Bernstein Functions: Theory and Applications [2nd rev. and ext. ed.] 9783110269338, 9783110252293

Table of contents :
Preface to the second edition
Preface
Index of notation
1 Completely monotone functions
2 Stieltjes functions
3 Bernstein functions
4 Positive and negative definite functions
5 A probabilistic intermezzo
6 Complete Bernstein functions
6.1 Representation of complete Bernstein functions
6.2 Extended complete Bernstein functions
7 Properties of complete Bernstein functions
8 Thorin-Bernstein functions
9 A second probabilistic intermezzo
10 Transformations of Bernstein functions
11 Special Bernstein functions and potentials
11.1 Special Bernstein functions
11.2 Hirsch’s class
12 The spectral theorem and operator monotonicity
12.1 The spectral theorem
12.2 Operator monotone functions
13 Subordination and Bochner’s functional calculus
13.1 Semigroups and subordination in the sense of Bochner
13.2 A functional calculus for generators of semigroups
13.3 Subordination and functional inequalities
13.4 Eigenvalue estimates for subordinate processes
14 Potential theory of subordinate killed Brownian motion
15 Applications to generalized diffusions
15.1 Inverse local time at zero
15.2 First passage times
16 Examples of complete Bernstein functions
16.1 Special functions used in the tables
16.2 Algebraic functions
16.3 Exponential functions
16.4 Logarithmic functions
16.5 Inverse trigonometric functions
16.6 Hyperbolic functions
16.7 Inverse hyperbolic functions
16.8 Gamma and related special functions
16.9 Bessel functions
16.10 Miscellaneous functions
16.11 CBFs given by exponential representations
16.12 Additional comments
Appendix
A.1 Vague and weak convergence of measures
A.2 Hunt processes and Dirichlet forms
Bibliography
Index

Citation preview

De Gruyter Studies in Mathematics 37 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Rene´ L. Schilling Renming Song Zoran Vondracˇek

Bernstein Functions Theory and Applications

2nd Edition

De Gruyter

Rene´ L. Schilling Institut für Stochastik TU Dresden 01062 Dresden Germany Renming Song Department of Mathematics University of Illinois Urbana, IL 61801 USA Zoran Vondracˇek Department of Mathematics University of Zagreb 10000 Zagreb Croatia Mathematics Subject Classification 2010: 26-02, 30-02, 31Bxx, 31Cxx, 44Axx, 47Dxx, 60Exx, 60Gxx, 60Jxx.

ISBN 978-3-11-025229-3 e-ISBN 978-3-11-026933-8 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. ” 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface to the second edition

We were pleasantly surprised by the positive responses to the first edition of this monograph and the invitation of our publisher, de Gruyter, to prepare a second edition. We took this opportunity to revise the text and to rewrite some portions of the book. More importantly, we have added a substantial amount of new material. Our aim was to retain the overall structure of the text with theoretical material in Chapters 1–11 and more specific applications in Chapters 12–15. The tables are now in Chapter 16. Let us highlight some of the most important additions: we included a new chapter on transformations of Bernstein functions, Chapter 10. This was motivated by the continuing interest in subclasses of infinitely divisible distributions. Our treatment focusses on the corresponding Laplace exponents. A new section on subordination and functional inequalities, Section 13.3, investigates the stability of Nashtype and related inequalities under subordination. Over the last decade this has been an active field of research. We have also updated the tables and added a new section on complete Bernstein functions which are given by an exponential representation, Section 16.11. These changes led to a shift in the numbering of the chapters. Among the many smaller changes and additions let us mention multiply monotone functions and their integral representation in Chapter 1, which allow us to characterize multiply self-decomposable completely monotone functions in Chapter 5. Chapter 6 now contains a proof of the integral representation of Nevanlinna–Pick functions as well as a complete treatment of extended complete Bernstein functions. Because of this we decided to divide Chapter 6 into two sections. We thank our readers for many comments and suggestions which helped us in the preparation of this new version. We are particularly grateful to Dave Applebaum, Christian Berg, Sonia Fourati, Wissem Jedidi, Mateusz Kwa´snicki, Michael Röckner, Ken-iti Sato and Thomas Simon. We also thank Christoph von Friedeburg and other de Gruyter staff for suggesting this new edition; and the copy-editors, typesetters and printers whose skills have impressed us. Having the continuing support and understanding of our families helped us to finish the book. Dresden, Urbana and Zagreb December 2011

René L. Schilling Renming Song Zoran Vondraˇcek

Preface

Bernstein functions and the important subclass of complete Bernstein functions appear in various fields of mathematics – often with different definitions and under different names. Probabilists, for example, know Bernstein functions as Laplace exponents, and in harmonic analysis they are called negative definite functions. Complete Bernstein functions are used in complex analysis under the name Pick or Nevanlinna functions, while in matrix analysis and operator theory, the name operator monotone function is more common. When studying the positivity of solutions of Volterra integral equations, various types of kernels appear which are related to Bernstein functions. There exists a considerable amount of literature on each of these classes, but only a handful of texts observe the connections between them or use methods from several mathematical disciplines. This book is about these connections. Although many readers may not be familiar with the name Bernstein function, and even fewer will have heard of complete Bernstein functions, we are certain that most have come across these families in their own research. Most likely only certain aspects of these classes of functions were important for the problems at hand and they could be solved on an ad hoc basis. This explains quite a few of the rediscoveries in the field, but also that many results and examples are scattered throughout the literature; the exceedingly rich structure connecting this material got lost in the process. Our motivation for writing this book was to point out many of these connections and to present the material in a unified way. We hope that our presentation is accessible to researchers and graduate students with different backgrounds. The results as such are mostly known, but our approach and some of the proofs are new: we emphasize the structural analogies between the function classes which we believe is a very good way to approach the topic. Since it is always important to know explicit examples, we took great care to collect many of them in the tables which form the last part of the book. Completely monotone functions – these are the Laplace transforms of measures on the half-line Œ0; 1/ – and Bernstein functions are intimately connected. The derivative of a Bernstein function is completely monotone; on the other hand, the primitive of a completely monotone function is a Bernstein function if it is positive. This observation leads to an integral representation for Bernstein functions: the Lévy–Khintchine formula on the half-line Z .1  e t / .dt /;  > 0: f ./ D a C b C .0;1/

Preface

vii

Although this is familiar territory to a probabilist, this way of deriving the Lévy– Khintchine formula is not the usual one in probability theory. There are many more connections between Bernstein and completely monotone functions. For example, f is a Bernstein function if, and only if, for all completely monotone functions g the composition g ı f is completely monotone. Since g is a Laplace transform, it is enough to check this for the kernel of the Laplace transform, i.e. the basic completely monotone functions g./ D e t , t > 0. A similar connection exists between the Laplace transforms of completely monotone functions, that is, double Laplace or Stieltjes transforms, and complete Bernstein functions. A function f is a complete Bernstein function if, and only if, for each t > 0 the composition .t C f .//1 of the Stieltjes kernel .t C /1 with f is a Stieltjes function. Note that .t C /1 is the Laplace transform of e t and thus the functions .t C /1 , t > 0, are the basic Stieltjes functions. With some effort one can check that complete Bernstein functions are exactly those Bernstein functions where the measure  in the Lévy–Khintchine formula has a completely monotone density with respect to Lebesgue measure. From there it is possible to get a surprising geometric characterization of these functions: they are non-negative on .0; 1/, have an analytic extension to the cut complex plane C n .1; 0 and preserve upper and lower half-planes. A familiar sight for a classical complex analyst: these are the Nevanlinna functions. One could go on with such connections, delving into continued fractions, continue into interpolation theory and from there to operator monotone functions . . . Let us become a bit more concrete and illustrate our approach with an example. The fractional powers  7! ˛ ,  > 0, 0 < ˛ < 1, are easily among the most prominent (complete) Bernstein functions. Recall that Z 1 ˛ ˛ .1  e t / t ˛1 dt: (1) f˛ ./ WD  D .1  ˛/ 0 Depending on your mathematical background, there are many different ways to derive and to interpret (1), but we will follow probabilists’ custom and call (1) the Lévy– Khintchine representation of the Bernstein function f˛ . At this point we do not want to go into details, instead we insist that one should read this formula as an integral representation of f˛ with the kernel .1  e t / and the measure c˛ t ˛1 dt . This brings us to negative powers, and there is another classical representation Z 1 1 ˇ  D e t t ˇ 1 dt; ˇ > 0; (2) .ˇ/ 0 showing that  7! ˇ is a completely monotone function. It is no accident that the reciprocal of the Bernstein function ˛ , 0 < ˛ < 1, is completely monotone, nor is it an accident that the representing measure c˛ t ˛1 dt of ˛ has a completely monotone density. Inserting the representation (2) for t ˛1 into (1) and working out the double integral and the constant, leads to the second important formula for the

viii

Preface

fractional powers, ˛ D

1 .˛/.1  ˛/

Z

1 0

 t ˛1 dt: Ct

(3)

We will call this representation of ˛ the Stieltjes representation. To explain why this is indeed an appropriate name, let us go back to (2) and observe that t ˛1 is a Laplace transform. This shows that ˛ , ˛ > 0, is a double Laplace or Stieltjes transform. Another non-random coincidence is that Z 1 1 1 f˛ ./ D t ˛1 dt  .˛/.1  ˛/ 0  C t is a Stieltjes transform and so is ˛ D 1=f˛ ./. This we can see if we replace t ˛1 by its integral representation (2) and use Fubini’s theorem: Z 1 1 1 1 D ˛ D t ˛ dt: (4) f˛ ./ .˛/.1  ˛/ 0  C t It is also easy to see that the fractional powers  7! ˛ D exp.˛ log / extend analytically to the cut complex plane C n .1; 0. Moreover, z ˛ maps the upper halfplane into itself; actually it contracts all arguments by the factor ˛. Apart from some technical complications this allows to surround the singularities of f˛ – which are all in .1; 0/ – by an integration contour and to use Cauchy’s theorem for the half-plane to bring us back to the representation (3). Coming back to the fractional powers ˛ , 0 < ˛ < 1, we derive yet another R representation formula. First note that ˛ D 0 ˛s .1˛/ ds and that the integrand s .1˛/ is a Stieltjes function which can be expressed as in (4). Fubini’s theorem and the elementary equality Z

 0

   1 ds D log 1 C t Cs t

yield ˛  D .˛/.1  ˛/

Z

˛

0

1

   t ˛1 dt: log 1 C t

(5)

This representation will be called the Thorin representation of ˛ . Not every complete Bernstein function has a Thorin representation. The critical step in deriving (5) was the fact that the derivative of ˛ is a Stieltjes function. What has been explained for fractional powers can be extended in various directions. On the level of functions, the structure of (1) is characteristic for the class BF of Bernstein functions, (3) for the class CBF of complete Bernstein functions, and (5) for the Thorin–Bernstein functions TBF. If we consider exp.tf / with f from BF,

Preface

ix

CBF or TBF, we are led to the corresponding families of completely monotone functions and measures. Apart from some minor conditions, these are the infinitely divisible distributions ID, the Bondesson class of measures BO and the generalized Gamma convolutions GGC. The diagrams in Remark 9.18 illustrate these connections. If we replace (formally)  by A, where A is a negative semi-definite matrix or a dissipative closed operator, then we get from (1) and (2) the classical formulae for fractional powers, while (3) turns into Balakrishnan’s formula. Considering BF and CBF we obtain a fully-fledged functional calculus for generators and potential operators. Since complete Bernstein functions are operator monotone functions we can even recover the famous Heinz–Kato inequality. Let us briefly describe the content and the structure of the book. It consists of three parts. The first part, Chapters 1–11, introduces the basic classes of functions: the positive definite functions comprising the completely monotone, Stieltjes and Hirsch functions, and the negative definite functions which consist of the Bernstein functions and their subfamilies – special, complete and Thorin–Bernstein functions. Two probabilistic intermezzi explore the connection between Bernstein functions and certain classes of probability measures. Roughly speaking, for every Bernstein function f the functions exp.tf /, t > 0, are completely monotone, which implies that exp.tf / is the Laplace transform of an infinitely divisible sub-probability measure. This part of the book is essentially self-contained and should be accessible to non-specialists and graduate students. In the second part of the book, Chapter 12 through Chapter 15, we turn to applications of Bernstein and complete Bernstein functions. The choice of topics reflects our own interests and is by no means complete. Notable omissions are applications in integral equations and continued fractions. Among the topics are the spectral theorem for self-adjoint operators in a Hilbert space and a characterization of all functions which preserve the order (in quadratic form sense) of dissipative operators. Bochner’s subordination plays a fundamental role in Chapter 13 where also a functional calculus for subordinate generators is developed. This calculus generalizes many formulae for fractional powers of closed operators. As another application of Bernstein and complete Bernstein functions we establish estimates for the eigenvalues of subordinate Markov processes. This is continued in Chapter 14 which contains a detailed study of excessive functions of killed and subordinate killed Brownian motion. Finally, Chapter 15 is devoted to two results in the theory of generalized diffusions, both related to complete Bernstein functions through Kre˘ın’s theory of strings. Many of these results appear for the first time in a monograph. The third part of the book is formed by extensive tables of complete Bernstein functions. The main criteria for inclusion in the tables were the availability of explicit representations and the appearance in mathematical literature. In the appendix we collect, for the readers’ convenience, some supplementary results.

x

Preface

We started working on this monograph in summer 2006, during a one-month workshop organized by one of us at the University of Marburg. Over the years we were supported by our universities: Institut für Stochastik, Technische Universität Dresden, Department of Mathematics, University of Illinois, and Department of Mathematics, University of Zagreb. We thank our colleagues for a stimulating working environment and for many helpful discussions. Considerable progress was made during the two week Research in Pairs programme at the Mathematisches Forschungsinstitut in Oberwolfach where we could enjoy the research atmosphere and the wonderful library. Our sincere thanks go to the institute and its always helpful staff. Panki Kim and Hrvoje Šiki´c read substantial parts of the manuscript. We are grateful for their comments which helped to improve the text. We thank the series editor Niels Jacob for his interest and constant encouragement. It is a pleasure to acknowledge the support of our publisher, Walter de Gruyter, and its editors Robert Plato and Simon Albroscheit. Writing this book would have been impossible without the support of our families. So thank you, Herta, Jean and Sonja, for your patience and understanding. Dresden, Urbana and Zagreb October 2009

René L. Schilling Renming Song Zoran Vondraˇcek

Contents

Preface to the second edition

v

Preface

vi

Index of notation 1 Completely monotone functions

xiii 1

2 Stieltjes functions

16

3 Bernstein functions

21

4 Positive and negative definite functions

35

5 A probabilistic intermezzo

48

6 Complete Bernstein functions

69

6.1 6.2

Representation of complete Bernstein functions . . . . . . . . . . Extended complete Bernstein functions . . . . . . . . . . . . . . .

7 Properties of complete Bernstein functions

69 80 92

8 Thorin–Bernstein functions

109

9 A second probabilistic intermezzo

117

10 Transformations of Bernstein functions

131

11 Special Bernstein functions and potentials 11.1 Special Bernstein functions . . . . . . . . . . . . . . . . . . . . . 11.2 Hirsch’s class . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 172

12 The spectral theorem and operator monotonicity

179

12.1 12.2

The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . Operator monotone functions . . . . . . . . . . . . . . . . . . . .

13 Subordination and Bochner’s functional calculus

179 187 200

13.1 13.2 13.3

Semigroups and subordination in the sense of Bochner . . . . . . . A functional calculus for generators of semigroups . . . . . . . . . Subordination and functional inequalities . . . . . . . . . . . . . .

200 216 233

13.4

Eigenvalue estimates for subordinate processes . . . . . . . . . . .

242

xii

Contents

14 Potential theory of subordinate killed Brownian motion

257

15 Applications to generalized diffusions 15.1 Inverse local time at zero . . . . . . . . . . . . . . . . . . . . . .

268 268

15.2

First passage times . . . . . . . . . . . . . . . . . . . . . . . . . .

285

16 Examples of complete Bernstein functions 16.1 Special functions used in the tables . . . . . . . . . . . . . . . . . 16.2 Algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 304

16.3 16.4 16.5

Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . Inverse trigonometric functions . . . . . . . . . . . . . . . . . . .

312 314 330

16.6 16.7 16.8

Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . Gamma and related special functions . . . . . . . . . . . . . . . .

330 336 340

16.9 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 Miscellaneous functions . . . . . . . . . . . . . . . . . . . . . . . 16.11 CBFs given by exponential representations . . . . . . . . . . . . .

350 358 366

16.12 Additional comments . . . . . . . . . . . . . . . . . . . . . . . .

371

Appendix A.1 Vague and weak convergence of measures . . . . . . . . . . . . .

374 374

A.2

Hunt processes and Dirichlet forms . . . . . . . . . . . . . . . . .

377

Bibliography

383

Index

406

Index of notation

This index is intended to aid cross-referencing, so notation that is specific to a single section is generally not listed. Some symbols are used locally, without ambiguity, in senses other than those given below; numbers following an entry are page numbers. Unless otherwise stated, binary operations between functions such as f ˙ g, f  g, j !1

f ^ g, f _ g, comparisons f  g, f < g or limiting relations fj ! f , limj fj , lim infj fj , lim supj fj , supj fj or infj fj are always understood pointwise. Operations and operators a_b a^b D, R L

maximum of a and b minimum of a and b domain and range Laplace transform, 1

Sets H" H# !  H N positive negative

¹z 2 C W Im z > 0º ¹z 2 C W Im z < 0º ¹z 2 C W Re z > 0º natural numbers: 1; 2; 3; : : : always in the sense > 0 always in the sense < 0

Spaces of functions B C H S BF CBF CM H PBF P S

Borel measurable functions continuous functions harmonic functions, 262 excessive functions, 261 Bernstein functions, 21 complete Bernstein fns, 69 completely monotone fns, 2 Hirsch functions, 172 61, 140, 155, 156 potentials, 64 Stieltjes functions, 16

SBF TBF

special Bernstein fns, 159 Thorin–Bernstein fns, 109

Sub- and superscripts C  ? b c f 0

sets: non-negative elements, functions: non-negative part non-trivial elements (6 0) orthogonal complement bounded compact support subordinate w.r.t. the Bernstein function f f .0/ D 0, resp., f .0C/ D 0

Spaces of distributions BO CE Exp GGC ID ME SD SD1

Bondesson class, 117 convolutions of Exp, 125 exponential distributions, 125 generalized Gamma convolutions, 121 infinitely divisible distr., 51 mixtures of Exp, 118 self-decomposable distr., 55 completely self-decomposable distributions, 59

Chapter 1

Laplace transforms and completely monotone functions In this chapter we collect some preliminary material which we need later on in order to study Bernstein functions. The (one-sided) Laplace transform of a function m W Œ0; 1/ ! Œ0; 1/ or a measure  on the half-line Œ0; 1/ is defined by Z Z 1 e t m.t / dt or L .I / WD e t .dt /; (1.1) L .mI / WD Œ0;1/

0

respectively, whenever these integrals converge. Obviously, L m D L m if m .dt / denotes the measure m.t / dt . The following real-analysis lemma is helpful in order to show that finite measures are uniquely determined in terms of their Laplace transforms. Lemma 1.1. We have for all t; x  0 lim e t

!1

X .t /k 1 D 1Œ0;x/ .t / C 1¹xº .t /: kŠ 2

(1.2)

kx

Proof. Let us rewrite (1.2) in probabilistic terms: if X is a Poisson random variable with parameter t , (1.2) states that lim P.X  x/ D 1Œ0;x/ .t / C

!1

1 1¹xº .t /: 2

Recall that the mean and variance of the Poisson random variable X are EX D t and VarX D E..X  t /2 / D t , respectively. For each n  1 the Poisson random variable X has the same law as Sn WD X1 C    C Xn where Xj are independent and identically distributed Poisson random variables with parameter t =n. Without loss of generality we may assume p that  D n is an integer. By the central limit theorem, the sequence .S  t /= t converges weakly to a standard normal random variable G. Therefore, P.X  x/ 8 ˆP.G < 1/ D 1; if x > t;   p x  t !1 < S  t 1 DP   p p ! P.G  0/ D 2 ; if x D t; ˆ t t : P.G D 1/ D 0; if x < t; and the claim follows.

2

Chapter 1 Completely monotone functions

Proposition 1.2. A measure  on Œ0; 1/ is finite if, and only if, L .I 0C/ < 1. The measure  is uniquely determined by its Laplace transform. Proof. The first part from monotone convergence since we R R of the assertion follows have Œ0; 1/ D Œ0;1/ 1 d D lim!0 Œ0;1/ e t .dt /. For the uniqueness part we use first the differentiation lemma for parameter dependent integrals to get Z .1/k L .k/ .I / D e t t k .dt /: Œ0;1/

Therefore, X

.1/k L .k/ .I /

kx

X Z k .t /k t D e .dt / kŠ Œ0;1/ kŠ kx Z X .t /k e t .dt / D kŠ Œ0;1/ kx

and we conclude with Lemma 1.1 and dominated convergence that lim

!1

X kx

.1/k L .k/ .I /

k D kŠ

Z Œ0;1/

 1Œ0;x/ .t / C

 1 1¹xº .t / .dt / 2

(1.3)

1 D Œ0; x/ C ¹xº: 2 This shows that  can be recovered from (all derivatives of) its Laplace transform. It is possible to characterize the range of Laplace transforms. For this we need the notion of complete monotonicity. Definition 1.3. A function f W .0; 1/ ! R is a completely monotone function if f is of class C 1 and .1/n f .n/ ./  0 for all n 2 N [ ¹0º and  > 0:

(1.4)

The family of all completely monotone functions will be denoted by CM. The conditions (1.4) are often referred to as Bernstein–Hausdorff–Widder conditions. The next theorem is known as Bernstein’s theorem. The version given below appeared for the first time in [45] and independently in [368]. Subsequent proofs were given in [116] and [102]. The theorem may be also considered as an example of the general integral representation of points in a convex cone by means of its extremal elements. See Theorem 4.8 and [85] for an elementary exposition. The following short and elegant proof is taken from [263].

3

Chapter 1 Completely monotone functions

Theorem 1.4 (Bernstein). Let f W .0; 1/ ! R be a completely monotone function. Then it is the Laplace transform of a unique measure  on Œ0; 1/, i.e. for all  > 0, Z e t .dt /: f ./ D L .I / D Œ0;1/

Conversely, whenever L .I / < 1 for every  > 0,  7! L .I / is a completely monotone function. Proof. Assume first that f .0C/ D 1 and f .C1/ D 0. Let  > 0. For any a > 0 and any n 2 N, we see by Taylor’s formula Z  .n/ n1 X f .k/ .a/ f .s/ .  a/k C .  s/n1 ds f ./ D kŠ a .n  1/Š D

kD0 n1 X kD0

.1/k f .k/ .a/ .a  /k C kŠ

Z

a 

.1/n f .n/ .s/ .s  /n1 ds: .n  1/Š

(1.5)

If a > , then by the assumption all terms are non-negative. Let a ! 1. Then Z 1 Z a .1/n f .n/ .s/ .1/n f .n/ .s/ n1 .s  / .s  /n1 ds ds D lim a!1  .n  1/Š .n  1/Š   f ./: This implies that the sum in (1.5) converges for every n 2 N as a ! 1. Thus, every term converges as a ! 1 to a non-negative limit. For n  0 let .1/n f .n/ .a/ .a  /n : a!1 nŠ This limit does not depend on  > 0. Indeed, for  > 0, n ./ D lim

.1/n f .n/ .a/ .a  /n a!1 nŠ .1/n f .n/ .a/ .a  /n .a  /n D lim D n ./: a!1 nŠ .a  /n

n ./ D lim

Let cn D

Pn1

kD0 k ./.

Then

f ./ D cn C

Z

1 

.1/n f .n/ .s/ .s  /n1 ds: .n  1/Š

Clearly, f ./  cn for all  > 0. Let  ! 1. Since f .C1/ D 0, it follows that cn D 0 for every n 2 N. Thus we have obtained the following integral representation of the function f : Z 1 .1/n f .n/ .s/ .s  /n1 ds: (1.6) f ./ D .n  1/Š 

4

Chapter 1 Completely monotone functions

By the monotone convergence theorem Z 1 .1/n f .n/ .s/ n1 s ds: 1 D lim f ./ D .n  1/Š !0 0

(1.7)

Let

.1/n .n/  n   n nC1 f : (1.8) nŠ s s Using (1.7) and changing variables according to s=t , it follows that for every n 2 N, fn is a probability density function on .0; 1/. Moreover, the representation (1.6) can be rewritten as  Z 1  n1 .1/n f .n/ .s/ n1 s 1 ds f ./ D s C .n  1/Š 0  Z 1 t n1 D fn .t / dt: (1.9) 1 n C 0 fn .s/ D

By Helly’s selection theorem, Corollary A.8, there exist a subsequence .nk /k1 and a probability measure  on .0; 1/ such that fnk .t / dt converges weakly to .dt /. Further, for every  > 0,   t n1 lim 1  D e t n!1 n C uniformly in t 2 .0; 1/. By taking the limit in (1.9) along the subsequence .nk /k1 , it follows that Z f ./ D e t .dt /: .0;1/

Uniqueness of  follows from Proposition 1.2. Assume now that f .0C/ < 1 and f .C1/ D 0. By looking at f =f .0C/ we see that the representing measure for f is uniquely given by f .0C/. Now let f be an arbitrary completely monotone function with f .C1/ D 0. For every a > 0, define fa ./ WD f . C a/,  > 0. Then fa is a completely monotone function with fa .0C/ D f .a/ < 1 and fa .C1/ D 0. By what has been already proved, there exists a unique finite measure a on .0; 1/ R such that fa ./ D .0;1/ e t a .dt /. It follows easily that for b > 0 we have e at a .dt / D e bt b .dt /. This shows that we can consistently define the measure  on .0; 1/ by .dt / D e at a .dt /, a > 0. In particular, the representing measure  is uniquely determined by f . Now, for  > 0, Z f ./ D f=2 .=2/ D e .=2/t =2 .dt / .0;1/ Z Z t .=2/t e e =2 .dt / D e t .dt /: D .0;1/

.0;1/

Chapter 1 Completely monotone functions

5

Finally, if f .C1/ D c > 0, add cı0 to . For the converse we set f ./ WD L .I /. Fix  > 0 and pick  2 .0; /. Since t n D  n .t /n  nŠ n e t for all t > 0, we find Z Z nŠ nŠ n t t e .dt /  n e ./t .dt / D n L .I   /  Œ0;1/  Œ0;1/ and this shows that we may use the differentiation lemma for parameter dependent integrals to get Z Z d n t .1/n f .n/ ./ D .1/n e .dt / D t n e t .dt /  0: n d  Œ0;1/ Œ0;1/ Remark 1.5. The last formula in the proof of Theorem 1.4 shows, in particular, that f .n/ ./ ¤ 0 for all n 2 N and all  > 0, unless f 2 CM is identically constant. Corollary 1.6. The set CM of completely monotone functions is a convex cone, i.e. sf1 C tf2 2 CM

for all s; t  0 and f1 ; f2 2 CM;

which is closed under multiplication, i.e.  7! f1 ./f2 ./ is in CM for all f1 ; f2 2 CM; and under pointwise convergence: CM D ¹L  W  is a finite measure on Œ0; 1/º (the closure is taken with respect to pointwise convergence). Proof. That CM is a convex cone follows immediately from the definition of a completely monotone function or, alternatively, from the representation formula in Theorem 1.4. If j denotes the representing measure of fj , j D 1; 2, the convolution “ 1Œ0;u .s C t / 1 .ds/2 .dt / Œ0; u WD 1 ? 2 Œ0; u WD Œ0;1/Œ0;1/

is the representing measure of the product f1 f2 . Indeed, Z Z Z e u .du/ D e .sCt/ 1 .ds/2 .dt / D f1 ./f2 ./: Œ0;1/

Œ0;1/

Œ0;1/

Write M WD ¹L  W  is a finite measure on Œ0; 1/º. Theorem 1.4 shows that M  CM  M . We are done if we can show that CM is closed under pointwise convergence. For this choose a sequence .fn /n2N  CM such that the limit

6

Chapter 1 Completely monotone functions

limn!1 fn ./ D f ./ exists for every  > 0. If n denotes the representing measure of fn , we find for every a > 0 Z n!1 e t n .dt /  e a fn ./ ! e a f ./ n Œ0; a  e a Œ0;a

which means that the family of measures .n /n2N is bounded in the vague topology, hence vaguely sequentially compact, see Appendix A.1. Thus, there exist a subsequence .nk /k2N and a measure  such that nk !  vaguely. For every function 2 Cc Œ0; 1/ with 0   1, we find Z Z .t /e t .dt / D lim .t /e t nk .dt /  lim inf fnk ./ Œ0;1/

k!1 Œ0;1/

k!1

D f ./: Taking the supremum over all such , we can use monotone convergence to get Z e s .dt /  f ./: Œ0;1/

On the other hand, we find for each a > 0 Z Z 1 1 e t nk .dt / C e  2 t e  2 t nk .dt / fnk ./ D Œ0;a/ Œa;1/   Z 1 t  12 a  : e nk .dt / C e fnk  2 Œ0;a/ If we let k ! R 1 and then a ! 1 along a sequence of continuity points of  we get f ./  Œ0;1/ e t .dt / which shows that f 2 CM and that the measure  is actually independent of the particular subsequence. In particular,  D limn!1 n vaguely in the space of measures supported in Œ0; 1/. The seemingly innocuous closure assertion of Corollary 1.6 actually says that on the set CM the notions of pointwise convergence, locally uniform convergence, and even convergence in the space C 1 .0; 1/ coincide. This situation reminds remotely of the famous Montel’s theorem from the theory of analytic functions, see e.g. Berenstein and Gay [31, Theorem 2.2.8]. Corollary 1.7. Let .fn /n2N be a sequence of completely monotone functions such that the limit limn!1 fn ./ D f ./ exists for all  2 .0; 1/. Then f 2 CM and .k/ limn!1 fn ./ D f .k/ ./ for all k 2 N [ ¹0º locally uniformly in  2 .0; 1/. Proof. From Corollary 1.6 we know already that f 2 CM. Moreover, we have seen that the representing measures n of fn converge vaguely in Œ0; 1/ to the representing

7

Chapter 1 Completely monotone functions

measure  of f . By the differentiation lemma for parameter dependent integrals we infer Z .k/ k t k e t n .dt / fn ./ D .1/ Œ0;1/ Z n!1 ! .1/k t k e t .dt / D f .k/ ./; Œ0;1/

since t 7! t k e t is a function that vanishes at infinity, cf. (A.3) in Appendix A.1. Finally, assume that j  j  ı for some ı > 0. Using the elementary estimate je t  e t j  j  j t e .^/t , ; ; t > 0, we conclude that for ;    and all >0 Z ˇ .k/ ˇ t ˇ ˇ ˇe ˇf ./  f .k/ ./ˇ   e t ˇ t k n .dt / n n .0;1/ Z e .^/t t kC1 n .dt / ı .0;1/

ˇ ˇ D ı ˇfn.kC1/ . ^ /ˇ: .kC1/

. ^ / D f .kC1/ . ^ /, we find for sufficiently large Using that limn!1 fn values of n ˇ .k/ ˇ ˇ ˇ ˇf ./  f .k/ ./ˇ  2ı sup ˇf .kC1/ . /ˇ: n n 

This proves that the functions fn.k/ are uniformly equicontinuous on Œ; 1/. Therefore, the convergence limn!1 fn.k/ ./ D f .k/ ./ is locally uniform on Œ; 1/ for every  > 0. Since  > 0 was arbitrary, we are done. Every family of completely monotone functions which is uniformly bounded at the origin is (weakly) sequentially compact. Corollary 1.8. Let .fn /n2N be a sequence of completely monotone functions such that supn2N fn .0C/ < 1. Then there exist a subsequence .fnk /k2N and a completely monotone function f W .0; 1/ ! Œ0; 1/ such that limk!1 fnk ./ D f ./ for all  2 .0; 1/. Proof. Set c WD supn2N fn .0C/ and denote for each n 2 N the representing measure of fn by n . Since fn .0C/  c, we have n Œ0; 1/  c, implying that the sequence of measures .n /n2N is vaguely bounded, hence vaguely compact by Theorem A.5. Therefore, there exist a subsequence .nk /k2N and a finite measure  on Œ0; 1/ such k!1

that nk !  vaguely. If we define f WD L , then limk!1 fnk ./ D f ./ for every  > 0 and, by Corollary 1.7, f 2 CM.

8

Chapter 1 Completely monotone functions

Remark 1.9. The representation formula for completely monotone functions given in Theorem 1.4 has an interesting interpretation in connection with the Kre˘ın–Milman theorem and the Choquet representation theorem. The set ® ¯ f 2 CM W f .0C/ D 1 is a basis of the convex cone CMb , and its extremal points are given by e t ./ D e t ;

0  t < 1;

and e1 ./ D 1¹0º ./;

see Phelps [291, Lemma 2.2], Lax [239, p. 139] or the proof of Theorem 4.8. These extremal points are formally defined for  2 Œ0; 1/ with the understanding that e1 j.0;1/  0. Therefore, the representation formula from Theorem 1.4 becomes a Choquet representation of the elements of CMb , Z Z e t .dt / D e t .dt /;  2 .0; 1/: Œ0;1/

Œ0;1

In particular, the functions e t j.0;1/ are prime examples of completely monotone functions. Theorem 1.4 and Corollary 1.6 tell us that every f 2 CM can be written as an ‘integral mixture’ of the extremal CM-functions ¹e t j.0;1/ W 0  t < 1º. In the remaining part of this chapter we study two generalizations of complete monotonicity: multiply monotone functions and completely monotone functions on the whole real line R. Definition 1.10. A function f W .0; 1/ ! R is 1-monotone (also monotone of order 1) if f ./  0 for all  > 0 and f is non-increasing and right-continuous. A function f W .0; 1/ ! R is n-monotone (also monotone of order n), n  2, if it is n  2 times differentiable, .1/j f .j / ./  0 for all j D 0; 1; : : : ; n  2 and  > 0, and .1/n2 f .n2/ is non-increasing and convex. It is not difficult to see from Definition 1.10 that an n times differentiable function f W .0; 1/ ! R is n-monotone if, and only if, .1/j f .j / ./ is non-negative for all j D 0; 1; : : : ; n. In particular, a function f W .0; 1/ ! R is completely monotone if, and only if, it is n-monotone for every n  1. In view of Theorem 1.4 it should not be surprising that n-monotone functions admit an integral representation. Theorem 1.11. Suppose that f W .0; 1/ ! R and n 2 N. Then the following statements are equivalent. (i) f is n-monotone.

9

Chapter 1 Completely monotone functions

(ii) There exist a unique constant c  0 and a unique measure  on .0; 1/ such that Z 1 .t  /n1 .dt /;  > 0: (1.10) f ./ D c C .n  1/Š .;1/ (iii) There exists a unique measure n on Œ0; 1/ such that Z f ./ D .1  t /n1

n .dt /; C

 > 0:

(1.11)

Œ0;1/

Proof. (i))(ii) Set c WD f .C1/ WD lim!1 f ./. For n D 1, f is a right-continuous, non-increasing and non-negative function such that the limit f .C1/ exists. Every such function can be written as f ./ D c C .; 1/ for a unique measure  on .0; 1/ and a unique constant c  0. For n D 2, f is non-negative, non-increasing, convex and such that the limit f .C1/ exists. A convex function has at each point in the interior of its domain a right-hand derivative fC0 which is non-decreasing and right-continuous. Since f is non-increasing it follows that fC0  0. Therefore, g./ WD fC0 ./ is rightR1 continuous, non-increasing and f ./ D f .C1/ C  g.t / dt: This proves that R1 f ./ D c C  .s; 1/ ds for the unique measure  on .0; 1/ given by .t; 1/ WD g.t / and the unique constant c WD f .C1/  0. We proceed by induction. Assume that n  2 and that the claim is proved for n, i.e. an n-monotone function admits the representation (1.10). Since the limit f .C1/ exists, it holds that lim!1 f 0 ./ D 0. Hence f 0 is n-monotone. By the induction hypothesis, there is a unique measure  on .0; 1/ such that Z 1 .t  s/n1 .dt /; s > 0: f 0 .s/ D .n  1/Š .s;1/ Using Fubini’s theorem we obtain for some integration constant c 2 R Z 1 f ./ D  f 0 .s/ ds C c   Z 1 Z 1 n1 .t  s/ DcC .dt / ds  .s;1/ .n  1/Š  Z t Z 1 .t  s/n1 ds .dt / DcC  .n  1/Š .;1/ Z 1 .t  /n .dt / DcC nŠ .;1/ proving the claim. Since c D f .C1/, we see that c is non-negative and unique.

10

Chapter 1 Completely monotone functions

(ii))(iii) Let  ? be the image measure of  with respect to the mapping t 7! t 1 , 1 and define the measure n on .0; 1/ by n .dt / D .n1/Š t 1n  ? .dt /. Then 1 .n  1/Š

Z 1.;1/ .t / .t  /n1 .dt / Z 1 D 1.;1/ .t 1 / .t 1  /n1  ? .dt / .n  1/Š .0;1/ Z 1.0;1 / .t / .1  t /n1 n .dt / D .0;1/ Z .1  t /n1

n .dt /: D C .0;1/

.0;1/

If we set n ¹0º D c, the claim follows. (iii))(i) Assume that f is given by (1.11). If n D 1, f ./ D n Œ0; 1=/ which is a non-negative, non-increasing and right-continuous function. In case n  2, the is n  2 times differentiable and for k D 0; 1; : : : ; n  2, function  7! .1  t /n1 C .1/k

dk .1  t /n1 D .n  1/.n  2/    .n  k/t k .1  t /nk1 C C d k

is non-negative, non-increasing and convex. Moreover, t 7! t k .1  t /nk1 , t  0, C is integrable with respect to n which allows us to differentiate in (1.11) under the integral sign. Hence, f is n  2 times differentiable and Z k .k/ .1/ f ./ D .n  1/.n  2/    .n  k/ t k .1  t /nk1

n .dt /: C Œ0;1/

Since non-negativity, monotonicity and convexity are preserved under integration, we conclude that f is n-monotone. Remark 1.12. (i) In the proof of Theorem 1.4 we have derived the representation (1.6) which requires only that f is n times differentiable and vanishes at infinity. If f is also n-monotone, we can compare this with (1.10), and conclude that the representing measure  is given by .dt / D .1/n f .n/ .t / dt ; thus,

n .dt / D

.1/n n1 .n/ 1 t f .t / dt: .n  1/Š

Let e

n be the image measure of n under the mapping t 7! nt . If f .C1/ D 0, (1.11) becomes   t n1 f ./ D 1 e

n .dt /: n C .0;1/ Z

11

Chapter 1 Completely monotone functions

Comparing this with (1.3) and (1.9), we see – if f is n times differentiable – that e

n .dt / D

.1/n .n/  n   n nC1 f : nŠ t t

(ii) The representation (1.10) allows an extension of the concept of monotonicity to non-integer order. A function f W .0; 1/ ! R is p-monotone, p > 0, (also monotone of order p) if there exist a constant c  0 and a measure  on .0; 1/ such that Z 1 f ./ D c C .t  /p1 .dt /;  > 0: .p/ .;1/ Equivalently, f is p-monotone, if (1.11) holds with n replaced by p. The conditions in the definition of n-monotonicity have redundancies; this was shown in [372]. Proposition 1.13. Let f W .0; 1/ ! R. Then f is n-monotone, n  2, if, and only if, (i) f .C1/ D lim!1 f ./ exists and is non-negative; (ii) .1/n2 f .n2/ is non-negative, non-increasing and convex. Proof. Assume that (i) and (ii) hold. Then f .n2/ is convex, hence continuous, and by Taylor’s formula, for every a > 0 f ./ D

n3 X kD0

f .k/ .a/ .  a/k C .1/n2 kŠ

Z

 a

.1/n f .n2/ .s/ .  s/n3 ds: .n  3/Š

If n  2 is even f ./ 

n3 X kD0

f .k/ .a/ .  a/k ; kŠ

while for n  2 odd f ./ 

n3 X kD0

f .k/ .a/ .  a/k : kŠ

Dividing by .  a/n3 , letting  ! 1, and by using that f ./ stays bounded, we arrive at f .n3/ .a/  0 if n  2 is even, and f .n3/ .a/  0 if n  2 is odd. Thus, .1/n3 f .n3/ .a/  0. By induction it follows that .1/k f .k/ .a/  0 for all k D 1; 2; : : : ; n  2. Since f .C1/  0, we see f .a/  0, which means that f is n-monotone.

12

Chapter 1 Completely monotone functions

It was pointed out in [326] that the conditions (1.4) have redundancies. This follows immediately from the preceding proposition. Corollary 1.14. Let f W .0; 1/ ! R be a C 1 function such that f  0, f 0  0 and .1/n f .n/  0 for infinitely many n 2 N. Then f is a completely monotone function. Corollary 1.15. The set of n-monotone functions is a convex cone which is closed under multiplication. Proof. It is clear from the definition that the set of n-monotone functions is a convex cone. We prove that it is closed under multiplication. The case n D 1 is immediate from definition. Let n  2 and assume that the claim is proved for n1. Let gj D fj0 where fj , j D 1; 2, is n-monotone. Then gj , j D 1; 2, is .n  1/-monotone. Noting that f1 f2 is the primitive of f10 f2 C f1 f20 D g1 f2  f1 g2 , we see Z 1  g1 .t /f2 .t / C f1 .t /g2 .t / dt: f1 ./f2 ./  f1 .C1/f2 .C1/ D 

By the induction hypothesis, g1 f2 and f1 g2 are .n  1/-monotone, and the claim follows. Note that Corollary 1.15 provides an alternative proof that CM is closed under multiplication. The concepts of n-monotonicity and complete monotonicity can be extended to functions defined on the whole real line. Let f W R ! R. Then f is 1-monotone if it is non-negative, non-increasing and right-continuous. If n  2 is an integer then f is n-monotone if it is n  2 times differentiable, .1/j f .j / ./  0 for all j D 0; 1; : : : ; n  2 and all  2 R, and .1/n1 f .n2/ is non-negative, nonincreasing and convex. The analogue of Theorem 1.11 – with the same proof – is still valid but with  2 R and  and n being measures on R. The function f W R ! R is completely monotone on R if it is n-monotone on R for all n 2 N; this is the same as to say that f 2 C 1 .R/ and .1/n f .n/ ./  0 for all n  0 and  2 R. Any such function f admits the representation Z f ./ D e t .dt /;  2 R; (1.12) .0;1/

where  is a measure on .0; 1/. This follows easily from Theorem 1.4 and the fact for each 0  0, the function  7! f . C 0 / is completely monotone (on .0; 1/, i.e. in the sense of Definition 1.3). The relationship between higher-order monotonicity on R and on .0; 1/ is explained in the following proposition.

13

Chapter 1 Completely monotone functions

Proposition 1.16. Suppose that f W R ! R is n-monotone, n 2 N, on R. Then  7! f .log /,  > 0, is n-monotone. In particular, if f W R ! R is completely monotone on R, then  7! f .log /,  > 0 is completely monotone. Proof. For n D 1 the claim follows from the definition and the fact that the logarithm maps .0; 1/ bijectively onto R. Let n  2 and assume that the claim is valid for 0 n  1. Assume that f is n-monotone R 1on R and set g./ D f ./. Then g is .n  1/-monotone on R and f ./ D  g.s/ ds. Hence, Z 1 Z 1 f .log / D g.s/ ds D g.log t / t 1 dt: log 



By the induction hypothesis, t 7! g.log t / is an .n  1/-monotone function, hence by Corollary 1.15, t 7! g.log t / t 1 is also .n  1/-monotone. Thus,  7! f .log / is n-monotone. Remark 1.17. The converse of Proposition 1.16 is not valid. Indeed, let n  2 and define f W R ! .0; 1/ by f ./ D .1  e  /n1 1.1;0/ ./. Then Z .1  t /n1 f .log / D .1  /n1 1.0;1/ ./ D C ı1 .dt / Œ0;1/

is n-monotone. On the other hand, f is not n-monotone on R since f 00 ./ D .n  1/.n  2/.1  e  /n3 e 2  .n  1/.1  e  /n2 e  < 0 for  <  log.n  1/. Similarly, let f ./ D exp.ce  / for some c > 0. Then f .log / D e c is completely monotone, while f is not completely monotone on R. These examples come from [317]. Comments 1.18. Standard references for Laplace transforms include D. V. Widder’s monographs [370, 371] and Doetsch’s treatise [96]. For a modern point of view we refer to Berg and Forst [39] and Berg, Christensen and Ressel [38]. The most comprehensive tables of Laplace transforms are the Bateman manuscript project [107] and the tables by Prudnikov, Brychkov and Marichev [296]. The concept of complete monotonicity seems to go back to S. Bernstein [43] who studied functions on an interval I  R having positive derivatives of all orders. If I D .1; 0 this is, up to a change of sign in the variable, complete monotonicity. In later papers, Bernstein refers to functions enjoying this property as absolument monotone, see the appendix première note, [44, pt. IV, p. 190], and in [45] he states and proves Theorem 1.4 for functions on the negative half-axis. Following Schur (probably [328]), Hausdorff [150, p. 80] calls a sequence .n /n2N total monoton – the literal translation totally monotone is only rarely used, e.g. in Hardy [145] – if all iterated differences .1/k k n , k  1, n  1, are non-negative.

14

Chapter 1 Completely monotone functions

As usual, n WD nC1  n . The modern terminology is completely monotone; for sequences the German equivalent vollmonoton was introduced by Jacobsthal [186], in connection with functions the notion of complete monotonicity appears for the first time in [368]. focusses in [150, 151] on the moment problem: the n are of the form R Hausdorff n .dt / for some measure  on .0; 1 if, and only if, the sequence . / t n n2N[¹0º .0;1 R is total monoton; moreover, he introduces the moment function  WD .0;1 t  .dt / which he also calls total monoton. A simple change of variables t e u , R u t 2 .0; 1, u 2 Œ0; 1/, shows that  D Œ0;1/ e .du/ Q for some suitable image measure Q of . This means that every moment sequence gives rise to a unique completely monotone function. The converse is much easier since Z e Œ0;1/

u

Z 1 X .1/n n .du/ Q D un .du/: Q nŠ Œ0;1/ nD0

Many historical comments can be found in the second part, pp. 29–44, of Ky Fan’s memoir [110]. For an up-to-date survey we recommend the scholarly commentary by Chatterji [78] written for Hausdorff’s collected works [152]. More general higher or multiple monotonicity properties for a sequence were already considered in [186]. A sequence is monotone of order m if .1/k k n  0 for all k D 0; 1; : : : ; m and n 2 N. This was extended by Knopp [211] to fractional orders and a series of papers by Fejér, Hausdorff, Kaluza and Knopp studies in depth the behaviour of power series and trigonometric series whose coefficients are completely monotone or monotone of some finite order. The paper [115] gives a detailed account on these developments. Royall [310] and, independently, Hartman [146] call a function f monotone of order m if it satisfies (1.4) only for n 2 ¹0; 1; : : : ; mº, m 2 N [ ¹0º. The present, more general, Definition 1.10 is due to Williamson [372]. Note that Löwner’s notion of higherorder monotonicity introduced in [248] (monoton nter Stufe) and [249] (monotonic of order n) is different; it will play an important role in Chapter 12.2, cf. Definition 12.9. Further applications of CM and related functions to ordinary differential equations can be found, e.g. in Lorch et al. [247] and Mahajan and Ross [258], see also the study by van Haeringen [359] and the references given there. The connection between integral equations and CM are extensively covered in the monographs by Gripenberg et al. [136] and Prüss [297]. A by-product of the proof of Proposition 1.2 is an example of a so-called real inversion formula for Laplace transforms. Formula (1.3) is due to Dubourdieu [102] and Feller [116], see also Pollard [295] and Widder [370, p. 295] and [371, Chapter 6]. Our presentation follows Feller [118, VII.6]. The proof of Bernstein’s theorem, Theorem 1.4, also contains a real inversion formula for the Laplace transform: (1.8) coincides with the operator Lk;y .f .// of Widder [371, p. 140] and, up to a constant, also [370, p. 288]. Since the proof of Theorem 1.4 relies on a compactness argument using subsequences, the weak limit

15

Chapter 1 Completely monotone functions

fnk .t / dt ! .dt / might depend on the actual subsequence .nk /k2N . If we combine this argument with Proposition 1.2, we get at once that all subsequences lead to the same  and that, therefore, the weak limit of the full sequence fn .t / dt ! .dt / exists. The representation (1.6) was also Robtained in [103] in the following way: as 1 f .C1/ D 0 we can write f ./ D  .1/f 0 .t1 / dt1 . Since f 0 is again com0 pletely monotone R 1 R 1 00and satisfies f .C1/ D 0, the same argument proves that f ./ D  t1 f .t2 / dt2 dt1 . By induction, for every n 2 N, Z f ./ D

1Z 1 

t1

Z 

1 tn1

.1/n f .n/ .tn / dtn    dt2 dt1 :

The representation (1.6) follows by using Fubini’s theorem and reversing the order of integration. The rest of the proof is now similar to our presentation. It is possible to avoid the compactness argument in Theorem 1.4 and to give an ‘intuitionistic’ proof, see van Herk [361, Theorem 33] who did this for the class S (denoted by ¹F º in [361]) of Stieltjes functions which is contained in CM; his arguments work also for CM. A proof of Theorem 1.4 using Choquet’s theorem or the Kre˘ın–Milman theorem can be found in Kendall [202], Meyer [265] or Choquet [85, 84]. A modern textbook version is contained in Lax [239, Chapter 14.3, p. 138], Phelps [291, Chapter 2], Becker [28] and also in Theorem 4.8 below. The somewhat surprising compactness result Corollary 1.8 seems to be new. Definition 1.10 and the results on multiply monotone functions are from [372], our proofs are adaptations of [317, Chapter 2]. The connection between complete monotonicity and complete monotonicity on R is also from [317]. Multiply monotone functions of non-integer order appear already in [372]; Sato [317] studies the interplay between fractional derivatives, fractional integrals and fractional-order monotone functions. Gneiting’s short note [131] contains an example showing that one cannot weaken the Bernstein–Hausdorff–Widder conditions beyond what is stated in Corollary 1.14. There is a deep geometric connection between completely monotone functions and the problem when a metric space can be embedded into a Hilbert space H. The basic result is due to Schoenberg [325]: a function f on Œ0; 1/ with f .0/ D f .0C/ is completely monotone if, and only if, 7! f .j j2 /, 2 Rd , is positive definite for all dimensions d  0, cf. Theorem 13.14.

Chapter 2

Stieltjes functions

Stieltjes functions are a subclass of completely monotone functions. They will play a central role in our study of complete Bernstein functions. In Theorem 7.3 we will see that f is a Stieltjes function if, and only if, 1=f is a complete Bernstein function. This allows us to study Stieltjes functions via the set of all complete Bernstein functions which is the focus of this tract. Therefore we restrict ourselves to the definition and a few fundamental properties of Stieltjes functions. Definition 2.1. A (non-negative) Stieltjes function is a function f W .0; 1/ ! Œ0; 1/ which can be written in the form Z a 1 f ./ D C b C  .dt / (2.1)  .0;1/  C t 0 are non-negative constants and  is a measure on .0; 1/ such that Rwhere a; b 1 .0;1/ .1 C t /  .dt / < 1. We denote the family of all Stieltjes functions by S. The integral appearing in (2.1) is also called the Stieltjes transform of the meaR1 sure  . Using the elementary relation . C t /1 D 0 e tu e u du and Fubini’s theorem one sees that it is also a double Laplace transform. In view of the uniqueness of the Laplace transform, see Proposition 1.2, a; b and  appearing in the representation (2.1) are uniquely determined by f . Since some authors consider measures  on the compactification Œ0; 1, there is no marked difference between Stieltjes transforms and Stieltjes functions in the sense of Definition 2.1. It is sometimes useful to rewrite (2.1) in the following form Z 1Ct N .dt / (2.2) f ./ D Œ0;1  C t where N WD aı0 C .1 C t /1  .dt / C bı1 is a finite measure on the compact interval Œ0; 1. Since for z D  C i  2 C n .1; 0 and t  0 ˇ ˇ ˇ 1 ˇ 1 1 ˇ ˇ ; ˇz C t ˇ D p 2 2 t C 1 . C t / C  i.e. there exist two positive constants c1 < c2 (depending on  and ) such that 1 c2 c1 p ;  t C1 t C1 . C t /2 C  2

17

Chapter 2 Stieltjes functions

we can use (2.2) to extend f 2 S uniquely to an analytic function on C n .1; 0. Note that Im z  Im

Im z 1 .Im z/2 D Im z  D  zCt jz C t j2 jz C t j2

which means that the mapping z 7! f .z/ swaps the upper and lower complex halfplanes. We will see in Corollary 7.4 below that this property is also sufficient to characterize f 2 S. (i) Every f 2 S is of the form     f ./ D L .a  dt I / C L b  ı0 .dt /I  C L L . I t / dt I 

Theorem 2.2.

for the measure  appearing in (2.1). In particular, S  CM and S consists of all completely monotone functions having a representation measure with completely monotone density on .0; 1/. (ii) The set S is a convex cone: if f1 ; f2 2 S, then sf1 C tf2 2 S for all s; t  0. (iii) The set S is closed under pointwise limits: if .fn /n2N  S and if the limit limn!1 fn ./ D f ./ exists for all  > 0, then f 2 S. R1 Proof. Since . C t /1 D 0 e .Ct/u du, assertion (i) follows from (2.1) and Fubini’s theorem; (ii) is obvious. For (iii) we argue as in the proof of Corollary 1.6: assume that the function fn is given by (2.2) where we denote the representing measure by N n D an ı0 C .1 C t /1 n .dt / C bn ı1 . Since n!1

N n Œ0; 1 D fn .1/ ! f .1/ < 1; the family .N n /n2N is uniformly bounded. By the Banach–Alaoglu theorem, Corollary A.6, we conclude that .N n /n2N has a weak* convergent subsequence .N nk /k2N such that N WD vague- limk!1 N nk is a bounded measure on the compact space Œ0; 1. Since t 7! .1 C t /=. C t / is in C Œ0; 1, we get Z Z 1Ct 1Ct N nk .dt / D N .dt /; f ./ D lim fnk ./ D lim k!1 k!1 Œ0;1  C t Œ0;1  C t i.e. f 2 S. Since the limit limn!1 fn ./ D f ./ exists – independently of any subsequence – and since the representing measure is uniquely determined by the function f , N does not depend on any subsequence. In particular,  D limn!1 n vaguely in the space of measures supported in .0; 1/. Remark 2.3. Let f; fn 2 S, n 2 N, where we write a; b;  and an ; bn ; n for the constants and measures appearing in (2.1) and N n ; N for the corresponding representation measures from (2.2). If limn!1 fn ./ D f ./, the proof of Theorem 2.2 shows that vague- lim n D  n!1

and vague- lim N n D N : n!1

18

Chapter 2 Stieltjes functions

Combining this with the portmanteau theorem, Theorem A.7, it is possible to show that   Z n .dt / a D lim lim inf an C !0 n!1 .0;/ 1 C t and



Z

b D lim lim inf bn C R!1 n!1

.R;1/

 n .dt / I 1Ct

in the above formulae we can replace lim inf n by lim supn . We do not give the proof here, but we refer to a similar situation for Bernstein functions which is worked out in Corollary 3.9. In general, it is not true that limn!1 an D a or limn!1 bn D b. This is easily seen from the following examples: fn ./ D 1C1=n and fn ./ D 1=n . Remark 2.4. Just as in the case of completely monotone functions, see Remark 1.9, we can understand the representation formula (2.2) as a particular case of the Choquet or Kre˘ın–Milman representation. The set ® ¯ f 2 S W f .1/ D 1 is a basis of the convex cone S, and its extremal points are given by e0 ./ D

1 ; 

e t ./ D

1Ct ; 0 < t < 1; Ct

and e1 ./ D 1:

To see that the functions e t , 0  t  1, are indeed extremal, note that the equality e t ./ D f ./ C .1  /g./;

f; g 2 S;

and the uniqueness of the representing measures in (2.2) imply that ı t D  Nf C .1  /N g (Nf ; N g are the representing measures) which is only possible if Nf D N g D ı t . Conversely, since every f 2 S is given by (2.2), the family ¹e t º t2Œ0;1 contains all extremal points. In particular, 1;

1 ; 

1 ; Ct

1Ct ; Ct

t > 0;

19

Chapter 2 Stieltjes functions

are examples of Stieltjes functions and so are their integral mixtures, e.g. ˛1 .0 < ˛ < 1/;

1 1 p arctan p ;  

1 log.1 C / 

p or 1 1.1;1/ .t / dt as which we obtain if we choose 1 sin.˛/ t ˛1 dt , 1.0;1/ .t / dt t 2 t the representing measures  .dt / in (2.1). The closure assertion of Theorem 2.2 says, in particular, that on the set S the notions of pointwise convergence, locally uniform convergence, and even convergence in the space C 1 coincide, cf. also Corollary 1.7.

Comments 2.5. The Stieltjes transform appears for the first time in the famous papers [340, 341] where T. J. Stieltjes investigates continued fractions in order to solve what we nowadays call the Stieltjes Moment Problem. For an appreciation of Stieltjes’ achievements see the contributions of W. van Assche and W. A. J. Luxemburg in [342, Vol. 1]. The name Stieltjes transform for the integral (2.1) was coined by RDoetsch [95] and, independently, Widder [369]. Earlier works, e.g. Perron [287], call . C t /1  .dt / Stieltjes integral (German: Stieltjes’sches Integral) but terminology has changed since then. Sometimes the name Hilbert–Hankel transform is also used in the literature, cf. Lax [239, p. 185]. A systematic account of the properties of the Stieltjes transform is given in [370, Chapter VIII]. Stieltjes functions are discussed by van Herk [361] as class ¹F º in the wider context of moment and complex interpolation problems, see also the Comments 6.22, 7.24. Van Herk uses the integral representation Z 1 .ds/ Œ0;1 1  s C sz which can be transformed by the change of variables t D s 1  1 2 Œ0; 1 for s 2 Œ0; 1 into the form (2.2); N .dt / and .ds/ are image measures under this transformation. Van Herk only observes that his class ¹F º contains S, but comparing [361, Theorem 7.50] with our result (7.2) in Chapter 7 below shows that ¹F º D S. In his paper [162] F. Hirsch introduces Stieltjes transforms into potential theory and identifies S as a convex cone operating on the abstract potentials, i.e. the densely defined inverses of the infinitesimal generators of C0 -semigroups. Hirsch establishes several properties of the cone S, which are later extended by Berg [32, 33]. A presentation of this material from a potential theoretic point of view is contained in the monograph [39, Chapter 14] by Berg and Forst, the connections to the moment problem are surveyed in [35]. Theorem 2.2 appears in Hirsch [162, Proposition 1], but with a different proof. In contrast to completely monotone functions, S is not closed under multiplication. This follows easily since the (necessarily unique) representing measure of the function

20

Chapter 2 Stieltjes functions

 7! . C a/1 . C b/1 , 0 < a < b, is .b  t /1 ıa .dt / C .a  t /1 ıb .dt / which is a signed measure. We will see in Proposition 7.13 that S is still logarithmically convex. It is known, Rsee Hirschman and Widder [167, VII the product R R 7.4], that 2 1  .dt / 1  .dt / is of the form .Ct / .Ct / .Ct /  .dt / for 1 2 .0;1/ .0;1/ .0;1/ some measure  . The latter integral is often called a generalized Stieltjes transform. The following related result is due to Srivastava and Tuan [335]: if f 2 Lp .0; 1/ and g 2 Lq .0; 1/ with 1 < p; q < 1 and r 1 D p 1 C q 1 < 1, then there is some h 2 Lr .0; 1/ such that L 2 .f I /L 2 .gI / D L 2 .hI / holds. Since Z 1 Z 1 g.u/ f .u/ du C g.t /  p.v. du; h.t / D f .t /  p.v. ut ut 0 0 h will, in general, change its sign even if f; g are non-negative.

Chapter 3

Bernstein functions

We are now ready to introduce the class of Bernstein functions which are closely related to completely monotone functions. The notion of Bernstein functions goes back to the potential theory school of A. Beurling and J. Deny and was subsequently adopted by C. Berg and G. Forst [39], see also [35]. S. Bochner [63] calls them completely monotone mappings (as opposed to completely monotone functions) and probabilists still prefer the term Laplace exponents, see e.g. Bertoin [47, 49]; the reason will become clear from Theorem 5.2. Definition 3.1. A function f W .0; 1/ ! R is a Bernstein function if f is of class C 1 , f ./  0 for all  > 0 and .1/n1 f .n/ ./  0 for all n 2 N and  > 0:

(3.1)

The set of all Bernstein functions will be denoted by BF. It is easy to see from the definition that, for example, the fractional powers  7! ˛ , are Bernstein functions if, and only if, 0  ˛  1. The key to the next theorem is the observation that a non-negative C 1 -function f W .0; 1/ ! R is a Bernstein function if, and only if, f 0 is a completely monotone function. Theorem 3.2. A function f W .0; 1/ ! R is a Bernstein function if, and only if, it admits the representation Z f ./ D a C b C .1  e t / .dt /; (3.2) .0;1/

R where a; b  0 and  is a measure on .0; 1/ satisfying .0;1/ .1 ^ t / .dt / < 1. In particular, the triplet .a; b; / determines f uniquely and vice versa. Proof. Assume that f is a Bernstein function. Then f 0 is completely monotone. According to Theorem 1.4, there exists a measure  on Œ0; 1/ such that for all  > 0 Z 0 e t .dt /: f ./ D Œ0;1/

22

Chapter 3 Bernstein functions

Let b WD ¹0º. Then Z f ./  f .0C/ D

 0

f 0 .y/ dy D b C

Z 0

Z D b C

Z

e yt .dt / dy .0;1/

.0;1/

1  e t .dt /: t

Write a WD f .0C/ and define .dt / WD t 1 j.0;1/ .dt /. Then the calculation from above shows that (3.2) is true. That a; b  0 is obvious, and from the elementary (convexity) estimate .1  e 1 /.1 ^ t /  1  e t ; we infer Z .0;1/

.1 ^ t / .dt / 

e e1

Z .0;1/

t  0;

.1  e t / .dt / D

e f .1/ < 1: e1

Conversely, suppose that f is given by (3.2) with .a; b; / as in the statement of the theorem. Since t e t  t ^ .e/1 , we can apply the differentiation lemma for parameter-dependent integrals for all  from Œ;  1  and all  > 0. Differentiating (3.2) under the integral sign yields Z Z e t t .dt / D e t .dt /; f 0 ./ D b C .0;1/

Œ0;1/

where .dt / WD t.dt / C bı0 .dt /. This formula shows that f 0 is a completely monotone function. Therefore, f is a Bernstein function. Because f .0C/ D a and because of the uniqueness assertion of Theorem 1.4 it is clear that .a; b; / and f 2 BF are in one-to-one correspondence. Remark 3.3. (i) The representing measure  and the characteristic triplet .a; b; / from (3.2) are often called the Lévy measure and the Lévy triplet of the Bernstein function f . The formula (3.2) is called the Lévy–Khintchine representation of f . (ii) A useful variant of the representation formula (3.2) can be obtained by an application of Fubini’s theorem. Since Z Z Z .1  e t / .dt / D e s ds .dt / .0;1/ .0;1/ .0;t/ Z 1Z e s .dt / ds D 0 .s;1/ Z 1 D e s .s; 1/ ds 0

23

Chapter 3 Bernstein functions

we get that any Bernstein function can be written in the form Z e s M.s/ ds f ./ D a C b C 

(3.3)

.0;1/

a non-increasing, right-continuous function. where M.s/ D M .s/ D .s; 1/ is R1 R1 Integration by parts and the fact that 0 se s ds D 2 and 0 e s ds D 1 yield Z e s k.s/ ds D 2 L .kI / (3.4) f ./ D 2 .0;1/

Rs

with k.s/ D as C b C 0 M.t / dt , compare with Theorem 6.2 (iii). Note that k is positive, non-decreasing and concave. R (iii) The integrability condition .0;1/ .1 ^ t / .dt / < 1 ensures that the integral in (3.2) converges for some, hence all,  > 0. This is immediately seen from the convexity inequalities t t  1  e t  1 ^ t  2 ; 1Ct 1Ct

t >0

and the fact that for   1 [respectively for 0 <  < 1] and all t > 0  1 ^ t  1 ^ .t /  .1 ^ t / respectively .1 ^ t /  1 ^ .t /  1 ^ t : In particular, if  is a measure on .0; 1/ such that for some  > 0, then  is a Lévy measure.

R

.0;1/ .1

 e t / .dt / < 1

(iv) A useful consequence of the above estimate and the representation formula (3.2) are the following formulae to calculate the coefficients a and b: a D f .0C/

f ./ : !1 

and b D lim

The first formula is obvious while the second follows from (3.2) and the dominated convergence theorem: 1  e t  1 ^ .t / and lim!1 .1  e t /= D 0. (v) Formula (3.3) shows, in particular, that Z

Z

1 0

.t; 1/ dt D

0

1

M.t / dt < 1:

(3.5)

Since a non-increasing function which is integrable near zero is o.1=t / as t ! 0, we conclude from (3.5) that lim t.t; 1/ D lim tM.t / D 0:

t!0C

t!0C

(3.6)

24

Chapter 3 Bernstein functions

(vi) Since the derivative of a Bernstein function f is completely monotone, Remark 1.5 shows that f .n/ ./ ¤ 0 for all n 2 N and  > 0, unless  vanishes. Lemma 3.4. Let f be a Bernstein function given by (3.2) where a D b D 0. Set R   I ./ WD 0  s; 1 ds. Then for  > 0, 1 1 e1  I  f ./   I : e   Proof. By Fubini’s theorem we find  I

1

Z

1=

D .s; 1/ ds  0 Z 1   t  ; 1 dt D  0 Z 1Z 1 D .dr/ dt 0 t= Z 1   D 1 ^ .r/ .dr/: 0

r  1 ^ r, r  0, we get Using the elementary inequalities e1 e .1 ^ r/  1  e Z 1 1 Z 1 e  1   e1  I D 1 ^ .r/ .dr/  .1  e r / .dr/ D f ./: e  e 0 0

The upper bound follows similarly. The derivative of a Bernstein function is completely monotone. The converse is only true, if the primitive of a completely monotone function is positive. This fails, for example, for the completely monotone function 2 whose primitive, 1 , is not a Bernstein function. The next proposition characterizes the image of BF under differentiation. R Proposition 3.5. Let g./ D b C .0;1/ e t .dt / be a completely monotone function. It has a primitive f 2 BF if, and only if, the representing measure  satisfies R 1 .dt / < 1. .0;1/ .1 C t / Proof. Assume that f is a Bernstein function given in the form (3.2). Then Z f 0 ./ D b C e t t .dt / .0;1/

is completely monotone and the measure .dt / WD t.dt / satisfies Z Z Z 1 t .dt / D .dt /  .1 ^ t / .dt / < 1: .0;1/ 1 C t .0;1/ 1 C t .0;1/

25

Chapter 3 Bernstein functions

R Retracing the above steps reveals that .0;1/ .1 C t /1 .dt / < 1 is also sufficient R to guarantee that g./ WD b C .0;1/ e t .dt / has a primitive which is a Bernstein function. Theorem 3.2 allows us to extend Bernstein functions analytically onto the right !  !  complex half-plane H WD ¹z 2 C W Re z > 0º and continuously onto the closure H. !  !  Proposition 3.6. Every f 2 BF has an extension f W H ! H which is continuous !  for Re z  0, holomorphic for Re z > 0 and satisfies f .z/ D f .Nz/ for all z 2 H. This extension has the following representation Z 2 1 z f .z/ D a C bz C Re f .i t / dt; Re z  0: (3.7) 2  0 z C t2 In particular, f preserves angular sectors, f .SŒ ;  /  SŒ ; 

with SŒ ;  D ¹z 2 C W j arg zj   º;  2 Œ0; /: (3.8)

Proof. The function  7! 1  e t appearing in (3.2) has a unique holomorphic extension. If z D  C i  is such that  D Re z > 0 we get ˇZ zt ˇ ˇ ˇ zt 

ˇ j1  e j D ˇ e d  ˇˇ  t jzj and j1  e zt j  1 C je zt j  2: 0

Therefore the integral appearing in (3.2) converges uniformly even if we replace the !  real variable  by z 2 H. This means that f .z/ is well defined and holomorphic on !  H such that f .z/ D f .Nz/. Moreover, Z Re f .z/ D a C bRe z C Re .1  e zt / .dt / Z .0;1/   D a C b C 1  e t cos.t / .dt / .0;1/

which is positive since  D Re z  0 and 1  e t cos.t /  1  e t  0. Continuity up to the boundary follows from the estimate Z jf .z/  f .w/j  bjz  wj C je wt  e zt j .dt / .0;1/ Z 2 ^ .t jw  zj/ .dt /  bjz  wj C .0;1/

!  for all z; w 2 H and the dominated convergence theorem. For z D 0 essentially the !  same calculation shows for all w 2 H ˇ ˇ Z ˇ f .w/   jwj!1 a ˇˇ 1 ˇ  b  2 ^ t jwj .dt / ! 0:  ˇ ˇ w w jwj .0;1/

26

Chapter 3 Bernstein functions

Without loss of generality we may assume that b D 0, otherwise we could consider !  f .z/  bz instead of f . Fix z 2 H and denote by h;R the closed integration contour consisting of a straight line from h C iR to h  iR which is closed by a semicircular arc of radius R centered at h. The point z is surrounded by h;R if 0 < h < Re z and R > jzj. This is shown in the picture. Since f is holomor!  phic in H we can use Cauchy’s theorem to get Z f ./ 1 d f .z/ D 2 i h;R   z 1 D 2

ZR R

1 C 2

6 h + iR −z  R

f .h  i t / dt h  it  z

Z=2

=2

-

h z

f .h C Re i! / Re i! d!: h C Re i!  z

h − iR

Since the integrands are continuous, we can let h ! 0 and find Z R Z =2 f .i t / f .Re i! / 1 1 dt C Re i! d!: f .z/ D 2 R i t C z 2 =2 Re i!  z The same argument yields for the point z which is not surrounded by the contour

h;R 1 0D 2

Z

R R

f .i t / 1 dt C it  z 2

Z

=2 =2

f .Re i! / Re i! d!: Re i! C z

If we subtract the last two equalities we arrive at Z Z z 1 R 1 =2 Re i! f .Re i! / z d!: f .z/ D f .i t / dt C  R z 2 C t 2  =2 Re i!  z Re i! C z !  For large values of R and fixed z 2 H, the integrand of the second integral is bounded.  ! f ./= D 0, we conclude that Since lim j j!1; 2 H

Z z 1 1 f .i t / dt f .z/ D 2  1 z C t 2 Z 1   1 z f .i t / C f .i t / dt: D 2 2 2 1 z C t

Finally, f .i t /C f .i t / D f .i t /C f .i t / D 2Re f .i t / and the formula (3.7) follows.

