N, while ∫X f dP = 1. (2) Convergence on sets is usually a very strong requirement for the convergence of measures. Indeed, the situation discussed in the previous remark indicates that convergence typically does not hold for all sets of F . Since, however, we only need to check convergence on elements of FT for finite T, the special structure of Gibbs measures allows to show local convergence of finite volume Gibbs measures in all cases we consider below. Proposition 5.48. Let P be a reference measure and V, W be admissible potentials. Let (Qn )n∈ℕ be a sequence of finite volume Gibbs measures for V, W and [−Tn , Tn ], with Tn → ∞ as n → ∞. Suppose that there exists a probability measure Q on X such that Qn → Q in the topology of local convergence. Suppose, moreover, that Q satisfies (1) in Definition 5.40 with respect to P. Then Q is a Gibbs measure for V, W. Proof. As (1) in Definition 5.40 holds by assumption, only DLR consistency remains to be shown. Each Qn is a finite volume Gibbs measure, thus for bounded f , g with TS -measurable g we have 𝔼Qn [g

𝔼P⋅ [fe−ℋΛ(S,Tn ) ] S

𝔼P⋅ [e−ℋΛ(S,Tn ) ]

(5.2.20)

] = 𝔼Qn [fg]

S

for every S < Tn . We show now that (5.2.20) remains valid in the n → ∞ limit. By a monotone class argument we may assume that both f and g are FR -measurable for some R > 0. In this case, the right-hand side of (5.2.20) converges by the definition of local convergence. At the left-hand side (5.2.12) implies that n→∞

∫ W(Bs , Bt , |s − t|) ds dt 󳨀→ ∫ W(Bs , Bt , |s − t|) ds dt Λ(S,Tn )

Λ(S)

uniformly in paths, and thus Fn (X(ω)) =

𝔼Pω [fe−ℋΛ(S,Tn ) ] S

𝔼Pω [e−ℋΛ(S,Tn ) ] S

n→∞

󳨀→

𝔼Pω [fe−ℋΛ(S) ] S

𝔼Pω [e−ℋΛ(S) ]

= F(X(ω))

S

uniformly in ω ∈ X. Thus for every ε > 0 there is an N ∈ ℕ such that ‖Fm − F‖∞ < ε, whenever m ≥ N. By the triangle inequality, n→∞

|𝔼Qn [gFn ] − 𝔼Q [gF]| ≤ |𝔼Qn [gFN ] − 𝔼Q [gFN ]| + 3‖g‖∞ ε 󳨀→ 3‖g‖∞ ε. Since ε is arbitrary, convergence of the left-hand side in (5.2.20) follows.

5.2 Gibbs measures on path space

| 491

The previous proposition shows that suitable limits of finite volume Gibbs measures over boundary conditions result in Gibbs measures. The next statement shows that this procedure actually yields all Gibbs measures supported on a specific set of paths. Proposition 5.49. Let X∗ ⊂ X be measurable and Q be a Gibbs measure supported on X∗ , with reference measure P. For N ∈ ℕ, ω ∈ X, define 𝔼Qω [f ] = N

𝔼Pω [fe−ℋΛ(N) ] N

𝔼Pω [e−ℋΛ(N) ] N

as in (5.2.15). Suppose that for every T > 0, A ∈ FT , and ω ∈ X∗ we have Qω N (A) → Q(A) as N → ∞. Then Q is the unique Gibbs measure associated with ℋ, supported on X∗ . Proof. Let Q̃ be a Gibbs measure supported by X∗ . For every T < N and A ∈ FT , ̃ T )(ω)̄ is a backward martingale in N, thus convergent almost everywhere to ω̄ 󳨃→ Q(A| N ̃ T )(ω). ̃ T )(ω)̄ = Qω̄ (A) Q-a. ̃ s., and thus for Q-a. ̃ e. ω̄ ∈ ̄ By the Gibbs property Q(A| Q(A| N N ∗ ω̄ ̃ ̃ X there exists Q(A|T )(ω)̄ = limN→∞ Q(A|TN )(ω)̄ = limN→∞ QN (A) = Q(A). Taking ̃ ̃ Q-expectations on both sides of the above equality shows Q(A) = Q(A), and since this holds for every A ∈ F and T > 0, we obtain Q̃ = Q. T

Finally, we give a useful criterion for a sequence of finite volume Gibbs measures to converge along a sequence of boundary conditions. Definition 5.50. We say that a family of probability measures (mT )T>0 is locally uniformly dominated by a family of probability measures (m̃ S )S>0 whenever (1) m̃ S is a probability measure on F[−S,S] ; (2) for every ε > 0 and S > 0 there exists δS > 0 such that lim supT→∞ mT (A) < ε when A ∈ F[−S,S] with m̃ S (A) < δS . The significance of uniform local domination is made clear by the following result. Proposition 5.51. Let (Tn )n∈ℕ be an increasing divergent sequence of positive numbers and assume that (QTn )n∈ℕ is a locally uniformly dominated family of finite volume Gibbs measures. Then (QTn )n∈ℕ has a convergent subsequence in the local strong topology and the limit is a Gibbs measure. Proof. Let S > 0 and write Q̃ S for the measure dominating Qn , i. e., suppose that for ε > 0 there exists δS > 0 such that lim supn→∞ Qn (A) < ε, for all A satisfying Q̃ S (A) < δS . Since Q̃ S is a measure we have limm→∞ QS (Am ) = 0. In particular, Q̃ S (Am ) < δS , for all m > M with some M. Thus lim supn→∞ Qn (Am ) < ε for all m > M by the local uniform domination property. This means that Q∞ (Am ) < ε for all m > M and hence limm→∞ Q∞ (Am ) = 0. Thus a limit measure on FS exists. For FS+1 we take a further subsequence of the subsequence Qn and find a measure Q󸀠∞ on FS+1 whose restriction to FS is Q∞ . Continuing inductively, we find a sequence of measures QS for each

492 | 5 Gibbs measures associated with Feynman–Kac semigroups S ∈ ℕ; they form a consistent family of probability measures and thus by the Kolmogorov extension theorem there is a unique probability measure Q on F such that all Q|FS = QS . This proves existence of a convergent subsequence. Since the DLR property holds in finite volume, due to convergence along a subsequence it carries over in the limit.

5.2.3 Gibbs measures on càdlàg paths When Brownian motion is replaced by a jump Lévy process, the constructions presented in the previous section need to be slightly modified due to measurability issues, which we will discuss next. We only consider the case with potentials V ≢ 0 and W ≡ 0. Recall the notation XD = D(ℝ; ℝd ) for two-sided càdlàg paths as defined in Section 3.2.2. Let F = ℬ(XD ) be the family of cylinder sets on XD , and consider a two-sided Lévy process (Xt )t∈ℝ with path measure P on the space (XD , ℬ(XD )). We assume that (Xt )t∈ℝ is such that it has transition probability densities pt (x, y) = pt (|y − x|), for all t > 0 and x, y ∈ ℝd . With no restriction of generality, we consider half-closed symmetric intervals I = [−T, T) ⊂ ℝ, with T > 0, and use the notations FT = F[−T,T) for the sub-σ-field generated by (Xt )t∈[−T,T) , FT = T[−T,T) = F[−T,T)c , and T = ⋂N∈ℕ TN for the tail field. For the projection of XD to I ⊂ ℝ we write XD |I , and FI for the related Borel σ-field. For x, y ∈ ℝd and s, t ∈ ℝ, s < t, we respectively denote by P x,y and 𝔼x,y the probability measure and expectation of the Lévy process (Xr )s≤r≤t starting in x ∈ ℝd at time s ∈ ℝ given by Xt = y. In fact, P x,y is a regular version of the family of conditional probability measures P x ( ⋅ | Xt = y), y ∈ ℝd , that is, if Y ≥ 0 is an F[s,t) -measurable random variable and g ≥ 0 is a Borel function on ℝd , then 𝔼x [Yg(Xt )] = ∫ 𝔼x,y [Y]g(y)pt−s (y − x)dy. ℝd

Under P x,y the process (Xr )s≤r 0 and x, y ∈ ℝd . Note that for Ψ-Kato decomposable or X-Kato decomposable potentials ZT (x, y) = K2T (x, y) < ∞, for all x, y ∈ ℝd , t ≥ 0, where Kt (x, y) is the integral kernel of the operator e−tH . For every T > 0 we then define the Gibbs specification QT (A, ω)̄ =

T 1 x,y ∫ 1A (ω)e− ∫−T V(Xt (ω))dt dℙT (ω), ̄ ̄ ZT (ω(−T), ω(T)−)

A ∈ F , ω̄ ∈ XD .

XD

We define Gibbs measures in the spirit of Remark 5.42.

(5.2.21)

Definition 5.52 (Gibbs measure on càdlàg paths). Let (Xt )t∈ℝ be a two-sided Lévy process on a given probability space (XD , F , P), and consider the Gibbs specification given by (5.2.21). We say that a probability measure Q on (XD , F ) is a Gibbs measure for the admissible potential V and reference measure P if for every A ∈ F and T > 0 the function ω̄ 󳨃→ QT (A, ω)̄ is a version of the conditional probability Q(A|TT ), i. e., ̄ Q(A|TT )(ω)̄ = QT (A, ω),

A ∈ F , T > 0, a. e. ω̄ ∈ XD .

Alternatively, we say that Q is a Gibbs measure on (XD , F ) for the admissible potential V and reference measure P if for every T > 0, every A ∈ F and B ∈ TT the equality Q(A ∩ B) = ∫ 1B (ω)QT (A, ω)dQ(ω) XD

holds.

(5.2.22)

494 | 5 Gibbs measures associated with Feynman–Kac semigroups

5.3 Gibbs measures for external potentials 5.3.1 Existence In this section we assume that no pair interaction potential is present and consider Gibbs measures for admissible external potentials only. Our goal is to prove that the path measures of the P(ϕ)1 -processes defined for Brownian motion in Theorem 5.15 and for jump Lévy processes in Theorem 5.31 are Gibbs measures. First we choose the case of continuous paths when the reference measure is a Brownian bridge measure. Recall the notation Πt (x) =

1 |x|2 exp (− ). d/2 2t (2πt)

For any interval I = [s, t] ⊂ ℝ define WI

x,y

x,y

= Πt−s (y − x)𝒲I ,

x,y

(5.3.1) x,y

where 𝒲I denotes the Brownian bridge measure. Although WI is a finite measure on C(I, ℝd ), it is not a probability measure. However, due to the normalizing factor it makes no difference in the definition of finite volume Gibbs measures which measure x,y is used, and we adopt WI as it has some advantages in calculations. Definition 5.53 (Itô bridge). Let V be an admissible potential. For any bounded I ⊂ ℝ and x, y ∈ ℝd define x,y

PI (A) =

1 x,y ∫ 1A e− ∫I V(Xt )dt dWI , ZI (x, y)

(5.3.2)

X

for every A ∈ FI , where x,y

ZI (x, y) = ∫ e− ∫I V(Xt )dt dWI .

(5.3.3)

X x,y

We call PI Itô bridge measure for interval I and potential V. The following gives a large class of admissible potentials V. Lemma 5.54. Let V ∈ 𝒦(ℝd ). Then for every bounded interval I ⊂ ℝ sup ∫ e− ∫I V(Xt )dt dWI

x,y∈ℝd

x,y

< ∞.

(5.3.4)

X

Proof. This follows from Khasminskii’s lemma. Let V ∈ 𝒦(ℝd ) be a potential such that the Schrödinger operator H = − 21 Δ + V has a unique strictly positive ground state φ0 at eigenvalue E0 . To simplify the notation,

5.3 Gibbs measures for external potentials |

495

we include E0 in the potential and denote V − E0 by just V. Recall the P(ϕ)1 -process (Xt )t∈ℝ with path measure 𝒩0 associated with H as obtained in Theorem 5.15. We will call 𝒩0 Itô measure for the potential V. By Theorem 5.21 we have 𝒩0 (A) = ∫ φ0 (x)dx ∫ φ0 (y)dy ∫ 1A e ℝd

ℝd

− ∫I V(Xt )dt

d WI

x,y

(5.3.5)

X

for every A ∈ FI . Lemma 5.55. For V ∈ 𝒦(ℝd ) and any I = [a, b] ⊂ ℝ the restriction 𝒩0 |FI is absolutely continuous with respect to 𝒲 |FI , and the density is hI = hI (ω) = φ0 (Xa )e− ∫I V(Xt )dt φ0 (Xb ).

(5.3.6)

Proof. This follows directly from (5.3.5) since by the definition of the Wiener measure 𝔼𝒲 [f (Xa )g(Xb )h] = ∫ f (x)dx ∫ g(y)dy ∫ hI dWI ℝd

ℝd

x,y

X

holds for bounded measurable f , g : ℝd → ℝ and bounded FI -measurable h. Write t

x,y

Kt (x, y) = ∫ e− ∫0 V(Xs )ds dW[0,t]

(5.3.7)

X

for the integral kernel of e−tH , and πt (x, y) =

Kt (x, y) φ0 (x)φ0 (y)

(5.3.8)

for the probability transition kernel of the P(ϕ)1 -process. Since (e−tL f )(x) = ∫ πt (x, y)f (y)φ20 (y)dy ℝd

holds, πt is the probability transition kernel of e−tL in L2 (ℝd , φ20 dx). Recall that L is given by (5.1.3) and (5.1.11). Then x,y

PI (A) =

ΠT2 −T1 (x − y)

φ0 (x)φ0 (y)πT2 −T1 (x, y)

𝔼W x,y [1A e− ∫I V(Xt )dt ]. I

(5.3.9)

We see now that an Itô measure indeed is a Gibbs measure in the sense of Definition 5.40. Theorem 5.56 (Existence of Gibbs measure: case of continuous paths). Let V be Katodecomposable and 𝒩0 be the Itô measure for the potential V. Then 𝒩0 is a Gibbs measure for V.

496 | 5 Gibbs measures associated with Feynman–Kac semigroups Proof. We need to check that (5.3.2) is a conditional probability 𝒩0 (A|TI ). This follows immediately from (5.3.9) and the representation (5.3.5) of 𝒩0 . We note that any two Itô measures are distinguished by their potentials. Proposition 5.57. Take V1 , V2 ∈ 𝒦(ℝd ) and their Itô measures 𝒩0(1) , 𝒩0(2) , respectively. If V1 − V2 is different from a constant outside of a set of Lebesgue measure zero, then 𝒩0(1) and 𝒩0(2) are mutually singular probability measures. Proof. If V1 and V2 differ by a nonconstant term, then the ground state ψ1 of H1 = − 21 Δ + V1 is different from the ground state ψ2 of H2 = − 21 Δ + V2 . As both are continuous functions, there is A ⊂ ℝd such that a1 = 𝒩0(1) (X0 ∈ A) = ∫ ψ21 (x)dx ≠ ∫ ψ22 (x)dx = 𝒩0(2) (X0 ∈ A) = a2 . A

A

Both 𝒩0(1) and 𝒩0(2) are stationary, and thus the ergodic theorem below implies 1 N ∑ 1A (Xn ) = ai N→∞ n n=0 lim

for 𝒩0(i) -almost all paths, i = 1, 2.