27

Chapter 3 Bernstein functions

From (3.7) we deduce immediately (3.8): if z D re i ; w D Re i 2 C, the argument of w C z lies between  and  . A similar statement is true for integrals: Z  arg f .t; z/ .dt / 2 inf t arg f .t; z/; sup t arg f .t; z/ : Note that for all t > 0 z2

re i re i z D 2 i2 D 0 i2 0 2 Ct r e Ct re

where 0 < j 0 j < j j:

z Thus, j arg. z 2 Ct 2 /j  j arg zj, and the same is true for f .z/ because of the representation (3.7).

The following structural characterization comes from Bochner [63, pp. 83–84] where Bernstein functions are called completely monotone mappings. Theorem 3.7. Let f be a positive function on .0; 1/. Then the following assertions are equivalent. (i) f 2 BF. (ii) g ı f 2 CM for every g 2 CM. (iii) e uf 2 CM for every u > 0. Proof. The proof relies on the following formula for the n-th derivative of the composition h D g ı f due to Faa di Bruno [109], see also [134, formula 0.430]: !ij ` X  Y f .j / ./ nŠ .n/ .m/ g f ./ (3.9) h ./ D i1 Š    i` Š jŠ where

P

j D1

.m;i1 ;:::;i` /

.m;i1 ;:::;i` / stands for summation over P P that j`D1 j  ij D n and j`D1 ij D m.

all ` 2 N and all i1 ; : : : ; i` 2 N [ ¹0º

such (i))(ii)Assume that f 2 BF and g 2 CM. Then h./ D g.f .//  0. Multiply P formula (3.9) by .1/n and observe that n D m C j`D1 .j  1/  ij . The assumptions f 2 BF and g 2 CM guarantee that each term in the formula multiplied by .1/n is non-negative. This proves that h D g ı f 2 CM. (ii))(iii) This follows from the fact that g./ WD gu ./ WD e u , u > 0, is completely monotone. P j j (iii))(i) The series e uf ./ D j1D0 .1/j Š u Œf ./j and all of its formal derivan

d uf ./ by termwise diftives (w.r.t. ) converge uniformly, so we can calculate d ne uf is completely monotone, we get ferentiation. Since e

0  .1/n

1 n  j d n uf ./ X uj nCj d f ./ : .1/ e D n n d jŠ d j D1

28

Chapter 3 Bernstein functions

Dividing by u > 0 and letting u ! 0 we see 0  .1/nC1

dn f ./: d n

Theorems 3.2 and 3.7 have a few important consequences. Corollary 3.8. (i) The set BF is a convex cone: sf1 Ctf2 2 BF for all f1 ; f2 2 BF and s; t  0. (ii) The set BF is closed under pointwise limits: if .fn /n2N  BF and if the limit limn!1 fn ./ D f ./ exists for every  > 0, then f 2 BF. (iii) The set BF is closed under composition: if f1 ; f2 2 BF, then f1 ı f2 2 BF. In particular,  7! f1 .c/ is in BF for any c > 0. (iv) For all f 2 BF the function  7! f ./= is in CM. (v) f 2 BF is bounded if, and only if, in (3.2) b D 0 and .0; 1/ < 1. (vi) Let f1 ; f2 2 BF and ˛; ˇ 2 .0; 1/ such that ˛Cˇ  1. Then  7! f1 .˛ /f2 .ˇ / is again a Bernstein function. (vii) Let f 2 BF. For every  > 0 the function f ./ WD f ./ C f ./  f . C /,  > 0, is a Bernstein function. In particular, f 2 BF is subadditive, i.e. f . C /  f ./ C f ./ for all ;  > 0. (viii) For any f 2 BF, there exists c > 0 such that f ./  c for all  > 1. Proof. (i) This follows immediately from Definition 3.1 or, alternatively, from the representation formula (3.2). (ii) For every u > 0 we know that e ufn is a completely monotone function and that e uf ./ D limn!1 e ufn ./ . Since CM is closed under pointwise limits, cf. Corollary 1.6, e uf is completely monotone and f 2 BF. (iii) Let f1 ; f2 2 BF. For any g 2 CM we use the implication (i))(ii) of Theorem 3.7 to get g ı f1 2 CM, and then g ı .f1 ı f2 / D .g ı f1 / ı f2 2 CM. The converse direction (ii))(i) of Theorem 3.7Rshows that f1 ı f2 2 BF. t (iv) Note that .1  e t /= D 0 e s ds is completely monotone. Therefore, a f ./ D CbC  

Z .0;1/

1  e t .dt / 

is the limit of linear combinations of completely monotone functions which is, by Corollary 1.6, completely monotone.

29

Chapter 3 Bernstein functions

(v) That b D 0 and .0; 1/ < 1 imply the boundedness of f is clear from the representation (3.2). Conversely, if f is bounded, b D 0 follows from Remark 3.3 (iv), and .0; 1/ < 1 follows from (3.2) and Fatou’s lemma. (vi) We know that the fractional powers  7! ˛ , 0  ˛  1, are Bernstein functions. Since h./ WD f1 .˛ /f2 .ˇ / is positive, it is enough to show that the derivative h0 is completely monotone. We have h0 ./ D ˛f10 .˛ /˛1 f2 .ˇ / C ˇf20 .ˇ /ˇ 1 f1 .˛ / D ˛Cˇ 1

! ˇ ˛ . / . / f f 2 1 C ˇf20 .ˇ / ˛f10 .˛ / : ˛ ˇ

Note that f10 ./; f20 ./ and, by part (iv),  1 f1 ./,  1 f2 ./ are completely monotone. By Theorem 3.7 (ii) the functions f10 .˛ /; f20 .ˇ /; f1 .˛ /=˛ and f2 .ˇ /=ˇ are again completely monotone. Since ˛ C ˇ  1,  7! ˛Cˇ 1 is completely monotone. As sums and products of completely monotone functions are in CM, see Corollary 1.6, h0 is completely monotone. (vii) Using the representation (3.2) of the Bernstein function f we get Z f ./ D a C b C .1  e t / .dt /; .0;1/

and so

Z f ./ D a C

.0;1/

.1  e t /.1  e t / .dt /:

It is easy to see that .1  e t / .dt / is, for each  > 0, a Lévy measure. The subadditivity of f follows now from f ./  0. (viii) Since f is concave, we have f ./  f .1/ for all   1. Just as for completely monotone functions, the closure assertion of Corollary 3.8 says that on the set BF the notions of pointwise convergence, locally uniform convergence, and even convergence in the space C 1 coincide. Corollary 3.9. Let .fn /n2N be a sequence of Bernstein functions such that the limit limn!1 fn ./ D f ./ exists for all  2 .0; 1/. Then f is a Bernstein function and for all k 2 N [ ¹0º the convergence limn!1 fn.k/ ./ D f .k/ ./ is locally uniform in  2 .0; 1/. If .an ; bn ; n / and .a; b; / are the Lévy triplets for fn and f , respectively, see (3.2), we have lim n D  vaguely in .0; 1/;

n!1

30

Chapter 3 Bernstein functions

and   a D lim lim inf an C n ŒR; 1/ ; R!1 n!1   Z t n .dt / : b D lim lim inf bn C !0 n!1

.0;/

In both formulae we may replace lim inf n by lim supn . Proof. From Corollary 3.8 (ii) we know that f is a Bernstein function. Obviously, limn!1 e fn D e f ; by Theorem 3.7, the functions e fn ; e f are completely monotone and we can use Corollary 1.7 to conclude that n!1

e fn ! e f

n!1

and .fn0 / e fn ! .f 0 / e f

converge locally uniformly on .0; 1/. In particular, limn!1 fn0 ./ D f 0 ./ for each  2 .0; 1/. Again by Corollary 1.7 and the complete monotonicity of fn0 ; f 0 we see .k/ that for k  1 the derivatives fn converge locally uniformly to f .k/ . By the mean value theorem, ˇ ˇ ˇ ˇ jfn ./  f ./j D ˇ log e f ./  log e fn ./ ˇ  C ˇe f ./  e fn ./ ˇ with C  e f ./Cfn ./ . The locally uniform convergence of e fn ensures that C is bounded for n 2 N and  from compact sets in .0; 1/; this proves locally uniform convergence of fn to f on .0; 1/. Differentiating the representation formula (3.2) we get Z Z   0 t te n .dt / D e t bn ı0 .dt / C t n .dt / ; fn ./ D bn C .0;1/

Œ0;1/

implying that bn ı0 .dt / C t n .dt / converge vaguely to bı0 .dt / C t .dt /. This proves at once that n !  vaguely on .0; 1/ as n ! 1. Since bn ı0 .dt / C t n .dt / converge vaguely to bı0 .dt / C t .dt /, we can use the portmanteau theorem, Theorem A.7, to conclude that   Z Z t n .dt / D b C t .dt / lim bn C n!1

.0;/

.0;/

at all continuity points  > 0 of . If j > 0 is a sequence of continuity points of  such that j ! 0, we get   Z b D lim lim bn C t n .dt / : j !1 n!1

.0;j /

31

Chapter 3 Bernstein functions

For a sequence of arbitrary j ! 0 we find continuity points ıj ; j , j 2 N, of  such that 0 < ıj  j  j and ıj ; j ! 0. Thus, Z Z Z bn C t n .dt /  bn C t n .dt /  bn C t n .dt /; .0;ıj /

.0;j /

and we conclude that  Z lim lim bn C j !1 n!1

.0;ıj /

.0; j /

  Z t n .dt /  lim lim inf bn C j !1 n!1



 t n .dt / .0;j /

Z

 lim lim sup bn C j !1 n!1

 Z  lim lim bn C j !1 n!1

 t n .dt /

.0;j /

 t n .dt / :

.0; j /

Since both sides of the inequality coincide, the claim follows. Using a D f .0C/ we find for each R > 1 a D lim f ./ !0

D lim lim fn ./ !0 n!1   Z D lim lim an C bn  C .1  e t / n .dt / !0 n!1 .0;1/  Z D lim lim an C .1  e t / n .dt / !0 n!1 ŒR;1/  Z .1  e t / n .dt / : C bn  C .0;R/

From the convexity estimate .1  e 1 /.1 ^ t /  .1  e t /;

t  0;

we obtain .1  e t / 1.0;R/ .t /  t 1.0;R/ .t /  .t ^ R/  R.1 ^ t /  R so that bn  C

Z .0;R/

.1  e t / n .dt /  bn  C

R .1  e 1 /

R fn .1/:  .1  e 1 /

Z .0;R/

.1  e t / .1  e 1 /

.1  e t / n .dt /

32

Chapter 3 Bernstein functions

Since limn!1 fn .1/ D f .1/, we get  Z a D lim lim an C !0 n!1

ŒR;1/

 .1  e t / n .dt / :

For any continuity point R > 1 of , we have by vague convergence Z Z !0 R!1 t lim e n .dt / D e t .dt / ! ŒR; 1/ ! 0: n!1 ŒR;1/

ŒR;1/

Letting R ! 1 through a sequence of continuity points Rj , j 2 N, of  we get   a D lim lim an C n ŒRj ; 1/ : j !1 n!1

That we do not need to restrict ourselves to continuity points Rj follows with a similar argument as for the coefficient b. Example 3.10. The proof of Corollary 3.9 shows that the vague limit  does not capture the accumulation of mass at  D 0 and  D 1 of the n as n ! 1. These effects can cause the appearance of a > 0 and b > 0 in the Lévy triplet of the limiting function f , even if an D bn D 0 for all functions fn . Here are two extreme cases: n!1

fn ./ D n.1  e =n / !  D f ./; i.e. .an ; bn ; n / D .0; 0; nı1=n / and .a; b; / D .0; 1; 0/, and n!1

fn ./ D 1  e n ! 1 D f ./ where .an ; bn ; n / D .0; 0; ın / and .a; b; / D .1; 0; 0/. Remark 3.11. Let .fn /n2N be a sequence of Bernstein functions and set gn WD e fn . By Theorem 3.7, .gn /n2N is a sequence of completely monotone functions satisfying gn .0C/  1. It follows from Corollary 1.8 that there exist a subsequence .gnk /k2N and some g 2 CM such that limk!1 gnk ./ D g./ for all  2 .0; 1/. Note that g may be identically equal to zero. Let f WD  log g (where we use the convention that log 0 D 1). Then f W .0; 1/ ! .0; 1 and if f ./ < 1 for some  > 0, we see that f 2 BF. There is a one-to-one correspondence between bounded Bernstein functions and bounded completely monotone functions. Proposition 3.12. If g 2 CM is bounded, then g.0C/  g 2 BF. Conversely, if f 2 BF is bounded, there exist some constant c > 0 and some bounded g 2 CM, lim!1 g./ D 0, such that f D c  g; then c D f .0C/ C g.0C/.

33

Chapter 3 Bernstein functions

Proof. Assume that g D L  2 CM is a bounded function. This means that .0; 1/ D g.0C/ D sup>0 g./ < 1. Hence, f ./ WD g.0C/  g./ D R t / .dt / is a bounded Bernstein function. .0;1/ .1  e R Conversely, if f 2 BF is bounded, f ./ D a C .0;1/ .1  e t /.dt / for some bounded measure , cf. Corollary 3.8. Thus f ./ D c g./ where g./ D L .I / is in CM, lim!1 g./ D 0 and c D a C .0; 1/ D f .0C/ C g.0C/ > 0. Remark 3.13. Just as in the case of completely monotone functions, see Remark 1.9, we can understand the representation formula (3.2) as a particular case of a Choquet or Kre˘ın–Milman representation. The set ² ³ Z f 2 BF W f ./e  d  D 1 .0;1/

is a basis of the convex cone BF, and its extremal points are given by e0 ./ D ;

e t ./ D

1Ct .1  e t /; t

0 < t < 1;

and e1 ./ D 1;

see Harzallah [149] or [320, Satz 2.9]. These functions are, of course, examples and building blocks for all Bernstein functions: every f 2 BF can be written as an ‘integral mixture’ of the above extremal Bernstein functions. For .dt / D ˛=.1  ˛/ t 1˛ dt , ˛ 2 .0; 1/, .dt / D e t , or .dt / D t 1 e t we see that the functions ˛ .0 < ˛ < 1/;

or

 1C

or

log.1 C /

are Bernstein functions. Comments 3.14. The name Bernstein function is not universally accepted in the literature. It originated in the potential theory school of A. Beurling and J. Deny, but the name as such does not appear in Beurling’s or Deny’s papers. The earliest mentioning of Bernstein functions as well as a nice presentation of their properties is Faraut [111]. Bochner [63] calls Bernstein functions completely monotone mappings but this notion was only adopted in the 1959 paper by Woll [375]. Other names include inner transformations of CM (Schoenberg [325]), Laplace or subordinator exponents (Bertoin [47, 49]), log-Laplace transforms or positive functions with completely monotone derivative (Feller [118]). Bochner pointed out the importance of Bernstein functions already in [62] and [63]; he defines them through the equivalent property (ii) of Theorem 3.7. The equivalence of (i) and (ii) in Theorem 3.7 is already present in Schoenberg’s paper [325, Theorem 8]; Schoenberg defines Bernstein functions as primitives of completely monotone functions and calls them inner transformations (of completely monotone functions); his notation is T for the class BF. Many properties as well as higher-dimensional

34

Chapter 3 Bernstein functions

analogues are given in [63, Chapter 4]. In particular, Theorems 3.2 and 3.7 can already be found there. An up-to-date account is given in the books by Berg and Forst [39, Chapter 9] (based on [111]) and Berg, Christensen and Ressel [38]. One can understand (3.2) as a Lévy–Khintchine formula for the semigroup .Œ0; 1/; C/. This justifies the name Lévy measure and Lévy triplet for  and .a; b; /, respectively; this point of view is taken in [38]. The representation (3.4) is taken from Prüss [297, Chapter I.4.1] who calls the functions k.s/ (of the type needed in (3.4), i.e. positive, non-decreasing and concave) creep functions. The estimate in Lemma 3.4 is from Bertoin [47, Proposition III.1.1, p. 74]. Bertoin’s proof mimics the arguments of the de-Haan–Stadtmüller Tauberian theorem [58, Theorem 2.10.2, p. 118]. Our short proof is taken from [324] where techniques from Ôkura [284, Theorem 3.2] are used. The representation formula (3.7) and the contraction property (3.8) were communicated to us by Sonia Fourati [119]. Our proof is based on techniques from the theory of harmonic functions in a half-plane, cf. Levin [241, Chapter V.§2, pp. 230–236] or Koosis [213, Chapter III.H.1, pp. 58–65]. Levin proves the following necessary !  !  and sufficient condition: Assume that g is harmonic1 in H, continuous in H and lim supjzj!1 jg.z/j=jzj < 1. Then g has the following representation g.z/ D

1 

Z

1

1

Re z g.i t / dt C k Re z jz  i t j2

with k D lim sup !0 g./= if, and only if, Z

1 1

jg.i t /j dt < 1: 1 C t2

Using the harmonic conjugate, cf. [213], this leads to (3.7). The recent preprint [120] contains an alternative proof of Proposition 3.6 based on negative definite functions. The assertions in Corollary 3.8 are mostly standard. We learned the nice argument in (vii) from Wissem Jedidi [119] who also pointed out the following converse of (vii): if for a smooth and increasing function f W .0; 1/ ! Œ0; 1/ we know that f 2 BF for all  > 0, then f 2 BF. The characterization of Bernstein functions using Choquet’s representation is due to Harzallah [147, 148, 149], a slightly different version is given in [320], reprinted in [183, Vol. 1, Theorem 3.9.20].

1

The English edition states holomorphic, but this is a misprint, see the German translation.

Chapter 4

Positive and negative definite functions

Positive and negative definite functions appear naturally in connection with Fourier analysis and potential theory. In this chapter we will see that (bounded) completely monotone functions correspond to continuous positive definite functions on .0; 1/ and that Bernstein functions correspond to continuous negative definite functions on the half-line. On Rd the notion of positive definiteness is familiar to most readers; for our purposes it is useful to adopt a more abstract point of view which includes both settings, Rd and the half-line Œ0; 1/. An Abelian semigroup with involution .S; C; / is a nonempty set equipped with a commutative and associative addition C, a zero element 0 and a mapping  W S ! S called involution, satisfying (a) .s C t / D s  C t  for all s; t 2 S ; (b) .s  / D s for all s 2 S . We are mainly interested in the semigroups .Œ0; 1/; C/ and .Rd ; C/ where ‘C’ is the usual addition; the respective involutions are the identity mapping s 7! s in Œ0; 1/ and the reflection at the origin 7!  in Rd . We will only develop those parts of the theory that allow us to characterize CM and BF as positive and negative definite functions on the half-line. Good expositions of the general case are the monographs by Berg, Christensen, Ressel [38] (for semigroups) and Berg, Forst [39] (for Abelian groups). Definition 4.1. A function f W S ! C is positive definite if n X

f .sj C sk / cj cNk  0

(4.1)

j;kD1

holds for all n 2 N, all s1 ; : : : ; sn 2 S and all c1 ; : : : ; cn 2 C. We need some simple properties of positive definite functions. Lemma 4.2. Let f W S ! C be positive definite. Then f .s C s  /  0 for all s 2 S and f is hermitian, i.e. f .s  / D f .s/. In particular, f .0/  0. Lemma 4.2 shows also that positive definite functions on Œ0; 1/ are non-negative: f  0.

36

Chapter 4 Positive and negative definite functions

Proof of Lemma 4.2. That f .s C s  /  0 follows immediately from (4.1) if we take n D 1 and s1 D s; f .0/  0 is now obvious. Again by (4.1) we see that the matrix .f .sj C sk // is (non-negative definite) hermitian. Thus we find with n D 2 and s1 D 0, s2 D s that 

  >   f .0/ f .s/ f .0/ f .s  / f .0/ f .s  / D D : f .s  / f .s C s  / f .s/ f .s C s  / f .s/ f .s C s  /

Comparing the entries of the matrices we conclude that f .s  / D f .s/. Definition 4.3. A function f W S ! C is negative definite if it is hermitian, i.e. f .s  / D f .s/, and if n X 

 f .sj / C f .sk /  f .sj C sk / cj cNk  0

(4.2)

j;kD1

holds for all n 2 N, all s1 ; : : : ; sn 2 S and all c1 ; : : : ; cn 2 C. Note that ‘f is negative definite’ does not mean that ‘f is positive definite’. The connection between those two concepts is given in the following proposition. Proposition 4.4 (Schoenberg). For a function f W S ! C the following assertions are equivalent. (i) f is negative definite. (ii) f .0/  0, f .s  / D f .s/ and f is conditionally positive definite, i.e. for all P n 2 N, all s1 ; : : : ; sn 2 S and all c1 ; : : : ; cn 2 C satisfying jnD1 cj D 0 one has n X f .sj C sk / cj cNk  0: (4.3) j;kD1

(iii) f .0/  0 and s 7! e tf .s/ is positive definite for all t > 0. Proof. (i))(iii) The Schur (or Hadamard) product of two n n matrices A D .aj k / and B D .bj k / is the n n matrix C with entries aj k bj k . If A; B are non-negative definite hermitian matrices, then C is non-negative definite hermitian. Indeed, writing B D PP  where P D .pj k / 2 Cnn and P  is the adjoint matrix of P , we have bj k D

n X `D1

pj ` pNk` :

37

Chapter 4 Positive and negative definite functions

For every choice of c1 ; : : : ; cn 2 C we get # " n n n X X X aj k bj k cj cNk D aj k .pj ` cj /.pk` ck /  0: j;kD1

`D1

j;kD1

In particular, if .aj k / is a non-negative definite hermitian matrix, so is .exp.aj k //. If we apply this to the non-negative definite hermitian matrices   f .sj / C f .sk /  f .sj C sk / j;kD1;:::;n ; n 2 N; s1 ; : : : ; sn 2 C; we conclude that



  exp f .sj / C f .sk /  f .sj C sk /

j;kD1;:::;n

is a non-negative definite hermitian matrix. This implies that for all c1 ; : : : ; cn 2 C n X

e

f .sj Csk /

cj cNk D

j;kD1

D

n X j;kD1 n X



e f .sj /Cf .sk /f .sj Csk / e f .sj / e f .sk / cj cNk 

e f .sj /Cf .sk /f .sj Csk / j Nk  0;

j;kD1

where j WD cj e f .sj / 2 C. This proves that e f is positive definite. Replacing f by tf , t > 0, the same argument shows that e tf is positive definite. That f .0/  0 follows directly from the definition if we take n D 1, s1 D 0 and c1 D 1. P (iii))(ii) Let n 2 N, s1 ; : : : ; sn 2 S and c1 ; : : : ; cn 2 C with jnD1 cj D 0. Because of (iii) we see that for all t > 0 n n X X  1 1 tf .sj Cs  / tf .sj Csk / k c c 1e e cj cNk D  j Nk  0: t t

j;kD1

j;kD1

Letting t ! 0 we obtain n X

f .sj C sk / cj cNk  0:

j;kD1

By Lemma 4.2, the functions e tf are hermitian and so is f D lim t!0 .1  e tf /=t . (ii))(i) We use (4.3) with P n C 1 instead of n and for 0; s1 ; : : : ; sn 2 S and c; c1 ; : : : ; cn 2 C where c D  nkD1 ck . Then f .0/jcj2 C

n X j D1

f .sj / cj cN C

n X j D1

f .sj / c cNj C

n X j;kD1

f .sj C sk / cj cNk  0:

38

Chapter 4 Positive and negative definite functions

Using the fact that f .sj / D f .sj / and inserting the definition of c, we can rewrite this expression and find n X 

 f .sj / C f .sk /  f .sj C sk / cj cNk  f .0/jcj2  0:

j;kD1

We will now characterize the bounded continuous positive definite and continuous negative definite functions on the semigroup .Œ0; 1/; C/ with involution  D ; it will turn out that these functions coincide with the families CMb and BF, respectively. Since we are working on the closed half-line Œ0; 1/, it is useful to extend f 2 CMb or f 2 BF continuously to Œ0; 1/. Because of the monotonicity of f this can be achieved by f .0/ WD f .0C/ D lim f ./: !0

Note that this extension is unique and that, for a completely monotone function f , f .0C/ < 1 if, and only if, f is bounded. Throughout this chapter we will tacitly use this extension whenever necessary. Lemma 4.5. Let f 2 CMb . Then f is positive definite in the sense of Definition 4.1. Proof. From Theorem 1.4 we know that f 2 CMb is the Laplace transform of a finite measure  on Œ0; 1/. Therefore, we find for all n 2 N, all 1 ; : : : ; n 2 Œ0; 1/ and all c1 ; : : : ; cn 2 C  n n Z X X j t k t f .j C k /cj cNk D e e .dt / cj cNk j;kD1

j;kD1

Z

D D

n X Œ0;1/

Z

Œ0;1/

! e

j t

cj e k t ck .dt /

j;kD1

ˇX ˇ ˇ n  t ˇ2 j ˇ e cj ˇˇ .dt /  0: ˇ

Œ0;1/ j D1

Corollary 4.6. Let f 2 BF. Then f is negative definite in the sense of Definition 4.3. Proof. Let f 2 BF. From Theorem 3.7 we know that e tf 2 CM for all t > 0. Since e tf is bounded, we can use Lemma 4.5 to infer that for each t > 0 (the unique extension of) the function e tf is positive definite, and we conclude with Proposition 4.4 that (the unique extension of) f is negative definite. We write a f ./ WD f . C a/  f ./ for the difference of step a > 0. The iterated differences of step sizes aj > 0, j D 1; : : : ; n, are defined by an    a1 f WD an . an1    a1 f /:

39

Chapter 4 Positive and negative definite functions

Theorem 4.7. Every bounded continuous positive definite function f on .Œ0; 1/; C/ satisfies (4.4) .1/n an    a1 f  0 for all n 2 N; a1 ; : : : ; an > 0: Proof. Assume that f is positive definite. Then f  0. Taking n D 2 in (4.1) we see that the matrix   f .2/ f . C a/ ; ; a  0; f . C a/ f .2a/ is positive definite. Therefore its determinant is non-negative which implies p f . C a/  f .2/ f .2a/; ; a  0:

(4.5)

Applying (4.5) repeatedly, we arrive at n

f ./  f 1=2 .0/ f 1=2 .2/  f 3=4 .0/ f 1=4 .4/      f 12

n

.0/ f 2

.2n /:

Since f is bounded, we can take the limit n ! 1 to get f ./  f .0/;

  0:

For N 2 N, c1 ; : : : ; cN 2 C and 1 ; : : : ; N > 0 we introduce an auxiliary function N X F .a/ WD f .a C ` C m / c` cNm : `;mD1

For all choices of n 2 N, b1 ; : : : ; bn 2 R and a1 ; : : : ; an  0 we find n X

F .aj C ak / bj bNk D

j;kD1

n X

N X

f .aj C ak C ` C m / bj bNk c` cNm

j;kD1 `;mD1

X

D

  f .aj C ` / C .ak C m / .bj c` /.bk cm /

.j;`/;.k;m/

 0: This proves that F is positive definite. Since f is bounded and continuous, so is F , and we conclude, as above, that F .a/  F .0/. This amounts to saying that N X 

 f .` C m /  f .a C ` C m / cj cNm  0;

`;mD1

i.e. the function  a f is continuous and positive definite. Repeating the above argument shows that .1/n an : : : a1 f is continuous and positive definite for all n 2 N and a1 ; : : : ; an > 0.

40

Chapter 4 Positive and negative definite functions

Theorem 4.8 (Bernstein). For a measurable function f W .0; 1/ ! R the following conditions are equivalent. (i) f  0 and .1/n an : : : a1 f  0 for all n 2 N and a1 ; : : : ; an > 0. (ii) There exists a unique non-negative measure  on Œ0; 1/ such that Z f ./ D e t .dt /: Œ0;1/

(iii) f is infinitely often differentiable and .1/n f .n/  0 for all n 2 N. Proof. (i))(ii) Denote by B.0; 1/ the real-valued Borel measurable functions on .0; 1/. Then ® C WD f 2 B.0; 1/ W f  0 and .1/n an : : : a1 f  0 ¯ for all n 2 N and a1 ; : : : ; an > 0 is a convex cone. If we equip B.0; 1/ with the topology of pointwise convergence, its topological dual separates points. Since every f 2 C is non-increasing and nonnegative, the limit f .0C/ exists, and f is bounded if, and only if, f .0C/ < 1. For f 2 C the second differences are non-negative; in particular a a f  0 which we may rewrite as f ./ C f . C 2a/  2f . C a/;

; a > 0:

This means that f is mid-point convex. Since f jŒc;d  is finite on every compact interval Œc; d   .0; 1/, we conclude that f j.c;d / is continuous for all d > c > 0, i.e. f is continuous, see for example Donoghue [99, pp. 12–13]. Define K WD ¹f 2 C W f .0C/ D 1º; this is a closed subset of C. Since it is contained in the set Œ0; 1.0;1/ \ B.0; 1/ which is, by Tychonov’s theorem, compact under pointwise convergence, it is also compact. Moreover, K is a basis of the cone Cb since for every f 2 Cb the normalized function f =f .0C/ 2 K. Now the Kre˘ın–Milman theorem applies and shows that K is the closed convex hull of its extreme points. For f 2 Cb we write   f ./ D f . C a/ C f ./  f . C a/ D f . C a/ C .1/ a f ./: It follows directly from the definition of the set C that f . C a/ and  a f are again in Cb . Thus, if f 2 K is extremal, both functions must be multiples of f . In particular, f . C a/ D c.a/f ./;   0; a > 0: Letting  ! 0 we see that f .a/ D c.a/.

41

Chapter 4 Positive and negative definite functions

Since f .0C/ D 1 and since f is continuous, all solutions of the functional equation f . C a/ D f .a/f ./ are of the form f ./ D e t ./ D e t where t 2 Œ0; 1/. This means that the extreme points of K are contained in the set ¹e t W t  0º. Since K is the closed convex hull of its extreme points, we conclude that there exists a measure  supported in Œ0; 1/ such that Œ0; 1/ D 1 and Z e t .dt /; f 2 K: f ./ D Œ0;1/

This proves (ii) for f 2 Cb . By the uniqueness of the Laplace transform, see Proposition 1.2, the representing measure  is unique. If f 2 C is not bounded, we argue as in the proof of Theorem 1.4. In this case, fa ./ WD f . C a/ is in Cb , and we find Z e t a .dt / for all a > 0: f . C a/ D Œ0;1/

By the uniqueness of the representation, it is easy to see that the measure e at a .dt / does not depend on a > 0. Writing .dt / for e at a .dt /, we get the representation claimed in (ii) for all f 2 Cb . (ii))(iii) This is a simple application of the dominated convergence theorem, see for example the second part of the proof of Theorem 1.4. (iii))(i) By the mean value theorem we have for all a > 0 .1/ a f ./ D f ./  f . C a/ D af 0 . C a/;

 2 .0; 1/;

hence .1/ a f  0. Iterating this argument yields .1/n an : : : a1 f ./ D .1/n a1    an f .n/ . C 1 a1 C    C n an / for suitable values of 1 ; : : : ; n 2 .0; 1/. This proves (i). Corollary 4.9. The family CMb and the family of bounded continuous positive definite functions on .Œ0; 1/; C/ coincide. Proof. From Lemma 4.5 we know already that (the extension of) every f 2 CMb is positive definite. Conversely, if f is continuous and positive definite, Theorem 4.8 shows that f j.0;1/ is completely monotone. As f is continuous, we see that f .0C/ D f .0/ < 1; this implies the boundedness of f , because f is non-increasing and non-negative on .0; 1/. Corollary 4.10. The family BF and the family of continuous negative definite functions on .Œ0; 1/; C/ coincide.

42

Chapter 4 Positive and negative definite functions

Proof. Lemma 4.6 shows that (the extension of) every f 2 BF is negative definite. Now let f be continuous and negative definite. Thus, for every t  0 the (extension of the) function e tf is bounded, continuous and positive definite; by Corollary 4.9 it is in CMb , and by Theorem 3.7 we get f 2 BF. Theorem 4.8 has the following counterpart for multiply monotone functions. Theorem 4.11. For a measurable function f W .0; 1/ ! R and n  2 the following conditions are equivalent. (i) f  0 and .1/j aj : : : a1 f  0 for all j D 1; 2; : : : n and a1 ; : : : ; an > 0. (ii) There exists a unique measure n on Œ0; 1/ such that Z f ./ D .1  t /n1

n .dt /; C

 > 0:

Œ0;1/

(iii) f is n-monotone. In order to prove this theorem we will use the following lemma. For a > 0 and m 2 N, let m a denote the m-times iterated operator a . Lemma 4.12. Let f W .0; 1/ ! R be a non-negative convex function satisfying .1/mC1 mC1 f ./  0 for all  > 0, a > 0 and some m 2 N. Then the right-hand a 0 side derivative fC0 exists and satisfies .1/m m a fC ./  0 for all  > 0 and a > 0. Proof. A convex function f has at each point in the interior of its domain a righthand side derivative fC0 which is non-decreasing. For each l  1 we find from a telescoping argument a f ./ D

l1 h   X a i a f Cj f  C .j C 1/ l l

j D0

D

l1 X j D0

 a : a= l f  C j l

By expanding it follows that m a f ./ D

l1 X



j1 D0

l1 X

jm D0

 a ; m f  C .j C    C j / 1 m a= l l

which implies .1/m a= l m a f ./ D

l1 X j1 D0

0



l1 X

 a .1/m mC1 f  C .j C    C j / 1 m a= l l

jm D0

43

Chapter 4 Positive and negative definite functions

since, by assumption, .1/m mC1 f  0. Therefore, a= l  a m m f ./  .1/ f  C .1/m m a a l for all l 2 N. By iterating this inequality we obtain for all k; l 2 N that   a a m m  : : :  .1/m m : .1/m m a f ./  .1/ a f  C af Ck l l Since f is continuous, we conclude that  7! .1/m m a f ./ is non-increasing. In particular, for all  > 0 and > 0,  1 m m .1/m m a f ./  .1/ a f . C /  0:

0 If we let ! 0, it follows that .1/m m a fC ./  0.

Proof of Theorem 4.11. (i))(iii) If n D 2, then we have seen in the first part of the proof of Theorem 4.8 that f is non-increasing and convex. Assume that n  3. Again, f is convex and therefore has left and right-hand side derivatives which satisfy fC0 . /  f0 ./  fC0 ./;

0 < <  < 1:

(4.6)

j

By Lemma 4.12, fC0 satisfies .1/j a .fC0 /./  0 for all  > 0, a > 0 and j D 0; 1; : : : ; n  1. Since n  1  2, fC0 is continuous. Together with (4.6) we j conclude that f 0 exists, f 0  0 and .1/j a .f 0 /./  0 for all  > 0, a > 0 and j D 0; 1; : : : ; n  1. Repeating this argument and using the assumption, we see that for j D 0; 1; : : : ; n  2, f .j / exists, .1/j f .j / ./  0 and .1/n2 f .n2/ is non-increasing and convex. R1 (iii))(i) If f is non-increasing and convex, then f ./ D c C  g.t / dt for some constant c  0 and some non-negative, right-continuous and non-increasing function g W .0; 1/ ! R, cf. proof of Theorem 1.11. This representation immediately implies that  a1 f  0 and a2 a1 f  0 for all a1 ; a2 > 0 proving the claim when n D 2. As in the proof of Theorem 4.8, we use the mean value theorem to see for all a > 0 .1/ a f ./ D f ./  f . C a/ D af 0 . C a/;

 2 .0; 1/;

hence .1/ a f  0. Iterating this argument yields for all j D 1; 2; : : : ; n  2, .1/j aj    a1 f ./ D .1/j a1    aj f .j / . C 1 a1 C    C j aj / for suitable 1 ; : : : ; j 2 .0; 1/. In particular, .1/n2 an2    a1 f ./ is nonincreasing and convex. Now we use the argument in the preceding paragraph to get the desired assertion.

44

Chapter 4 Positive and negative definite functions

(i))(ii) For any m  1, we define ° Cm WD f 2 B.0; 1/ W f  0; .1/j ja f  0

± for all j D 1; : : : ; m and a > 0 ;

and Km WD ¹f 2 Cm W f .0C/ D 1º. Then Km is a closed convex subset of Cm with respect to pointwise convergence. Let f be an extreme point in Kn and set  WD inf¹ > 0 W f ./ D 0º with the usual convention that inf ; D 1. Fix some ˛ 2 .0; /. By Taylor’s formula with integral remainder term, f ./ D f .˛/ C

n2 X j D1 Z˛

C 

.1/j .j / f .˛/.˛  /j jŠ

.1/n1 .n1/ f .t /.t  /n2 dt; .n  2/Š C

where fC.n1/ is the right-hand side derivative of f .n2/ . Note that fC.n1/ exists

because .1/.n2/ f .n2/ is non-increasing and convex. Since .1/n1 fC.n1/ is non-increasing, we see for all  2 .0; ˛/ that Z ˛ Z ˛ .1/n1 .n1/ .1/n1 .n1/ fC fC .t /.t  /n2 dt  .˛/.t  /n2 dt .n  2/Š .n  2/Š   .1/n1 .n1/ f D .˛/.˛  /n1 : .n  1/Š C Define a new function g W .0; 1/ ! R by gjŒ˛;1/ D f jŒ˛;1/ and, for all  2 .0; ˛/, g./ D f .˛/ C

n2 X j D1

.1/n1 .n1/ .1/j .j / f .˛/.˛  /j C f .˛/.˛  /n1 : jŠ .n  1/Š C

Then g  f and, by construction, both g and f  g are n-monotone, hence in Cn . Set q D g.0C/ and observe that f D g C .f  g/ D q

f g g C .1  q/ : g.0C/ 1  g.0C/

Since f is extremal, this means that g D qf for some q 2 Œ0; 1. But g D f on Œ˛; 1/ implying q D 1, and thus g D f on .0; ˛/. Therefore, f j.0;˛/ is a polynomial of degree at most n  1. Since ˛ <  was arbitrary, we conclude that f j.0; / is a polynomial of degree at most n  1.

45

Chapter 4 Positive and negative definite functions

If  < 1, then f .0C/ D 1 and f .j / ./ D 0 for j D 0; 1; : : : ; n  2, imply that f ./ D .1  =/n1 for 0 <  < . By setting t D 1= > 0 we see that f ./ D en;t ./ WD .1  t /n1 C . If  D 1, then clearly f  1 DW en;0 . This means that the extreme points of Kn are contained in the set ¹en;t W t  0º. As in the proof of Theorem 4.8 we can use the Kre˘ın–Milman theorem to conclude that for any f 2 Kn there exists a probability measure n supported in Œ0; 1/ such that Z .1  t /n1 f ./ D C n .dt /: Œ0;1/

For a general f 2 Cn , we can use the argument in the last part of the proof of Theorem 4.8 or Theorem 1.4 to arrive at the above integral representation. The uniqueness of the measure n follows once we know that every function in the family ¹en;t W t  0º D ¹.1  t /n1 W t  0º is an extremal point of Kn . We C are going to show that the en;t are extremal even for the larger set Cn . Assume that en1;t is an extremal point of Cn1 . Then en;t is extremal for Cn . Otherwise, there are g; h 2 Cn not proportional to each other and q 2 .0; 1/ such that en;t D qg C .1  q/h;

hence, t .n  1/en1;t D q.g 0 / C .1  q/.h0 /:

Note that g 0 ; h0 2 Cn1 are not proportional to each other, otherwise g; h would be proportional. This implies that neither t .n  1/en1;t nor its multiple en1;t are extremal in Cn1 , contradicting our assumption. Thus, it is enough to prove that e1;t D .1  t /0C D 1.0;1=t/ ./, t  0, is extremal for C1 . Assume that 1.0;1=t/ is not extremal. Then there are g; h 2 C1 ; g ¤ h and q 2 .0; 1/ such that 1.0;1=t/ D qg C .1  q/h. Since g and h are nonnegative and non-increasing, we conclude that g and h are constant in .0; 1=t /. Thus 1.0;1=t/ is indeed extremal. (ii))(iii) This is proved in Theorem 1.11. Remark 4.13. Let f W .0; 1/ ! Œ0; 1/, and for each a > 0, fa W .0; 1/ ! Œ0; 1/. Suppose that a f ./ D fa ./ for all a > 0. Then f is n-monotone, n  2, if, and only if, fa is .n  1/-monotone for all a > 0. This fact is a direct consequence of Theorem 4.11. Analogues of Theorem 4.11 and Remark 4.13 are also valid for multiply monotone functions on R. Let us briefly review the notion of positive and negative definiteness on the group .Rd ; C/. Our standard reference is the monograph [39] by Berg and Forst. On Rd the involution is given by  D  , 2 Rd . This means that f is positive definite (in the sense of Bochner) if n X j;kD1

f . j  k / cj cNk  0;

(4.7)

46

Chapter 4 Positive and negative definite functions

and negative definite (in the sense of Schoenberg) if n X 

 f . j / C f . k /  f . j  k / cj cNk  0

(4.8)

j;kD1

hold for all n 2 N, 1 ; : : : ; n 2 Rd and c1 ; : : : ; cn 2 C. The Rd -analogue of Lemma 4.5 and Corollary 4.9 is known as Bochner’s theorem. Theorem 4.14 (Bochner). A function  W Rd ! C is continuous and positive definite if, and only if, it is the Fourier transform of a finite measure  on Rd , Z . / D b . / D e ix .dx/; 2 Rd : Rd

The measure  is uniquely determined by , and vice versa. Corollaries 4.6 and 4.10 yield on Rd the Lévy–Khintchine representation. Theorem 4.15 (Lévy; Khintchine). A function W Rd ! C is continuous and negative definite if, and only if, there exist a number ˛  0, a vector ˇ 2 Rd , a symmetric and matrix Q 2 Rd d and a measure  on Rd n ¹0º satisfying R positive semi-definite 2 y¤0 .1 ^ jyj / .dy/ < 1 such that . / D ˛ C iˇ  C

1

 Q C 2

  i  y 1  e i y C .dy/: 1 C jyj2 y¤0

Z

The quadruple .˛; ˇ; Q; / is uniquely determined by

(4.9)

, and vice versa.

Comments 4.16. Standard references for positive and negative definite functions are the monographs [38] by Berg, Christensen and Ressel and [39] by Berg and Forst. For a more probabilistic presentation we refer to Dellacherie and Meyer [93, Chapter X]. The natural notion of positive and negative definiteness of a (finite-dimensional) matrix was extended to functions by Mercer [264]. A function (or kernel) of positive type is a continuous and symmetric function  W Œa; b Œa; b ! R satisfying Z a

b

Z a

b

.s; t / .s/.t / ds dt  0 for all  2 C Œa; b:

This condition is originally due to Hilbert [159] who refers to it as Definitheit (definiteness). It is shown in [264] that this is equivalent to saying that for all n 2 N and s1 ; : : : ; sn 2 R the matrices ..sj ; sk // are non-negative definite hermitian, which is in accordance with Definition 4.1. Mercer says that  is of negative type if  is of positive type. Note that this does not match our definition of negative definiteness which appears first in Schoenberg [325] and, independently, in Beurling [52];

Chapter 4 Positive and negative definite functions

47

the name negative definite was first used by Beurling [53] in connection with bounded negative definite functions – these are all of the form c f where f is positive definite and c is a constant c  f .0/, see e.g. [39, 7.11–7.13] – while Definition 4.3 seems to appear first in Beurling and Deny [54]. The notion of a positive definite function was introduced by Mathias [261] and further developed by Bochner [61, Chapter IV.20] culminating in his characterization of all positive definite functions, Theorem 4.14. Note that, in general, a negative definite function is not the negative of a positive definite function – and vice versa. In order to avoid any confusion, many authors nowadays use ‘positive definite (in the sense of Bochner)’ and ‘negative definite (in the sense of Schoenberg)’. The abstract framework of positive (and negative) definiteness on semigroups is due to Ressel [302] and Berg, Christensen and Ressel [37]. Ressel observes that positive definiteness (in a semigroup sense) establishes the connection between two seemingly different topics: Fourier transforms and Bochner’s theorem on the one hand and, on the other, Laplace transforms, completely monotone functions and Bernstein’s theorem. Note that in the case of Abelian groups there are always two choices for involutions: s  WD s, which is natural for groups, and s  D s which is natural for semigroups. Each leads to a different notion of positive and negative definiteness. The proof of Proposition 4.4 is adapted from [39, Chapter II.7] and [38, Chapters 4.3, 4.4]; for Theorems 4.7 and 4.8 we used Dellacherie–Meyer [93, X.73 and X.75] as well as Lax [239, Chapter 14.3]. The best references for continuous negative definite functions on Rd are [39, Chapter II] and [93, Chapter X]. Further proofs of the Bochner–Bernstein theorem based on extreme point methods are discussed in Lukes et al. [252, Chapter 14.7] and Simon [330, Theorem 9.10]. Lemma 4.12 and the proof of the implication (i))(iii) in Theorem 4.11 are taken from Wendland [366, Theorem 7.4]. The extreme point argument used in the proof of Theorem 4.11 is a simplified version of Pestana and Mendonça [288]. A more classical proof is contained, as a by-product, in Wendland’s proof of the integral representation of completely monotone functions, cf. [366, Sections 7.1 and 7.2]. Note that Theorem 4.11 gives an independent proof of part of Theorem 1.11. Since limn!1 je nx  .1  x/n1 C j D 0 uniformly for all x 2 Œ0; 1/, Theorem 4.11 can be easily made into a further proof of Theorem 1.4 and 4.8, cf. [288, Theorem 4].

Chapter 5

A probabilistic intermezzo

Completely monotone functions and Bernstein functions are intimately connected with vaguely continuous convolution semigroups of sub-probability measures on the half-line Œ0; 1/. The probabilistic counterpart of such convolution semigroups are subordinators – processes with stationary independent increments with state space Œ0; 1/ and right-continuous paths with left limits, i.e. Lévy processes with values in Œ0; 1/. In this chapter we are going to describe this connection which will lead us in a natural way to the concept of infinite divisibility and the central limit problem. Recall that a sequence .n /n2N of sub-probability measures on Œ0; 1/ converges vaguely to a measure  if Z Z lim f .t / n .dt / D f .t / .dt / n!1 Œ0;1/

Œ0;1/

holds for every compactly supported continuous function f W Œ0; 1/ ! R, see Appendix A.1. The convolution of the sub-probability measures  and  on Œ0; 1/ is defined to be the sub-probability measure  ?  on Œ0; 1/ such that for every bounded continuous function f W Œ0; 1/ ! R, Z Z Z f .t /. ? /.dt / D f .t C s/ .dt /.ds/: Œ0;1/

Œ0;1/

Œ0;1/

Definition 5.1. A vaguely continuous convolution semigroup of sub-probability measures on Œ0; 1/ is a family of measures . t / t0 satisfying the following properties: (i)  t Œ0; 1/  1 for all t  0; (ii)  tCs D  t ? s for all t; s  0; (iii) vague- lim t!0  t D ı0 . It follows from (ii) and (iii) that vague- lim t!0  tCs D s for all s  0. Unless otherwise stated we will always assume that a convolution semigroup is vaguely continuous. For all 2 Cc Œ0; 1/ satisfying 0   1Œ0;1/ and .0/ D 1 we find Z Z t!0 .s/  t .ds/ ! .s/ ı0 .ds/ D 1: 1   t Œ0; 1/  Œ0;1/

Œ0;1/

This proves lim t!0  t Œ0; 1/ D 1 D ı0 Œ0; 1/; from Theorem A.4 we conclude that in this case weak and vague continuity coincide.

49

Chapter 5 A probabilistic intermezzo

Theorem 5.2. Let . t / t0 be a convolution semigroup of sub-probability measures on Œ0; 1/. Then there exists a unique f 2 BF such that the Laplace transform of  t is given by L  t D e tf for all t  0: (5.1) Conversely, given f 2 BF, there exists a unique convolution semigroup of subprobability measures . t / t0 on Œ0; 1/ such that (5.1) holds true. Proof. Suppose that . t / t0 is a convolution semigroup of sub-probability measures on Œ0; 1/. Fix t  0. Since L  t > 0, we can define a function f t W .0; 1/ ! R by f t ./ D  log L . t I /. In other words, L . t I / D e f t ./ : By property (ii) of convolution semigroups, it holds that f tCs ./ D f t ./Cfs ./ for all t; s  0, i.e. t 7! f t ./ satisfies Cauchy’s functional equation. Vague continuity ensures that this map is continuous, hence there is a unique solution f t ./ D tf ./ where f ./ D f1 ./. Therefore, L  t D e tf

for all t  0I

in particular, e tf 2 CM for all t  0. By Theorem 3.7, f 2 BF. Conversely, suppose that f 2 BF. Again by Theorem 3.7 we have that for all t  0, e tf 2 CM . Therefore, for every t  0 there exists a measure  t on Œ0; 1/ such that L  t D e tf . We check that the family . t / t0 is a convolution semigroup of sub-probability measures. Firstly,  t Œ0; 1/ D L . t I 0C/ D e tf .0C/  1. Secondly, L . t ? s / D L  t L s D e tf e sf D e .tCs/f D L  tCs . By the uniqueness of the Laplace transform we get that  tCs D  t ? s . Finally, lim t!0 L  t ./ D lim t!0 e tf ./ D 1 D L ı0 ./ for all  > 0 and, by Lemma A.9, vague- lim t!0  t D ı0 . Remark 5.3. Because of formula (5.1), probabilists often use the name Laplace exponent instead of Bernstein function. Definition 5.4. A stochastic process S D .S t / t0 defined on a probability space .; F ; P/ with state space Œ0; 1/ is called a subordinator if it has independent and stationary increments, S0 D 0 a.s., and for almost every ! 2 , t 7! S t .!/ is a right-continuous function with left limits. The measures . t / t0 defined by  t .B/ D P.S t 2 B/;

B  Œ0; 1/ Borel;

are called the transition probabilities of the subordinator S . Almost all paths t 7! S t .!/ of a subordinator S are non-decreasing functions.

50

Chapter 5 A probabilistic intermezzo

Let ea be an exponentially distributed random variable with parameter a  0, i.e. P.ea > t / D e at , t > 0. We allow that a D 0 in which case ea D C1. Assume S D .b S t / t0 by that ea is independent of the subordinator S . We define a process b ´ t < ea ; St ; b (5.2) S t WD C1; t  ea : The process b S is the subordinator S killed at an independent exponential time. Any process with state space Œ0; 1 having the same distribution as b S will be called a killed subordinator. The connection between (killed) subordinators and convolution semigroups of subprobability measures on Œ0; 1/ is as follows. For t  0 let  t .B/ WD P.Sbt 2 B/;

B  Œ0; 1/ Borel;

be the transition probabilities of .b S t / t0 , and note that Z St D e y  t .dy/ D L . t I /: EŒe b Œ0;1/

Proposition 5.5. The family . t / t0 is a convolution semigroup of sub-probability measures. Proof. We check properties (i)–(iii) in Definition 5.1. It is clear that  t Œ0; 1/  1. Let g W Œ0; 1/ ! R be a bounded continuous function. Then by the right-continuity of the paths, lim t!0 g.S t / D g.S0 / D g.0/ a.s. By the dominated convergence theorem it follows that lim t!0 EŒg.S t / D g.0/, i.e. Z Z g d t D g.0/ D g d ı0 ; lim t!0 Œ0;1/

Œ0;1/

thus proving property (iii). In order to show property (ii), assume first that a D 0, that is there is no killing. Then for s; t  0, L .sCt I / D EŒe SsCt  D EŒe .SsCt Ss / e .Ss S0 /  D EŒe S t EŒe Ss  D L . t I /L .s I /; which is equivalent to sCt D s ?  t . If a > 0, we find by independence St EŒe b  D EŒe S t 1¹t 0 if, and only if,  t Œ0; 1/ < 1 for all t > 0; in this case, the associated stochastic process has a.s. finite lifetime. This is why a is usually called the killing term of the Laplace exponent f . Let . t / t0 be a convolution semigroup of sub-probability measures on Œ0; 1/ and let f be the corresponding Bernstein function. Then the completely monotone function g./ WD L .1 I / D e f ./ has the property that for every t > 0, g t is again completely monotone. This follows easily from g t ./ D e tf ./ D L . t I /: From the probabilistic point of view, thisP means that S t is an infinitely divisible random variable. To be more precise S t D jnD1 .Sjt=n  S.j 1/t=n / for every n 2 N, and the random variables .Sjt=n  S.j 1/t=n /1j n are independent and identically is the n-fold convolution. This motivates the following distributed. Hence  t D ?n t=n definition. Definition 5.6. A completely monotone function g is said to be infinitely divisible if for every t > 0 the function g t is again completely monotone. If g is infinitely divisible and g.0C/  1, the sub-probability measure  on Œ0; 1/ satisfying L  D g is said to be an infinitely divisible distribution; we write  2 ID. Remark 5.7. In Definition 5.6 it is enough to require that g t 2 CM for small t . In fact, g 2 CM is infinitely divisible if, and only if, g 1=n 2 CM for all n  n0 . The necessity is obvious. Assume that g 1=n 2 CM for all n  n0 . For any t > 0 there is a sequence rm D km =`m 2 QC such that `m  n0 and limm!1 rm D t . Observe that g t ./ D limm!1 g km =`m ./ for all  > 0. Since CM is closed under pointwise limits, cf. Corollary 1.6, it is enough to show that g km =`m 2 CM for each m  1. This, however, follows since by assumption g 1=`m 2 CM and since CM is closed under pointwise multiplication: g km =`m D g 1=`m    g 1=`m , cf. Corollary 1.6.

52

Chapter 5 A probabilistic intermezzo

The discussion preceding Definition 5.6 shows that g WD e f is completely monotone and infinitely divisible if f 2 BF. Moreover, g.0C/  1. This is already one direction of the next result. Lemma 5.8. Suppose that g W .0; 1/ ! .0; 1/. Then the following statements are equivalent. (i) g 2 CM, g is infinitely divisible and g.0C/  1. (ii) g D e f where f 2 BF. Proof. Suppose that (i) holds. Since g t 2 CM and clearly g t .0C/  1, there exists a sub-probability measure  t on Œ0; 1/ such that g t ./ D L . t I /. Since lim t!0 g t ./ D 1, it follows that vague- lim t!0  t D ı0 . For s; t  0 it holds that g t g s D g tCs , and consequently L . t I / L .s I / D L . tCs I /: By the uniqueness of the Laplace transform this means that . t / t0 is a convolution semigroup of sub-probability measures on Œ0; 1/. By Theorem 5.2, there exists a unique f 2 BF such that L . t I / D e tf ./ ; in particular, g D e f . Lemma 5.8 admits an extension to completely monotone functions g which need not satisfy g.0C/  1. For the purpose of stating this extension, let us say that a C 1 function f W .0; 1/ ! R is an extended Bernstein function if .1/n1 f .n/ ./  0 for all n 2 N and  > 0: Remark 5.9. Note that a non-negative extended Bernstein function is actually a Bernstein function, and a non-positive extended Bernstein function is the negative of a completely monotone function. In fact, f is an extended Bernstein function if, and only if, f D g  h for some g 2 BF; h 2 CM; i.e. f 2 BF  CM: Indeed, if f D g  h where g 2 BF and h 2 CM, it is obvious that for n 2 N  .1/n1 f .n/ D .1/n1 g .n/  h.n/ D .1/n1 g .n/ C .1/n h.n/  0; i.e. g  h is an extended Bernstein function. Conversely, assume that f is an extended Bernstein function. If f .c/  0 for some c > 0, we set g./ WD f . C c/ and h./ WD f . C c/  f ./. By construction, f D g  h and g; h  0; for all n 2 N .1/n1 g .n/ ./ D .1/n1 f .n/ . C c/  0;

and

.1/n h.n/ ./ D .1/n f .n/ . C c/  .1/n f .n/ ./  0 where we used that .1/n f .nC1/  0 which shows that .1/n f .n/ is increasing. If f .c/  0 for all c > 0, we use g  0 and h D f .

53

Chapter 5 A probabilistic intermezzo

Definition 5.10. A C 1 function g W .0; 1/ ! .0; 1/ is said to be logarithmically completely monotone if (5.4) .log g/0 2 CM: Theorem 5.11. Suppose that g W .0; 1/ ! .0; 1/. Then the following assertions are equivalent. (i) g 2 CM and g is infinitely divisible. (ii) g D e f where f is an extended Bernstein function. (iii) g is logarithmically completely monotone. Proof. (i))(ii) For c > 0 let gc ./ WD g. C c/,  > 0. Then  7!

gc ./ gc ./ D gc .0C/ g.c/

is again in CM, infinitely divisible and tends to 1 as  ! 0. By Lemma 5.8, there exists fc 2 BF such that g. C c/ D gc ./ D g.c/e fc ./ for all  > 0. This can be written as g./ D g.  c C c/ D g.c/e fc .c/ D e fc .c/Clog g.c/ ;

 > c:

(5.5)

If 0 < b < c the same formula is valid with b replacing c; for 0 < b < c <  it holds that fc .  c/ C log g.c/ D fb .  b/ C log g.b/: This implies that one can define a function f W .0; 1/ ! R by f ./ WD fc .  c/  log g.c/;

 > c:

Clearly, f is C 1 on .0; 1/. For any  > 0 find 0 < c < . By (5.5), g./ D e f ./ , and for n 2 N, .1/n1 f .n/ ./ D .1/n1 fc.n/ .  c/  0. Thus, f is an extended Bernstein function. (ii))(iii) If g D e f , then .log g/0 D f 0 and clearly f 0 2 CM. (iii))(i) Suppose g is logarithmically completely monotone. For c > 0, Z fc ./ WD

Cc c

.log g/0 .t / dt D  log g. C c/ C log g.c/;

 > 0;

belongs to BF. Since g. C c/ D g.c/e fc ./ , it follows from Theorem 3.7 that  7! g. C c/ is completely monotone for all c > 0. Since g is continuous, we get that g./ D limc!0 g. C c/, hence g 2 CM. Note that g t is also logarithmically completely monotone for every t > 0, hence by the already proven fact, g t 2 CM. Thus, g is infinitely divisible.

54

Chapter 5 A probabilistic intermezzo

Remark 5.12. (i) The concept of infinite divisibility appears naturally in the central limit problem of probability theory. This problem can be stated in the following way: Let .Xn;j /j D1;:::;k.n/; n2N be a (doubly indexed) family of real random variables such that the random variables in each row, Xn;1 ; : : : ; Xn;k.n/ , are independent. When do the probability distributions   Xn;1 C    C Xn;k.n/  an law ; bn for suitably chosen sequences of real .an /n2N and positive .bn /n2N numbers, have a non-trivial weak limit as n ! 1? The classical example is, of course, the central limit theorem where a sequence of independent and identically distributed random variables .Xj /j 2N is arranged in a triangular array of the form X1 I

X1 ; X2 I

X1 ; X2 ; : : : ; Xn I : : : p where k.n/ D n, an D n EX1 is the mean value and bn D nVar X1 the standard deviation of the partial sum of the variables in the nth row. Clearly, every limiting random variable X is infinitely divisible in the sense that for every n 2 N there are independent and identically distributed random variables Y1 ; : : : ; Yn such that X and Y WD Y1 C  CYn have the same probability distribution. The corresponding probability distributions are also called infinitely divisible. It can be shown, see e.g. Gnedenko and Kolmogorov [130], that every infinitely divisible distribution can be obtained as the limiting distribution of a triangular array .Xn;j /j D1;:::;k.n/;n2N , where each row Xn;1 ; : : : ; Xn;k.n/ consists of independent random variables which are asymptotically negligible, i.e. lim

X1 ; X2 ; X3 I

max

n!1 1j k.n/

:::

P.jXn;j j  / D 0 for all  > 0:

(ii) Specializing to one-sided distributions  supported in Œ0; 1/ we can express the property  2 ID in terms of the Laplace transform. By Lemma 5.8, g 2 CM with g.0C/  1 is infinitely divisible if, and only if, for every n 2 N there is some gn 2 CM such that g D gnn . In probabilistic terms this means that the law  2 ID has an nth convolution root n such that L n D gn and  D n?n . If X denotes a random variable with distribution  and Yj , 1  j  n, denote independent and identically distributed random variables with the common distribution n , this means that X and Y1 C    C Yn have the same probability distribution. Depending on the structure of the triangular array, the limiting distributions may have different stability properties. We will discuss this for one-sided distributions supported in Œ0; 1/. Often it is easier to describe them in terms of the corresponding Laplace transforms.

55

Chapter 5 A probabilistic intermezzo

Definition 5.13. A completely monotone function g 2 CM with g.0C/  1 is said to be (weakly) stable if for all a > 0 there are b  0; c > 0 such that g./a D g.c/  e b I

(5.6)

if b D 0 we call g strictly stable. The corresponding random variables and sub-probability distributions are again called (weakly or strictly) stable. We will see below in Proposition 5.15 that the constant c appearing in (5.6) is necessarily of the form c D a1=˛ for some ˛ 2 .0; 1; ˛ is often called the index of stability. It can be shown, cf. [130], that every stable  is the limiting distribution of a sequence of independent and identically distributed random variables .Xj /j 2N and suitable centering and norming sequences .an /n2N , an  0, and .bn /n2N , bn > 0: 11 0 0 n X 1 weak Xj  an AA ! : (5.7) law @ @ n!1 bn j D1

All (weakly) stable distributions on Œ0; 1/ have Laplace transforms which are given by g./ D exp.ˇ˛  / with ˛ 2 .0; 1 and ˇ;  0. The distributions are (strictly) stable if D 0. If the Xj are independent (but not necessarily identically distributed) and limn!1 bn D 1 and limn!1 bnC1 =bn D 1, then the limit (5.7) characterizes all self-decomposable (also: class L) distributions. Again we focus on one-sided distributions where we can express self-decomposability conveniently in terms of Laplace transforms. We will denote the set of all self-decomposable distributions on Œ0; 1/ by SD. Definition 5.14. A completely monotone function g 2 CM with g.0C/  1 is said to be self-decomposable, if  7!

g./ D gc ./ is completely monotone for all c 2 .0; 1/: g.c/

(5.8)

We will see in Proposition 5.17 that the Laplace exponent f of a  2 SD, i.e. L  D e f , is a Bernstein function with a Lévy triplet of the form .a; b; m.t / dt / such that t 7! t m.t / is non-increasing. The rather deep limit theorems – or the equally difficult structure results on the completely monotone functions – of Remark 5.12 show that ¹stable lawsº  SD  ID: The following two results contain a simple, purely analytic proof for this.

56

Chapter 5 A probabilistic intermezzo

Proposition 5.15. Let g 2 CM, g.0C/  1, be (weakly) stable. Then there exist ˛ 2 .0; 1 and ˇ;  0 such that g./ D e ˇ

˛ 

:

In particular, g.0C/ D 1. In the strictly stable case, D 0. Proof. The function  7! g.c/e b appearing in (5.6) is a product of two completely monotone functions, hence itself completely monotone. Thus, g a 2 CM for every a > 0 which means that g is infinitely divisible. By Lemma 5.8, there exists some f 2 BF such that g D e f . The structure equation (5.6) becomes, in terms of f, for all a > 0 there are c > 0; b  0 such that af ./ D f .c/ C b:

(5.9)

This implies that f .0C/ D 0. Differentiating this equality twice yields af 00 ./ D c 2 f 00 .c/:

(5.10)

In the trivial case where f 00  0 we have f ./ D  for a suitable integration constant  0, and we are done. In all other cases, (5.10) shows that c is a function of a only and that it does not depend on b. Indeed, if both .a; c/ and .a; c/ Q satisfy (5.9), we find from (5.10) with  D 1=cQ c 2 00  c  D f 00 .1/: f cQ 2 cQ As f 00 is increasing and non-trivial, it is easy to see that c D cQ and c D c.a/. Setting  D 1 in (5.9) we see that b D af .1/  f .c/ is also unique. Since the inverse f 1 of a non-trivial, i.e. non-constant, Bernstein function is again a continuous function, we get from (5.9)   1 c D c.a/ D f 1 af ./  b I  this proves that c D c.a/ depends continuously on a > 0. For any two a; aQ > 0 we can use (5.10) twice to find     00 Q D c 2 .a/af c.a/ Q Q 00 c.a/ Q aaf Q 00 ./ D ac 2 .a/f   D c 2 .a/c Q : Q 2 .a/f 00 c.a/c.a/ On the other hand, aaf Q 00 ./ D c 2 .aa/f Q 00 .c.aa//, Q so that c.aa/ Q D c.a/c.a/. Q Since c.a/ is continuous, there exists some ˛ 2 R n ¹0º such that c.a/ D a1=˛ . Inserting this into (5.10) we have 2

1

f 00 ./ D a ˛ 1 f 00 .a ˛ /

for all a;  > 0:

Chapter 5 A probabilistic intermezzo

57

Now take a D ˛ to get f 00 ./ D ˛2 f 00 .1/ and integrate twice. Using that f .0C/ D 0 we find f 00 .1/ ˛  C f 00 .1/ C  f ./ D ˛.˛  1/ with some integration constant C 2 R. Since f 2 BF is non-trivial, we know that 0 < ˛  1. Finally, we can use (5.9) to work out that f 00 .1/ C D b=.a  a1=˛ /. Corollary 5.16. We have ¹stable laws on Œ0; 1/º  SD  ID. Proof. The first inclusion follows from Proposition 5.15. If g is (weakly) stable, f ./ WD  log g./ D ˇ˛ C . Thus, for 0 < c < 1, f ./  f .c/ D ˇ.1  c ˛ / ˛ C .1  c/  is a Bernstein function, and we get that gc ./ D g./=g.c/ D exp..f ./f .c/// is completely monotone. If (5.8) holds for g and all 0 < c < 1, we have   g 0 ./ g./ g.c/  g./ 1  D lim D lim 1 : c!1 .1  c/g.c/ c!1 .1  c/ g./ g.c/ Since g./=g.c/ is bounded by 1 and completely monotone, the expression inside the brackets is a Bernstein function by Proposition 3.12. Dividing by  makes it completely monotone, cf. Corollary 3.8 (iv). Since limits of completely monotone functions are again completely monotone, cf. Corollary 1.7, we know that g is logarithmically completely monotone, hence infinitely divisible by Theorem 5.11. Proposition 5.17. Let f 2 BF satisfy L  D e f and let  be the Lévy measure of f . Then the following assertions are equivalent. (i)  has a density m.t / such that t 7! t  m.t / is non-increasing; (ii)  2 SD. Proof. (ii))(i) Assume that  2 SD. Set g WD L  and gc ./ WD g./=g.c/ for c 2 .0; 1/. Since g./ D e f ./ , we have that gc ./ D exp..f ./  f .c///. We will now show that gc is for each c 2 .0; 1/ logarithmically completely monotone. For this we have to check that gc0 =gc 2 CM. A small change in the proof of Corollary 5.16 shows that   g 0 ./ g./ g.c/  g./ 1 ./ WD  D lim D lim 1 c!1 .1  c/g.c/ c!1 .1  c/ g./ g.c/

58

Chapter 5 A probabilistic intermezzo

is a Bernstein function and, therefore,  0 is completely monotone. Moreover, d

g./

g 0 ./g.c/  g./cg 0 .c/ g.c/ g 0 ./ d g.c/ D  g./ D    c gc ./ g 2 .c/ g./ g.c/

g 0 ./ cg 0 .c/ C g./ g.c/ Z  Z 1 1 0 D  .s/ ds D  0 .t / dt:  c c

D

Thus, gc0 =gc is an integral mixture of completely monotone functions and as such, see Corollary 1.6, itself completely monotone. Now define fc ./ WD f ./  f .c/; by Theorem 5.11, fc 2 BF. Let Z 1 Z t .1  e / .dt / D a C b C  e t M.t / dt; f ./ D a C b C .0;1/

0

where M.t / D .t; 1/. Then fc ./ D f ./  f .c/   Z t D a C b C .1  e / .dt / .0;1/   Z ct  a C bc C .1  e / .dt / .0;1/ Z Z .1  e t / .dt /  D b.1  c/ C .0;1/

.0;1/

(5.11) .1  e t / .c/ .dt /;

where .c/ ./ WD .c 1 /. On the other hand, fc being a Bernstein function, it has a representation Z Q fc ./ D b C .1  e t / c .dt /: (5.12) .0;1/

implying in particular that for every Borel set B  .0; 1/ it Hence, c D   1 holds that .B/  .c B/. For B D .s; t , this gives .s; t   .c 1 s; c 1 t  for every c 2 .0; 1/. For s 2 R define h.s/ WD M.e s / D .e s ; 1/. Clearly, h is non-negative and non-decreasing. Moreover, we have for all u > 0 and c 2 .0; 1/ .c/ ,

h.s C u/  h.s/ D M.e su /  M.e s / D .e su ; e s   .e sulog c ; e slog c  D h.s C u C log c/  h.s C log c/:

59

Chapter 5 A probabilistic intermezzo

This shows that for all u > 0 the function s 7! h.s C u/  h.s/ is non-decreasing. Therefore, h is convex. Being non-negative, non-decreasing and convex, h can be written in the form Z s h.s/ D

`.u/ du 1

where ` is non-negative and non-decreasing. Define a function k W .0; 1/ ! R by k.v/ WD `.e v /. Then k is non-negative and non-increasing. For t > 0 a change of variables shows Z 1 Z 1 Z  log t `.e v / k.v/ dv D dv; `.u/ du D M.t / D h. log t / D v v 1 t t and the proof is finished by defining m.t / WD k.t /=t . (i))(ii) To prove the converse, let c 2 .0; 1/ and define fc ./ WD f ./  f .c/. Then   Z 1 .1  e t / m.t / dt fc ./ D a C b C 0   Z 1  a C bc C .1  e ct / m.t / dt Z 10   .1  e t / m.t /  c 1 m.t =c/ dt: D b.1  c/ C 0

Since t 7! t  m.t / is non-increasing, nc .t / WD m.t /  c 1 m.t =c/  0 is non-negative for all t  0. Hence, fc 2 BF. Let g WD L  D e f . Then gc ./ D

e f ./ g./ D f .c/ D e fc ./ ; g.c/ e

proving that g is completely monotone. Since g.0C/  1, we see that g is selfdecomposable, i.e.  2 SD. Remark 5.18. In the proof one can choose m.t / so that t  m.t / is right-continuous. Hence, Proposition 5.17 can be stated as  2 SD if, and only if, t m.t / is 1-monotone. Alternatively,  2 SD if, and only if, t  m.t / D n.log t / where n is 1-monotone on R. The concept of self-decomposability can be extended to multiple and complete selfdecomposability. Definition 5.19. (i) A completely monotone function g 2 CM with g.0C/  1 is said to be p-times self-decomposable, p  2, if it is self-decomposable and if  7!

g./ D gc ./ is .p  1/-times self-decomposable for all c 2 .0; 1/: g.c/

60

Chapter 5 A probabilistic intermezzo

(ii) A completely monotone function g 2 CM with g.0C/  1 is said to be completely self-decomposable if it is p-times self-decomposable for all p  1. In analogy with the self-decomposable case we denote by SDp (respectively SD1 ) the family of infinitely divisible sub-probability measures  on Œ0; 1/ such that L  is a p-times self-decomposable (respectively a completely self-decomposable) completely monotone function. We have the following analogue of Proposition 5.17. Proposition 5.20. Let f 2 BF satisfy L  D e f and let  be the Lévy measure of f . Then the following assertions are equivalent. (i)  2 SDp (respectively  2 SD1 ); (ii)  has a density m.t / such that t  m.t / D n.log t / for some p-monotone (respectively completely monotone) function n on R. Proof. We prove the result by induction in p. For p D 1 the claim is true by Proposition 5.17 (cf. Remark 5.18). Suppose that it is true for p  1, p  2. Let  2 SDp , define g D L , and gc ./ D g./=g.c/. Then by definition of p-times selfdecomposability, gc is a .p  1/-times self-decomposable function for all c 2 .0; 1/. Let gc D e fc . By the induction hypothesis, the Lévy measure c of fc has a density mc .t / such that t  mc .t / D nc .log t / where nc is .p  1/-monotone on R. Since f corresponds to a self-decomposable distribution, its Lévy measure has a density m.t /. Comparing the formulae (5.11) and (5.12) for fc in the proof of Proposition 5.17 we see that m.t /  c 1 m.c 1 t / D mc .t /, hence t m.t /  c 1 t m.c 1 t / D t mc .t /:

(5.13)

Define n W R ! R by n.s/ WD e s m.e s / so that t m.t / D n.log t /. Then (5.13) becomes n.log t /  n.log t  log c/ D nc .log t / for all t 2 R; or, equivalently,  . log c/ n.t / D n.t /  n.t  log c/ D nc .t / for all t > 0; c 2 .0; 1/: Together with the characterization from Theorem 4.11, cf. Remark 4.13, this shows that n is p-monotone on R. R1 Conversely, assume that f ./ D b C 0 .1  e t / m.t / dt and the density m.t / satisfies t m.t / D n.log t / where n is p-monotone on R. For c 2 .0; 1/ define nc .t / WD n.t /  n.t  log c/, t 2 R, and mc .t / WD t 1 nc .log t /. Then (5.13) is valid for all t > 0 and all c 2 .0; 1/. Let fc ./ WD f ./  f .c/.Then the computation in

61

Chapter 5 A probabilistic intermezzo

the proof of Proposition 5.17 shows that Z fc ./ D b.1  c/ C

1 0

.1  e t / mc .t / dt:

Again by Theorem 4.11 we see that nc is .p  1/-monotone on R for every c 2 .0; 1/. The part of the proposition related to SD1 is now clear. Definition 5.21. A Bernstein function f belongs to the class PBF if the Lévy measure  from (3.2) has a density m.t / such that t 7! t  m.t / D n.log t / where n is completely monotone on R, i.e. Z .1  e t / m.t / dt; (5.14) f ./ D a C b C .0;1/

R

where a; b  0, .0;1/ .1 ^ t / m.t / dt < 1 and t 7! t  m.t / D n.log t / where n is completely monotone on R. Theorem 5.22. For a function f W .0; 1/ ! .0; 1/ the following assertions are equivalent. (i) f D L  where  2 SD1 . (ii) f 2 PBF. (iii) f 2 BF and the Lévy measure of f has a density m.t / such that for every

 1, there is a function n D n which is completely monotone on R such that m.t / D t  n.log t /. (iv) f is of the form Z f ./ D a C b C

1 0

.1  e t /

Z

 t ˛1 .d˛/ dt

(5.15)

.0;1/

where a; b  0 and  is a measure on .0; 1/ satisfying Z  1  ˛ C .1  ˛/1 .d˛/ < 1:

(5.16)

.0;1/

(v) f is of the form

Z f ./ D

˛ .d˛/;

(5.17)

Œ0;1

where  is a finite measure on Œ0; 1. Additionally, SD1 is the smallest class of sub-probability measures on Œ0; 1/ which contains all stable laws and is closed under convolution and vague limits.