In other words, the set {ω ∈ X | limN→∞ n1 ∑Nn=0 1A (Xn (ω)) = a1 } is of 𝒩0(1) -measure 1 and of 𝒩0(2) -measure 0, i. e., 𝒩0(1) and 𝒩0(2) are mutually singular. Definition 5.58 (Ergodic map). A map T on a probability space (Ω, F , P) is called ergodic whenever T is measure preserving and TA = A for A ∈ F implies P(A) = 1 or 0. Proposition 5.59 (Ergodic theorem). Let T be a measure preserving map on a probability space (Ω, F , P). Take f ∈ L1 (Ω, dP). Then the limit 1 n−1 ∑ f (T j ω) n→∞ n j=0

g(ω) = lim

exists for almost every ω ∈ Ω, and 𝔼P [g] = 𝔼P [f ]. If, moreover, T is ergodic, then g = 𝔼P [f ] is constant. Next we consider càdlàg P(ϕ)1 -processes with path measure M0 as obtained in Theorem 5.31. Recall that Theorem 5.35 gives M0 (A) = ∫ φ0 (x)dx ∫ φ0 (y)dy ∫ 1A e ℝd

ℝd

XD

T

− ∫S V(Xt )dt

x,y

dℙ[S,T) ,

0 < S < T, A ∈ F[S,T) . (5.3.10)

Theorem 5.60 (Existence of Gibbs measure: case of càdlàg paths). Let Ψ ∈ ℬ0 be a Bernstein function, and V a Ψ-Kato decomposable potential. Denote by P the path

5.3 Gibbs measures for external potentials |

497

measure of the two-sided Lévy process generated by −Ψ(− 21 Δ), and consider the probability measure M0 of the càdlàg P(ϕ)1 -process for the nonlocal Schrödinger operator H = Ψ(− 21 Δ) + V. Then M0 is a Gibbs measure with respect to the reference measure P and potential V. Proof. Let 0 < S < T, A1 ∈ F[−T,−S) , A2 ∈ F[−S,S) , A2 ∈ F[S,T) , B1 ∈ F[−T,−S) , B2 ∈ F[S,T) , and A = A1 ∩ A2 ∩ A2 , B = B1 ∩ B2 . For simplicity, in the proof we write Q = M0 . By a monotone class argument, it suffices to consider sets of the form A ∩ B. Note that since ξ ,η ξ ,η ̄ ̄ ℙS (ω(−S) ≠ ξ ) = ℙS (ω(S)− ≠ η) = 0, we have S

S

ξ ,η

ξ ,η

̄ ∫ e− ∫−S V(Xs (ω)) ds QS (A2 , ω)̄ dℙS (ω)̄ = ∫ e− ∫−S V(Xs (ω)) ds 1A2 (ω)̄ dℙS (ω). ̄

XD

̄

XD

Then the Markov property of (Xt )t∈ℝ yields T

̄ x,y ∫ e− ∫−T V(Xs (ω)) ds QS (A, ω)̄ dℙT (ω)̄ B

−S

=

̄ x,ξ ̄ ∫ ( ∫ e− ∫−T V(Xs (ω)) ds 1A1 ∩B1 (ω)̄ dℙ[−T,−S) (ω))

ℝd ×ℝd

XD S

T

ξ ,η

̄ × ( ∫ e− ∫−S V(Xs (ω)) ds QS (A2 , ω)̄ dℙS (ω))( ∫ e− ∫S ̄

XD

= ∫e

̄ ds V(Xs (ω))

η,y

̄ dξdη 1A3 ∩B2 (ω)̄ dℙ[S,T) (ω))

XD T

̄ ds − ∫−T V(Xs (ω))

x,y 1A∩B (ω)̄ dℙT (ω)̄

XD

for all x, y ∈ ℝd . By (5.3.10) we then obtain ∫ 1B QS (A ∩ B, ω)̄ dQ(ω)̄ = Q(A ∩ B)

(5.3.11)

XD

verifying (5.2.22) as required.

5.3.2 Uniqueness The next natural question is whether only a single or multiple Gibbs measures exist for a given V. We start by an illuminating example. Example 5.61 (Nonuniqueness). Take the potential Vosc (x) = 21 (x 2 − 1), x ∈ ℝ, of the one-dimensional harmonic oscillator, and consider the corresponding Schrödinger 2 operator Hosc = − 21 Δ + Vosc . Recall that its ground state is φosc (x) = π −1/4 e−x /2 , and the

498 | 5 Gibbs measures associated with Feynman–Kac semigroups ground state-transformed process for Hosc is a one-dimensional Ornstein–Uhlenbeck process. Fix α, β ∈ ℝ and define for s, x ∈ ℝ ψls (x) = ψrs (x) =

1

π 1/4 1

π 1/4

2

1 αe−s exp (− (x + αe−s )2 ) exp ( ) , 2 2 2

βes 1 exp (− (x + βes )2 ) exp ( ) . 2 2

A simple calculation yields e−tHosc ψls = ψls+t ,

e−tHosc ψrs = ψrs−t ,

(ψls , ψrs ) = eαβ/2 .

Hence n

∫ ∏ fj (Xtj )dPα,β = e−αβ/2 (f1 ψlt1 , e−(t2 −t1 )Hosc f2 ⋅ ⋅ ⋅ fn−1 e−(tn −tn−1 )Hosc fn ψrtn )

X j=1

gives rise to the finite-dimensional distributions of the probability measure Pα,β on X of a Gaussian Markov process. This is stationary if and only if α = β = 0. It can be checked by straightforward computation that Pα,β is a Gibbs measure for every α, β ∈ ℝ. In this case thus uncountably many Gibbs measures exist for the same potential. The above example shows that a Gibbs measure for a Kato-decomposable potential may not be unique. However, Lemma 5.49 gives a simple criterion allowing to check if a Gibbs measure is the only one supported on a given subset. In order to apply x,y ̄ it we need to know for which ω̄ ∈ X it is true that PN (A) → P(A) (with ω(−N) = x, ̄ ω(N) = y). The next lemma gives a sufficient condition for this in terms of the transition densities. Lemma 5.62. (1) Let V be a Kato-decomposable potential, such that the Schrödinger operator H = − 21 Δ + V has a ground state φ0 . Suppose that for some ω̄ ∈ X 󵄨󵄨 π 󵄨󵄨 ̄ ̄ (ω(−N), x)πN−T (y, ω(N)) 󵄨 󵄨 lim sup 󵄨󵄨󵄨 N−T − 1󵄨󵄨󵄨φ0 (x)φ0 (y) = 0 󵄨󵄨 ̄ ̄ N→∞ x,y∈ℝd 󵄨󵄨 πN (ω(−N), ω(N))

(5.3.12)

x,y

for all T > 0. Then for every T > 0 and A ∈ FT , PN (A) → P(A) as N → ∞. (2) Let Ψ ∈ ℬ0 be a Bernstein function, and V a Ψ-Kato-decomposable potential, such that the nonlocal Schrödinger operator H = Ψ(− 21 Δ) + V has a ground state φ0 . Suppose that for some ω̄ ∈ XD 󵄨󵄨 π 󵄨󵄨 ̄ ̄ (ω(−N), x)πN−T (y, ω(N)−) 󵄨 󵄨 lim sup 󵄨󵄨󵄨 N−T − 1󵄨󵄨󵄨φ0 (x)φ0 (y) = 0 󵄨󵄨 ̄ ̄ N→∞ x,y∈ℝd 󵄨󵄨 πN (ω(−N), ω(N)−) x,y

for all T > 0. Then for every T > 0 and A ∈ FT , PN (A) → P(A) as N → ∞.

(5.3.13)

5.3 Gibbs measures for external potentials |

499

Proof. We only prove part (1), the proof of (2) is similar with slight modifications as seen in Section 5.2.3. Let A ∈ FT , and take A ⊂ {ω ∈ X | |ω(±T)| < M} for some M > 1. ̄ ̄ Fix N > T and ω̄ ∈ X, and put ω(−N) = ξ and ω(N) = η. The Markov property of Brownian motion and the Feynman–Kac formula give ξ ,η

PN (A) =

−T T N 1 ξ ,x y,η x,y ∫ dx ∫ 𝔼[−N,−T] [e− ∫−N V(Xs )ds ]𝔼[−T,T] [e− ∫−T V(Xs )ds 1A ]𝔼[T,N] [e− ∫T V(Xs )ds ]dy ZN (ω)̄

ℝd

= ∫ dx ∫ ℝd

ℝd

ℝd

T KN−T (ξ , x)KN−T (y, η) x,y 𝔼[−T,T] [e− ∫−T V(Xs )ds 1A ]dy. K2N (ξ , η)

By the selection of A and boundedness of K2T (x, y), the last factor in the above formula is a bounded function of x and y with compact support, thus integrable over ℝd × ℝd . The claim follows for A once we show that ̄ ̄ KN−T (ω(−N), x)KN−T (y, ω(N)) N→∞ 󳨀→ φ0 (x)φ0 (y), ̄ ̄ K2N (ω(−N), ω(N)) uniformly in x, y ∈ ℝd . This is equivalent to (5.3.12). For general A ∈ FT , consider BM = {ω ∈ X | max{|ω(T)|, |ω(−T)|} < M} with M ∈ ℕ, and AM = A ∩ BM . Since BM ↗ X as M → ∞, for given ε > 0 we may take M ∈ ℕ with P(BcM ) < ε. Moreover, for AM and BM we find N0 ∈ ℕ such that for all N > N0 we have |PN (AM ) − P(AM )| < ε resp. |PN (BM ) − P(BM )| < ε. Hence PN (BcM ) < 2ε for all N > N0 , and thus |PN (A) − P(A)| = |PN (AM ) + PN (A \ BM ) − P(AM ) − P(A \ BM )| ≤ |PN (AM ) − P(AM )| + PN (BcM ) + P(BcM ) ≤ 4ε.

This shows that PN (A) → P(A). Now we state and prove our results on uniqueness. Recall asymptotic intrinsic ultracontractivity (AIUC) introduced in Definition 5.3. Theorem 5.63 (Case of AIUC). (1) Let H = − 21 Δ + V such that the Schrödinger semigroup {e−tH : t ≥ 0} is asymptotically intrinsically ultracontractive. Then the Itô measure 𝒩0 for the related P(ϕ)1 -process is the unique Gibbs measure for V supported on X. (2) Let Ψ ∈ ℬ0 be a Bernstein function, H = Ψ(− 21 Δ) + V a nonlocal Schrödinger operator, and suppose that the potential V is such that the semigroup {e−tH : t ≥ 0} is asymptotically intrinsically ultracontractive. Then the path measure M0 of the related càdlàg P(ϕ)1 -process is the unique Gibbs measure for V supported on XD . Proof. By Proposition 5.9 it follows that limN→∞ |πN (x, y) − 1| = 0 uniformly in x, y. Thus (5.3.12) holds. The proof for the càdlàg case is similar.

500 | 5 Gibbs measures associated with Feynman–Kac semigroups The above result shows that AIUC is a very strong regularity property, allowing a unique Gibbs measure to exist on the full set of paths. When AIUC does not hold, we can find a full-measure subset of paths on which Gibbs measures are unique. Recall that for an operator H the nonnegative number Λ = inf{Spec(H) \ {inf Spec(H)}} − inf Spec(H) is called spectral gap of H. Theorem 5.64 (Case of Kato-class). (1) Suppose that V is Kato-decomposable, consider H = − 21 Δ + V, and let its unique strictly positive ground state be φ0 ∈ L2 (ℝd ) ∩ L1 (ℝd ). Define 󵄨󵄨 e−Λ|N| 󵄨 X∗ = {ω ∈ X 󵄨󵄨󵄨 lim = 0} . 󵄨󵄨N→±∞ φ0 (ω(N))

(5.3.14)

Then 𝒩0 is the unique Gibbs measure for V supported on X0 . (2) Let Ψ ∈ ℬ0 be a Bernstein function, consider H = Ψ(− 21 Δ) + V, and let its unique strictly positive ground state be φ0 ∈ L2 (ℝd ) ∩ L1 (ℝd ). Define 󵄨󵄨 e−Λ|N| 󵄨 X∗D = {ω ∈ XD 󵄨󵄨󵄨 lim = 0} . 󵄨󵄨N→±∞ φ0 (ω(N))

(5.3.15)

Then M0 is the unique Gibbs measure for V supported on X∗D . Proof. With the projection 1φ0 used in the proof of Proposition 5.9, write Lt = e−tH −1φ0 . This is an integral operator with kernel K̃ t (x, y) = Kt (x, y) − φ0 (x)φ0 (y). As Λ > 0 by assumption, we have ‖Lt ‖ 2,2 = e−Λt . In order to estimate K̃ t note that sup |K̃ t (x, y)| =

x,y∈ℝd

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ K̃ t (x, y)f (y)dy󵄩󵄩󵄩 = ‖Lt ‖ 1,∞ , 󵄩 󵄩󵄩∞ f ∈L1 ,‖f ‖ 1 =1 󵄩 d sup

ℝ

and since e−tH 1φ0 = 1φ0 e−tH = Pφ0 for all t > 0, we get ‖Lt ‖ 1,∞ = ‖e−H (e−(t−2)H − 1φ0 )e−H ‖1,∞ ≤ ‖e−H ‖2,∞ ‖Lt−2 ‖2,2 ‖e−H ‖1,2 . By Theorem 4.107, both ‖e−H ‖2,∞ and ‖e−H ‖1,2 are finite, thus it follows that for every t ≤ N, |KN−t (x, y) − φ0 (x)φ0 (y)| ≤ Ct e−ΛN , where Ct = ‖e−H ‖2,∞ ‖e−H ‖1,2 eΛ(2+t) is independent of x, y and N. In terms of πN this implies that for all ω̄ ∈ X0 , ̄ |πN−T (ω(−N), x) − 1|φ0 (x) ≤

CT e−ΛN → 0, ̄ φ0 (ω(−N))

5.3 Gibbs measures for external potentials |

̄ |πN−T (y, ω(N)) − 1|φ0 (y) ≤ ̄ ̄ |π2N (ω(−N), ω(N)) − 1| ≤

501

CT e−ΛN → 0, ̄ φ0 (ω(N))

C0 e−2ΛN →0 ̄ ̄ (φ0 (ω(−N))φ 0 (ω(N)))

as N → ∞. From this (5.3.12) can be deduced. It remains to show that P is actually supported on X0 . By time reversibility it suffices to show that P (lim sup N→∞

e−ΛN > n) = 0, φ0 (ω(N))

n ∈ ℕ.

To prove this, note that by the stationarity of P, P(

e−ΛN e−ΛN > n) = ∫ 1{φ0 1,

(5.3.16)

outside of a compact set of ℝd ; we say in this case that V is a super-quadratic potential. Then there exist constants b1 , b2 , D1 , D2 such that D2 e−b2 |x|

α+1

α+1

≤ φ0 (x) ≤ D1 e−b1 |x| ,

(5.3.17)

see Corollaries 4.177 and 4.182. Also, for super-quadratic potentials V the corresponding Schrödinger semigroup is intrinsically ultracontractive (see Theorem 5.5). It can be checked directly that all potentials considered in Lemma 4.176 are Katodecomposable. We complete this section by discussing some examples for standard and fractional Schrödinger operators and the supports of the related P(ϕ)1 -processes. Example 5.66. Let V be a nonconstant polynomial bounded below. This in particular implies that the degree of V is even, so H = − 21 Δ + V has a unique ground state and a positive spectral gap. From Theorem 5.63 and Example 5.61 it follows that a Gibbs measure for V is unique if and only if the degree of V is greater than 2.

502 | 5 Gibbs measures associated with Feynman–Kac semigroups Example 5.67. Let V(x) = |x|2β with β > 0. Again, H = − 21 Δ + V has a unique ground state φ0 and a spectral gap Λ. In the case of β ≥ 1, Theorem 5.63 and Example 5.61 completely solve the question of uniqueness. In the case of 0 < β < 1, (4.3.81) implies that Theorem 5.64 applies, and thus uniqueness holds on a set X∗ = X∗ (β) given by (5.3.14). By Corollaries 4.178 and 4.182 sharp estimates can be obtained on this supporting set. Indeed, 󵄨󵄨 |ω(±N)| {ω ∈ X 󵄨󵄨󵄨 lim sup 󵄨 N→±∞ N

β+1


1. What happens for β < 1 results from (5.3.14), which turns out to be too crude for this range of exponents. Example 5.68. Let H = (−Δ)α/2 + V be a fractional Schrödinger operator with potential V(x) = C0 |x|δ +

C1 C2 − β |x − x1 | 1 |x − x2 |β2

where C0 > 0, C1 , C2 ≥ 0, x1 , x2 ∈ ℝd and δ > 0, β1 , β2 ≥ 0. It is straightforward to check that if 0 < β1 , β2 < α < d or 0 < β1 , β2 < 1 = d ≤ α, then V is fractional Katodecomposable. An immediate consequence of Theorem 5.63 is that the P(ϕ)1 -measure M0 is the only Gibbs measure corresponding to the fractional P(ϕ)1 -process and the potential V supported on XD . By some extra work it is possible to show that M0 is in fact supported by a subset XD consisting of all path functions ω such that for every θ>0 1+θ

|ω(N)| = o (|N| δ+d+α ) . Example 5.69. Let H = (−Δ)α/2 + V, with a fractional Kato-decomposable potential V whose positive part V + ≡ 0 and negative part V − (x) → 0 as |x| → ∞, such that a spectral gap Λ > 0 exists. Theorem 5.64 implies that M0 is the unique Gibbs measure supported on the subspace of paths such that |ω(N)| = o (exp (

Λ |N|)) . 1+α

Again, with some extra work it is possible to show that M0 is actually supported on a subset of paths given by the growth condition 1+θ

|ω(N)| = o (|N| 1+α ) ,

θ > 0.

5.4 Gibbs measures for external and pair interaction potentials: direct method |

503

5.4 Gibbs measures for external and pair interaction potentials: direct method In this section we discuss the more involved situation when the pair interaction potential W is nonzero. Throughout this section we consider Gibbs measures corresponding to them only on Brownian paths. Let X󸀠 be the space of functions ω : ℝ → ℝd , t 󳨃→ ω(t), that are continuous in every t ≠ 0, and for which a left and a right limit exist in t = 0. For τ > 0 define the shrinking operator θτ : X → X󸀠 by (θτ (ω))(t) = {

ω(t − τ) ω(t + τ)

if t ≥ 0, if t < 0.

(5.4.1)

Let V : ℝd → ℝ be an external potential, and W : ℝd × ℝd × ℝ → ℝ be a pair interaction potential in the sense of Definition 5.38. Write T T

WT (X(ω)) = ∫ ∫ W(Xs (ω), Xt (ω), s − t)dsdt

(5.4.2)

−T −T

whenever the right-hand side is well defined. We consider the following regularity conditions on the potentials V and W. Assumption 5.70. (1) V is Kato-decomposable such that the Schrödinger operator H = − 21 Δ + V has a unique ground state φ0 ∈ L2 (ℝd ) ∩ L1 (ℝd ) with the property Hφ0 = 0.

(5.4.3)

(The ground state eigenvalue E0 is included by a shift in V.) (2) There exists C∞ ≥ 0 such that +∞

sup ∫ |W(Xt (ω), Xs (ω), t)|dt ≤ C∞ . ω∈X

(5.4.4)

−∞

(3) There exist C, D ≥ 0 such that with C < lim inf V(x),

(5.4.5)

−WT (X(ω)) ≤ −WT (X(θτ ω)) + Cτ + D

(5.4.6)

|x|→∞

and the bound

holds for all T, τ > 0 and all ω ∈ X.

504 | 5 Gibbs measures associated with Feynman–Kac semigroups Recall that 𝒩0 denotes the path measure of the P(ϕ)1 -process with potential V and stationary measure φ20 (x)dx. This measure will be used as a new reference measure whose modifications by densities dependent on W will be next introduced. For the remainder of this section we prove the following theorem. Theorem 5.71 (Existence of Gibbs measure). Under Assumption 5.70 there exists a Gibbs measure for the potentials V and W. We begin by the following result. Proposition 5.72. With Assumption 5.70, let QhT be the finite volume Gibbs measure for the potentials V and W, with stochastic boundary condition h = hT = φ0 (X−T )φ0 (XT ). Then on FT we have dQhT = T

T T 1 − ∫−T ds ∫−T W(Xs ,Xt ,s−t)ds e d 𝒩0 , ZT

T

with ZT = 𝔼𝒩0 [e− ∫−T ds ∫−T W(Xs ,Xt ,s−t)dt ]. Proof. By the Feynman–Kac formula (5.1.45) we find for all −T < s1 < . . . < sn < T that T

𝔼𝒩0 [f1 (Xs1 ) ⋅ ⋅ ⋅ fn (Xsn )] = ∫ 𝔼x [φ0 (X−T )f1 (Xs1 ) ⋅ ⋅ ⋅ fn (Xsn )φ0 (XT )e− ∫−T V(Xs )ds ]dx. ℝd

By an approximation argument the above equality can be extended to all bounded T

T

FT -measurable functions. Since e− ∫−T ds ∫−T W(Xs ,Xt ,s−t)dt is such a function, we conclude

that

T

T

𝔼Qh [g1 (Xs1 ) ⋅ ⋅ ⋅ gn (Xsn )] = 𝔼𝒩0 [e− ∫−T ds ∫−T W(Xs ,Xt ,s−t)dt g1 (Xs1 ) ⋅ ⋅ ⋅ gn (Xsn )], T

for all bounded functions g1 , . . . , gn , and all −T < s1 < . . . < sn < T, and the claim follows. In order to prove the existence of a Gibbs measure for the potentials V and W, we apply Proposition 5.48. To be able to use this result we need to prove that the family dQT =

T T 1 − ∫−T ds ∫−T W(Xs ,Xt ,s−t)dt e d 𝒩0 ZT

(5.4.7)

from Proposition 5.72 is relatively compact. By Proposition 5.51 it will suffice to show local uniform domination. We start by showing that the one-point distributions QT are tight as T → ∞. Proposition 5.73. Define QT as in (5.4.7), and let Assumption 5.70 hold. Then for every ε > 0 and S > 0 there exists R > 0 such that sup QT (|Xu | > R) ≤ ε T>0

for all |u| ≤ S.

5.4 Gibbs measures for external and pair interaction potentials: direct method |

505

Proof. First we relate QT (|Xu | > R) to QT (|X0 | > R). Assume that u > 0; the case u < 0 can be treated similarly. Using a formal, though suggestive, notation we obtain T T

T+u

−T+u

T+u

2

∫ ∫ =( ∫ + ∫ − ∫) −T −T

−T

−T+u

T+u T+u

= ∫

∫ + ∫

−T+u −T+u

T

−T+u −T+u

−T

T+u T+u

−T+u T+u

∫ + ∫ ∫ −2 ∫ −T

T

T

−T

−T+u T+u

∫ +2 ∫ T

T+u T+u

∫ −2 ∫

−T −T+u

∫ .

T −T+u

Inserting the pair interaction potentials in the double integrals, we see that the last integral in the first line is bounded in absolute value by ‖W‖∞ u2 , and the same is true for the first two terms in the second line, with a factor 2 in front of the second term. The last two terms in the second line are bounded by 2C∞ u, where we used (5.4.4). Thus, sup |WT (X(ω)) − WT+u (X(ω))| = 4‖W‖∞ u2 + 4C∞ u = C1 (u) ω∈X

(5.4.8)

and we find

ZT QT (|Xu | > R) = 𝔼𝒩0 [1{|Xu |>R} e−WT (X) ] ≤ eC1 (u) 𝔼𝒩0 [1{|Xu |>R} e−WT+u (X) ] = eC1 (u) 𝔼𝒩0 [1{|X0 |>R} e−WT (X) ] = eC1 (u) ZT QT (|X0 | > R).

The equality of the first and second lines above follows from the stationarity of the measure 𝒩0 . Thus the claim will be shown once we show it for u = 0. We proceed in several steps. Step 1: By Theorem 5.15 we have QT (|X0 | > R) =

1 ∫ φ20 (y)𝔼𝒩 y [e−WT (X) ]dy. 0 ZT

(5.4.9)

|y|>R

In the next steps we will show that there exist K, r > 0 such that for every T > 0 and y ∈ ℝd , 𝔼𝒩 y [e−WT (X) ] ≤ 0

K inf 𝔼 z [e−WT (X) ]. φ0 (y) |z|≤r 𝒩0

(5.4.10)

Assume for the moment that (5.4.10) holds. We can estimate 1 ≥ QT (|X0 | ≤ r) =

1 1 C inf 𝔼 z [e−WT ] inf φ20 (z), ∫ 𝔼𝒩 y [e−WT ]φ20 (y)dy ≥ 0 |z|≤r ZT ZT r |z|≤r 𝒩0 |y|≤r

where Cr is the volume of a ball with radius r. Thus inf|z|≤r 𝔼𝒩0z [e−WT ] ≤ C̃ r ZT , where the constant only depends on r but not on T. Using (5.4.9) and (5.4.10) gives QT (|X0 | > R) ≤ K C̃ r ∫ φ0 (y)dy.

(5.4.11)

|y|>R

Since φ0 ∈ L1 (ℝd ), the right-hand side above vanishes when R → ∞, and the claim follows.

506 | 5 Gibbs measures associated with Feynman–Kac semigroups Step 2: To prove (5.4.10) we change the probability space. Recall the definition of X󸀠 at the beginning of the section, and consider J : X󸀠 → X 2 = C([0, ∞), ℝd × ℝd ), (ω(t))t∈ℝ 󳨃→ {

(ω(t), ω(−t)),

(limt→0+ ω(t), limt→0− ω(t)),

t > 0, t = 0.

(5.4.12)

On fixing the value of ω ∈ X󸀠 at t = 0 the map J becomes a bijection; this value can be chosen, for instance, equal to the left limit. Write ω = (ω󸀠 , ω󸀠󸀠 ) for the elements 󸀠 󸀠 of X 2 . When we omit the variable ω, we write X = (X 󸀠t , X 󸀠󸀠 t )t≥0 , where X t (ω) = X t (ω ) 󸀠󸀠 󸀠󸀠 and X t (ω) = X t (ω ). The image measure of 𝒩0z under J can be described explicitly. For z = (z 󸀠 , z 󸀠󸀠 ) ∈ d ℝ × ℝd denote by 𝒩0z the measure of the ℝd × ℝd -valued P(ϕ)1 -process with poteñ y) = V(x) + V(y), conditional on ω0 = z. Write FT̃ for the σ-field over X 2 tial V(x, generated by point evaluations at points inside [0, T]. Then for every FT̃ -measurable bounded function f 𝔼𝒩0z [f ] =

T 1 − ∫0 (V(X 󸀠s )+V(X 󸀠󸀠 z s ))ds e f (ω)φ0 (X 󸀠T )φ0 (X 󸀠󸀠 ∫ T ) d𝒲 (ω). φ0 (z 󸀠 )φ0 (z 󸀠󸀠 )

(5.4.13)

X2

Here, 𝒲 z denotes a (2d)-dimensional Wiener measure starting at z. Equation (5.4.13) follows from the fact that the ground state of the Schrödinger operator 1 1 H ⊗ 1 + 1 ⊗ H = − Δz 󸀠 − Δz 󸀠󸀠 + V(z 󸀠 ) + V(z 󸀠󸀠 ) 2 2

(5.4.14)

is given by φ0 (z 󸀠 )φ0 (z 󸀠󸀠 ), combined with (5.1.43). Note that the bottom of the spectrum of (5.4.14) is zero by shifting by a constant. The Markov property and time reversibility of X imply that for every z ∈ ℝd , 𝒩0(z,z) is the image of 𝒩0z under J, i. e., 𝔼𝒩0z [f ] = 𝔼𝒩 (z,z) [f ∘ J −1 ], for all bounded functions f . 0 Write W̃ T (X) = WT (J −1 (X)). Then, in particular, 𝔼𝒩0z [e−WT (X) ] = 𝔼𝒩 (z,z) [e−WT (X) ]. ̃

0

(5.4.15)

From (5.4.10) it can be seen that the quantity we want to estimate is the expectation of ̃ e−WT (X) with respect to 𝒩0(z,z) . It is easy to check that T

T

󸀠󸀠 W̃ T (X) = − ∫ ds ∫ {W(X 󸀠s − X 󸀠t , |s − t|) + W(X 󸀠󸀠 s − X t , |s − t|) 0

0

󸀠󸀠 󸀠 + W(X 󸀠s − X 󸀠󸀠 t , |s + t|) + W(X s − X t , |s + t|)}dt.

(5.4.16)

5.4 Gibbs measures for external and pair interaction potentials: direct method |

507

Step 3: Next note that the shrinking operator is mapped into the time shift on X 2 under J. In other words, θ̃τ = Jθτ J −1 is the usual time shift that maps (ω(t))t≥0 into (ω(t + τ))t≥0 . Thus in the new representation (5.4.6) becomes −W̃ T (X(ω)) ≤ −W̃ T (X(θ̃τ ω)) + Cτ + D

(5.4.17)

for all X ∈ X 2 and all T, τ > 0. Our strategy is to use (5.4.17) together with the strong Markov property of 𝒩0z . For r > 0 let τr = inf{t ≥ 0 | |X t | ≤ r} be the first hitting time of the centered ball with radius r. This is a stopping time and we write Fτr for the associated σ-field. Then for every z ∈ ℝd × ℝd , 𝔼𝒩0z [e−WT (X) ] = 𝔼𝒩0z [𝔼𝒩0z [e−WT (X) |Fτr ]] ≤ 𝔼𝒩0z [𝔼𝒩0z [e−WT (X∘θτr ) eCτr +D |Fτr ]] ̃

̃

̃

̃

= 𝔼𝒩0z [eCτr +D 𝔼𝒩0z [eWT (X∘θτr ) |Fτr ]] = 𝔼𝒩0z [eCτr +D 𝔼 ̃

̃

≤ sup 𝔼𝒩 y [e−WT ]𝔼𝒩0z [eCτr +D ].

[e−WT ]] ̃

(5.4.18)

̃

0

|y|≤r

X τr

𝒩0

To obtain (5.4.10) we need a good estimate of the second factor at the right-hand side of (5.4.18) and estimate the supremum in the first factor against an infimum. This will be done in the next two steps below. Step 4: We show that there exist r, γ > 0, such that for all z ∈ ℝd × ℝd 𝔼𝒩0z [eCτr ] ≤ 1 +

C‖φ0 ‖∞ 1 1 ( + ). γ φ0 (z 󸀠 ) φ0 (z 󸀠󸀠 )

(5.4.19)

Above C is the constant from (5.4.6). To show (5.4.19), we choose 0 < γ < lim inf|x|→∞ V(x) − C, and we take r so large that V(x) > C + γ for all x ∈ ℝd with |x| > r/√2. This is possible by the definition of C. Clearly, {z ∈ ℝd × ℝd | |z| > r} ⊂ {z ∈ ℝd × ℝd | |z 󸀠 | > r/√2} ∪ {z ∈ ℝd × ℝd | |z 󸀠󸀠 | > r/√2}, and with (5.4.13) it follows that t

z φ0 (z 󸀠 )φ0 (z 󸀠󸀠 )𝔼𝒩0z [τr > t] = ∫ e− ∫0 (V(X s )+V(X s ))ds 1{|X s |>r, s≤t} φ0 (X 󸀠t )φ0 (X 󸀠󸀠 t ) d𝒲 (ω)

≤ ∫ e

t − ∫0 (V(X 󸀠s )+V(X 󸀠󸀠 s ))ds

X2 t

󸀠

󸀠󸀠

X2

z 󸀠 z 󸀠󸀠 1{|X 󸀠 |>r/√2, s≤t}∪{|X 󸀠󸀠 |>r/√2, s≤t} φ0 (X 󸀠t )φ0 (X 󸀠󸀠 t ) d𝒲 (ω ) d𝒲 (ω ) 󸀠

s

s

= φ0 (z 󸀠󸀠 ) ∫ e− ∫0 V(X s )ds 1{|X 󸀠 |>r/√2,s≤t} φ0 (X 󸀠t ) d𝒲 z (ω󸀠 ) X2

󸀠

󸀠

s

󸀠󸀠

508 | 5 Gibbs measures associated with Feynman–Kac semigroups t

z 󸀠󸀠 + φ0 (z 󸀠 ) ∫ e− ∫0 V(X s )ds 1{|X 󸀠󸀠 |>r/√2,s≤t} φ0 (X 󸀠󸀠 t ) d𝒲 (ω ) 󸀠󸀠

󸀠󸀠

s

X2

≤ (φ0 (z ) + φ0 (z 󸀠󸀠 ))‖φ0 ‖∞ e−(C+γ)t . 󸀠

The second equality above is due to the eigenvalue equation e−tH φ0 = φ0 and the Feynman–Kac formula. It follows that 𝔼𝒩0z [τr > t] ≤ (