62

Chapter 5 A probabilistic intermezzo

Proof. (i),(ii) This is proved in Proposition 5.20. (ii))(iv) LetR m.t / D t 1 n.log t / where n is completely monotone on R. By (1.12), n.t / D .0;1/ e ts .ds/ for all t 2 R where  is a measure on .0; 1/. In particular, Z m.t / D t 1 n.log t / D

t 1s .ds/:

.0;1/

R1 Since m.t / is the density of the Lévy measure of f , it holds that 0 t m.t / dt < 1 R1 and 1 m.t / dt < 1. By Fubini’s theorem, the first condition gives that  Z 1 Z s t dt .ds/ < 1; .0;1/

0

implying that Œ1; 1/ D 0 (otherwise the inner integral would diverge), as well R as .0;1/ .1  s/1 .ds/ < 1 (by computing the inner integral). Similarly, from R the second condition on m.t / we conclude that .0;1/ s 1 .ds/ < 1. Therefore, R R m.t / D .0;1/ t s1 .ds/ and .0;1/ .s 1 C .1  s/1 / .ds/ < 1. R (iv))(iii) Note first that (5.16) guarantees that m.t / WD .0;1/ t ˛1 .d˛/ is the density of a Lévy measure thus implying that f 2 BF. Fix any  1 and let  be the Rimage measure of  with respect to theR mapping ˛ 7! ˛ C C 1 so that m.t / D .C1;C2/ t s .ds/. Define n.t / WD .C1;C2/ e ts .ds/, t 2 R. Then m.t / D t  n.log t / for all t > 0 implying that n is well defined, and hence completely monotone on R. (iii))(ii) This follows immediately by taking D 1. (iv),(v) Suppose that (v) holds and write Z f ./ D ˛ .d˛/ Œ0;1 Z ˛ .d˛/ D ¹0º C ¹1º C .0;1/  Z 1 Z ˛ t ˛1 t .1  e / dt .d˛/ D ¹0º C ¹1º C .1  ˛/ .0;1/ 0 Z 1 D ¹0º C ¹1º C .1  e t / m.t / dt 0

Z

with

m.t / D Let .d˛/ WD 

Z .0;1/

˛ .1˛/

.0;1/

˛ t ˛1 .d˛/: .1  ˛/

.d˛/. Since lim˛!1 .1  ˛/.1  ˛/ D 1, it follows that

   Z 1 1 ˛ 1 1 .d˛/ D C C .d˛/ < 1: ˛ 1˛ 1  ˛ .1  ˛/ .0;1/ ˛

The converse direction follows by retracing the above steps.

63

Chapter 5 A probabilistic intermezzo

To prove the last statement of the theorem, consider the family of all Bernstein functions f which are of the form (5.17) with a finite measure  on Œ0; 1. This family is a convex cone containing ˛ for Rall ˛ 2 Œ0; 1. We show now that it is closed under pointwise limits. Let fn ./ D Œ0;1 ˛ n .d˛/ and assume that f ./ D limn!1 fn ./ for all  > 0; so, f .1/ D limn!1 fn .1/ D limn!1 n Œ0; 1. Hence, the family of measures .n /n1 is bounded. Without loss of generality we may assume that .n /n1 converges weakly to a bounded measure . Since the function ˛ 7!R ˛ is bounded and continuous on Œ0; 1 for every  > 0, it follows that f ./ D Œ0;1 ˛ .d˛/. This proves that the family of Bernstein functions of the form (5.17) contains PBF. The converse inclusion follows from the fact that every probability measure on Œ0; 1 is a weak limit of a sequence of convex combinations of point masses. Remark 5.23. (i) By taking D 0 we see that L  D e f with  2 SD1 if, and only if, the Lévy measure of f has a density of the form m.t / D n.log t / where n is a completely monotone function on R. (ii) The equivalence (iii) and (iv) from Theorem 5.22 can be, with a small modification, extended to a slightly larger range of parameters . Indeed, almost the same proof gives that f satisfies condition (iii) with 2 .2;1/ if, is of the form (5.15) where  is a measure on .  1; 1/ satisfying R and only if, f 1 .1  ˛/ .d˛/ < 1. .1;1/ (iii) The representation (5.17) is called the power representation of f . The representing measure  and the Lévy triplet of f are related by a D ¹0º, b D ¹1º and Z ˛ t ˛1 .d˛/: (5.18) m.t / D .0;1/ .1  ˛/ Now we define the potential measure of a vaguely continuous convolution semigroup . t / t0 of sub-probability measures on Œ0; 1/. Notice that Z 1Z Z 1 1 s : (5.19) e  t .ds/ dt D e tf ./ dt D f ./ 0 0 Œ0;1/ Since every continuous function u.s/ with compact support in Œ0; 1/ Rcan be domi1 nated by a multiple of e s , this implies that the vague integral U WD 0  t dt exists, see Remark A.3. U is called the potential measure of the convolution semigroup . t / t0 or, equivalently, of the corresponding killed subordinator .S t / t0 . Moreover, Z 1 Z 1 U.A/ D  t .A/ dt D E 1¹S t 2Aº dt for all Borel sets A  Œ0; 1/: 0

0

It is a simple consequence of Fubini’s theorem and (5.19) that  Z 1 Z s L .U I / D e  t .ds/ dt D Œ0;1/

0

1 ; f ./

(5.20)

64

Chapter 5 A probabilistic intermezzo

where f 2 BF is the Bernstein function corresponding to the convolution semigroup . t / t0 . This shows, in particular, that for all  > 0 Z Z 2 t 1 > L .U I / D e U.dt /  e t U.dt /  e  U Œ0; ; Œ0;1/

Œ0;

proving that U is finite on bounded sets. In a similar way one defines for  > 0 the -potential measure as the vague integral Z 1 e t  t dt: (5.21) U WD 0

Definition 5.24. A function f W .0; 1/ ! .0; 1/ is said to be a potential if f D 1=g where g 2 BF  D BF n ¹0º. The set of all potentials will be denoted by P. By (5.20), P consists of Laplace transforms of potential measures. In particular, P  CM, and P consists of exactly those completely monotone functions f which have the property that 1=f 2 BF, i.e. ² ³ ³ ² 1 1  PD W f 2 BF D g 2 CM W 2 BF : (5.22) f g Proposition 5.25. Let g 2 P be a potential. Then g is logarithmically completely monotone and hence there exists an extended Bernstein function f such that g D e f . Proof. Let g 2 P. Then h D 1=g 2 BF and .log g/0 D .log h/0 D h0

1 D h0 g: h

Since both h0 and g are in CM, it follows that their product .log g/0 2 CM. The second part of the statement follows from Theorem 5.11. The converse does not hold. Indeed, f ./ D  is a Bernstein function and, therefore, g./ WD e f ./ D e  is logarithmically completely monotone. Since 1=g./ D e  … BF, g is not a potential. Remark 5.26. Proposition 5.25 can be restated as follows. For every h 2 BF  , there exists an extended Bernstein function f such that h D e f . In other words, log h is an extended Bernstein function. In Corollary 3.8 (iii) we have proved that the set BF is closed under composition. Now we are going to give an alternative proof of this fact by explicitly producing the corresponding convolution semigroup. Suppose that . t / t0 and . t / t0 are two convolution semigroups on Œ0; 1/ with the corresponding Bernstein functions Z f ./ D a C b C .1  e t / .dt / .0;1/

65

Chapter 5 A probabilistic intermezzo

and

Z g./ D ˛ C ˇ C

.0;1/

.1  e t / .dt /:

Let us define a new family of measures . t / t0 by Z  t .dr/ D s .dr/  t .ds/

(5.23)

Œ0;1/

where the integral is a vague integral in the sense of Remark A.3. Theorem 5.27. The family . t / t0 is a convolution semigroup of sub-probability measures on Œ0; 1/ whose corresponding Bernstein function is equal to f ı g. Moreover, Z .f ı g/./ D f .˛/ C ˇb C .1  e t / .dt / (5.24) .0;1/

and the Lévy measure is given by the vague integral Z s .dt / .ds/: .dt / D b.dt / C

(5.25)

.0;1/

Remark 5.28. (i) Once R formula (5.24) is established, it follows immediately that  is a Lévy measure, i.e. .0;1/ .1 ^ t / .dt / < 1. (ii) The convolution semigroup . t / t0 is called subordinate to . t / t0 by . t / t0 . If .S t / t0 and .T t / t0 are independent subordinators corresponding to . t / t0 and . t / t0 , respectively, then the subordinator corresponding to . t / t0 is the process .T .S t // t0 . This process, called a subordinate to T by the subordinator S , is a particular form of a stochastic time change: T .S t /.!/ D TS t .!/ .!/. We will extend the concept of subordination to strongly continuous semigroups on a Banach space in Section 13.2. Proof of Theorem 5.27. We check first that each  t is a sub-probability measure. Indeed, Z Z s Œ0; 1/  t .ds/   t .ds/  1:  t Œ0; 1/ D Œ0;1/

Œ0;1/

Let us compute the Laplace transform of the measure  t :  Z Z Z u u L . t I / D e  t .du/ D e s .du/  t .ds/ Œ0;1/ Œ0;1/ Z ZŒ0;1/ L .s I /  t .ds/ D e sg./  t .ds/ D Œ0;1/

Œ0;1/

  D L  t I g./ D e tf .g.// :

66

Chapter 5 A probabilistic intermezzo

This proves also that L . tCs I / D e .tCs/f .g.// D e tf .g.// e sf .g.// D L . t I /L .s I /; as well as vague continuity since t 7! L . t I / D e tf .g.// is continuous. In order to obtain formula (5.24), first note that e ˛s D s Œ0; 1/, and Z Z t .1  e / s .dt / .ds/ .0;1/ .0;1/ Z  Z t D .1  e / s .dt / .ds/ .0;1/ .0;1/ Z  ˛s  e  L .s I / .ds/ D Z.0;1/ .e s˛  e sg./ / .ds/ D .0;1/ Z Z sg./ .1  e / .ds/  .1  e ˛s / .ds/ D .0;1/ .0;1/ Z .1  e sg./ / .ds/  .f .˛/  a  b˛/: D .0;1/

Every function u 2 Cc .0; 1/ can be estimated by ju.t /j  c;u .1  e t /, t > 0, with a suitable constant c;u > 0. Therefore, the above calculation shows that the vague integral (5.25) exists and defines a measure  on .0; 1/, cf. Remark A.3. Now we have Z   f g./ D a C bg./ C .1  e tg./ / .dt / .0;1/   Z D a C b ˛ C ˇ C .1  e t / .dt / .0;1/ Z Z   t C .1  e / s .dt / .ds/ C f .˛/  a  b˛ .0;1/ .0;1/ Z .1  e t / .dt /: D f .˛/ C ˇb C .0;1/

Comments 5.29. A good treatment of convolution semigroups and their potential theory is the monograph by Berg and Forst [39]; the classic exposition by Feller, [118], covers the material from a probabilistic angle and is still readable. Subordinators are treated in most books on Lévy processes, for example in Bertoin [47] and his St.-Flour course notes [49].

Chapter 5 A probabilistic intermezzo

67

Infinitely divisible and self-decomposable functions can be found in many books on probability theory. The monograph by Steutel and van Harn [339, Sections I.5, V.2] is currently the most comprehensive study and has dedicated chapters for one-sided laws, but Gnedenko and Kolmogorov [130], Lukacs [251], Loève [246, vol. 1, Chapter VI], Petrov [289, pp. 82–87], [290, Chapters 3,4], Rossberg, Jesiak, Siegel [309, Sections 7, 11–13] and Sato [314, Chapter 3] contain most of the material mentioned in Remark 5.12. That all infinitely divisible functions in CM are of the form exp.BF/ was discovered by Schoenberg [325, Theorem 9, p. 835]. An alternative short proof is in Horn [169, Theorem 4.4]; note that one has to have Œ0; 1 as support of the measure d and not, as stated in [169], .0; 1/. The name logarithmically completely monotone appears in 2004 for the first time in [299] and [300], see also the comments in [34]. Theorem 5.11 (ii) and the characterization of extended Bernstein functions in Remark 5.9 seem to be new. Our presentation of the stable distributions is from Sato [314, Examples 24.12, 21.7], the representation of the Laplace exponents of self-decomposable distributions is from Steutel and van Harn [339, Theorem 2.11, p. 231]. The short proofs of Proposition 5.15 and Corollary 5.16 seem to be new. The proof of Proposition 5.17, which is due to Lévy [243, Section VII.55], is adapted from Sato [314, p. 95]. Self-decomposable distributions were characterized by P. Lévy as limit distributions of normed partial sums of sequences of independent random variables, see [242] and [243, pp. 192–193]; Lévy mentions that Khintchine drew his attention to this problem in a letter in 1936. The name self-decomposable appears for the first time in 1955 in the first edition of Loève’s book on probability theory [246], the notation L goes back to Khintchine [207], see the comment in Gnedenko and Kolmogorov [130, Section 29]. Multiple and complete self-decomposability was introduced by Urbanik [357, 358] for distributions on R; Sato [315] reformulated and extended it to Rd . The standard notation for p-times and completely self-decomposable distributions is Lp1 and L1 , respectively. In the 1980s Nguyen [277, 278, 279] obtained the continuous parameter extension of these classes. The names multiple and complete self-decomposability are from [276]. The multidimensional version of Proposition 5.20 is from [315] where it is shown that almost every (w.r.t. to the spherical part of the Lévy measure) radial part of the Lévy measure has the given representation. The analogue of the representation (5.15) is from [315]. The last part of Theorem 5.22 appears in [357] and [315]. The best sources for potential theory of convolution semigroups are Berg and Forst [39, Chapter III] and the original papers by F. Hirsch from the early 1970s. A more probabilistic approach can be found in Bertoin [47, Chapter II, III]. A somewhat hard to read but otherwise most comprehensive treatment is the monograph by Dellacherie and Meyer [93, 94]. M. Itô [181, Section 2], see also [180], gives the following interesting characterization of potential measures and potentials: U is a potential measure on Œ0; 1/ if, and only if, for all vaguely continuous convolution semigroups . t / t0 of measures on,

68

Chapter 5 A probabilistic intermezzo

R1 R1 say Rd , the measure 0  t U.dt / is of the form 0  t dt for some other vaguely continuous convolution semigroupR . t / t0 of measures on Rd . Since the potential 1 measures on Rd are of the form 0  t dt , this means that P or the potential measures on Œ0; 1/ are characterized by the fact that they operate on the potentials in Rd . It is also shown in [181, pp. 118–119] that U is a potential measure on Œ0; 1/ if, and only if, it satisfies the domination principle, i.e. if for all continuous, positive and compactly supported v; w 2 CcC Œ0; 1/ U ? v  U ? w on supp v

implies

U ? v  U ? w on Œ0; 1/:

Subordination was introduced by Bochner in the short note [62], see also his monograph [63, Chapter 4.4]. A rigorous functional analytic and stochastic account is in the paper by Nelson [270].

Chapter 6

Complete Bernstein functions

The class of complete Bernstein functions has been used throughout the literature in many branches of mathematics but under various names and for very different reasons, e.g. as Pick or Nevanlinna functions in (complex) interpolation theory, Löwner or operator monotone functions in functional analysis, or as class .S / in the Russian literature on complex function theory in a half-plane.

6.1 Representation of complete Bernstein functions Many explicitly known Bernstein functions are actually complete Bernstein functions, which is partly due to the fact that the measure  in the representation formula (3.2) for complete Bernstein functions has a nice density. We take this as our starting point and show that our definition coincides with other known approaches. Definition 6.1. A Bernstein function f is said to be a complete Bernstein function if its Lévy measure  in (3.2) has a completely monotone density m.t / with respect to Lebesgue measure, Z f ./ D a C b C .1  e t / m.t / dt: (6.1) .0;1/

We will use CBF to denote the collection of all complete Bernstein functions. Denote by H" and H# the open upper and lower complex half-planes, i.e. the sets of all z 2 C with Im z > 0 and Im z < 0, respectively. Theorem 6.2. Suppose that f is a non-negative function on .0; 1/. Then the following conditions are equivalent. (i) f 2 CBF. (ii) The function  7! f ./= is in S. (iii) There exists a Bernstein function g such that f ./ D 2 L .gI /;

 > 0:

(iv) f has an analytic continuation to H" such that Im f .z/  0 for all z 2 H" and such that the limit f .0C/ D lim.0;1/3!0 f ./ exists and is real.

70

Chapter 6 Complete Bernstein functions

(v) f has an analytic continuation to the cut complex plane C n .1; 0 such that Im z  Im f .z/  0 and such that the limit f .0C/ D lim.0;1/3!0 f ./ exists and is real. (vi) f has an analytic continuation to H" which is given by Z z  .dt / f .z/ D a C bz C .0;1/ z C t where R a; b  0 are non-negative constants and  is a measure on .0; 1/ such that .0;1/ .1 C t /1  .dt / < 1. Proof. (i),(ii) If m is completely monotone, then we know from Theorem 1.4 that m D L  for some measure  on Œ0; 1/. By Fubini’s theorem Z Z Z t .1  e /m.t / dt D .1  e t /e ts dt .ds/ .0;1/ Œ0;1/ .0;1/   Z 1 1  D .ds/ Cs Œ0;1/ s Z  s 1 .ds/: D Œ0;1/  C s The expression on the right-hand side converges R R if, and only if, ¹0º D 0, i.e. if  is supported in .0; 1/, and if .0;1/ s 1 .ds/ C .1;1/ s 2 .ds/ < 1. This shows that a f ./ D CbC  

Z .0;1/

1 s 1 .ds/ Cs

is indeed a Stieltjes function. The converse direction follows easily by retracing the above steps. (ii))(iii) Note that h./  a has L .aI / D a as Laplace transform. Moreover, we have L .hI / D h.0C/ C L .h0 I /. Thus,      a f ./ D b C C L 2 s 1 .ds/I  D b C L a C L s 1 .ds/ I    D L .gI /; where we set g 0 WD aCL .s 1 .ds// 2 CM so that g 2 BF, cf. the remark following Definition 3.1. R (iii))(i) If g.t / D ˛ C ˇt C .0;1/ .1  e ts / .ds/, we find  2 L .gI / D 2

ˇ ˛ C 2C  

Z

1 0

e t

Z .0;1/

.1  e ts / .ds/ dt



Section 6.1 Representation of complete Bernstein functions

71

 1 1  .ds/  Cs .0;1/   Z 1 2 1  D ˛ C ˇ C s .ds/ s Cs .0;1/ Z Z 1 .1  e t / e ts s 2 .ds/ dt D ˛ C ˇ C 

Z

2

D ˛ C ˇ C

0

.0;1/

and (i) follows. R   .dt /, the natural way to extend f (ii))(iv) Since f ./ D a C b C .0;1/ Ct " is to replace  by z 2 H . If we write z D  C i, then z zN C zt 2 C t  C 2 t z D D Ci : 2 2 2 zCt jz C t j . C t / C  . C t /2 C 2 Moreover, for any fixed z D  C i 2 H" we have 1 2 C t  C 2 2 2 . C t / C  1Ct

and

t t 2 2 . C t / C  1 C t2

(.t / .t / means that there exist two absolute constants c2 > c1 > 0 such that c1 .t /  .t /  c2 .t /) which entails that the integral Z Z z 2 C t  C 2  .dt / D  .dt / 2 2 .0;1/ z C t .0;1/ . C t / C  Z t  .dt / Ci 2 2 .0;1/ . C t / C  converges locally uniformly and defines an analytic function on H" with positive imaginary part whenever  D Im z  0. All these remain true if we add a C bz. The above estimates and the integration properties of the measure  also show that Z Z z  lim  .dt / D lim  .dt / D 0 .0;1/3!0 .0;1/  C t H" 3z!0 .0;1/ z C t so that the limit lim.0;1/[H" 3z!0 f .z/ D a exists and is real. (iv))(v) This is an easy consequence of the Schwarz reflection principle: since f j.0;1/ is real-valued, we may use f .Nz/ WD f .z/, z 2 H" , to extend f across .0; 1/ to the lower half-plane. By the reflection principle this gives an analytic function on C n .1; 0 D H" [ .0; 1/ [ H# . Since Im f .z/ D Im f .z/ < 0 for z 2 H" , we get Im f .z/ > 0 or Im f .z/ < 0 for z 2 H" or z 2 H# respectively, which can be expressed by Im z  Im f .z/  0. Since limH" 3z!0 f .z/ exists and is real, the above construction automatically shows that the limit limCn.1;03z!0 f .z/ exists.

72

Chapter 6 Complete Bernstein functions

(v))(vi) Note that for p; q > 0 the function fp;q .z/ WD

p.f .z/ C i q/ p C .f .z/ C i q/

is analytic in H" . Write f .z/ D u.z/C iv.z/ and observe that v.z/  0 on H" . Then fp;q .z/ D

p.u.z/ C p/u.z/ C p.v.z/ C q/2 i p 2 .v.z/ C q/ C : .u.z/ C p/2 C .v.z/ C q/2 .u.z/ C p/2 C .v.z/ C q/2

This shows that fp;q maps H" into itself. Moreover, on H" jfpq j D

pjp C f C i qj C p 2 pjf C i qj  jp C .f C i q/j jp C .f C i q/j p2 p2 pC : pC jIm .p C .f C i q//j q

Now set p D n and q D 1=n, n 2 N, and write fn .z/ WD fn;1=n .z/ D

f .z/ C 1C

1 n

i n

f .z/ C

i n2

:

Thus, fn is analytic on H" , bounded 6 by 2n3 and limn!1 fn .z/ D f .z/. iR We want to apply Cauchy’s theorem. For this we fix z 2 H" and consider z + ih the following integration contour R ih (in counterclockwise orientation) consisting of a circular arc of radius R θ θ extending from ! D  to ! D    . −R R The arc is closed by a straight line, parallel to the x-axis; the distance between this line and the x-axis is h D h.; R/. The parameters R and  are chosen in such a way that z Cih lies inside R and zN Cih outside of R but not on the real axis. Using Cauchy’s formula we find Z 2 fn ./ fn00 .z C ih/ D d 2 i R .  z  ih/3 Z R cos Z 1 1  Re i! fn .Re i! / fn .t C ih/ D dt C d!:  i R cos .t  z/3  .Re i!  z  ih/3 Since fn is bounded, the second integral converges to 0 as R ! 1. Therefore, Z 1 1 fn .t C ih/ 00 dt: fn .z C ih/ D  i 1 .t  z/3

73

Section 6.1 Representation of complete Bernstein functions

A similar calculation where we replace z C ih by zN C ih which is not surrounded by R shows 1 0N D i

Z

1 1

1 fn .t C ih/ dt D  3 .t  z/ N i

Z

1 1

fn .t C ih/ dt: .t  z/3

Adding these two equalities yields Z Z 2 1 Im fn .t C ih/ 2 1 Im fn .s C ih/ fn00 .z C ih/ D dt D ds:  1 .t  z/3  1 .s C z/3 Since

1 d2 2 dz 2 .s

(6.2)

C z/1 D .s C z/3 we find by integrating twice that for every  > 0

1 fn .i C ih/  fn .i  C ih/ D 

Z

1

1

i.1  / Im fn .s C ih/ ds: .i C s/.i  C s/

Therefore, Im fn .i C ih/  Im fn .i  C ih/ Z 1 1 .1  /.s 2  / Im fn .s C ih/ ds D  1 .1 C s 2 /. 2 C s 2 /

Z 1 2  Z 1 1 s Im fn .s Cih/ 1 Im fn .t Cih/ ds  D dt 2 2 2 2  1C.t /2 1 .1Cs /. Cs / 1 .1Ct / where we used the change of variables s D t in the last step. Using dominated convergence we get, as  ! 0, Im fn .i C ih/  Im fn .ih/ Z Z 1 1 Im fn .s C ih/ 1 1 Im fn .ih/ D ds  dt  1 1 C s2  1 1 C t 2 Z 1 1 Im fn .s C ih/ ds  Im fn .ih/: D  1 1 C s2 This and an application of Fatou’s lemma show that Im f .i C ih/ D lim Im fn .i C ih/ n!1 Z 1 1 Im fn .s C ih/ ds D lim n!1  1 1 C s2 Z 1 1 Im f .s C ih/ ds:   1 1 C s2

(6.3)

We conclude that the family of measures ¹.1 C s 2 /1 Im fn .s C ih/ dsºn2N is vaguely bounded, hence relatively compact in the vague topology of measures, cf.

74

Chapter 6 Complete Bernstein functions

Theorem A.5; thus, every subsequence has a vaguely convergent subsequence. Note that limn!1 Im fn .s C ih/ D Im f .s C ih/ boundedly as is easily seen from n2 .Im f .s C ih/ C n1 / .Re f .s C ih/ C n/2 C .Im f .s C ih/ C n1 /2 Im f .s C ih/ C 1  1 .n Re f .s C ih/ C 1/2  CK .Im f .s C ih/ C 1/

Im fn .s C ih/ D

as long as s is in a compact set K  R and n is sufficiently large. This means that by dominated convergence Z 1 Z 1 Im fn .s C ih/ Im f .s C ih/ .s/ ds D .s/ ds lim n!1 1 1 C s2 1 C s2 1 for all  2 Cc .R/. Thus, the limiting measure is .1 C s 2 /1 Im f .s C ih/ ds and we get from (6.2) Z 1 2 1 Im f .s C ih/ ds: f 00 .z C ih/ D  1 .z C s/3 From (6.3) we deduce with a similar argument that ¹Im f .s C ih/ dsºh>0 is a vaguely relatively compact family of measures. For some subsequence hk ! 0 we get 2 Im f .s C ihk / k!1 ….ds/ ds ! (6.4) 2  1Cs 1 C s2 R1 1 for a suitable limiting measure … satisfying 1 1Cs 2 ….ds/  Im f .i / < 1. We will see below in Corollary 6.3 that … does actually not depend on the subsequence. Thus, Z 1 00 ….ds/: f .z/ D 3 R .z C s/ Integrating twice, we find for w; z 2 H" , with a suitable constant b 2 C, Z zw 1 ….ds/; f .z/  f .w/ D b.z  w/ C 2 R .z C s/.w C s/ and it is clear that this formula extends to all z; w 2 C n .1; 0. In particular, we get for z D  > 0 and w D  > 0 Z 1  ….ds/: (6.5) f ./  f ./ D b.  / C 2 R . C s/. C s/ Since limh!0 Im f .s C ih/ D 0 locally boundedly on the negative half-axis s < 0, the measure … is supported in Œ0; 1/. If we set in (6.5)  D 1 and let  ! 1, we

Section 6.1 Representation of complete Bernstein functions

75

find that b  0. By Fatou’s lemma and the finiteness of f .0C/ D lim!0 f ./ we conclude that Z 1  ….ds/ : f ./  f .0C/  2 Œ0;1/  C s s R This shows that …¹0º D 0 and .0;1/ . C s/1 s 1 ….ds/ < 1; by the dominated convergence theorem we get Z z f .z/ D a C bz C  .ds/; z C s .0;1/ where a D b  0 and  .ds/ D 12 s 1 ….ds/ is a measure on .0; 1/ R f .0C/  0, 1 such that .0;1/ .1 C s/  .ds/ < 1. (vi))(ii) This is obvious if we take z D  2 .0; 1/. Theorem 6.2 and its proof allow us to draw a few interesting conclusions. Corollary 6.3. The representation measure  in Theorem 6.2 (vi) is given by Z 1 1  .u; v D lim Im f .s C ih/ ds h!0C  .u;v s Z 1 f .s C ih/ D lim ds Im s  ih h!0C  .u;v

(6.6)

which holds for all positive continuity points u; v > 0 of the distribution function t 7!  .1; t . Proof. The proof of Theorem 6.2 (v))(vi) shows that the limits appearing in (6.6) exist along some sequence .hk /k2N , 0 < hk ! 0. Since f is analytic in H" , the limit does not depend on the particular sequence and we conclude that  (hence, ) is uniquely determined by f . Note that a D  ¹0º, a D f .0C/ D lim!0 f ./ and b D lim!1 f ./=. Note that (6.6) remains valid for u < v < 0 and shows that  .1; 0/ D 0. Remark 6.4. It follows from Theorem 6.2 and Corollary 6.3 that every complete Bernstein function f 2 CBF can be uniquely represented in the following form Z   .dt /;  > 0; (6.7) f ./ D a C b C .0;1/  C t where a; b  0 are non-negative constants and  is a measure on .0; 1/ such that R .1 C t /1  .dt / < 1. A comparison of the representation formulae (2.1) and .0;1/ (6.7) shows that for all g 2 S the function g./ is a complete Bernstein function. The formula (6.7) is called the Stieltjes representation of the complete Bernstein function f , and the measure  is called the Stieltjes measure of the complete Bernstein function f .

76

Chapter 6 Complete Bernstein functions

Corollary 6.5 (Angular limits; G. Julia et al.). Let f be (the analytic extension of) a complete Bernstein function. Then the limit f .Re i! / Db R!1 Re i! lim

exists uniformly for all ! 2 .;    / and all fixed  2 .0; =2/. Proof. Let z D Re i! with  < ! <    for some fixed  2 .0; =2/. From the formula in Theorem 6.2 (vi) we get Z f .Re i! / a  .dt / : D CbC i! i! i! C t Re Re .0;1/ Re An elementary but somewhat tedious calculation shows that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 RCt Re i! C t ˇ ˇ ˇDˇ ˇ Re i! C t ˇ ˇ R2 C 2Rt cos ! C t 2 ˇ  R2  2Rt cos  C t 2 5 1 :  1  cos  R C t From dominated convergence we conclude f .Re i! / Db R!1 Re i! lim

uniformly for all ! 2 .;    /. In fact, our consideration shows ˇ ˇ

 Z ˇ ˇ f .Re i! / 5 5 1 f .R/ a ˇ ˇ  .dt /  b ˇ Re i!  b ˇ  R C 1  cos  1  cos  R .0;1/ R C t which is valid for all ! 2 . C ;    /. Corollary 6.6. Let f be (the analytic extension of) a complete Bernstein function. Then f .SŒ0;  /  SŒ0; 

with SŒ0;  D ¹z 2 C W 0  arg z   º;  2 .0; /:

(6.8)

Proof. This follows again from the formula in Theorem 6.2 (vi). Indeed, if we add z D re i ; w D Re i 2 C, the argument of w C z lies between  and  ; a similar statement is true for integrals: Z  arg f .t; z/ .dt / 2 inf t arg f .t; z/; sup t arg f .t; z/ : Note that for all t > 0 re i re i z D i D 0 i 0 zCt re C t re

where 0 <  0 < :

z /  arg z and the same is true for f .z/ because of the formula in Thus, arg. zCt Theorem 6.2 (vi).

Section 6.1 Representation of complete Bernstein functions

77

The proof of Theorem 6.2 (v))(vi) uses essentially the fact that the function f is analytic and preserves the upper half-plane H" . This class of functions is well known in the literature. Definition 6.7. A Pick function or Nevanlinna function or Nevanlinna–Pick function is an analytic function f defined on H" or C n R such that f preserves the upper half-plane H" , i.e. f .H" /  H" . If the analytic function f is only defined on H" , then we can extend it onto C n R by setting ´ f .Nz/ if z 2 H# I F .z/ WD f .z/ if z 2 H" : Although F is analytic, it is not a proper analytic extension since H" and H# are disjoint. If, however, f has a real-valued, continuous extension on some interval .a; b/  R, then Schwarz’ reflection principle applies and F is a proper analytic extension. Corollary 6.8. Let f be a Nevanlinna–Pick function. Then f has a unique integral representation Z zt  1 f .z/ D ˛ C ˇz C .dt / (6.9) R zCt  Z  1 t D ˛ C ˇz C .1 C t 2 / .dt /  (6.10) 2 1 C t z C t R for all z 2 C n .1; 0. Here  is a finite measure on R, and ˛; ˇ are constants satisfying ˇ  0 and ˛ 2 R. If f has a continuous extension with f W .0; 1/ ! R, then supp   Œ0; 1/, i.e. Z zt  1 f .z/ D ˛ C ˇz C .dt / (6.11) Œ0;1/ z C t   Z 1 t D ˛ C ˇz C .1 C t 2 / .dt /:  (6.12) 2 1 C t z C t Œ0;1/ If f .0C/ is finite, then ¹0º D 0. Proof. The representations (6.9) and (6.10) are clearly equivalent. When deriving (6.5) in the proof of Theorem 6.2 (v))(vi) we used only the fact that f preserves the upper half-plane. Therefore we can use (6.5) in the present context

78

Chapter 6 Complete Bernstein functions

and rewrite it in the following form Z  1 ….ds/ f ./  f ./ D b.  / C 2 R . C s/. C s/  Z  1 1 s ….ds/ D b C  2 R 1 C s2  C s  Z  1 1 s   b  ….ds/: 2 R 1 C s2  C s Since we have ˇ ˇ   ˇ ˇ s 1 1 1 s C 1 ˇ ˇ    C ; ˇ 1 C s 2  C s ˇ .1 C s 2 /. C s/ 1 C s2  both integrals appearing in the formula above are convergent. This already shows that  Z  1 s 1 ….ds/  f ./ D c C b C 2 R 1 C s2  C s for some constant c 2 R. Setting ˛ WD c, ˇ WD b and .dt / WD 12 .1 C t 2 /1 ….dt / gives (6.10). If f has a continuous extension to f W .0; 1/ ! R we see, just as in the proof of Theorem 6.2 (v))(vi), that supp   Œ0; 1/ and (6.12) follows. It is clear that (6.11) and (6.12) are equivalent. Again as in the proof of Theorem 6.2 (v))(vi) we find that f .0C/ < 1 implies ¹0º D 0. The following theorem shows that the family of complete Bernstein functions coincides with the family of Nevanlinna–Pick functions which are non-negative on the positive real axis and admit a right limit at the origin. Theorem 6.9. The set CBF consists of all Nevanlinna–Pick functions which are nonnegative on .0; 1/. In particular, the following integral representations hold for all f 2 CBF and z 2 C n .1; 0 Z zt  1 .dt / (6.13) f .z/ D ˛ C ˇz C .0;1/ z C t   Z 1 t D ˛ C ˇz C  (6.14) .1 C t 2 / .dt /: 2 zCt .0;1/ 1 C t R R Here  is a measure on .0; 1/ such that .0;1/ t 1 .dt / C Œ1;1/ .dt / < 1 and R ˛; ˇ  0 are constants with ˛  .0;1/ t 1 .dt /.

79

Section 6.1 Representation of complete Bernstein functions

Proof. The equivalence of the two formulae (6.13) and (6.14) is obvious. Assume now that f is the function given by (6.13). Then f .z/  ˇz  ˛ Z zt  1 .dt / D .0;1/ z C t   Z Z Z z 1 1 1 D t .dt /   .dt /  .dt / t .0;1/ z C t .0;1/ z C t .0;1/ t   Z Z 1 z 1 D tC .dt /  .dt / z C t t .0;1/ .0;1/ t R which becomes the formula in Theorem 6.2 (vi) if we set a WD ˛  .0;1/ t 1 .dt /, b WD ˇ and  .dt / WD .t C t 1 / .dt / D Z .0;1/

 .dt / D 1Ct

Z .0;1/

t 2 C1 t

.dt /. By assumption, a; b  0 and

t2 C 1 .dt /  2 t .1 C t /

Z .0;1/

.dt / C t

Z Œ1;1/

.dt / < 1:

The same calculation, run through backwards, shows that the formula in Theorem 6.2 (vi) entails (6.13), and the proof is complete. Remark 6.10. (i) The (unique) representing measure  appearing in Corollary 6.8 and Theorem 6.9 is called the Pick measure or Nevanlinna measure or Nevanlinna– Pick measure. (ii) Just as in Corollary 6.3 we can calculate the coefficients ˛; ˇ and the Nevanlinna–Pick measure  appearing in (6.11) and (6.13) from f . Clearly, ˛ D Re f .i /; while

Z

ˇ D lim

1 .1 C t / .dt / D lim h!0  .u;v

!1

f .i / i

Z

2

.u;v

Im f .s C ih/ ds

(6.15)

holds for all u < v which are continuity points of . To see this, recall from the proof of Theorem 6.9 that t  .dt / D .1 C t 2 / .dt / where  is the representing measure from the formula in Theorem 6.2 (vi). The vague convergence in (6.15) follows now from (6.4).

80

Chapter 6 Complete Bernstein functions

(iii) For further reference let us collect the relations between the various representing measures and densities appearing in Definition 6.1, Theorem 6.2 and Theorem 6.9: Z   e ts s  .ds/ D L s .ds/I t m.t / D Z.0;1/   e ts .s 2 C 1/.ds/ D L .s 2 C 1/ .ds/I t ; D  .dt / D

.0;1/ 2 t C1

t

.dt /:

6.2 Extended complete Bernstein functions In analogy with the notion of extended Bernstein functions and their characterization given in Remark 5.9, we introduce the concept of extended complete Bernstein functions. Definition 6.11. A function f W .0; 1/ ! R is called an extended complete Bernstein function if there exist some g 2 CBF and h 2 S such that f D g  h. We write CBF e for the set of all extended complete Bernstein functions. Using the representation of Nevanlinna–Pick functions which are real on .0; 1/, cf. Corollary 6.8, we can prove the following characterization of extended complete Bernstein functions. Proposition 6.12. The set of extended complete Bernstein functions consists of all Nevanlinna–Pick functions which are real on .0; 1/. In other words, f is an extended complete Bernstein function, if f W .0; 1/ ! R extends analytically to H" and preserves it. Equivalently f 2 CBF e if, and only if, Z zt  1 .dt / (6.16) f .z/ D ˛ C ˇz C Œ0;1/ z C t   Z 1 t D ˛ C ˇz C  (6.17) .1 C t 2 / .dt / 2 zCt Œ0;1/ 1 C t for all z 2 C n .1; 0. Here  is a finite measure on Œ0; 1/, and ˛ 2 R and ˇ  0 are constants. Proof. Suppose that f is given by (6.16). Then Z Z  1 t.dt /  .dt / f ./ D ˛ C ˇ C .0;1/  C t Œ0;1/  C t for all  > 0. Since  is a finite measure, we see that Z Z 1 1 t.dt / < 1 and .dt / < 1: .0;1/ 1 C t Œ0;1/ 1 C t

81

Section 6.2 Extended complete Bernstein functions

If ˛  0 we define

Z

g./ WD ˛ C ˇ C while for ˛ < 0 we set Z g./ WD ˇ C

.0;1/

.0;1/

 t.dt /; Ct

 t.dt /; Ct

Z h./ WD

Œ0;1/

1 .dt /; Ct

Z h./ WD ˛ C

Œ0;1/

1 .dt /: Ct

In both cases g 2 CBF, h 2 S and f D g  h. Conversely, suppose that f D g  h with g 2 CBF and h 2 S. Then Z  .dt / g./ D a C b C  C t .0;1/ R with a; b  0 and a measure  on .0; 1/ such that .0;1/ .1 C t /1 .dt / < 1, and Z

1  .dt / Œ0;1/  C t R with c  0 and a measure  on Œ0; 1/ such that Œ0;1/ .1 C t /1  .dt / < 1. Define R the measure  on Œ0; 1/ by .dt / D t .dt /C .dt /. As Œ0;1/ .1Ct 2 /1 .dt / < 1 all integrals below are well defined. Then   Z 1 t  .dt / 2 Ct Œ0;1/ 1 C t     Z Z 1 1 t t t .dt / C  .dt /   D 2 2 Ct Ct .0;1/ 1 C t Œ0;1/ 1 C t     Z Z 1 t 1 1  D  t .dt / C t .dt / 2 t Ct .0;1/ 1 C t .0;1/ t   Z 1 t  .dt /  C 2 Ct Œ0;1/ 1 C t Z Z Z .dt /  t  .dt / D .dt / C C 2 2 1 C t  C t .0;1/ .0;1/ Œ0;1/ 1 C t Z 1  .dt /  Œ0;1/  C t Z Z    .dt / t  .dt /  D C g./  a  b C  h./  c 2 2 .0;1/ 1 C t Œ0;1/ 1 C t h./ D c C

D f ./  ˇ  ˛;

82

Chapter 6 Complete Bernstein functions

where ˇ D b  0 and ˛ D aC Therefore,

R

.0;1/ .1Ct



Z f ./ D ˛ C ˇ C

Œ0;1/

2 /1 .dt /

R

t 1  2 1Ct Ct

Œ0;1/ t .1Ct

2 /1  .dt / 2

R.

 .1 C t 2 / .dt /;

where .dt / D .1 C t 2 /1 .dt / is a finite measure on Œ0; 1/. A typical example of an extended complete Bernstein function is obtained by taking .dt / D .1 C t 2 /1 dt : Z 1 t  1 dt ;  > 0: (6.18) log  D  C t 1 C t2 0 The following results contain further connections between the classes of extended complete Bernstein functions and Nevanlinna–Pick functions. Corollary 6.13. Let f W H" ! H" be a Nevanlinna–Pick function. Then the following statements are equivalent. (i) f 2 CBF e . (ii) g.z/ WD

z .Re f .i / 1Cz 2

C zIm f .i /  f .z// is in CBF.

Proof. Let f be a Nevanlinna–Pick function. Using the representation (6.9) we find   Z Z it  1 .dt / D ˛ C i ˇ C f .i / D ˛ C ˇi C .dt / : R t Ci R Therefore,  z  Re f .i / C zIm f .i /  f .z/ 2 1Cz  Z  tz  1 z .dt / z D 1 C z2 R t Cz Z z D .dt /: R t Cz R If g is in CBF, then g.z/ D aCbz C .0;1/ z=.t Cz/ .dt /. By the representation we have obtained above, limjzj!1 g.z/=z D 0, so that b D 0. Now a D ¹0º, implying that supp   Œ0; 1/ and Proposition 6.12 shows that f 2 CBF e . Conversely, if f 2 CBF e , we know from Proposition 6.12 that supp   Œ0; 1/; thus g is a complete Bernstein function. g.z/ D

Corollary 6.14. Let f W .0; 1/ ! R be a function. Then the following statements are equivalent.

Section 6.2 Extended complete Bernstein functions

83

(i) f 2 S; (ii) f has an extension which is a Nevanlinna–Pick function and f j.0;1/ is continuous and non-positive. Proof. Assume that (i) holds. Since f 2 CBFe , Proposition 6.12 shows that f has an extension which is a Nevanlinna–Pick function; moreover f j.0;1/ is non-positive and continuous. Conversely, by Proposition 6.12 every function satisfying (ii) is in CBF e . By (6.16) and the fact that f ./  0 for all  > 0, we get Z t  1 .dt /  0: ˛ C ˇ C .0;1/  C t Therefore, we find for every R > 0 and  > 0 Z Z t 1 .dt /  .dt /: ˛ C ˇ C  C t  C t .0;R/ .0;1/ Divide both sides of this inequality by  and let  ! 1. Since  is a finite measure, we find by dominated convergence that ˇ D 0. Taking this into account and using again dominated convergence,  ! 1 shows Z t .dt /  0: 1 < ˛ C .0;R/

Since R > 0 is arbitrary, we see that   Z Z t  1  t .dt / t .dt / C f ./ D ˛ C Ct .0;1/ .0;1/ Z Z 1 .1 C t 2 /.dt /: D˛C t .dt /  .0;1/ .0;1/  C t Proposition 6.15. Let f W H" ! H" be a Nevanlinna–Pick function. Then the following statements are equivalent. (i) f has an extension to H" [ Œ0; 1/ such that f j.0;1/ is continuous and nonpositive; (ii) zf .z/ is a complete Bernstein function; (iii) zf .z/ is a Nevanlinna–Pick function. Proof. (i))(ii) By Corollary 6.14, every Nevanlinna–Pick function f which is continuous and non-positive on .0; 1/ is of the form a  g where a  0 and g 2 S.

84

Chapter 6 Complete Bernstein functions

Therefore, zf .z/ D az C zg.z/, and this is a complete Bernstein function by (6.7), cf. Remark 6.4 (ii))(iii) This follows from Theorem 6.9. (iii))(i) Since zf .z/ is a Nevanlinna–Pick function, we know from (6.11) and the remark following the proof of Corollary 6.8 that Z zt  1 .dt / (6.19) zf .z/ D ˛ C ˇz C R zCt for some constants ˛ 2 R, ˇ  0 and a finite measure  on R. Since f is also a Nevanlinna–Pick function, it has a similar representation with ˛; O ˇO and . O Therefore, Z zt  1 O 2 z .dt O / zf .z/ D ˛z O  ˇz R zCt  Z  1  zt O 2 D ˛z O  ˇz t C zt  1 .dt O / zCt R  Z Z  zt  1 2 O D ˛z O  ˇz C  z t .dt O /: .dt O /C zCt R R Note that .zt 1/=.z Ct /z D .z 2 C1/=.z Ct /. If we subtract both representations for zf .z/ we get     Z Z Z 2  z C 1 2 O   t O .dt /: .dt O / C ˇ C ˛O C .dt / z  0 D ˇz C ˛  R R R zCt (6.20) (a) Divide the equality (6.20) by z 2 , pick z D iy, y > 0, and let y ! 1. By dominated convergence we find that ˇO D 0. (b) Once we know that ˇO D 0, we can divide the thus modified equality (6.20) by z, set z D iy, y > 0, and let y ! 1. This gives   Z Z  iy    t O .dt /: .dt /  lim 0 D ˇ C ˛O C y!1 R R iy C t R R Since, by dominated convergence, limy!1 R iy=.iy C t / .dt / D R .dt / < 1, we conclude that Z Z  t .dt O / < 1 and ˇ C ˛O D  t O dt /: R

(c) Setting z D i in (6.20), we see that ˛ D

R

O / R .dt

R

and ˇ C ˛O D 

R

R .dt /.

If we combine (a)–(c), we get that (6.20) entails Z Z 2   z C 1 z    t O .dt / D 0; hence,   t O .dt / D 0: R zCt R zCt

Section 6.2 Extended complete Bernstein functions

85

Since the representing measure of a Nevanlinna–Pick function is unique, cf. (6.6) and Corollary 6.8, we conclude that .dt / D t .dt O / on R. The measures ; O are positive which means that supp   Œ0; 1/. Using this and (a)–(c) in the representation (6.19) gives   Z 1 1 ˛ zt  1 1 C  .dt / f .z/ D   ˇ  z zCt z tz tz Œ0;1/ Z  Z 1 .dt / 1 t2 C 1 D ˛ ˇ .dt / z t t Œ0;1/ Œ0;1/ z C t and, because of (c), the coefficient of z 1 vanishes. An analytic function ˆ on a domain  is said to be of bounded type (or bounded character) if ˆ can be written as a quotient ˆ D G=H where G and H are bounded analytic functions in . If ˆ is of bounded type in the upper half-plane, it satisfies the canonical or Nevanlinna factorization, i.e. ˆ.z/ D cb.z/d.z/s.z/ where jcj D 1, b.z/ is a Blaschke product, see e.g. [259, Vol. 2, Section 36],    Z  t 1 1  q.t / dt d.z/ D exp i R z C t 1 C t2 R with jq.t /j dt < 1, and Z 1 1  zt !.dt / g.z/ D 2 i R z C t R for some signed measure ! such that R .1 C t 2 / j!j.dt / < 1, see e.g. Krylov [226, Theorem XX] and the comments below. Lemma 6.16. Let ˆ W H" ! H" be an analytic function such that Re ˆ  0. Then ˆ is of bounded type. Proof. Note that j1 C ˆj  Re .1 C ˆ/  1. Therefore  WD log j1 C ˆj is a nonnegative harmonic function on H" . Since H" is simply connected,  is the real part of an analytic function h, i.e. Re h D . Define H WD e h and G WD ˆ e h . Clearly, G and H are analytic, jH j D je h j D e Re h D e   1 and, since jˆj2  jˆj2 C 2Re ˆ C 1 D jˆ C 1j2 , log jGj D log jˆj  log jˆ C 1j  0I thus, jGj  1. By definition ˆ D G=H , and the assertion follows.

86

Chapter 6 Complete Bernstein functions

If f 2 CBF, ˆ.z/ WD if .z/ is analytic in H" and Re ˆ D Im f  0. Lemma 6.16 tells us that if is of bounded type and has therefore a Nevanlinna factorization. This is the background for the next theorem. For CBF functions the factorization turns out to be rather easy and we can give a direct proof without resorting to the elaborate machinery of complex function theory in a half-plane. Theorem 6.17. For f 2 CBF, there exist a real number and a function  on .0; 1/ taking values in Œ0; 1 such that    Z 1 1 t .t / dt ;  > 0: (6.21)  f ./ D exp C 1 C t2  C t 0 Conversely, any function of the form (6.21) is in CBF. Thus there is a one-to-one correspondence between CBF and the set ± ° measurable  D . ; / W 2 R and  W .0; 1/ ! Œ0; 1 : (6.22) Proof. Assume first that f is given by (6.21). From the structure of the integral kernel it is clear that f extends to an analytic function for all z D  C i  2 H" . Moreover, Z 1    Im log f .z/ D .t / dt . C t /2 C  2 Z 01   dt . C t /2 C  2 Z0 1  dt D   2 2 1  C t shows that f preserves the upper half-plane. Since f is positive on .0; 1/, f 2 CBF by Theorem 6.2. Conversely, assume that f 2 CBF. Using the principal branch of the logarithm we get g.z/ WD log f .z/ D log jf .z/j C i arg f .z/ and arg f .z/ 2 Œ0; /. This means that g maps z 2 H" to H" ; since g is clearly analytic in H" and real (but not necessarily positive) on .0; 1/, it is an extended complete Bernstein function. Therefore, Proposition 6.12 applies and (6.17) shows that   Z 1 t 0 0 g.z/ D ˛ C ˇ z C .1 C t 2 / 0 .dt /  2 zCt .0;1/ 1 C t where ˛ 0 2 R, ˇ 0  0 and 0 is a finite measure on .0; 1/. We will now determine the possible values of .˛ 0 ; ˇ 0 ; 0 /. Set WD ˛ 0 . Since f 2 CBF we know that f ./,  > 0, grows at most linearly and g./ at most logarithmically as  ! 1, i.e. ˇ 0 D 0. Using Remark 6.10 (ii) we find that Z Z 1 .1 C t 2 / 0 .dt / D lim Im g.s C ih/ ds h!0C  .u;v .u;v Z 1 D lim arg f .s C ih/ ds  v C  uC ; h!0C  .u;v

Section 6.2 Extended complete Bernstein functions

87

where we used that 0 is supported on .0; 1/ and that arg f .s C ih/ 2 Œ0; /. This means that .1 C t 2 / 0 .dt / is absolutely continuous with respect to Lebesgue measure on .0; 1/ and that the density .t / takes almost surely values in Œ0; 1. Remark 6.18. If f .0C/ > 0, we see that  Z 1 Z 1 1 1 t .t / dt D .t / dt > 1  2 1 C t t t .1 C t 2/ 0 0 R1 which is the same as to say that 0 .t /t 1 dt < 1. In this case we can rewrite the exponent in the representation (6.21) in the following form:  Z 1 1 t .t / dt 

C 1 C t2  C t

0   Z 1   Z 1 1 1 t 1 D C .t / dt C  .t / dt  1 C t2 t t Ct 0 0 Z 1  .t / dt I DˇC Ct t 0 this is a function of the form RCCBF, i.e. an extended (complete) Bernstein function, see Remark 5.9 and Proposition 6.12. Therefore, we get the following multiplicative representation: ® ¯ CBF \ f W f .0C/ > 0  e RCCBF D .0; 1/ e CBF : Remark 6.18 has two immediate consequences. The first result is the counterpart of Remark 5.26: Corollary 6.19. If f is a complete Bernstein function with f .0C/ > 0, then log f is an extended complete Bernstein function. If f .0C/  1, then log f 2 CBF. The second result is useful if we want to construct concrete examples of complete Bernstein functions, see Section 16.11. Corollary 6.20. Suppose that f is a complete Bernstein function. If the Stieltjes measure of f has a density given by  .t / D .t /=t with 0  .t /  1 for all t > 0, then e f is a complete Bernstein function and  is the representing function appearing in the formula (6.21). We conclude this chapter with an application which is important in the fluctuation theory of Lévy processes. Proposition 6.21. If f is a complete Bernstein function, then the function  Z 1  1 log f .2  2 / g./ D exp d ;  > 0;  0 1 C 2

88

Chapter 6 Complete Bernstein functions

is again a complete Bernstein function. Moreover, for all  > 0 q q =2 =2 2 e f . /  g./  e f .2 / :

(6.23)

Proof. By Theorem 6.17, there exist some 2 R and a function  W .0; 1/ ! Œ0; 1 such that  Z 1 1 t .t / dt:  log f ./ D C 1 C t2  C t 0 Thus we have  Z Z 

1 1 1 d 1 t log g./ D C .t / dt  2 2 : 2 2  0 0 1Ct   Ct 1 C 2 From 0  .t /  1 we get ˇ ˇ ˇ t ˇ 1 1 2  2 t  1 1 1 ˇ .t / ˇˇ   1 C t 2 2  2 C t ˇ 1 C  2 1 C t 2 1 C  2 2  2 C t   2  2 t 1 1 1 C 2 2 :  1 C t 2 1 C  2 2  2 C t   Ct Since

Z

1 0

d  2 2   C t 1 C 2 1

Z

1

d Ct 0 Z d 1 1 D 2 2 t 0   t 1 C 1 p Z 1 t 1 d

 D  p t  0 2 C 1 2 t 2  2

p Z 1  t 2  2 t d d 2  ;  2  2 C t 1 C  2 2  2 t 1 C 1 2 0 0 we can use Fubini’s theorem to get ! Z 1

1 t log g./ D C  p  p  .t / dt 2 2.1 C t 2 / 2 t  C t 0  Z 1 1 t

 D C .t / dt 2 2.1 C t 2 / 2.1 C t / 0 ! Z 1 1 1  p  p  .t / dt C 2.1 C t / 2 t  C t 0  Z 1 1 s D 1 C .s 2 / ds:  1 C s2  C s 0 and

Z

1

Applying Theorem 6.17 again we get that g is a complete Bernstein function.

Section 6.2 Extended complete Bernstein functions

89

For the second part we compute ˇ ˇ ˇ ˇ ˇ log g./  1 log f .2 /ˇ ˇ ˇ 2 ˇZ " ˇ #   ˇ 1 1 t 1 ˇˇ 1 t ˇ   .t / dt p  p  D ˇ  ˇ 2 2Ct ˇ 2ˇ 0 1 C t2 1 C t  t C t p  Z  t C 1 1 p  p  dt  2 2 0 . C t / t  C t Z  1 1  D p dt D : 2 2 0 . C t / t 2 This implies that =2  log g./  (6.23)

1 2

log f .2 /  =2 for every  > 0, which is

Comments 6.22. Section 6.1: The notion of complete Bernstein function seems to appear first in the book by Prüss [297] in connection with Volterra-type integral equations. Prüss uses (iii) of Theorem 6.2 as definition of CBF: f ./ D 2 L .gI / where g is a Bernstein function. The specification complete derives from the fact that every Bernstein function can be written in the form 2 L .kI / where k is positive, nondecreasing and concave, cf. (3.4), whereas k 2 BF if f 2 CBF. Stieltjes functions and (complete) Bernstein functions were introduced into integral equations by G. E. H. Reuter [303] who found necessary and sufficient conditions so that certain Volterra integral equations have positive solutions; the starting point for Reuter’s investigations was a problem from probability theory. The equivalent integral representations for complete Bernstein functions from Theorems 6.2 and 6.9 have a long history. This is due to the fact that the class CBF has been used under different names and in different contexts. One of the earliest appearances of CBF is in the papers by Pick [293, 294] and, independently, R. Nevanlinna [272, 273] in connection with interpolation problems in C and the Stieltjes moment problem. Pick gives a characterization of all analytic functions in the unit disk or the upper half-plane which attain the values wj at the points jzj j  1 and Im zj  0, j D 1; 2; : : : ; n, respectively; the class of functions on the half-plane that admit a unique solution of this interpolation problem satisfy (a discrete version of) the representation (6.13). Independently of Pick, Nevanlinna arrives in [272] at the same result for finitely and infinitely many interpolation pairs .zj ; wj /. In [273] Nevanlinna shows that Hamburger’s solution to the (extended) Stieltjes’ moment problem [140] can be equivalently stated in terms of analytic functions f in the upper complex half-plane with negative imaginary part. Nevanlinna proves the equivalence of conditions (iv) and (vi) of Theorem 6.2; note that he does not assume that f has a positive continuous extension to .0; 1/ (as we do). The main tool are methods from the theory of continued fractions and Schur’s algorithm [327]. Later, he gives in [274] a different proof for this representation. He solves the above described

90

Chapter 6 Complete Bernstein functions

interpolation problem in the unit disk for infinitely many pairs .zj ; wj /. For the case where the zj are on the boundary of the disk or the upper half-plane, respectively, he employs earlier results of Julia [190] and Carathéodory [75] on non-tangential limits (also angular limits, Winkelderivierte in German). By the Cayley transform the open disk is mapped onto H" and the zj ’s are mapped into points j 2 R. Letting the j ’s collapse into a single point, the problem becomes the Stieltjes moment problem and one is back in the situation of [273]. Note that this approach using non-tangential limits also yields (6.6) which is nowadays called Stieltjes inversion formula. It is interesting to note that Nevanlinna did not use Herglotz’ seminal paper [156] on the representation of analytic functions in the unit disk having positive real part. Using Herglotz’ result and the Cayley transform, Cauer [77] gives a short proof of the equivalence of Theorem 6.2 (iv) and Theorem 6.9, and refers to [274] for Theorem 6.2 (vi). Cauer’s approach is used by most modern presentations of the material, see e.g. Akhiezer [2, Chapter 3] or Akhiezer and Glazman [3, Vol. II, Sect. 59] for particularly nice proofs. Akhiezer [2] contains many interesting notes on the Russian school of complex function theory and operator theory. In a series of papers [219, 220, 221] M. G. Kre˘ın introduced the classes .S / and .S 1 / in connection with investigations on the theory of generalized resolvents and the theory of strings – see Chapter 15 below; a survey of Kre˘ın’s result is contained in Kac and Kre˘ın [195]. In our notation f 2 .S / if f .z/ 2 S and f 2 .S 1 / if f .z/ 2 CBF. Kre˘ın’s work focusses on the connection between .S /; .S 1 / and the Nevanlinna–Pick functions which are denoted by .R/. The paper [212] by Komatu gives a modern presentation of the material from the viewpoint of complex function theory in a half-plane. Hirsch [162, 163, 165] introduces complete Bernstein functions – in [165] they are referred to as family H – through a variant of (6.7). His intention was to find a cone of functions operating on Hunt kernels and more general abstract potentials [166]. Hirsch shows, among other things, the equivalence of (6.1) and (6.7) and that CBF are exactly those functions which operate on all densely defined, closed operators V on a Banach space such that .1; 0/ is in the resolvent set and sup>0 k.id CV /1 k is finite. This set includes the abstract potentials in the sense of Yosida [379, XIII.9], as well as all infinitesimal generators of strongly continuous contraction semigroups. The present rather complete form of Theorems 6.2 and 6.9 appeared in [320] – this proof is reproduced in [183, Vol. 1, pp. 193–202] – see also [321] and [322] as well as the survey paper by Berg [35]. The elementary proof presented here seems to be new. The result contained in Corollary 6.5 is originally due to Julia [190], Wolff [374], Carathéodory [75] and one of the most elegant direct proofs is by Landau and Valiron [237]. Some of the above equivalences are part of the mathematical folklore; see e.g. Hirsch [163], Berg [32] or Nakamura [269] for corresponding remarks and statements. Corollary 6.8 and the remark following this corollary provides a new proof of the representation theorem for general Nevanlinna–Pick functions, cf. Donoghue [100, Chapter II, Theorem I and Lemma 2] for the standard proof.

Section 6.2 Extended complete Bernstein functions

91

The Stieltjes-type inversion formulae appearing in Corollary 6.3 and Remark 6.10 are classical and can be found, e.g. in Akhiezer [2, p. 124], de Branges [92, Chapter 1.3] or Berg [33, 35]. Section 6.2: Our definition of extended complete Bernstein function is formally inspired by the results on extended Bernstein functions, see Remark 5.9, Theorem 5.11 or Remark 5.26; all these results have their counterparts for CBF e but the underlying theory is much deeper. As far as we are aware, Akhiezer [2, Notes and additions to Chapter 3] is one of the few places in the literature where CBF e are (indirectly) studied as a particular subclass of Nevanlinna–Pick functions, cf. Proposition 6.12. The Corollaries 6.13 and 6.14 and Proposition 6.15 are adapted from Akhiezer [2, pp. 126–128]; Corollary 6.13 is originally due to Akhiezer and Krein [4]. The proof of the canonical or Nevanlinna factorization of bounded type in H" goes back to Krylov [226]. His paper is still highly readable and the formulation here is taken directly from [226, Theorem XX]. Other sources include Duren [104, Chapter 11.3], Hoffmann [168, pp. 132-3], de Branges [92, Chapter I.8] and Rosenblum and Rovnyak [308, Chapter 5]; in [308, Chapter 2] there is also a proof of Theorem 6.9 using methods from operator theory as well as a discussion of the Nevanlinna–Pick interpolation problem. Aronszajn and Donoghue [12] give a thorough discussion of exponential representations of Nevanlinna–Pick functions. The short proof of Theorem 6.17 is inspired by Kre˘ın and Nudelman [225, Appendix, Theorems A.3, A.8]. The function g appearing in Proposition 6.21 is the Laplace exponent of the ladder height process of a one-dimensional subordinate Brownian motion whose subordinator has the Laplace exponent f . Proposition 6.21 is from [208], the first part is independently shown in [233], a result similar to the second part appears in [234].

Chapter 7

Properties of complete Bernstein functions

The representation results of Chapter 6 enable us to give various structural results characterising complete Bernstein and Stieltjes functions. Our first aim is to make a connection between Stieltjes and complete Bernstein functions. For this the following result is useful. Proposition 7.1. f 2 CBF; f 6 0; if, and only if, the function f ? ./ WD =f ./ is in CBF. Proof. If f 2 CBF we may use the representation (6.7) and get Z a f .z/ 1 D CbC  .dt /; z 2 C n .1; 0: z z z C t .0;1/ Since Im z  Im

Im z .Im z/2 1 D Im z  D 0 zCt jz C t j2 jz C t j2

we see that f .z/=z maps H" into H# . As 1=z switches the upper and lower halfplanes, z=f .z/ D .z 1 f .z//1 maps H" into itself. Further, =f ./ 2 .0; 1/ for  > 0, and since a D lim!0C f ./, we have 8 0 and lim!0C  f .˛ / D 0, we find by Theorem 6.2 (iv) that g.z/ WD z  f .z ˛ / is a complete Bernstein function. The proof of Proposition 7.22 is easily adapted to show that for every f 2 BF and !   2 CBF satisfying  W H" ! H" \ H the function z 7! .z/f ..z// is in CBF. Moreover,

 z arg f ..z// D arg z C arg f ..z//  arg .z/  arg z < ; .z/ z which shows that also z 7! .z/ f ..z// is in CBF. A sufficient condition such that !  " "  W H ! H \ H is, for example, that  2 2 CBF.

Proposition 7.23. Let f 2 CM such that f .0C/ D 1 and let ˛ 2 .0; 1=2. Then the function g./ WD ˛ .1 C f .˛ // is in CBF. !  Proof. Fix t > 0 and ˛ 2 .0; 1=2. For every z 2 H" we have z ˛ 2 H" \ H, i.e. z ˛ D x C iy with x D x.˛/ > 0 and y D y.˛/ > 0. Therefore,    ˛ z ˛ 1 C e tz D .x C iy/ 1 C e tx cos.ty/  i e tx sin.ty/ : Using the elementary inequalities e tx  1  tx and sin.ty/  ty we see    ˛  Im z ˛ .1 C e tz / D e tx ye tx C y cos.ty/  x sin.ty/    e tx y.e tx  1/  x sin.ty/  e tx .ytx  xty/ D 0:

Chapter 7 Properties of complete Bernstein functions

107

This means that the function z 7! z ˛ .1 C e tz / is holomorphic and preserves the ˛ upper half-plane. Since ˛ .1 C e t /,  > 0, is real and tends to 0 as  ! 0, it is a complete Bernstein function, cf. Theorem 6.2 (iv). From f .0C/ D 1 and Bernstein’s theorem, Theorem 1.4, we conclude that there exists a probability measure  on Œ0; 1/ such that Z f ./ D L .I / D e t .dt /: ˛

Œ0;1/

Z

Then g./ D ˛ .1 C f .˛ // D

Œ0;1/

 ˛ ˛ 1 C e t .dt /;

and the claim follows because g is a mixture of CBF functions, cf. Corollary 7.6. Comments 7.24. Proposition 7.1 appears in [269, 35] and, independently, in [322]. The second equality of Theorem 7.3 is already mentioned in [269] and it is implicitly contained in [162, 163]. Stability properties of CBF and S under particular compositions have been studied by Hirsch [162, 163], see also [33, 35]: the ‘if’ directions of (7.1)–(7.4) appear in [162], (7.5) is due to Reuter [303] and (7.2) and the Stieltjes version of (7.3) can be found in van Herk [361, Theorems 7.49, 7.50]. The general statement of Corollary 7.9 and the characterizations of CBF and S in terms of their stability properties, Theorem 7.5, seem to be new. The calculations for the examples in Remark 7.8 are the same as those of the corresponding Remark 2.4 for Stieltjes functions. An easier way to arrive there, without duplicating all calculations, would be to use the examples of Remark 2.4 and to observe that f 2 CBF if, and only if, f ./= 2 S, see Theorem 6.2 (ii). Corollary 7.10 is an extension of Teke and Deshmukh [346, Theorem 2.1] which contains the assertion 7.10 (iii). The case n D 1 of Proposition 7.12 is due to [354], while the case n > 1 seems to be new. The log-convexity of CBF and S, Proposition 7.13, was originally shown by Berg [32] using a rather complicated function-theoretic proof. Our first argument is from Nakamura [269] while the second is due to Berg, see [33]. The cones CBF ˛ and S˛ were introduced by Nakamura [269] for ˛ ¤ 0, and Proposition 7.14 along with Corollary p 7.15 can be found there for positive ˛ > 0. 1 ˛ 1 ˛ 1=˛ D fg (use, e.g. de l’Hospital’s rule), the limiting Since lim˛!0 . 2 f C 2 g / case of Corollary 7.15 (i) contains yet another proof of the log-convexity of CBF. Part (ii) of Corollary 7.15 seems to be due to Ando [9]; for general j˛j  1 it appears, with a different proof using the mapping property of CBF, in [320, 321]. Proposition 7.16 is from Uchiyama [353, Lemma 2.1], but his proof of part (ii) has a gap. Proposition 7.17 is mainly from [269], the function p appears in [233, Lemma 2] and q is from [355, Theorem 2.1], both with different proofs. Proposition 7.18 is from [132], the given proof is different. The concept of symmetric complete Bernstein functions can be traced back to [227], the equivalence of (i)

108

Chapter 7 Properties of complete Bernstein functions

and (iii) in Proposition 7.20 is from [127], our proof is new. It is also shown in [127] sym that ƒf ./ WD f .0/1 .1C/f ./1 .1/2 is a bijection from CBF sym nCBF 0 sym to CBF0 . The statement of Proposition 7.22 was communicated to us by Wissem Jedidi and Sonia Fourati [119], see also [120] for a different proof, the related Proposition 7.23 seems to be new. In some sense, they are generalizations of the entries 18 and 19 in the Tables 16.3, page 312. Complete Bernstein functions can also be characterized in the following way: a continuously differentiable function f W .0; 1/ ! R is in CBF if, and only if, the kernel 8 < f .s/  f .t / ; if s; t > 0; s ¤ t; st K.s; t / D : 0 f .t /; if s; t > 0; s D t is positive definite in the sense that Z 1Z 1 K.s; t / .s/.t / ds dt  0 0

0

for all compactly supported continuous functions  W .0; 1/ ! C, cf. also the Comments 4.16. A proof can be found in Rosenblum and Rovnyak [308, Chapter 2.12]. The discrete version requiring for all n 2 N, s1 ; : : : ; sn 2 R and c1 ; : : : ; cn 2 C n X

K.sj ; sk /cj c k  0

j;kD1

can be found in Korányi [214] and Korányi and Nagy [215]. These are variants of Löwner’s original result that a C 1 -function f W .0; 1/ ! R is in CBF if, and only if, the Pick matrix     ; z1 ; : : : ; zn 2 H" f .zj /  f .zk / = zj  zNk j;kD1;:::n

is non-negative hermitian, cf. [248], see also Comments 12.18.