1 1 + ) ‖φ0 ‖∞ e−(C+γ)t , φ0 (z 󸀠 ) φ0 (z 󸀠󸀠 )

and using the equality 𝔼𝒩0z [e

Cτr

∞

] = 1 + ∫ CeCt 𝔼𝒩0z [τr > t]dt,

(5.4.20)

0

we arrive at (5.4.19). Step 5: Let r > 0 be as in Step 4. We show that there exists M > 0 such that sup 𝔼𝒩0z [e−WT ] ≤ M inf 𝔼𝒩0z [e−WT ] ̃

̃

|z|≤r

|z|≤r

(5.4.21)

uniformly in T > 0. Denote by pt (z, y) the probability transition kernel from y to z in time t under 𝒩0 , i. e., define pt through the relation z

𝒩0 (X t ∈ A) =

∫ pt (z, y)1A (y)dy ℝd ×ℝd

for all z ∈ ℝd × ℝd and measurable A ⊂ ℝd × ℝd . By (5.4.13) and the Feynman–Kac formula we have pt (z, y) =

φ0 (z 󸀠 )φ0 (z 󸀠󸀠 ) K (y󸀠 , z 󸀠 )Kt (y󸀠󸀠 , z 󸀠󸀠 ), φ0 (y󸀠 )φ0 (y󸀠󸀠 ) t

(5.4.22)

where Kt is the integral kernel of e−tH . Note that φ0 and Kt are both strictly positive and uniformly bounded. The latter statement follows from the boundedness from L1 to L∞ of the operator e−tH ; see Theorem 4.107. Thus they are bounded and nonzero on compact sets, and for every R > 0 St (R, r) = sup {

pt (x, z) 󵄨󵄨󵄨 󵄨 |x| ≤ r, |y| ≤ r, |z| ≤ R} pt (y, z) 󵄨󵄨

is finite. Defining W̃ T1 like in (5.4.16) with the integrals starting at 1 rather than at 0, we see that there is a constant C1 independent of T such that −W̃ T (X(ω)) − C1 ≤ −W̃ T1 (X(ω)) ≤ −W̃ T (X(ω)) + C1

5.4 Gibbs measures for external and pair interaction potentials: direct method |

509

for all ω ∈ X 2 and T > 0. Define B = {|X 1 | < R}. Then for every x with |x| < r we have ̃1

𝔼𝒩0x [e−WT ] ≤ eC1 𝔼𝒩0x [1B e−WT ] + eC+D 𝔼𝒩0x [1Bc e−WT ∘θ1 ]. ̃

(5.4.23)

̃

̃

Defining W̄ T as in (5.4.16) with |s + t + 2| appearing instead of |s + t| everywhere, in the first term at the right-hand side of (5.4.23) we find ̃1

𝔼𝒩0x [1B e−WT ] = ∫ p1 (x, z)𝔼𝒩0z [e−WT−1 ]dz ≤ S1 (R, r) ∫ p1 (y, z)𝔼𝒩0z [e−WT−1 ]dz ̄

|z|R

≤ sup 𝔼𝒩0x [e

−W̃ T

|x|≤r

] ∫ p1 (z, y)𝔼𝒩 y [eCτr +D ]dy 0

|y|>R

≤ sup 𝔼𝒩0x [e−WT ]eD ∫ p1 (z, y) (1 + ̃

|x|≤r

|y|>R

C‖φ0 ‖∞ 1 1 ( + )) dy. γ φ0 (y󸀠 ) φ0 (y󸀠󸀠 )

By (5.4.22) and the eigenvalue equation we have ∫ p1 (z, y) ( ℝd ×ℝd

=

1 1 + ) dy φ0 (y󸀠 ) φ0 (y󸀠󸀠 )

1 1 ∫ K1 (z 󸀠󸀠 , y󸀠󸀠 )dy󸀠󸀠 + ∫ K1 (z 󸀠 , y󸀠 )dy󸀠 . 󸀠 φ0 (z ) φ0 (z 󸀠󸀠 ) ℝd

(5.4.25)

ℝd

We know by Theorem 4.107 that e−tH is bounded from L∞ to L∞ . In particular, the image of f (x) = 1 is bounded, and we conclude that supz 󸀠󸀠 ∈ℝd | ∫ℝd K1 (z 󸀠󸀠 , y󸀠󸀠 )dy󸀠󸀠 | < ∞. Thus the right-hand side of (5.4.25) is uniformly bounded on {z ∈ ℝd × ℝd | |z| < r}. This implies that for every δ < 1 there is R̄ > 0 large enough such that sup ∫ p1 (z, y) (1 + |z|R̄

C‖φ0 ‖∞ 1 1 ( + )) dy ≤ e−(C+2D) δ. γ φ0 (y󸀠 ) φ0 (y󸀠󸀠 )

Inserting this and (5.4.24) into (5.4.23) we arrive at 𝔼𝒩0z [e−WT ] ≤ S1 (R,̄ r)eC1 𝔼𝒩0x [e−WT ] + δ sup 𝔼𝒩 y [e−WT ], ̃

̃

̃

|y|≤r

0

(5.4.26)

510 | 5 Gibbs measures associated with Feynman–Kac semigroups which holds for all x, z with |x|, |z| ≤ r. By taking the supremum over z and the infimum over x in (5.4.26), after rearranging we find sup 𝔼𝒩0x [e−WT ] ≤ ̃

|x|≤r

S1 (R,̄ r)eC1 ̃ inf 𝔼 z [e−WT ], 1 − δ |z|≤r 𝒩0

which concludes Step 5 and completes the proof of the theorem. The next step is to show local uniform domination. Proposition 5.74. Let Assumption 5.70 hold, and define QT as in (5.4.7). Then for every S > 0 the restrictions of the family (QT )T≥0 to FS are uniformly dominated by the restrictions of 𝒩0 to FS . Proof. Fix S > 0 and ε > 0. Using Proposition 5.73, choose R > 0 large enough so that QT (|Xs | > R) < ε/8 for all |s| < S + 1 and T > 0. Put B = {|X−S−1 | < R, |X−S | < R, |XS | < R, |XS+1 | < R}. Then we have QT (Bc ) < ε/2 for all T > 0. For every A ∈ ℱS QT (A) = QT (A ∩ Bc ) + QT (A ∩ B) ≤ ε/2 + 𝔼QT [QT (A ∩ B|TS+1 )].

(5.4.27)

We are only interested in the lim sup of this as T → ∞, thus we choose T > S + 1. Since QT is a finite volume Gibbs measure, (5.2.16) gives QT (A ∩ B|TS+1 )(ω)̄ =

𝔼(𝒩0 )ω̄

S+1

≤e

8C(S+1)2

1 −ℋΛ(S+1,T) 𝔼 1A∩B ] ω̄ [e [e−ℋΛ(S+1,T) ] (𝒩0 )S+1 𝔼(𝒩0 )ω̄ [1A∩B ]. S+1

The inequality above follows from (5.4.4) by a similar reasoning as the one leading to (5.4.8), and the constant C above does not depend on T. Moreover, by the choice of B we have 𝔼(𝒩0 )ω̄ [1A∩B ] = 1{|ω(−S−1)|≤R} 1{|ω(S+1)|≤R} ∫ ̄ ̄ S+1

∫

|z|≤R |y|≤R

̄ K1 (ω(−S − 1), y) φ0 (y)

̄ ̄ × 𝔼𝒩0 [1A |ω(−S) = y, ω(S) = z]

̄ + 1)) K1 (z, ω(S dy dz, φ0 (z)

where, as in (5.4.22), Kt is the kernel of e−tH . As in the argument following (5.4.22), it ̄ can be seen that there exists D > 0 such that K1 (ω(−S−1),y) ≤ Dφ0 (y) for all |y| ≤ R and φ0 (y) ̄ |ω(−S − 1)| ≤ R. Thus, 𝔼(𝒩0 )ω̄ [1A∩B ] ≤ D2 ∫ S+1

̄ ̄ = y, ω(S) = z]φ0 (z)dy dz ∫ φ0 (y)𝔼𝒩0 [1A |ω(−S)

|z|≤R |y|≤R 2

̄ ̄ ≤ D ∫ ∫ φ0 (y)𝔼𝒩0 [1A |ω(−S) = y, ω(S) = z]φ0 (z)dy dz = D2 𝔼𝒩0 [1A ]. ℝd ℝd

5.5 Gibbs measures for external and pair interaction potentials: cluster expansion

| 511

We have obtained 2

QT (A) ≤ ε/2 + D2 eC(S+1) 𝒩0 (A) 2

for every A ∈ FS . Thus for given ε > 0, we can choose δ < D−2 e−C(S+1) ε/2, and then for every A ∈ FS with 𝒩0 (A) < δ we have QT (A) < ε uniformly in T. This shows the claim. Finally, we complete the proof of Theorem 5.71. Proof. By Propositions 5.74 and 5.72 below, the family PTh of finite volume Gibbs measures is locally uniformly dominated. By Proposition 5.51, the sequence (Pnh )n∈ℕ then has a convergent subsequence, and the limit point is a Gibbs measure.

5.5 Gibbs measures for external and pair interaction potentials: cluster expansion 5.5.1 Cluster representation Beside existence a number of properties of Gibbs measures for densities dependent on a pair interaction potential W can be derived by using a special technique called cluster expansion. Some of these properties have an immediate application to the understanding of spectral properties of the related operators. In lack of space we can only summarize the key elements of this method and the main results. We refer the reader to the literature for a more detailed discussion. Cluster expansion is a technique developed originally in classical statistical mechanics. It relies on the existence of a small parameter so that a probability measure for the interaction switched on can be constructed in terms of a convergent perturbation series around the interaction-free case. In our case, the Gibbs measure for W ≡ 0 is a path measure of a Markov process, while the Gibbs measure we want to construct for W ≢ 0 is a path measure of a random process which is not Markovian. The cluster expansion method has two distinct aspects. One is a combinatorial framework allowing to make series indexed by graphs and derive bounds on them by series indexed by trees. The other part is analytic and more dependent on the details of the model, and consists of basic estimates on the entries of these sums leading to the convergence of the expansion. In this section we make the following assumptions on the potentials. Assumption 5.75. (1a) The external potential V : ℝd → ℝ is Kato-decomposable and the Schrödinger operator H = − 21 Δ + V has a unique ground state φ0 ∈ L2 (ℝd ) with the property Hφ0 = 0; and

512 | 5 Gibbs measures associated with Feynman–Kac semigroups (1b) the Schrödinger semigroup generated by H = − 21 Δ + V is intrinsically ultracontractive (see Definition 5.3). (2) The pair interaction potential W : ℝd × ℝd × ℝ → ℝ has the symmetry properties W(⋅, ⋅, s−t) = W(⋅, ⋅, |s−t|), W(x, y, ⋅) = W(y, x, ⋅) and satisfies either of the following regularity conditions: (W1) There exist R > 0 and β > 2 such that |W(x, y, s − t)| ≤ R

|x|2 + |y|2 1 + |s − t|β

(5.5.1)

for every x, y ∈ ℝd and t, s ∈ ℝ. (W2) There exist R > 0 and β > 1 such that |W(x, y, s − t)| ≤

R 1 + |s − t|β

(5.5.2)

for every x, y ∈ ℝd and t, s ∈ ℝ. In particular, if V is a super-quadratic potential with exponent α > 1 as defined by (5.3.16), then conditions (1a) and (1b) above hold. Let P be the Itô measure for V, which we will use as reference measure. Similarly to (5.4.7) we define dQT =

T T 1 −λ ∫−T ds ∫−T W(Xs ,Xt ,s−t)dt e dP, ZT

(5.5.3)

where λ > 0 is a parameter. Theorem 5.76 (Existence of Gibbs measure). Let (Tn )n∈ℕ ⊂ ℝ be an increasing sequence such that Tn → ∞, and suppose 0 < |λ| ≤ λ∗ with λ∗ small enough. Then under Assumption 5.75 the local limit limn→∞ Q[−Tn ,Tn ] = Q exists and is a Gibbs measure on (X, F ). Moreover, Q does not depend on the sequence (Tn )n∈ℕ . In order to prove existence of the Gibbs measure Q we use a cluster expansion controlled by the parameter λ which we choose to be suitably small. In what follows we outline the main steps of the proof of this theorem. For simplicity and without restricting generality we consider the family of symmetric bounded intervals of ℝ for the index set of measures. We partition [−T, T] into disjoint intervals τk = (tk , tk+1 ), k = 0, . . . , N − 1, with t0 = −T and tN = T, each of length b, i. e., we fix b = 2T/N; for convenience we choose N to be an even number so that the origin is endpoint to some intervals. We break up a path X into pieces Xτk by restricting it to τk . The total energy contribution of the pair interaction can be written in terms of the sum T

T

WT = ∫ ds ∫ W(Xs , Xt , s − t)dt = −T

−T

∑

0≤i 1. { G∈Gn ij∈G {Γi ∩Γj =0} By using a combinatorial function calculus it is possible to show that n

∞

log ZT = ∑

n=1

∑

{Γ1 ,...,Γn }∈𝕂N 0∈Γ∗ 1

ϕT (Γ1 , . . . , Γn ) ∏ KΓl . l=1

(5.5.14)

516 | 5 Gibbs measures associated with Feynman–Kac semigroups Also, consider for any subset A ⊂ ℝ the set function n

∞

ZTA = 1 + ∑

∑

n=1

Γ1 ,...,Γn ∈𝕂N Γ∗ ∩Γ∗ =0, i=j̸ i j A∩(∪i Γ∗ )=0 i

∏ K Γi i=1

and write ZTΓ = ZTΓ∪Γ . Then by combinatorial function calculus it is furthermore possible to derive the expressions ∗

n

∞

log ZTΓ = 1 + ∑

n=1

∑

Γ1 ,...,Γn ∈𝕂N Γ∗ ∩(∪i Γ∗ )=0 i

ϕT (Γ1 , . . . , Γn ) ∏ KΓi i=1

and fTΓ =

∞ ZTΓ = exp ( − ∑ ZT n=1

n

∑

Γ1 ,...,Γn ∈ℂN Γ∗ ∩(∪i Γ∗ )=0̸ i

ϕT (Γ1 , . . . , Γn ) ∏ KΓi ). i=1

(5.5.15)

The function fTΓ is a correlation function associated with cluster Γ. 5.5.2 Basic estimates and convergence of cluster expansion There are two crucial estimates on which the cluster expansion depends. The first estimate is on the energy terms coming from the contributions of the intervals in the partition. Recall the notation 𝒥ij in (5.5.5). Lemma 5.77 (Energy estimates). We have 2

2

∫ Xτ (t)dt+∫τ Xτ (s)ds j { j { C 1 b τi i , (|j−i−1|b)β +1 |𝒥ij | ≤ { { C2 b2 , { (|j−i−1|b)β +1

(W1) case, (W2) case,

(5.5.16)

with some C1 , C2 > 0 in each respective case.

As a consequence, Wτi ,τj is bounded below for all i, j, which is a stability condition ensuring that the variables indexed by the subintervals are well defined and there are no short-range (i = j) singularities. The second estimate for the cluster expansion is given as follows. Lemma 5.78 (Cluster estimates). There exist some constants c1 , c2 > 0 and δ > 1 such that for every cluster Γ ∈ 𝕂N |KΓ | ≤ ∏(c1 |λ|1/3 )|ϱ| ∏ ∏ ̄

ϱ∈Γ

γ∈Γ (τi ,τj )∈γ

c2 |λ|1/3 , (|i − j − 1|b)δ + 1

where |ϱ|̄ denotes the number of intervals contained in ϱ.

(5.5.17)

5.5 Gibbs measures for external and pair interaction potentials: cluster expansion

| 517

In estimate (5.5.17) the factor accounting for the contribution of chains comes from the upper bound Ce−Λb on |πb (x, y) − 1| uniform in x, y (see the second factor in (5.5.10)), where Λ is the spectral gap of the Schrödinger operator H of the underlying P(ϕ)1 -process, and C > 0. This bound results from intrinsic ultracontractivity of e−tH ; see Lemma 5.9. The factor accounting for the contribution of contours comes from a suitable use of the Hölder inequality and optimization applied to the products over −λW e τi τj − 1 (see the first factor in (5.5.10)), taken together with Lemma 5.77; b is then chosen in such a combination with λ and Λ that the expression (5.5.17) results. Proposition 5.79 (Convergence of cluster expansion). If there exists η(λ) > 0 such that limλ→0 η(λ) = 0 and for every n ∈ ℕ ∑

Γ∈𝕂N ̄ Γ∗ ∋0,|Γ|=n

|KΓ | ≤ cηn ,

(5.5.18)

then the series at the right-hand sides of (5.5.13) and (5.5.14), respectively, are absolutely convergent as N → ∞. In order to prove this statement we consider the generating function H(z, λ) = lim ∑ KΓ z |Γ| , N→∞

Γ∈𝕂N Γ∗ ∋0 1

in which λ is considered a parameter and |Γ| is the number of intervals contained by Γ. We show that H(z, λ) is analytic in z in a circle of radius r(λ) and that inside this circle H(z, λ) is uniformly bounded in λ. Choosing η(λ) = 1/r(λ) then gives the bound in (5.5.18). This estimate follows through a procedure of translating the sums over 𝕂N at the left-hand side of (5.5.18) to sums indexed by connected graphs. For every Γ ∈ 𝕂N we define a graph GΓ whose vertex set are the contours {γ1 , . . . , γr } in Γ and whose edge set are those pairs of contours which are appropriately matched with the chains ρ1 , . . . , ρs of Γ. For r = 1 we have s = 0 and therefore the empty graph GΓ1 . For r ≥ 2 we have s ≥ 1 and the sums can be rewritten by sums over connected graphs following a specific enumeration. By using a graph-tree bound, the graph-indexed sums can be bounded by tree-indexed sums, which can be better further estimated. An inductive procedure and some nontrivial combinatorics then yield (5.5.18). By an application of Proposition 5.79 to (5.5.15) it can be shown that f Γ = lim fTΓ T→∞

exists, and |fTΓ | ≤ 2|Γ| holds uniformly in T for λ small enough. This can be used to prove the following statement.

518 | 5 Gibbs measures associated with Feynman–Kac semigroups Proposition 5.80. The local limit P = limT→∞ PT exists and satisfies the equality n

∞

𝔼P [FS ] = 𝔼χ [FS ]f S + ∑

n=1

∑

Γ1 ,...,Γn ∈ℂ Γ∗ ∩Γ∗ =0, i=j̸ i j i:S∩Γ∗ =0̸ i

𝔼χ [FS ∏ κΓi ]f ∪Γ i=1

(5.5.19)

for any bounded FS -measurable function FS , where S is a finite union of intervals of the partition considered in the cluster expansion. Moreover, the measure P is invariant with respect to the time shift. Theorem 5.76 follows directly from this proposition. Equality (5.5.19) implies that P satisfies the DLR equations; in particular, it is a Gibbs measure. 5.5.3 Further properties of the Gibbs measure An important aspect in understanding the Gibbs measure is to characterize almost sure properties of paths under this measure. This is done in the following theorem. Theorem 5.81 (Typical path behavior). If V is a super-quadratic potential with exponent α > 1, then |Xt | ≤ C (log(|t| + 1))1/(α+1) ,

P-a. s.

(5.5.20)

with a suitable number C > 0. The strategy of proving Theorem 5.81 goes by comparing the typical behaviors of the reference process and of the process with the pair interaction potential (which is not a Markov process). Lemma 5.82. Suppose there exist some numbers C, θ > 0 such that for any x > 0 α+1

P(max |Xt | ≥ x) ≤ Ce−θx . 0≤t≤1

(5.5.21)

Then there exist C 󸀠 > 0 and θ󸀠 > 0 such that for any x > 0 󸀠 α+1

Q(max |Xt | ≥ x) ≤ C 󸀠 e−θ x . 0≤t≤1

(5.5.22)

The proof of this lemma requires once again the use of cluster expansion. The assumption of the lemma can be verified by using (5.23) for the underlying P(ϕ)1 -process. Another important property of Q is its uniqueness in DLR sense. This means that for any increasing sequence of real numbers (Tn )n∈ℕ , Tn → ∞, and any corresponding sequence of boundary conditions (Yn )n∈ℕ ⊂ X0 , limn→∞ 𝔼QT [FB |Yn ] = 𝔼Q [FB ], for n every bounded B ⊂ ℝ and every bounded FB -measurable function FB . Here X0 is the subspace given by (5.3.14). In other words, DLR uniqueness means that the limit measure Q is independent of the boundary paths.

5.5 Gibbs measures for external and pair interaction potentials: cluster expansion

| 519

Theorem 5.83 (Uniqueness of Gibbs measure). Suppose that V satisfies the conditions in Assumption 5.75 and W satisfies (W2). Then the following cases occur: (1) If β > 2, then whenever the Gibbs measure Q exists, it is unique in DLR sense. (2) If β > 1, then for sufficiently small |λ| the limiting Gibbs measure Q is unique in DLR sense whenever the reference measure is unique. If β > 2 the energy functional given by (5.2.14) is uniformly bounded in T and in the restrictions of paths over [−T, T] and R \ [−T, T], respectively. This then implies that only one Gibbs measure can exist as the limit does not depend on the choice of the boundary conditions. This argument requires no restriction on the values of λ. For 1 < β ≤ 2 this uniform boundedness does not hold any longer and cluster expansion is to date the only method allowing to derive these results. To conclude, we list some additional properties of Gibbs measures for (W2)-type pair interaction potentials, useful in various contexts. This case in particular covers the Nelson model. Theorem 5.84. Let P be a Gibbs measure for W satisfying (W2), and suppose V satisfies Assumption 5.75. Then the following hold: (1) (Invariance properties) With respect to time shift and time reflection, Q is invariant. (2) (Univariate distributions) The distributions QT under QT of positions x at time t = 0 are equivalent to P, i. e., there exist C1 , C2 ∈ ℝ, independent of T and x, such that C1 ≤

dQT (x) ≤ C2 dP

(5.5.23)

T

for every x ∈ ℝd and T > 0. Moreover, limT→∞ dQ exists pointwise. dP (3) (Univariate conditional probability distributions) The conditional distributions QT ( ⋅ |X0 = x) converge locally weakly to Q( ⋅ |X0 = x), for all x ∈ ℝd . (4) (Mixing properties) For any bounded functions F, G on ℝd |covQ (Fs ; Gt )| ≤ const

sup ‖F‖∞ sup ‖G‖∞ , 1 + |s − t|γ

(5.5.24)

where γ > 0 and the constant prefactor is independent of s, t and F, G. The invariance properties are inherited trivially from the underlying P(ϕ)1 -process. The proof of the remaining properties makes extensive use of cluster expansion.

6 Notes and references Notes to the Preface Classic books of this early period include H. Poincaré: Les méthodes nouvelles de mécanique céleste (Gauthier-Villars, 1893, English translation: New Methods of Celestial Mechanics), 3 vols.; see also C. Siegel and J. Moser: Vorlesungen über Himmelsmechanik (1956, English translation: Lectures on Celestial Mechanics, Springer, 1971), J. C. Maxwell: A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1873, reprinted by Dover, 1954, 2003), and Theory of Heat (Longmans, 1872, reprinted by Dover, 2001). A comprehensive account of methods of mathematical physics with a strong focus on partial differential equations which stood as a standard work for generations is D. Hilbert and R. Courant: Methoden der mathematischen Physik, vols. 1–2 (Springer, 1924, 1937, with subsequent English translation). The intellectual biographies and autobiographies of some of the main protagonists of early quantum mechanics capture the excitement of those days and convey the spirit how these ideas were born; see, for instance, W. Heisenberg: Der Teil und das Ganze (Piper, 1969), D. Cassidy: Uncertainty: The Life and Science of Werner Heisenberg (W. H. Freeman, 1993), A. Pais: Niels Bohr’s Times: In Physics, Philosophy, and Polity (Oxford UP, 1994), and G. Farmelo: The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (Basic Books, 2009). For an account from a major figure in the development of stochastic analysis and Feynman–Kac formulae, see R. Bhatia: A conversation with S. R. S. Varadhan, Math. Intelligencer 2008, pp. 24–42. E. Wigner’s essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences appeared in Commun. Pure Appl. Math., 13 (1960). He mentions the idea attributed to Galileo that the laws of nature are written in the language of mathematics. The concept of the “book of nature” is a topos that can be traced back to antiquity and the middle ages; see the early and illuminating account given by the great German literary scholar E. R. Curtius in Europäische Literatur und lateinisches Mittelalter (Francke, 1954, 350ff) (especially the concept of “chiffre”), and more modern discussions in The Book of Nature, A. Vanderjagt, K. van Berkel (eds.), vol. 1: Antiquity and the Middle Ages, vol. 2: Early Modern and Modern History (Peeters, Leuven, 2005–2006). On the quoted passage of Feynman, see R. Dijkgraaf, Rapporteur Talk: Mathematical structures, in: The Quantum Structure of Space and Time, Proceedings of the 23rd Solvay Conference on Physics (D. Gross, M. Henneaux, A. Sevrin, eds., World Scientific, 2007, p. 91). Apparently a mathematician has replied that the time by which mathematics would be set back is “precisely the week in which God created the world”; a version of this anecdote names Mark Kac to be this mathematician.

https://doi.org/10.1515/9783110330397-006

522 | 6 Notes and references

Notes to Chapter 1 Feynman’s path integral method appeared first in the papers [117–119]; see also [120]. Dirac’s contact transformation theory was published in [89–91]. For some historic background see [323]. Modern references on Feynman integrals include [4, 57, 62, 67, 87, 88, 143, 149, 191, 193, 222, 305, 370]. A first paper by Kac using analytic continuation of the time variable is [194]; see [56, 274] for later developments. The disadvantage of Kac’s approach is that it deals with the diffusion equation with dissipation instead of the time-dependent Schrödinger equation, and therefore the Feynman–Kac formula gives no direct information on the quantum dynamics. However, the timeindependent solutions of both equations are the same and coincide with the eigenfunctions of the Schrödinger operator, which justifies the use of the Feynman–Kac formula in quantum mechanics. An early but little known paper taking a different approach is [116].

Notes to Chapter 2 2.1 Measure theory has been developed from the late 19th century by E. Borel, H. Lebesgue, J. Radon, and M. Fréchet, among others. Some applications of measure theory are in the theory of integration, probability and stochastic processes, ergodic theory, geometric measure theory, or topological measure theory. Integration theory developed in the line of defining integrals on spaces more general than subsets of the Euclidean space ℝd , arriving to integration with respect to Lebesgue measure on ℝd and futher concepts, leading to a richer theory than Riemann–Stieltjes integrals. For instance, Lp -spaces defined with respect to the Lebesgue measure are important Banach spaces, with many applications in functional analysis. Probability theory uses finite measures, and considers measurable subsets to be events whose probabilities are given by their measures, and ergodic theory considers measures that are kept invariant under a dynamical system. Stochastic analysis has been developed to extend the concept of integral to irregular functions, and apart for its intrinsic interest in mathematics, a major modern source of motivation was provided by financial mathematics. We refer to [310, 29] for a general analysis and measure theory background, to [103, 11, 26, 38, 140, 349, 209, 101] for probability theory also emphasizing an analytic point of view, and to [230, 136, 219] for random processes. A π-system, a λ-system and the π − λ theorem are basic facts in measure theory, which are reviewed in, e. g., [265, Appendix B]. Measures on a metric space are reviewed in, e. g., [224, Section 13], where the existence of regular conditional probabilty measure is also discussed in [224, Section 8]. In Definition 2.5 weak topology and vague topology on the set of measures are defined. The weak topology τw on the set of finite measures Pf (S) is investigated in,

Notes to Chapter 2

| 523

e. g., [293, p. 167] and the vague topology τv on the set of Radon measures Pr (S) in, e. g., [208, Section 15.7]. If S is a locally compact Polish space, then (Pf (S), τw ) and (Pf (S), τv ) are also Polish spaces. 2.2 The concept of a martingale in probability theory was introduced by P. Lévy in 1934, and the term “martingale” was introduced later by J. Ville in 1939, who also extended the definition to continuous-time martingales. Much of the original development of martingale theory was done by J. Doob and D. Burkholder. We refer to e. g., [152, 103, 263, 264, 304, 306, 307] for continuous time martingales. Martingales and stopping times are useful tools to study stochastic analysis, and we refer to e. g., [265], [182, Chapter I. 6] and references therein. Convergence properties of martingales (Theorem 2.65) are studied in, e. g., [182, Theorem 6.5]. The optional sampling theorem (Theorem 2.70) is taken from [182, Theorem 6.11]. The classes DL and D were introduced in e. g., [211, Definition 4.8] and [182, Definition 6.4]. Increasing processes are also introduced in e. g., [211, Definition 4.4] and [182, Definition 6.2]. The definition of integrability of increasing process is taken from [211, Definition 4.4]. The Doob–Meyer decomposition (Theorem 2.78) is proven in, e. g., [211, Theorem 4.10] and [182, Theorem 6.12]. A. A. Markov was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as Markov processes. He studied Markov chains in the early 20th century, and was interested in studying an extension of independent random sequences. A. Kolmogorov developed in 1931 a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by a work of L. Bachelier on fluctuations in the stock market as well as that of N. Wiener on Einstein’s model of Brownian motion. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov’s work, S. Chapman derived in 1928 the Chapman–Kolmogorov equation. Markov processes include W. Feller and then later E. Dynkin, starting in the 1950s. We refer to [45, 97, 105, 110, 306, 307] for Markov processes, to [188] for Markov generators, to [37, 190] for limit theorems. We refer to the detail of Feller transition kernels and generators to [188, Chapter 3.4] and [262, Section 4.1]. Further applications of an invariant measure and the martingale derived from the generator of a Markov process mentioned in Theorem 2.93 are investigated in [220], where the functional central limit t theorem is discussed, i. e., the asymptotic behavior of 1t ∫0 (Lf )(Xs )ds as t → ∞ is studied, where L is the generator of Markov process (Xt )t ≥ 0. In the proof of Theorem 2.91 it is mentioned that for each t > 0 and each g ∈ C ∞ (ℝd ) there exists f ∈ D(L) with f − tLf = g. This is due to [247].

524 | 6 Notes and references 2.3 The Scottish botanist Robert Brown is credited with the observation reported in 1827 that pollen grains suspended in water perform an irregular motion, however, he was preceded by more than a century by the Dutch natural scientist van Leeuwenhoek by a similar observation. Also less known is that Brown enthusiastically continued his observations by replacing pollen with minerals and even a piece of the Sphinx. Brownian motion has been first modeled in terms of a random process by Bachelier [21] as early as 1900 in order to describe stock prices. Einstein [107] and von Smoluchowski [343] studied in 1905–1906 the fluctuations of thermal motion of particles and related them with Brownian motion, which Perrin tested in 1909 experimentally, noting the rough shape of particle paths. Langevin proposed in 1908 an equation to describe the equation of motion of a particle performing Brownian motion. In 1923 Wiener constructed Brownian motion as a random process in a mathematically rigorous way, and Lévy made important early contributions in the study of some fine details of Brownian motion. For a history see [102] as well as [226, 275]. We single out [182, 186, 223, 211, 224, 275, 277, 304, 324, 351] for fundamental facts on Brownian motion and general stochastic analysis. Brownian motion in many ways stands out as a random process having a distinguished role. It is a Gaussian process, a Markov process, and a martingale. It is the only Lévy process allowing a version with continuous paths. It is a scaling limit of simple random walk, and it is intimately related with the Laplace operator in that − 21 Δ is its infinitesimal generator, and thus its transition probability density is the fundamental solution of the heat equation. There are a number of equivalent ways of defining and constructing Brownian motion, of which we have presented one and commented on others. The literature on Brownian motion is vast due to the fact that it is an object in the overlap zone of the fields and approaches mentioned above. A wealth of facts and explicit formulae can be found in [49] and the references therein, and we single out [261, 365] for further developments. In the Lévy martingale characterization theorem we need the momentum estimate 𝔼[(Xt − Xs )4 |ℱs ] ≤ 4|t − s|2 , which is given in [324, Lemma 9.10]. In Proposition 2.55 we discuss a right-continuous filtration which is given in [265, Chapter II, 2.f.0]. The basic idea for the reflection principle for Brownian motion (Bt )t≥0 is familiar from simple symmetric random walk. This results from the strong Markov property of (Bt )t≥0 and is stated in Corollary 2.116. We also refer to [306, I.13]. 2.4 In 1944 Kiyosi Itô [184] proposed a notion of stochastic integral using the paths of Brownian motion as integrator, and he also established the chain rule for stochastic differentials, which is known today as the Itô formula. While the construction of the Itô integral starts from the classic notion of integral proceeding through approximate Riemann–Stieltjes sums over finite divisions, it successfully copes with the difficulty that these sums do not stand a chance to converge pathwise since Brownian motion

Notes to Chapter 3

| 525

has paths of almost surely unbounded variation. However, by requiring less Itô has shown that convergence holds in L2 sense (see Definition 2.129), which extends to convergence in probability (see Definition 2.139). Since the original ideas of Itô, stochastic integration theory has further developed by using continuous martingales as integrators instead of Brownian motion. We refer to [150, 186, 259, 211, 263, 264, 277, 304, 306, 307, 351] for stochastic integration theory with Brownian or more general integrators, and to [70, 182, 277] for stochastic differential equations.