Chapter 8

Thorin–Bernstein functions

Proposition 7.11 contains a characterization of those complete Bernstein functions f whose derivative f 0 is a Stieltjes function. For probabilists this subclass of CBF is quite important, see Proposition 9.11 and Remark 9.18. We begin with a definition which reminds of the definition of complete Bernstein functions, Definition 6.1, based on the representation result for general Bernstein functions, cf. Theorem 3.2. Definition 8.1. A Bernstein function f is called a Thorin–Bernstein function if the Lévy measure  from (3.2) has a density m.t / such that t 7! t  m.t / is completely monotone, i.e. Z f ./ D a C b C

.0;1/

.1  e t / m.t / dt;

(8.1)

R where a; b  0, .0;1/ .1 ^ t / m.t / dt < 1 and t 7! t  m.t / is completely monotone. We will use TBF to denote the family of all Thorin–Bernstein functions. Observe that the complete monotonicity of t 7! t  m.t / implies immediately that m.t / D t 1  .t  m.t // is completely monotone. Therefore, TBF  CBF. Theorem 8.2. For a function f W .0; 1/ ! .0; 1/ the following assertions are equivalent. (i) f 2 TBF. (ii) f 0 2 S and f .0C/ D lim!0C f ./ exists. (iii) f is of the form    log 1 C  .dt /; t .0;1/

Z f ./ D a C b C

(8.2)

where a; b  0 and with representing measure  on .0; 1/ satisfying R an unique R 1  .dt / < 1. j log t j  .dt / C t .0;1/ Œ1;1/ (iv) f 2 CBF and f 0 2 S. (v) f is of the form Z f ./ D a C b C

0

1

 w.t / dt Ct t

(8.3)

110

Chapter 8 Thorin–Bernstein functions

where a; b  0 are positive Rconstants and w W .0; 1/ ! Œ0; 1/ is a non1 decreasing function such that 0 .1 C t /1 t 1 w.t / dt < 1. In order to assure the uniqueness of w we can, and will, assume that w is left-continuous. Proof. (i))(ii) Let f 2 TBF. Then f .0C/ exists and is finite. Moreover, Z 1 f ./ D a C b C .1  e t / m.t / dt; 0

where t  m.t / is completely monotone. Write t  m.t / D L .I t / with a suitable measure . Differentiation gives Z 1   0 f ./ D b C e t t m.t / dt D b C L L .I t / dt I  ; 0

and by Theorem 2.2 (i) we find f 0 2 S. (ii))(iii) By assumption, ˛ f ./ D C ˇ C  0

Z .0;1/

1  .dt /: Ct

Since the primitive f ./ exists with f .0C/ finite, we see that necessarily ˛ D 0 as well as Z   f ./ D a C ˇ C log. C t /  log t  .dt / .0;1/   Z   .dt /: D a C ˇ C log 1 C t .0;1/ Since f is non-negative, the integration constant a has to be non-negative and if we set b D ˇ we get (8.2). Since f .1/ is finite,  integrates t 7! log.1 C t 1 /; equivalently, R R 1 see Proposition 7.11, .0;1/ j log t j  .dt / C Œ1;1/ t  .dt / < 1. The uniqueness of  follows immediately from the uniqueness of the representing measure of Stieltjes functions, see the comment following Definition 2.1. (iii))(iv) This is the converse direction in Proposition 7.11. (iv))(v) By Proposition 7.11 every f 2 CBF with f 0 2 S is of the form (7.7). Using Fubini’s theorem we get   Z 1 Z  d ds  .dt / .1/ log 1 C f ./ D f .0C/ C b C ds s .0;1/ t Z Z 1 1  D f .0C/ C b C ds  .dt / s Cs .0;1/ t Z 1 Z  1 D f .0C/ C b C  .dt / ds:  C s s .0;s/ 0

111

Chapter 8 Thorin–Bernstein functions

This shows (8.3) with the left-continuous density w.t / D  .0; t / and a D f .0C/; the integrability condition on w.t / is a consequence of Fubini’s theorem and the fact that f .1/ < 1. (v))(i) We may use Fubini’s theorem to get Z 1  w.s/ f ./ D a C b C ds Cs s Z0 1  1  1 w.s/ ds  D a C b C s Cs Z0 1 Z 1 D a C b C .e st  e .Cs/t / w.s/ dt ds 0 0 Z 1  Z 1 t ts D a C b C .1  e / e w.s/ ds dt: 0

0

Since w is non-decreasing, we have Z 1 Z 1 1 w.0C/ 1 w.s/ ds  ds < 1 1Cs s 1Cs s 0 0 which is only possible if w.0C/ D 0. Thus, w.s/ D .0; s/ for a suitable measure  defined on .0; 1/, and thus Z 1 Z 1 Z ts ts m.t / WD e w.s/ ds D e .du/ ds 0 0 .0;s/ Z Z 1 D e ts ds .du/ .0;1/ u Z 1 D e tu .du/: t .0;1/ R Hence, t m.t / D .0;1/ e tu .du/ is completely monotone, and (i) follows. A comparison with the formulae in the second half of the proof of Proposition 7.11 allows us, because of the uniqueness of the measure in (8.2), to identify .ds/ with  .ds/. Remark 8.3. (i) (8.2) is called the Thorin representation of f 2 TBF and the (unique) representing measure  appearing in (8.2) is called the Thorin measure of f . (ii) For further reference let us collect the relations between the various representing measures and densities from Definition 8.1 and Theorem 8.2: Z e ts w.s/ ds D L .wI t /I m.t / D .0;1/ Z Z ts e dw.s/ D e ts  .ds/ D L . I t /I t  m.t / D .0;1/

.0;1/

112

Chapter 8 Thorin–Bernstein functions

w.t / D  .0; t /I w.t / dt:  .dt / D t (iii) The Thorin measure  of f 2 TBF is the Stieltjes measure of the complete Bernstein function f 0 ./. This useful observation can be seen in the following way: Let f 2 TBF be given by   Z   .dt /: log 1 C f ./ D a C b C t .0;1/ Then f 0 2 S and f 0 ./ D b C

Z

1 .0;1/

Thus, f 0 ./ D b C

1C

Z .0;1/

 t

1  .dt /: t

  .dt /: Ct

This shows that whenever we have an explicit representation for f 2 TBF in terms of the Thorin measure, then we have an explicit representation of the complete Bernstein function f 0 ./ in terms of the Stieltjes measure. (iv) With formula (8.3) it is easy to find examples of complete Bernstein functions which do not have derivatives in S. For example, p Z 1 Z p 1 1  1  t dt dt; p D f ./ WD  arctan p D  C t 2  C t t 2 t  0 0 p

cf. Remark 7.8; but the density 12 t t 1.0;1/ .t / is clearly not of the form R 1 t .0;t/  .ds/ which shows that f is of class CBF having a derivative which is not a Stieltjes function; in other words, f 2 CBF n TBF. The next two theorems describe the structure of the family TBF. Theorem 8.4. Let f 2 TBF. The following statements are equivalent. (i) g ı f 2 TBF for every g 2 TBF. (ii) f 0 =f 2 S. Proof. (ii))(i) Assume that f 2 TBF and f 0 =f 2 S. By Theorem 8.2, f 0 2 S. From 1=S D CBF we conclude that 1 f 0 ./ f ./Ct

D

f ./ t f ./ C t D 0 C 0 D 0 f ./ f ./ f ./

1 f 0 ./ f ./

C

t f

0 ./

113

Chapter 8 Thorin–Bernstein functions

is a complete Bernstein function. This means that f 0 ./=.f ./ C t / is a Stieltjes function. For all g 2 TBF we know that g 0 2 S and Z 1  .dt /: g 0 ./ D b C  C t .0;1/ Therefore   .g ı f /0 ./ D g 0 f ./ f 0 ./ D bf 0 ./ C

Z .0;1/

f 0 ./  .dt /: f ./ C t

The integrand is a Stieltjes function, thus .g ı f /0 2 S. Since .g ı f /.0C/ < 1, Theorem 8.2 shows that g ı f 2 TBF. (i))(ii) As g./ D log.1C=t / 2 TBF for all t > 0, we see that log.1Cf ./=t / is a Thorin–Bernstein function for all t > 0. This implies that   d f ./ f 0 ./  7! log 1 C D 2S d t t C f ./ for all t > 0. By letting t ! 0, we get that f 0 =f 2 S. Theorem 8.5. The set TBF is a convex cone which is closed under pointwise limits. Proof. That TBF is a convex cone follows immediately from the representation (8.1). Suppose that .fn /n2N is a sequence of functions in TBF such that limn!1 fn D f pointwise. It follows from Corollary 7.6 (iii) that f 2 CBF and by Corollary 3.9 we find that limn!1 fn0 D f 0 . Since fn0 2 S, we get from Theorem 2.2 that f 0 2 S. Hence, f 2 TBF, cf. Theorem 8.2. Corollary 8.6. If .fn /n2N  TBF with Thorin measures .n /n2N , and f D limn fn pointwise, then limn!1 n D  vaguely in .0; 1/, where  is the Thorin measure of f . Proof. This is proved similarly to the proof of Corollary 7.6 (ii). The following assertion should be compared with Theorem 5.11. Proposition 8.7. Suppose that g W .0; 1/ ! .0; 1/. Then the following statements are equivalent. (i) g D e f where f 2 TBF. (ii) .log g/0 2 S and g.0C/  1. Proof. Suppose that g D e f where f 2 TBF. Then g D e f where f 0 2 S, and since f D  log g we get (ii). The converse assertion follows by retracing the steps backwards.

114

Chapter 8 Thorin–Bernstein functions

Remark 8.8. Just as in the case of (complete) Bernstein functions, see Remarks 3.13 and 7.8, we can understand the representation formula (8.2) as a particular case of the Kre˘ın–Milman or Choquet representation. For this we rewrite (8.2) as     Z log 1 C t 1  .dt / f ./ D a C b C  log 1 C  1 t .0;1/ log 1 C t   Z log 1 C t D  .dt N /;  1 Œ0;1 log 1 C t where N D aı0 C log.1 C t 1 /  C bıR1 is a finite  on Œ0; 1. Here we used,  measure cf. the proof of Proposition 7.11, that .0;1/ log 1 C t 1  .dt / < 1 and that     log 1 C t log 1 C t lim  D 1 and lim  D :   t!1 log 1 C 1 t!0 log 1 C 1 t t The set

®

¯ f 2 TBF W f .1/ D 1

is a basis of the convex cone TBF, and its extremal points are given by   log 1 C t e0 ./ D 1; e t ./ D  ; 0 < t < 1; and e1 ./ D :  log 1 C 1t Their extremal property follows exactly as in Remark 2.4. Thus,   log 1 C t 1; ;  ; .0 < t < 1/;  log 1 C 1t are typical functions in TBF. Note that these are the logarithms of the Laplace transforms of the following (degenerate) Gamma distributions

0;0 ;

1;1 ;

and 1= log.1C 1 /; t ; t

As usual,

˛;ˇ .x/ D

.0 < t < 1/;

ˇ ˛ ˛1 ˇ x x e ; .˛/

respectively:

x > 0:

The following generalization of the class TBF is due to Bondesson. Definition 8.9. A Bernstein function f is said to be in TBF  for some > 0 if the Lévy measure  from (3.2) has a density m.t / such that t 7! t 2 m.t / is completely monotone, i.e. Z f ./ D a C b C where a; b  0, monotone.

R

.0;1/ .1

.0;1/

.1  e t / m.t / dt;

(8.4)

^ t / m.t / dt < 1 and t 7! t 2  m.t / is completely

115

Chapter 8 Thorin–Bernstein functions

Obviously, TBF D TBF1 and CBF D TBF 2 . For ;  > 0 we see that t 2 m.t / D t 2 m.t /  t  I since t  is completely monotone and since the product of two completely monotone functions is again in CM we conclude that TBF   TBF C for all > 0 and  > 0. Proposition 8.10. Let > 1. A function f W .0; 1/ ! Œ0; 1/ is in TBF if, and only if, f is differentiable, f .0C/ exists, and Z 1  .dt / (8.5) f 0 ./ D b C . C t / .0;1/ R for some b  0 and a measure  on .0; 1/ such that .0;1/ .1 C t /  .dt / < 1. Proof. Assume that f is such that f 0 is of the form (8.5). By Fubini’s theorem we can calculate the primitive   Z 1 1 f ./ D a C b C   .dt / 1 . C t /1 .0;1/ t  Z 1 Z 1 Z  .dt / 2 yt 2 y.tC/ y e dy  y e dy D aCbC .  1/ .0;1/ 0 0 Z 1 Z .1  e y / y 2 e yt  .dt / dy: D a C b C 0

.0;1/

In the second equality we used the elementary identity Z 1 1 y ˛1 e yt dy: t ˛ D .˛/ 0

y  2 R yt Since f is non-negative, a  0; setting m.y/ WD .1/  .dt / shows .0;1/ e 2 m.y/ is, up to a constant, the Laplace transform of the measure  , hence that y completely monotone. Conversely, if f 2 TBF we can differentiate (8.9) and use that t 2 m.t / is the Laplace transform of a measure Q : Z 1 0 e t t m.t / dt f ./ D b C 0 Z Z 1 t 1 e t e yt Q .dy/ dt DbC Œ0;1/ 0 Z Z 1 t 1 e t.Cy/ dt Q .dy/ DbC Œ0;1/ 0 Z 1 DbC .  1/ Q .dy/: . C y/ Œ0;1/ R Since f 0 has a primitive with f .0C/ < 1, we see that Œ0;1/ .1 C t / Q .dt / < 1 and Q ¹0º D 0. This proves (8.5).

116

Chapter 8 Thorin–Bernstein functions

Comments 8.11. The class TBF appears for the first time in Thorin’s 1977 papers [347, 348] as the family of Laplace exponents of a class of probability distributions, the so-called generalized Gamma convolutions. This class of distributions will be discussed in the next chapter, Chapter 9. Nowadays the standard reference are the comprehensive lecture notes [67] by Bondesson and the monograph [339] by Steutel and van Harn. The exposition in Bondesson [67] focusses on probability distributions which are generalized Gamma convolutions, GGC for short. Bondesson calls, in honour of O. Thorin, the family of (sometimes: Laplace exponents of) distributions from GGC the Thorin class, T . It is easy to see that for f 2 T the reflected function  7! f ./ is a Thorin–Bernstein function TBF in the sense of Definition 8.1. Taking this reflection into account, the class TBF  can be found in [66, 67] where the assertion of Proposition 8.10 is used as the definition of TBF  . The integral appearing in (8.5) is sometimes called generalized Stieltjes transform, see e.g. Hirschman and Widder [167]. Bondesson shows in [67, §9.5, pp. 150–1] that the S closure of the set  TBF  are the Bernstein functions. The above mentioned reflection in the argument of the log-Laplace exponent makes Bondesson sometimes difficult to read. The presentation in Steutel and van Harn [339] uses our convention but does not always contain (complete) proofs. Theorem 8.2 is, in less comprehensive form, essentially contained in [66], see also [67]. Note that Thorin [347] uses Theorem 8.2 (iii) as definition of GGC or TBF, respectively. Since we want to emphasize the ‘(complete) Bernstein’ nature of TBF, we prefer (8.1) as the basic definition. Part (iv) of Theorem 8.2 is the main characterization theorem of GGC in [67]. It appears in [347] and in the present form in [66]; both references use the connection with complex analysis that complete Bernstein (respectively, Stieltjes) functions preserve (respectively, switch) the upper and lower half-planes, see Theorem 6.2. The stability result of Theorem 8.4 is the counterpart of [67, Theorem 3.3.1], the closure result Theorem 8.5 is due to Thorin [348]; among probabilists it is often referred to as Thorin’s continuity theorem. Theorem 9.13 was the starting point of Thorin’s investigations [347, 348]. Proposition 8.7 is due to Bondesson [66, Remark 3.2]. It provides, in particular, another proof for the fact that all probability distributions in GGC are infinitely divisible; this will again be proved in Proposition 9.11 of the next chapter. The interpretation of (8.2) in the context of Choquet theory seems to be new.

Chapter 9

A second probabilistic intermezzo

In Chapter 5 we discussed the relation between (extended) Bernstein functions BF and the infinitely divisible elements in CM, see Lemma 5.8 and Theorem 5.11. This allowed us to give a probabilistic characterization of the cone BF. A similar characterization is possible for complete Bernstein functions. To do this we use Lemma 5.8 as starting point. Definition 9.1. A measure  on Œ0; 1/ belongs to the Bondesson class if L .I / D e f ./ for some f 2 CBF. We write  2 BO. Note that  2 BO is a sub-probability measure: Œ0; 1/ D L .I 0C/  1. A probabilistic characterization of the class BO will be given in Theorem 9.7 below. Let us first collect a few elementary properties of the Bondesson class BO. Lemma 9.2. (i) A sub-probability R measure  on Œ0; R1/ is in BO if, and only if, there exist a measure  on .0; 1/ with .0;1/ s 1 .ds/ C .1;1/ s 2 .ds/ < 1 and constants a; b  0 such that     Z 1 1  .ds/ : (9.1) L .I / D exp a  b  Cs .0;1/ s (ii) Every  2 BO is infinitely divisible. (iii) BO is closed under convolutions: if ;  2 BO, then  ?  2 BO. (iv) BO is closed under vague convergence. Proof. (i) Because of Theorem 6.2, see also Remark 6.4, every f 2 CBF has a representation   Z 1 1  .ds/; f ./ D a C b C Cs .0;1/ s with a; b and  as claimed. (ii) Since CBF  BF, Lemma 5.8 tells us that each  2 BO is infinitely divisible.

118

Chapter 9 A second probabilistic intermezzo

(iii) Since CBF is a convex cone, this follows immediately from the convolution theorem for the Laplace transform: L . ? / D L   L . (iv) Let .n /n2N  BO be a sequence of sub-probability measures and denote by .fn /n2N  CBF the complete Bernstein functions such that L .n I / D e fn ./ . If limn!1 n D  vaguely, then limn!1 L .n I / D L .I / for every  > 0, and the claim follows from the fact that CBF is closed under pointwise limits. Since CBF  BF, Theorem 5.2 guarantees the existence of a unique semigroup of sub-probability measures . t / t0 on Œ0; 1/ such that L . t I / D e tf ./ , t  0; in particular,  t 2 BO. Clearly, the converse is also true and we arrive at the analogue of Theorem 5.2. Theorem 9.3. Let . t / t0  BO be a convolution semigroup of sub-probability measures on Œ0; 1/. Then there exists a unique f 2 CBF such that the Laplace transform of  t is given by L  t D e tf

for all t  0:

(9.2)

Conversely, given f 2 CBF, there exists a unique convolution semigroup of subprobability measures . t / t0  BO such that (9.2) holds true. Note that the set e BF D ¹g W 9 f 2 BF; g D e f º is a relatively simple subset of CM: it consists of all infinitely divisible functions from CM \ ¹g W g.0C/  1º, cf. Lemma 5.8. The structure of e CBF  CM is more complicated. If we restrict our attention to g 2 CBF such that g.0C/  1, we find that   f D log g D log .g  1/ C 1 2 CBF and e f D 1=g 2 S with 1=g.0C/  1. This motivates the following definition. Definition 9.4. A measure  on Œ0; 1/ is a mixture of exponential distributions if L 2 S

and L .I 0C/  1:

(9.3)

We write  2 ME. Note that  2 ME is a sub-probability measure: Œ0; 1/ D L .I 0C/  1. The reason for the name mixture of exponential distributions is given in part (iii) of the following theorem. Theorem 9.5. Let  be a measure on Œ0; 1/. Then the following conditions are equivalent. (i)  2 ME.

119

Chapter 9 A second probabilistic intermezzo

(ii) There exist ˇ  0 and a measurable function  W .0; 1/ ! Œ0; 1 satisfying R1 1 dt < 1, such that 0 .t / t    Z 1 1 1  .t / dt : (9.4) L .I / D exp ˇ  t Ct 0 (iii) There exists a sub-probability measure  on .0; 1 such that Z Œ0; t  D .1  e ts / .ds/:

(9.5)

.0;1

Proof. (i))(ii) Since L .I / D g./ with g 2 S and g.0C/  1 we know from Theorem 7.3 that f WD 1=g 2 CBF and f ./  f .0C/  1. From the representation (6.21) of Theorem 6.17, we find that  Z 1 1 t log f ./ D C .t / dt  1 C t2  C t 0 for some 2 R and a measurable function  W .0; 1/ ! Œ0; 1. As 1 > f .0C/  1, we see that Z 1 1 .t / dt < 1

C t .1 C t 2/ 0 R1 which entails that 0 .t / t 1 dt < 1. Because of this integrability condition, we have   Z 1 Z 1 1 1 t 1 .t / dt   .t / dt  log f ./ D C 1 C t2 t Ct t 0 0  Z 1 1 1  .t / dt: DˇC t Ct 0 Since f .0C/  1, we conclude that ˇ  0. (ii))(iii) In view of Theorem 6.17, formula (9.4) means that L .I / is of the form 1=f where f 2 CBF and f .0C/  1; hence, h./ is a Stieltjes R D L .I /1 function satisfying h.0C/  1. But then h./ D b C .0;1/ . C s/  .ds/ and R b C .0;1/ s 1  .ds/ D h.0C/  1. Define a sub-probability measure  on .0; 1 by .ds/ D s 1  .ds/ on .0; 1/, and ¹1º D b. By Fubini’s theorem Z s .ds/ L .I / D b C  C s .0;1/  Z 1 Z .Cs/t DbC e dt s.ds/ 0 .0;1/ Z Z Z e t bı0 .dt / C e t se ts .ds/ dt D Œ0;1/ Œ0;1/ .0;1/   Z Z t ts e se .ds/ dt bı0 .dt / C D Œ0;1/

which is equivalent to saying that Œ0; t  D

R

.0;1/

.0;1 .1

 e ts / .ds/.

120

Chapter 9 A second probabilistic intermezzo

R (iii))(i) If Œ0; t  D .0;1 .1  e ts / .ds/ we can perform the calculation of the previous step in reverse direction and arrive at Z s .ds/: L .I / D b C  C s .0;1/ R R Set  .ds/ D s.ds/. Since .0;1/ .1 C s/1  .ds/ D .0;1/ s.1 C s/1 .ds/ < 1, we see that L  2 S. Moreover, L .I 0C/ D b C .0; 1/ D .0; 1  1. Corollary 9.6. We have ME  BO. Moreover, ME is closed under vague limits. Proof. Let  2 ME. By Theorem 9.5, the Laplace transform L  is of the form e f where f 2 CBF  BF. This shows that  2 BO. If .n /n2N  ME converges vaguely to , we see for the Laplace transforms limn!1 L .n I / D L .I /. Since Stieltjes functions are closed under pointwise limits, cf. Theorem 2.2 (iii), L  2 S is a Stieltjes function. Moreover, we have L .I 0C/ D lim!0C L .I / D lim!0C limn!1 L .n ; /  1. Thus,  is a mixture of exponential distributions. We are now ready for the probabilistic interpretation of the class BO. Theorem 9.7. BO is the vague closure of the set ® ¯ MEF WD 1 ?    ? k W j 2 ME; j D 1; 2; : : : ; k; k 2 N ; i.e. BO is the smallest class of sub-probability measures on Œ0; 1/ which contains ME and which is closed under convolutions and vague limits. Proof. We have already seen in Corollary 9.6 that ME  BO. That BO is vaguely closed and stable under convolutions has been established in Lemma 9.2. It is therefore enough to show that the extremal elements of CBF,  7! 1;

 7!

 ; Cs

s > 0;

 7! ;

can be represented as pointwise limits of Laplace exponents of measures in MEF . For  7! 1 there is nothing to show. Note that Z 1 1  dt lim D : r !1 log r C1 r  C t t r 1 2 N, we If we set r .t / WD 1Œr;1/ .t / and pick a sequence of r such that .log r C1 r / see that  7!  is the limit of Laplace exponents of convolutions of measures from ME. Further, Z sC1=r  r;s .t /  D lim r dt; r !1 Cs t s1=r  C t

121

Chapter 9 A second probabilistic intermezzo

where r 2 N, t 1 r;s .t / is a tent-function centered at s with basis Œs  r 1 ; s C r 1  and height 1, i.e. r;s .t / D rt Œ.t s Cr 1 /C ^.s Cr 1 t /C . Therefore  .Cs/1 is the limit of Laplace exponents of convolutions of measures from ME. This proves that BO  vague-closure.MEF /. Remark 9.8. The proof of Theorem 9.7 uses implicitly the representation of CBF in terms of its extremal points es ./ D  .1 C s/=. C s/, 0  s  1, cf. Remark 7.8. It shows that we can approximate the corresponding representing measures ıs as vague limits of probability measures of the form c .1Ct /1 t 1 .t / dt with a function  W Œ0; 1/ ! Œ0; 1. Remark 9.9. Suppose that L  D e f where  2 SD1 and f is given by the representation (5.17). From the Stieltjes representation of ˛ we see that Z  Z 1 dt  sin.˛/ ˛ t .d˛/ : f ./ D ¹0º C ¹1º C  C t  t 0 .0;1/ If also  2 ME, then by Theorem 9.5

Z

f ./ D ˇ C

1 0

 .t / dt Ct t

where ˇ  0 and  W .0; 1/ ! Œ0; 1. By comparing the last two expressions for f , we see that  2 SD1 \ ME if, and only if, ˇ D .0/ and .0; 1 D 0. Thus, SD1 \ ME D ¹e ˇ ı0 W ˇ  0º. Let us introduce a class of sub-probability measures which is related to the family TBF in the same way as BO is related to CBF, see Definition 9.1. Definition 9.10. A measure  on Œ0; 1/ is a generalized Gamma convolution if L .I / D e f ./ for some f 2 TBF. We write  2 GGC. Note that  2 GGC is a sub-probability measure: Œ0; 1/ D L .I 0C/  1. A probabilistic characterization of the class GGC will be given below. Let us first collect a few elementary properties of this class. Proposition 9.11. (i) A sub-probability R if, and only if, there R measure  is in GGC exist a measure  on .0; 1/ with .0;1/ j log t j  .dt / C .1;1/ t 1  .dt / < 1 and constants a; b  0 such that     Z   .dt / ; log 1 C L .I / D exp a  b  t .0;1/

122

Chapter 9 A second probabilistic intermezzo

or a non-decreasing function w.t / on .0; 1/ with such that  Z L .I / D exp a  b  .0;1/

R1 0

.1 C t /1 t 1 w.t / dt < 1

  w.t / dt : Ct t

(ii) SD1  GGC  SD  ID. (iii) GGC  BO  ID. (iv) GGC contains all Gamma distributions ˛;ˇ , ˛; ˇ > 0. (v) GGC is closed under convolutions: if ;  2 GGC, then  ?  2 GGC. (vi) GGC is closed under vague convergence. Proof. Since TBF  CBF  BF, and since TBF is a convex cone which is closed under pointwise limits, (iii), (v) and (vi) follow as in Lemma 9.2. To see (ii), note that the Lévy measure of a Thorin–Bernstein function has a density m.t / such that t  m.t / is completely monotone, cf. Definition 8.1; as such, t  m.t / is non-increasing and the inclusion GGC  SD follows from the structure result for Laplace exponents of SD distributions, cf. Proposition 5.17. Since the Laplace exponent of any SD1 distribution has a Lévy density m such that t  m.t / D n.log t / with n completely monotone on R, cf. Proposition 5.20, t  m.t / is completely monotone by Proposition 1.16 proving that SD1  GGC. For (i) we have to use the representations (8.2) and (8.3) of Thorin–Bernstein functions. Property (iv) can be directly verified: suppose that ˛;ˇ is a Gamma distribution with parameters ˛; ˇ > 0, then     ˛ D e ˛ log.1C ˇ / D e f ./ : L . ˛;ˇ I / D 1 C ˇ The Thorin measure  of f is equal to ˛ıˇ . Thus, GGC contains all Gamma distributions. Since TBF  BF, Theorem 5.2 guarantees the existence of a unique semigroup of sub-probability measures . t / t0 on Œ0; 1/ such that L . t I / D e tf ./ , t  0; in particular,  t 2 GGC. Clearly, the converse is also true and we arrive at the analogue of Theorem 5.2: Theorem 9.12. Let . t / t0  GGC be a convolution semigroup of sub-probability measures on Œ0; 1/. Then there exists a unique f 2 TBF such that the Laplace transform of  t is given by L  t D e tf

for all t  0:

(9.6)

Conversely, given f 2 TBF, there exists a unique convolution semigroup of subprobability measures . t / t0  GGC such that (9.6) holds true.

123

Chapter 9 A second probabilistic intermezzo

We are now ready for the probabilistic interpretation of the class GGC. Theorem 9.13. GGC is the vague closure of the set ¹ ˛;ˇ W ˛; ˇ > 0ºF of all finite convolutions of Gamma distributions. In particular, GGC is the smallest class of subprobability measures on Œ0; 1/ which contains all Gamma distributions and which is closed under convolutions and vague limits. Proof. We have seen that ˛;ˇ 2 GGC in Proposition 9.11. Moreover, 

L . ˛;ˇ I / D e ˛ log.1C ˇ / which shows that the extreme points e t ./ D log.1 C t /= log.1 C 1t /; 0 < t < 1, of TBF are the Laplace exponents of 1= log.1C 1 /; t , 0 < t < 1. t Since TBF consists of integral mixtures of its extreme points, cf. Remark 8.8, we conclude that GGC is the vague closure of finite convolutions of Gamma distributions. There seems to be no deeper relation between the classes ME and GGC. Indeed, the Laplace exponent of  2 ME is of the form Z 1  .t / dt ˇC Ct t 0 with  taking values in Œ0; 1; on the other hand, the Laplace exponent of a distribution in GGC looks like Z 1  w.t / a C b C dt; Ct t 0 where w is a non-decreasing function. The intersection GGC \ ME is characterized in Proposition 9.14 (iii). We have seen in Proposition 9.11 that GGC distributions are self-decomposable, i.e. they satisfy (5.8). In fact, GGC contains all distributions  that are S-self decomposable in the following sense: g D L , g.0C/  1 and  7!

g./ g.c/

is a Stieltjes function for all c 2 .0; 1/:

(9.7)

Proposition 9.14. Let  be a sub-probability measure and g D L . The condition (9.7) is equivalent to any of the following assertions. (i)  2 GGC with Laplace exponent f 2 TBF having no linear part, i.e. b D 0 in (8.3), and with a bounded density w.t / 2 Œ0; 1. (ii)  2 GGC with Laplace exponent f 2 TBF having no linear part, i.e. b D 0 in (8.1) and such that t  m.t / D L . I t / for some sub-probability measure  on .0; 1/.

124

Chapter 9 A second probabilistic intermezzo

(iii)  2 GGC \ ME. Proof. The equivalence of (i) and (ii) follows immediately from Remark 8.3 (ii). Assume that (9.7) holds true; we want to show (i). Let us begin with the proof that  2 GGC. By (9.7), we get   g 0./ g./ g.c/  g./ 1  D lim D lim 1 : c!1 .1  c/g.c/ c!1 .1  c/ g./ g.c/ Since g./=g.c/ is bounded by 1 and since it is a Stieltjes function, the expression inside the brackets is a complete Bernstein function by Proposition 7.7; dividing by  transforms it into a Stieltjes function. Since limits of Stieltjes functions are again Stieltjes functions, cf. Theorem 2.2, we conclude that g 0 =g 2 S and so  2 GGC by Proposition 8.7. Consequently, L .I / D e f ./ where f 2 TBF; since f can be written as Z 1  w.t / dt; f ./ D a C b C t C t 0 we see that

  Z 1 g.c/  w.t /  w.ct / D exp b.1  c/ C dt : g./ t C t 0

By assumption,  7! g./=g.c/ is a Stieltjes function so that, by Theorem 7.3, the function  7! g.c/=g./ is in CBF. In view of the uniqueness of the representation in Theorem 6.17 and Remark 6.18 this is only possible, if in the above representation b D 0 and 0  w.t /  w.ct /  1 for all t  0 and c 2 .0; 1/; letting c ! 0 proves w.t / 2 Œ0;R 1. Here we used that w.0C/ D 0, which follows from the integrability 1 condition 0 .1 C t /1 t 1 w.t / dt < 1. Conversely, assume that g D e f where f 2 TBF can be represented by (8.3) where b D 0 and w.t / takes values in Œ0; 1 for all t  0. For any 0 < c < 1 we find Z 1 Z 1  w.t / c w.t / f ./  f .c/ D dt  dt t C t t C c t 0 0 Z 1  w.t /  w.ct / dt: D t C t 0 Since w.t / is non-decreasing and bounded by 1, we get 0  w.t /  w.ct /  1. Thus, Z 1    g.c/  w.t /  w.ct / D exp f ./  f .c/ D exp dt ; g./ t C t 0 and we conclude from Theorem 6.17 and Remark 6.18 that  7! g.c/=g./ is for all c 2 .0; 1/ a complete Bernstein function. Together with Theorem 7.3 this proves (9.7). Since S is closed under pointwise limits, cf. Theorem 2.2, and since g.0C/ > 0, (iii) follows from (9.7) if we let c ! 0. On the other hand, we infer (i) from (iii) by Theorem 6.17 and Remark 6.18.

125

Chapter 9 A second probabilistic intermezzo

We will now discuss another class of probability measures on Œ0; 1/. This class is related to the exponential distributions in the same way as GGC is related to the Gamma distributions. Definition 9.15. Let CE be the smallest class of sub-probability measures on Œ0; 1/ which contains all exponential laws and which is closed under convolutions and weak limits. The class CE is called the class of convolutions of exponential distributions. It is straightforward to see that  2 CE if, and onlyP if,  D ıc for some c  0, or if there exist c  0 and a sequence .bn /n2N satisfying n bn1 < 1 such that L .I / D e c

 Y  1 : 1C bn n

(9.8)

The sequence .bn /n2N will be called the representing sequence of the measure . Corollary 9.16.

(i) CE  GGC.

(ii) CE \ ME D Exp, where Exp denotes the family of all exponential distributions. Here the point mass at zero ı0 is interpreted as an exponential distribution with mean zero. (iii) CE \ SD1 D ¹ıc W c  0º Proof. (i) Let  2 CE. It follows from the representation (9.8) that L  D e f with     Z X    .dt /; (9.9) D c C log 1 C log 1 C f ./ D c C bn t .0;1/ n P where  D n ıbn . Thus f 2 TBF, and hence  2 GGC. (ii) Assume that  2 CE \ ME and  ¤ ı0 . Let L  D e f where f has the representation (9.9). Then the Stieltjes measure  of f is given by  .dt / D

 .0; t / #¹n W bn < t º dt D dt; t t

cf. Remark 8.3 (ii). On the other hand, comparing this representation with (9.4), it follows that t 7!  .0; t / is bounded by 1. Hence,  D ıb for some b > 0. Again by comparing (9.4) and (9.9), it follows that c D 0 and ˇ D 0. Hence  is an exponential distribution.

126

Chapter 9 A second probabilistic intermezzo

(iii) Assume that  2 CE \ SD1 and let L  D e f where f is given by (9.9). Since  2 SD1 , the Lévy measure of f has a density m of the form (5.18). Therefore, Z ˛ t ˛ .d˛/ t m.t / D .0;1/ .1  ˛/  Z Z 1 ˛ s ˛1 .d˛/ ds: e ts D 0 .0;1/ .1  ˛/.˛/ By Remark 8.3 (ii), the expression in parentheses is the density of the Thorin measure of f . Comparing with (9.9) it follows that the Thorin measure is equal to zero. Hence f ./ D c proving the claim. Suppose that 0 < b < a  1 and define a measure  on .0; 1 by   b b ıb C ı1 ;  D 1 a a and a measure  2 ME by Z Œ0; t  D

.0;1

.1  e ts / .ds/:

Note that  is the mixture of a point mass at 0 and the exponential distribution with parameter b. By integration it follows that      1 L .I / D 1 C 1C : a b In the case when a D 1,  is the exponential law with parameter b. Corollary 9.17. Let .an /n2N and .bn /n2N be two sequences P of non-negative real numbers such that 0 < bn < an  1 for all n 2 N and n bn1 < 1. Then the function   Y   1 g./ WD ;  > 0; 1C 1C an bn n is well defined, strictly positive, and it is the Laplace transform of a probability measure from the Bondesson class. If all an D 1, then g is the Laplace transform of a probability measure from CE. Proof. By the assumptions on the sequences .an /n2N and .bn /n2N the product converges to a strictly positive function. Since the factors are Laplace transforms of probability measures in BO, the claim follows from Theorem 9.7. The case when an D 1 for all n 2 N follows from the definition of CE.

127

Chapter 9 A second probabilistic intermezzo Table 9.1. Overview of the classes of distributions, Laplace transforms and exponents. measure

Laplace transform

Laplace exponent

references

ID

log-CM

extended BF

5.6, 5.10, 5.11

ID, Œ0; 1/  1

log-CM, f .0C/  1

BF

5.8

SD

—

BF, .dt / D m.t / dt ,

5.17

t  m.t / non-increasing SD1

—

PBF

5.21, 5.22

BO

—

CBF

9.1, 9.2

ME

S, f .0C/  1

CBF, .dt / D

.t / t

dt ,

9.4, 9.5

0  .t /  1 GGC CE Exp

log-S, f .0C/  1 1 Q  e c n 1 C bn  1 1 C b

TBF

8.7, 9.10, 9.11

TBF;  D

P

n ıbn

TBF;  D ıb

(9.8), 9.16 9.16

Another way to prove the corollary is to notice that        7! log 1 C  log 1 C 2 CBF an bn and to apply the closure property of CBF, Corollary 7.6. Remark 9.18. Let us briefly give an overview of all classes of distributions and their Laplace exponents we have encountered so far. The numbers n.m in the ‘references’ column of Table 9.1 refer to the corresponding statements in this tract. We use logCM as a shorthand for logarithmically completely monotone (see Definition 5.10) and, analogously, log-S as a shorthand for logarithmically Stieltjes in the sense that the condition in Proposition 8.7 (ii) holds. We follow Bondesson [67] and use Venn diagrams to illustrate the relations among the classes of distributions and their Laplace exponents. The non-inclusions shown in Figures 9.1 and 9.2 are easily verified by the following examples: SD 6 BO, in particular SD 6 ME. Consider the Lévy measure with density m.t / D

1 1.0;1/ .t /; t

t > 0:

The corresponding Bernstein function is f ./ D C log./ R 1  Ei./ where

D 0:5772 : : : is Euler’s constant and Ei./ D p:v:  t 1 e t dt stands

128

Chapter 9 A second probabilistic intermezzo

BO ME Exp

SD∞

CE

ID

GGC

SD

Figure 9.1. Relations between various classes of distributions. . .

CBF t dσ dt ∈ [0, 1] τ =δb  τ = n δbn

BF

t dμ dt

PBF

TBF

non-increasing

Figure 9.2. . . . and their Laplace exponents.

for the exponential integral. Clearly, t  m.t / is non-increasing, but m.t / is not completely monotone. ME 6 SD, in particular BO 6 SD. This is shown by Entry 3 in the Tables 16.2. The corresponding Bernstein function is f ./ D 1  .1 C /˛1 , 0 < ˛ < 1.

Chapter 9 A second probabilistic intermezzo

129

GGC 6 ME. This follows from Entry 1 in the Tables 16.2. The corresponding Bernstein function is f ./ D ˛ , 0 < ˛ < 1. ME 6 GGC. This follows from Entry 3 in the Tables 16.2. The corresponding Bernstein function is f ./ D 1  .1 C /˛1 , 0 < ˛ < 1. GGC 6 SD1 . This follows from Entry 26 in the Tables 16.4 and Remark 1.17. Indeed, the density of the Lévy measure is m.t / D t 1 e at so that t  m.t / is at completely monotone. If t  m.t / D n.log t /, then n.t / D e e , t 2 R. But this function is not completely monotone on R. Comments 9.19. The class of Bondesson distributions, BO, appears first in Bondesson’s 1981 paper [66] under the name g.c.m.e.d. distributions. In his lecture notes [67] he also calls them class T2 distributions. The best source for the history is the following quote from Bondesson [67, p. 151] (with references pointing to our bibliography): The T2 [here: BO] distributions were introduced in [66] for the purpose of finding a class of ID densities containing the MED’s [here: ME] as well as the B-densities [B denotes the hyperbolically completely monotone functions]. The attempt was not successful and I [i.e.: L. Bondesson] have often considered the class as a failure. Others have been much more interested in it and my name was even attached to it by Kent [205]. Unfortunately, some people have also called it B, e.g. Küchler & Lauritzen [232]. Earlier, Ito & McKean [178, pp. 214–217] encountered the class in connection with first passage times for diffusion processes [see Chapter 15]. Thorin introduced the notion of generalized Gamma convolutions in his two 1977 papers [347, 348] in order to show the infinite divisibility of certain probability distributions. Subsequently, a large number of papers appeared on the subject, many of them in the Scandinavian Actuarial Journal. Nowadays the standard texts in the field are the survey paper [66] and the comprehensive lecture notes [67] by Bondesson. Chapter VI of the monograph [339] by Steutel and van Harn contains a digest of this material. Most of the results of this section are already in the original papers by Thorin [347, 348] and Bondesson [66]. New is the presentation, which emphasizes the analogy with the material of Chapter 5, in particular with infinite divisibility and selfdecomposability. Theorem 9.3 is from [66], Theorem 9.5 and its Corollary 9.6 are due to Steutel [338]. Our proof of Theorem 9.7 seems to be new. Alternatively, it is possible to construct  2 vague-closure.MEF / such that L .I / D exp.=. C t // resp. L .I / D exp.b/. Typically, such distributions  are vague limits of convolutions of elementary measures of the form .t C /1 ı t C t .t C /1 ı1 . This is the usual way to prove Theorem 9.7, see e.g. Steutel and van Harn [339, Chapter VI.3] and Bondesson [67, p. 141]. The first part of Proposition 9.11 appears as Theorem 5.2 in [66], while all other assertions seem to be general knowledge. Theorems 9.12 and 9.13 are due to Thorin [347, 348], while Proposition 9.14 and the notion of S-self decomposability seem to be new. Probabilistically, (9.7) can be interpreted in

130

Chapter 9 A second probabilistic intermezzo

the following way: (the distribution of) a random variable X is S-self decomposable if, and only if, X has a GGC distribution and if for every c 2 .0; 1/ there is a further random variable X .c/ with distribution in GGC such that X D cX C X .c/ . The class of convolutions of exponential distributions, CE, is introduced by Yamazato in [376], see also [377], in the context of first passage distributions. The notion itself appears in the same context much earlier, see Keilson [201] and Kent [203, 204]. Corollary 9.17 is taken from Kent [203, 204]; he refers to this class of distributions as infinite convolutions of elementary MEDs, i.e. mixtures of exponential distributions. Figures 9.1 and 9.2 appearing in Remark 9.18 owe much to the presentation in Bondesson [67, p. 4].

Chapter 10

Transformations of Bernstein functions

Suppose that S D .S t /Rt0 is a subordinator and  W .0; 1/ ! Œ0; 1/ a non-negative function. The integral .0;1/ .t / dS t defines a non-negative, possibly infinite, random variable. If this random variable is finite, its distribution is infinitely divisible. Various subfamilies of infinitely divisible sub-probability measures on Œ0; 1/ can be obtained by integrating the same function  against all subordinators. In this chapter we will consider general properties of such transformations and also the properties of three concrete families of transformations. We will identify the domains and ranges of these transformations and their iterations. It turns out that familiar classes of infinitely divisible distributions on Œ0; 1/ such as BO and GGC can be obtained as ranges of particular transformations or their iterates. In the spirit of the monograph we will focus on transformations of the corresponding Laplace exponents, i.e. Bernstein functions. To be more specific, let  W .0; A/ ! .0; B/, 0 < A; B  1, be a strictly decreasing function, and let S D .S t / t0 be a (possibly killed) subordinator defined on a probability space .; F ; P/ with Laplace exponent f 2 BF. We define a random variable X by Z X WD

.t / dS t :

(10.1)

.0;A/

Then X is well defined and takes values in Œ0; 1. We start by computing the Laplace transform of X. Lemma 10.1. Let X be defined by (10.1). Then ´ Z μ h i A E e X D exp  f ..t // dt ;

 > 0:

(10.2)

0

Proof. Assume first that A < 1. Since  is decreasing we have by the monotone convergence theorem that Z .0;A/

.t / dS t D lim

n!1

n 1 2X

j D1



j 2n

A

  S jA  S .j 1/A : 2n

2n

132

Chapter 10 Transformations of Bernstein functions

By the dominated convergence theorem, the independence and stationarity of the increments of S , and then the monotone convergence theorem, we get that " ´ μ# n 1 2X h i   j  X E e D E exp   lim  n A S jA  S .j 1/A n!1 2n 2n 2 j D1 μ# " ´ n 1 2Y   j  E exp   n A S jA  S .j 1/A D lim n!1 2n 2n 2 j D1 ´ n 1    μ 2Y A j exp  n f  A D lim n!1 2 2n j D1 ´ 2n 1    μ X j A f  A D lim exp  n n!1 2 2n j D1 μ ´ Z A

D exp



f ..t // dt : 0

The case of the infinite interval follows by letting A ! 1. Remark 10.2. Lemma 10.1 remains valid for general non-negative measurable functions  . To see this we use that  is an increasing limit of step functions and the fact that each subordinator generates a Poisson random measure on Œ0; 1/ .0; 1/. Motivated by the expression on the right-hand side of (10.2), for a function f 2 BF we define Z A

‚f ./ WD

f ..t // dt;

 > 0:

(10.3)

0

Clearly, ‚f W .0; 1/ ! .0; 1. From now on we will assume that  is a bijection from .0; A/ onto .0; B/, which is equivalent to assuming that  is continuous. Let  1 W .0; B/ ! .0; A/ be the inverse function of  . The function  1 is strictly decreasing, hence, almost everywhere differentiable. Define # W .0; B/ ! .0; 1/ by #.t / WD . 1 /0 .t / at all points t where  1 is differentiable, and #.t / D 0 elsewhere. With this definition we have Z B #.s/ ds; 0 < t < B: (10.4)  1 .t / D t

A change of variables in (10.3) yields an alternative representation of ‚f : Z B f .t /#.t / dt;  > 0: ‚f ./ D

(10.5)

0

Alternatively, we could start with a measurable function # W .0; B/ ! .0; 1/ such that the integral in (10.4) is finite for all t 2 .0; B/, and define a continuous, strictly

133

Chapter 10 Transformations of Bernstein functions

decreasing function  as the inverse of that integral. In fact, this is the most common way of prescribing the mapping ‚. For a given mapping ‚ we denote by D.‚/ WD ¹f 2 BF W ‚f 2 BFº its domain and by R.‚/ WD ¹‚f W f 2 D.‚/º its range. Since the prescription (10.3) defining ‚ is linear, we see that D.‚/ is a convex cone. Recall that there is a one-to-one correspondence between the family BF of all Bernstein functions and the family ID of all infinitely divisible sub-probability measures on Œ0; 1/ which is realized as follows: For f 2 BF, let . t / t0 be the unique convolution semigroup of sub-probability measures on Œ0; 1/ such that L  t D e tf for all t  0. This correspondence allows to define an analogue of the mapping ‚ on ID which we will denote again by ‚. Let 1 2 ID and let f 2 BF be the unique function such that L 1 D e f . If ‚f 2 BF, there exists a unique 1 2 ID such that L 1 D e ‚f . In this case we define ‚1 WD 1 . Note that 1 is the distribution of the random variable X defined by (10.1). If ‚f D 1, then we define ‚1 to be the zero measure on Œ0; 1/. In this case X D C1 a.s. We use the same notation D.‚/ to denote the family of all 1 2 ID such that ‚1 is not the zero measure – the domain of ‚ acting on measures. We will focus on the three families of mappings ‚ which are defined by the following two-parameter functions, where ˛ 2 .1; 1/, ˇ; p > 0: #˛;ˇ W .0; 1/ ! .0; 1/; b # ˛;p W .0; 1/ ! .0; 1/;

#˛;ˇ .t / D c˛;ˇ e t t ˛1 ; b # ˛;p .t / D b c ˛;p .1  t /p1 t ˛1 ;

# ˛;p W .0; 1/ ! .0; 1/;

# ˛;p .t / D c ˛;p . log t /p1 t ˛1 :

ˇ

(10.6)

c ˛;p and c ˛;p are either chosen to be 1 (if # is not integrable) or TheR constants c˛;ˇ , b 1= #.t / dt (if # is integrable). Thus c˛;ˇ D

ˇ ; .˛=ˇ/

b c ˛;p D

1 ; B.˛; p/

c ˛;p D

.˛/p ; .p/

1 < ˛ < 0;

(10.7) c ˛;p D c ˛;p D 1 if 0  ˛ < 1. The corresponding mappings ‚ will be and c˛;ˇ D b b ˛;p and ‚˛;p respectively. Note that we have ‚ b ˛;1 D ‚˛;1 for denoted by ‚˛;ˇ , ‚ all ˛ 2 .1; 1/. In order to describe the domains of these mappings, we need to introduce subfamilies of Bernstein functions whose Lévy measures have additional integrability properties at infinity. For p > 0 and ˛ > 0, let ² ³ Z p .log t / .dt / < 1 ; BFlogp WD f 2 BF W .1;1/ ² ³ Z BF ˛ WD f 2 BF W t ˛ .dt / < 1 ; .1;1/ ² ³ Z p1 ˛ .log t / t .dt / < 1 : BF˛;logp1 WD f 2 BF W .1;1/

134

Chapter 10 Transformations of Bernstein functions

Table 10.1. Domains and ranges of ‚˛;ˇ , b ‚˛;p and ‚˛;p . #

˛2 .1; 0/

#˛;ˇ

BFlog;0

.0; 1/

BF˛;0 BF

R.‚/ t .˛C1/=ˇ  e m.t 1=ˇ /

completely monotone

t 1=ˇ  e m.t 1=ˇ /

completely monotone

t

.˛C1/=ˇ

e m.t

1=ˇ

p-monotone

t e m.t /

.0; 1/

BF˛;0

t ˛C1  e m.t /

¹0º .0; 1/

BFlogp ;0 BF˛;logp1 ;0

completely monotone and vanishes at 1 p-monotone

BFlog;0

BF

/

t ˛C1  e m.t /

¹0º

.1; 0/ # ˛;p

BF

¹0º

.1; 0/

b # ˛;p

D.‚/

p-monotone and vanishes at 1

t ˛C1  e m.t / D n.log t /

n p-monotone on R

t e m.t / D n.log t /

n p-monotone on R

t ˛C1  e m.t / D n.log t /

n p-monotone on R and vanishes at 1

For a family of Bernstein functions, the subscript 0 denotes the subfamily consisting of those functions f satisfying f .0C/ D 0, e.g. BFlog;0 D ¹f 2 BFlog W f .0C/ D 0º. Table 10.1 summarizes some of the main results of this chapter. It gives the domains b ˛;p and ‚˛;p . In each case, the Lévy measure and ranges of the mappings ‚˛;ˇ , ‚ of the Bernstein function in the range has a density denoted by m e . The ranges are completely characterized by monotonicity properties of m e. Example 10.3. In this example we illustrate the results stated in Table 10.1 by conb ˛;p and ‚˛;p , and identify sidering eight particular choices of the mappings ‚˛;ˇ , ‚ their ranges with some of the known subfamilies of Bernstein functions. b 1;1 . Then b # 1;1 .t / D 1 and b  1;1 .t / D 1  t . By the fourth line of Table (i) ‚ b1;1 / consists of all Bernstein functions whose Lévy measure 10.1, the range R.‚ b 1;1 is has a non-increasing density. The multidimensional version of the mapping ‚ b 1;1 .ID/ is known as the Jurek known in the literature as U, while the analogue of ‚ class or the class of s-self-decomposable distributions. (ii) ‚1;1 . Then #1;1 .t / D e t and 1;1 .t / D log 1t . By Definition 6.1 and the first line of Table 10.1, we have that ‚1;1 .BF/ D CBF and ‚1;1 .ID/ D BO. Let BF SD denote the family of Bernstein functions f such that the corresponding subprobability  is self-decomposable. By Proposition 5.17, f 2 BF SD if, and only if, the Lévy measure  of f has a density m.t / such that t 7! t  m.t / is non-increasing.

Chapter 10 Transformations of Bernstein functions

135

It follows from Definition 8.1 and Corollary 10.13 that ‚1;1 .BF SD / D TBF, and consequently ‚1;1 .SD/ D GGC. The mapping ‚1;1 is known in the literature as the ‡ (upsilon) transformation; it is introduced in [21, 22] in the one-dimensional case, and [20] in the multidimensional case. p 2 (iii) ‚1;2 . Then #1;2 .t / D 2= e t and 1;2 W .0; 1/ ! .0; 1/ is the inR1p 2 verse function of t 7! t 2= e s ds. By the first line of Table 10.1, the range of ‚1;2 consists p of all Bernstein functions whose Lévy measure has a density m e such that t 7! m e . t / is completely monotone. This class of Lévy measures is related to subordinate Brownian motions: A rotationally invariant Lévy process in Rd is a subordinate Brownian motionpif, and only if, the radial part of its Lévy measure has a density n such that t 7! n. t / is completely monotone, cf. Bochner [63, Theorem 4.3.3] or Example 13.16 below. The multidimensional version of the mapping ‚1;2 is known in the literature as G, while the analogue of ‚1;2 .ID/ is denoted by G. b 0;1 . Then b # 0;1 .t / D t 1 and b  0;1 .t / D e t . By the fifth line of Table 10.1 (iv) ‚ b 0;1 .BF log;0 / D BFSD;0 . The multidimensional version of the and Proposition 5.17, ‚ b b 0;1 .ID/ is the mapping ‚0;1 is known in the literature as ˆ, while the analogue of ‚ class of all self-decomposable distributions. . Then #0;1 .t / D t 1 e t and 0;1 W .0; 1/ ! .0; 1/ is the inverse function (v) ‚0;1 R1 of t 7! t s 1 e s ds. By the second line of Table 10.1 and Definition 8.1 it follows that ‚0;1 .BF log;0 / D TBF 0 . This can also be deduced from (ii) and (iv) by using b 0;1 , cf. Proposition 10.11. The multidimensional version of that ‚0;1 D ‚1;1 ı ‚ the mapping ‚0;1 is known in the literature as ‰. (vi) ‚0;2 . Then #0;2 .t / D t 1 e t and 0;2 W .0; 1/ ! .0; 1/ is the inverse R1 2 function of t 7! t s 1 e s ds. By the second line of Table 10.1, the family ‚0;2 .BF log;0 / consistspof all functions in BF0 whose Lévy measure has a density m e such that t 7! t  m e. t / is completely monotone. The multidimensional version of the mapping ‚0;2 is known in the literature as M. 2

(vii) ‚0;p . Then # 0;p .t / D . log t /p1 t 1 and 0;p .t / D exp..pt /1=p /. Let BFSDp denote the family of Bernstein functions f such that the corresponding subprobability measure is in SDp . By Proposition 5.20, f 2 BFSDp if, and only if, the Lévy measure  of f has a density m.t / such that t 7! t  m.t / D n.log t / where n is p-monotone on R. It follows from the eighth line in Table 10.1 that ‚0;p .BF logp ;0 / D BFSDp ;0 . (viii) ‚1;1 . Then #1;1 .t / D c1;1 e t t 2 . By the first three lines of Table 10.1 we see that R.‚1;1 / is equal to TBF  for > 1 and TBF ;0 for 2 .0; 1, see Definition 8.9.

136

Chapter 10 Transformations of Bernstein functions

Table 10.2. Overview of the classical cases. D.‚/

‚

R.‚/

b ‚1;1

U

‚1;1

‡

‚1;1

‡

‚1;2

G

b ‚0;1

ˆ

BF log;0

BFSD;0

‚0;1

‰

BF log;0

TBF0

‚0;2

‚0;p

BF

.t /

#.t /

1t

1

e m.t / 1 R t

M

BF

CBF

BF SD

TBF

log

log

1 t

e t

1 t

e t

1

.ds/ s

1 R

e t s

1

.ds/ s

0

q

e t

BF log;0

BFSDp ;0

e t s

0

BF

BF logp ;0

1 R

.ds/ s

e .pt /

1=p

2 t 2 e

q

1 R

2 

e t

1 t

1 t .t; 1/

1 t te

1 t

1 t 2 te

1 t

1 p t . log t /

1 t

1 R

2 s 2

0

e t s

1

.ds/ s

.ds/

0 1 R

e t

2 s 2

.ds/

0

R .t;1/

.log st /p1 .ds/

Table 10.2 summarizes the discussion in Example 10.3. The last column, the form of the Lévy measure of ‚f , follows from (10.12). The notation BFSD;0 and BF SDp ;0 is explained in Example 10.3 (ii) and (vii), respectively. Most of the remaining part of this chapter is devoted to proving the results listed in Table 10.1. We start with a discussion of the properties of the general mapping ‚. Proposition 10.4. Let f 2 BF be given by Z f ./ D a C b C

.0;1/

.1  e t / .dt /:

(10.8)

(i) If ‚f is defined by (10.3), then Q C ‚f ./ D aQ C b

Z

1 0

.1  e t / m e .t / dt;

(10.9)

137

Chapter 10 Transformations of Bernstein functions

where Z aQ D a

#.t / dt ; 0

Z

bQ D b

(10.10)

!

B

t #.t / dt ;

(10.11)

0

Z m e .t / D

!

B

#.t s 1 / s 1 .ds/;

(10.12)

.t=B;1/

with the convention that t =B D 0 if B D 1, and 0  1 D 0. (ii) If ‚f .1/ < 1, then ‚f 2 BF. In particular, aQ < 1, bQ < 1 and m e is the density of a Lévy measure. (iii) ‚f 2 BF if, and only if, the following three conditions hold: Z B #.t / dt < 1; a D 0 or

(10.13)

0 Z B

b D 0 or Z

Z

1 1=B

0

1=t

Proof. (i) From (10.5) we get that Z Z B a C bt C ‚f ./ D Z

Da

0

Z

.0;1/

Q C D aQ C b Q C D aQ C b Q C D aQ C b

Z

0

.1  e

B

ts

0 ts

Z

 / .ds/ #.t / dt

 /#.t / dt .ds/

Bs 0

(10.15)

 t #.t / dt 

.1  e Z .1  e t /

t

 /#.t s /s dt .ds/  #.t s 1 /s 1 .ds/ dt 1

1

.t=B;1/

0 1 0

.1  e

Z

B

.0;1/ Z 1

Z

.0;1/

Cb

#.t / dt Z

C



B

(10.14)

s#.s/ ds .t; 1/ dt < 1:

0

0

t #.t / dt < 1; !

.1  e

t

/m e .t / dt;

with a, Q bQ and m e as in the statement. R1 e .t / dt < 1. (ii) If ‚f .1/ < 1, then clearly aQ < 1, bQ < 1, and 0 .1  e t / m Remark 3.3 (iii) shows that the measure m e .t / dt is a Lévy measure.

138

Chapter 10 Transformations of Bernstein functions

(iii) It is clear that aQ < 1 is equivalent to (10.13), and bQ < 1 is equivalent to (10.14). Since ‚ is additive we may from now on assume that a D b D 0. Note that ‚f 2 BF is equivalent to Z ‚f .1/ D

B 0

f .s/#.s/ ds < 1:

By Lemma 3.4 e tI .1=t /  f .t /  tI .1=t / for all t > 0; e1 Rs where I .s/ D 0 .t; 1/ dt . We write this fact as f .s/ sI .1=s/. Assume that B < 1. Then by Fubini’s theorem Z ‚f .1/

B

sI .1=s/#.s/ ds

Z 1=s  Z B s#.s/ .t; 1/ dt ds D 0

0

0

Z D

.t; 1/ dt

0

Z C

 Z

1=B

1 1=B

Z

1=t 0



B

s#.s/ ds 0



s#.s/ ds .t; 1/ dt:

If ‚f .1/ < 1, then clearly (10.15) holds. Conversely, suppose that the integral in (10.15) is finite. Then s 7! s#.s/ is Lebesgue integrable near zero implying that RB R 1=B R .t; 1/ dt D .0;1/ ..1=B/ ^ t / .dt / < 1 we see 0 s#.s/ ds < 1. Since 0 that ‚f .1/ < 1. The case when B D 1 is proved analogously. Remark 10.5. (i) We have given necessary and sufficient conditions for ‚f 2 BF in terms of the function #. Alternatively, we can give conditions using the function  : a D 0 or A < 1; Z A .t / dt < 1; b D 0 or 0  Z AZ .t .s/ ^ 1/ .dt / ds < 1: 0

.0;1/

(10.16) (10.17) (10.18)

139

Chapter 10 Transformations of Bernstein functions

To see (10.18) we assume that a D b D 0, and note that ‚f .1/ < 1 if, and only if,  Z AZ t .s/ .1  e / .dt / ds < 1: 0

.0;1/

We conclude by invoking the inequality 12 .1 ^ t /  1  e t  1 ^ t valid for all t > 0. (ii) Let  be the Lévy measure of f 2 BF and define the measure e  on .0; 1/ by  Z AZ 1C .t .s// .dt / ds (10.19) e .C / WD Z D

0 B

0

Z

.0;1/

.0;1/

for any Borel set C  .0; 1/. Then Z Z .1  e t / e .dt / D .0;1/

 1C .t u/ .dt / #.u/ du;

AZ 0

.0;1/

(10.20)

 .1  e t .s/ / .dt / ds;

showing that e  is the Lévy measure of ‚f . Suppose now that  is any measure on .0; 1/ and define e  by (10.19). If e  is a Lévy measure, then so is . Indeed, there exist 2 .0; 1 and C  .0; 1/ with 0 < Leb.C / < 1 such that .s/  for s 2 C . Then  Z Z 1 Z .t ^ 1/ e .dt / D .t .s/ ^ 1/ .dt / ds 0 .0;1/ .0;1/  Z Z  .t ^ 1/ .dt / ds C .0;1/ Z .t ^ 1/ .dt /:  Leb.C / .0;1/

(iii) Suppose that ˛  1 and ˇ > 0. If we define #˛;ˇ .t / D t ˛1 exp.t ˇ /, then R 1=t s #˛;ˇ .s/ ds D 1. Hence, for every Lévy measure , the integral in (10.15) will 0 be infinite and the domain of ‚˛;ˇ consists only of the function identically equal to zero. The same argument works for t ˛1 .1  t /p1 and t ˛1 . log t /p1 . This explains the range of the parameter ˛ in (10.6). The family PBF of Laplace exponents of completely self-decomposable sub-probability measures on Œ0; 1/ plays a distinguished role in studying the mappings ‚. In the next proposition we prove that it is invariant under all mappings ‚. We will later see that it naturally appears as the intersection of iterates of certain families of mappings ‚, cf. Theorem 10.22. Recall that every f 2 PBF admits the power R representation (5.17), f ./ D Œ0;1 ˛ .d˛/. For any measurable set E  Œ0; 1 let Z ° ±   ˛ .d˛/;  Œ0; 1 n E D 0 : (10.21) PBF E WD f 2 BF W f ./ D Œ0;1

140

Chapter 10 Transformations of Bernstein functions

Proposition 10.6. For 0  ˛  1, let c.˛/ WD

RB 0

t ˛ #.t / dt 2 .0; 1.

(i) If f ./ D ˛ , then ‚f D c.˛/f . (ii) If E  Œ0; 1 and f 2 PBF E \ D.‚/, then ‚f 2 PBFE . (iii) If E  Œ0; 1 is a measurable set and c.˛/ 2 .0; 1/ for all ˛ 2 E, then ‚.PBF E \ D.‚// D PBF E . R  RB B Proof. (i) ‚f ./ D 0 f .t /#.t / dt D 0 t ˛ #.t / dt ˛ D c.˛/f ./: (ii) By (10.21), there exists a finite measure  on Œ0; 1 such that .Œ0; 1 n E/ D 0 R and f ./ D Œ0;1 ˛ .d˛/. Therefore  Z B Z Z B ˛ f .t /#.t / dt D .t / .d˛/ #.t / dt ‚f ./ D Z D

0

0

Z

Œ0;1

B



Œ0;1

t ˛ #.t / dt ˛ .d˛/:

0

R Let e  .d˛/ WD c.˛/.d˛/. Then ‚f ./ D Œ0;1 ˛ e  .d˛/. Since ‚f .1/ < 1 by the assumption, it follows that e  is a finite measure on Œ0; 1. Moreover, e  .Œ0; 1 n E/ D 0; thus ‚f 2 PBF E . R (iii) Let fQ./ D Œ0;1 ˛ e  .d˛/ 2 PBF E , and define a finite measure  on Œ0; 1 R B^1 ˛ R B^1  .d˛/. Since c.˛/  0 t #.t / dt  0 t #.t / dt > 0, by .d˛/ D c.˛/1e we have that c.˛/1 is a boundedR function on Œ0; 1. Thus  is a finite measure and .Œ0; 1 n E/ D 0. Let f ./ WD Œ0;1 ˛ .d˛/ 2 PBF E . The computation in (ii) shows that ‚f D fQ. The next proposition states that, if # satisfies certain integrability conditions, D.‚/ and R.‚/ are closed under pointwise limits, and ‚ is continuous with respect to pointwise convergence. In Remark 10.8 we will show that continuity need not be true without the integrability conditions. R1 RB Proposition 10.7. Assume that 0 #.t / dt < 1 if B < 1, and that 0 #.t / dt < 1 R1 and 1 t #.t / dt < 1 if B D 1. (i) If .fn /n2N  D.‚/ is such that limn!1 fn ./ D f ./ for all  2 .0; 1/, and f .1/ < 1, then f 2 D.‚/ and limn!1 ‚fn ./ D ‚f ./ for all  2 .0; 1/. (ii) If .fn /n2N  D.‚/ is such that limn!1 ‚fn ./ D fQ./ 2 .0; 1/ for all  2 .0; 1/, then there exists f 2 BF such that fQ D ‚f 2 BF. Moreover, if ‚ is one-to-one, then limn!1 fn D f . Proof. (i) Since f .1/ < 1, we have that f 2 BF. Suppose first that B < 1. Then Z B Z B ‚f ./ D f .t /#.t / dt  f .B/ #.t / dt < 1; 0

0

141

Chapter 10 Transformations of Bernstein functions

implying that f 2 D.‚/. Let  > 0. Since limn!1 fn .B/ D f .B/, there exists n0 2 N such that fn .t /  fn .B/  2f .B/ for all t 2 .0; B and all n  n0 . By the dominated convergence theorem it follows that Z B Z B lim ‚fn ./ D lim fn .t /#.t / dt D f .t /#.t / dt D ‚f ./: n!1

n!1 0

0

If B D 1, then the same argument as above shows that Z 1 Z 1 fn .t /#.t / dt D f .t /#.t / dt < 1: lim n!1 0

0

Fix  > 0 and let t  1. Then

Z

f .t / D a C bt C t

1 0

 t a C t .b/ C t 

Z

e ts .s; 1/ ds 1

0

e s .s; 1/ ds D tf ./:

Hence, there exists some n0 2 N such that fn .t /  fn ./t  2f ./t for all n  0 and all t  1. Again by the dominated convergence theorem we conclude that Z 1 Z 1 lim fn .t /#.t / dt D f .t /#.t / dt < 1: n!1 1

1

(ii) Because of Remark 3.11 we can find a subsequence .nk /k2N and a function f W .0; 1/ ! Œ0; 1 such that limk!1 fnk ./ D f ./ for all  2 .0; 1/. By Fatou’s lemma Z B Z B f .t /#.t / dt  lim inf fnk .t /#.t / dt D lim ‚fnk ./ D fQ./ < 1; 0

k!1

0

k!1

implying that f ./ < 1 for all  2 .0; 1/. Thus f 2 BF. From (i) we conclude that ‚f D fQ. If ‚ is one-to-one, then all subsequential limits are equal to f , hence the whole sequence .fn /n2N converges to f . RB Remark 10.8. Let ˛ 2 Œ0; 1/ and suppose that c.˛/ D 0 t ˛ #.t / dt D C1, but RB c. / D 0 t  #.t / dt < 1 for all 2 .˛; 1. Examples that satisfy these conditions are #˛;ˇ , b # ˛;p and # ˛;p . Consider the sequence .fn /n0 where fn ./ WD ˛C1=n . By Proposition 10.6 (i), ‚fn D c.˛ C 1=n/fn , hence fn 2 D.‚/. On the other hand, with f ./ D ˛ we have that limn!1 fn ./ D f ./ for all  > 0. Since ‚f D c.˛/f  C1 we conclude that f … D.‚/. Therefore, D.‚/ is not closed with respect to pointwise convergence. Let gn ./ D c.˛ C 1=n/1 ˛C1=n , n  1. Then gn 2 D.‚/, ‚gn ./ D ˛C1=n , and limn!1 ‚gn ./ D ˛ for all  > 0. Setting g./  0 we see that g 2 D.‚/, ‚g./ D 0, and limn!1 gn ./ D g./ for all  > 0. This shows that ‚ is not continuous with respect to pointwise convergence.

142

Chapter 10 Transformations of Bernstein functions

The next example shows that the mapping ‚ need not be one-to-one. Example 10.9. Let ˛ > 0 and define # W .0; 1/ ! .0; 1/ by #.t / D ˛t ˛1 . Then Z 1 #.s/ ds D t ˛ ; t

implying that  W .0; 1/ ! .0; 1/ is given by .t / D t 1=˛ . Let f 2 BF with the generating triplet .a; b; /. If a > 0 (respectively b > 0), then aQ D 1 (respectively bQ D 1). Assume that a D b D 0. By using (10.12) we get that Z m e.t / D t ˛1 s ˛ .ds/; t > 0: .0;1/

R

If .0;1/ s ˛ .ds/ D 1, then m e is identically infinite, hence ‚f D 1. On the other R ˛ hand, if .0;1/ s .ds/ < 1, m e is proportional to t ˛1 . If ˛  1, then since R1 ˛1 D 1, m e.t / dt is not a Lévy measure and again, Thus, for 0 .t ^ 1/t R 1‚f D 1. ˛1 ˛  1, D.‚/ D ¹0º. On the other hand, if 0 < ˛ < 1, then 0 .t ^ 1/t < 1, and m e is a Lévy measure. Hence ³ ² Z ˛ s .ds/ < 1 D.‚/ D f 2 BF W a D b D 0; .0;1/

(this can be also computed by using (10.15)). RBy using the formula for m e we obtain that ‚f ./ D c;˛ ˛ 2 BF, where c;˛ D .0;1/ s ˛ .ds/. There exist different R R Lévy measures  and  such that .0;1/ s ˛ .ds/ D .0;1/ s ˛ .ds/ < 1. Hence, ‚ is not one-to-one. We show now that any two operators ‚1 and ‚2 commute and give the formula for their composition. Proposition 10.10. Let #j .t / D .j1 /0 .t /, j D 1; 2, define Z #.t / D

B2

#1 .t s t=B1

1

/#2 .s/s

1

Z ds D

B1

#2 .t s 1 /#1 .s/s 1 ds;

(10.22)

t=B2

RB B and let  be the inverse of t 7! t 1 2 #.s/ ds. Then ‚ D ‚2 ı ‚1 D ‚1 ı ‚2 . In particular, ‚1 and ‚2 commute.

143

Chapter 10 Transformations of Bernstein functions

Proof. By a change of variables and Fubini’s theorem we have that Z ‚2 .‚1 f /./ D

B2

‚1 f .s/#2 .s/ ds  Z B2  Z B1 f .su/#1 .u/ du #2 .s/ ds D 0

Z

0

Z

0

Z

0

D

Z

0

B1 s

f .t /#1 .t s

D D

B2

0

B1 B2

Z f .t /

B2

1

/s

1

 dt #2 .s/ ds

 #1 .t s 1 /#2 .s/s 1 ds dt

t=B1 B1 B2

f .t /#.t / dt; 0

proving the first equality in (10.22). The second equality follows by another change of variables. Proposition 10.10 applied to the three two-parameter families of mappings defined in (10.6) reveals certain relations among those families. Proposition 10.11.

(i) Assume that p; q > 0 and 1 < ˛ < 1. Then b ˛;p ı ‚ b ˛p;q D b b ˛;pCq ; ‚ c.˛; p; q/ ‚

(10.23)

where b c.˛; p; q/ > 0 is a positive constant. If 1 < ˛  0, b c.˛; p; q/ D 1. (ii) Assume that ˇ > 0, 1 < ˛ < 0 and ˛ C ˇ < 1. Then b ˛Cˇ;1 ı ‚˛;ˇ D c.˛; ˇ/ ‚˛Cˇ ;ˇ ; ‚

(10.24)

where c.˛; ˇ/ > 0 is a positive constant. If ˛ C ˇ  0, c.˛; ˇ/ D 1. (ii0 ) Assume that ˛; ˇ < 1 and ˛ > ˇ. Then b ˛;˛ˇ ı ‚ˇ;1 D c 0 .˛; ˇ/ ‚˛;1 : ‚

(10.25)

where c 0 .˛; ˇ/ > 0 is a positive constant. If ˛  0, then c 0 .˛; ˇ/ D 1. (iii) Assume that p; q > 0 and 1 < ˛ < 1. Then ‚˛;p ı ‚˛;q D c.˛; p; q/ ‚˛;pCq ;

(10.26)

where c.˛; p; q/>0 is a positive constant. If 1 0 and 0 < ˛ < 1. Then D.‚0;ˇ / D BF˛;0 and ‚˛;ˇ W BF ˛;0 ! BF 0 is one-to-one. Moreover, R.‚˛;ˇ / consists of all Bernstein functions in BF 0 whose Lévy measure has a density m e such that t 7! t .˛C1/=ˇ  m e .t 1=ˇ / is a completely monotone function vanishing at infinity.

Chapter 10 Transformations of Bernstein functions

147

Proof. Let  be the image measure of s ˛1 .ds/ under the mapping .s/ WD s ˇ . Then Z ˇ e t s .ds/ D c˛;ˇ t ˛1 L .I t ˇ /: m˛;ˇ .t / D c˛;ˇ t ˛1 .0;1/

This shows that t 7! t .˛C1/=ˇ  m˛;ˇ .t 1=ˇ / D c˛;ˇ L .I t / is a completely monotone function vanishing at infinity (since ¹0º D 0), and that the measure  is uniquely determined by m˛;ˇ . R1 (i)R We check that ‚˛;ˇ f 2 BF for every f 2 BF. Clearly, 0 #˛;ˇ .t / dt < 1 1 and 0 t #˛;ˇ .t / dt < 1, showing that (10.13) and (10.14) are satisfied. Further, 1  Z 1=t

Z 0

0

 s#˛;ˇ .s/ ds .t; 1/ dt  Z 1Z 1 ˛ s ˇ s e ds .t; 1/ dt  c˛;ˇ 0

0 1  Z 1=t

 s ˛ ds .t; 1/ dt C c˛;ˇ 1 0  Z 1  Z 1 ˛ s ˇ .t; 1/ dt s e ds dt  c˛;ˇ 0 0 Z 1 c˛;ˇ C t ˛1 .t; 1/ dt: 1˛ 1 Z

Since this is finite, (10.15) holds as well. Further, a, Q bQ and m˛;ˇ are uniquely determined by the triplet .a; b; /, i.e. ‚˛;ˇ is one-to-one from BF onto ‚˛;ˇ .BF/. Since #˛;ˇ satisfies the integrability conditions of Proposition 10.7, ‚˛;ˇ is continuous with respect to pointwise convergence. We have already seen that every fQ 2 R.‚˛;ˇ / has a Lévy measure with density m e such that t 7! t .˛C1/=ˇ  m e.t 1=ˇ / is completely monotone. Conversely, supQ Q /, pose that f 2 BF with the generating triplet .a; QR b; Q where e .dt / D m e .t / dt 1 .˛C1/=ˇ 1=ˇ such that t 7! t m e .t / 2 CM. Since 1 m e .t / dt < 1, it follows that e .t 1=ˇ / D lims!1 s ˛C1 m e .s/ D 0. Therefore, there exists a mealim t!1 t .˛C1/=ˇ m sure  on .0; 1/ such that t .˛C1/=ˇ  m e .t 1=ˇ / D L .I t /. Define a measure  on .0; 1/ by .ds/ WD s ˛C1 . ı /.ds/ (recall that .s/ D s ˇ ). Then Z   exp  t ˇ s ˇ s ˛1 .ds/ D t ˛1 L .I t ˇ / D m e .t /: t ˛1 .0;1/

Let a D aQ and b D

R1 0

t #˛;ˇ .t / dt

1

Q If f is given by (10.8), then ‚˛;ˇ f D fQ. b.

R 1(ii) We check that ‚0;ˇ f is a Bernstein function if, and only if, f 2 BF log;0 . Since 0 #0;ˇ .t / dt D 1, the condition (10.13) holds if, and only if, a D 0; (10.14) is

148

Chapter 10 Transformations of Bernstein functions

R1 always satisfied because 0 t #0;ˇ .t / dt < 1. Finally, # Z 1 "Z 1=t s#0;ˇ .s/ ds .t; 1/ dt 0

0

Z D

1

"Z

0

0

Z C

1 1



1=t

exp  s

"Z

1=t 0



 ˇ

# ds .t; 1/ dt

exp  s

 ˇ

# ds .t; 1/ dt:

The first term is always finite. In the second term, we use that e 1  exp.s ˇ /  1 for s 2 .0; 1/. This shows that the second term is comparable to Z Z 1 1 t .t; 1/ dt D log t .dt /: 1

.1;1/

Thus, ‚0;ˇ f 2 BF if, and only if, f 2 BF log;0 . The rest follows as in (i). (iii) The proof is similar to the proof of (ii). Note that for the converse we have to assume that t 7! t .˛C1/=ˇ  m e .t 1=ˇ / vanishes at infinity, because for ˛ 2 .0; 1/ this does not come for free. Let us record two consequences of Theorem 10.12. Corollary 10.13. Let ˇ > 0 and 1 < ˛ < 1. Suppose that the Lévy measure  has a density m. Then, under the transformation ‚˛;ˇ , t 7! t .˛C1/=ˇ C1  m.t 1=ˇ / is decreasing if, and only if, e .t 1=ˇ / is completely monotone: t 7! t .˛C1/=ˇ C1  m Proof. Assume that t 7! t .˛C1/=ˇ C1  m.t 1=ˇ / is decreasing. By a change of variable u D s ˇ in (10.27), we get that Z c˛;ˇ 1 tu .˛1/=ˇ 1 .˛C1/=ˇ 1=ˇ t m e .t / D e u m.u1=ˇ / du: ˇ 0 Let n.u/ WD u.˛1/=ˇ 1 m.u1=ˇ /, u > 0. By the assumption, n is an increasing function. Denote by N the measure corresponding to the increasing function n. Then Z c˛;ˇ 1 tu .˛C1/=ˇ C1 1=ˇ t m e .t / D t e n.u/ du ˇ 0 Z c˛;ˇ c˛;ˇ L .N I t /: e tu N.du/ D D ˇ .0;1/ ˇ The converse follows by retracing the steps backwards.

149

Chapter 10 Transformations of Bernstein functions

Corollary 10.14. (i) Let ˇ > 0 and 1 < ˛1 < ˛2 < 0. Then R.‚0;ˇ /  R.‚˛2 ;ˇ /  R.‚˛1 ;ˇ /: (ii) Let 0 < ˇ1 < ˇ2 and ˛  0. Then R.‚˛;ˇ1 /  R.‚˛;ˇ2 /. Proof. (i) We show the second inclusion, the first follows similarly. If f 2 R.‚˛2 ;ˇ /, then the Lévy measure of f has a density n such that t 7! t .˛2 C1/=ˇ  n.t 1=ˇ / is completely monotone. Since t 7! t .˛1 ˛2 /=ˇ is also completely monotone, the product t 7! t .˛1 C1/=ˇ  n.t 1=ˇ / is completely monotone. Therefore, f 2 R.‚˛1 ;ˇ /. (ii) Let ˛  0. If f 2 R.‚˛;ˇ1 /, then the Lévy measure of f has a density n such that t 7! t .˛C1/=ˇ1  n.t 1=ˇ1 / is completely monotone. The function t 7! t ˇ1 =ˇ2 is a Bernstein function. Since by Theorem 3.7 CM ı BF  CM, we conclude that t 7! t .˛C1/=ˇ2 n.t 1=ˇ2 / is a completely monotone function. Thus f 2 R.‚˛;ˇ2 /. Theorem 10.15. Let p 2 N. b ˛;p / D BF, ‚ b ˛;p W BF ! BF is one-to(i) Let 1 < ˛ < 0. Then D.‚ b˛;p / one and continuous with respect to pointwise convergence. Moreover, R.‚ consists of all Bernstein functions whose Lévy measure has a density m e such that e .t / is p-monotone. t 7! t ˛C1  m b 0;p W BFlog;0 ! BF 0 is one-tob0;p / D BF log;0 and ‚ (ii) Let ˛ D 0. Then D.‚ b0;p / consists of all Bernstein functions in BF 0 whose Lévy one. Moreover, R.‚ measure has a density m e such that t 7! t  m e .t / is p-monotone. b ˛;p W BF ˛;0 ! BF 0 is oneb ˛;p / D BF ˛;0 and ‚ (iii) Let 0 < ˛ < 1. Then D.‚ b˛;p / consists of all Bernstein functions in BF 0 whose Lévy to-one. Further, R.‚ e.t / is p-monotone and vanishes measure has a density m e such that t 7! t ˛C1  m at infinity. Proof. We first compute the domains by checking (10.15). Since ! Z 2 Z 2 Z 1=t p1 ˛ .1  s/ s ds .t; 1/ dt  B.p; 1  ˛/ .t; 1/ dt < 1; 1

0

1

it suffices to check that Z 1 Z 2

!

1=t 0

p1 ˛

.1  s/

s

ds .t; 1/ dt < 1:

For s 2 .0; 1=2/ we have 21p  .1s/p1  1 if p  1, and 1  .1s/p1  21p if 0 < p < 1. Thus, (10.15) is valid if, and only if, ! Z Z 1

2

1=t

0

s ˛ ds .t; 1/ dt < 1 :

150

Chapter 10 Transformations of Bernstein functions

If 1 < ˛ < 0 we see by a straightforward calculation and Fubini’s theorem ! Z 1 Z 1 Z 1=t ˛ ˛ s ds .t; 1/ dt D t ˛1 .t; 1/ dt 1˛ 2 2 0 Z 1 ˛  t ˛1 .t; 1/ dt 1˛ 1 Z 1 .1  s ˛ / .ds/ < 1: D 1  ˛ .1;1/ If ˛ D 0, then Z

1

Z

2

1=t 0

!

Z

ds .t; 1/ dt 

1

t 1

1

Z .t; 1/ dt D

Finally, let 0 < ˛ < 1. Then ! Z Z 1

1=t

s 2

0

˛

1 ds .t; 1/ dt  1˛

Z

1 1

1 D ˛.1  ˛/

.1;1/

log s .ds/ < 1:

t ˛1 .t; 1/ dt

Z .1;1/

.s ˛  1/ .ds/:

b ˛;p / D BF ˛;0 for 0 < ˛ < 1, b ˛;p / D BF for 1 < ˛ < 0, D.‚ Therefore, D.‚ b 0;p / D BFlog;0 . and D.‚ Now we prove thatRthe mappings are one-to-one on their domains. If p D 1, b ˛;1 .t / D b c ˛;1 .t;1/ s ˛ .ds/ for every t > 0, implying that the measure t ˛C1 m b ˛;1 is one-tob ˛;p . Hence, the transformation ‚ s ˛ .ds/ is uniquely determined by m one for all ˛ 2 .1; 1/. b ˛;p1 is one-to-one for all ˛ 2 .1; 1/. From Let p  2 and suppose that ‚ b ˛C1p;1 . b ˛;p D b b ˛;p1 ı ‚ c.˛; p  1; 1/1 ‚ Proposition 10.11 (i) we know that ‚ b ˛;p is also one-to-one for all ˛ 2 .1; 1/. From Proposition 10.7 it Therefore, ‚ b ˛;p is continuous with respect to pointwise convergence for ˛ < 0. follows that ‚ R Finally, from (10.28) we see that t ˛C1 m b ˛;p .t /  b c ˛;p .t;1/ s ˛ .ds/, hence it vanishes at infinity. b ˛;p .t / is p(i) It is clear from (10.28) and Theorem 1.11 that t 7! t ˛C1  m Q monotone. Conversely, suppose that f is a Bernstein function with the generating Q /, triplet .a; Q b; .dt / D m e .t / dt and t 7! t ˛C1  m e .t / is p-monotone. MoreR Q1 where e e .t / dt < 1, we have that lim t!1 t ˛C1 m e .t / D 0. Hence, there over, since 1 m exists a measure  on .0; 1/ such that Z t ˛C1 m e .t / D b c ˛;p .s  t /p1 .ds/: .t;1/

151

Chapter 10 Transformations of Bernstein functions

Let  be the measure on .0; 1/ defined by .dt / WD t ˛Cp1 .dt /. Then for u > 0,  Z Z 1 Z 1 ˛1 p1 ˛C1p m e .t / dt D b c ˛;p t .s  t / s .ds/ dt .t;1/ u u  Z s Z Db c ˛;p s ˛C1p .s  t /p1 t ˛1 dt .ds/ Z Db c ˛;p

Z

.0;1/

Z

.0;1/ 1Z

D D

0

Z

.u;1/

Z

Z D

.u;1/

Z

u

1

p1 ˛1

u=s

.1  v/

v

 dv .ds/

 b # ˛;p .v/ dv .ds/

1

(10.30)

1^.u=s/ 1

 1.u;1/ .sv/b # ˛;p .v/ dv/ .ds/

0

 1.u;1/ .sv/b # ˛;p .v/ .ds/ dv:

.0;1/

This shows that the measure with density m e is of the form (10.20). Since by assumption this is a Lévy measure, it follows by Remark 10.5 (ii) that  is also a Lévy ıR1 # ˛;p .t / dt . If f is given by (10.8), then measure. Finally, let a D aQ and b D bQ 0 t b Q b b ‚˛;p f D f . This shows that ‚˛;p maps BF onto the family of all Bernstein funce .t / is p-monotone. tions whose Lévy measure has a density m e such that t 7! t ˛C1  m (ii) The proof is analogous to (i). The only additional ingredient is the part of Rthe converse claiming that the Lévy measure  constructed in the proof satisfies .1;1/ log t .dt / < 1. To prove this we use (10.30) with u D 1: Z 1>

1 1

m e .t / dt D

.0;1/ 1

Z

0

Z

.0;1/

1^s 1 Z 1

.1;1/

Z

D D Z 

 b 1.1;1/ .sv/# 0;p .v/ .ds/ dv

1Z

Z

.2;1/ 1p

 .2

Z

p1 1

.1  u/

 .1  u/p1 u1 du .ds/

1=s 1=2

1=s

 .1  u/p1 u1 du .ds/

Z

^ 1/

R

.2;1/ log t

Z

1=2

u .2;1/

Z D .21p ^ 1/ Therefore,

u

 du .ds/

.2;1/

1

 du .ds/

1=s

.log s  log 2/ .ds/:

.dt / < 1 which is equivalent to

R

.1;1/ log t

.dt / < 1.

152

Chapter 10 Transformations of Bernstein functions

the proof is analogous to the proof of (i). We only need to check that R (iii) Again, ˛ .dt / < 1. The same calculation as in (ii) yields t .1;1/ Z 1> 

1

1 Z

Z m e.t / dt D

1 1p

2

˛

^1

0

Z .2;1/

.0;1/

 b 1.1;1/ .sv/# ˛;p .v/ .ds/ dv

.s ˛  2˛ / .ds/;

proving the claim. Remark 10.16. Although the theorem is stated for integer values of p, it is true for all p > 0 (see [317, Theorem 2.10]). The only place in the proof where we used the b ˛;p are one-to-one, assumption that p is an integer was to show that the mappings ‚ b ˛;p is one-to-one b more precisely that ‚˛;1 is one-to-one. It is shown in [317] that ‚ for all p 2 .0; 1/ which suffices to prove Theorem 10.15 for all values p > 0. Corollary 10.17.

(i) Let 1 < ˛ < 1 and 0 < p < q. Then b˛;q /  R.‚ b˛;p /: R.‚

(ii) Let 1 < ˛ < 1. Then

T

b

p>0 R.‚˛;p /

D R.‚˛;1 /.

Proof. (i) This is a direct consequence T of Proposition 10.11 T (i). b ˛;p / D b (ii) First note that because of (i), p>0 R.‚ p2N R.‚˛;p /. Suppose that T b f 2 p2N R.‚˛;p /. By Theorem 10.15, for every p 2 N, the Lévy measure of of e.t / is p-monotone. Hence t 7! t ˛C1  m e .t / f has a density m e such that t 7! t ˛C1  m is completely monotone. By Theorem 10.12, f 2 R.‚˛;1 /. The other direction is proved by retracing the step backwards. Theorem 10.18. Let p 2 N. (i) Let 1 < ˛ < 0. Then D.‚˛;p / D BF, ‚˛;p W BF ! BF is one-toone and continuous with respect to pointwise convergence. Moreover, R.‚˛;p / consists of all Bernstein functions whose Lévy measure has a density m e of the ˛1 form t n.log t / where n is p-monotone on R. (ii) Let ˛ D 0. Then D.‚0;p / D BF logp ;0 and ‚0;p W BFlogp ;0 ! BF0 is one-toone. Moreover, R.‚0;p / consists of all Bernstein functions whose Lévy measure has a density m e of the form t 1 n.log t / where n is p-monotone on R. (iii) Let 0 < ˛ < 1. Then D.‚˛;p / D BF logp1 ;˛;0 and ‚0;p W BF logp1 ;˛;0 ! BF 0 is one-to-one. Moreover, R.‚˛;p / consists of all Bernstein functions whose Lévy measure has a density m e of the form t ˛1 n.log t / where n is p-monotone on R and vanishes at infinity.

Chapter 10 Transformations of Bernstein functions

153

Proof. We first determine the domains by checking (10.15). For ˛ < 0, let  > 0 be such that ˛ C  < 0. There is a constant c > 0 such that . log s/p1 s ˛  cs ˛ for all s 2 .0; 1/. Hence,  Z 1  Z 1=t c ˛;p . log s/p1 s ˛ ds .t; 1/ dt 1

0

Z

Z  c c ˛;p

.1;1/

1=t

0

 s ˛ ds .t; 1/ dt < 1:

This shows that D.‚˛;p / D BF for 1 < ˛ < 0. If ˛ D 0, after the change of variables u D log.1=s/, we arrive at  Z 1  Z 1=t . log s/p1 s ˛ ds .t; 1/ dt 1 0  Z 1 Z p1 u D u e du .t; 1/ dt .1;1/ log t Z .p; log t / .t; 1/ dt: D .1;1/

(.p; x/ is the incomplete Gamma function, cf. Section R 16.1.) p1 x As limx!1 .p; x/=.x e / D 1, the integral .2;1/ .p; log t /.t; 1/ dt is comparable with  Z s Z Z 1 p1 1 p1 t .log t / .t; 1/ dt D t .log t / dt .ds/ .2;1/ .2;1/ 1 Z .log s/p .ds/: D .2;1/

Thus, D.‚0;p / D BF logp ;0 . A similar analysis shows that D.‚˛;p / D BF˛;logp1 ;0 for 0 < ˛ < 1. b ˛;1 , it is We prove that the mappings are one-to-one on their domains. As ‚˛;1 D ‚ one-to-one, cf. Theorem 10.15. For integers p  2, the proof is the same as the proof of Theorem 10.15, the only difference being that instead of Proposition 10.11 (i) we use 10.11 (iii). Let  be the image measure of  under the map s 7! log s, and define n W R ! .0; 1/ by Z .s  t /p1 e ˛s .ds/: n.t / WD c ˛;p .t;1/

Then n is p-monotone on R, vanishes at infinity, and Z ˛C1 t m˛;p .t / D c ˛;p .log s  log t /p1 s ˛ .ds/ D n.log t /: .t;1/

154

Chapter 10 Transformations of Bernstein functions

Q /, (i) Let fQ 2 BF with the generating triplet .a; Q b; Q where .dt / D m e .t / dt and R1 e m e .t / D n.log t / where n is p-monotone on R. Since 1 m e.t / dt < 1, we have t that lim t!1 t ˛C1 m e .t / D 0. Hence, there exists a measure  on .0; 1/ such that Z e .t / D c ˛;p .s  log t /p1 .ds/: t ˛C1 m ˛C1

.log t;1/

Let  be the image measure of  under the map s 7! e s , and .ds/ D s ˛ .ds/. Then for u > 0,  Z Z 1 Z 1 ˛1 p1 m e .t / dt D c ˛;p t .log s  log t / .ds/ dt u u .t;1/  Z s Z D c ˛;p . log.t =s//p1 t ˛1 dt .ds/ Z D c ˛;p

u 1

p1 ˛1

. log v/ u=s

v



dv s ˛ .ds/



1

(10.31)

# ˛;p .v/ dv .ds/ Z

.0;1/

Z

.0;1/ 1Z

D D

Z

.u;1/

Z

Z D

.u;1/

0

1^.u=s/ Z 1

 1.u;1/ .sv/ # ˛;p .v/ dv/ .ds/

0

.0;1/

 1.u;1/ .sv/ # ˛;p .v/ .ds/ dv:

This shows that the measure with density m e is of the form (10.20). By our assumption, this is a Lévy measure, and Remark 10.5 (ii) shows that  is also a Lévy meaR Q 1 t # ˛;p .t / dt . If f is given by (10.8), then sure. Finally, let a D aQ and b D b= 0 ‚˛;p f D fQ. This shows that ‚˛;p maps BF onto the family of all Bernstein funce .t / D n.log t / where n is tions whose Lévy measure has a density m e such that t ˛C1 m p-monotone on R. (ii) The proof is analogous to (i). The only additional ingredient is the part of the R conversepclaiming that the Lévy measure  constructed in the proof satisfies .1;1/ .log t / .dt / < 1. To prove this we use (10.31) with u D 1: Z 1>

1

1

1Z

Z m e .t / dt D Z

0

Z

.0;1/

1^s 1 Z 1

.1;1/

1=s

D D

.0;1/ 1

 1.1;1/ .sv/ # 0;p .v/ .ds/ dv

Z

p1 1

. log u/

u

 du .ds/

 . log u/p1 u1 du .ds/

155

Chapter 10 Transformations of Bernstein functions

Z

Z D D



log s

t .1;1/

1 p

Z

.1;1/

p1

dt .ds/

1



 .log s/p  .log 2/p .ds/:

R

Therefore, .1;1/ .log t /p .dt / < 1. (iii) Again, the proof is similar to the proof of Theorem 10.15. Corollary 10.19. (i) Let 1 < ˛ < 1 and 0 < p < q. Then R.‚˛;q /  R.‚˛;p /. b ˛;p /. (ii) Let 1 < ˛ < 1 and let p 2 N. Then R.‚˛;p /  R.‚ T T b˛;p /. (iii) Let 1 < ˛ < 1. Then p>0 R.‚˛;p /  p>0 R.‚ Proof. (i) This is a direct consequence of Proposition 10.11 (iii). (ii) Suppose f 2 R.‚˛;p /. By Theorem 10.18 the Lévy measure of f has a density e .t / D n.log t / where n is p-monotone on R. By Proposition 1.16, m e such that t ˛C1 m t 7! n.log t / is p-monotone on .0; 1/. The claim follows from Theorem 10.15. (iii) This follows immediately from (i), (ii) and Corollary 10.17 (i). Remark 10.20. The inclusions in (ii) and (iii) are strict, see Remark 1.17. T Theorem 10.21. (i) Let 1 < ˛ < 0. Then p>0 R.‚˛;p / D PBF. T (ii) Let ˛ D 0. Then p>0 R.‚0;p / D PBF 0 . T .˛;1 (iii) Let 0 < ˛ < 1. Then p>0 R.‚0;p / D PBF 0 . T T Proof. (i) Let f 2 p>0 R.‚˛;p / D p2N R.‚˛;p /. By Theorem 10.18, the Lévy measure of f has a density m e such that m e .t / D t ˛1 n.log t / where n is p-monotone on R for every p 2 N, hence completely monotone on R. Since ˛  1  1, Theorem 5.22 implies that f 2 PBF. Conversely, if f 2 PBF, then again by Theorem 5.22, the Lévy measure of f has a density m e such that m e .t / D t ˛1 n.log t / for some function n which is completely monotone on TR and, in particular, p-monotone on R for every p 2 N. By Theorem 10.18, f 2 p2N R.‚˛;p /. (ii) This is proved as (i). T in the same wayT (iii) Let f 2 p>0 R.‚˛;p / D p2N R.‚˛;p /. By Theorem 10.18, the Lévy measure of f has a density m e such that m e .t / D t ˛1 n.log t / where n is p-monotone on R for every p 2 N, hence completely monotone on R. By Remark 5.23 (ii), .˛;1 f 2 PBF with a representing measure  such that Œ0; ˛ D 0. Thus f 2 PBF 0 . The converse is proved similarly as in (i) by using Remark 5.23 (ii) instead of Theorem 5.22.

156

Chapter 10 Transformations of Bernstein functions

For a mapping ‚ and m 2 N, let ‚m D ‚ ı    ı ‚ be the m-fold composition of ‚ with itself. T m (i) Let 1 < ˛ < 0. Then 1 mD1 R.‚˛;1 / D PBF. T bm (ii) Let 1 < ˛ < 0. Then 1 mD1 R.‚˛;1 / D PBF. T m (iii) Let 1 < ˛ < 0 and ˇ > 0. Then 1 mD1 R.‚˛;ˇ / D PBF. Theorem 10.22.

m

Proof. (i) By Proposition 10.11 (iii), ‚˛;1 D ‚˛;m , and the claim follows from Theorem 10.21. b ˛;1 D ‚˛;1 . (ii) This immediately follows from (i) since ‚ b ˛;1 ı ‚˛ˇ ;ˇ . Since the (iii) Recall from Proposition 10.11 (ii) that ‚˛;ˇ D ‚ various transformations ‚ commute, we find for every m 2 N, m m m m b ˛;1 bm D‚ ı ‚˛ˇ;ˇ D ‚˛ˇ ‚˛;ˇ ;ˇ ı ‚˛;1 ; m b m /. Hence, /  R.‚ implying that R.‚˛;ˇ ˛;1 1 \

m R.‚˛;ˇ /

mD1

1 \

m b˛;1 R.‚ / D PBF:

mD1

The other direction is simpler. For 0  s  1, we see that Z 1 Z 1 ˇ t s #˛;ˇ .t / dt D c˛;ˇ t s˛1 e t dt < 1: c.s/ D 0

0

Hence, by Proposition 10.6 (iii), ‚˛;ˇ .PBF/ D PBF. Since BF  PBF, it follows that R.‚˛;ˇ / D ‚˛;ˇ .BF/  ‚˛;ˇ .PBF/ D PBF. By induction we get that m R.‚˛;ˇ /  PBF for every m 2 N. Therefore, 1 \

m R.‚˛;ˇ /  PBF:

mD1

Remark 10.23. Theorem 10.22 remains valid for ˛ D 0 if we replace PBF by the family PBF 0 . Comments 10.24. The study of representations of infinitely divisible distributions in Rd by means of stochastic integrals of deterministic functions was started by Wolfe in [373]. He considered an Ornstein–Uhlenbeck stochastic differential equation dX t D  X t d Y t driven by a centered one-dimensional Lévy process Y D .Y t / t0 t!1

and showed that X t ! X (in law) if, and only if, EŒlogC jY1 j < 1, in which case X has self-decomposable distribution on R. In this way he could determine the

Chapter 10 Transformations of Bernstein functions

157

R1 domain and the range of the transformation law.Y1 / 7! law. 0 e  t d Y t /. This relation of self-decomposable distributions to Ornstein–Uhlenbeck processes and stochastic integrals driven by a Lévy process was discovered around the same time (or slightly later) by Jurek and Vervaat, [193], for Banach space valued random variables, Sato and Yamazato, [319], for an Ornstein–Uhlenbeck process in Rd , and Gravereaux [135], cf. [305] for a more R 1detailed historical account. Jurek [191] introduced another integral transformation, 0 t d Y t (the Jurek transformation), showed that its domain is the family of all infinitely divisible distributions and that its range is the class U of s-self-decomposable distributions which had been introduced slightly earlier. The interest in integral representations of various subfamilies of ID distributions was revived in the last ten years, starting in [20] and culminating with the definitive [317]. The focus was to determine the domains, ranges and limits of ranges of nested subfamilies. The present chapter is an exposition of these results for the case of distributions on Œ0; 1/. Although in many respects simpler, the presentation encompasses most of the main features of the multivariate case. This comes from the fact that the ranges of integral transformations are described in terms of the radial part of the d -dimensional Lévy measure, cf. [20, Lemma 2.1] for the polar decomposition of the Lévy measure. Lemma 10.1 and Remark 10.2 are quite standard. In a similar context they apb ˛;p and ‚˛;p pear already in Lukacs [250]. The families of transformations ‚˛;ˇ , ‚ have been introduced over the years for certain ranges of parameters and with varying notation. Transformations ‚1;1 (known as ‡ , notation introduced in [22]) and ‚0;1 (denoted by ‰ in [316]) are from [20], ‚1;2 (denoted by G) from [254, 255], and ‚0;2 (denoted by M) from [14]. The family ‚˛;1 is from [316] (denoted by ‰˛ ), ‚˛;˛ from [13] (denoted by E˛ ), and the general ‚˛;ˇ from [253] (denoted by J˛;ˇ ). b 0;1 is the earliest, cf. [135, 193, 319, 373], and is known as ˆ The transformation ‚ b 1;1 is from [191], now known as U. The fam(the notation already used in [305]), ‚ b ily ‚˛;p is introduced in [316] with different parametrization and denoted by ˆˇ ;˛ (ˇ D ˛  p); in [317] the notation ˆp;˛ is used. Finally, ‚˛;p is from [317], denoted by ƒp;˛ . Propositions 10.4 and 10.7 – in the context of Bernstein functions and in this generality – seem to be new; various versions for infinitely divisible distributions on Rd are known. The question of the domain of ‚ is particularly simple here. Various versions of Proposition 10.6 have been proved over the years, cf. [193, 191] for ˆ, [191] for U, [20] for ‡ . If ‚ is one-to-one, then Proposition 10.6 (i) has an obvious converse: if ‚f ./ D ˛ , then f ./ D c.˛/1 ˛ . The fact that any two transformations commute, cf. Proposition 10.10, is particularly simple in the context of Bernstein functions, the formula for the composition seems to be new. Concrete examples have appeared in the literature several b 0;1 D ‚ b0;1 ı ‚1;1 D ‚0;1 was proved in [20], times: for instance, ‚1;1 ı ‚ b b and ‚2;2 ı ‚0;1 D ‚0;1 ı ‚2;2 D ‚0;2 appeared in [13], (10.25) is from [316], (10.23) and (10.26) from [317], while (10.24) seems to be new.

158

Chapter 10 Transformations of Bernstein functions

Theorem 10.12 appears in [20] for ‚1;1 and ‚0;1 , in [255] for ‚1;2 , in [14] for ‚0;2 , in [316] for ‚˛;1 and in [253] for the general ‚˛;ˇ . Theorem 10.15 has a long b 0;1 see [373, 193], for ‚ b 1;1 cf. [191], the general case is discussed history – for ‚ in [316, 317]. Theorem 10.18 is from [317], Corollaries 10.14, 10.17 and 10.19 are simple consequences of the corresponding theorems and can be found in the papers mentioned earlier. Corollary 10.13 is contained in [20] for ˛ D 1, ˇ D 1, in [316] for ˇ D 1 and general ˛, and in Œ253 in the general case. The first systematic study of the limits of nested subfamilies of infinitely divisib 1;1 , ‚ b 0;1 , ble distributions seems to be [256] where Theorem 10.22 is proved for ‚ ‚1;1 , ‚1;2 and ‚0;1 . For earlier results cf. [20, 192, 255]. The general result is b ˛;p and ‚˛;p , and from [257] for ‚˛;ˇ . For Theorem 10.21 from [318] for ‚˛;1 , ‚ we refer to [317].

Chapter 11

Special Bernstein functions and potentials

In the first section of this chapter we will focus our attention on a less known family of Bernstein functions, the special Bernstein functions. This family, denoted by SBF, contains CBF and inherits some of its good properties. The main interest in special Bernstein functions lies in the fact that the potential measure, restricted to .0; 1/, of the convolution semigroup corresponding to a special Bernstein function admits a non-increasing density. On the other hand, a non-increasing function on .0; 1/ need not be a potential density of any convolution semigroup. The best known sufficient condition for this to be true is that the function is non-increasing and log-convex. The Laplace transforms of such functions form the so-called Hirsch class of functions which is between the families S and CM. The Hirsch class is studied in the second part of this chapter.

11.1 Special Bernstein functions It was shown in Proposition 7.1 that f 6 0 belongs to CBF if, and only if, the function  7! =f ./ belongs to CBF. Generalizing this property leads to the larger class of special Bernstein functions which enjoy certain desirable properties. On the other hand, the class itself does not have good structural properties. Definition 11.1. A function f 2 BF is said to be a special Bernstein function if the function f ? ./ WD =f ./ is again a Bernstein function, i.e. f ? 2 BF. We will use SBF to denote the collection of all special Bernstein functions. A subordinator S whose Laplace exponent f belongs to SBF will be called a special subordinator. Note that if f 2 SBF, then also f ? 2 SBF, where f ? ./ WD =f ./. We call f and f ? a conjugate pair of special Bernstein functions. It is clear from Proposition 7.1 that CBF  SBF. We will show later that CBF ¤ SBF ¤ BF, cf. Propositions 11.16, 11.17 and Example 11.18. Remark 11.2. Suppose that f 2 BF. Then =f ./ is a Bernstein function if, and only if, f ./= 2 P. Hence f 2 SBF if, and only if, f 2 BF and f ./= 2 P.

160

Chapter 11 Special Bernstein functions and potentials

Let

Z f ./ D a C b C

.0;1/

.1  e t / .dt /;

Z

f ? ./ D a? C b ?  C

.0;1/

(11.1)

.1  e t / ? .dt /

be representations of f and its conjugate f ? . Then, using the convention 8 0;  D a? D lim f ? ./ D lim 1 : R !0 !0 f ./ ; a D 0; bC .0;1/ t .dt/ ´ 0; b > 0; f ? ./ 1 ? D lim D b D lim 1  !1 !1 f ./ ; b D 0: aC.0;1/

(11.2) 1 1

D 0, (11.3)

(11.4)

The following theorem gives a characterization of a special subordinator in terms of its potential measure. Roughly speaking, it says that a subordinator is special if, and only if, its potential measure restricted to .0; 1/ has a non-increasing density. Theorem 11.3. Let S be a subordinator with potential measure U . Then S is special if, and only if, (11.5) U.dt / D cı0 .dt / C u.t / dt some c  0 and some non-increasing function u W .0; 1/ ! .0; 1/ satisfying Rfor 1 u.t / dt < 1. 0 Remark 11.4. (i) In general, the measure U given by (11.5) – with c  0 and a nonR1 increasing function u satisfying 0 u.t / dt < 1 – need not be the potential measure of any subordinator. For example, if U.dt / D u.t / dt with u.t / D 1 ^ e 1t , then L .U I / D

1 C   e  : .1 C /

If U were the potential measure of a subordinator, then f ./ WD 1=L .U I / would be a Bernstein function, see (5.22). A direct calculation shows that, for example, f 00 .1/ > 0, hence f 62 BF. (ii) Let f 2 BF be the Laplace exponent of the subordinator S . Theorem 11.3 can be equivalently stated in the following way: f 2 SBF if, and only if, Z 1 1 DcC e t u.t / dt (11.6) f ./ 0 for some c  0 and some non-increasing function u W .0; 1/ ! .0; 1/ satisfying R1 0 u.t / dt < 1.

161

Section 11.1 Special Bernstein functions

Proof of Theorem 11.3. Suppose that S is a special subordinator, and let f be its Laplace exponent. Then f and its conjugate f ? have representations (11.1) and (11.2) where the coefficients a? and b ? are given by (11.3) and (11.4). Define u.t / WD a? C ? .t; 1/;

t > 0;

(11.7)

and note that

Z 1 f ? ./ a? 1 ? D D Cb C e t ? .t; 1/ dt L .U I / D f ./   Z 1 0 e t u.t / dt: D b? C 0

D b ? ı0 .dt / C u.t / dt , with R1 and that 0 u.t / dt < 1.

This shows that U.dt / u given by (11.7). It is clear that u is non-increasing Conversely, suppose that (11.5) holds with a density u W .0; 1/ ! .0; 1/ which is R1 non-increasing and satisfies 0 u.t / dt < 1. Then Z 1 1 D L .U I / D c C e t u.t / dt: f ./ 0 Let be the measure on .0; 1/ defined by .t; 1/ D u.t /  u.1/, where we define u.1/ D lim t!1 u.t /. Then Z Z Z 1Z Z s s .ds/ D dt .ds/ D

.ds/ dt .0;1

.0;1

0

Z D

R

0

0

.t;1 1

 u.t /  u.1/ dt < 1

by the assumption on u. Hence, .0;1/ .1 ^ s/ .ds/ < 1. It follows from (11.6) by Fubini’s theorem that Z 1 Z 1    D c C e t u.1/ dt C e t u.t /  u.1/ dt f ./ 0 0 Z 1 D c C u.1/ C e t .t; 1/ dt 0 Z .1  e t / .ds/: (11.8) D c C u.1/ C .0;1/

Therefore, =f ./ is a Bernstein function and S is a special subordinator. Remark 11.5. In case c D 0, we will call u the potential density of the subordinator S or of the Laplace exponent f . The following remark provides yet another characterization of complete Bernstein functions, this time in terms of the corresponding potential measures.

162

Chapter 11 Special Bernstein functions and potentials

Remark 11.6. Let S be a subordinator with Laplace exponent f and potential measure U . Then f is a complete Bernstein function if, and only if, U restricted to .0; 1/ has a completely monotone density u. This follows immediately from Theorem 7.3 and Theorem 2.2. If we compare the expressions (11.2) and (11.8) for =f ./, and use formulae (11.3) and (11.4), we get immediately ?

cDb D u.1/ D a? D

´ 0;

b > 0;

1 ; aC.0;1/

b D 0;

8 0; 1

R

.0;1/

t .dt/

;

a D 0;

u.t / D a? C ? .t; 1/:

(11.9)

In particular, it cannot happen that both a and a? are positive, or that both b and b ? are positive. Moreover,Rit is clear from the definition of a? and b ? that a? > 0 if, and only if, a D 0 and .0;1/ t .dt / < 1, and b ? > 0 if, and only if, b D 0 and .0; 1/ < 1. The following two corollaries follow immediately, if we combine Theorem 11.3 and Remark 3.3 (v) with the formulae for a? ; b ? and (11.9). Corollary 11.7. Suppose that S D .S t / t0 is a subordinator whose Laplace exponent Z .1  e t / .dt / f ./ D a C b C .0;1/

is a special Bernstein function with b > 0 or .0; 1/ D 1. Then the potential measure U of S has a non-increasing density u satisfying Z t s du.s/ D 0: (11.10) lim t u.t / D 0 and lim t!0

t!0 0

Corollary 11.8. Suppose that S D .S t / t0 is a special subordinator with the Laplace exponent given by Z .1  e t / .dt / f ./ D a C .0;1/

where  satisfies .0; 1/ D 1. Then f ? ./ WD

 D a? C f ./

Z .0;1/

.1  e t / ? .dt /

where the Lévy measure ? satisfies ? .0; 1/ D 1.

(11.11)

163

Section 11.1 Special Bernstein functions

Let T be a subordinator with Laplace exponent f ? . If u and v denote the potential densities of S and T , respectively, then v.t / D a C .t; 1/:

(11.12)

In particular, a D v.1/ and a? D u.1/. Assume that f is a special Bernstein function of the form (11.1) where b > 0 or .0; 1/ D 1. Let S be a subordinator with Laplace exponent f , and let U denote its potential measure. By Corollary 11.7, the measure U has a non-increasing density u W .0; 1/ ! .0; 1/. Let T be a subordinator with Laplace exponent f ? ./ D =f ./ and let V denote its potential measure. Then V .dt / D bı0 .dt / C v.t / dt where v W .0; 1/ ! .0; 1/ is a non-increasing function. If b > 0, the potential measure V has an atom at zero, and hence the subordinator T is a compound Poisson process. If b D 0, we require that .0; 1/ D 1, and then, by Corollary 11.8, f ? has the same structure as f , namely b ? D 0 and ? .0; 1/ D 1. In this case, the subordinators S and T play symmetric roles. The defining property of special Bernstein functions says that the identity function is factorized into the product of two Bernstein functions. The following theorem is the counterpart of this property in terms of potential densities. Theorem 11.9. Let f be a special Bernstein function with representation (11.1) satisfying b > 0 or .0; 1/ D 1. Then Z t Z t b u.t / C u.s/v.t  s/ ds D b u.t / C v.s/u.t  s/ ds D 1; t > 0: (11.13) 0

0

Proof. Since for all  > 0 we have 1 D L .uI /; f ./

f ./ D b C L .vI /; 

we get after multiplying 1 D bL .uI / C L .uI /L .vI /  D bL .uI / C L .u ? vI /: By Laplace inversion we see Z 1 D b u.t / C

t 0

u.s/v.t  s/ ds;

t > 0:

Our next goal is to describe a sufficient condition on the Lévy measure of the subordinator which guarantees that the subordinator is special. Since logarithmic convexity will play a major role, we recall the definition.

164

Chapter 11 Special Bernstein functions and potentials

Definition 11.10. (i) A function f W .0; 1/ ! .0; 1/ is said to be logarithmically convex (or log-convex) if log f is convex. (ii) A sequence .an /n0 of non-negative real numbers is logarithmically convex (or log-convex) if an2  an1 anC1 for all n 2 N. Let f be a Bernstein function with the representation (11.1). It will be convenient to extend the measure  onto .0; 1 by defining ¹1º D a. Let .x/ N WD .x; 1, x > 0, be the tail of . Then one can write, as in Remark 3.3 (ii), Z 1 f ./ D b C  e t .t N / dt: (11.14) 0

Theorem 11.11. Suppose that f is given by (11.14) and that x 7! .x/ N is log-convex on .0; 1/. (i) If b > 0 or .0; 1/ D 1, then the potential measure U has a non-increasing density u. (ii) If b D 0 and .0; 1/ < 1, then the restriction U j.0;1/ has a non-increasing density u. In both cases, f 2 SBF. Before we give the proof of the theorem we note some immediate consequences. Corollary 11.12. R 1 Suppose that u W .0; 1/ ! .0; 1/ is non-increasing, log-convex and satisfies 0 u.t / dt < 1, and let c  0. Then the measure U defined by U.dt / D cı0 .dt / C u.t / dt is the potential measure of a special subordinator S D .S t / t0 . ? ? ? Proof. Define a measure ? on .0; WD u.C1/. R 1/ by ? .t; 1/ WD u.t /a with a ? ? Then  is a Lévy measure, i.e. .1 ^ t /  .dt / < 1, and the tail  .t; 1 D u.t / is log-convex. R Define f ? ./ WD a? Cb ?C .0;1/ .1e t / ? .dt /, where b ? D c. By Theorem 11.11, f ? is a special Bernstein function. Let f ./ WD =f ? ./ with corresponding special subordinator S D .S t / t0 . It follows from Theorem 11.3 and (11.9) that the potential measure of S is given by

b ? ı0 .dt / C a? C ? .t; 1/ dt D cı0 .dt / C u.t / dt D U.dt /: Hence, U is the potential measure of the special subordinator S .

165

Section 11.1 Special Bernstein functions

Remark 11.13. As a consequence of the above corollary, we can conclude that the Laplace transform L U 2 P. In this form, Corollary 11.12 is due to [164]. See also [39, Corollary 14.9], and [180, 181]. Corollary 11.12 will be strengthened in the next section, see Theorem 11.24. In order to prove Theorem 11.11 we need two preliminary lemmas. Lemma 11.14. Let .vn /n0 be a sequence satisfying v0 D 1 and 0 < vn  1, n 2 N. Assume that .vn /n0 is a log-convex sequence. Then there exists a non-increasing sequence .rn /n0 such that rn  0 for all n  0, and 1D

n X

for all n  0:

rj vnj

(11.15)

j D0

Proof. Note that the inequality vn2  vn1 vnC1 is equivalent to vn =vn1  vnC1 =vn . Therefore, the sequence .vn =vn1 /n2N is non-decreasing. Define f1 WD v1 and recursively fn WD vn 

n1 X

fj vnj ;

n D 2; 3; : : :

(11.16)

j D1

We claim that fn  0 for all n 2 N. This is clear for f1 . Assume that fk  0 for 1  k  n. Then X vnC1j vnC1 D vn  fj vnj vn vnj j D1 ! n X vnC1  fj vnj vn  vn n

fnC1

j D1

D 0; where the inequality follows since .vn =vn1 /n2N is non-decreasing; the last equality comes from and the fact that v0 D 1. P1 P (11.16) n n v z and F .z/ WD Let V .z/ WD 1 nD0 n nD1 fn z . Since .vn /n0 is a bounded sequence and fn  vn for all n 2 N, we Pnget that V .z/ and F .z/ are well defined for z 2 .0; 1/. The renewal equation vn D j D1 fj vnj implies V .z/ D 1CF .z/V .z/, hence V .z/  1 F .z/ D  1: V .z/ P By letting z ! 1, it follows that 1 nD1 fn  1.

166

Chapter 11 Special Bernstein functions and potentials

P Define the sequence .rn /n0 by r0 WD 1 and rn WD 1  jnD1 fk , n  1. Clearly, .rn /n0 is a non-increasing sequence of non-negative numbers. Since rn1 rn D fn for all n  1, we see vn D

n X

fj vnj D

j D1

D D

n X

.rj 1  rj / vnj

j D1 n X j D1 n1 X

rj 1 vnj  rj vn1j 

j D0

n X j D1 n X

rj vnj

rj vnj :

j D1

P P This implies that jnD0 rj vnj D jn1 D0 rj vn1j for all n  1. But for n D 1 we P Pn1 have that j D0 rj vn1j D r0 v0 D 1. This proves that jnD0 rj vnj D 1 for all n  0. Lemma 11.15. Suppose that x 7! .x/ N is absolutely continuous on .0; 1/. If the total mass .0; 1/ D 1 or if b > 0, then the potential measure U is absolutely continuous. If the total mass .0; 1/ < 1 and b D 0, then U j.0;1/ is absolutely continuous. Proof. Assume first that a D 0. If b > 0, then it is well known that U is absolutely continuous, see e.g. [47, Section III.2, Theorem 5]. Assume that .0; 1/ D 1. Since .x/ N is absolutely continuous, by [314, Theorem 27.7] the transition probabilities of S are absolutely continuous and therefore U is absolutely continuous. If a > 0, then the potential measure of the killed subordinator is equal to the apotential measure of the (non-killed) subordinator, hence again absolutely continuous, see e.g. [314, Remark 41.12]. Assume now that .0; 1/ < 1 and b D 0. Since x 7! .x/ N is absolutely continuous, we have .dx/ D m.x/ dx for some function m. Let c WD .0; 1/. The transition probability at time t of the non-killed subordinator is given by t D

1 X

e tc

kD0

t k ?k  : kŠ

Here and in the rest of this proof ?k stands for the k-fold convolution of , and ?0 D ı0 . Therefore, the potential measure of the killed subordinator is equal to Z 1 e at  t dt U D 0 ! Z 1 1 k X t e at e tc ?k dt D kŠ 0 kD0

167

Section 11.1 Special Bernstein functions

Z 1 X ?k 1 .aCc/t k D e t dt kŠ 0 kD0 ?k 1  1 X  1 ı0 C : D aCc aCc aCc kD1

This shows that U j.0;1/ is absolutely continuous with the density ?k 1  1 X m .x/: u.x/ D aCc aCc kD1

Proof of Theorem 11.11. The log-convexity of .x/ N implies that .x/ N is absolutely continuous on .0; 1/. By Lemma 11.15 we know that the density of U exists if .0; 1/ D 1 or b > 0. We choose a version of u such that lim sup h!0

U.x; x C h/ D u.x/ h

for all x > 0:

(11.17)

Because of the log-convexity, N is strictly positive everywhere. This excludes the case where a C .x; 1/ D 0 for some x > 0. Fix c > 0 and define a sequence .vn .c//n0 by b=c C .c/ N .nc N C c/ v0 .c/ WD D 1; vn .c/ WD ; n  1: b=c C .c/ N b=c C .c/ N Clearly 0 < vn .c/  1 for all n  0. Moreover, vn .c/2  vn1 .c/vnC1 .c/ for all n  1. Indeed, for n  2 this is equivalent to     .nc N C c/2  N .n  1/c C c  N .n C 1/c C c which is a consequence of the log-convexity of . N For n D 1 we have   .2c/ N  .c/ N .3c/ N  b=c C .c/ N .3c/: N By Lemma 11.14, there exists a non-increasing sequence .rn .c//n0 such that n X

rj .c/vnj .c/ D 1 for all n  0:

(11.18)

j D0

Define

rn .c/ ; n  0: b=c C .c/ N Inserting this expression into (11.18) we get for all n  0 un .c/ WD

X   b un .c/ C uj .c/N .n  j /c C c D 1: c n

j D0

(11.19)

168

Chapter 11 Special Bernstein functions and potentials

Multiplying (11.19) by ce .nC1/c and summing over all n  0, we obtain b

1 X

e .nC1/c un .c/ C

n 1 X X

  ce .nC1/c uj .c/N .n  j /c C c

nD0 j D0

nD0

D

1 X

ce .nC1/c :

nD0

This can be simplified to be

c

1 X

e

nc

un .c/ C

nD0

1 X

! e

nc

un .c/

c

nD0

1 X

! e

nc

.nc/ N

nD1

D

ce c : 1  e c

P Define a measure Uc on .0; 1/ by Uc WD 1 nD0 un .c/ ınc . Then the above equation becomes   Z 1 1 X ce c t c nc e d Uc .t / be C ce .nc/ N D : 1  e c 0 nD1 Let c # 0. The right-hand side converges to 1, while   Z 1 X ce nc .nc/ N D b C lim be c C c#0

nD1

Z

Therefore lim

c#0 Œ0;1/

e t d Uc .t / D

1 D f ./

1 0

e t .t N / dt D f ./:

Z

e t d U.t / Œ0;1/

which means that Uc converges vaguely to U . Since U is absolutely continuous, this implies that for all x > 0 and all h > 0, Z xCh u.t / dt: lim Uc .x; x C h/ D c#0

x

Now suppose that 0 < x < y and choose h > 0 such that x < x C h < y. Moreover, let c be such that none of the points x; x C h; y; y C h is an integer multiple of c. By the monotonicity of .un .c//n0 , it follows that Uc .y; y C h/  Uc .x; x C h/: Let c go to zero along values such that the points x; x C h; y; y C h are not integer multiples of c. It follows that U.y; y C h/  U.x; x C h/ and (11.17) guarantees that u.y/  u.x/ for all x < y.

169

Section 11.1 Special Bernstein functions

We give now two sufficient conditions such that a Bernstein function is not special. R Proposition 11.16. Let f ./ D b C .0;1/ .1  e t / .dt /, b  0, be a Bernstein function. If  has bounded support, then f cannot be special. Proof. Assume that f 2 SBF and .t0 ; 1/ D 0 for some t0 > 0. Let T D .T t / t0 be the subordinator with Laplace exponent f ? ./ WD =f ./. Denote by V the potential measure of T and by v the density of V j.0;1/ ; then v.t / D .t; 1/ D 0 for all t  t0 . But this implies V .t0 ; 1/ D 0 which is impossible. R Proposition 11.17. Let f ./ D a C .0;1/ .1  e t / .dt /, a  0, be a Bernstein function. Assume that the Lévy measure  is nontrivial and that .0; t0  D 0 for some t0 > 0. Then f is not special. Proof. Suppose, on the contrary, that f is a special Bernstein function. Then the conjugate function f ? ./ D =f ./ is of the form (11.2). The potential density of the corresponding subordinator T is given by the formula v.t / D a C .t; 1/. In particular, v.t / D a C .t0 ; 1/ DW for 0 < t < t0 . It follows from (11.4) that b ? D 1= > 0. On the other hand, from Theorem 11.9 (with the roles of u and v reversed), we have that Z t u.s/v.t  s/ ds D 1 for all t > 0: b ? v.t / C 0

For 0 < t < t0 we have v.t / D , so Z t u.s/ ds D 1; 1C

0 < t < t0 :

0

Rt It follows that 0 u.s/ ds D 0 for all 0 < t < t0 , and since u is non-increasing we have u  0. Thus U.dt / D b ? ı0 , implying that f D 1=b ? . But this contradicts the assumption that the Lévy measure  is not trivial. The last two propositions clearly show that SBF is strictly smaller than BF. A typical example of a Bernstein function which is not special is f ./ D 1  e  . The corresponding subordinator is a Poisson process with rate > 0. The next example shows that the family SBF is strictly larger than CBF. Example 11.18. (i) Define ´ v.t / WD

t ˛ ; 0 < t < 1; t ˇ ; 1  t < 1

170

Chapter 11 Special Bernstein functions and potentials

where 0 < ˇ < ˛ < 1. Obviously, v is non-increasing, log-convex and satisfies R1 0 v.t / dt < 1. By Corollary 11.12, v is the potential density of a special subordinator. Since v is not completely monotone – it is clearly not C 1 –, the corresponding Laplace exponent is not a complete Bernstein function. (ii) For 0 < ˛ < 1 define ´ t ˛ ; 0 < t < 1; v.t / WD 1; 1  t < 1: R1 Again, v is non-increasing, log-convex and satisfies 0 v.t / dt < 1. Hence, there exists a special subordinator T D .T t / t0 with potential measure V such that v is the of T , and define f ./ WD =f ? ./. density of V . Let f ? be the Laplace exponent R Since v.1/ D 1, we have that f ./ D 1C .0;1/ .1e t / .dt / with Lévy measure .dt / D m.t / dt , where ´ ˛ t ˛1 ; 0 < t < 1; m.t / WD 0; 1  t < 1: Note that the Lévy measure  has bounded support. This does not contradict Proposition 11.16 since we have a non-zero killing term of f equal to 1. This example shows that for f 2 BF it may happen that  7! a C f ./ 2 SBF for some a > 0, while f … SBF. (iii) This example is similar to the previous one, but the Lévy measure is finite. Let ´ e 1t ; 0 < t < 1; v.t / WD 1; 1  t < 1: R1 Again, v is non-increasing, log-convex and satisfies 0 v.t / dt < 1. Hence, there exists a special subordinator T D .T t / t0 with potential measure V such that v is the density of V . Let f R? be the Laplace exponent of T , and define f ./ WD =f ? ./. Then f ./ D 1 C .0;1/ .1  e t / .dt / with Lévy measure .dt / D m.t / dt , where ´ e 1t ; 0 < t < 1; m.t / WD 0; 1  t < 1: Remark 11.19. In the third example above, the density of the Lévy measure .x/ is discontinuous at x D 1. Put ´ e x ; 0 < x < 1; f .x/ WD 0; 1  x < 1:

171

Section 11.1 Special Bernstein functions

Clearly,

8 x ˆ 0 < x < 1; 0. (iii) f 2 SBF if, and only if,

tf tCf

2 SBF for all t > 0.

(iv) f 2 SBF if, and only if,  7! f . C c/ 2 SBF for all c > 0. Proof. (i) follows immediately from the fact that BF is a cone which is closed under pointwise convergence. (ii) Let f 2 SBF and b > 0. Define h./ WD

 : 1 C b

Then h 2 BF, and by the composition result for Bernstein functions, cf. Corollary 3.8 or Theorem 5.27, it follows for the conjugate f ? ./ D =f ./ that h ı f ? ./ D

f ? ./  D 2 BF: 1 C bf ? ./ b C f ./

172

Chapter 11 Special Bernstein functions and potentials

Since  7! b C f ./ 2 BF, we know that  7! b C f ./ 2 SBF, too. The converse statement follows from (i) by letting b ! 0. (iii) Let f 2 SBF and t > 0. As in (ii), tf =.t C f / 2 BF. Further,  tf ./ tCf ./

D

  t  C f ./ D C 2 BF; tf ./ f ./ t

implying that tf =.t C f / 2 SBF. The converse follows from (i) by letting t ! 1. (iv) First note that if f 2 BF, g 2 CM and f  g  0, then f  g 2 BF which is easily seen from the definition of BF and CM. Let now f 2 SBF and c > 0. Then Cc c  D   0: f . C c/ f . C c/ f . C c/ Since  7! . C c/=f . C c/ 2 BF and  7! c=f . C c/ 2 CM, we conclude that =f . C c/ is a Bernstein function which proves the claim. The converse follows by letting c ! 0. Remark 11.21. We do not know whether SBF is a convex cone. It is even unclear whether a C f 2 SBF, a > 0, for f 2 SBF. Let f 2 SBF. Then a C f 2 SBF if, and only if,  7! .a C f .//= 2 P. We can also use potential densities to characterize when a C f 2 SBF. Assume, for simplicity, that f 2 SBF is of the form Z 1 f ./ D 

0

e t .t N / dt:

Then v.t / D .t N / is the potential density of the conjugate function f ? ./ D =f ./. For a > 0, Z 1 a C f ./ D e t .v.t / C a/ dt;  0 showing that a C f 2 SBF if, and only if, v.t / C a is a potential density.

11.2 Hirsch’s class In this section we describe another family of completely monotone functions which was introduced by F. Hirsch in [164]. It turns out that this class, H, is a rich intermediate class between the Stieltjes functions S on the one side, and the completely monotone functions of the form 1=f where f 2 SBF on the other side. In some sense H is the largest known family of concrete examples for potentials. Definition 11.22. We call a function f 2 CM a Hirsch function if for all  > 0 the sequence   .1/n .n/ f ./ nŠ n0

173

Section 11.2 Hirsch’s class

is log-convex in the sense of Definition 11.10. The set of all Hirsch functions will be denoted by H. Proposition 11.23. The set H is a convex cone closed under pointwise convergence. Proof. The convexity of H follows from the fact that log-convex sequences form a convex cone. Indeed, let .an /n0 and .bn /n0 be two log-convex sequences. Then it holds for all n  1 that an2  an1 anC1 and bn2  bn1 bnC1 . Therefore .an C bn /2 D an2 C bn2 C 2an bn

p  an1 anC1 C bn1 bnC1 C 2 an1 anC1 bn1 bnC1  an1 anC1 C bn1 bnC1 C an1 bnC1 C anC1 bn1 D .an1 C bn1 /.anC1 C bnC1 /:

Since complete monotonicity and log-convexity are preserved under pointwise convergence, H is closed under pointwise convergence. Theorem 11.24. R 1 Suppose that u W .0; 1/ ! .0; 1/ is non-increasing, log-convex and satisfies 0 u.t / dt < 1, and let c  0. Define a measure U by U.dt / D cı0 .dt / C u.t / dt:

(11.20)

Then L U 2 H. Conversely, every f 2 H is the Laplace transform of a measure U of the form (11.20). Proof. Suppose that U is given by (11.20) and that the function u is twice continuously differentiable. For every  > 0, the function g defined by g .s/ D e s u.s/;

s > 0;

is also logarithmically convex and of class C 2 , hence .g0 /2  g g00 : Z

Let f ./ WD L .uI / D

1

e s u.s/ ds;

 > 0:

0

Then for n  1, .1/n .n/ f ./ D nŠ

Z

1 0

sn g .s/ ds D  nŠ 

Z

1 0

s nC1 g 0 .s/ ds: .n C 1/Š 

174

Chapter 11 Special Bernstein functions and potentials

As g0  

q g g00 , we find by the Cauchy–Schwarz inequality that

2 Z 1 nC1 Z 1 nC1 s s .1/n .n/ f ./  g .s/ ds g 00 .s/ ds nŠ .n C 1/Š .n C 1/Š  0 0 Z 1 nC1 Z 1 n1 s s g .s/ ds g .s/ ds D .n C 1/Š .n  1/Š  0 0    .1/nC1 .nC1/ .1/n1 .n1/ f f D ./ ./ ; .n C 1/Š .n  1/Š

showing that f 2 H. Every non-increasing log-convex function u W .0; 1/ ! .0; 1/ is the supremum of a sequence of C 2 functions un W .0; 1/ ! .0; 1/ which are nonincreasing and log-convex. For such functions we have proved that L un 2 H. By the monotone convergence theorem it follows that L un increases pointwise to L u. By Proposition 11.23 it follows that L u 2 H. Since positive constants are in H, and since H is a convex cone, we see that L U 2 H. Now we prove the converse statement. Let f 2 H  CM and define fn ./ WD

.1/n  n nC1 .n/  n  ; f nŠ  

 > 0;

(compare with (1.8)). Then fn is C 1 and strictly positive. We are going to prove that it is log-convex, by showing that .log fn /00 ./ > 0. We first compute   nC1 2n f .nC1/ n   .log f / ./ D C 3 2  f .n/ n  n 2 f .nC2/  n   n   C 2 .n/  f  00

  !2 n f .nC1/ n   : 2 f .n/ n

By the inequality f .n/

n 

f .nC2/

n 



n C 2  .nC1/  n 2 f ; nC1 

we obtain     !2 n f .nC1/ n 1 nC1 2n f .nC1/ n   C   .log f / ./  C 3 2  f .n/ n n C 1 2 f .n/ n !2   p n C 1 f .nC1/ n n   C D  0: p  f .n/ n 2 n C 1 00

175

Section 11.2 Hirsch’s class

The next step is to show that fn is non-increasing. Since f 2 CM, there exists a measure U on Œ0; 1/ such that f D L U . Assume that U is bounded. Differentiating n times under the integral sign yields Z .n/ .1/n e t U.dt /: f ./ D Œ0;1/

Using the elementary inequality e t .t /n  .n=e/n , ; t  0, and dominated convergence we see lim f .n/ ./nC1 D 0: !0

Hence

.1/n .n/  n   n nC1 f D 0:   !1 nŠ

lim fn ./ D lim

!1

Since fn is convex, this implies that it is non-increasing. For a general measure U , R pick c > 0 and set fc ./ WD f . C c/ D Œ0;1/ e t e ct U.dt /. Then fc 2 H with a bounded measure e ct U.dt /. By the proof above, fc;n is non-increasing. As fn ./ D limc!0 fc;n ./,   0, it follows that fn is non-increasing. Let a WD lim!1 f ./. Then U D aı0 C U0 where U0 D U j.0;1/ . Moreover, if fQ WD f  a, then the proof of Theorem 1.4 shows that fQn .x/ dx converges weakly to U0 .dx/ as n ! 1. Since fn D fQn , n  1, we get that fn .x/ dx converges weakly to U0 .dx/. Since fn0  0 and fn00  0, it follows that DU0  0 and D 2 U0  0 where we take derivatives in the distributional sense. Hence, there exists a convex nonincreasing function u W .0; 1/ ! .0; 1/ such that U0 .dx/ D u.x/ dx. In particular, u is continuous. By the result in [370, p. 288] it follows that limn!1 fn .x/ D u.x/ for all x > 0. Since the pointwise limit of log-convex functions is again log-convex, this finishes the proof. We record several simple consequences of the preceding theorem. Corollary 11.25.

(i)

1 H

 SBF

and

H  P.

(ii) If h 2 H, then h./ 2 SBF. (iii) If f 2 BF and N is log-convex, then f ./= 2 H. (iv) S  H. Proof. (i) Let g 2 H . By Theorem 11.24, g D L U for a measure U of the form (11.20). By Corollary 11.12, U is the potential measure of a special subordinator, i.e. L U D 1=f for f 2 SBF. Thus, 1=g D f 2 SBF and, in particular, g 2 P. (ii) Let f WD 1= h. By (i), f 2 SBF, and therefore h./ D =f ./ 2 SBF.

176 (iii) By (11.14) we have Z f ./ D b C 

Chapter 11 Special Bernstein functions and potentials

1 0

  e t .t N / dt D  L bı0 .dt / C .t N / dt I 

which proves the claim. (iv) Let g 2 S. By Theorem 7.3, g D 1=f where f 2 CBF. By Remark 11.6, the potential measure U of f is of the form U.dt / D cı0 .dt / C u.t / dt with u 2 CM. Since completely monotone functions are logarithmically convex, Theorem 11.24 yields L U 2 H. But L U D g. Corollary 11.26. CBF ı H  CM Proof. Let h 2 H. We know from Corollary 11.25 (i) that h D Consider for every t > 0 1

h f D t Ch tC

1 f

D

1 f

for some f 2 BF.

1 : tf C 1

Since  7! tf ./ C 1 is a Bernstein function, we find 1 h 2  CM: t C h BF Since CM is a closed convex cone, we conclude that Z h  .dt / 2 CM .0;1/ t C h whenever the integral exists. Using the Stieltjes representation (6.7) of a complete Bernstein function, we find that g ı h 2 CM for all g 2 CBF. Corollary 11.27. Assume that .dx/ WD cı0 .dx/Cp.x/ dx is a probability measure where c  0 and p.x/ is a non-increasing, log-convex density. Then  2 ID. Proof. By Theorem 11.24 we see that L  2 H. By Corollary 11.26 we get that .L /˛ 2 CM for all 0 < ˛ < 1 and this means that L  is infinitely divisible, i.e.  2 ID, see Definition 5.6 and Remark 5.7. Remark 11.28. In [39] there is a proof of Corollary 11.25 (i), H  P, which does not rely on Theorems 11.11 and 11.24. By using this fact and Theorem 11.24 one can give an alternative proof of Theorem 11.11. R Indeed, suppose that f 2 BF, f ./ D a C b C .0;1/ .1  e t / .dt / and that N /. N is log-convex. Define the measure V .dt / WD bı0 .dt / C v.t / dt where v.t / D .t Then L V 2 H  P, and f ./ D L V ./. Therefore, =f ./ D 1=L V ./ 2 BF, proving that f 2 SBF.

177

Section 11.2 Hirsch’s class

Remark 11.29. (i) It is shown in Remark 11.19 that there exists a special Bernstein function f such that the density u of U j.0;1/ is not continuous. This shows that 1=H is strictly contained in SBF. (ii) Suppose that f 2 SBF such that

f 

2 H. Let a > 0. Then

f ./ a f ./ C a D C 2 H  P;    implying that =.f ./ C a/ 2 BF, i.e. f C a 2 SBF. Compare with Remark 11.21. At the end of this section we point out a unified way to describe some of the considered classes of functions. Consider the family ®

¯ f W f D L U; U.dt / D cı0 C u.t / dt  CM;

R1 where c  0 and u W .0; 1/ ! .0; 1/ such that 0 u.t / dt < 1. Different choices of u result in different subfamilies of completely monotone functions. More precisely, (i) u 2 CM if, and only if, L U 2 S; (ii) u is non-increasing and log-convex if, and only if, L U 2 H; (iii) u is non-increasing and is a potential density if, and only if, L U 2

1 SBF ;

(iv) u is non-increasing if, and only if, L .U I / 2 BF. Statements (i), (ii) and (iii) are proved, respectively, in Remark 11.6, Theorem 11.24 and Theorem 11.3. For the last statement note that if u is non-increasing, then it is the tail of a measure , u.t / D .t N /. Thus Z 1 Z 1 t e u.t / dt D c C  e t .t N / dt 2 BF: L .U I / D c C  0

0

The converse is proved similarly. Hence,

1 SBF

CM; 



SH





BF 

1 BF

and all inclusions are strict. For the second inclusion see Remark 11.29 (i), and for the third notice that SBF is strictly contained in =BF and BF, respectively.