Notes to Chapter 3 3.1 Lévy processes are random processes with stationary and independent increments, in many ways generalizing Brownian motion and Poisson process. In particular, Lévy processes are Markov processes and semimartingales, with right-continuous paths allowing left limits, which in general contain continuous pieces with jump discontinuities at random times. The generators of Lévy processes are in general nonlocal pseudo-differential operators. We refer to [7, 24, 30, 32, 47, 262, 322, 325] for general results on Lévy processes, to [96, 229] for their fluctuation theory, and to the threevolume set [188], which is a comprehensive study emphasizing the context of semigroups and generators. The work [47] is devoted to the potential theory of stable random processes such as subordinate Brownian motions, from which we take estimates of pXt . The asymptotic behavior lim|x|→∞ pX1 (x) in (3.1.27) is given in [44], and the estimate of pXt (x) in (3.1.28) by [48] and [47, (2.30)]. Subordinators are an example of random time change. The work [25] is a comprehensive reference of time changes and discusses processes of the form (XT(θ) ̂ )θ≥0 with change of time (T(θ))θ≥0 . 3.2 The material on path properties of Lévy processes is taken from [322, Chapter 2.11]. The space XD of ℝd -valued càdlàg paths on [0, ∞) is investigated in [110, Section 3.5], for the separability and completeness of (XD , dD ) see [110, Theorem 5.6]. We also refer to [37, Section 16] and [37, Theorem 16.3]. These properties depend on the measure defined on XD . Let D1 = D([0, 1]) be the set of real-valued functions on [0, 1] that are right-continuous with left-limits, and Λ be the class of strictly increasing continuous functions λ : [0, 1] → [0, 1]. We then have λ(0) = 0 and λ(1) = 1. For X, Y ∈ D1 one can define d(X, Y) = inf { sup |γ(t) − 1| ∨ supt∈[0,1] |Xt − Yλ(t) |} λ∈Λ t∈[0,1]

and d∘ (X, Y) = inf (‖λ‖∘ ∨ supt∈[0,1] |Xt − Yλ(t) |). λ∈Λ

526 | 6 Notes and references Here

λ(s) − λ(t) 󵄨󵄨 󵄨 ‖λ‖∘ = sup 󵄨󵄨󵄨 log 󵄨. s−t 󵄨 0≤t 0 | Bt ∉ D} from D of standard Brownian motion (Bt )t≥0 . From the definition we have Bt ∈ D for t < τD , and BτD ∈ 𝜕D if τD < ∞. It is possible to prove that τD < ∞ 𝒲 -a. s. Then the Feynman–Kac formula becomes τD

ψ(x) = 𝔼x [e− ∫0

V(Bt )dt

ϕ(BτD )]

and thus it offers a probabilistic representation of the solution to the Dirichlet problem. In this case the spectrum of the Laplace operator is discrete consisting of eigenvalues 0 < λ0 (D) < λ1 (D) < . . . (a) Isoperimetric inequalities: One line of research is addressing the question how the various properties of the spectrum depend on geometric properties of D. Let D be a convex domain and V be a convex function. In this case also the Schrödinger operator has a purely discrete spectrum 0 < λ0 (D, V) < λ1 (D, V) < . . . Without restricting generality suppose that 0 ∈ D, and let 𝔹 denote a ball centered in the origin such that |D| = |𝔹|, where the bars denote Lebesgue measure. Then it can be shown that the first exit time of Brownian motion is maximized by the ball, i. e., supx∈D 𝒲 x (τD > t) ≤ 𝒲 0 (τ𝔹 > t), for all t > 0. This is an example of an isoperimetric inequality. Since 1 lim log 𝒲 x (τD > t) = −λ0 (D), t

t→0

the above inequality implies the Faber–Krahn inequality λ0 (𝔹) ≤ λ0 (D), which is an isoperimetric property of the eigenvalues of the Dirichlet Laplacian [23, 64]; see

Notes to Chapter 4

| 529

also [279]. The Faber–Krahn inequality has been similarly shown for Schrödinger operators for convex domains [58, 153, 286]; see the conditions on V in the referred papers. Another isoperimetric inequality concerns the ratio of consecutive eigenvalues, originating from the Payne–Pólya–Weinberger conjecture [282] which originally stated that for planar domains λ1 (D)/λ0 (D) is maximized when D is chosen to be a disk. The conjecture has been proven to hold in any dimension in [13, 14]. (b) Spectral gaps: The difference Λ(D) = λ1 (D) − λ0 (D) is called fundamental gap for the Dirichlet Laplacian, and a similar object can be defined for the Dirichlet Schrödinger operator. Based on specific cases in which the eigenvalues can be 2 computed it has been conjectured [31] that Λ(D) ≥ 3π , where d is the diameter of 2d2 D, provided both D and V are convex. An upper bound can be obtained as follows. Since by the Payne–Pólya–Weinberger conjecture λ1 (D)/λ0 (D) ≤ λ1 (𝔹)/λ0 (𝔹), it follows that Λ(D) ≤ Λ(𝔹)(λ0 (D)/λ0 (𝔹)), in which the second factor at the righthand side can be further estimated by using the Faber–Krahn inequality. For a discussion see [12] and the references therein. For Schrödinger operators with convex V the fundamental gap conjecture was proven recently in [6]. (c) Harnack inequalities: In its classical form, the boundary Harnack inequality states that for any open sets K ⊂ K ∗ ⊂ ℝd with K ⊂ K ∗ , there exists C > 0 such that for all functions f harmonic and nonnegative on K ∗ we have f (x) ≤ Cf (y), for all x, y ∈ K. In other words, just by knowing that a nonnegative f solves Δf = 0 we can already conclude that infx∈K f (x) ≥ C1 supx∈K f (x), where C depends on K and K ∗ but not on f . A version of this inequality for Schrödinger operators was first proven by [360] using analytic methods. The work [3] realized that the Feynman–Kac formula can be used to obtain these results. They used the fact that when stopped at the boundary of K ∗ , a Feynman–Kac-type formula just gives the solution of the Dirichlet problem for H and K ∗ . Further references include [67, 154, 368, 369]. Further interesting aspects include heat kernel estimates [20, 74, 76, 141, 142, 285], functional inequalities [320], and intrinsic ultracontractivity [19, 75, 77]. For the general theory of how the Feynman–Kac formula can be used to study various classes of partial differential equations we also refer to [26, 27, 103, 125, 350]. Kato-class has been introduced in [215]. Simon [334] studies Schrödinger semigroups with Kato-class potentials in great detail. In particular, here it is proven that e−tH is a bounded operator from Lp to Lq for 1 ≤ p ≤ q ≤ ∞. A natural question is how the norm of this operator depends on t for large t. It is shown in [332] that α = limt→∞ (1/t) ln ‖e−tH ‖p,q is independent of p and q; it turned out [333] that the same α is obtained by taking pointwise limits of images of nonnegative functions. Since then there has been considerable research activity on the topic; see [74, 76, 226, 269, 270, 285, 366, 367], as well as [227, 352, 353]. Yet another, more recent, line of research is concerned with cases when the potential V is replaced by more singular objects such

530 | 6 Notes and references as distributions, or when Brownian motion is replaced by a Bessel process. For research in this direction see [311–315] and the references therein. 4.3 Theorem 4.116 can be improved by showing that e−tK (x, y) is even jointly continuous in x, y and t [334]. The Kallianpur–Robbins laws stated in Proposition 4.137 estimate the occupation time of a Brownian motion, which is studied in [210]. It can be extended to a fractional Brownian motion in [212]. The discussion in Theorems 4.138 and 4.139 of the existence of bound states of low-dimensional Schrödinger operators follows [319]. The proof of Lemma 4.144 is due to [68]. Estimates of the number of negative eigenvalues of Schrödinger operators are due to [238, 239], where the Birman–Schwinger principle and the Feynman–Kac formula are used; see also [244, 245]. The proof of Lemma 4.153 is taken from [338] and [331, Theorems 8.1-8.2]. In [68, Theorem] it is shown that sup k k∈ℕ

1/p

(k

−1

k

1/2

2

≤ Kp ‖u‖∗p,w ‖g‖p ,

∑ μm (Bu,g ) )

m=1

where Bu,g is defined by (4.1.28) and the constant Kp is explicitly given by Kp =

1−2/p

p 4 (p ) 2 2 −1

(1 +

2 1/p ) . p−2

The most important corollary derived from this inequality is an upper bound on the number of nonpositive eigenvalues N̄ 0 (V) which coincides with that of eigenvalues of the operator (−Δ)−1/2 |V|1/2 greater than or equal to 1; (−Δ)−1/2 |V|1/2 is the operator B(2π|ξ |)−1 ,|V(x)|1/2 followed by inverse Fourier transform. It is known that u(ξ ) = (2π|ξ |)−1 ∈ Ldw (ℝd ) with ‖u‖∗d,w = ω1/d /2π, where ωd = π d/2 /Γ(1 + d/2) ded d notes the volume of the unit ball in ℝ . Let k = N̄ 0 (V) and p = d. Then we have N̄ (V)

0 1 1≤( ∑ μm ((−Δ)−1/2 |V|1/2 )2 ) N̄ 0 (V) m=1

1/2

≤

Kd ω1/d d 2π

1/d

(

∫ℝd |V(x)|d/2 dx ) N̄ 0 (V)

and so N̄ 0 (V) ≤ (Kd /2π)d ωd ∫ |V(x)|d/2 dx ℝd

follows. Here numerically (Kd /2π)d ωd = 27/(2π 2 ) ≈ 1.37 for d = 3. The method in [238] gives the same estimate in this context, with the much better constant 0.116 when d = 3.

Notes to Chapter 4

| 531

Moments of negative eigenvalues N α (V) = ∑Nj=1 |Ej |α for d = 1, 2 are also investigated. The cases α > 1/2, d = 1, and α > 0, d ≥ 2, are proven by [246]. For α = 0, d ≥ 3, proofs are given independently in [68, 238, 309]. The remaining critical case α = 1/2, d = 1, is proven in [363]. An interesting problem is to determine sharp values of the constants ad and ad,α in Theorems 4.159 and 4.162, respectively. This is found for the cases in which α ≥ 3/2 and d ≥ 3 in [2, 234, 246], and α = 1/2 and d = 1 in [172]. The Lieb–Thirring inequality can be extended to a wide class of Schrödinger operators, including those with spin and magnetic field; we refer to [165, 241, 268] and references therein. The extension of the Birman–Schwinger principle is shown in Theorem 4.149, which is due to [268, 355]. Theorem 4.149 gives a sufficient condition such that − 21 Δ + V has no nonpositive eigenvalues. For d = 1 an upper bound on N0 (V) for V < 0 is studied in [43, 221], which is of the form N0 (λV) ≤ 1 + ∫ |x|λV(x)dx. ℝ

Whenever d = 1, 2 and under some conditions on V < 0, the operator − 21 Δ + λV has a negative eigenvalue no matter how small λ is. In [328] bound states of − 21 Δ + V in one and two dimensions are studied. Proposition 4.120 is from [299, Theorem XIII.52]. The absence of zero eigenvalues of Schrödinger operators is a problem of fundamental importance. Proposition 4.168 is due to [123], proven by using the continuity of solutions of Schrödinger operator for d ≥ 3, which is shown in [192]. For a more general case the absence of zero eigenvalues of Schrödinger operator is shown in [361]. The critical order of the absence of zero eigenvalues is V(x) = −r −2 , studied in [342]. There is a long history of confusion and controversy for time operators. The origin of this may come from Pauli [281, p. 63, footnote 2] and [280], from 1933. In this argument a time operator satisfying [H, T] = −i1 with Hamiltonian H having a discrete spectrum is basically forbidden. It should be noted however that the argument is very formal and no attention was paid to the domain of operators involved. In a rigorous manner the time operator associated with a self-adjoint operator with purely discrete spectrum is discussed in [9, 128]. In particular, Lemma 4.165 is due to [9]. Strong time operators associated with an abstract self-adjoint operator with purely absolutely continuous spectrum are initiated by [267] and developed in, e. g., [8]. For Lemma 4.163, see [8, 267]. A generalization of Theorems 4.169 and 4.172 is discussed in 2 [10, Theorem 5.4], where ∑∞ j=1 Ej < ∞ is not assumed and an ultraweak time operator associated with a Hamiltonian with purely discrete spectrum is constructed by an infinite direct sum of ultraweak time operators in [10, Theorem 5.2]. On Agmon potential, Lemma 4.170, we learn from [299, Theorem XIII.33, Agmon–Kato–Kuroda theorem]. Conditions in Theorem 4.172 are discussed in [299, Theorems XIII.6(a) and XIII56]. The modified wave operator in Example 4.174 is taken from [300, Theorem XI.71]. The Perron–Frobenius theorem first appeared in [126, 283] for matrices, and it was applied to quantum field theory in [137, 144, 145]. The Klauder phenomenon is discussed in [111, 112, 115, 327]. Proposition 4.126 is proven in [207]. Due to the Klauder

532 | 6 Notes and references phenomenon one can construct examples of Schrödinger operators with degenerate ground states. Exponential decay in L2 -sense, i. e., ‖ea|x| ϕ‖ < ∞ for eigenfunctions, is studied in [1], where operators in the divergence form −𝜕μ Aμν (x)𝜕ν + V are also considered. Pointwise exponential decay of eigenfuntions of Schrödinger operators is investigated in [59], and in [60] for the relativistic Schrödinger operator. In the massless relativistic case, i. e., for √−Δ + V, eigenfunctions decay only polynomially. For the divergence form −𝜕μ Aμν (x)𝜕ν + V with the uniform elliptic bound c ≤ |Aμν (x)| ≤ C and a further regularity condition on Aμν the pointwise exponential decay of eigenfunctions can be proven in a similar way to Carmona’s estimates with respect to the diffusion process associated with the divergence form. Fromula (4.3.85) gives the probability of the set of paths P(a, [c, d], t): 𝒲 (P(a, [c, d], t)) =