178

Chapter 11 Special Bernstein functions and potentials

Comments 11.30. Section 11.1: The term special Bernstein function was introduced in [332], although the notion itself is encountered in the literature earlier. Theorem 11.3 is essentially contained in Bertoin [48, Corollaries 1 and 2] who attributes the result to Van Harn and Steutel [360]. Our presentation follows [332] where the result was rediscovered. Theorem 11.9 also comes from [332]. Special Bernstein functions were used in [235] to characterize the scale functions of one-dimensional Lévy processes having only negative jumps, and in [236] to study de Finetti’s optimal dividend problem. Log-convexity of the tail N of the Lévy measure was studied by J. Hawkes in [153, Theorem 2.1]. Theorem 11.11, cf. [333], is a slight extension as it allows for the killing term, the drift term, and finite Lévy measure. Lemma 11.14 P appears also in [153] with a proof having a minor gap (it works only for the case j1D1 fj D 1). The conclusion of the lemma, equation (11.15), is a discrete version of Chung’s equation, see [86]. Examples 11.18 and Propositions 11.16 and 11.17 are from [333], the proof of the latter is new. One direction of Proposition 11.20 (iv) is proved in [235] where also the corresponding Lévy measure is identified. Section 11.2: The class H was introduced by F. Hirsch in [163] and [164]. The starting point of his investigations is the problem to find a structurally wellbehaved family of functions which operate on the (abstract) potential operators in the sense of Yosida [379, Chapter XIII.9]; these are densely defined, injective operators V such that A D V 1 is the generator of a strongly continuous contraction semigroup, cf. Chapter 13 below. Although the Laplace transform induces a bijection between Hunt kernels on the half-line Œ0; 1/ (leading to potential operators) and the family of potentials P, see e.g. [162, page 175], the set P lacks nice structural properties. Hirsch shows that the family H  P is a convex cone which allows to define a symbolic calculus on (abstract) potential operators along the lines of Faraut’s symbolic calculus [112]. See also the Comments 2.5, 5.29 and 6.22. Theorem 11.24 is from [164, Théorème 2], our proof follows the presentation in Berg and Forst [39, Section 14]. Corollary 11.27 appears, without point mass at the origin, for the first time in Steutel [338, Theorem 4.2.6], see also [339, Theorem III.10.2, p. 117].

Chapter 12

The spectral theorem and operator monotonicity

In this chapter we look at applications of complete Bernstein functions in operator theory on Hilbert spaces. First we give a proof of the spectral theorem for self-adjoint operators on a Hilbert space which uses the Stieltjes representation of complete Bernstein functions. Then we prove Löwner’s theorem which says that the family of complete Bernstein functions coincides with the family of non-negative operator monotone functions on .0; 1/.

12.1 The spectral theorem A surprising consequence of the representation formulae for complete Bernstein functions is a simple proof of the spectral theorem for self-adjoint operators on Hilbert spaces. This was first observed by Doob and Koopman [101] and Lengyel [240]. The paper by Nevanlinna and Nieminen [275] contains a full account of this approach and its extensions. Our exposition follows these lines; the monograph by Kato [200] will be our standard reference for spectral analysis in Hilbert spaces. Let H be a complex Hilbert space with scalar product h; i and norm k  k. We assume that the scalar product is linear in the first, and skew-linear in the second argument. A linear operator A W D.A/ ! H is said to be self-adjoint if D.A/  H is dense and .A; D.A// D .A ; D.A //. The resolvent set %A consists of all  2 C such that the operator   A WD  id A has a bounded inverse R WD .  A/1 ; the operators .R / 2%A are the resolvent of A. It is well known that the resolvent set %A is open. The set A WD C n %A is called the spectrum of A. All self-adjoint operators A have real spectrum A  R and their resolvents satisfy kR uk 

1 1 kuk  kuk; dist.; A / jIm j

 2 %A ; u 2 H:

(12.1)

For ; z 2 %A the following resolvent equation holds R  Rz D .z  /Rz R

(12.2)

which is easily seen from R D R .z  A/Rz D R ..z  / C .  A//Rz . Since  7! hR u; vi is continuous, the resolvent equation tells us

R  Rz u; v D  lim hRz R u; vi D hR R u; vi: (12.3) lim z z!

z!

180

Chapter 12 The spectral theorem and operator monotonicity

This means that hR u; vi, u; v 2 H, is complex differentiable for all  2 %A , hence it is analytic. Recall that a self-adjoint operator .A; D.A// is dissipative if hAu; ui  0 for all u 2 D.A/:

(12.4)

Lemma 12.1. Let .A; D.A// be a self-adjoint operator on H. Then A is dissipative if, and only if, the following holds ku  Auk   kuk

for all u 2 D.A/;  > 0:

(12.5)

Proof. Clearly, k.  A/uk2 D 2 kuk2  2hAu; ui C kAuk2 : If hAu; ui  0 we see that k.  A/uk2  2 kuk2 and (12.5) follows. Conversely, under (12.5) we have 2 kuk2  2hAu; ui C kAuk2  2 kuk2 or 2hAu; ui  kAuk2 . Since  can be arbitrarily large, this is only possible if hAu; ui  0 and (12.4) follows. Proposition 12.2. Let .A; D.A// be a self-adjoint operator on H. Then A is dissipative if, and only if, the spectrum A is contained in .1; 0. Proof. First we suppose that A  .1; 0. Since  > 0 is in the resolvent set, we may use (12.1) with  D  and u D .  A/w, w 2 D.A/, to get kwk D kR .  A/wk 

1 1 kw  Awk  kw  Awk: dist.; A / 

This proves dissipativity. Now we suppose that A is dissipative and we want to show that A  .1; 0. For any  > 0, let v 2 ..A  /.D.A///? . Then v 2 D.A/ and Av D v since A is selfadjoint. Hence kvk2 D hAv; vi  0, which gives v D 0. Thus .A  /.D.A// D H and consequently A  .1; 0. From now on we will assume that A is a dissipative self-adjoint operator. Lemma 12.3. Let .A; D.A// be a dissipative self-adjoint operator on H. For every u 2 H the function f ./ WD hR u; ui,  2 C n .1; 0, is (the extension of) a complete Bernstein function such that f ./ D hR u; ui  kuk2

for all  > 0; u 2 H:

181

Section 12.1 The spectral theorem

Moreover, lim hR u; vi D hu; vi;

!1

u; v 2 H;

(12.6)

and there exists a uniquely determined finite measure E D Eu;u supported in .1; 0 such that Eu;u .1; 0 D kuk2 and Z 1 Eu;u .dt /;  2 C n .1; 0: (12.7) hR u; ui D .1;0   t Proof. It is clear that f ./ is analytic on %A . Note that ˝ ˛ f ./ D .  A C A/R u; u D hu; ui C hAR u; ui:

(12.8)

Setting u D .  A/ for a suitable  D R u 2 D.A/ and  D  C i , we conclude that ˛ ˝ ˛ ˝ N f ./ D R .  A/; .  A/ D ; .  A/ D  h; i  h; Ai: Thus Im f ./ D Im h; Ai  0 for Im   0, and for  D  > 0 we see f ./ D 2 h; i  h; Ai  0: From (12.3) and (12.8) we get d d d f ./ D hAR u; ui D hR u; Aui D hR R u; Aui d d d D hR u; AR ui  0: Therefore, f .0C/ D inf >0 f ./ < 1 exists, and by Theorem 6.2 (iv), f 2 CBF; in particular, Z   .dt / f ./ D a C b C .0;1/  C t R where a; b  0 and  is a measure on .0; 1/ such that .0;1/ .1 C t /1  .dt / < 1. The Cauchy–Schwarz inequality and the resolvent estimate (12.1) show for any self-adjoint operator .A; D.A// on H and u 2 D.A/   1 lim jhARi u; uij  lim kRi Auk  kuk  lim kAuk  kuk D 0: !1 !1 !1  Thus, lim f .i / D hu; ui C lim hARi u; ui D kuk2

!1

!1

and we find from the integral representation of f 2 CBF that Z 2 Re f .i / D a C  .dt /: 2 2 .0;1/  C t

182

Chapter 12 The spectral theorem and operator monotonicity

Monotone convergence now shows that kuk2 D lim f .i / D a C  .0; 1/: !1

On the other hand,

Z Im f .i / D b C

.0;1/

2

t  .dt /; C t2

so that lim!1 f .i / D kuk2 combined with the dominated convergence theorem proves that b D 0. This shows Z   .dt /; a C  .0; 1/ D kuk2 : f ./ D a C .0;1/  C t Moreover,

Z f ./ D a C

.0;1/

  .dt /  a C  .0; 1/ D kuk2 ; Ct

and we can use monotone convergence to get lim hR u; ui D lim f ./ D kuk2 ;

!1

!1

u 2 H:

A standard polarization argument proves then (12.6). To see (12.7), set Eu;u .I / WD ı0 .I / C  .I / for all Borel sets I  .1; 0. The uniqueness of the measure Eu;u follows from the uniqueness of the measure  in the Stieltjes representation of complete Bernstein functions. With a simple polarization argument we can extend (12.7) to Z 1 hR u; vi D Eu;v .dt /;  2 C n .1; 0; u; v 2 H: .1;0   t Since Eu;u is uniquely determined by u, the complex-valued measure Eu;v is uniquely determined by u and v, and Eu;v depends linearly on u and skew-linearly on v; moreover, Eu;v D E v;u is skew-symmetric and Eu;v has total variation less or equal than kuk  kvk. Using the inversion formula (6.6) we can actually work out the measure Eu;v . For ˛ < ˇ  0 we get the Hellinger–Stone formula Eu;v Œ˛; ˇ/ D  .ˇ; ˛   Z h.s  ih/RsCih u; vi 1 ds D lim Im s  ih h!0C  .ˇ;˛ Z 1 D  lim Im hR tCih u; vi dt h!0C  Œ˛;ˇ /

(12.9)

183

Section 12.1 The spectral theorem

at all continuity points ˛; ˇ of  7! Eu;v .1; /. Note that this formula is still valid for ˛; ˇ > 0 with Eu;v Œ˛; ˇ/ D 0; this allows to extend Eu;v Œ˛; ˇ/ trivially by 0 onto the positive real line. If I  .1; 0 is a Borel set, this shows that .u; v/ 7! Eu;v .I / is a skew symmetric sesquilinear form in the Hilbert space H. By a standard argument, see Lengyel [240] or Lax [239, Chapter 32], we deduce that there exists a bounded self-adjoint operator E.I / such that ˝ ˛ Eu;v .I / D E.I /u; v ;

u; v 2 H; I  .1; 0 Borel:

This shows that (12.7) can be written as Z ˛ 1 ˝ hR u; vi D E.dt /u; v ; .1;0   t

 2 C n .1; 0; u; v 2 H:

From this point onwards we can argue as usual: one concludes that .E.1; //2R is a spectral resolution of the operator A and proves the spectral theorem for (dissipative) self-adjoint operators. Theorem 12.4. Let .A; D.A// be a dissipative self-adjoint operator on H. Then there exists an orthogonal projection-valued measure E on the Borel sets of R with support A such that for all Borel sets I; J  R (i) E.;/ D 0, E.R/ D id; (ii) E.I \ J / D E.I /E.J /; (iii) E.I / W D.A/ ! D.A/ and AE.I / D E.I /A; R R (iv) Au D A  E.d /u for u 2 D.A/ D ¹u 2 H W A 2 hE.d /u; ui < 1º. R Note that kAuk2 D A 2 hE.d /u; ui. For ˆ W .1; 0 ! R one can define the function ˆ.A/ of the operator A by Z ˆ./ E.d /u; (12.10) ˆ.A/u WD .1;0 9 8 > ˆ Z = < ˝ ˛   2 2 jˆ./j E.d /u; u < 1 : (12.11) D ˆ.A/ WD u 2 H W kˆ.A/uk D > ˆ ; : .1;0

Example 12.5. Let .A; D.A// be a dissipative self-adjoint operator on H. Then A generates a semigroup of linear operators .T t / t0 which is strongly continuous, i.e. t 7! T t u is continuous from Œ0; 1/ to H, and contractive, i.e. kT t uk  kuk.

184

Chapter 12 The spectral theorem and operator monotonicity

Indeed: Let E.d / be the spectral resolution of the operator A. Then Z tA Tt u D e u D e t E.d /u; u 2 H; t > 0; .1;0

and by Theorem 12.4 we get for all u; v 2 H and all s; t > 0 Z

hT t Ts u; vi D T t e s E.d /u; v .1;0 Z Z ˝ ˛ D e t e s E.d /u; E.d /v .1;0 .1;0 Z ˝ ˛ D e .tCs/ E.d /u; v .1;0

D hTsCt u; vi which shows that .T t / t0 is a semigroup. That t 7! T t u, u 2 H, is continuous in the Hilbert space topology follows directly from the continuity of t 7! e t ,  2 .1; 0, and the fact that Z ˝ ˛ kT t u  uk2 D .e t  1/2 E.d /u; u : .1;0

In a similar way we deduce from the fact that e t  1, t > 0 and  2 .1; 0, that kT t uk  kuk which proves that T t is a contraction operator. Example 12.6. Let .A; D.A// be a dissipative self-adjoint operator on H and denote by .T t / t0 the strongly continuous contraction semigroup generated by A. Let f be a Bernstein function given by the Lévy–Khintchine representation (3.2), Z f ./ D a C b C .1  e t / .dt /; R

.0;1/

^ t / .dt / < 1. Then D.A/  D.f .A// and Z .T t u  u/ .dt /; u 2 D.A/: (12.12) f .A/u D au C bAu C

where a; b  0 and

.0;1/ .1

.0;1/

Indeed: Let E.d / be the spectral resolution of the operator A. The elementary inequality 1 ^ .t /  .1 C /.1 ^ t /, ; t  0, shows that Z .1 ^ t / .dt /  c .1 C / f ./  a C b C .1 C / .0;1/

for some suitable constant c > 0. Therefore Z Z ˝ ˛ 2 jf ./j E.d /u; u  c .1;0

.1;0

˝ ˛ .1  /2 E.d /u; u

185

Section 12.1 The spectral theorem

which means that D.A/  D.f .A//. The representation (12.12) follows now directly from the integral formula (3.2) for f . For u 2 D.A/ we get from Fubini’s theorem Z f ./ E.d /u f .A/u D .1;0   Z Z t .1  e / .dt / E.d /u D a  b C .1;0 Z.0;1/ Z .1  e t / E.d /u .dt / D au  bAu C .0;1/ .1;0 Z .u  T t u/ .dt /: D au  bAu C .0;1/

Example 12.7. Let .A; D.A// be a dissipative self-adjoint operator on H, denote by .R / 2%A its resolvent, and let f be a complete Bernstein function. We use the Stieltjes representation (6.7) Z   .dt /; f ./ D a C b C  C t .0;1/ where a; b  0 and

R

.0;1/ .1

C t /1  .dt / < 1. Then D.A/  D.f .A// and Z

f .A/u D au C bAu C

AR t u  .dt /; .0;1/

u 2 D.A/:

(12.13)

Indeed: Since CBF  BF, the inclusion D.A/  D.f .A// follows from Example 12.6. Let E.d / be the spectral resolution of the operator A. Then Z  E.d /u; u 2 H; t > 0; AR t u D t .1;0   defines a bounded operator. Since ./=.t  /  1 ^ .=t / for   0 and t > 0, we conclude from this that ³ ² 1 kAR t uk  min kuk; kAuk ; t > 0: t From the elementary inequality min¹1; sº  2s=.1 C s/, s  0, we get kAR t uk  2

1

1 kAuk t kuk C 1t kAuk kuk

kuk D 2

t

kAuk kuk C kAuk kuk

kuk:

(12.14)

186

Chapter 12 The spectral theorem and operator monotonicity

Let us now determine f .A/. Z ˝ ˛ f ./ E.d /u; u hf .A/u; ui D .1;0 Z Z ˝ ˛   .dt / E.d /u; u D ahu; uiCbhAu; uiC .1;0 .0;1/ t   Z Z ˛  ˝ D ahu; uiCbhAu; uiC E.d /u; u  .dt / t  Z.0;1/ .1;0 D ahu; uiCbhAu; uiC hAR t u; ui  .dt /: .0;1/

The change in the order of integration is possible since all integrands are non-negative. The estimate (12.14) tells us that for u 2 D.A/ Z kf .A/uk  akuk C bkAuk C kAR t uk  .dt / .0;1/

Z  akuk C bkAuk C 2   2f

 kAuk kuk: kuk

.0;1/

t

kAuk kuk C kAuk kuk

 .dt / kuk

Remark 12.8. Let .A; D.A// be a dissipative self-adjoint operator on H with resolvent .R />0 . By Lemma 12.3 the function  7! hR u; ui is for all u 2 H a complete Bernstein function. Therefore, f ./ WD hR u; ui is a Stieltjes function. Consider the special situation where H D L2 .Rd ; dx/ and where resolvent operators R are integral operators given by kernels R .x; y/, i.e. Z R u.x/ D R .x; y/u.y/ dy; u 2 L2 : Rd

If x 7! R .x; x/ is continuous at x D x0 , then f ./ WD R .x0 ; x0 / is a Stieltjes function. Without loss of generality we can assume that x0 D 0. Consider a sequence of type n!1 delta, i.e. un 2 Cc1 .Rd / such that un ! ı0 weakly, then “ n!1 hR un ; un i D R .x; y/ un .x/ un .y/ dx dy ! R .0; 0/ since R is continuous at x D 0. The assertion follows since S is stable under pointwise limits, cf. Theorem 2.2. We have seen in Example 12.5 that A generates a strongly continuous contraction semigroup .T t / t0 . Obviously Z ˝ ˛ s 7! hTs u; ui D e s E.d /u; u ; u 2 L2 ; .1;0

Section 12.2 Operator monotone functions

187

is completely monotone. If Ts is an integral operator with kernel ps .x; y/, we can show as above that s 7! ps .x; x/ is completely monotone for all x such that x 7! ps .x; x/ is continuous. A particularly interesting example is given by strongly continuous convolution semigroups or, equivalently, Lévy processes; for details we refer to Jacob [183, Vol. 1, Section 3.6] or [184]. In this case, A D  .D/ is a pseudo differential operator with symbol which is negative definite in the sense of Schoenberg, cf. (4.8), and Z Z s . / ix Ts u.x/ D e b u. / e d D u.x C y/ ps .dy/; u 2 Cc1 .Rd /; where b u denotes the Fourier transform and b p s D e s . The resolvent operators are given by Z 1 R u.x/ D b u. / e ix d ; u 2 Cc1 .Rd /:  C . / Since b p s . / D e s . / , it is clear that the measure ps .dy/ is for all s > 0 absolutely continuous with density ps 2 L1 if, and only if, e s 2 L1 for all s > 0. In this case, the Riemann–Lebesgue lemma applies and shows that y 7! ps .y/ is continuous. Sufficiency is obvious, necessity follows from the fact that ps=2 2 L1 \L1 is already an L2 -function. Therefore, 1

e 2 s or e s

Db p s=2 2 L2

2 L1 . This proves that the semigroup and resolvent densities exist and that    7! R .0/ 2 S t 7! p t .0/ 2 CM and

whenever e s

2 L1 for all s > 0.

12.2 Operator monotone functions Let .H; h; i/ be a finite or infinite-dimensional real Hilbert space and let A and B be bounded self-adjoint operators on H. If dim H D n, we will identify H with Rn and A; B with symmetric n n matrices. We write B  A if B  A is dissipative. This is equivalent to saying that hBu; ui  hAu; ui for all u 2 H:

(12.15)

As usual we define functions of A by the spectral theorem (in finite or infinite dimension).

188

Chapter 12 The spectral theorem and operator monotonicity

Definition 12.9. A function f W .0; 1/ ! R is said to be matrix monotone of order n if for all symmetric matrices A; B 2 Rnn with A ; B  .0; 1/ B  A implies f .B/  f .A/: We write Pn for the set of matrix monotone functions of order n. If f is matrix monotone of all orders n 2 N, we call f operator monotone. In the remaining part of this section we will assume that the operators .A; D.A// and .B; D.B// are dissipative self-adjoint operators on H which are not necessarily bounded. It may happen that the domain D.A/ \ D.B/ of the difference B  A of two unbounded operators is not dense in H; this means that (12.15) might not be well defined. To circumvent this, we use quadratic forms. Denote by .A/1=2 the unique positive square root of .A/ defined via the spectral theorem. Then ´˝ ˛ if u 2 D.QA / D D..A/1=2 /I .A/1=2 u; .A/1=2 u ; QA .u; u/ WD C1; for all other u 2 H defines the quadratic form generated by the operator A. If u 2 D.A/, then we have QA .u; u/ D hAu; ui; this is, in particular, the case for bounded operators. Therefore, 0  QA .u; u/  QB .u; u/  C1; u 2 H; (12.16) is the proper generalization of (12.15) for arbitrary dissipative operators. Definition 12.10. Let A and B be dissipative self-adjoint operators on H. If (12.16) holds for the quadratic forms, we say that .B/ dominates .A/ in quadratic form sense and write .A/ qf .B/. Note that .A/ qf .B/ entails     D .B/1=2 D D.QB /  D.QA / D D .A/1=2 and that 0  .A/ amounts to saying that A is dissipative. Obviously, the condition for f W .0; 1/ ! R T that f .A/ qf f .B/ whenever .A/ qf .B/ seems to be stronger than f 2 n2N Pn . But we will see later in Theorem 12.17 that these properties are equivalent. A hint in this direction is given by the following lemma. Lemma 12.11. Let .A; D.A// and .B; D.B// be two dissipative self-adjoint operators on H and denote by .R A / 2%A and .R B / 2%B their resolvents. Then the following assertions are equivalent (i) 0 qf .A/ qf .B/;

Section 12.2 Operator monotone functions

189

A (ii) 0 qf RB t qf R t for all t > 0; B (iii) 0 qf .A/RA t qf .B/R t for all t > 0.

Proof. (i))(ii) Fix t > 0. For any u 2 H set  WD .t  A/1 u 2 D.A/  D.QA /  D.QB /; WD .t  B/1 u 2 D.B/  D.QB /: Since A  t is dissipative, we can use the spectral theorem, Theorem 12.4, to define the square root .t  A/1=2 ; this is again a self-adjoint operator and D.A/  D..A/1=2 / D D..t  A/1=2 /: Since by assumption .t  A/ qf .t  B/, we get by the Cauchy–Schwarz inequality ˝ ˛2 ˝ ˛2 .t  B/1 u; u D .t  B/1 u; .t  A/ ˝ ˛2 D .t  A/1=2 .t  B/1 u; .t  A/1=2  ˝ ˛ ˝ ˛  .t  A/1=2 ; .t  A/1=2  ; .t  A/ ˛ ˝ ˛ ˝  .t  B/1=2 ; .t  B/1=2  .t  A/1 u; u ˝ ˛˝ ˛ D .t  B/1 u; u .t  A/1 u; u : This proves .t  B/1 qf .t  A/1 . (ii))(iii) We have for all t > 0 and u 2 H ˝

˛ ˝ ˛ A A .A/RA t u; u D .t  A/R t u  tR t u; u D hu; ui  htRA t u; ui  hu; ui  htRB t u; ui ˝ ˛ D .B/RB t u; u :

(iii))(i) Lemma 12.3 shows that ˝ ˝ ˛ ˛ 1=2 lim tRA lim tRA u; .A/1=2 u t .A/u; u D t!1 t .A/ ˛ ˝ D .A/1=2 u; .A/1=2 u

t!1

for all u 2 D.QA /. Using a similar argument for tRB t .B/ yields ˝

˛ ˝ ˛ .A/1=2 u; .A/1=2 u  .B/1=2 u; .B/1=2 u :

190

Chapter 12 The spectral theorem and operator monotonicity

Theorem 12.12. Let f 2 CBF and let .A; D.A// and .B; D.B// be two dissipative self-adjoint operators on H; denote by .R A / 2%A and .R B / 2%B the corresponding resolvents. Then .A/ qf .B/ implies f .A/ qf f .B/:

(12.17)

In particular, f is operator monotone in the sense of Definition 12.9. Proof. We have seen in Example 12.7 that ˝ ˛ ˝ ˛ f .A/u; u D ahu; ui C b .A/u; u C

Z

˝ .0;1/

˛ .A/RA t u; u  .dt /

holds for all u 2 D.A/ – and even u 2 H if we understand the expression on the right as a quadratic form which is C1 if u 62 D.Qf .A/ /. The operators tR t A are bounded, see Example 12.7, and  .0; n/ < 1 for all n 2 N. Therefore, Z hARA t u; ui  .dt / < 1 for all u 2 H; n 2 N: .0;n/

˝ ˛ ˝ ˛ Since limn!1 nRnA Au; u D Au; u on D.A/, we have for u 2 D.A/   Z ˝ ˝ ˛ ˛ ˛ ˝ .A/RA f .A/u; u D lim ahu; ui C b nRnA .A/u; u C u; u  .dt / t n!1

.0;n/

where the expression inside the bracket defines a bounded operator. If we understand this in the sense of quadratic forms, the right-hand side converges to a finite value if, and only if, u 2 D.Qf .A/ /. Analogous assertions hold for the operators B, f .B/ and the quadratic forms generated by them. B Lemma 12.11 shows that h.A/RA t u; ui  h.B/R t u; ui for all u 2 H. If we apply this to the limit expression above, we get hf .A/u; ui  hf .B/u; ui or, more precisely, 0  Qf .A/ .u; u/  Qf .B/ .u; u/  1;

u 2 H;

which means that f .A/ qf f .B/. Corollary 12.13. Let f 2 CBF and let .A; D.A// be a dissipative self-adjoint operator on H. Then for all c > 0 it holds that         D f .cA/ D D f .A/ and D f .c  A/ D D f .A/ : Proof. Assume first that c  1. Observe that for all ; t > 0 c  c t C c t C

and

cC c   C : t C .c C / t Cc t C

191

Section 12.2 Operator monotone functions

Since f 2 CBF is non-decreasing we conclude from this and the Stieltjes representation (6.7) of f that f ./  f .c/  cf ./

and f ./  f .c C /  f .c/ C f ./:

As both  7! f .c/ and  7! f .cC/ are complete Bernstein functions, the assertion follows from (12.11). The argument for 0 < c < 1 is similar. Corollary 12.14. Let f 2 CBF and let .A; D.A// and .B; D.B// be two dissipative self-adjoint operators on H. If D.B/  D.A/, then D.f .B//  D.f .A//. Proof. Since A and B are closed operators with D.B/  D.A/ we know from an inequality due to Hörmander, see [379, Chapter II.6, Theorem 2], that there exists a constant c > 0 such that   kAuk  c kBuk C kuk ; u 2 D.B/: Using the triangle inequality and dissipativity, (12.5), we find on D.B/     .1 C / kAuk  c k.B  /uk C .1 C /kuk  c k.B  /uk C k.B  /uk :  With a suitable constant c > 0 we can recast this inequality as .A/2 qf c2 .  B/2 : p Setting f D g and ˛ D 1=2 in Corollary 7.15 (iii), we see that h./ WD f 2 . / is again a complete Bernstein function. By spectral calculus and Theorem 12.12 we get q  q    2 2 2 2 2 2 f .A/ D f .A/ qf f c .  B/ D f 2 c .  B/ ; hence, kf .A/uk  kf .c .  B//uk. Therefore, the claim follows from Corollary 12.13. If we use the complete Bernstein functions f ./ D ˛ , 0 < ˛ < 1, we obtain the famous Heinz–Kato-inequality. Corollary 12.15. Let .A; D.A//, .B; D.B// be two dissipative self-adjoint operators on H. Then   k.A/1=2 uk  k.B/1=2 uk; u 2 D .B/1=2 implies for all ˛ 2 .0; 1/ k.A/˛=2 uk  k.B/˛=2 uk;

  u 2 D .B/˛=2 :

192

Chapter 12 The spectral theorem and operator monotonicity

In the remaining part of this section we will prove that the family of complete Bernstein functions and the family of positive operator monotone functions coincide. Recall that f 2 Pn if for all positive semi-definite symmetric matrices A; B 2 Rnn , B qf A implies that f .B/ qf f .A/. Let ˛1 ; : : : ; ˛n be the (non-negative) eigenvalues of A and denote by DŒ˛j  the n n diagonal matrix with diagonal entries ˛1 ; : : : ; ˛n . By choosing a suitable orthonormal basis we can assume that A D DŒ˛j ; then f .A/ D DŒf .˛j /. Let ˇ1 ; : : : ; ˇn be the eigenvalues of B. There exists an orthogonal n n matrix Q such that B D Q> DŒˇj Q and f .B/ D Q> DŒf .ˇj /Q. The conditions hBx; xi  hAx; xi and hf .B/x; xi  hf .A/x; xi, x 2 Rn , can be equivalently restated as n X

.Qx/j2 ˇj 

j D1

and

n X

xj2 ˛j ;

x 2 Rn ;

j D1

.Qx/j2 f .ˇj / 

j D1

n X

n X

xj2 f .˛j /;

x 2 Rn ;

j D1

respectively. P Define the weighted norm kxk.˛/ WD . jnD1 xj2 ˛j /1=2 and the corresponding operator norm kQxk.ˇ / : kQk.˛;ˇ / WD sup x¤0 kxk.˛/ Then f 2 Pn if, and only if, for every orthogonal n n matrix Q and all x 2 Rn , kQk.˛;ˇ /  1

implies that

n X

.Qx/j2 f .ˇj / j D1



n X

xj2 f .˛j /:

(12.18)

j D1

We say that a function f W .0; 1/ ! R is in the set MP n , n 2 N, if for all distinct 1 ; : : : ; 2n > 0 and any choice of a1 ; : : : ; a2n 2 R with j2n D1 aj D 0 2n X

aj

j D1

j t  1 0 j C t

for all t > 0 implies that

2n X

aj f .j /  0:

(12.19)

j D1

Clearly, Mn  MnC1 . Since     2n  2n 2n X X j 1 X j t  1 j 1 1 D D tC aj aj  aj ; tC j C t t t C j t t t C j j D1

j D1

j D1

we have f 2 Mn if, and P2nonly if, for all distinct 1 ; : : : ; 2n > 0 and any choice of a1 ; : : : ; a2n 2 R with j D1 aj D 0 2n X j D1

2n X j aj  0 for all t > 0 implies that aj f .j /  0: t C j j D1

(12.20)

193

Section 12.2 Operator monotone functions

We explain now yet another way to characterize when f 2 Mn . Denote by ….n/ the family of polynomials of degree less than or equal to n. ForQdistinct 1 ; : : : ; 2n > 0 Q 0 put .t / WD j2n D1 .t C j / and observe that  .k / D j ¤k .j  k /. For any p 2 ….2n  1/ set ak D ak .p/ WD

p.k / ; 0  .k /k

k D 1; : : : ; 2n;

(12.21)

and let a D a.p/ D .ak .p//1k2n 2 R2n . Then 2n X j p.t / D aj .t / t C j

(12.22)

j D1

holds true and shows that p 7! a.p/ defines a linear bijection from ….2n  1/ P onto R2n . By observing that p.0/ D 0 is equivalent to j2n D1 aj .p/ D 0, it follows from (12.20) that f 2 Mn if, and only if, for all p 2 ….2n  1/ with p.0/ D 0, p.t /  0 for all t > 0 implies that

2n X

aj .p/f .j /  0:

(12.23)

j D1 C  MnC . Lemma 12.16. For every n 2 N, PnC1

Proof. Every polynomial p 2 ….2n  1/ with p.0/ D 0 and p.t /  0 for all t > 0 can be written as p.t / D t q12 .t / C q22 .t / where q1 ; q2 2 ….n  1/ and q2 .0/ D 0, cf. Akhiezer [2, p. 77]. Because of the linearity of the map (12.21) it suffices to consider two cases: (i) p.t / D t q 2 .t /; (ii) p.t / D q 2 .t /;

q 2 ….n  1/; q 2 ….n  1/; q.0/ D 0.

Relabel 1 ; 2 ; : : : ; 2n > 0 as 0 < ˇ1 < ˛1 < ˇ2 <    < ˇn < ˛n . Then  0 .ˇj / > 0 and  0 .˛j / < 0, j D 1; 2; : : : ; n. (i) Given a polynomial q 2 ….n  1/ the partial fractions representation (12.22) of the polynomial t q 2 .t / reads X X t q 2 .t / ˇj ˛j D yj2 C xj2 .t / t C ˇj t C ˛j n

n

j D1

j D1

xj D

q.˛j / ; . 0 .˛j //1=2

(12.24)

where yj D

q.ˇj / ; 0 . .ˇj //1=2

j D 1; : : : ; n:

(12.25)

194

Chapter 12 The spectral theorem and operator monotonicity

Observe that (12.25) defines two linear bijections: ˆ W Rn ! ….n  1/, ˆ.x/ D q and ‰ W ….n  1/ ! Rn , ‰.q/ D y. Let Q WD ‰ ı ˆ be the composition of these two maps. Then Q is a linear bijection from Rn to Rn . Moreover, for q D ˆ.x/ and y D Qx, the partial fractions representation (12.24) is valid. By taking t D 0 in (12.24) it follows that n n X X yj2 C xj2 D 0;  j D1

j D1

implying By multiplying (12.24) by t and letting t ! 1 we see P P that Q is orthogonal. that  jnD1 yj2 ˇj C jnD1 xj2 ˛j  0, i.e. kQk.˛;ˇ /  1. Let f 2 PnC . By (12.18), 

n X

yj2 f .ˇj / C

j D1

n X

xj2 f .˛j /  0

(12.26)

j D1

and y D Qx, which is precisely (12.23). So far we have only used for all x 2 C . that f 2 PnC but not the stronger assumption f 2 PnC1 (ii) For q 2 ….n  1/ with q.0/ D 0 the partial fractions representation (12.22) of the polynomial q 2 .t / is Rn

n n X X ˇj ˛j q 2 .t / D yj2  xj2 .t / t C ˇj t C ˛j j D1

(12.27)

j D1

where yj D

q.ˇj / ; 0 . .ˇj /ˇj /1=2

xj D

q.˛j / ; 0 . .˛j /˛j /1=2

j D 1; : : : ; n:

(12.28)

In order to verify (12.23) for p.t / D q 2 .t / we have to show that 

n X

xj2 f .˛j / C

j D1

n X

yj2 f .ˇj /  0:

(12.29)

j D1

Q / D .t C ˛0 /.t /.t C ˇnC1 /. Then Let 0 < ˛0 < ˇ1 , ˇnC1 > ˛n and .t ˇnC1 t q 2 .t / q 2 .t / ˇnC1 t D t C ˛0 .t / t C ˇnC1 .t Q / n nC1 X X ˛j ˇj xQj2 C yQj2 D t C ˛j t C ˇj j D0

(12.30)

j D1

with xQj , j D 0; : : : ; n, and yQj , j D 1; : : : ; n C 1, given by the formulae analogous C . From part (i) of the proof we conclude that to (12.28). Let f 2 PnC1 xQ 02 f .˛0 / 

n X j D1

xQj2 f .˛j /

C

n X j D1

2 yQj2 f .ˇj / C yQnC1 f .ˇnC1 /  0:

(12.31)

195

Section 12.2 Operator monotone functions

Comparing the rational functions in (12.27) and (12.30) we get for j D 1; : : : ; n, lim xQj D xj ;

˛0 !0

and

lim

ˇnC1 !1

yQj D yj :

Since q.0/ D 0, xQ 02 D

˛0 q 2 .˛0 / D O.˛02 / as ˛0 ! 0 .˛0 / ˛0  ˇnC1

and since the degree of q is less than or equal to n  1, 2 yQnC1 D

ˇnC1 q 2 .ˇnC1 / 2 D O.ˇnC1 / ˇnC1  ˛0 .ˇnC1 /

as ˇnC1 ! 1:

C All f 2 PnC1 are non-decreasing and non-negative, hence lim˛0 !0 ˛02 f .˛0 / D 0 2 implying that lim˛0 !0 xQ 02 f .˛0 / D 0. If also limˇnC1 !1 f .ˇnC1 /=ˇnC1 D 0, then we can let ˛0 ! 0 and ˇnC1 ! 1 in (12.31) to get (12.29). C Since PnC1  P2C , we can use formulae (12.26) and (12.25) of part (i) of the proof with n D 2, ˇ1 D 1, ˛1 D 2, ˇ2 D ˇ, ˛2 D 2ˇ and q.t / D t C 2ˇ to arrive at



ˇ 2ˇ  2 2ˇ  1 f .1/  f .ˇ/ C f .2/  0: ˇ1 .ˇ  1/.ˇ  2/ ˇ2

Multiplying with .ˇ  1/.ˇ  2/=ˇ 3 and rearranging yields 0

f .ˇ/ .2ˇ  1/.ˇ  2/ 2.ˇ  1/2   f .1/ C f .2/: ˇ2 ˇ3 ˇ3

Since the right-hand side tends to 0 as ˇ ! 1, we find limˇ !1 f .ˇ/=ˇ 2 D 0. T It remains to verify that n2N MnC  CBF. This is now mainly a functional analytic argument. Theorem 12.17. The families of complete Bernstein and positive operator monotone functions coincide: \ CBF D PnC : n2N

In particular, 0 qf .A/ qf .B/ implies 0 qf f .A/ qf f .B/ if, and only T if, f 2 n2N PnC . Proof of Theorem 12.17. Combining Theorem 12.12 and Lemma 12.16 shows that \ \ CBF  PnC  MnC : n2N

n2N

It is, therefore, enough to show that the right-hand side is contained in CBF.

196

Chapter 12 The spectral theorem and operator monotonicity

Write  .t / WD .t  1/=. C t / and consider the set μ ´ X X aj j .t /; aj 2 R; j > 0; aj D 0 : G WD g W g.t / D finite

finite

Clearly, G  C Œ0; 1 and for each f 2 ƒf W G ! R;

ƒf

T

X

C n2N Mn

!

aj j

finite

WD

X

aj f .j /

finite

defines a linear functional on G. The defining property (12.19) of the sets Mn just says that ƒf is positive. Moreover,  5.t 2 C 1/ 5.t 2 C 1/  2 1 g0 .t / WD 5 2 .t /  1 .t / D 2 t C 3t C 2 t C 3t 2 C 3 C 2 is an element from G. Let us show that ƒf is bounded. Pick g 2 G and define s WD sup g. Obviously, g.t /  s C  g0 .t / where s C WD max¹s; 0º. Thus we see that for all h 2 C C Œ0; 1 ƒf .g/  ƒf .s C  g0 / D s C ƒf .g0 /  sup jg.t / C h.t /j  ƒf .g0 / : t>0

Since the expression g 7! sup t>0 jg.t / C h.t /j  ƒf .g0 / appearing on the right-hand side is a seminorm on C Œ0; 1, we can use the Hahn–Banach theorem to find an extension of ƒf as a continuous linear functional on C Œ0; 1 bounded by the same seminorm. Now evaluate ƒf for h. Then ƒf .h/  kh  hk1 ƒf .g0 / D 0 which shows that ƒf is a bounded, positive linear functional on C Œ0; 1. We can now appeal to the Riesz representation theorem to get Z ƒf .h/ D h.t / .dt /; h 2 C Œ0; 1: Œ0;1

for some finite positive measure  on Œ0; 1. Thus, for g.t / D  .t /  1 .t /, f ./  f .1/ D ƒf .g/ D ƒf . /  ƒf .1 /: This shows

Z

t  1 .dt / C f .1/  ƒf .1 / Œ0;1  C t Z t  1 .dt / C f .1/  ƒf .1 /: D ¹1º C Œ0;1/  C t

f ./ D

Section 12.2 Operator monotone functions

197

Observe that the kernel  7! .t  1/=. C t / is monotonically increasing. Therefore, monotone convergence shows that f .0C/ exists. Since it is positive, we see that R 1 .dt /  f .1/  ƒ . / < 1; in particular, ¹0º D 0. As  is finite, this t 1 f Œ0;1/ shows that Z t  1 .dt / f ./ D ˛ C ˇ C .0;1/  C t satisfies all conditions of Theorem 6.9, i.e. f 2 CBF. Comments 12.18. Section 12.1: Standard references for the spectral theorem for normal and self-adjoint operators are the monographs by Yosida [379] and Kato [200]. Our exposition owes a lot to Akhiezer and Glazman [3] and, in particular, to Lax [239, Chapter 32]. Lax [239] is one of the few textbook-style presentations of the Doob and Koopman [101] and Lengyel [240] approach to the spectral theorem. The first proof of the spectral theorem (for bounded operators) is due to Hellinger [155] and Hahn [139]. These papers already contain a version of the Hellinger–Stone formula (12.9). The case of unbounded self-adjoint operators was treated by von Neumann [271] and Stone [343]. We follow Doob and Koopman [101], Lengyel [240] and the beautiful paper by Nevanlinna and Nieminen [275]; see also Akhiezer and Glazman [3] or Lax [239]. This approach works also for general, not necessarily spectrally negative, self-adjoint operators if one uses the version of Theorem 6.2 for CBF-like functions preserving the upper half-plane – these functions are not necessarily positive on the positive real line – see Remark 6.18. The Examples 12.5–12.7 also have an interpretation in connection with Bochner’s subordination which will be discussed in Chapter 13. It turns out that in the Hilbert space setting the generator of the subordinate semigroup, Af , f is a Bernstein function, can be identified with the operator f .A/ defined by spectral calculus. An interesting application is the subordination of Dirichlet forms: the subordinate Dirichlet form .Ef ; D.Ef // is generated by f .A/; using the spectral representation of f .A/, the fact that a Bernstein function is non-decreasing and convex and Jensen’s inequality for integrals, it is straightforward to show that   E.u; u/ Ef .u; u/  kuk2 f for all u 2 D.E/: kuk2 For related results we refer to Ôkura [284] and Section 13.4 below. Section 12.2: The best reference on quadratic forms is still Kato [200], in particular Sections VI.2–6. In order to define B  A using quadratic forms one does not need that the ˝ actually ˛ quadratic form is bounded from below QB .u; u/  c u; u ; it is also not necessary to require that A  B is densely defined, see Davies [88, Section 4.2]. Lemma 12.11

198

Chapter 12 The spectral theorem and operator monotonicity

is from [88, Theorem 4.17] and [320, Lemma 5.19]. Theorem 12.12 appears for the first time in Heinz [154], our proof is from [320, Satz 5.20]. The Heinz–Kato inequality, Corollary 12.15, goes back to [154]; we give a different proof based on [88, Chapter 4.2] and [320]. The characterization of matrix monotone and operator monotone functions goes back to Löwner [248] who establishes a determinant criterion (in terms of divided differences) for matrix monotonicity. Section 6 of that paper also contains the representation of matrix monotone functions f of finite P order by a discrete Stieltjes-like representation of the form f ./ D a C a0  C jn1 D1 aj =.aj C /; the limit n ! 1 is discussed and Löwner actually shows that a matrix monotone function (of all orders n) preserves the upper and lower half-planes. Löwner’s technique is similar to the interpolation technique of Pick [293] – this paper is the only one quoted in [248] – but he seems to be unaware of the progress made by Nevanlinna on the integral representation of such functions, see Comments 6.22 and 7.24. Löwner’s paper had a huge impact on the literature and Theorem 12.17 is nowadays called Löwner’s theorem. Several books are devoted to this topic, among them the monographs [100] by Donoghue, [55] and [56] by Bhatia, and the standard treatise by Horn and Johnson [170, 171]. Several quite different proofs of Löwner’s theorem are known: Bendat and Sherman [29] use the Hamburger moment problem; Hansen and Pedersen [144] show, using extreme-point methods, that operator monotone functions have a Nevanlinna–Pick-type representation, see also Bhatia [55]. Both Hansen– Pedersen and Bhatia derive the integral representation formula for functions which preserve order relations of operators with spectrum in Œ1; 1; the case of operator monotone functions in the sense of Definition 12.10, i.e. for operators with spectrum in Œ0; 1/, can be recovered by mapping Œ1; 1 to Œ0; 1/ by an affine-linear transformation (which is itself operator monotone). It is possible to determine the extreme points of operator monotone functions (or complete Bernstein functions) directly: if f is operator monotone, then f ./ D

f ./  f .1/ f .1/  f ./ C D g1 ./ C g2 ./; 1 1

 > 0:

Using results from [55] it is possible to show that g1 ; g2 are again operator monotone and that operator monotone functions are two times continuously differentiable. Consider now the set of operator monotone functions with f 0 .1/ D 1. This is a basis of the cone of all operator monotone functions. One can show that jf 00 .1/j  2f 0 .1/ D 2 and ˛ WD f 00 .1/=2 2 .0; 1/. Since g10 .1/ D f 0 .1/ C f 00 .1/=2 D 1  ˛ and g20 .1/ D f 00 .1/=2 D ˛, we find that f ./ D .1  ˛/g1 ./=.1  ˛/ C ˛g2 ./=˛. If f is extremal, we conclude that f ./ D g1 ./=.1  ˛/ D g2 ./=˛. This shows that the extremal points of the basis are necessarily of the form f ./ D

f .1/ C

1˛ ˛

;

 > 0; ˛ 2 Œ0; 1:

Section 12.2 Operator monotone functions

199

The proof by Korányi [214], see also Sz.-Nagy’s remarks [344] and their joint work [215], uses Hilbert space methods and the theory of reproducing kernel spaces. A nice presentation can be found in [100, Chapters X, XI] and [308, Addenda to Chapter 2]. Our proof is a slight simplification of Sparr’s elementary approach [334] which itself is inspired by the theory of interpolation spaces; a related account is given by Ameur, Kaijser and Silvestrov [7]. Simon [330, Chapters 6,7 and Theorem 9.10] contains a new treatment of the Bendat–Sherman [29] proof as well as the extreme point argument of Hansen and Pedersen [144]. There is a deep relation between operator monotone functions and operator means. Ando [10] proves that there is a one-to-one relation between the class of operator monotone functions on Œ0; 1/ with f .1/ D 1 and the class of operator means. The latter are extremely useful in the theory of electrical networks, see e.g. Anderson and Trapp [8]. Other recent papers on operator means and operator monotone functions include [158] and [182]. Operator inequalities are an active area of research with many contributions by Ando, Furuta [126], Hansen and Pedersen [144], Uchiyama [350, 351, 352, 353, 356] and others. A recent monograph is Zhan [380].

Chapter 13

Subordination and Bochner’s functional calculus In this chapter we consider strongly continuous contraction semigroups of operators on a Banach space. Our focus will be on subordination of such semigroups in the sense of Bochner. We give a proof of Phillips’ theorem describing the infinitesimal generator of the subordinate semigroup and discuss the related functional calculus for generators of semigroups and applications to functional inequalities. In the final section of this chapter we discuss the probabilistic counterpart of subordination and use it to obtain eigenvalue estimates for subordinate processes.

13.1 Semigroups and subordination in the sense of Bochner In this section we will briefly discuss a method to generate new semigroups from a given one. We assume that the reader is familiar with the theory of semigroups on a Banach space .B; k  k/; our standard references are the monographs by Davies [88], Pazy [286] and Yosida [379]. Throughout this section .B; kk/ denotes a Banach space, .B ; kk / its topological dual and hu; i stands for the dual pairing between u 2 B and  2 B . Recall that a strongly continuous or C0 -semigroup of operators on B is a family of bounded linear operators T t W B ! B, t  0, which is strongly continuous (at t D 0), i.e. lim kT t u  uk D 0 for all u 2 B;

t!0

and has the semigroup property T t Ts D Ts T t D T tCs ;

for all s; t  0:

If kT t uk  kuk for all u 2 B and t  0, we call .T t / t0 a C0 -contraction semigroup. The (infinitesimal) generator of the semigroup is the operator Tt u  u ; Au WD strong- lim t!0 t ² ³ Tt u  u D.A/ WD u 2 B W lim exists as strong limit : t!0 t

(13.1)

Since A is the (strong) right derivative of T t at zero, and since .T t / t0 is a semigroup, we have Z t Z t Tt u  u D Ts Au ds D ATs u ds; u 2 D.A/; t > 0: (13.2) 0

0

Section 13.1 Semigroups and subordination in the sense of Bochner

This shows, in particular, the following elementary but useful inequality ® ¯ kT t u  uk  min t kAuk; 2kuk ; u 2 D.A/; t > 0:

201

(13.3)

The generator .A; D.A// of a C0 -contraction semigroup is a densely defined linear operator which is dissipative and closed. Dissipative means that one of the following equivalent conditions is satisfied Re hAu; i  0 for all u 2 D.A/;  2 F .u/

(13.4)

where F .u/ D ¹ 2 B W kk2 D kuk2 D hu; iº, ku  Auk   kuk for all u 2 D.A/;  > 0; ku  Auk  Re  kuk for all u 2 D.A/;  2 C; Re  > 0:

(13.5) (13.6)

Generators are, in general, unbounded operators. By the Hille–Yosida theorem an operator A generates a C0 -contraction semigroup if, and only if, D.A/ is dense in B, A is dissipative and the range .  A/.D.A// D B for some (hence, all)  > 0. The resolvent set %A of a closed operator A consists of all  2 C such that the operator   A WD  id A has a bounded inverse denoted by R WD .  A/1 ; the family .R / 2%A is the resolvent of A. It is well known that %A is open and that for C0 -contraction semigroups ¹Re  > 0º  %A ; in particular, .0; 1/  %A . The set A WD C n %A is called the spectrum of A. The resolvent operators satisfy the resolvent estimate kR uk 

1 kuk 

for all u 2 B;  > 0;

(13.7)

and the following resolvent equation holds for all ; z 2 %A R  Rz D .z  / Rz R :

(13.8)

This follows exactly as in the Hilbert space setting, cf. page 179, and one sees immediately that  7! hR u; i is analytic for all u 2 B and  2 B . There is a one-to-one correspondence between a C0 -contraction semigroup .T t / t0 and its generator .A; D.A// which can formally be expressed by writing T t D e tA . If A is bounded, the exponential e tA may be defined by the exponential series; in the general case one has to use the Yosida approximation, A WD AR ,  > 0. By the resolvent estimate (13.7) kA uk D  kAR uk D  kR u  uk  2 kuk which shows that A is bounded. Therefore, exp.tA / can be defined as an exponential series and one can show that T t u D strong- lim e tA u: !1

In this sense, we may indeed write T t D e tA .

202

Chapter 13 Subordination and Bochner’s functional calculus

The following method to generate a new C0 -semigroup from a given one is due to S. Bochner. It uses vaguely continuous convolution semigroups introduced in Definition 5.1 and Bernstein functions which appear as Laplace exponents, cf. Theorem 5.2. Proposition 13.1. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B and let . t / t0 be a vaguely continuous convolution semigroup of sub-probability measures on Œ0; 1/ with corresponding Bernstein function f . Then the Bochner integral Z f

T t u WD

Ts u  t .ds/; Œ0;1/

t  0;

(13.9)

defines again a C0 -contraction semigroup on the Banach space B. Proof. Since t 7! T t u is strongly continuous, and since Z  Z Z     Ts u  t .ds/  kTs uk  t .ds/   Œ0;1/

Œ0;1/

Œ0;1/

kuk  t .ds/  kuk;

the operators (13.9) are well defined and contractive. In particular, every T tf , t  0, is a bounded linear operator and we find, using the semigroup property of .T t / t0 , for all s; t  0 Z f f Ts T t u D Tsf Tr u  t .dr/ ZŒ0;1/ Z T Tr u s .d/  t .dr/ D Œ0;1/ Œ0;1/ Z Z TCr u s .d/  t .dr/ D Œ0;1/ Œ0;1/ Z T u s ?  t .d /: D Œ0;1/

f Since . t / t0 is a convolution semigroup, this proves that Tsf T tf D TsCt . The

strong continuity of .T tf / t0 finally follows from Z   f   T u  u  kTs u  uk  t .ds/ C 1   t Œ0; 1/ kuk t Œ0;1/

which tends to zero as t ! 0 since vague- lim t!0  t D ı0 and lim t!0  t Œ0; 1/ D 1, see the discussion following Definition 5.1.

Section 13.1 Semigroups and subordination in the sense of Bochner

203

Definition 13.2. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B and let . t / t0 be a vaguely continuous convolution semigroup of sub-probability measures on Œ0; 1/ with corresponding Bernstein function f . Then the semigroup f .T t / t0 defined by (13.9) is called subordinate (in the sense of Bochner) to the semigroup .T t / t0 with respect to the Bernstein function f . The infinitesimal generator of .T tf / t0 will be denoted by .Af ; D.Af //; it is called the subordinate generator. From now on, we will indicate all operators related to the subordinate semigroup .T tf / t0 by the superscript f . Subordination of subordinate semigroups corresponds to the composition of Bernstein functions. Lemma 13.3. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B and let . t / t0 and . t / t0 be vaguely continuous convolution semigroups of subprobability measures on Œ0; 1/ with corresponding Bernstein functions f and g. Then f ıg .T tg /f u D T t u; t  0; u 2 B: Proof. From Theorem 5.27 we know that the sub-probability measures  t given by R the vague integral Œ0;1/ s  t .ds/ form a vaguely continuous convolution semigroup with corresponding Bernstein function f ı g. Therefore, Z Z g f .T t / u D Tr u s .dr/  t .ds/ Œ0;1/ Œ0;1/ Z Z Tr u s .dr/  t .ds/ D Œ0;1/ Œ0;1/ Z Tr u  t .dr/ D T tf ıg u: D Œ0;1/

Remark 13.4. It is instructive to consider Bochner’s subordination in the context of the spectral calculus of a dissipative operator .A; D.A// Ron the Hilbert space H. Denote by E.d / the spectral resolution of A and by T t u D .1;0 e t E.d /u the C0 -semigroup generated by A, cf. Example 12.5. Let f 2 BF and let . t / t0 be the convolution semigroup associated with it. Then we find Z Z f e s E.d /u  t .ds/ Tt u D Œ0;1/ .1;0 Z Z e s  t .ds/ E.d /u D Z.1;0 Œ0;1/ e tf ./ E.d /u D e tf .A/ u: D .1;0

204

Chapter 13 Subordination and Bochner’s functional calculus

This implies that .Af ; D.Af // coincides with .f .A/; D.f .A/// and that Z f .T t u  u/ .dt /; A u D f .A/u D au C bAu C .0;1/

as we have seen in Example 12.6. The formula for Af from Remark 13.4 remains valid in Banach spaces. This result is due to R. S. Phillips [292, Theorem 4.3]. As usual, we set .A0 ; D.A0 // D .id; B/, Ak WD A ı A ı    ı A (k times), and ® ¯ D.Ak / WD u 2 D.Ak1 / W Ak1 u 2 D.A/ : With this definition .Ak ; D.Ak // is a closed operator if .A; D.A// is a closed operator. Proposition 13.5. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f be a Bernstein function. Then for all k 2 N the set D.Ak / is an operator core for the subordinate generator .Af ; D.Af //. Proof. It is well known that a dense subset D  D.Af / is an operator core for the generator Af if it is invariant under the semigroup, i.e. if T tf D  D, see e.g. Davies f [88, Theorem 1.9]. Let us show that T t .D.Ak //  D.Ak / for all k 2 N. Note that for all u 2 D.Ak / Z Z f Ts u  t .ds/ D Ts u  t .ds/ D T t u; u 2 D.Ak /: strong- lim n!1 Œ0;n/

Œ0;1/

Since .A; D.A//, hence .Ak ; D.Ak //, is a closed operator, we find for all m; n 2 N with m < n and all u 2 D.Ak /  Z  Z   k    k A   Ts u  t .ds/ D  Ts A u  t .ds/   Œm;n/ Œm;n/ Z kTs Ak uk  t .ds/  Œm;n/ Z kTs k  t .ds/ kAk uk;  Œm;n/

which shows that .A

R k

Œ0;n/ Ts u  t .ds//n2N k

is for all u 2 D.Ak / a Cauchy sef

quence. The closedness of A now shows that T t u 2 D.Ak / for all u 2 D.Ak /. Theorem 13.6 (Phillips). Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f be a Bernstein function with Lévy triplet

205

Section 13.1 Semigroups and subordination in the sense of Bochner

.a; b; /. Denote by .T tf / t0 and .Af ; D.Af // the subordinate semigroup and its infinitesimal generator. Then D.A/ is an operator core for .Af ; D.Af // and Af jD.A/ is given by Z Af u D au C bAu C

.0;1/

.Ts u  u/ .ds/;

u 2 D.A/:

(13.10)

The integral is understood as a Bochner integral. Proof. That D.A/ is an operator core of Af follows from Proposition 13.5. Using (13.3) we see that the integral term in formula (13.10) converges for all u 2 D.A/, Z Z Z kTs u  uk .ds/ C kTs u  uk .dt / kTs u  uk .ds/ D .0;1/ .0;1/ Œ1;1/ Z s .ds/ kAuk C 2Œ1; 1/ kuk;  .0;1/

which means that the operator given by (13.10) is defined on D.A/. Assume first that f .0C/ D 0 and that .T t / t0 satisfies kT t uk  e t kuk for all t > 0 and some  > 0. We write f 2 BF with f .0C/ D 0 as Z f ./ D b C .1  e s / .ds/; .0;1/

and denote the corresponding vaguely continuous convolution semigroup of probabilR tf ./ s ity measures by . t / t0 . By definition, e D Œ0;1/ e  t .ds/, and we see that Z fn ./ WD

1

Œ0;1/

.1  e

s

/ n1=n .ds/ D

1  e  n f ./ 1 n

n!1

! f ./

defines a sequence of Bernstein functions fn 2 BF approximating f 2 BF. Obviously, the Lévy triplet of fn is .0; 0; n1=n / and we see from Corollary 3.9 that vague- lim n1=n D ;

(13.11)

0 D lim lim inf n1=n ŒC; 1/; C !1 n!1 Z s n1=n .ds/: b D lim lim inf

(13.12)

n!1

c!0 n!1

(13.13)

Œ0;c/

Throughout the proof we assume, for simplicity, that c and C are always continuity points of the measure  and that the limits in c and C are taken along sequences of

206

Chapter 13 Subordination and Bochner’s functional calculus

continuity points of . The proof of Corollary 3.9 shows that we may then replace lim inf n by limn . Since s 7! Ts u  u is strongly continuous, the function s 7! hTs u  u; i, u 2 B,  2 B , is continuous and we conclude that for all continuity points 0 < c < C < 1 Z Z lim hTs u  u; i n1=n .ds/ D hTs u  u; i .ds/: n!1 .c;C /

.c;C /

Because of (13.3) we know that jhTs u  u; ij  kTs u  ukkk  min¹s kAuk; 2 kukº kk for all u 2 D.A/ and s > 0. The function 1 ^ s is -integrable, and by dominated convergence we see that for all u 2 D.A/ Z Z lim lim hTs u  u; i n1=n .ds/ D hTs u  u; i .ds/: (13.14) C !1 n!1 .c;C / c!0

.0;1/

For u 2 D.A/ and all c > 0 we find Z Z Z s .Ts u  u/ n1=n .ds/ D Tr Au dr n1=n .ds/ Œ0;c/ Œ0;c/ 0 Z s Z .Tr Au  Au/ dr n1=n .ds/ D Œ0;c/ 0 Z s n1=n .ds/ Au: C Œ0;c/

R

Because of (13.13), Œ0;1/ s n1=n .ds/, n 2 N, is a bounded sequence. Using the strong continuity of the semigroup we find for c 2 .0; 1/ Z  Z s    sup  .T Au  Au/ dr n .ds/ r 1=n   n2N Œ0;c/ 0 Z c!0 s n1=n .ds/ ! 0  sup kTr Au  Auk sup r c

n2N Œ0;1/

which implies that Z strong- lim lim

c!0 n!1 Œ0;c/

.Ts u  u/ n1=n .ds/ D bAu;

Finally, we get Z Z .Ts u  u/ n1=n .ds/ D ŒC;1/

ŒC;1/

u 2 D.A/:

(13.15)

Ts u n1=n .ds/  n1=n ŒC; 1/ u:

207

Section 13.1 Semigroups and subordination in the sense of Bochner

Because of (13.12) the sequence n1=n Œ1; 1/, n 2 N, is bounded. Therefore we get for C > 1 Z  Z    T u n .ds/ kTs uk n1=n .ds/ sup   sup s 1=n   n2N n2N ŒC;1/ ŒC;1/ Z e s n1=n .ds/kuk  sup n2N ŒC;1/

C !1

 e C  sup n1=n Œ1; 1/ ! 0; n2N

and we conclude that Z strong- lim

lim

C !1 n!1 ŒC;1/

.Ts u  u/ n1=n .ds/ D 0;

u 2 D.A/:

(13.16)

If we combine (13.14)–(13.16) we find that for all u 2 D.A/ and  2 B

Z ˝ ˛ f f hA u; i D lim n.T1=n u  u/;  D bAu C .Ts u  u/ .ds/;  n!1

.0;1/

holds, and this implies (13.10). If f .0C/ D 0 and if .T t / t0 is a general C0 -contraction semigroup on B, then for any  > 0, TIt WD e t T t is a C0 -contraction semigroup which satisfies the additional assumption used above. It is not hard to check that the generator of this semigroup is .A  ; D.A//. From the first part of the proof we know that Z .e s Ts u  u/ .ds/; u 2 D.A/: .A  /f u D b.A  /u C .0;1/

Note that the expression on the right-hand side still makes sense if  D 0. For every u 2 D.A/ we have   Z    f  .Ts u  u/ .ds/  .A  / u bAu C   .0;1/   Z   s .Ts u  e Ts u/ .ds/ D bu C  .0;1/ Z .1  e s / .ds/ kuk  b kuk C .0;1/

and this converges to 0 as  ! 0. This means that Z strong- lim .A  /f u D bAu C .Ts u  u/ .ds/; !0

.0;1/

u 2 D.A/:

208

Chapter 13 Subordination and Bochner’s functional calculus

On the other hand,       f   f f f u  u u  u u  T u T T T   1=n  1=n I1=n I1=n       D   1 1 1     n n n  Z   s  .Ts u  e Ts u/ n1=n .ds/ D  Œ0;1/ Z .1  e s / n1=n .ds/ kuk  Œ0;1/

1

D

1  e  n f ./ 1 n

n!1

!0

kuk ! f ./ kuk ! 0:

This shows that Af u D strong- lim!0 .A  /f u and that Af is, on D.A/, given by (13.10). Finally consider h 2 BF with h.0C/ D a  0. We write h D f C a where f 2 BF and f .0C/ D 0. As above we denote by . t / t0 the convolution semigroup corresponding to f ; then at WD e ta  t , t  0, is the vaguely continuous convolution semigroup associated with h. Indeed, L at D e ta L  t D e ta e tf . Therefore, Z Ts u at .ds/ D e ta T tf u; u 2 B; t  0; T th u D Œ0;1/

and thus Ah D Af  a id. This completes the proof of the theorem. Corollary 13.7. Let .A; D.A// be the generator of a C0 -contraction semigroup on the Banach space B and let f be a Bernstein function. Then D.Af / D D.A/ if, and only if, A is a bounded operator or if b D lim!1 f ./= > 0. If f is a bounded Bernstein function or if A is a bounded operator, the subordinate generator Af is bounded, i.e. D.Af / D B. Proof. Using (13.3) we can estimate the integral term in Phillips’ formula (13.10) in the following way: Z  Z Z     .Ts u  u/ .ds/  kTs u  uk .ds/ C kTs u  uk .dt /  .0;1/ .0;/ Œ;1/ Z s .ds/ kAuk C 2Œ; 1/ kuk:  .0;/

Therefore we get kAf uk  .a C d /kuk C .b C c /kAuk; u 2 D.A/; R where a; b  0 are from (13.10), c D .0;/ s .ds/ and d D 2Œ; 1/.

(13.17)

Section 13.1 Semigroups and subordination in the sense of Bochner

209

Assume that D.A/ D D.Af / and that b D 0. Since .Af ; D.A// and .A; D.A// are closed operators we have by a theorem of Hörmander, see [379, Chapter II.6, Theorem 2],   kAuk  c kAf uk C kuk ; u 2 D.A/; with a suitable constant c > 0. Choosing  in (13.17) so small that c < 1=.2c/, we obtain   1 kAuk  c kAf uk C kuk  kAuk C c .d C a/ kuk; u 2 D.A/; 2 which shows that the operator A is bounded. Conversely, if A is bounded D.A/ D B and by (13.17) Af is bounded, therefore D.Af / D B, too. If b > 0, we get from (13.10) using the estimate leading to (13.17) kAf uk  .b  c / kAuk  .a C d / kuk;

u 2 D.A/:

Pick  > 0 such that b  c > 0 and observe that then kAuk  c kAf uk C c 0 kuk;

u 2 D.A/;

holds for some constants c; c 0 > 0. Since .A; D.A// is closed, this immediately implies the closedness of the operator .Af ; D.A//. The second part of the assertion follows from (13.17) with  D 1, if A is bounded, and with  D 0 if f is bounded; in the latter case we used b D 0 and that  is a finite measure, cf. Corollary 3.8 (v). Corollary 13.8. Let f 2 BF and let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A//. Then the following estimate holds   2e kAf uk 1 kAuk  f ; u 2 D.A/; u ¤ 0: (13.18) kuk e1 2 kuk For the fractional powers .A/˛ , ˛ 2 .0; 1/, one has k.A/˛ uk  4 kAuk˛ kuk1˛ ;

u 2 D.A/:

Proof. By Phillips’ theorem, Theorem 13.6, we have Z Af u D au C bAu C .T t u  u/ .dt /; .0;1/

u 2 D.A/:

If we combine the elementary convexity estimate e .1  e ct /; c; t  0; min¹1; ct º  e1 with the operator inequality (13.3), ² ³ kAuk kT t u  uk  min t ; 2 ; u 2 D.A/; u ¤ 0; t > 0; kuk kuk

(13.19)

210

Chapter 13 Subordination and Bochner’s functional calculus

we find for c D kAuk=.2 kuk/ and u ¤ 0 Z kAf uk kT t u  uk kAuk aCb C .dt / kuk kuk kuk .0;1/ ³ ² Z kAuk kAuk C ; 2 .dt / aCb min t kuk kuk .0;1/  

Z kAuk 2e t kAuk aCb C .dt / 1  exp  kuk e  1 .0;1/ 2 kuk   1 kAuk 2e  f : e1 2 kuk Note that 2e=.e  1/  4; if we take f ./ D ˛ , ˛ 2 .0; 1/, (13.18) becomes (13.19). Corollary 13.9. Let f 2 BF and let .T tA / t0 , .T tB / t0 be C0 -contraction semigroups on the Banach space B with generators .A; D.A// and .B; D.B//, respectively. Assume that the operators A; B commute. Then the following estimate holds for u 2 D.A/ \ D.B/, u ¤ 0, 2e kAf u  B f uk  ' kuk e1



1 kAu  Buk 2 kuk

 (13.20)

where './ D f ./  f .0C/. Proof. By Phillips’ formula (13.10) we see for u 2 D.A/ \ D.B/ Z f f .T tA u  T tB u/ .dt /: A u  B u D b.A  B/u C .0;1/

From the theory of operator semigroups, see e.g. [286, Chapter I, Theorem 2.6, p. 6], we know that Z t Z t d A B B A B Tt u  Tt u D .T ts Ts /u ds D T ts .A  B/TsA u ds: 0 ds 0 Since B and A commute, so do the semigroups generated by them, and we find Z t A B B kT ts TsA .A  B/uk ds  t k.A  B/uk kT t u  T t uk  0

on D.A/ \ D.B/. On the other hand, we have always kT tA u  T tB uk  2 kuk. From now onwards we can argue exactly as in the proof of Corollary 13.8. Without proof we state the following result on the spectrum of a subordinate generator. It is again from Phillips [292, Theorem 4.4].

Section 13.1 Semigroups and subordination in the sense of Bochner

211

Theorem 13.10. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f be a Bernstein function. Then Af  f .A /. The following improvement of Theorem 13.10 is due to Hirsch [163, Théorème 16] and [162, pp. 195–196]. The extended spectrum of an operator A on B is the set N A contained in the one-point compactification C [ ¹1º of C such that N A D A if A is densely defined and bounded and N A D A [ ¹1º otherwise. Further, we extend f 2 CBF to C n .0; 1/ [ ¹1º by setting f .0/ WD f .0C/ 2 Œ0; 1/ and f .1/ WD lim f ./ 2 .0; 1: !1

Theorem 13.11. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f be a complete Bernstein function. Then it holds that N Af D f .N A /. Remark 13.12. If .T t / t0 gives rise to a transition function of a Markov process .X t / t0 , i.e. if T t u.x/ D P t u.x/ D Ex u.X t /;

u 2 Bb .E/;

then Bochner’s subordination has a probabilistic interpretation which parallels the more specialized situation in Remark 5.28 (ii). We know from Definition 5.4 and Proposition 5.5 that every f 2 BF defines a convolution semigroup . t / t0 of sub-probability measures on Œ0; 1/ or, equivalently, a (killed) subordinator b S D .b S t / t0 . Without loss of generality we can assume that b the processes X and S are defined on the same probability space and that they are independent. It is not difficult to check that the subordinate process f

.!/ WD Xb .!/ X t .!/ WD Xb S S .!/ t

t

is again a Markov process. Because of independence, the associated operator semigroup is given by Z P tf u.x/ D Ex u.Xb / D Ex u.Xs / P0 .b S t 2 ds/ St Œ0;1/ Z Ps u.x/  t .ds/: D Œ0;1/

f

f

Since T t D P t , we have T t D P t , and Bochner’s subordination can be interpreted as a stochastic time change with respect to an independent subordinator. An interesting special case is the subordination of Lévy processes.

212

Chapter 13 Subordination and Bochner’s functional calculus

Example 13.13. A stochastic process .X t / t0 with values in Rd is called a Lévy process if it has independent and stationary increments and if almost all sample paths t 7! X t .!/ are right-continuous and have left limits. It is well known that the transition function of a Lévy process is characterized by its Fourier transform which is of the form E0 e i X t D e t . / ; 2 Rd : The characteristic exponent W Rd ! C is continuous and negative definite in the sense of Schoenberg, cf. Chapter 4, in particular (4.8). Every continuous and negative definite function is uniquely determined by its Lévy–Khintchine formula, see (4.9) in Theorem 4.15 Denote by b S D .b S t / t0 and . t / t0 the (killed) subordinator and the convolution semigroup given by the Bernstein function f . Because of independence, we see Z Z f i X t i Xs D Ee  t .ds/ D e s . /  t .ds/ D e tf . . // : Ee Œ0;1/

Œ0;1/

A calculation similar to the one in the proof of Theorem 5.27 shows that f ı is again given by a Lévy–Khintchine formula. This means that f ı is a continuous and negative definite function and that a subordinate Lévy process is still a Lévy process. There is an interesting converse to the last remark of Example 13.13. This is due to Schoenberg [325]; we follow Bochner’s presentation [63, p. 99]. Theorem 13.14 (Schoenberg; Bochner). A function f W .0; 1/ ! Œ0; 1/ is a Bernstein function if, and only if, for all d 2 N the function 7! f .j j2 /, 2 Rd , is continuous and negative definite. Proof. Since 7! j j2 is continuous and negative definite in all dimensions, sufficiency follows from Example 13.13. Conversely, fix d 2 N and assume that f .j j2 / is continuous and negative definite. By Proposition 4.4 this is equivalent to saying that F t .j j2 / WD exp.tf .j j2 // is continuous and positive definite for all t > 0, and by Bochner’s theorem, Theorem 4.14, we know that F t .j j2 / is for fixed t > 0 a Fourier transform. This means that Z 2 e i y dt .dy/; t > 0; (13.21) F t .j j / D Rd

with a measure dt on Rd which is invariant under rotations and whose total mass dt .Rd / D F t .0/ D exp.tf .0// does not depend on the dimension d .