1 ∫ ∑∞ (−1)k √2πt [c,d] k=−∞

2

exp (− (u−2ka) )du, which 2t

is taken from [49, 1.15.8, p. 174]. We refer [59] for the pointwise lower bound of positive bound states discussed in Lemma 4.181 and Theorem 4.182. 4.4 References on Schrödinger operators with magnetic field include [15–17]; they are comprehensively studied in [297]. Self-adjointness of the Schrödinger operator with magnetic field H(a) is established in [236]. The alternative proof of the Feynman– Kato–Itô formula of the Schrödinger operator with magnetic field H(a) is taken from [331, Section V], where ∇ ⋅ a = 0 is assumed. The applications of functional integration with magnetic field and the diamagnetic inequality are from [52, 108, 109, 124, 173, 276]. 4.5 This material is taken from [336]. For some further cases of interest of unbounded Feynman-Kac semigroups see [53, 364]. 4.6 For results on the spectrum of the relativistic Schrödinger operator, see [22, 72, 73, 158, 237, 243]. The Feynman–Kac formula for relativistic Schrödinger operators HR (a) with a = 0 is studied in [60, 78]. The case of nonzero vector potential is discussed in [81, 162]. The function f (x) = √x 2 + m2 − m is an example of Bernstein functions. The relativistic Schrödinger operator is then realized as f ( 21 (−i∇ − a)2 ) + V. The relativistic Schrödinger operator can also be defined through the Weyl quantization. This is given by H w = hw (x, D) + V(x), where (hw u)(x) =

∫ ei(x−y)⋅ξ h( ℝd ×ℝd

x+y , ξ )u(y)(2π)−d dydξ , 2

Notes to Chapter 4

| 533

with symbol h(x, ξ ) = √|ξ − a(x)|2 + m2 . The work [271] shows that in general (hw (x, D))2 ≠ (−i∇ − a)2 + m2 . w

The Feynman–Kac formula of e−tH is given in [177]. 4.7 In quantum mechanics the notion of spin is introduced as an internal degree of freedom of electrons. Spin plays a central role in the statistics of electrons. Another aspect of the influence of spin is in the behavior of electrons in a magnetic field. The Feynman–Kac formula for the Schrödinger operator with spin is discussed in [79, 80, 129, 130, 162]. The diamagnetic inequality (4.7.35) is due to [80, 162]. Paramagnetism of Schrödinger operators with spin 1/2 is also an interesting subject. It has been conjectured in [168] that paramagnetism is universal, but a counterexample has been given in [18], while paramagnetism in the classical limit is discussed in [167]. For a two-dimensional space-time Dirac operator, [175–178] propose a path integral representation in terms of a S 󸀠 (ℝ2 ; M2 (ℂ))-valued countably additive measure on C([0, t]; ℝ). In particular, in [178], the measure is concentrated on paths having differential coefficients equal to “±1 × speed of light” in every finite time interval except at finitely many instants of time. This relates with the Zitterbewegung of Dirac particles. A similar result is also given in [79] with respect to a Poisson process. Furthermore, [169–171, 248] treat quantum probability and derive a noncommutative Feynman–Kac formula. The Rabi Hamiltonian is a model of a harmonic oscillator coupled to a twolevel atom, and it is given by Δσz + ωa∗ a + gσx (a + a∗ ), where Δ ≥ 0 and ω > 0 are positive constants, g ∈ ℝ is a parameter, and a=

1 1 d ( + √ωx), √2 √ω dx

a∗ =

1 1 d (− + √ωx). √2 √ω dx

This was introduced in [296] and the integrability is studied in [51] from a physical point of view. Applications of the Rabi model range from quantum optics to molecular physics, and recently it plays an important role in cavity QED and circuit QED. It is a self-adjoint operator on ℂ2 ⊗ L2 (ℝ). Let {En (g)}∞ n=0 be the discrete spectrum of the Rabi Hamiltonian. Then we can define spectral curves g 󳨃→ En (g). It is interesting to consider crossings of En (g) and En+1 (g) under varying g. We can construct a Feynman– Kac-type formula of the semigroup generated by the Rabi Hamiltonian and show that the ground state is unique [160], which implies that E0 (g) has no crossing at least. The infinite-dimensional version of the Rabi model is the so-called spin-boson model. This is also studied in [161] and the P(ϕ)1 process associated with a spin-boson model is constructed.

534 | 6 Notes and references 4.8 A Feynman–Kac-type formula for relativistic Schrödinger operators with spin HRS (a, b) = √(−i∇ − a)2 − σ ⋅ b + m2 − m + V is from [81, 162, 163], where a combination of a stopping time (subordinator) and a Poisson process is applied. The Schrödinger operator with spin can be transformed to the operator on L2 (ℝ3 × ℤ2 ), which is from [81]. In [163] unbounded magnetic fields b ∈ ̸ (L∞ (ℝ3 ))3 are also studied, and by extending the strategy developed in [60] we can establish decay properties of the eigenfunctions of HRS (a, b). 4.9 Nonlocal equations, and related operators and jump random processes are currently much investigated in analysis and probability, attracting a wide range of applications. Nonlocal Schrödinger operators can be defined in the spirit of classical Schrödinger operators by adding a nonlocal kinetic term to a multiplication operator playing the role of a potential. In [162] a class of nonlocal Schrödinger operators has been introduced by choosing the kinetic term to be a Bernstein function of the Laplacian, and adding further possible terms like vector potentials and generalized spin leading to 1 Ψ( (−i∇ − a)2 ) + V 2 or 1 Ψ( (σ ⋅ (−i∇ − a))2 ) + V 2 with Bernstein function Ψ. The operator Ψ( 21 (σ ⋅ (−i∇ − a))2 ) + V can be realized as a self-adjoint operator in L2 (ℝ3 × ℤ2 ). A general spin can be defined on L2 (ℝd × {eθp , e2θp , . . . , epθp }),

θp = 2πi/p.

The idea to construct Feynman–Kac formulae is from [78–81]. In [165] Lieb–Thirring inequalities for nonlocal Schrödinger operators (Theorem 4.314) are given, see also [72] for the relativistic Schrödinger operator. Proposition 4.278 is a modification of [60, Theorem III.1] and Proposition 4.279 is taken from [60, Lemma III.3]. We refer to [189] for volume doubling in Lemma 4.315. Bounds on pΨ t (x), see (4.9.78), are given in [66], and (4.9.79) in [65]. The kernel of (−Δ)−α/2 and Riesz potentials are studied in, e. g., [348, Chapter V], [242, Chapter 5], and [47, 139]. We refer to [340, Section 6.8] and [231, Chapter I] for the case of α = d. The massless relativistic Schrödinger operator in Section 4.9.9 is due to [255]. Eigenfunction decay for more general nonlocal Schrödinger operators with confining or decaying potentials discussed in Section 4.9.8 has been investigated in the

Notes to Chapter 5

| 535

papers [200, 201, 203]. For a discussion of the qualitative regime change (phase transition in the decay rates) see also [202]. The framework of jump-paring processes has been first proposed in [201] and further developed in [203]. The explicit solution of the eigenvalue problems for the one-dimensional harmonic relativistic oscillator is presented in [255], and of the quartic anharmonic relativistic oscillator in [104]. Embedded eigenvalues for non-local Schrödinger operators just started to be studied. A standard account for classical Schrödinger operators in [106]. In [257] relativistic counterparts of the classical von Neumann-Wigner and Moses-Tuan potentials have been obtained, showing that in the nonrelativistic limit the potentials obtained for the relativistic case converge to the classically known potentials. Here also two families of potentials have been constructed for which the related massless relativistic Schrödinger operators have zero eigenvalues. For the delicate case of zero-eigenvalues a probabilistic study of eigenfunction decay has been done in [205] also throwing light on the mechanisms of bound state formation at the continuum edge. For a study of some local properties of eigenfunctions of nonlocal Schrödinger operators with potential wells see [40]. The same reference also discusses the location of extrema of eigenfunctions of nonlocal Schrödinger operators for bounded domains and its multiple consequences such as a generalized Faber-Krahn inequality; for further developments, see also [41, 39].

Notes to Chapter 5 5.1 Ground state transform is a case of Doob’s h-transform, making the relationship between a classical Schrödinger operator and a diffusion. For basic references in this context see [331, 287]. An extension to fractional Schrödinger operators and stable processes has been first made in [199, 200], and for more general nonlocal Schrödinger operators in [258]. For intrinsic ultracontractivity properties, see the basic references [74, 75, 77]. Asymptotic intrinsic ultracontractivity has been introduced in [200]. For a comprehensive study of intrinsic ultracontractivity in the context of nonlocal Schrödinger operators we refer to [201], where also the literature is discussed. The relationship between these and other contractivity properties (supercontractivity, hypercontractivity) has been explored in [206]. For P(ϕ)1 -processes a first source is [308]. Such processes have an interesting property deriving from connections with statistical mechanics and Gibbs measures. Formally such a process is the stationary solution of the stochastic partial differential equation (SPDE) ̇ t) = 1 Δu X(u, t) − (𝜕V)(X(u, t)) + 𝒲 , X(u, 2

(6.0.1)

536 | 6 Notes and references where 𝒲 now is space-time white noise. This was shown rigorously for a restricted class of potentials V in [187]. An interesting application appears in [151], where the SPDE above is used in a Markov chain Monte Carlo algorithm for sampling the conditional P(ϕ)1 -measure in various situations. In the context of SPDEs the result of [35] is also of interest: when the potential grows sufficiently rapidly, it implies uniqueness of those stationary measures of the SPDE above that are Gibbs measures, i. e., satisfy the Dobrushin–Lanford–Ruelle (DLR) equations. It is expected, but not proven, that all stationary measures of the above SPDE satisfy the DLR equations. The P(ϕ)1 -process associated with the divergence form d

− ∑ 𝜕i aij (x)𝜕j + V(x) i,j=1

with a confining potential can be constructed, and also that of −Δ + V(x). We refer to, e. g., [135]. Lemma 5.24 is taken from [135], which is a modification of [220]. The proof of Mehler’s formula through analytic continuation t 󳨃→ −it has been first obtained in [251]. Lemma 5.26 can be found in [261]. Càdlàg P(ϕ)1 -processes can be further constructed for jump processes and nonlocal Schrödinger operators. Fractional P(ϕ)1 -processes associated with fractional Schrödinger operators have been investigated in [200], and with more general nonlocal Schrödinger operators in [258]. For the long-time fluctuation properties of such processes see [204], and for refined results on their path regularity (multifractal spectrum of local Hölder-exponents) we refer again to [258]. 5.2 The seminal papers introducing Gibbs measures for lattice spin systems are [92, 93] and [233]. Further details on the prescribed conditional probability kernels (called specifications) can be found in [94, 122] and [138, 291, 345]. Surveys and books on Gibbs measures include the early [318], covering also classical continuous and quantum lattice systems, and [131, 132, 232, 290, 292, 295, 317, 346, 347]. In [291] a general framework is developed, while the approach of [183] is more from the perspective of the variational principle in which equilibrium states appear as tangents to the free energy functional. Reference [133] is a highly authoritative comprehensive monograph with a strong stress on DLR theory, covering a wealth of material. The work [260] is a systematic text on cluster expansion methods applied to Gibbs measures. Other relevant books on rigorous statistical mechanics include [146], focusing more on Ising models, [341], giving an early account of contour techniques, and [335], giving an account of lattice gases. A reappraisal of Gibbsian theory was presented in the book-length paper [362], especially in the context of renormalization group transformations. For Gibbs measures on lattice spin systems with noncompact space existence of a Gibbs measure is harder to prove: see the superstability conditions in [316], tempered Gibbs measures [63, 235], [249, 253, 254] for cases when the potential is not uniformly but

Notes to Chapter 5

| 537

pointwise or almost surely summable, and [134] for Gibbs measures indexed by subsets of an uncountable index set. For other approaches to and applications of Gibbs measures on path space(s) we refer to [5] and the references therein. 5.3 Some of the basic notions, such as local uniform domination, are adapted from discrete models; see [133]. The discussion of uniqueness of Gibbs measures follows [35]. The case α = 1 in the uniqueness criterion suggests that a maximal configuration space on which μ is the unique Gibbs measure, a set consisting of exponentially growing paths is closer to truth than X0 ; however, establishing this is beyond the method of our proof. In [187] sets of exponentially growing paths as sets of uniqueness for the Gibbs measure were obtained, however, assuming convexity of the function V(x) − κ|x|2 for some κ > 0; therefore the case α < 1 is not covered in this work. The advantage of our method is that it is not limited to polynomials, nor even to continuous or semibounded potentials; in particular, local singularities and other perturbations of V in the above examples will not alter the conclusions. This corresponds to the intuition that only the behavior of the potential at infinity should determine whether uniqueness of the Gibbs measure holds on a given set. For existence, uniqueness and support properties of Gibbs measures on càdlàg paths we refer to [200, 204, 258]. 5.4 An early variant of the existence proof for Gibbs measures with pair interaction potential comes from [33]. For further literature on Gibbs measures with pair interaction potential we refer to [33, 36, 157, 278, 34]. 5.5 There are several versions of the cluster expansion method. A systematic monograph is [260] and useful reviews are [54, 114]. For the general combinatorial scheme, see [113, 225, 288]. We refer to [28, 50, 55, 95, 266, 284] for improvements on convergence; however, these methods depend on the specific models at hand. For Brownian paths a cluster expansion technique has been developed first in [250, 256]. More singular objects, such as densities dependent on double stochastic integrals are treated in [147, 252] using techniques of rough paths.

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Index α-stable process 157, 158, 164, 274, 401, 484 π-λ theorem 11 π-system 10 σ-field 9 σ-finite measure 9 A-compact 232 A-form-bounded 237 A-form-compact 239 absence of ground state – criterion 279 absolutely continuous spectrum 225 abstract Kato’s inequality 527 action functional 2 adapted process 50 adjoint operator 221 Agmon potential 312 Aharonov–Bohm operator 308 Airy differential equation 432 Airy functions 433 – fourth order 436 algebra of bounded continuous functions 145 algebra of events 16 asymptotically ultracontractive semigroup 450 augmented filtration 53 – Brownian motion 92 Bernstein function 209, 532 – characterization 213 Bessel function – first kind 393 – modified – first kind 286, 289 – third kind 256, 273, 289, 291, 352 Bessel function of the first kind 393 Bessel potential 257 Bessel process 138, 285 bipolaron model 484 Birman–Schwinger kernel 291, 293 Birman–Schwinger principle 291, 294 – extended 296, 413 Blumenthal’s zero–one law 92 Bochner’s theorem 145 Borel σ-field 9 Borel isomorphic 12 Borel measurable function 12

Borel measurable space 9 Borel–Cantelli lemma 17 bound state 201, 289 bounded operator 218 bridge measure – Brownian 136 – Itô 494 – Lévy process 178 Brownian bridge 134, 136 Brownian bridge measure 88, 136 Brownian motion 4, 77 – d-dimensional 78 – local time 206 – occupation time 283 – over ℝ 84 – standard 78 Brownian path 78 Burkholder–Davis–Gundy inequality 127 C0 -semigroup 73, 74, 76, 162, 163, 243 – generator 243 – on C∞ (ℝd ) 162 – on L2 (ℝd ) 162 càdlàg P(ϕ)1 -process 481 càdlàg paths 165 càdlàg version 168 càglàd paths 165 càglàd version 168 Calderón norm 230 Calkin algebra 228 Cameron–Martin formula 142 canonical commutation relation 307 – weak 307 capacitary measure 203 capacity 203 Carmona’s estimate 315 – divergence form 532 Cauchy distribution 144 CCR 307 chain 514 Chapman–Kolmogorov identity 68, 70 characteristic function 143 Chung–Fuchs criterion 205 class D 66 class DL 66 closable operator 219