Section 13.1 Semigroups and subordination in the sense of Bochner

213

Write !d for the canonical surface measure on the unit sphere S d 1 in Rd and j!d j for the surface volume of S d 1 . Set G t . / WD F t .j j2 /; then G t .j j/ D G t . / for all  2 S d 1 . Therefore, we can radialise the integral in (13.21) to get Z 1 2 F t .j j / D G t . / D G t .j j/ !d .d/ j!d j j jD1 Z Z 1 D e ij j y dt .dy/ !d .d/ j!d j j jD1 Rd Z Z 1 e ij j y !d .d/ dt .dy/ D d j! j R j jD1 d Z H 1 .d 2/ .j j  jyj/ dt .dy/: D 2

Rd

The function H appearing in the last line is essentially a Bessel function of the first kind, see e.g. Stein and Weiss [337, p. 154], H .x/ D c J .x/ x  ;

(13.22)

where the constant c D limx!0 J .x/x  D 2 . C 1/ is chosen in such a way that H .0/ D 1, cf. [134, 8.440]. Therefore, H .x/ D

1 X

.1/j

j D0

. C 1/ .x=2/2j : .j C  C 1/ jŠ

Let  D .d / D .d  2/=2 and let td be the image measure of dt under the map ˆd W y 7! jyj2 =.4/. Then Z Z p p t H 1 .d 2/ .j j  jyj/ d .dy/ D H .2  j j r/ td .dr/ Rd

2

Œ0;1/

Œ0; 1// D dt .Rd / D F t .0/ is independent of d . This and td Œ0; 1/ D dt .ˆ1 d shows that for fixed t > 0 the measures .td /d 2N are vaguely bounded and by Theorem A.5 vaguely compact. Consequently, there are a subsequence .dk /k1 and a measure t such that vague- limk!1 td.k/ D t . Since all measures d.k/ have the same mass, we get from Theorem A.4 that weak- lim tdk D t k!1

and t Œ0; 1/ D F t .0/:

Without loss of generality we may assume that the whole sequence converges. Then Z p p H .2  j j r/ td .dr/ Œ0;1/ Z Z  p p 2 2  H .2  j j r/  e j j r td .dr/: e j j r td .dr/ C D Œ0;1/

Œ0;1/

214

Chapter 13 Subordination and Bochner’s functional calculus

From Lemma 13.15 below we see that the integrand of the second integral converges uniformly to 0 as d ! 1, hence  ! 1, while td .0; 1/ is independent of d . Since td converges weakly to t we get Z lim

d !1 Œ0;1/

p p H .2  j j r/ td .dr/ D

Z

e j j

2r

t .dr/

Œ0;1/

R

which shows that F t ./ D .0;1/ e r t .dr/, i.e. F t ./ is completely monotone for all t > 0. Since by our construction F t ./ D e tf ./ , we can use Theorem 3.7 to conclude that f is a Bernstein function. Lemma 13.15. (i) There exists a positive constant C > 0 such that for all  > 1 and all r > 0, p ˇ ˇ ˇH .2 2 r/ˇ  C : (13.23) r (ii) lim!1

p 2 H .2  r /e r r2

D 0 locally uniformly in r.

p 2 (iii) lim!1 supr 0 jH .2  r/  e r j D 0. Proof. (i) By (13.22) and [134, 8.411.9] the following integral formula is valid H .y/ D

2 . C 1/       C 12  12

Z 0

1

.1  t 2 /1=2 cos yt dt;

y > 0:

(13.24)

The second mean value theorem for integrals, cf. e.g. [323, Theorem E.22], shows that there exists  2 Œ0; 1 such that ˇ ˇ ˇˇZ ˇZ 1 ˇ 1 ˇ ˇ ˇ ˇ 2 1=2 ˇDˇ ˇ .1  t / cos yt dt cos yt dt ˇ ; ˇ ˇ ˇ ˇ y 0 0 implying that jH .y/j  Hence, for r > 0,

Since lim!1

2 . C 1/ 1     :   C 12  12 y

ˇ  p ˇ 1 . C 1/ ˇH 2  r ˇ  p p :  r   C 12  

.C1/ p .C 21 / 

D 1, see [134, 8.328.2], we get (13.23).

215

Section 13.1 Semigroups and subordination in the sense of Bochner

(ii) We have

!ˇˇ ˇ . C ˇ  . C 1 C j / j Š j Š ˇˇ ! 1 X r 2j . C 1/  j 1  . C 1 C j / j Š j D1 ! 1 j C1 X r 2j . C 1/   r2 : 1 . C 2 C j / .j C 1/Š

ˇ ˇ1 ˇX p r 2 jH .2  r/  e j D ˇˇ .1/j ˇj D1

1/  j

r 2j

r 2j

j D0

j C1

 Since 1  .C1/ is positive and converges to zero as  ! 1, the claim follows .C2Cj / from the dominated convergence theorem. 2 (iii) Fix A > 1. Because of (i) and the elementary inequality e r  e=r 2 we get ˇ ˇ ˇC ˇ p p e r 2 r 2 sup jH .2  r/  e j  sup jH .2  r/  e j C sup ˇˇ C 2 ˇˇ : r r 0 r >A r r 2Œ0;A

By (ii) the first term tends to zero as  ! 1. Letting A ! 1 proves the claim. Example 13.16. Let W Rd ! C be the characteristic exponent of a Lévy process .X t / t0 with values in Rd . It is given by the Lévy-Khintchine formula   Z 1 i  y i y . / D iˇ  C  Q C C 1e .dy/: (13.25) 2 1 C jyj2 y¤0 Assume that . / depends on the radial part j j only, and that the Lévy measure  is of the form .dy/ D n.y/ dy with Z 2 e sjyj .ds/; (13.26) n.y/ D .0;1/

where  is a measure on .0; 1/. In particular, is real valued and we have ˇ D 0, 1 2 2  Q D bj j for some b  0. Therefore, Z  Z   2 1  cos.  y/ e sjyj .ds/ dy . / D b j j2 C Rd .0;1/  Z Z   2 2 sjyj 1  cos.  y/ dy .ds/ D b j j C e .0;1/ Rd  Z Z   2 sjyj2 i y 1e dy .ds/ e D b j j C Re .0;1/ Rd Z   jj2  d=2 s d=2 1  e  4s .ds/ D b j j2 C Z.0;1/   2 2 .dt /; D b j j C .4 t /d=2 1  e tj j b .0;1/

216

Chapter 13 Subordination and Bochner’s functional calculus

where b  is the image measure of  with respect t D .4s/1 . Let R to the mapping d=2 t .dt / D .4 t / b .dt / and set f ./ D b C .0;1/ .1  e / .dt /,  > 0. Then f is a Bernstein function and . / D f .j j2 / for all ¤ 0 and .0/ D f .0C/ D 0. Conversely, if . / D f .j j2 / with f 2 BF, then by retracing the steps we conclude that the Lévy measure  is of the form .dy/ D n.jyj/dy where n is given by (13.26). The discussion above can be summarized as follows. Let be given by (13.25). 2 Then . / p D f .j j / for f 2 BF if, and only if, Q D b id and .dy/ D n.jyj/ dy where n. / is completely monotone. In probabilistic terms this means that X t is f a subordinate Brownian motion, X t D Bp t , if, and only if, Q D b id and the Lévy measure has a density n.jyj/ such that n. / is completely monotone. In dimension d D 1 we have the following similar statement for complete Bernstein functions: . / D f .j j2 / for f 2 CBF if, and only if, ˇ D 0, Q D b id and .dy/ D n.jyj/ dy where n./ is completely monotone. This follows from Z R

  e sjyj 1  cos. y/ dy D

2 2 ; . 2 C s 2 / s

hence, Z 2

. / D b C

.0;1/

2 2 .ds/ D b 2 C 2 2

Cs s

Z .0;1/

2  .dt /

2 C t

for some measure  .

13.2 A functional calculus for generators of semigroups We continue our study of C0 -contraction semigroups .T t / t0 on a Banach space B. In this section we will develop a functional calculus for the generators f .Af ; D.Af //, f 2 BF, of the subordinate semigroup .T t / t0 . This calculus is a natural extension of the spectral calculus in Hilbert spaces, and in many cases it is possible to interpret the subordinate generator Af as the operator f .A/. In Hilbert spaces and for self-adjoint semigroups this follows immediately from the familiar spectral calculus for self-adjoint operators, see Examples 12.5, 12.6 and 12.7. In Banach spaces, one can use the Dunford–Riesz functional calculus to express Af as an unbounded Cauchy integral and to identify this operator with f .A/, see [36, 298] and [320, 321]. The particularly interesting case of fractional powers of dissipative operators is included in this calculus if we take the (complete) Bernstein functions f˛ ./ D ˛ , ˛ 2 .0; 1/. Phillips’ formula (13.10) reads Z 1 ˛ ˛ f˛ .T t u  u/ t ˛1 dt; u 2 D.A/; (13.27) .A/ u D A u D .1  ˛/ 0

Section 13.2 A functional calculus for generators of semigroups

217

and, if we use the Stieltjes representation (6.7) of f˛ ./, we get from Corollary 13.22 below Balakrishnan’s famous formula for the fractional power of a dissipative operator Z sin.˛/ 1 AR t u t ˛1 dt; u 2 D.A/: (13.28) .A/˛ u D Af˛ u D  0 The main result for the functional calculus, Theorem 13.23, shows that Af˛ Afˇ D Af˛Cˇ ;

that is

.A/˛ .A/ˇ D .A/˛Cˇ ;

holds on D.A/ for ˛; ˇ  0 such that ˛ C ˇ  1; Corollary 13.28 extends this to all ˛; ˇ  0. We begin with a few preparations. Lemma 13.17. Let f be a Bernstein function, gk ./ WD k.k C /1 , k 2 N, and set fk WD gk ı f . Then fk 2 BF and k fk D D L k f kCf k!1

for a sub-probability measure k on Œ0; 1/ such that k ! ı0 weakly. Proof. Since gk ; f 2 BF, the composition fk D gk ı f is in BF. Moreover, kf

k fk kCf D D f f kCf which is completely monotone since it is the composition of the completely monotone function  7! k.k C /1 with f 2 BF, cf. Theorem 3.7 (ii). Therefore, there exists a measure k on Œ0; 1/ with L k D fk =f . By monotone convergence, k Œ0; 1/ D lim L .k I / D lim !0

!0

fk ./ k D lim f ./ !0 k C f ./ k D  1: k C f .0C/

Weak convergence follows from limk!1 fk ./=f ./ D 1 and Lemma A.9. Denote by . t / t0 the convolution semigroup of measures on Œ0; 1/ associated with the Bernstein function f , and write Z 1 e kt  t dt; k 2 N; (13.29) Uk D 0

for the k-potential measure, cf. (5.21).

218

Chapter 13 Subordination and Bochner’s functional calculus

Lemma 13.18. Let fk ; f and k be as in Lemma 13.17 and let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and resolvent .R />0 . Then Z Ts u k .ds/; k 2 N; u 2 B; Ik u WD Œ0;1/

defines a family of bounded operators on B and Ik D kRkf ;

k 2 N;

f

where Rk are the subordinate resolvent operators. In particular, we have for all u 2 B that strong- limk!1 Ik u D u and Ik u 2 D.Af /. Proof. Observe that (5.20) with f replaced by k C f shows L Uk D

1 : kCf

Thus, L .kUk / D k.k C f /1 D L k , and we conclude that kUk D k . Since t 7! T t u is strongly continuous, this proves for all u 2 B Z Z 1Z Ts u k .ds/ D k Ts u e kt  t .ds/ dt Ik u D 0 Œ0;1/ Œ0;1/ Z 1 f e kt T t u dt Dk D

0 f kRk u:

R1

e kt dt D 1 we find Z 1 Z f ke kt kT t u  uk dt D kIk u  uk 

Using k

0

0

0

1

e s kTs=k u  uk ds: f

k!1

Because of the strong continuity of the subordinate semigroup we get Ik u ! u in the strong sense, and a similar calculation proves that kIk k  1. That Ik u 2 D.Af / f follows from the mapping properties of the resolvent Rk . Proposition 13.19. Let fk ; f 2 BF be as in Lemma 13.17, let Ik be the operator from Lemma 13.18 and denote by Afk and Af the corresponding subordinate generf ators and by .R />0 the resolvent of Af . Then Afk is a bounded operator, f

Afk u D Af Ik D kAf Rk u; D Ik Af D kRkf Af u;

k 2 N; u 2 B;

(13.30)

k 2 N; u 2 D.Af /;

(13.31)

219

Section 13.2 A functional calculus for generators of semigroups

and u 2 D.Af / if, and only if, .Afk u/k2N converges strongly; if this is the case, then Af u D limk!1 Afk u. Moreover, I` Afk D Afk I`

and Afk I` u D Af` Ik u;

k; ` 2 N; u 2 B:

(13.32)

Proof. The boundedness of Afk follows from the boundedness of the function fk and Corollary 13.7. By Lemma 13.17 Z   kf ./ fk ./ D D k 1  L .k I / D ak C .1  e s / kk .ds/ k C f ./ .0;1/ with ak D k.1  k Œ0; 1//. A combination of Phillips’ theorem, Theorem 13.6, for fk and Lemma 13.18 gives for u 2 D.A/ Z .Ts u  u/ kk .ds/ Afk u D ak u C .0;1/ (13.33) 2 f f f D ku C kIk u D ku C k Rk u D kA Rk u: Since the left and right-hand sides define bounded operators, this equality extends f to all u 2 B, and (13.30) follows. Using the commutativity of the operators Rk f

k!1

and Af on D.Af /, we get (13.31). By Lemma 13.18, kRk u ! u strongly. If u 2 D.Af /, (13.31) shows that limk!1 Afk u D Af u strongly. Conversely, if limk!1 Afk u converges in the strong sense, (13.30) and the closedness of the operator Af prove that u 2 D.Af /. f For (13.32) we use I` D `R` and (13.30) to get f

f

Afk I` D `kAf R` Rk D Af` Ik where the second equality follows from the symmetric roles of k and ` in the middle f f term. Since Af , R` and Rk commute, we also get Afk I` D I` Afk . Corollary 13.20. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with resolvent .R />0 and generator .A; D.A//. For every Bernstein function f it holds that ² ³ f fk D.A / D u 2 B W lim A u exists strongly k!1 ² ³ Z 2 D u 2 B W lim .T t u  u/ k Uk .dt / exists strongly k!1 .0;1/ ³ ² D u 2 B W lim .kIk u  ku/ exists strongly ; k!1

f

where fk D kf =.k C f /, Ik D kRk , and Uk is the k-potential measure (13.29) associated with the Bernstein function f .

220

Chapter 13 Subordination and Bochner’s functional calculus

Proof. From Proposition 13.19 we know that u 2 D.Af / if, and only if, the limit limk!1 Afk u exists strongly. Moreover, the k-potential measure Uk is k 1 k and Z   fk .Ts u  u/ k 2 Uk .ds/: A u D k 1  kUk Œ0; 1/ u C .0;1/

Since Uk Œ0; 1/ D L .Uk I 0C/ D .k C f .0C//1 , we see     kf .0C/ k!1 k D k 1  kUk Œ0; 1/ D k 1  ! f .0C/: k C f .0C/ k C f .0C/ This proves the second characterization of D.Af /; the last one follows immediately from (13.33). Remark 13.21. A close inspection of the proof of Proposition 13.19 shows that we f can replace the approximating operator Ik D kRk by any other bounded operator Jk which is of the form Z Ts u k .ds/; u 2 B; Jk u D Œ0;1/

where k is a family of (signed) measures on Œ0; 1/ which have uniformly bounded variation and satisfy L k D fk =f where fk 2 BF b with limk!1 fk ./ D f ./. Any such Jk satisfies strong- limk!1 Jk u D u for all u 2 B as well as (13.32), which is all we need for the proofs of Proposition 13.19 and Corollary 13.20. The first assertion follows since .T t / t0 is strongly continuous and since the k converge weakly to ı0 , the second assertion is immediate from the fact that each T t , t  0, commutes with A, R and the subordinate analogues. An example of such a situation is given in [322] for f 2 CBF. Let us rewrite Corollary 13.20 for a complete Bernstein function f . In this case R we have f ./ D a C b C .0;1/ .s C /1  .ds/, with a; b  0 and the Stieltjes R representation measure  on .0; 1/ satisfying .0;1/ .1 C s/1  .ds/ < 1, see (6.7); the Lévy measure .dt / has a density m.t / which is given by m.t / D L .s  .ds/I t /. Corollary 13.22. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with resolvent .R />0 and generator .A; D.A//, and let f 2 CBF with Stieltjes representation given above. Then ² ³ Z f D.A / D u 2 B W lim ARs u  .ds/ exists strongly k!1 .0;k/

and we have for u 2 D.A/

Z

Af u D au C bAu C Z D au C bAu C

.0;1/

.0;1/

.sRs u  u/  .ds/

(13.34)

ARs u  .ds/:

(13.35)

Section 13.2 A functional calculus for generators of semigroups

221

Proof. Applying Phillips’ formula (13.10) for f 2 CBF we see for u 2 D.A/ Z 1 f .T t u  u/ m.t / dt A u D au C bAu C Z0 1 Z .T t u  u/e st s  .ds/ dt D au C bAu C .0;1/ 0  Z 1 Z se st T t u dt  u  .ds/ D au C bAu C .0;1/ 0 Z .sRs u  u/  .ds/ D au C bAu C .0;1/

and this proves (13.34), and (13.35) follows as sRs u  u D ARs u. In order to justify the interchange of integrals in the calculation above, we use the estimate (13.3) and observe that the iterated integrals are finite for u 2 D.A/: Z Z 1 kT t u  uk e st dt s  .ds/ .0;1/ 0 Z Z 1 ® ¯  min t kAuk; 2kuk e st dt s  .ds/ .0;1/ 0 Z Z Z 1 Z 1 st se dt  .ds/kuk C st e st dt  .ds/kAuk 2 .0;1/ 0 Œ1;1/ 0 Z Z 1  .ds/ kAuk < 1:  .ds/ kuk C D2 .0;1/ Œ1;1/ s In order to see the assertion for the domain, we use Remark 13.21 with Z  fk ./ D  .ds/: .0;k/ s C  From this it is possible to compute I.z/ D Im Œ.f .z/  fk .z//=f .z/ and an elementary but lengthy calculation shows for z D x C iy 2 H" 

Z Z 1 .t s/y.x 2 Cy 2 /  .ds/  .dt /: I.z/ D jfk .z/j t2.k;1/ s2.0;k/ ..t Cx/2 Cy 2 /..s Cx/2 Cy 2 / The integrand is positive since Im z D y > 0. Moreover, by monotone convergence we see R R 1 1 f ./  fk ./ Œk;1/ sC  .ds/ !0C Œk;1/ s  .ds/ D R       ! 2 Œ0; 1/: R 1 1 fk ./ .0;k/ sC  .ds/ .0;k/ s  .ds/ Therefore, Theorem 6.2 (iv) shows that .f  fk /=fk is in CBF, and consequently f =fk D 1 C .f  fk /=fk 2 CBF, and fk =f 2 S by Theorem 7.3. This proves that fk =f D L k for a sub-probability measure k on Œ0; 1/. The form of D.Af / follows now from Remark 13.21 and Corollary 13.20.

222

Chapter 13 Subordination and Bochner’s functional calculus

We can use the knowledge of the structure of the domain D.Af / to construct a functional calculus for semigroup generators. Theorem 13.23. Let .A; D.A// be the generator of a C0 -semigroup .T t / t0 on the Banach space B, and let f; g 2 BF. Then we have (i) Acf D cAf for all c > 0; (ii) Af Cg D Af C Ag ; (iii) Af ıg D .Ag /f ; (iv) AcCid Cf D c id CA C Af for all c  0; (v) if fg 2 BF, then Afg D Af Ag D Ag Af . The equalities in (i)–(v) are identities in the sense of closed operators, including their domains which are the usual domains for sums, compositions etc. of closed operators. Before we proceed with the proof of Theorem 13.23 let us mention the following useful corollaries. Both of them are immediate consequences of Theorem 13.23 (v) in conjunction with Proposition 7.13 and Definition 11.1, respectively. Corollary 13.24. Let .A; D.A// be as in Theorem 13.23 and assume that f and g are in CBF. Then f ˛ g 1˛ 2 CBF for all ˛ 2 .0; 1/ and Af

˛ g 1˛

˛

D Af Ag

1˛

D Ag

1˛

˛

Af :

Corollary 13.25. Let .A; D.A// be as in Theorem 13.23 and assume that f; f ? are conjugate functions from SBF, cf. Definition 11.1, i.e. f ? ./ D =f ./. Then ?

?

A D Af Af D Af Af : Proof of Theorem 13.23. For f; g 2 BF we know that cf; f C g; f ı g 2 BF. Since D.A/  D.Af / \ D.Ag /, all of the above operators are densely defined. By the linearity of the definition of .Af ; D.Af //, see Corollary 13.20, (i) and (ii) are immef ıg diate, (iii) is a consequence of the transitivity of the subordination: T t D .T tg /f , see Lemma 13.3. For (iv) we note that Af is A-bounded in the sense that kAf uk  ckAuk C c 0 kuk for c; c 0 > 0 and all u 2 D.A/, cf. (13.17). Thus, .c C A C Af ; D.A// is a closed operator, and the assertion follows from Phillips’ formula (13.10). For (v) we assume that fg 2 BF. Write h for any of the Bernstein functions f; g or fg and set, as in Lemma 13.17, hk WD

kh kCh

and

hk D L h;k ; h

k 2 N;

(13.36)

Section 13.2 A functional calculus for generators of semigroups

223

for a suitable sub-probability measure h;k on Œ0; 1/. Note that fk g` L fg;m D .fg/m L f;k L g;`

(13.37)

holds for all k; `; m 2 N. By (13.36),     k hk D k 1  D L kı0  kh;k kCh allows to rewrite the identity (13.37) in terms of convolutions of (signed) measures k`.f;k  ı0 / ? .g;`  ı0 / ? fg;m D m.fg;m  ı0 / ? f;k ? g;` :

(13.38)

We want to show that f

fg mAfk Ag` Rm u D k`A.fg/m Rk R`g u:

(13.39)

Applying Phillips’ formula (13.10) to Ahk shows Z   .Ts u  u/ kh;k .ds/ Ahk u D k 1  h;k Œ0; 1/ u C .0;1/ Z Ts u kh;k .ds/  ku D Œ0;1/ Z Ts u .h;k  ı0 /.ds/; Dk Œ0;1/

and from Lemma 13.18 we know that Z h kRk u D

Ts u h;k .ds/:

Œ0;1/

Since the operators Ahk and Rkh , where h stands for f; g or fg, are bounded we can freely interchange the order of integration in the calculation below. fg mAfk Ag` Rm u Z Z Z D k` Tr u .f;k  ı0 /.dr/ Ts u .g;`  ı0 /.ds/ T t u fg;m .dt / Œ0;1/ Œ0;1/ Œ0;1/ Z Z Z Tr CsCt u .f;k  ı0 /.dr/ .g;`  ı0 /.ds/ fg;m .dt / D k` Œ0;1/ Œ0;1/ Œ0;1/ Z Ts u .f;k  ı0 / ? .g;`  ı0 / ? fg;m .ds/: D k` Œ0;1/

A very similar calculation leads to f

g

k`A.fg/m Rk R` u D m

Z Œ0;1/

Ts u .fg;m  ı0 / ? f;k ? g;` .ds/

224

Chapter 13 Subordination and Bochner’s functional calculus

and the convolution identity (13.38) shows that (13.39) holds. Since (13.39) is an equality between bounded operators, it extends from D.A/ to all u 2 B. Using (13.39) we can now prove (v). Recall from operator theory that Af Ag is a closed operator on D.Af Ag / D ¹u 2 B W u 2 D.Ag / and Ag u 2 D.Af /º. If u 2 D.Af Ag /, we find by letting ` ! 1, k ! 1 and then m ! 1 in (13.39) that u 2 D.Afg /, see Corollary 13.20. This shows that Afg extends Af Ag . Conversely, if f 2 D.Afg /, see Corollary 13.20, we first let m ! 1 in (13.39) and then ` ! 1 and k ! 1. This proves that u 2 D.Af Ag / and that Af Ag extends Afg . Therefore Af Ag D Afg and, since f and g play symmetric roles, also Ag Af D Afg . Remark 13.26. The proof of Theorem 13.23 (v) shows that the functional calculus is also an operational calculus. In fact, (13.39) is derived from the identity (13.37) between functions from BFb and CMb or from the corresponding identity for (signed) measures (13.38). In the proof of Theorem 13.23 we see that it is sufficient to verify the operator identity at the level of CMb and BFb , i.e. for the trivial semigroup .e t / t0 rather than the operator semigroup .T t / t0 . The requirement that fg 2 BF for f; g 2 BF in Theorem 13.23 (v) is quite restrictive. We can overcome this difficulty if f; g are complete Bernstein functions by using the log-convexity of CBF, see Proposition 7.13, i.e. f ˛ g 1˛ 2 CBF

for all f; g 2 CBF; ˛ 2 .0; 1/:

(13.40)

This, together with Theorem 13.23 enables us to write down a consistent formula for the operator .fg/.A/ and all f; g 2 CBF – even if Afg has no longer any meaning since it is not the generator of a subordinate semigroup if fg 62 BF. Definition 13.27. For f1 ; f2 ;    ; fn 2 CBF, n 2 N, we set  1=n 1=n 1=n n .f1  f2    fn /.A/ D .Af1 /.Af2 /    .Afn /

(13.41)

on the natural domain for compositions of closed operators. Let A D ¹f1 f2    fn W n 2 N; fj 2 CBF; 0  j  nº denote the set of all finite products of complete Bernstein functions. We want to show that the above definition extends Theorem 13.23 (v) to the set A. First, however, we have to check that (13.41) is independent of the representation of F 2 A. Assume that F D f1 f2    fn and F D g1 g2    gm , m  n, fj ; gj 2 CBF, are two representations of F . Extending (13.40) to n-fold products we see by Theorem 13.23 (v)  1=n 1=n 1=n n 2 1=n 1=n .Af1 /.Af2 /    .Afn / D .1/.n / ŒAF n D ŒAF n ;

225

Section 13.2 A functional calculus for generators of semigroups 2

where we used that .1/n D .1/.n / . A similar formula holds for the other representation of F . In particular, F 1=n ; F 1=m 2 CBF and it remains to show that 1=n 1=m m  . This, however, follows from ŒAF n D ŒAF 1=m  1=n 1=n 1=m D ŒAF n=m D AF ; ŒAF n where the first equality is well known for (arbitrary) powers of closed operators, while the second one comes from Theorem 13.23 (iii). Thus, (13.41) is a well defined closed operator for all F 2 A and the following corollary shows that Definition 13.27 extends Theorem 13.23 (v). Corollary 13.28. Let f; g 2 CBF and let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A//. For H D fg 2 A it holds that H.A/ D Af Ag D Ag Af :

(13.42)

More generally, if H D F G with F; G 2 A, then H.A/ D F .A/G.A/ D G.A/F .A/:

(13.43)

The above equalities are identities between closed operators. p

Proof. p Put B D Ap fg p . By definition, H.A/ D B 2 , and by Theorem 13.23 (v) p p B D A fg D A f A g D A g A f . Since p p p p p p ¯ ® D.B 2 / D u 2 B W u 2 D.A f g / and A f g u 2 D.A f g / p p p ® D u 2 B W u 2 D.A g /; A g u 2 D.A f /; p

A

f

A

p g

u 2 D.A

p

g

/ and A

p

g

p

A

f

A

p g

u 2 D.A

p

f

¯ / ;

we get B 2 D .A

p

f

A

p g

p

/.A

f

A

p g

/ D .A

p

f

A

p f

/.A

p

g

A

p

g

/ D Af Ag D Ag Af :

This proves (13.42). Iterating this identity we get (13.43) first for F 2 CBF and G D g1 g2    gm with gj 2 CBF, and then for F; G 2 A. We continue with a study of the domain D.Af / of the subordinate generator. For this it is important to understand the asymptotic behaviour of kT t u  uk for t ! 0 and u 2 D.Af /. Let f 2 BF and write U for the potential measure, i.e. L U D 1=f . Then ˇs WD f .s/.U  ı1=s ? U /;

s > 0;

(13.44)

226

Chapter 13 Subordination and Bochner’s functional calculus

is a family of (signed) measures and L .ˇs I / D

f .s/ .1  e =s /: f ./

The next lemma is stated for special Bernstein functions; recall that CBF  SBF. Lemma 13.29. Let .ˇs /s>0 be as in (13.44) and assume that f 2 SBF. Then the total variation of the measures ˇs is uniformly bounded by 2.e C 1/. a nonProof. We know from Theorem 11.3 that U.dt R 1 / D c ı0 .dt / C u.t / dt with ac increasing density u.t /, t > 0, such that 0 u.t / dt < 1. Writing ˇs for the absolutely continuous part of ˇs we get kˇs kT V  f .s/kcı0  cı1=s kT V C kˇsac kT V  2cf .s/ C kˇsac kT V and

ˇ ˇ ˇ ˇ dˇs .t / ˇ dt ˇ dt ˇ 0 "Z Z 1=s u.t / dt C D f .s/ Z

kˇsac kT V D

1ˇ

Z D 2f .s/

0

1 1=s

 u.t  1=s/  u.t / dt

#

1=s

u.t / dt 0

Z

 2f .s/ e  2 ef .s/

1=s

e st u.t / dt

0

1 D 2e: f .s/

As f ./ D 1=L .cı0 .dt /Cu.t / dt I /, we have sup>0 f ./ D lim!1 f ./ D c 1 , i.e. f is bounded if, and only if, c > 0. This shows that the total variation is bounded by 2 C 2e or by 2e, depending on f being bounded or unbounded. Lemma 13.30. Let f 2 SBF with potential measure U , set fk D kf =.kCf / and let .ˇs /s>0 be as in (13.44). For every C0 -contraction semigroup .T t / t0 on the Banach space B with generator .A; D.A// the operators Z T t u ˇs .dt /; u 2 B; Œ0;1/

are uniformly bounded, and the following identity holds Z Z T t u k .dt / D Afk T t u ˇs .dt /; f .s/ .T1=s  id/ Œ0;1/

where fk =f D L k as in Lemma 13.17.

Œ0;1/

(13.45)

227

Section 13.2 A functional calculus for generators of semigroups

Proof. The uniform boundedness follows directly from Lemma 13.29, while (13.45) is a consequence of Remark 13.26 and the identity fk ./ f ./ D fk ./ f .s/ L .ı1=s ? U  U I /

f .s/ .e =s  1/L .k I / D f .s/ .e =s  1/

D fk ./ L .ˇs I /: We will now combine Lemma 13.30 with the technique developed in Proposition 13.19 and the proof of Theorem 13.23 (v) to derive the next result. Theorem 13.31. Let f 2 SBF, let .ˇs /s>0 be as above and let .T t / t0 be a C0 contraction semigroup on the Banach space B with generator .A; D.A//. Then Z f T t u ˇs .dt / s > 0; u 2 B; (13.46) f .s/ .T1=s u  u/ D A Œ0;1/ Z T t Af u ˇs .dt / s > 0; u 2 D.Af /: (13.47) D Œ0;1/

Proof. Recall that limk!1 Afk v existsRif, and only if, v 2 D.Af /. Therefore, the limit k ! 1 in (13.45) shows that v D Œ0;1/ T t u ˇs .dt / 2 D.Af / and that (13.46) holds. This implies (13.47) for all u 2 D.Af /, since Af is a closed operator. Corollary 13.32. Let f 2 SBF and let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A//. Then we have kT t u  uk 

2.e C 1/ kAf uk; f .t 1 /

u 2 D.Af /:

(13.48)

Proof. If u 2 D.Af /, (13.47) gives for t D 1=s, kT t u  uk 

1 kˇ1=t kT V kAf uk; f .t 1 /

and the assertion follows because sups>0 kˇs kT V  2.e C 1/, see Lemma 13.29. The following corollary contains the converse of Corollary 13.7. Corollary 13.33. Let f 2 SBF and let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A//. The subordinate generator Af is bounded if, and only if, A is bounded or f is bounded.

228

Chapter 13 Subordination and Bochner’s functional calculus

Proof. The sufficiency has already been established (under less restrictive assumptions) in Corollary 13.7. In order to see the necessity, let Af be a bounded operator, i.e. kAf uk  c kuk for some c > 0 and all u 2 B. Suppose that lim!1 f ./ D 1. By Corollary 13.32, kT u  uk 2.e C 1/ kAf uk 2.e C 1/c   kuk f . 1 / kuk f . 1 / for all u 2 B, u ¤ 0. Thus, lim sup

!0 0¤u2B

kT u  uk 2.e C 1/c 2.e C 1/c  lim D lim D 0: !0 f . 1 / kuk f ./ !1

which shows that .T t / t0 is continuous in the uniform operator topology. This means that the generator A of .T t / t0 is bounded, cf. [286, p. 2, Theorem 1.2]. Our next corollary is a generalization of the well known relation for fractional powers D..A/ˇ /  D..A/˛ / if 0 < ˛  ˇ  1. Corollary 13.34. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f 2 SBF, g 2 BF with Lévy triplets .a; b; f / and .˛; 0; g /. If Z 1 g .dt / < 1 for some  > 0; (13.49) 1 / .0;/ f .t then D.Af /  D.Ag /. Proof. If u 2 D.A/  D.Af /, we can use Corollary 13.32 and get for sufficiently small  > 0   Z   g  .T u  u/  .dt / ˛u C kAg uk D  t   .0;1/   Z Z g g kT t u  uk  .dt / C ˛ C 2  .dt / kuk  .0;/ Œ;1/ Z   1 g .dt / kAf uk C ˛ C 2g Œ; 1/ kuk:  2.e C 1/ 1 / .0;/ f .t Because of our assumption, the integral above converges, and since D.A/ is an operator core for Af and Ag , the inequality extends to all u 2 D.Af /. This shows that u 2 D.Ag /.

Section 13.2 A functional calculus for generators of semigroups

229

Corollary 13.35. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A// and let f 2 SBF, g 2 BF with Lévy triplets .a; b; f / and .˛; 0; g /. If Z 1 0 f ./g./ g./ d  < 1 and D O.1/ as  ! 1; (13.50) 2 ./ f f ./ 1 then D.Af /  D.Ag /. Proof. In the notation of Corollary 13.34 it is easy to see that Z 1 e g g.ı 1 / .1  e s=ı / g .ds/   Œı; /  .1  e 1 / Œı;1/ e1 holds for all 0 < ı < . Note that Z Z 1 1 g  .dt / D g .dt /; lim 1 1 / f .t / f .t ı!0 Œı;/ .0;/ whenever the integral on the right-hand side exists. Integrating by parts we get Z Z g Œı; / 1 t 2 f 0 .t 1 / g g  C  Œt; / dt .dt / D 1 / 2 1 / f .ı 1 / Œı;/ f .t Œı;/ f .t   Z t 2 f 0 .t 1 /g.t 1 / g.ı 1 / e C dt  e  1 f .ı 1 / f 2 .t 1 / Œı;/ for  > 0 and ı 2 .0; /. This shows that (13.50) implies (13.49) and the assertion follows from Corollary 13.34. Corollary 13.35 can be further simplified. Since Bernstein functions are nondecreasing and concave, we have 1 f 0 ./  : f ./  Therefore the integrand appearing in (13.50) can be estimated by g./ 1 f 0 ./g./   : 2 f ./  f ./ The expression on the right-hand side is the product of two completely monotone functions, see Corollary 3.8 (iv) and Theorem 3.7 (ii) for the first and the second factor, respectively. In particular, g./=.f .// isRnon-increasing. 1 Using a standard Abelian argument shows that 1 g./=.f .// d  < 1 always entails that, for some constant C > 0, C g./  f ./ 

for all  > 1:

230

Chapter 13 Subordination and Bochner’s functional calculus

Thus, g./=f ./ D O.1/ as  ! 1. If .˛; ˇ; g / is the Lévy triplet of g, then by Corollary 3.8 (viii), there exists c > 0 such that g./ ˛ C ˇ ˇ  ;  2 f ./ c c

 > 1:

From this we conclude that ˇ D 0, and we have proved the following result. Corollary 13.36. Let f 2 SBF, g 2 BF and let .T t / t0 be a C0 -semigroup on the Banach space B with generator .A; D.A//. If Z 1 g./ d  < 1; (13.51) f ./ 1 then D.Af /  D.Ag /. In particular, if for some  > 0 the ratio g./=f ./ D O. / as  ! 1, the condition (13.51) holds. The following result shows that the functional calculus is stable under pointwise limits of Bernstein functions. Theorem 13.37. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B with generator .A; D.A//. Assume that .f n /n2N is a sequence of Bernstein functions such that the limit f ./ D limn!1 f n ./ exists. Then n

strong- lim Af u D Af u; n!1

for all u 2 D.A2 /:

Proof. Let f .0C/ D a > 0 and write fk D kf =.k C f / and fkn D kf n =.k C f n /. Each k=.kCf n / is a completely monotone function which is bounded by 1; therefore, it is the Laplace transform of a sub-probability measure kn and we see    fkn k k kf n k C f D C1 1 D fk k C f n kf f kCfn     k k C 1  L kn  C1 : D f f Since 1=f D L U 2 CM, cf. (5.20), we see that fkn =fk D L kn with the signed measure kn D .kU C ı0 /  kn ? .kU C ı0 /. The total variations of the signed measures . kn /n2N satisfy   k n C 1 < 1 for every k 2 N: sup k k kT V  2 f .0C/ n2N Now Remark 13.21 applies to the sequence .fkn /n2N and shows that n

strong- lim Afk u D Afk u for all k 2 N; u 2 D.A/: n!1

231

Section 13.2 A functional calculus for generators of semigroups

Let .an ; b n ; n / denote the Lévy triplet of f n . Recall from the proof of Phillips’ theorem, Theorem 13.6, the estimate   Z   fn n n kA uk  b C s  .ds/ kAuk C an C n Œ1; 1/ kuk: .0;1/

Combining this with the elementary convexity inequality min¹1; sº  s > 0, we get n

kAf uk 

2 e f n .1/ .kAuk C kuk/  c .kAuk C kuk/ ; e1

e e1

.1  e s /,

n 2 N; u 2 D.A/;

where c is independent of n 2 N. Because of Proposition 13.19 we have n

n

fn

n

n

fn

n

n

Af u  Afk u D Af u  kRk Af u D Rk Af Af u which implies that for all u 2 D.A2 /  1 C  2 n n kAf Af uk  kA uk C kAuk C kuk : k k

n

n

kAf u  Afk uk 

The same argument shows that the inequality is valid for kAf u  Afk uk with the same constant C . Therefore, n

n

n

n

kAf u  Af uk  kAf u  Afk uk C kAfk u  Afk uk C kAfk u  Af uk  2C  2 n kA uk C kAuk C kuk C kAfk u  Afk uk:  k n

Letting n ! 1 and then k ! 1 proves that limn!1 Af u D Af u on D.A2 /. If f .0C/ D 0, we replace f and f n in the above calculations by f C and f n C, respectively. Because of Theorem 13.23 n

n

n!1

u C Af u D ACf u ! ACf u D u C Af u;

u 2 D.A2 /;

and the general case follows. Remark 13.38. It is straightforward to extend the functional calculus to Stieltjes functions. Let .A; D.A// be the generator of a C0 -contraction semigroup .T t / t0 on the Banach space B and denote by .R />0 the resolvent. We define the zero-resolvent and the potential operator by Z n R0 u WD lim R u; and V u WD lim T t u dt; !0

n!1 0

232

Chapter 13 Subordination and Bochner’s functional calculus

with domains D.R0 / and D.V /, respectively, comprising all u 2 B where the limits above exist in the strong sense. The following facts are well known, see e.g. [39, Chapter II.11]. The zero-resolvent .R0 ; D.R0 // is an extension of the potential operator .V; D.V // and R0 is densely defined if, and only if, lim!0 R u D 0 for all u 2 B. Moreover, the range R.R0 /  D.A/, and on D.R0 / we have AR0 u D u. Since ² ³ R.A/ D u 2 B W lim R u D 0 ; !0

R0 is densely defined if, and only if, the range R.A/ is dense in B. If R0 is densely defined we have R0 D A1 . Assume, therefore, that D.R0 / is dense in B. R For every  2 S given by ./ D a1 C b C .0;1/ . C t /1  .dt / we define Z .A/u WD aR0 u C bu C

R t u  .dt /; .0;1/

u 2 R.A/:

Since for u D Av, v 2 D.A/, it holds that R0 u D R0 Av D v and R t u D R t Av, we see that .A/u D Af v; for all u 2 R.A/; u D Av; where f is the complete Bernstein function given by Z  f ./ D a C b C  .dt /: .0;1/ t C  Denote by f ? the conjugate of the complete Bernstein function f , i.e. the unique function f ? 2 CBF such that f ./f ? ./ D . Then we see for all u 2 R.A/ ?

?

u D Av D Af Af v D Af .A/u ?

which means that .A/ is the right-inverse of Af on R.A/. ? ? ? ? Now let v D Af w for w 2 D.AAf /. Then u D Av D AAf w D Af Aw and ? ? .A/Af Aw D Af Af w D Aw ?

?

which shows that .A/ is the left-inverse of Af on R.A/\D.Af /. This proves the following result: Let .A; D.A// be the generator of some C0 -contraction semigroup on the Banach space B such that the range R.A/ is dense in B and let f; f ? 2 CBF be conjugate, i.e. f ./f ? ./ D . Then ./ WD f ./= is a Stieltjes function and .Af /1 u D .A/u ?

?

for all u 2 R.A/ \ D.Af /:

233

Section 13.3 Subordination and functional inequalities

13.3 Subordination and functional inequalities In this section we show that certain functional inequalities are preserved under subordination. Functional inequalities are important tools to prove regularity results for semigroups. The following deep result, taken from [362], illustrates this nicely. Let m be a Radon measure on a locally compact separable metric space E such that supp m D E. Assume that .T t / t0 is a symmetric sub-Markovian C0 -contraction semigroup on L2 .E; m/ with generator .A; D.A//. For p > 2 and n WD 2p=.p  2/ the following inequalities are equivalent:

2 kukL 2

2 for all u 2 D.A/I kukL p  c hAu; uiL2 ! 2=n 2 kukL 2  c 0 hAu; uiL2 for all u 2 D.A/ \ L1 I 2 kukL1

kT t kL1 !L1  c 00 t n=2

for all t > 0:

(13.52) (13.53) (13.54)

The inequality (13.52) is often called a Sobolev-type inequality, (13.53) a Nash-type inequality and (13.54) is the ultracontractivity of the semigroup .T t / t0 . If the semigroup is ultracontractive, then there exists a family of kernels p t .x; y/, t  0, such R that T t u.x/ D u.y/p t .x; y/ dy and we can interpret (13.54) as an on-diagonal estimate: ess supx;y p t .x; y/  c 00 t n=2 . Functional inequalities can also be used to study spectral properties of A and .T t / t0 . For a state-of-the art survey we refer to the monograph [364]. Throughout this section .B; kk/ denotes a Banach space, .B ; kk / its topological dual and hu; i stands for the dual pairing between u 2 B and  2 B . By u we denote a normalized tangent functional (or duality map) of u 2 B, i.e. any element u 2 B such that kuk2 D ku k2 D hu; u i. By the Hahn–Banach theorem, each u 2 B has at least one normalized tangent functional. The map u 7! u is singlevalued if, and only if, B is smooth; this is the same as to say that u 7! kuk is Gateaux-differentiable, see e.g. [218, §26]. A sufficient, and in the reflexive case also necessary, condition is that B is strictly convex. Note that the duality mapping u 7! u is homogeneous. As in the previous sections, .T t / t0 is a C0 -contraction semigroup on the Banach space .B; k  k/, the generator is denoted by .A; D.A// and ˆ is a (not necessarily linear) functional ˆ W B ! Œ0; 1 such that ˆ.u/ D 0 H) u D 0;

ˆ.cu/ D jcjˆ.u/;

and ˆ.T t u/  ˆ.u/

(13.55)

hold for all c 2 R, t  0 and u 2 B. Functional inequalities are often considered in the Hilbert space L2 .E; m/. In this case the normalized tangent functional u is unique and can be identified with u, and ˆ.u/ is usually taken to be the norm in L1 .E; m/ or L1 .E; m/.

234

Chapter 13 Subordination and Bochner’s functional calculus

For the main result, Theorem 13.41 below, we need a few preparations. We begin with a differential inequality which is an abstract version of the well known Gronwall– Bellman–Bihari inequality. Recall that for an increasing function G W .0; 1/ ! R the generalized (right continuous) inverse G 1 W R ! Œ0; C1 is defined as ® ¯ G 1 .y/ WD inf t > 0 W G.t / > y ;

inf ; WD 1:

If G is bijective, e.g. strictly increasing and onto, then G 1 is just the usual inverse. Lemma 13.39. Let h W Œ0; 1/ ! Œ0; 1/ be a differentiable function. Suppose that there exists an increasing function ' W Œ0; 1/ ! Œ0; 1/ such that '.t / > 0 for t > 0, R 0C 1='.t / dt D 1 and h0 .t /  '.h.t // Then, we have

for all t  0:

h.t /  G 1 .G.h.0//  t /

(13.56)

for all t  0;

(13.57)

where G 1 is the (generalized right continuous) inverse of 8 Z t dr ˆ ˆ ; if t  1; ˆ < 1 '.r/ G.t / D Z 1 ˆ dr ˆ ˆ : ; if t  1: t '.r/ Proof. Since h0 .t /  '.h.t //  0, the set I D ¹t W h.t / > 0º is a (bounded or unbounded) interval; for t 2 I , the function h.t / is strictly decreasing. With the Ra Rb convention a D  b , we see for all t 2 I that Z G.h.t // D

h.t/ 1

1 dr D G.h.0// C '.r/

Z

h.t/

Z

h.0/ t 0

D G.h.0// C

0

1 dr '.r/

h .r/ dr '.h.r//

 G.h.0//  t: If t 62 I , G.h.t // D G.0/ D 1, and the above inequality is trivial. The claim follows from the definition of the generalized inverse G 1 . An abstract Nash inequality can be characterized in terms of estimates for the underlying semigroup. This is the essence of the next proposition.

235

Section 13.3 Subordination and functional inequalities

Proposition 13.40. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B. Denote by .A; D.A// its generator and let ˆ W B ! Œ0; 1 be a (not necessarily linear) functional satisfying (13.55). Then the following abstract Nash inequality   (13.58) kuk2 B kuk2  Re hAu; u i; u 2 D.A/; ˆ.u/ D 1; with some increasing function B W .0; 1/ ! .0; 1/ holds if, and only if,   for all t  0; u 2 D.A/; ˆ.u/ D 1; kT t uk2  G 1 G.kuk2 /  t where G.t / D

8 Z ˆ ˆ ˆ < Z ˆ ˆ ˆ :

t

1 1 t

dr ; 2rB.r/

(13.59)

if t  1;

dr ; if t  1: 2rB.r/

Proof. Assume that (13.58) holds. Then,   kuk2 2  Re hAu; u i; kuk B ˆ2 .u/

(13.60)

u 2 D.A/:

For all u 2 D.A/ with ˆ.u/ D 1 we have d d kT t uk2 D 2kT t uk kT t uk dt dt   1 hT t u; .T t u/ i hT th u; .T th u/ i D 2kT t uk lim  kT t uk kT th uk h!0C h  1 hT t u; .T t u/ i  Re hT th u; .T t u/ i  2 lim h!0C h

T t u  T th u  ; .T t u/ D 2 lim Re h h!0C ˛ ˝ D 2 Re AT t u; .T t u/ : The inequality in the above calculation follows from the definition of a normalized tangent functional, Re hT th u; .T t u/ i  kT th uk  k.T t u/ k D

hT th u; .T th u/ i kT t uk: kT th uk

Since the function B is increasing and ˆ.T t u/  ˆ.u/ D 1, we have   d kT t uk2  2 kT t uk2 B kT t uk2 : dt This, together with Lemma 13.39, proves (13.59).

236

Chapter 13 Subordination and Bochner’s functional calculus

Conversely, assume that (13.59) holds. Then, for all u 2 D.A/ with ˆ.u/ D 1,   kuk C kT t uk   Re hAu; u i 2 Re hAu; u i D lim t!0C kuk 

 1 kuk C kT t uk D lim Re .u  T t u/; u t!0C kuk t   kuk C kT t uk hu; u i  Re hT t u; u i D lim t!0C kuk t   2 kuk C kT t uk kuk  kT t ukkuk  lim t!0C kuk t kuk2  kT t uk2 D lim t!0C t   2 kuk  G 1 G.kuk2 /  t  lim t!0C t ˇ  ˇ d D  G 1 G.kuk2 /  t ˇˇ dt tD0 ˇ h     i ˇ 1 2 1 2 ˇ G.kuk /  t  B G G.kuk /  t D2 G ˇ tD0

2

2

D 2 kuk B.kuk /; which is just the abstract Nash inequality (13.58). Theorem 13.41. Let .T t / t0 be a C0 -contraction semigroup on the Banach space B. Denote by .A; D.A// its generator and let ˆ W B ! Œ0; 1 be a (not necessarily linear) functional satisfying (13.55). Assume that the following abstract Nash inequality holds   (13.61) kuk2 B kuk2  Re hAu; u i; u 2 D.A/; ˆ.u/ D 1; where B W .0; 1/ ! .0; 1/ is an increasing function. Then, for any Bernstein function f , the generator Af of the subordinate semigroup satisfies    kuk2 kuk2 f 2B  Re hAf u; u i; u 2 D.Af /; ˆ.u/ D 1: (13.62) 4 2 Proof. For the Bernstein function a C b it is obvious that (13.61) implies (13.62). Therefore, R it is enough to prove the theorem for Bernstein functions of the form f ./ D .0;1/ .1  e t / .dt /. Since D.A/ is an operator core for .Af ; D.Af //, cf. Theorem 13.6, we only need to consider (13.62) for u 2 D.A/. We know from (13.61) and Proposition 13.40 that for all t  0, u 2 D.A/ with ˆ.u/ D 1 and for G given by (13.60)   kT t uk2  G 1 G.kuk2 /  t :

Section 13.3 Subordination and functional inequalities

From the definition of a normalized tangent functional u we see q      G 1 G.kuk2 /  t jhT t u; u ij kT t uk  ku k Re hT t u; u i :    kuk2 kuk2 kuk2 kuk Phillips’ formula (13.10) for Af yields for any u 2 D.A/ with ˆ.u/ D 1, Z f  Re hu  Ts u; u i .ds/ Re hA u; u i D .0;1/   Z Re hTs u; u i 2 1 .ds/ D kuk kuk2 .0;1/ 0 q  1 Z 1 G.kuk2 /  s G C B  kuk2 A .ds/ @1  kuk .0;1/ 0

1   1 .G kuk2 /  s G B 1 C Z B C 2 kuk B C D kuk2 q .ds/ B  C B C .0;1/ G 1 G.kuk2 /  s A @ 1C kuk  ! Z 1 G.kuk2 /  s G kuk2 1 .ds/;  2 kuk2 .0;1/ since G 1 is increasing. Thus, Re hAf u; u i  g.kuk2 /; where

  Z G 1 .G.r/  s/ r .ds/: 1 g.r/ D 2 .0;1/ r For all r > 0 we have Z    1 g.r/ D r  G 1 G.r/  s .ds/ 2 .0;1/ ! Z Z G.r / 1 1 D dG .t / .ds/ 2 .0;1/ G.r /s Z Z 1 1.1;G.r // .t /1.G.r /t;1/ .s/ dG 1 .t / .ds/ D 2 .0;1/ R Z   1  G.r/  t; 1 dG 1 .t / D 2 .1;G.r // Z  1 r  D  G.r/  G.t /; 1 dt: 2 0

237

238

Chapter 13 Subordination and Bochner’s functional calculus

In the last equality we used that B is increasing, G.t / > 1 for all t > 0 and G.0/ D 1; this follows from Z G.0/ D 

1 0

1 dr  2rB.r/ 2B.1/

Z

1 0

dr D 1: r

Using again the monotonicity of B, we find from the mean value theorem 1 G.r/  G.t /  r t 2tB.t /

for all 0 < t < r:

(13.63)

Therefore, g.r/    D D

  Z 1 r 1 .r  t /; 1 dt  2 0 2tB.t /  Z r  1 1 .r  t /; 1 dt  2 r=2 2tB.t /  Z r  1 1 .r  t /; 1 dt  2 r=2 rB.r=2/   Z 1 r=2 v ; 1 dv  2 0 rB.r=2/ Z 1=2B.r=2/ 1 rB.r=2/ .s; 1/ ds: 2 0

Now we can use Lemma 3.4 to deduce that r   r  g.r/  f 2B 4 2

for all r > 0;

which implies (13.62). Remark 13.42. If .T t / t0 is a self-adjoint C0 -contraction semigroup on a Hilbert space, then we can replace the left-hand side of (13.62) by kuk2 f 2

   kuk2 B : 2

Since the Bernstein function f is subadditive, the estimate 12 f .2/  f ./,  > 0, shows that this improves (13.62). We can use Theorem 13.41 to show that super-Poincaré and weak Poincaré inequalities are also preserved under subordination.

239

Section 13.3 Subordination and functional inequalities

Corollary 13.43. Let .A; D.A// be the generator of a C0 -contraction semigroup on the Banach space B, let ˆ W B ! Œ0; 1 be a (not necessarily linear) functional satisfying (13.55), and let f be a Bernstein function. Assume that A satisfies the following super-Poincaré inequality kuk2  r Re hAu; u i C ˇ.r/ ˆ2 .u/;

r > 0; u 2 D.A/;

(13.64)

where ˇ W .0; 1/ ! .0; 1/ is a decreasing function such that limr !1 ˇ.r/ D 0 and ˇ.0/ WD limr !0 ˇ.r/ D 1. Then the generator Af of the subordinate semigroup also satisfies a super-Poincaré inequality kuk2  r Re hAf u; u i C ˇf .r/ ˆ2 .u/;

r > 0; u 2 D.Af /;

(13.65)

where ˇf .r/ D 4ˇ.1=f 1 .4=r// and f 1 .y/ D 1 if y 62 R.f /. Proof. Set B.x/ WD sups>0 1s .1 ˇ.s/ x /. Then we can rewrite (13.64) in the following form kuk2 B.kuk2 /  Re hAu; u i; u 2 D.A/; ˆ.u/ D 1: Clearly, B.x/ is an increasing function on .0; 1/. Since ˇ1 W .0; 1/ ! .0; 1/, we infer from   ˇ ˇ 1 . x 2/ 1  ˇ x.s/ 1 1 1 x  D xx   1 (13.66)  B.x/ D sup 1 1 s ˇ .x/ 2ˇ ˇ sˇ 1 .x/ 2 2 that B W .0; 1/ ! .0; 1/. Theorem 13.41 shows that C.kuk2 /  Re hAf u; u i;

u 2 D.A/; ˆ.u/ D 1;

   2ˇ.s/ x   x  x 2 D sup f 1 : C.x/ D f 2B 4 2 4 s>0 s x ® ¯ For r > 0 we define e ˇ.r/ WD sups>0 C 1 .s/  rs . Then where

ˇ.r/ ˆ2 .u/; kuk2  rRe hAf u; u i C e Let us now estimate e ˇ.r/. By (13.66),   x 1 WD C0 .x/; C.x/  f 4 ˇ 1 .x=4/

r > 0; u 2 D.A/:

hence, C 1 .y/  C01 .y/:

By its very definition, C0 W .0; 1/ ! .0; 1/ is a strictly increasing function such that limx!0 C0 .x/ D 0 and limx!1 C0 .x/ D 1. Therefore, " !#1 1 1 C0 .y/ D 4y f : (13.67) ˇ 1 .C01 .y/=4/

240

Chapter 13 Subordination and Bochner’s functional calculus

On the other hand, ¯ ® e ˇ.r/  sup C01 .y/  ry D y>0

sup y>0; C01 .y/ry

C01 .y/;

and from (13.67) we see that C01 .y/  ry leads to 1 f

1 .4=r/

   ˇ 1 C01 .y/=4 :

Since ˇ is decreasing, we can rewrite this as   1 1 C0 .y/  4ˇ ; f 1 .4=r/ and so e ˇ.r/ 

sup y>0; C01 .y/4ˇ .1=f 1 .4=r //

C01 .y/

 1 :  4ˇ f 1 .4=r/ 

This finishes the proof. Corollary 13.44. Let .A; D.A// be the generator of a C0 -contraction semigroup on the Banach space B, let ˆ W B ! Œ0; 1 be a (not necessarily linear) functional satisfying (13.55), and let f be a Bernstein function. Assume that A satisfies the following weak Poincaré inequality kuk2  ˛.s/Re hAu; u i C s ˆ2 .u/;

s > 0; u 2 D.A/;

(13.68)

where ˛ W .0; 1/ ! .0; 1/ is a decreasing function such that lims!1 ˛.s/ D 0 and ˛.0/ WD lims!0 ˛.s/ D 1. Then the generator Af of the subordinate semigroup also satisfies a weak Poincaré inequality kuk2  ˛f .s/ Re hAf u; u i C s ˆ2 .u/;

s > 0; u 2 D.A/;

(13.69)

1 /. where ˛f .s/ D 4=f . ˛.s=4/

Proof. Since the proof is similar to the proof of Corollary 13.43, we only provide a formal argument for the form of ˛f .s/. Note that s D ˛ 1 .r/ D ˇ.r/ transforms the weak Poincaré inequality (13.68) into the super-Poincaré inequality (13.64) and vice versa. Thus, we can use Corollary 13.43 to infer that the weak Poincaré inequality remains valid under subordination with ˛f .s/ D ˇf1 .s/. Since   1 ˇf .r/ D 4ˇ f 1 .4=r/ 1 1 we may insert r D 4=f . ˛.s=4/ / and get ˇf .4=f . ˛.s=4/ // D s. This shows that 1 1 ˛f .s/ D ˇf .s/ D 4=f . ˛.s=4/ /.

241

Section 13.3 Subordination and functional inequalities

We close this section with a few applications which illustrate how contractivity and decay properties of a semigroup are influenced by subordination. Let E be a locally compact separable metric space and m a Radon measure on E with full support. A symmetric sub-Markovian semigroup .T t / t0 on L2 .E; m/ is said to be hypercontractive, if kT t kL2 !L4 < 1 for some t > 0; supercontractive, if kT t kL2 !L4 < 1 for all t > 0; ultracontractive, if kT t kL2 !L1 < 1 for all t > 0. Corollary 13.45. Let f be a Bernstein function and .T t / t0 be an ultracontractive symmetric sub-Markovian semigroup on L2 .X; m/. If there are constants c > 0 and p > 1 such that   kT t kL2 !L1  exp c t 1=.p1/ for all t > 0; then the following assertions hold: R 1 d f (i) If 1 f . p / < 1, then .T t / t0 is ultracontractive. (ii) If lim!1

f 1 ./ p

D 0, then .T tf / t0 is supercontractive.

(iii) If lim!1

f 1 ./ p

2 .0; 1/, then .T t / t0 is hypercontractive.

f

Proof. Denote by A and Af the generators of the semigroups .T t / t0 and .T tf / t0 , respectively. By [364, Proposition 3.3.16, p. 157], we know that the following super-Poincaré inequality holds, 2 2 kukL 2  rhAu; uiL2 C ˇ.r/ kukL1 ;

where

r > 0; u 2 D.A/ \ L1 .E; m/;

i h   ˇ.r/ D c1 exp c2 r 1=p  1

for some c1 ; c2 > 0. By Corollary 13.43, 2 f 2 kukL 2  r hA u; ui C ˇf .r/ kukL1 ;

where

r > 0; u 2 D.A/;

° h  ±  1=p i ˇf .r/ D 4c1 exp c2 f 1 4=r 1 :

Therefore, the assertions follow from [364, Theorem 3.3.14, p. 156; Theorem 3.3.13, p. 155].

242

Chapter 13 Subordination and Bochner’s functional calculus

Corollary 13.46. Let .T t / t0 be a symmetric sub-Markovian semigroup on L2 .X; m/ and f a Bernstein function. Assume that there is a constant p > 0 such that ˆ2 .u/ 2 for all t > 0; u 2 L2 .E; m/; kT t ukL 2  tp where ˆ W L2 .E; m/ ! Œ0; 1 is a (not necessarily linear) functional satisfying (13.55). If Z 1 d .t / WD < 1 for all t > 0; f ./ t then there are constants c1 ; c2 > 0 such that  1 p 2 kT tf ukL .c2 t / ˆ2 .u/: 2  c1  f

Proof. Denote by A and Af the generators of .T t / t0 and .T t / t0 , respectively. From [364, Corollary 4.1.8 (1), p. 189; Corollary 4.1.5 (2), p. 186] we know that the following weak Poincaré inequality holds 2 2 kukL 2  ˛.r/ hAu; uiL2 C r ˆ .u/;

r > 0; u 2 D.A/;

where ˛.r/ D c3 r 1=ı for some constant c3 > 0. Corollary 13.44 shows that 2 f 2 kukL 2  ˛f .r/hA u; uiL2 C r ˆ .u/;

r > 0; u 2 D.Af /;

 ı  where ˛f .r/ D 4 f c4 r 1=p for some constant c4 > 0. Therefore, the assertion follows from [364, Theorem 4.1.7, p. 188].

13.4 Eigenvalue estimates for subordinate processes For ˛ 2 .0; 2, the generator of a killed symmetric ˛-stable process in an open subset D of Rd is the Dirichlet fractional Laplacian . /˛=2 jD . The spectrum of . /˛=2 jD is very important both in theory and in applications. Although a lot is known in the case ˛ D 2, until recently very little was known for ˛ < 2. In this section we discuss the spectrum of the generator of a general killed subordinate process. The main tools of this section are subordinate killed processes and the theory of Dirichlet forms. Let E be a locally compact separable metric space, B D B.E/ the Borel  -algebra on E, and m a positive Radon measure on .E; B/ with full support supp m D E. Let X D .X t ; Px / t0;x2E be an m-symmetric Hunt process on E. Adjoined to the state space E is a cemetery point @ … E; the process X retires to @ at its lifetime  WD inf¹t  0 W X t D @º. Denote by P t .x; dy/ the transition function of X. The

243

Section 13.4 Eigenvalue estimates for subordinate processes

transition semigroup .P t / t0 and the resolvent .Gˇ /ˇ >0 of X are defined by   P t g.x/ WD Ex g.X t / D Ex g.X t /1¹t< º ; 

Z 1 Z 1 ˇ t ˇ t Gˇ g.x/ WD e P t g.x/ dt D Ex e g.X t / dt 0 "Z0 # D Ex



e ˇ t g.X t / dt :

0

Here g is a non-negative Borel measurable function on E and we are using the convention that any function g defined on E is automatically extended to E@ D E [ ¹@º by setting g.@/ D 0. The L2 -generator of .P t / t0 will be denoted by A and its domain by D.A/, see Appendix A.2 for details. Let .E; D.E// be the Dirichlet form associated with X, i.e. ´ ˛ ˝ E.u; v/ D .A/1=2 u; .A/1=2 v L2 ; (13.70) D.E/ D D..A/1=2 /: In general, .D.E/; E/ is not a Hilbert space. Therefore, one considers the forms Eˇ .u; v/ WD E.u; v/ C ˇ hu; viL2 ;

u; v 2 D.E/; ˇ > 0;

which make .D.E/; Eˇ / into a Hilbert space. For suitable constants cˇ ; Cˇ > 0 we have cˇ E1 .u; u/  Eˇ .u; u/  Cˇ E1 .u; u/; this means that the spaces .D.E/; Eˇ / coincide. Let .T t / t0 and .Rˇ /ˇ >0 be the semigroup and resolvent associated with the operator A or, equivalently, the Dirichlet form .E; D.E//. It follows from [123, Theorem 4.2.3] that .T t / t0 can be identified with .P t / t0 and that .Rˇ /ˇ >0 can be identified with .Gˇ /ˇ >0 . So in this section, we will use .T t / t0 , .Rˇ /ˇ >0 and .P t / t0 , .Gˇ /ˇ >0 interchangeably. For any open subset D of E, let D D inf¹t > 0 W X t … Dº be the first exit time of X from D. The process X D obtained by killing the process X upon exiting from D is defined by ´ X t ; t < D ; X tD WD @; t  D : The killed process X D is again an m-symmetric Hunt process on D, see [123] or Appendix A.2. The associated Dirichlet form is .ED ; D.ED // where ED D E and ® ¯ (13.71) D.ED / D u 2 D.E/ W u D 0 E-q.e. on E n D : Here E-q.e. stands for quasi everywhere with respect to the capacity associated with the Dirichlet form .E; D.E//. Let .P tD / t0 and .GˇD /ˇ >0 denote the semigroup and resolvent of X D , and let AD be the L2 -generator of X D . The following result will play an important role in this section. Note that the semigroup version of this result is not true, see [91].

244

Chapter 13 Subordination and Bochner’s functional calculus

Lemma 13.47. For any ˇ > 0 and every g 2 L2 .E; m/ with g D 0 m-a.e. on E n D, we have hGˇD g; giL2  hGˇ g; giL2 : (13.72) Proof. For every ˇ > 0 and every g 2 L2 .E; m/ with g D 0 m-a.e. on E n D, the strong Markov property of X shows Gˇ g.x/ D GˇD g.x/ C HˇD Gˇ g.x/

for all x 2 D:

This is the Eˇ -orthogonal decomposition of Gˇ g into GˇD g 2 D.ED / and its orthogonal complement, see (A.20) in Appendix A.2. Here  HˇD g.x/ WD Ex e ˇ D g.XD / is the ˇ-hitting operator of X. Hence hGˇ g  GˇD g; giL2 D hHˇD Gˇ g; giL2 1 Eˇ .HˇD Gˇ g; Gˇ g/ ˇ 1 D Eˇ .HˇD Gˇ g; HˇD Gˇ g/  0: ˇ

D

This establishes (13.72). Suppose that S D .S t / t0 is a subordinator with Laplace exponent f given by Z .1  e t /.dt /;  > 0; f ./ D b C .0;1/

with b > 0 or .0; 1/ D 1. Let . t / t0 be the convolution semigroup corresponding to S . Throughout this section we will assume that X and S are independent. The f f process X f D .X t / t0 defined by X t WD X.S t / is called the subordinate process of X with respect to S ; this is again an m-symmetric Hunt process on E. The f subordinate transition function P t .x; dy/ of X f is given by P tf .x; B/ D

Z

1 0

Ps .x; B/  t .ds/ for t > 0; x 2 E and B 2 B.E/: (13.73)

Let .Ef ; D.Ef // be the Dirichlet form corresponding to X f . Then D.E/  D.Ef /. When b > 0, D.E/ D D.Ef / (13.74)

245

Section 13.4 Eigenvalue estimates for subordinate processes

and for u 2 D.E/, Z Ef .u; u/ D bE.u; u/ C

.0;1/

hu  Ps u; uiL2 .ds/

Z  2 u.x/  u.y/ J f .dx; dy/ D bE.u; u/ C E E Z 2 C u .x/  f .dx/;

(13.75)

E

where J f .dx; dy/ WD

1 2

Z

Z

and  f .dx/ WD

Ps .x; dy/ .ds/ m.dx/ .0;1/



.0;1/

 1  Ps .x; E/ .ds/ m.dx/:

When b D 0, ² Z D.Ef / D u 2 L2 .E; m/ W

³ .0;1/

hu  Ps u; uiL2 .ds/ < 1

and for any u 2 D.Ef /, Z hu  Ps u; uiL2 .ds/ Ef .u; u/ D .0;1/ Z Z  2 f u.x/  u.y/ J .dx; dy/ C u2 .x/  f .dx/; D E E

(13.76)

(13.77)

E

with J f and  f from above. For the above facts regarding .Ef ; D.Ef // we refer to [284]. Let Af be the L2 -infinitesimal generator of X f . From Theorem 13.6, see also Example 12.6, it follows that Z Af u D bAu C

.0;1/

D.A/  D.Af /;

(13.78)

.Ps u  u/ .ds/ for u 2 D.A/:

(13.79)

By Proposition 13.5 we know that D.A/ is a core for the generator Af ; in particular, D.A/ is dense in .D.Ef /; Ef /. For every open subset D of E let X f;D denote the process obtained by killing X f f;D upon exiting from D. We will use .P t / t0 to denote the transition semigroup of f;D f;D X and A to denote the generator of X f;D . The Dirichlet form of X f;D is f;D f;D .E ; D.E // where Ef;D D Ef and ® ¯ D.Ef;D / D u 2 D.Ef / W u D 0 Ef -q.e. on E n D :

246

Chapter 13 Subordination and Bochner’s functional calculus

For u 2 D.Ef;D / we can rewrite Ef .u; u/ as Z  2 f u.x/  u.y/ J f .dx; dy/ E .u; u/ D bE.u; u/ C DD Z u.x/2  f;D .x/ m.dx/; C

(13.80)

D

Z

where  f;D .x/ D

.0;1/

Ps .x; E n D/ .ds/ C  f .x/

for x 2 D:

Let X D;f denote the process obtained by subordinating X D with the subordinator S . The semigroup of X D;f will be denoted by .P tD;f / t0 and its generator by AD;f . The Dirichlet form of X D;f is .ED;f ; D.ED;f //, where ED;f is defined in the same way through X D as Ef is defined through X. For any u 2 D.ED;f /, Z  2 u.x/  u.y/ J D;f .dx; dy/ ED;f .u; u/ D bE.u; u/ C DD Z (13.81) C u2 .x/  D;f .x/ m.dx/; D

where J D;f .dx; dy/ D

1 2

Z .0;1/

Z

and 

D;f

.x/ D

PsD .x; dy/ .ds/ m.dx/

 .0;1/

for x; y 2 D;

 1  PsD 1.x/ .ds/ for x 2 D:

Theorem 13.48. Suppose that the Laplace exponent f of S is a complete Bernstein function. Then for any open subset D of E, we have D.ED;f /  D.Ef;D / and Ef .u; u/  ED;f .u; u/

for every u 2 D.ED;f /:

Proof. Since f is a complete Bernstein function, it has the following representation Z 1 .1  e t / m.t / dt; f ./ D b C 0

R

where m.t / D .0;1/ e ts s  .ds/ and  is the Stieltjes measure of f , see Remark 6.10. From Corollary 13.22 we know that Af has the following representation in terms of the L2 -generator A and the resolvent .Gˇ /ˇ >0 of X: Z f .ˇGˇ u  u/  .dˇ/ for u 2 D.A/: (13.82) A u D bAu C .0;1/

247

Section 13.4 Eigenvalue estimates for subordinate processes

For u 2 D.A/, it follows from (13.82) that Z Ef .u; u/ D hAf u; uiL2 D bE.u; u/ C Since D.A/ is E1 -dense in D.E/, Z Ef .u; u/ D bE.u; u/ C

.0;1/

.0;1/

ˇhu  ˇGˇ u; uiL2  .dˇ/:

ˇhu  ˇGˇ u; uiL2  .dˇ/

for u 2 D.E/:

Similarly, we have Z ED;f .u; u/ D bED .u; u/ C

.0;1/

ˇhu  ˇGˇD u; uiL2  .dˇ/

for u 2 D.ED /:

As D.ED /  D.E/, we deduce from these formulae and (13.72) that for u 2 D.ED /, Z ˇhGˇ u  GˇD u; uiL2  .dˇ/  0: ED;f .u; u/  Ef .u; u/ D ˇ .0;1/

f

Since D.AD /  D.ED / is E1 -dense in D.ED;f / – see the comments after (13.79) –, we conclude that D.ED;f /  D.Ef / and, therefore, that D.ED;f /  D.Ef;D /. Moreover Ef .u; u/  ED;f .u; u/ holds for every u 2 D.ED;f /. The semigroup .P t / t0 is ultracontractive if each P t , t > 0, is a bounded operator from L2 .E; m/ to L1 .E; m/. Under this condition, for all t > 0 the transition operator P t has an integral kernel p.t; x; y/ with respect to the measure m, which satisfies (13.83) 0  p.t; x; y/  c t a.e. on E E with t 7! c t being a non-increasing function on .0; 1/, see e.g. [89, (2.1.1) and Lemma 2.1.2]. Conversely, if for every t > 0 the transition operator P t has a bounded density function, i.e. if (13.83) holds for some c t > 0, then for a.e. x 2 E and every f 2 L2 .E; m/ Z 2 p.t; x; y/ f 2 .y/ m.dy/  c t kf kL jP t f .x/j2  2 .E;m/ : E

So .P t / t0 is ultracontractive. Therefore the semigroup .P t / t0 is ultracontractive if, and only if, P t has a bounded integral kernel for every t > 0. In the remainder of this section we will assume that the semigroup .P t / t0 of the process X is ultracontractive. Assume further that D is an open subset of E with m.D/ < 1. It is well known that under these assumptions the semigroup .P tD / t0 of X D has a density with respect to m. We will denote this density by p D .t; x; y/.