554 | Index

closed operator 219 cluster 514 cluster expansion 511 cluster representation 515 compact operator 227, 241 – |V |1/2 (−Δ + m2 )−1 |V |1/2 291 – A- 232 compensated Poisson process 160 compensator 160, 188 completely monotone function 209 – characterization 209 compound Poisson distribution 144 compound Poisson process 157, 186, 187 compound Poisson random variable 144 conditional expectation 38 conditional probability 41 confining potential 240, 316 conjugate 223 consistency relation 42, 43 – distribution functions 46 – Kolmogorov 80 constant drift 187 continuous spectrum 221 continuous time random process 45 continuous version 48, 113, 460 contour 514 contraction operator 218 convergence – almost sure 32 – in distribution 32 – in Lp 32 – in law 32 – in probability 32 – weakly 32 convergence of submartingale 56 convolution 30 coordinate process 78 core 219 correlation function associated with a cluster 516 countable product of probability measures 27 countably determined 17 counting measure 180, 188, 361 covariance 29 cylinder set 27 decay property – lower bound 320, 355 – upper bound 314, 316, 353, 373, 386, 387

decaying potential 242, 316 densely defined operator 218 diamagnetic inequality – nonlocal Schrödinger operator 390 – Ψ-Kato-decomposable potential 400 – generalized spin 411 – relativistic Schrödinger operator 344 – relativistic Kato-decomposable potential 348 – spin 1/2 379 – Schrödinger operator 333 – Kato-decomposable potential 334 – spin 1/2 370 diffusion matrix 156 diffusion process 132, 458 – scale function 139 diffusion term 118 Dirac operator – two-dimensional space-time 533 Dirichlet principle 465 discrete spectrum 224 discrete-time random process 45 distribution 29 distribution function 14, 45 DLR equation 536 Dobrushin–Lanford–Ruelle construction 483 Dobrushin–Lanford–Ruelle equation 488 dominated convergence theorem 34 Doob–Kolmogorov inequality 63 Doob–Meyer decomposition 67, 188 Doob’s h-transform 450 Doob’s martingale 59 double stochastic integral 485 doubly substochastic 231 drift term 118, 156 drift transformation 141 dyadic rationals 49, 99 Dynkin–Kinney condition 167 Dynkin’s formula 133 eigenfunction 221 eigenvalues 221 eigenvector 221 embedded eigenvalue 225 ergodic map 496 ergodic theorem 496 essential spectrum 224 expectation 29 extension of operator 218 external potential 486

Index | 555

Feller transition kernel 72, 162 Feynman–Kac formula 3 – càdlàg P(ϕ)1 -process 481 – e−E(−i∇) 273 – noncommutative 533 – nonlocal Schrödinger operator 389 – Ψ-Kato-decomposable potential 398 – Ψ-Kato-decomposable potential and vector potential 399 – singular external potential and singular vector potential 392 – singular vector potential 390 – vector potential and generalized spin 410 – P(ϕ)1 -process 463 – relativistic Schrödinger operator – relativistic Kato-decomposable potential and vector potential 346 – singular external potential, singular vector potential and spin 379 – singular external potential and singular vector potential 343 – singular vector potential and spin 378 – vector potential 340 – vector potential and spin 376 – Schrödinger operator 246 – Kato-decomposable potential 269 – Kato-decomposable potential and vector potential 333 – singular external potential 249 – singular external potential, singular vector potential and spin 371 – singular external potential and singular vector potential 331 – singular vector potential and spin 369 – vector potential 321 – vector potential and spin 363 – unbounded semigroup 337 Feynman–Kac semigroup 248 – intrinsic 449 Feynman–Kac–Itô formula 321 Feynman’s integral 3 field 10 filtered space 50 filtration 50 – natural 50 finite measure 9 form core 237 form domain 237 fractional Kato-class 401

fractional Kato-decomposable 401 fractional Laplacian 226 – relativistic 226 fractional Schrödinger operator 484 full Wiener measure 88 functional central limit theorem 77 Gaussian distribution 143 Gaussian random variable 30, 469 – multivariate 30 – standard 30 generalized spin operator 406 generator – α-stable process 164 – Brownian motion 96 – Brownian motion with drift 164 – C0 -semigroup 243 – Lévy process 163 – Markov process 74 – Poisson process 164 Gibbs measure 487 – boundary condition 487 – existence 504, 512 – external potential 486 – finite volume 487 – P-admissible external potential 486 – pair interaction potential 486 – partition function 487 – reference measure 485 – sharp boundary condition 489 – stochastic boundary condition 489 – uniqueness 519 Girsanov theorem 140 graph 219 Gronwall inequality 131 ground state 5, 271 – degenerate 277 – domination 453 – multiplicity 271 – uniqueness 271 ground state energy 271 ground state transform 450, 468 ground state-transformed process 454 Hamilton operator 1 Hamiltonian 2 Hardy–Littlewood–Sobolev inequality 228, 292, 402

556 | Index

harmonic oscillator 134, 242, 467, 497 – fermionic 363 – relativistic 432 Harnack inequality 529 heat equation with dissipation 4 heat kernel 5, 78, 243 Helly’s second theorem 14 Helly’s selection theorem 14 Hermite polynomial 467 Hilbert–Schmidt class 229 Hille–Yosida theorem 163, 244 hitting time 53 Hölder constant 48 Hölder continuity 48, 97 Hölder exponent 48 Hölder inequality for trace ideals 230 Hopf’s extension theorem 42 hydrogen atom 242 identically distributed 29 image measure 29 increasing process 66 – integrable 66 independent – events 17 – random variables 30 – sub-σ-fields 17 infinitely divisible – probability 143 – process 143 infinitesimally small 233 initial distribution 458, 477 – Markov process 69 injective 219 integrable increasing process 66, 188 integral kernel 1, 227 – (−Δ + λ)−1 256 – (−Δ + m2 )−1 291 – e−tE(−i∇) 273 intensity measure 181 intersection local time 485 intrinsic ultracontractivity 499, 512 intrinsically ultracontractive semigroup 450 invariant measure 74, 463 inverse Gaussian subordinator 208 inverse operator 219 invertible operator 219 Itô bridge measure 494 Itô diffusion 132

Itô formula – Brownian motion 119 – Itô process 119 – product 125 – rules 125 – semimartingale 195, 362 Itô integral 111, 117 Itô isometry 110, 112 Itô measure 495 Itô process 118, 119 Jensen’s inequality 51, 417 jump process 361 Kallianpur–Robbins laws 283, 285 Kato–Rellich theorem 234 Kato-class 251, 477, 500 – Ψ 397 – fractional 401 – local 252 – relativistic 345 Kato-decomposable 252 – Ψ 397 – fractional 401 – relativistic 345 Kato’s inequality 236 – abstract 527 – relativistic Schrödinger operator 527 Khasminskii’s lemma 261 Klauder phenomenon 277 KLMN theorem 237 Kolmogorov consistency relation 24 Kolmogorov extension theorem 25, 47 Kolmogorov–Čentsov theorem 48 Kolmogorov’s zero–one law 18 Lagrangian 2 λ-system 10 Langevin equation 133 Laplace–Beltrami operator 285 Laplacian 225, 226 law of large numbers 36 law of the iterated logarithm 105 Lebesgue–Stieltjes measure 46 Lévy bridge measure 178 Lévy measure 149, 156 – Bernstein functions 213 – conditional 178

Index | 557

Lévy process 154, 361 – bridge 492 – characteristics 156 – generator 163 – jump-paring 423, 424 – occupation time 204 – over ℝ 169 – potential measure 204 – recurrent 204 – transient 204 Lévy symbol 156 Lévy–Itô decomposition 187 Lévy–Khintchine formula 149 Lévy’s upward theorem 58 Lieb–Thirring inequality 303, 305, 310, 534 – fractional relativistic Schrödinger operator 420 – fractional Schrödinger operator 419 – jump-diffusion operator 422 – nonlocal Schrödinger operator 416 – relativistic Schrödinger operator 421 – sums of different stable generators 421 linear SDE 133 local convergence of measures 489 local Kato-class 252 local linear operator 388 local martingale 118 local time 206 locally γ-Hölder continuity 48 locally finite measure 9 locally uniform topology 15 locally uniformly dominated probability measures 491 Lp -convergence 32 Lp -Lq boundedness 265 – nonlocal Schrödinger operator with Ψ-Kato-decomposable potentials and vector potentials 400 – relativistic Schrödinger operator with relativistc Kato-decomposable potentials and vector potentials 348 – Schrödinger operator with Kato-decomposable potentials and vector potentials 333 magnetic field 360 Markov process 67 – generator 74 – reversible 72 – stationary 72, 161

martingale 50, 112, 351, 374, 383 – compensated Poisson process 161 – inequality 62 – local 118 martingale inequality 104 martingale transformation 55 massive relativistic quantum particle 484 massless relativistic quantum particle 484 measurable 12 measurable space 9 measure 9 – σ-finite 9 – finite 9 – locally finite 9 measure space 9 Mehler’s formula 467, 473 min-max principle 226 Mittag-Leffler function 263 modified Bessel function of the first kind 286 modified Bessel function of the third kind 256, 273, 291, 352 modulus of continuity 22 moments of negative eigenvalues 531 monotone convergence theorem 34 monotone convergence theorem for forms 238 monotone family 10 multiplication operator 225 multiplicity 221, 271 natural filtration 50 natural integrable increasing process 188 natural process 66 Nelson model 484 Neumann–Wigner potential 437 Newton potential 404 non-relativistic limit 356 noncommutative Feynman–Kac formula 533 nonlocal Schrödinger operator 389 – Bernstein function of the Laplacian 389 – Ψ-Kato-decomposable potential and vector potential 399 – Ψ-Kato-decomposable potential 399 – singular external potential and singular vector potential 391 – vector potential 389 – vector potential and generalized spin 407 nowhere differentiability 99 occupation time 204

558 | Index

operator – A-bounded 233 – bounded 218 – closable 219 – closed 219 – closure 219 – conjugate 223 – contraction 218 – core 219 – densely defined 218 – extension 218 – injective 219 – inverse 219 – invertible 219 – product 220 – relatively compact 232 – restriction 218 – strongly continuous 218 – sum 220 – unbounded 218 optional sampling theorem – continuous 60 – discrete 55, 59 Ornstein–Uhlenbeck process 134, 456, 468, 498 P-admissible external potential 486 p-variation 100 pair interaction potential 486, 512 partition function 487 Pauli matrix 360 Pauli–Fierz model 485 Perron–Frobenius theorem 274 P(ϕ)1 -process 459, 483, 495, 535 – càdlàg 481 – divergence form 536 – fractional 481 point function 188 point process 188 point spectrum 221, 225 Poisson distribution 144 Poisson point process 188 Poisson process 157, 182, 361 – compensated 160, 183 Poisson random measure 179, 188 polarization identity 224 polaron model 484 Polish space 14 positive function 272

positivity improving 272 – relativistic Schrödinger operator 344 – Schrödinger operator 274 positivity preserving 272 potential 225, 232 – Bessel 257 – confining 240 – decaying 240 – Newton 404 – Riesz 257, 404 potential measure 204 potential well 282 predictable 191 predictable process 189 principle of least action 2 probability measure 16 probability space 16 probability transition kernel 68, 158, 161, 392 product Itô formula 125, 200 projection-valued measure 224 quadratic form 237 – semibounded 237 – symmetric 237 quadratic variation 100 quantum probability 533 quasi-left-continuous 188 Rabi Hamiltonian 533 radially nonincreasing 393 Radon measure 13 Radon–Nikodým derivative 39 random measure 178 random process 45 – continuous-time 45 – discrete time 45 random step process 109 random variable 29 Rayleigh–Ritz method 227 recurrent 204, 412 reference measure 485 Reflection Principle 96, 105 reflection symmetry 84, 458, 477 regular conditional probability measure 41 – existence 41, 44 – given G 41 – given X 41 regular measure 15 relative bound 233

Index | 559

relative compactness of probability measures 20 relatively compact operator 232 relatively form-bounded 237 relatively form-compact 239 relativistic fractional Laplacian 226 relativistic Kato-class 345 relativistic Kato-decomposable 345 relativistic Schrödinger operator – relativistic Kato-decomposable potential and vector potential 346 – singular external potential, singular vector potential and spin 380 – singular external potential and singular vector potential 343 – vector potential 339 – vector potential and spin 375 Rellich’s criterion 240 residual spectrum 221 resolvent 220 resolvent kernel 203 resolvent operator 203 resolvent set 220 restriction of operator 218 reversible Markov process 72 Riemann integral 107 Riesz potential 257, 404 Riesz theorem 74 Riesz–Markov representation theorem 20 Riesz–Schauder theorem 227 Riesz–Thorin theorem 266 right-continuous filtration 50 rough paths 537 rules of Itô differential calculus 125 sample path 45, 97 sample space 16 scale function 139 Schatten class 229 Schrödinger equation 1 Schrödinger operator 5, 232, 233 – Kato-decomposable potential 269 – Kato-decomposable potential and vector potential 333 – singular external potential, singular vector potential and spin 371 – singular external potential and singular vector potential 327, 331 – vector potential 320

– vector potential and generalized spin 407 – vector potential and spin 360 Schrödinger semigroup 245 Schwartz space 16 self-adjoint operator 221 – essential 221 – positive 223 self-similarity 84 semi-ring 11 semibounded quadratic form 237 semigroup – asymptotically ultracontractive 450 – hypercontractive 453 – intrinsically hypercontractive 453 – intrinsically supercontractive 453 – intrinsically ultracontractive 450 – supercontractive 453 – ultracontractive 450 semimartingale 194, 195 separable 9 separating points 145 set-algebra 10 sharp boundary condition 489 shift invariance – fractional p(ϕ)1 -process 477 – p(ϕ)1 -process 459 shrinking operator 503 singular continuous spectrum 225 singular spectrum 225 Skorokhod topology 169 smoothing effect 266, 347, 399 Sobolev space 419 spatial homogeneity 84 spectral gap 434, 453, 500 spectral resolution 223, 224 spectral theorem 223 spectrum 220 – absolutely continuous 225 – continuous 221 – discrete 224 – essential 224 – point 221 – residual 221 – singular 225 – singular continuous 225 spin process 174 spin-boson model 533 stable process 157, 274 – index 157

560 | Index

stable random variable 157 stable subordinator 208 standard Brownian motion 78 standard measurable space 12, 41 Stark Hamiltonian 336 state space 188, 485, 488 stationary 188 Stieltjes integral 107 stochastic boundary condition 489, 504 stochastic continuity 154 – uniform 168 stochastic differential equation 128 – strong solution 128 – weak solution 136 stochastic differentials 118 stochastic integral 111 – integration by parts 125 stochastic partial differential equation 535 Stone–Weierstrass theorem 145, 298 Stone’s theorem 244 – semigroup version 244 stopped process – continuous time 63 – discrete time 55 stopping time 53 Stratonovich integral 115, 126, 322 strictly positive function 272 strong convergence 219 strong law of large numbers 37, 104 strong Markov property – Brownian motion 94 – Lévy process 164 strong resolvent convergence 223 strong solution 128 strong time operator 308 strongly continuous operator 218 strongly continuous semigroup 73 Stummel-class 260 submartingale 50 subordinate Brownian motion 209 subordinator 206 – characterization 206 – inverse Gaussian 208 – stable 208 super-quadratic potential 501 supermartingale 50 symmetric difference 11 symmetric operator 221 symmetric quadratic form 237

tail σ-field 17 Tanaka equation 137 telegraph process 174 thermodynamic limit 483 tightness 20, 148 time inversion 84 time operator 307 – strong 308 – ultrastrong 308 – ultraweak 307 – weak 307 time reversibility 84 time-shift invariance – two-sided Brownian motion 86 – two-sided Lévy process 170 total variation 100 trace 229 trace class 229 trace ideal 229 – duality 230 transient 204 Trotter product formula 249 turbulent fluids 485 ultracontractive semigroup 450 ultrastrong time operator 308 ultraweak CCR domain 308 ultraweak time operator 307 unbounded operator 218 uncertainty principle 238 uniform convergence 219 uniform integrability 32 uniform resolvent convergence 223 uniformly bounded sequene of measures 13 uniformly locally integrable 345, 394 unitary operator 223 vague convergence 13 version of random process 48 volume doubling 418 von Neumann uniqueness theorem 308 wave function 1 weak canonical commutation relation 307 weak convergence – linear operators 219 – probability measures 13 weak Lp -space 228 weak partial derivative 225

Index |

weak solution 136 weak time operator 307 weak trace ideal 230 weak Weyl relation 308 weak Young inequality 229 Weyl quantization 532 Weyl relation 308 – weak 308 Weyl’s law 434 white noise 536 Wick product 126

Wiener measure 4, 78 – conditional 88 – full 88 Wiener process 77 Young’s inequality 228 – weak 229 zero eigenvalue 296, 311, 531 – absence 296, 311

561

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