248

Chapter 13 Subordination and Bochner’s functional calculus

Under the assumption (13.83), it is easy to see that the subordinate process X f has a transition density p f .t; x; y/ given by Z 1 f p .t; x; y/ D p.s; x; y/  t .ds/: 0

Under the assumptions stated in the paragraph above, P tD is a Hilbert–Schmidt operator for t > 0. Thus for any t > 0, both P tD and AD have discrete spectra. We will use .e n t /n2N and .n /n2N to denote the eigenvalues of P tD and AD respectively, arranged in decreasing order and repeated according to multiplicity. f;D There are examples, see [83, Section 3], where for every t > 0, P t is not a f;D Hilbert–Schmidt operator. But, for every t > 0, P t is still a compact operator. f;D

Theorem 13.49. For every t > 0, P t therefore it has discrete spectrum.

is a compact operator in L2 .D; m/ and

Proof. For every s > 0 we define the operator Qs on L2 .D; m/ by Z p.s; x; y/ g.y/ m.dy/; g 2 L2 .D; m/: Qs g.x/ WD D

For g 2 L2 .D; m/ with kgkL2 .D;m/  1 and any Borel subset A of D, we have Z Z  2 Qs g.x/ m.dx/  Qs .g 2 /.x/ m.dx/ A A Z g 2 .x/ Ps 1A .x/ m.dx/ D E Z  kPs 1A k1 g 2 .x/ m.dx/ E

 kPs 1A k1 ® ¯  min 1; cs m.A/ :

(13.84)

Now we fix t > 0. For g 2 L2 .D; m/ with kgkL2 .D;m/  1 and all Borel subsets A of D, we have 2 Z Z 1 Z Z 1 Qs g.x/ t .ds/ m.dx/  jQs g.x/j2  t .ds/ m.dx/ 0 A A 0 Z 1Z jQs g.x/j2 m.dx/  t .ds/: D 0

A

For every  > 0 we find some s0 > 0 such that  t Œ0; s0  < Z Z A

1 0

1 2

. Then by (13.84),

2  Qs g.x/  t .ds/ dx  C cs0 m.A/: 2

249

Section 13.4 Eigenvalue estimates for subordinate processes

This shows that for Borel sets A  E with m.A/ < =2cs0 , we have Z Z

1 0

A

2 Qs g.x/  t .ds/ m.dx/  

which entails that the family of functions μ ´Z 2 1 Qs g./  t .ds/ W kgkL2 .D;m/  1 0

is uniformly integrable. Let p f;D .t; x; y/ be the density function of P tf;D . Clearly, p f;D .t; x; y/  p f .t; x; y/;

.t; x; y/ 2 .0; 1/ D D:

Thus for all t > 0 and all Borel functions g on D, we have Z Z 1 f;D p f .t; x; y/ jgj.y/ m.dy/  Qs jgj.x/  t .ds/: jP t g.x/j  0

D

Consequently, the family of functions ®

f;D

.P t

¯ g/2 W kgkL2 .D;m/  1

is uniformly integrable on .D; B.D/; m/. By using this one can show that, for each f;D is a compact operator in L2 .D; m/ and hence has discrete spectrum. t > 0, P t Indeed: assume that P tf;D is not compact. Then there exist some  > 0 and a f;D sequence .gn /n2N in A WD ¹P t=2 g W kgkL2 .D;m/  1º such that f;D

f;D

kP t=2 gn  P t=2 gm kL2 .D;m/  ;

n ¤ m:

Since we know that ¹g 2 W g 2 Aº is uniformly integrable, we have that Z lim sup g 2 .x/ 1¹jgj>Kº .x/ m.dx/ D 0; K!1 g2A D

f;D

and consequently, by the boundedness of P t=2 in L2 .D; m/, Z lim sup

K!1 g2A D



2 f;D P t=2 .g 1¹jgj>Kº /.x/ m.dx/ D 0:

Therefore there exists some K > 0 such that  f;D f;D kP t=2 gn;K  P t=2 gk;K kL2 .D;m/  ; 2

n ¤ k;

(13.85)

250

Chapter 13 Subordination and Bochner’s functional calculus

where gn;K D gn 1¹jgn jKº . Let C be the  -algebra generated by .gn;K /n2N . Since L1 .D; C ; m/ is separable, the set .gn;K /n2N  L1 .D; C ; m/ is weakly compact and metrizable with respect to the weak topology  .L1 .D; C ; m/; L1 .D; C ; m//, cf. [70, IV.36, Lemma 1]. Therefore there exist g 2 L1 .D; C ; m/ and a subsequence .gnj ;K /j 2N such that for h 2 L1 .D; C ; m/ we have Z

Z lim

j !1 D

gnj ;K .x/ h.x/ m.dx/ D

g.x/ h.x/ m.dx/: D

Without loss of generality, we will assume in the rest of this proof that m.D/ D 1, that is, m is a probability measure on D. For any h 2 L1 .D; m/ we will use m.h j C / to denote the conditional expectation of h with respect to C . Noting that for any h 2 L1 .D; m/ and k 2 L1 .D; C ; m/ we have Z Z k.x/ h.x/ m.dx/ D k.x/ m.h j C /.x/ m.dx/; D

D

we obtain that Z Z gnj ;K .x/ h.x/ m.dx/ D g.x/ h.x/ m.dx/ lim j !1 D

D

for all h 2 L1 .D; m/:

On the other hand, for every x 2 D, we have Z   f;D f;D p f;D .t =2; x; y/ g.y/  gnj ;K .y/ m.dy/: P t=2 g.x/  P t=2 gnj ;K .x/ D D

The fact that p f;D .t =2; x; / is integrable and the weak convergence of the sequence j !1

f;D f;D gnj ;K .x/ ! P t=2 g.x/ for all x 2 D. Since the family .gnj ;K /j 2N imply P t=2 j !1

f;D f;D f;D gnj;K /2 /j2N is uniformly integrable, we get P t=2 gnj;K ! P t=2 g in ..P t=2 2 L .D; m/. This contradicts (13.85) and the proof is complete. Q In the remainder of this section, we will use .Q n /n2N and .e n t /n2N to denote f;D the eigenvalues of Af;D and P t , respectively, arranged in decreasing order and repeated according to multiplicity.

Theorem 13.50. If the Laplace exponent f of the subordinator S is a complete Bernstein function, then Q n  f .n / for every n 2 N: Proof. Since AD has discrete spectrum .n /n2N , the subordinate generator AD;f has discrete spectrum .f .n //n2N with the same eigenfunctions as AD . By the min-max

251

Section 13.4 Eigenvalue estimates for subordinate processes

principle for eigenvalues, see [301, Section XIII.1], and by Theorem 13.48 we get for every n 2 N ® ¯ Q n D inf sup Ef .u; u/ W u 2 L and hu; uiL2 .D;m/ D 1 L: subspace of L2 .D; m/ with dim L D n



inf

L: subspace of L2 .D; m/ with dim L D n

® ¯ sup ED;f .u; u/ W u 2 L and hu; uiL2 .D;m/ D 1

D f .n /: Here we are using the convention that for a Dirichlet form .E; D.E//, E.u; u/ D 1 when u 2 L2 .D; m/ n D.E/. This proves the theorem. Proposition 13.51. Suppose that S is a subordinator with Laplace exponent f . If there exists some C 2 .0; 1/ such that Px .X t 2 D/  C;

for every t > 0 and every x 2 E n D;

(13.86)

then .1  C /  D;f .x/   f;D .x/   D;f .x/

for x 2 D:

(13.87)

Proof. Recall that  is the Lévy measure of S . For any x 2 D we have Z Z Z    f;D .x/ D 1  Ps .x; E/ .ds/ Ps .x; dy/ .ds/ C .0;1/ E nD .0;1/ Z   1  Ps 1D .x/ .ds/ D .0;1/

Z

and 

D;f

.x/ D

 .0;1/

 1  PsD 1E .x/ .ds/:

Thus, the second inequality of (13.87) follows because p D .s; x; /  p.s; x; / for all .s; x/ 2 .0; 1/ D. In order to prove the first inequality, let D be the first exit time of D for the process X and let u.x; s/ D Px .Xs 2 D/. According to the assumption (13.86), u.x; t /  C for every x 2 E n D and t > 0. By the strong Markov property of X we have for all x 2 D and t > 0  Px .D  t; X t 2 D/ D Ex 1¹D tº u.XD ; D /  C Px .D  t /: Therefore for all s > 0 and x 2 D, 1  Ps 1D .x/ D 1  PsD 1.x/  Px .D  t; X t 2 D/  1  PsD 1.x/  C Px .D  t /   D .1  C / 1  PsD 1.x/ : This proves the first inequality of (13.87).

252

Chapter 13 Subordination and Bochner’s functional calculus

A symmetric ˛-stable process in Rd , 0 < ˛ < 2, is a Lévy process X D .X t /0 taking values in Rd and with the characteristic exponent . / D j j˛ , cf. Example 13.13. It is well known that the semigroup of this process is ultracontractive. Proposition 13.52. Let X be a symmetric ˛-stable process in Rd and let D be a bounded convex open and connected subset of Rd . Then 1 D;f  .x/   f;D .x/   D;f .x/: 2 Proof. Because of the spherical symmetry of X we find for every bounded convex open connected subset D  Rd 1 Px .X t 2 D/ < ; 2

for every t > 0 and x 2 E n D:

The assertion now follows from Proposition 13.51. It follows from Theorem 13.48 that D.ED;f /  D.Ef;D /. From the expressions for the subordinate and killed subordinate jump measures J f and J D;f one easily concludes that (13.88) J D;f .  ;  /  J f .  ;  /: Theorem 13.53. Suppose that the Laplace exponent f of S is a complete Bernstein function. If X is a symmetric ˛-stable process in Rd and D is a bounded convex open and connected subset of Rd , then D.ED;f / D D.Ef;D / and 1 D;f E .u; u/  Ef .u; u/  ED;f .u; u/; 2

u 2 D.Ef;D /:

(13.89)

Proof. We know from Theorem 13.48 that D.ED;f /  D.Ef;D /. On the other hand, by (13.74)–(13.77), (13.80)–(13.81) and Proposition 13.52, D.Ef;D /  D.ED;f /. Thus D.ED;f / D D.Ef;D /. The inequality (13.89) follows immediately by combining Proposition 13.52 with (13.88) and Theorem 13.48. Theorem 13.53 together with the min-max principle for eigenvalues, see the proof of Theorem 13.50, immediately gives the following result for the eigenvalues f .n / of Af and Q n of Af;D . Theorem 13.54. Suppose that the Laplace exponent f of S is a complete Bernstein function. If X is a symmetric ˛-stable process in Rd and D is a bounded convex open and connected subset of Rd , then 1 f .n /  Q n  f .n / for every n 2 N: 2

Section 13.4 Eigenvalue estimates for subordinate processes

253

Comments 13.55. Sections 13.1 and 13.2: Subordination was introduced by Bochner in the short note [62] in connection with diffusion equations and semigroups. Using Hilbert space spectral calculus, Bochner points out that the generator of a subordinate semigroup is a function of the original generator. The name subordination seems to originate in Bochner’s monograph [63, Chapters 4.3, 4.4] where subordination of stochastic processes, their transition semigroups and generators is developed. A rigorous functional analytic and stochastic account is in the paper by Nelson [270] where, for the first time, the interpretation of subordination as stochastic time-change can be found; this was, independently, also pointed out by Woll [375] and Blumenthal [59]. Nelson is also the first to make the link between Bochner’s subordination and the rapidly developing theory of operator semigroups pioneered by Hille and Phillips. Phillips’ paper [292] contains the modern formulation of subordination of semigroups on abstract Banach spaces as well as the famous result on the form of the subordinate generator. An operational calculus for bounded generators of semigroups, based on the Dunford–Riesz functional calculus, is described in Hille and Phillips [160, Chapter XV]. For C0 -semigroups and unbounded generators this calculus is further developed in Balakrishnan [16] and [17]. The origins of subordination in potential theory can be traced back to the papers by Feller [117] and Bochner [62] where Riesz potentials of fractional order are discussed, and to Faraut [111] who studies fractional powers of Hunt kernels and a symbolic calculus induced by subordination and Bernstein functions, see also [112]. This line of research is continued by Hirsch who develops in a series of papers, [162, 163, 164, 165], a Bochner-type functional calculus for Hunt kernels and abstract potentials; his main motivation is the question which functions operate on Hunt kernels which was raised in the note by M. Itô [179] and generalized in [161]. In functional analysis, the study of fractional powers and functions of infinitesimal generators was continued by Nollau [280, 281, 282], Westphal [367], Pustyl’nik [298] and later by Berg, Boyadzihev and deLaubenfels [36] and Schilling [320, 321, 322]. Up-to-date accounts of subordination, mainly for Lévy processes and convolution semigroups, can be found in Sato [314, Chapter 6] and Bertoin [49]; fractional powers and related functional calculi are discussed in Martínez-Carracedo and Sanz-Alix [260]. The material on C0 -semigroups is mostly standard and can be found in many books on semigroup theory and functional analysis. Our sources are Davies [88], Ethier and Kurtz [108], Pazy [286] and Yosida [379]. An excellent exposition of dissipativity and fractional powers is in Tanabe [345, Chapter 2]. Remark 13.4 is modeled after Bochner [62], the proof of Proposition 13.5 is taken from [320], see also [183, Vol. 1]. Phillips’ theorem appears for the first time in [292] where it is proved in the context of Banach algebras. Our proof of this theorem seems to be new. The results for the domains of subordinate generators are from [322]; Corollaries 13.8 and 13.9 are generalizations of the moment inequality in Pustyl’nik [298, Theorem 6] and Berg–Boyadzhiev–deLaubenfels [36, Prop. 5.14]; our method of proof is an improvement of [113, Prop. 1.4.9]. Observe that a combination of Corollary 7.15 or

254

Chapter 13 Subordination and Bochner’s functional calculus

Proposition 7.16 and Theorem 13.23 (v) with the estimate (13.19) leads to interesting new interpolation-type estimates. Take, for example f; g 2 CBF and ˛ 2 .0; 1/. Then f ˛ g 1˛ 2 CBF and we have for all u 2 D.A/ kAf

˛ g 1˛

˛

uk D kAf Ag

1˛

uk  4kAf uk˛ kAg uk1˛ ;

(13.90)

or, with f ˛ .ˇ /g 1˛ .1ˇ / and ˛; ˇ 2 Œ0; 1, we have for all u 2 D.A/               f .A/ˇ ˛ g .A/1ˇ 1˛ u  4f .A/ˇ u˛ g .A/1ˇ u1˛ : (13.91) Theorem 13.10 on the spectrum of the subordinate generator is from [292], the CBFversion is from Hirsch [163]. A predecessor of this is Balakrishnan [16, 17]. The proof of Theorem 13.14 follows Schoenberg’s original argument [325] who considers completely monotone functions; the statement of the theorem is due to Bochner, but his proof in [63, pp. 99–100] is, unfortunately, incorrect. Lemma 13.15 is taken from [325]. The Bochner–Schoenberg theorem still holds on Abelian groups if we cast it in the following form: determine all functions f which operate on the negative (resp. positive) definite functions, i.e. f ı is negative (resp. positive) definite for all negative (resp. positive) definite functions . Note that . / D j j2 is indeed negative definite in the sense of Schoenberg, cf. Chapter 4. This is due to Herz [157] for positive definite functions and Harzallah [147, 148, 149] for negative definite functions; further generalizations are in Kahane [197]. Example 13.16 is due to Bochner [63, Theorem 4.3.3, p. 94], see also [185, Lemma 2.1]; note that the integrability condition in [185] is automatically satisfied. The second part of Example 13.16 appears in [233, Proposition 2.14] who hints to Rogers [306]. The section on the functional calculus is a simplified and generalized version of [320, 322], see also [183, Vol. 1, Chapter 4.3]. The presentation owes a lot to F. Hirsch who pointed out that the operator Ik appearing in Lemma 13.18 coincides with kRkf . The idea to reduce the functional calculus to an operational calculus on the level of the trivial semigroup t 7! e ta , cf. Remark 13.26, is from Westphal [367] where this is worked out for fractional powers. That paper also contains the fractional power analogues of Theorem 13.31 and Corollary 13.32. This approach is heavily influenced by (abstract) approximation theory, see e.g. Butzer and Berens [73], in particular Chapter 2.3. Corollary 13.34 is from [322], and Corollary 13.36, with a different proof and for complete Bernstein functions, is due to Pustyl’nik [298, Theorem 8]. The remarkable pointwise stability assertion for the functional calculus, Theorem 13.37, is an improvement of a similar result in [322]. The extension of the functional calculus sketched in Remark 13.38 is in line with the calculus developed in [298], and links our results with Hirsch’s theory on abstract potential operators: note that .A/ is essentially the inverse of the generator of a C0 -contraction semigroup, hence an abstract potential operator.

Section 13.4 Eigenvalue estimates for subordinate processes

255

Section 13.3: Functional inequalities have become indispensable in the analysis of Markovian semigroups, especially in estimates for semigroups and the corresponding heat kernels. Among the earliest contributions in this area are the papers by Federbush [114], Gross [138], and [122] by Fukushima who exploits ideas by Stampaccia [336] and uses the Sobolev-type inequality (13.52) to derive Lp , p > 2, estimates for the resolvent of a symmetric Markovian semigroup on L2 . In the hands of Varopoulos, Saloff-Coste and Coulhon – cf. [362] and the references given there – and Carlen, Kusuoka and Stroock – see, e.g. [74] – functional inequalities became a most powerful tool. For the history we refer to the surveys in Jacob [183, Vol. 2, Chapter 3.6] and Varopoulos et al. [362, p. 25]. The monographs [362], [313] and [311] are very good introductions to the field, while [364] is currently the most comprehensive treatise on functional inequalities and their applications. Our presentation follows the paper [324] where all results are proved in a Hilbert space context. This paper extends earlier results by Bendikov and Maheux [30] who consider only fractional powers and the Bernstein functions f ./ D ˛ , 0 < ˛ < 2. Lemma 13.39 is a generalization of Bihari’s inequality [57, Section 3], see also Walter [363, Chapter I, 1.VI and 6.IX] and Beckenbach-Bellman [27, Chapter 4, §4 and §5]. The idea of the proof comes from [80, Appendix A, Lemma A.1, A.3 pp. 193-4]; note, however, that the right hand side of the inequality (13.56) may be negative. This is the main difference compared to the earlier papers. For symmetric semigroups on a Hilbert space, Theorem 13.41 has a converse: if (13.62) holds for some f 2 BF, then (13.61) is also true, see [324, Proposition 11]. Similar converse statements hold for Corollaries 13.43 and 13.44. Therefore, it is possible to extend Corollary 13.45: .T tf / t0 is not hypercontractive if lim!1 f 1 ./=p D 1. Section 13.4: The recent study on the spectrum of the Dirichlet fractional Laplacian was initiated by Bañuelos and Kulczycki in [19]. Suppose that D is a bounded open .˛/ subset of Rd , let .Q n /n2N be the eigenvalues of . /˛=2 jD , arranged in decreasing order and repeated according to multiplicity. Using a mixed Steklov problem for the Laplacian in Rd C1 it was shown in [19] that for bounded Lipschitz domains D  Rd q Q.1/  Q .2/ for all n 2 N: n n By considering a higher order Steklov problem, DeBlassie shows in [91] that for bounded Lipschitz domains D  Rd and for any rational ˛ 2 .0; 2/ Q .2/ ˛=2 Q .˛/ n  .n /

for all n 2 N:

The upper bound above is shown for general subordinate Markov processes in [81], where a lower bound is also established. The exposition of this section is a combination of [81] and [83]. The last part of the proof of Theorem 13.49 is adapted from the proofs of [133, Lemma 3.1, Theorem 3.2]. For a general subordinator S whose Laplace exponent f is not necessarily a complete Bernstein function we have a weaker version of Theorem 13.48: that is, we

256

Chapter 13 Subordination and Bochner’s functional calculus

always have D.ED;f /  D.Ef;D / and Ef .u; u/  4ED;f .u; u/ for all u 2 D.ED;f /:

(13.92)

This is due to the fact that PsD u  Ps u for all non-negative u 2 D.ED;f /, which allows to conclude from (13.73)–(13.77) that u 2 D.Ef / and Ef .u; u/  ED;f .u; u/. For a general u 2 D.ED;f /, uC and u are both in D.ED;f /, and so we have that both uC and u are in D.Ef / and that Ef .uC ; uC /  ED;f .uC ; uC /;

Ef .u ; u /  ED;f .u ; u /:

Thus u D uC  u 2 D.Ef /. The contraction property of Dirichlet forms yields ED;f .uC ; uC /  ED;f .u; u/;

ED;f .u ; u /  ED;f .u; u/:

Therefore we have D.ED;f /  D.Ef;D / as well as (13.92). Using (13.92), we can see that the conclusions of Theorems 13.53 and 13.54 hold for a general subordinator S , with the exception that the upper bounds for Ef .u; u/ in (13.89) and for Q n have an additional multiplicative factor 4. Using the following formula for the first eigenvalue ® ¯ Q 1 D inf Ef .juj; juj/ W u 2 D.Ef;D / with kukL2 .D;m/ D 1 and the analogous formula for the first eigenvalue f .1 / for .ED;f ; D.ED;f //, we get that Q 1  f .1 / for a general subordinator. One can also prove that the eigenvalues of the generator of a killed subordinate Markov process depend continuously on the Laplace exponent of the subordinator. For this and related results, we refer to [82].

Chapter 14

Potential theory of subordinate killed Brownian motion Recall that a Bernstein function f is a special Bernstein function if its conjugate function f ? ./ WD =f ./ is again a Bernstein function. Thus, the identity function factorizes into the product of two Bernstein functions. In this chapter we will use this factorization and show, as a consequence of Theorem 11.9, that the potential kernel of a strong Markov process Y is the composition of the potential kernels of two subordinate processes. As before, we call the subordinators associated with special Bernstein functions special subordinators. If the underlying process Y is a killed Brownian motion with an intrinsically ultracontractive semigroup, the factorization of the potential kernel implies a representation of excessive functions of Y as potentials of one of the subordinate processes. Let E be a locally compact separable metric space with the Borel  -algebra B.E/. By Y D .Y t ; Px / t0;x2E we denote a strong Markov process with state space E and semigroup .P t / t0 . We will assume that the process Y is transient, R 1 i.e. there exists a positive measurable function g W E ! .0; 1/ such that x 7! 0 P t g.x/ dt is not identically infinite. The potential operator of the process Y is defined by Z 1 P t g.x/ dt; (14.1) Gg.x/ D 0

where x 2 E and g W E ! Œ0; 1/ is a measurable function. The function Gg is called the potential of g. Let f 2 SBF with the representation (11.1) and let f ? be its conjugate function given by (11.2). Further, let S D .S t / t0 and T D .T t / t0 be the special subordinators with Laplace exponents f; f ? 2 SBF and assume that S; T and Y are independent. The potential measures of S and T will be denoted f by U and V respectively. We define two subordinate processes Y f D .Y t / t0 and ? ? f Y f D .Y t / t0 by ?

Y tf D Y .S t / and Y tf D Y .T t /; Then Y f and Y f [69].

?

t  0:

are again transient strong Markov processes on .E; B.E//, see

258

Chapter 14 Potential theory of subordinate killed Brownian motion ?

?

Lemma 14.1. The potential operators G f and G f of Y f and Y f are given by Z P t g.x/ U.dt /; (14.2) G f g.x/ D Œ0;1/ Z ? P t g.x/ V .dt /: (14.3) G f g.x/ D Œ0;1/

Proof. We will only show (14.2), the proof of (14.3) is similar. Let . t / t0 be the convolution semigroup on Œ0; 1/ corresponding to f . The transition operators f .P t / t0 of Y f are given by f

f

P t .x; A/ D Px .Y t 2 A/ D Px .Y .S t / 2 A/ Z Px .Ys 2 A/  t .ds/ D Œ0;1/ Z Ps .x; A/  t .ds/; D Œ0;1/

where x 2 E and A 2 B.E/. For every non-negative Borel function g W E ! Œ0; 1/ it follows that Z 1 Z 1Z f f P t g.x/ dt D Ps g.x/  t .ds/ dt G g.x/ D Œ0;1/ 0 0 Z Ps g.x/ U.ds/: D Œ0;1/

Throughout this chapter we will assume that b > 0 or .0; 1/ D 1. This implies that U.dt / D u.t / dt and V .dt / D bı0 .dt / C v.t / dt where u; v W .0; 1/ ! .0; 1/ ? are non-increasing functions. Hence, the potential operators G f and G f can be written as Z 1 G f g.x/ D P t g.x/ u.t / dt; (14.4) 0 Z 1 ? P t g.x/ v.t / dt: (14.5) G f g.x/ D bg.x/ C 0

Moreover, by Theorem 11.9, we have Z t Z t u.s/v.t  s/ ds D bu.t / C v.s/u.t  s/ ds D 1; bu.t / C 0

t > 0:

0

If the transition kernels .P t .x; dy// t0 have densities p.t; x; y/ with respect to a reference measure m.dy/ on .E; B.E//, then the potential operator G will also have a density G.x; y/ given by Z 1 p.t; x; y/ dt: G.x; y/ D 0

259

Chapter 14 Potential theory of subordinate killed Brownian motion

In this case one can define the potential of a measure on .E; B.E// by Z 1 Z G.x; y/ .dy/ D P t .x/ dt: G .x/ D 0

E

Note that the potential of a function g can be regarded as the potential of the measure g.x/ m.dx/. Further, the potential operator G f will have a density G f .x; y/ given by the formula Z G f .x; y/ D

1

p.t; x; y/u.t / dt: 0

The factorization in the next proposition is similar in spirit to Corollary 13.25. Proposition 14.2.

(i) For any non-negative Borel function g on E we have ?

?

G f G f g.x/ D G f G f g.x/ D Gg.x/;

x 2 E:

(ii) If the transition kernels .P t .x; dy// t0 admit densities, then for any Borel measure on .E; B.E// we have ?

G f G f .x/ D G .x/; ?

x 2 E:

Proof. (i) We are only going to show that G f G f g.x/ D Gg.x/ for all x 2 E. For ? the proof of G f G f g.x/ D Gg.x/ we refer to part (ii). Let g be any non-negative Borel function on E. Using (14.4), (14.5), Fubini’s theorem and Theorem 11.9 we get Z 1 ? f f? P t G f g.x/ u.t / dt G G g.x/ D  Z0 1  Z 1 D P t bg.x/ C Ps g.x/ v.s/ ds u.t / dt 0  Z 1 0Z 1 f Pt Ps g.x/ v.s/ds u.t / dt D bG g.x/ C Z0 1 Z 1 0 P tCs g.x/ v.s/u.t / ds dt D bG f g.x/ C 0  Z01 Z 1 f D bG g.x/ C Pr g.x/ v.r  t /dr u.t / dt 0 t  Z Z 1 r f u.t /v.r  t /dt Pr g.x/ dr D bG g.x/ C 0 0  Z 1 Z r D u.t /v.r  t /dt Pr g.x/ dr bu.r/ C 0 Z0 1 Pr g.x/ dr D 0

D Gg.x/:

260

Chapter 14 Potential theory of subordinate killed Brownian motion

(ii) Similarly as above, ?

Z

G f G f .x/ D bG f .x/ C

1

P t G f .x/ v.t / dt  Z0 1 Z 1 f Pt Ps .x/ u.s/ds v.t / dt D bG .x/ C Z0 1 Z 1 0 f D bG .x/ C P tCs .x/ u.s/v.t / ds dt 0  Z0 1 Z 1 f D bG .x/ C Pr .x/ u.r  t /dr v.t / dt  Z0 1 Z t r f D bG .x/ C u.r  t /v.t /dt Pr .x/ dr 0  Z r0 Z 1 u.r  t /v.t /dt Pr .x/ dr bC D 0 Z0 1 Pr .x/ dr D 0

D G .x/: In the rest of this chapter, we will always assume that the underlying Markov process Y is a Brownian motion killed upon exiting a bounded open connected subset D  Rd . To be more precise, let X D .X t ; Px / t0;x2Rd be a Brownian motion in Rd with transition density   jx  yj2 d=2 p.t; x; y/ D .4 t / ; t > 0; x; y 2 Rd ; exp  4t and let D D inf¹t > 0 W X t … Dº be the first exit time of X from D. The process X D D .X tD ; Px / t0;x2D defined by ´ X t ; t < D ; D Xt D @; t  D ; where @ is the cemetery point, is called the Brownian motion killed upon exiting D. The process X D is a Hunt process, and since D is bounded, it is transient. It will play the role of the process Y from the beginning of this chapter. The semigroup of X D will be denoted by .P tD / t0 , and its transition density with respect to Lebesgue measure by p D .t; x; y/, t > 0, x; y 2 D. The potential operator of X D is given by Z 1 D P tD g.x/ dt G g.x/ D 0

and has a density G D .x; y/ D

R1 0

p D .t; x; y/ dt , x; y 2 D.

261

Chapter 14 Potential theory of subordinate killed Brownian motion

Since the transition density p D .t; x; y/ is strictly positive, the eigenfunction '1 of the operator  jD corresponding to the smallest eigenvalue 1 can be chosen to be positive, see for instance [89]. We assume that '1 is normalized so that R strictly 2 ' .x/ dx D 1. D 1 From now on we will assume that .P tD / t0 is intrinsically ultracontractive, that is, for each t > 0 there exists a constant c t such that p D .t; x; y/  c t '1 .x/'1 .y/;

x; y 2 D:

Intrinsic ultracontractivity of .P tD / t0 is a very weak geometric condition on D. It is well known, cf. [90], that for a bounded Lipschitz domain D the semigroup .P tD / t0 is intrinsically ultracontractive. For every intrinsically ultracontractive .P tD / t0 there is some cQt > 0 such that p D .t; x; y/  cQt '1 .x/'1 .y/;

x; y 2 D:

Let S D .S t / t0 and T D .T t / t0 be two special subordinators with conjugate Laplace exponents f; f ? 2 SBF given by (11.1) and (11.2), respectively. We retain the assumption that b > 0 or .0; 1/ D 1, and keep the notation for the potential measures U.dt / D u.t / dt and V .dt / D bı0 .dt / C v.t / dt . Suppose that X, S and D;f T are independent and define the subordinate processes X D;f D .X t / t0 and ? ? D;f / t0 by X D;f D .X t D;f

Xt

D;f ?

D X D .S t / and X t

D X D .T t /;

t  0:

?

Then X D;f and X D;f are strong Markov processes on D. We call X D;f and ? X D;f subordinate killed Brownian motions. Their semigroups and potential op? ? erators will be denoted by .P tD;f / t0 , G D;f and .P tD;f / t0 , G D;f , respectively. Clearly, Z Z 1 D;f D g.x/ D P t g.x/ U.dt / D P tD g.x/ u.t / dt; G Œ0;1/ 0 Z Z 1 ? G D;f g.x/ D P tD g.x/ V .dt / D bg.x/ C P tD g.x/ v.t / dt; Œ0;1/

0

where x 2 D and g is a non-negative Borel measurable function on D. By Proposi? ? tion 14.2 we have G D;f G D;f g D G D;f G D;f g D G D g for every Borel function ? g W D ! Œ0; 1/, and G D;f G D;f D G D for every measure on .D; B.D//. D;f Recall that a Borel function s W D ! Œ0; 1 is excessive for X D;f (or .P t / t0 ), if P tD;f s  s for all t  0 and s D lim t!0 P tD;f s. We will denote the family of ? all excessive function for X D;f by S.X D;f /. The notation S.X D / and S.X D;f / is now self-explanatory.

262

Chapter 14 Potential theory of subordinate killed Brownian motion

A Borel function h W D ! Œ0; 1 is harmonic for X D;f if h is not identically infinite in D and if for every relatively compact open subset O  O  D,    h.x/ D Ex h X D;f .O /

for all x 2 O;

® ¯ D;f … O is the first exit time of X D;f from O. We dewhere O D inf t > 0 W X t note the family of all non-negative harmonic functions for X D;f by HC .X D;f /. Similarly, HC .X D / denotes the family of all non-negative harmonic functions for X D . These are precisely those non-negative functions in D which satisfy h D 0 in D. Recall that if h 2 HC .X D / then both h and P tD h are continuous functions. It is easy to show that HC .  /  S.  /, see the proof of Lemma 2.1 in [331]. D;f D;f D;f Let .G />0 be the resolvent of the semigroup .P t / t0 . Then G is given by a kernel which is absolutely continuous with respect to Lebesgue measure. Moreover, one can easily show that for a bounded Borel function g vanishing outside a D;f compact subset of D, the functions x 7! G g.x/,  > 0, and x 7! G D;f g.x/ are continuous. This implies, see e.g. [60, p. 266], that all functions in S.X D;f / are lower semicontinuous. ?

Proposition 14.3. If g 2 S.X D;f /, then G D;f g 2 S.X D /. Proof. First observe that if g 2 S.X D;f /, then g is the increasing limit of potentials G D;f gn for suitable non-negative Borel functions gn , cf. [87, p. 86]. Hence it follows from Proposition 14.2 that ?

?

G D;f g D lim G D;f G D;f gn D lim G D gn : n!1

n!1

?

Therefore, G D;f g is either in S.X D / or identically infinite since it is an increasing ? limit of excessive functions of X D . We prove now that G D;f g is not identically infinite. In fact, since g 2 S.X D;f /, there exists some x0 2 D such that for every t > 0, Z D;f

1 > g.x0 /  P t

g.x0 / D

1

0

PsD g.x0 /  t .ds/;

where  t is the distribution of S t . Thus there is some s > 0 such that PsD g.x0 / is finite. Hence Z Z D D 1 > Ps g.x0 / D p .s; x0 ; y/g.y/ dy  cQs '1 .x0 / '1 .y/g.y/ dy; D

D

263

Chapter 14 Potential theory of subordinate killed Brownian motion

so we have Z G D

R D

D;f ?

'1 .y/g.y/ dy < 1. Consequently Z

?

g.x/ G D;f '1 .x/ dx D   Z Z 1 D D g.x/ b'1 .x/ C P t '1 .x/ v.t /dt dx D 0   Z Z 1 D g.x/ b'1 .x/ C e 1 t '1 .x/ v.t / dt dx  0 Z 1  ZD D '1 .x/g.x/ dx b C e 1 t v.t / dt < 1:

g.x/ '1 .x/ dx D

D

0

?

Therefore G D;f g is not identically infinite in D. ?

Remark 14.4. Note that the proposition above is valid with X D;f and X D;f inter? changed: if g 2 S.X D;f /, then G D;f g 2 S.X D /. Proposition 14.2 (ii) shows that if s D G D is the potential of a measure, then ? s D G D;f g where g D G D;f is in S.X D;f /. The function g can be written in the following way: Z 1 PrD .x/ u.r/ dr g.x/ D   Z0 1 Z   D D  du.t / dr Pr .x/ u.1/ C .r;1/ 0  Z Z 1 Z 1   D D  du.t / dr Pr .x/u.1/ dr C Pr .x/ D 0

D u.1/s.x/ C Z D u.1/s.x/ C

0

Z

Z .0;1/

.0;1/



t 0



PrD .x/ dr

.r;1/



  du.t /

 P tD s.x/  s.x/ du.t /:

(14.6)

In Proposition 14.8 below we will show that every s 2 S.X D / can be represented as ? a potential G D;f g, where g, given by (14.6), is in S.X D;f /. In order to accomplish this, we will first show that g is continuous when s 2 HC .X D /. Before we do this, we prove two preliminary results about non-negative harmonic functions of X D . For any x0 2 D, choose r > 0 such that B.x0 ; 2r/  D. Put B D B.x0 ; 2r/. Lemma 14.5. If h 2 HC .X D /, then there is a constant c > 0 such that h.x/  P tD h.x/  ct;

x 2 B; t > 0:

264

Chapter 14 Potential theory of subordinate killed Brownian motion

Proof. For any x 2 B, h.X t^B / is a Px -martingale. Therefore,   0  h.x/  P tD h.x/ D Ex h.X t^B /  Ex h.X t /1¹t 0, n 2 N and y 2 B, we have Z t PrD kn .y/ dr sn .y/  P tD sn .y/ D 0 Z t D PrD .h  P1=n h/.y/ dr Dn Z Dn

0

t 0

Z PrD h.y/ dr

n

1 tC n 1 n

PrD h.y/ dr:

If t < 1=n, then it follows from Lemma 14.5 that Z sn .y/ 

P tD sn .y/

Dn Z n

1 tC n



1 n 1 tC n 1 n

Z

 c1 n



Z dr n

 h.y/  PrD h.y/ dr

1 tC n 1 n



h.y/PrD h.y/

r dr  c2 t:

0

t



 h.y/PrD h.y/ dr

265

Chapter 14 Potential theory of subordinate killed Brownian motion

If t  1=n, then again by Lemma 14.5 we have Z sn .y/  P tD sn .y/ D n Z Dn Z

1 n

0

Z PrD h.y/ dr  n

1 tC n 

t 1 tC n

1 tC n

t

PrD h.y/ dr

 h.y/PrD h.y/ dr n

Z

1 n



0

 h.y/PrD h.y/ dr



 h.y/  PrD h.y/ dr n t   1 D  c4 t:  h.y/  P tC h.y/  c t C 3 1 n n Lemma 14.7. If h 2 HC .X D /, the function g defined by Z  D  P t h.x/  h.x/ du.t /; g.x/ D u.1/h.x/ C .0;1/

x 2 D;

(14.7)

is continuous. Proof. For x0 2 D choose r > 0 such that B.x0 ; 2r/  D, and let B WD B.x0 ; r/. It follows from the continuity and harmonicity of h that there exists a constant c1 > 0 such that 0  h.x/  P tD h.x/  c1 on B. Using Lemma 14.5 we know that there exists some c2 > 0 such that 0  h.x/  P tD h.x/  c2 t;

x 2 B:

R Since .0;1/ .1 ^ t /.du.t // < 1, we can apply the dominated convergence theorem to get the continuity of g. Proposition 14.8. If s 2 S.X D /, then ?

s.x/ D G D;f g.x/;

x 2 D;

where g 2 S.X D;f / and is given by the formula Z 1  D  P t s.x/  s.x/ du.t /: g.x/ D u.1/s.x/ C

(14.8)

0

Proof. We know that the result is true when s is the potential of a measure. Let s 2 S.X D / be arbitrary. By the Riesz decomposition theorem, s D G D Ch, where

is a measure on D and h is in HC .X D /, see e.g. [60, Chapter 6]. By linearity, it is enough to prove the result for s 2 HC .X D /. In the rest of the proof we assume therefore that s 2 HC .X D /. Define the function g by formula (14.8). We have to prove that g is an excessive function for X D;f ? and s D G D;f g. By Lemma 14.7, we know that g is continuous.

266

Chapter 14 Potential theory of subordinate killed Brownian motion

D s.x// and s WD G D k .x/. It follows from Lemma 14.6 Define kn WD .s.x/ P1=n n n n!1

that sn increases to s. Thus P tD sn " P tD s and sn  P tD sn ! s  P tD s. If Z    sn .x/  P tD sn .x/  du.t / ; gn .x/ D u.1/sn .x/ C .0;1/

D;f ?

then we know that sn D G gn and gn is excessive for X D;f . It follows from Lemma 14.6 that, for every x 2 D, 0  sn .x/  P tD sn .x/  s.x/. Again by Lemma D 14.6 there R exists for every x 2 D some c > 0 such that 0  sn .x/  P t sn .x/  c t . Since .0;1/ .1 ^ t /.du.t // < 1, we can apply the dominated convergence theorem to get Z    s.x/  P tD s.x/  du.t / g.x/ D u.1/s.x/ C .0;1/ Z    lim sn .x/  P tD sn .x/  du.t / D lim u.1/sn .x/ C n!1

.0;1/ n!1

D lim gn .x/: n!1

By Fatou’s lemma and the fact that gn is G D;f -excessive, we get for any x 2 D and  > 0   D;f D;f D;f G g.x/ D G lim gn .x/  lim inf G gn .x/ n!1

n!1

 lim inf gn .x/ D g.x/I n!1

this means that g is supermedian. Since it is well known that a supermedian function which is lower semicontinuous is excessive, cf. [87, p. 81], this proves that g ? is excessive for X D;f . By Proposition 14.3 we then have that G D;f g is excessive for X D . R1 Set G1D h.x/ D 0 e t P tD h.x/ dt for any non-negative measurable function h and define s 1 WD s  G1D s. Using an argument similar to that of the proof of Proposition 14.3, we can show that G D s is not identically infinite. By the resolvent equation G D s 1 D G D s  G D G1D s D G1D s, or equivalently, s.x/ D s 1 .x/ C G1D s.x/ D s 1 .x/ C G D s 1 .x/;

x 2 D: ?

Formula (14.6) for the potential G D s 1 , Fubini’s theorem and the fact that G D;f and G1D commute, show   Z D D 1 D;f ? D 1 D D 1 D 1 G1 s D G s D G .P t G s  G s / du.t / u.1/G s C .0;1/   Z D;f ? D D D D .P t G1 s  G1 s/ du.t / DG u.1/G1 s C .0;1/   Z ? D G1D G D;f u.1/s C .P tD s  s/ du.t / : .0;1/

Chapter 14 Potential theory of subordinate killed Brownian motion

267

By the uniqueness principle, see e.g. [87, p. 140], it follows that   Z 1 ? D;f ? D u.1/s C sDG .P t s  s/ du.t / D G D;f g a.e. in D: 0

?

Since both s and G D;f g are excessive for X D , they are equal everywhere. Propositions 14.2 and 14.8 can be combined in the following theorem containing additional information on harmonic functions. ?

Theorem 14.9. For s 2 S.X D /, there is a function g 2 S.X D;f / with s D G D;f g. The function g is given by the formula (14.6). Furthermore, if s 2 HC .X D /, then g 2 HC .X D;f /. ? Conversely, if g 2 S.X D;f /, then the function s defined by s D G D;f g is in S.X D /. If, moreover, g 2 HC .X D;f /, then s 2 HC .X D /. Finally, every non-negative harmonic function for X D;f is continuous. Proof. It remains to show the statements about harmonic functions. First note that every g 2 S.X D;f / admits the Riesz decomposition g D G D;f C h where is a Borel measure on D and h 2 HC .X D;f /, see [60, Chapter 6]. We have already mentioned that functions in S.X D / admit such a decomposition. Since S.X D / and S.X D;f / are in one-to-one correspondence, and since potentials of measures of X D and X D;f are in one-to-one correspondence, the same must hold for HC .X D / and HC .X D;f /. The continuity of non-negative harmonic functions for X D;f follows from Lemma 14.7 and Proposition 14.8. Comments 14.10. The study of the potential theory of subordinate killed Brownian motions was initiated in [128] for stable subordinators and carried out satisfactorily in this case in [129]. The general case of special subordinators was dealt with in [332]. Our exposition follows that of [64, Section 5.5.2], but we provide a more elementary and direct proof of Proposition 14.8 here. Using Theorem 14.9, one can generalize some deep and important potential theoretic results for killed Brownian motions, like the Martin boundary identification and the boundary Harnack principle, to subordinate killed Brownian motions. For these results we refer to [129], [64, Section 5.5] and [332] where the underlying process X is a killed rotationally invariant ˛-stable process. Intrinsic ultracontractivity was introduced by Davies and Simon in [90]. It is a very weak regularity condition on D. For example, see [90], the killed semigroup .P tD / t0 is intrinsically ultracontractive whenever D is a bounded Lipschitz domain. For weaker conditions on D guaranteeing that .P tD / t0 is intrinsically ultracontractive, we refer to [18] and [25].

Chapter 15

Applications to generalized diffusions

Complete Bernstein functions and the corresponding Bondesson class of sub-probability measures play a significant role in various parts of probability theory. In this chapter we will explain their appearance in the theory of generalized one-dimensional diffusions (also called gap diffusions or quasi-diffusions). We have two main objectives: firstly, we want to characterize the Laplace exponents of the inverse local times at zero of generalized diffusions. It turns out that these are precisely the complete Bernstein functions. Secondly, we want to study the first passage distributions for generalized diffusions and to explain their connection with the Bondesson class and its subclasses. The crucial tool in these investigations will be Kre˘ın’s theory of strings which we will describe in some detail. In this chapter we will use Dirichlet forms and Hunt processes. The basic definitions and results are collected in Appendix A.2.

15.1 Inverse local time at zero We start by introducing the family MC of the so-called strings. Definition 15.1. A non-decreasing and right-continuous function m W R ! Œ0; 1 is a string, if (a) m.x/ D m.0/ D 0 for all x < 0, (b) m.x0 / < 1 for some x0  0, (c) m.x/ > 0 for all x > 0. The family of all strings will be denoted by MC . Slightly abusing notation, we will use the same letter m to denote the measure on R induced by the non-decreasing function m 2 MC . Clearly, m.1; 0/ D 0. Let r WD sup¹x W m.x/ < 1º be the length of the string, and let Em denote the support of the measure m restricted to Œ0; r/. Note that the point r of a possibly infinite jump is excluded from the support while 0 2 Em . Finally, let l WD sup Em  r. Next we introduce generalized diffusions. For this let B C D .B tC ; Px / t0;0x t º. Define a process X D .X t / t0 via the time-change of B C by the process  , X t WD B C . t /. By [123, Theorem 6.2.1] X is an m-symmetric Hunt process on Em which is associated with a regular Dirichlet form on L2 .Em ; m/. Note that the lifetime  of X is equal to C1 . Definition 15.2. The process X is called a generalized diffusion on Œ0; r/ (or, more precisely, on Em ). Clearly, X t D @ for t  . The class of such processes includes regular diffusions on Œ0; r/ and Œ0; l (in the sense of Itô–McKean), as well as birth-and-death processes. We will use the standard convention that any function g on the state space is automatically extended to @ by setting g.@/ D 0. The generalized diffusion X admits the local time process LX D .LX .t; x// t0;x2Em which can be realized as a time-change of the local time of B C . More precisely, in the same way as in [307, Theorem V.49.1], it follows that LX .t; x/ D L. t ; x/ and Z Z s g.X t / dt D g.x/LX .s; x/ m.dx/; (15.2) 0

Œ0;r /

for every bounded measurable function g, and every s  0. In particular, it follows from the joint continuity of L that x 7! LX .s; x/ is continuous. For simplicity, let us denote the local time at zero for X, LX .t; 0/, by `X .t /, and let .`X /1 .t / WD inf¹s > 0 W `X .s/ > t º be the inverse local time at zero. The inverse local time is a (possibly killed) subordinator, see, e.g. [47, p. 114]. Therefore we may assume that the Laplace exponent f of .`X /1 has the representation Z .1  e t / .dt /: f ./ D a C b C .0;1/

270

Chapter 15 Applications to generalized diffusions

An interesting problem is to characterize the family of Bernstein functions that can arise as Laplace exponents of inverse local times at zero of generalized diffusions. This question was raised in [178, p. 217] and solved, independently, in [209] and [217]. The answer is given in the following theorem. Theorem 15.3. Let X be a generalized diffusion corresponding to m 2 MC and let .`X /1 be its inverse local time at zero. Then the Laplace exponent f of .`X /1 belongs to CBF. Conversely, given any function f 2 CBF, there exists a generalized diffusion such that f is the Laplace exponent of its inverse local time at zero. The proof of this theorem relies heavily on Kre˘ın’s theory of strings. A full exposition of this theory would lead us far away from the topics of this book. Therefore, we will only outline the main ideas, and give more details for the parts of the proof where the methods are close to our subject. The first ingredient of the proof is probabilistic: for  > 0 the -potential operator of X is defined by Z G g.x/ WD Ex

1

e t g.X t / dt;

(15.3)

0

where x 2 Em and g is a non-negative Borel function on Œ0; r/. By Fubini’s theorem and (15.2), the -potential can be rewritten as  Z s Z 1 s e g.X t / dt ds G g.x/ D Ex  Z0 Z0 1 s X D Ex e g.y/L .s; y/ m.dy/ ds 0 Œ0;r /  Z 1 Z D Ex e s LX .s; y/ ds g.y/ m.dy/ Œ0;r / 0  Z Z s X Ex e L .ds; y/ g.y/ m.dy/ D Œ0;r / Œ0;1/ Z G .x; y/g.y/ m.dy/: (15.4) D Œ0;r /

Here G .x; y/ is the kernel of the -potential operator G defined by Z  s X e L .ds; y/ ; x; y 2 Em : G .x; y/ WD Ex

(15.5)

Œ0;1/

Note that for each x 2 Em , the function y 7! G .x; y/ is continuous on Em . From now on, each function u defined on Em is linearly extended to Œ0; r/ n Em ; note that this does not uniquely determine the extension on .l; r/. This applies also to G .x; /.

271

Section 15.1 Inverse local time at zero

Proposition 15.4. For every  > 0 it holds that G .0; 0/ D

1 : f ./

Proof. From (15.5) we have Z Z 1 X 1 G .0; 0/ D E0 e t `X .dt / D E0 e  .` / .t/ dt 0 Œ0;1/ Z 1 Z 1 1  .`X /1 .t/ : E0 .e /dt D e tf ./ dt D D f ./ 0 0 By Proposition 15.4 and Theorem 7.3, f 2 CBF if and only if  7! G .0; 0/ 2 S. In order to show that  7! G .0; 0/ 2 S, we are going to identify G .0; 0/ with the so-called characteristic function of the string and show that the latter is a Stieltjes function. To do so we need the analytic counterpart of generalized diffusions, namely d the second-order differential operator A D ddm dx which is formally defined in the 0 following way. The domain D .A/ of A consists of functions u W R ! C such that Z u.x/ D ˛ C ˇx C .x  y/g.y/ m.dy/ Œ0;x Z xZ (15.6) D ˛ C ˇx C g.w/ m.dw/ dy; Œ0;y

0

where ˛; ˇ 2 C and g W R ! C. If x < 0 we interpret Œ0; x as the empty set ;, therefore u.x/ D ˛ Cˇx when x < 0. Then A is defined by Au WD g for u 2 D0 .A/. Every function u in D0 .A/ is absolutely continuous, linear outside of Em , and has right u0C and left derivatives u0 . More precisely, Z u.y/  u.x/ DˇC g.y/ m.dy/; (15.7) u0C .x/ D lim yx y#x Œ0;x Z u.y/  u.x/ DˇC u0 .x/ D lim g.y/ m.dy/: (15.8) yx y"x Œ0;x/ In particular, ˇ D u0 .0/. Moreover, for 0  x  l, u0C .x/  u0 .x/ D m¹xºAu.x/: For z 2 C, let  D .x; z/ and tions for x 2 Œ0; r/: .x; z/ D 1 C z

D

(15.9)

.x; z/ satisfy the following integral equa-

Z Œ0;x

.x  y/ .y; z/ m.dy/;

(15.10)

.x  y/ .y; z/ m.dy/:

(15.11)

Z .x; z/ D x C z

Œ0;x

272

Chapter 15 Applications to generalized diffusions

It can be shown that both equations above have unique solutions. We sketch the construction of . Let 0 .x/  1 and define for n 2 N Z .x  y/ n1 .y/ m.dy/: (15.12) n .x/ WD Œ0;x

Expressing n as an n-fold integral, one can prove the following estimate, see [196, p. 32] or [105, p. 162] for details: n .x/ 

.2x m.x//n .x m.x//n  : nn nŠ .2n/Š

Then .x; z/ WD

1 X

n .x/z n

(15.13)

(15.14)

nD0

converges and satisfies (15.10). Note that the equations (15.10) and (15.11) can be rewritten as A.; z/ D z.; z/, 0 A .; z/ D z .; z/, with the boundary conditions .0; z/ D 1,  .0; z/ D 0, 0 .0; z/ D 0 and  .0; z/ D 1. Here, and in the remaining part of this chapter, the derivatives are with respect to the first variable. In the remainder of this chapter we will use the following notational convention when working with the functions  and : if the second argument z is a real number, we will write  instead of z. The functions  and have the following properties: (a) for each fixed x 2 Œ0; r/, z 7! .x; z/ and z 7! .x; z/ are real entire functions, i.e. they are analytic functions on C which are real if z 2 R; 0 (b) for all x 2 .0; r/ and  > 0 we have .x; / > 0, C .x; / > 0, 0 and C .x; / > 0;

.x; / > 0

0 .x; z/ ¤ 0, (c) for every z 2 C with Im z ¤ 0 we have .x; z/ ¤ 0, C 0 and C .x; z/ ¤ 0.

.x; z/ ¤ 0

Property (a) follows from (15.14) and the analogous equation for , and (b) follows from (15.12) and the analogous equations for ; (c) can be deduced from the following Lagrange identity: if u; v 2 D0 .A/, then for all x 2 Œ0; r/, Z   v.y/Au.y/  u.y/Av.y/ m.dy/ (15.15) Œ0;x  0   0  0 0 D uC .x/v.x/  u.x/vC .x/  u .0/v.0/  u.0/v .0/ ; cf. [196, p. 24]. It follows from this that for any x 2 Œ0; r/ and z 2 C, .x; z/

0 C .x;z/

0  C .x; z/ .x; z/ 0 .x; z/ .x; z/ D 1: D .x; z/ 0 .x; z/  

(15.16)

273

Section 15.1 Inverse local time at zero

Proposition 15.5. For x 2 Œ0; r/ and z 2 C n .1; 0 define h.x; z/ WD

.x; z/ : .x; z/

Then h.x; / restricted to .0; 1/ belongs to S. Moreover, for 0  x < r, Z x 1 h.x; z/ D dy: .y; z/2 0

(15.17)

Proof. If x D 0, then h.0; /  0, so there is nothing to prove. Thus we assume that x > 0. Since the denominator does not vanish, h.x; z/ is well defined for all z 2 C n .1; 0 and it is analytic on C n .1; 0. We will show that h.x; / switches H" and H# , which proves the claim because of Corollary 7.4. For 0  y < r define x .y; z/ WD  .y; z/ C h.x; z/.y; z/;

(15.18)

and note that A x .; z/ D z x .; z/ on Œ0; x. A straightforward calculation yields that 1 ; . x /0C .x; z/ D  x .x; z/ D 0; .x; z/ (15.19) x .0; z/ D h.x; z/; . x /0 .0; z/ D 1: Let Im z ¤ 0. By the Lagrange identity (15.15) applied to x .; z/ and x .; z/, N and by (15.19), we get Z x .y; z/ x .y; z/ N m.dy/ .z  z/ N Œ0;x   N  . x /0C .x; z/ N x .x; z/ D . x /0C .x; z/ x .x; z/   N  . x /0 .0; z/ .0; N z/  . x /0 .0; z/ x .0; z/     D  h.x; z/  h.x; z/ N D  h.x; z/  h.x; z/ : Z

Hence Im z R

Œ0;x

  j x .y; z/j2 m.dy/ D Im h.x; z/ :

Since Œ0;x j x .y; z/j m.dy/ > 0, we have that Im z  Im h.x; z/  0. The last statement of the proposition is a consequence of 2

.x; z/ h.x; z/ D D .x; z/

Z 0

x



.y; z/ .y; z/

0

where the last equality follows from (15.16).

Z dy D

0

x

1 dy; .y; z/2

(15.20)

274

Chapter 15 Applications to generalized diffusions

Let  > 0 and 0 < x  y < r. Then Z

y

m.w/ dw .y; /  1 C 1 .y/ D 1 C  0 Z y y x  :  m.w/ dw   m 2 2 x=2 Therefore,

Z x

r

4 dy  2 .y; /2  m.x=2/2

Z

r x

(15.21)

dy < 1: y2

Since .; / is bounded away from zero on Œ0; x, it follows that Z r dy < 1; 2 0 .y; /

(15.22)

and it can be easily shown that Z lim

!1 0

r

dy D 0: .y; /2

(15.23)

Definition 15.6. The function h W .0; 1/ ! Œ0; 1/ defined by Z r .x; / dy D h./ WD lim h.x; / D lim 2 x!r x!r .x; / 0 .y; /

(15.24)

is called the characteristic function of the string m 2 MC . It follows from Proposition 15.5 and (15.22) that h is well defined. The next theorem is one of the fundamental results of Kre˘ın’s theory of strings. Theorem 15.7. (i) Let m 2 MC be a string. Then its characteristic function h is a Stieltjes function such that h.C1/ D 0, and hence it has the representation Z 1  .dt /; h./ D Œ0;1/  C t where  is a measure on Œ0; 1/ such that

R

.0;1/ .1

C t /1  .dt / < 1.

(ii) Conversely, for any h 2 S such that h.C1/ D 0 there exists a unique string m 2 MC such that h is the characteristic function of m. Proof. (i) It was shown in Proposition 15.5 that h.x; / 2 S for every x 2 Œ0; r/. By Definition 15.6, h is a pointwise limit of Stieltjes functions which is, by The-

275

Section 15.1 Inverse local time at zero

orem 2.2 (iii), again a Stieltjes function. Formula (15.23) implies that h.C1/ D 0, whence the form of the representation. (ii) The proof of the converse is beyond the scope of this book and we refer the interested reader to [105] and [196]. Remark 15.8. If we drop the requirement that m.x/ > 0 for every x > 0 in the definition of m 2 MC , then c WD inf Em may be strictly positive. The discussions leading to Theorem 15.7 remain almost literally the same, except that the limit in (15.23) turns out to be equal to c. Theorem 15.7 is then slightly extended to encompass all Stieltjes functions. Proposition 15.9. Let m 2 MC be a string and define for 0  x < r and  > 0 a function .x; / WD  .x; / C h./.x; /. Then Z r dy ; (15.25) .x; / D .x; / 2 x .y; / A .; / D  .; / and 0 .0; / D 1. Proof. By (15.24) and (15.20) Z

r dy .x; / D  .x; / C .x; / 2 0 .y; / Z x Z r dy dy D .x; / C .x; / 2 2 .y; / 0 .y; / Z r0 dy D .x; / : 2 x .y; /

That A .; / D  .; / follows immediately from the same property of .; / and .; /, while 0 .0; / D 1 is a straightforward computation. For  > 0 let

´ R .x; y/ WD

.x; / .y; /; .x; /.y; /;

0  x  y < r; 0  y  x < r;

(15.26)

and note that by Proposition 15.9 R .0; 0/ D .0; / .0; / D h./: Thus,  7! R .0; 0/ 2 S, see also Remark 12.8. For g 2 L2 ..1; r/; m/ define the -resolvent operator Z R .x; y/g.y/ m.dy/: R g.x/ WD Œ0;r /

(15.27)

(15.28)

276

Chapter 15 Applications to generalized diffusions

It is easy to prove that R is a bounded linear operator from L2 ..1; r/; m/ to itself, see [105, p. 168]. In what follows we will need the L2 -domain of the operator A that appears in [105, p. 151]. Definition 15.10. Let L2 WD L2 ..1; r/; m/. R (i) If l C m.l/ D 1 and Œ0;r / x 2 m.dx/ D 1, let ® ¯ D.A/ WD D0 .A/ \ u 2 L2 W Au 2 L2 ; u0 .0/ D 0 : (ii) If l C m.l/ D 1 and

R

Œ0;r / x

2 m.dx/

< 1, let

® ¯ D.A/ WD D0 .A/ \ u 2 L2 W Au 2 L2 ; u0 .0/ D u0C .l/ D 0 : (iii) If l C m.l/ < 1, let ® ¯ D.A/ WD D0 .A/ \ u 2 L2 W Au 2 L2 ; u0 .0/ D .r  l/u0C .l/ C u.l/ D 0 : R Note, that in case (ii) of Definition 15.10, the assumption Œ0;r / x 2 m.dx/ < 1 implies m.l/ < 1, hence r D l D 1. Therefore, u0C .l/ D 0 means that limx!1 u0C .x/ D 0. In case (iii) it is more convenient to write the assumption in terms of the function u on Œ0; l. Note that by (15.9), u0C .l/ D u0 .l/ C m¹lºAu.l/. Thus, if r < 1, the condition .r  l/u0C .l/ C u.l/ D 0 can be restated as m¹lºAu.l/ D u0 .l/ C

1 u.l/: r l

(15.29)

If r D 1, we may interpret the condition .r  l/u0C .l/ C u.l/ D 0 as u0C .l/ D 0, which is equivalent to m¹lºAu.l/ D u0 .l/: (15.30) Lemma 15.11. Let g 2 L2 ..1; r/; m/. Then u D R g is a solution to the equation u  Au D g. Moreover, R g 2 D.A/, and it is the unique solution of u  Au D g belonging to D.A/. A proof of this lemma can be found in [105, Theorem on p. 167] and [196]. The next step is the identification of the probabilistic concept of the -potential with the analytic concept of the -resolvent. Let Cb .Em / denote the space of bounded continuous functions on Em  Œ0; r/. For x 2 Em , and g 2 Cb .Em /, let Z 1 e t g.X t / dt: G g.x/ D Ex 0

277

Section 15.1 Inverse local time at zero

Since G 1  1=, G g is well defined for all g 2 Cb .Em /. For x 2 Em we define C TxB D inf¹t > 0 W B tC D xº and TxX D inf¹t > 0 W X t D xº. Then TxX D CT B C . x

Lemma 15.12. For all x 2 Em we have lim

Em 3y!x

TyX D TxX ;

Px  a.s.

(15.31)

Proof. Note that by the monotone convergence theorem, C tn ! C t as tn ! t . This proves that (15.31) is true as y ! x from the left. Suppose now that y ! x from the right. Then Z Z C C C L.TyB ; w/ m.dw/ D L.TyB ; w/ m.dw/ TyX D C.TyB / D Œ0;l

Œ0;y

C

Px -a.s. where the last equality uses L.TyB ; w/ D 0 Px -a.s. for all w > y. Since m is finite on compact intervals, the claim follows by the dominated convergence theorem C C and the fact that TyB ! TxB Px -a.s. as y ! x. Lemma 15.13. G W Cb .Em / ! Cb .Em / and G W Bb .Em / ! Cb .Em /. Proof. Let g 2 Cb .Em / and x 2 Em . Assume that x is not isolated from the right within Em . We will prove the right-continuity of G g at x. Pick y > x; by the strong Markov property we find for all x; y 2 Em that ! Z TyX X t G g.x/ D Ex e g.X t / dt C Ex .e Ty /G g.y/: (15.32) 0

Letting y ! x we can use Lemma 15.12 to see that the first term on the right-hand X side of (15.32) converges to zero, while Ex .e Ty / converges to 1; therefore, we have G g.y/ ! G g.x/. In the same way one checks the left-continuity at points in Em which are not isolated from the left. The above argument uses only the fact that g 2 Bb .Em /, i.e. the second assertion of the lemma follows at once. Because of the resolvent equation, the range D WD G .Cb .Em // of G does not depend on  > 0. Moreover,   Z 1 lim G g.x/ D Ex lim  e t g.X t / dt !1 !1 0   Z 1 D Ex lim e s g.Xs= / ds !1  Z 1 0 D Ex e s g.X0 / ds 0

D g.x/;

278

Chapter 15 Applications to generalized diffusions

implying that G is one-to-one, hence invertible. Define AQ WD I  G1 W D ! Cb .Em / and note that, again by the resolvent equation, AQ does not depend on  > 0. For 0 < y < l and x; w 2 Œ0; y/ let C

GyB .x; w/ D .y  x/ ^ .y  w/

(15.33)

be the Green function of B C killed upon hitting y. Lemma 15.14. Let g W Œ0; y ! R be a bounded Borel function. Then for every x 2 Œ0; y/ Z Z TyX C g.X t / dt D GyB .x; w/g.w/ m.dw/: Ex 0

Œ0;y

Proof. By a change of variables and Fubini’s theorem Z Ex

TyX 0

Z g.X t / dt D Ex Z D Ex Z D Ex Z D Z D Z D

C

C.TyB / 0

C g.B.t/ /dt

C

g.B tC / C.dt /

C

g.B tC /

Œ0;TyB 

Œ0;TyB 

Z

Œ0;y

Œ0;y

Ex

C

Œ0;TyB 

Z L.dt; w/ m.dw/ Œ0;r /

!

g.B tC / L.dt; w/

m.dw/

   C g.w/ Ex L.TyB ; w/ m.dw/ C

Œ0;y

g.w/ GyB .x; w/ m.dw/:

In the penultimate line we used that L.dt; w/ is supported on ¹B tC D wº, and in the last line we used the well-known formula relating the Green function and the local time for (reflected) Brownian motions. Let x 2 Em be positive and apply Dynkin’s formula for the stopping time TxX , see [178, pp. 98–99]. For u 2 D ! Z TxX    X Q E0 u X.Tx /  u.0/ D E0 Au.X t / dt : 0

279

Section 15.1 Inverse local time at zero

By Lemma 15.14 and the obvious equality X.TxX / D x we get Z C Q GxB .0; y/ Au.y/ m.dy/ u.x/ D u.0/ C Z D u.0/ C

Œ0;x

Œ0;x

Q .x  y/ Au.y/ m.dy/:

Q and u0 .0/ D 0. This formula clearly shows that u 2 D0 .A/, Au D Au Take g 2 Cb .Em / \ L2 ..1; r/; m/ and  > 0. Then G g 2 D  D0 .A/, and Q  g D G g  g; AG g D AG i.e. G g  AG g D g. Moreover, .G g/0 .0/ D 0. Proposition 15.15. For every g 2 Cb .Em / \ L2 ..1; r/; m/ we have G g D R g. Proof. Let g 2 Cb .Em / \ L2 ..1; r/; m/. In the paragraph preceding Lemma 15.14 it was shown that G g solves the equation u  Au D g. If G g 2 D.A/, the assertion follows by the uniqueness of the solution, cf. Lemma 15.11. Observe that .G g/0 .0/ D 0, G g 2 L2 ..1; r/; m/ and AG g D G gg 2 L2..1; r/; m/. that we are in case (i) of Definition 15.10, that is l C m.l/ D 1 and R Suppose 2 x m.dx/ D 1. Then there is nothing else to show, i.e. G g 2 D.A/. Œ0;r / Suppose now that we are in case (ii) of Definition 15.10, that is l C m.l/ D 1 R and Œ0;r / x 2 m.dx/ < 1. By the remark following Definition 15.10, it remains to show that limx!1 u0C .x/ D 0. Suppose that g  0 and g vanishes on Œy; 1/ for some y 2 Em . By the strong Markov property we see for all x 2 Em that ! Z X G g.x/ D Ex

Ty

e t g.X t / dt

0

C Ex .e Ty /G g.y/; X

(15.34)

and if x > y it follows that G g.x/ D Ex .e Ty /G g.y/: X

(15.35)

Thus G g./ is proportional to E:.e Ty / on Œy; 1/ \ Em . Let y; x; v 2 Em such that y < x < v. Again by the strong Markov property, X

   X X X Ev e Ty D Ev e Tx Ex e Ty ;

280

Chapter 15 Applications to generalized diffusions

showing that x 7! Ex .e Ty / is decreasing. Furthermore, let w 2 Em be such that y < w < x. Then X

Ex .e Ty / D Ex .e Ty 1¹TwX 0 W `.s/ > t º be the inverse local time of B C at zero. The inverse local time of X at zero, .`X /1 , has the following representation: Z   L `1 .t /; x m.dx/: (15.39) .`X /1 .t / D C`1 .t/ D Œ0;1/

C

Note that inf¹t > 0 W .`X /1 .t / D 1º D `X ./ D `.TrB / where  is the lifetime C C of X and TrB the lifetime of B C . Recall also the well-known fact that `.TrB / is

283

Section 15.1 Inverse local time at zero

an exponential random variable with parameter r 1 , cf. [304, Proposition VI 4.6]. Hence, the lifetime of .`X /1 is exponentially distributed with parameter r 1 , that is a D r 1 . Now we get from (15.39) and L.`1 .t /; 0/ D t that Z   X 1 .` / .t / D m¹0º t C L `1 .t /; x m.dx/: .0;1/

The second term is a pure jump subordinator, hence the drift of .`X /1 is m¹0º and b D m¹0º. Finally, we use the fact that .L.`1 .t /; x// t0 is for every x  0 a subordinator with the Lévy density x 2 e =x , see [178, Problem 4, p. 73]. This implies that E0 ŒL.`1 .1/; x/ D 1 for every x  0. Hence, by (15.39) Z   E0 .`X /1 .1/ D E0 L.`1 .1/; x/ m.dx/ D mŒ0; 1/: Œ0;1/

The connection between the Lévy triplet .a; b; / and the measure  is straightforward. Indeed,   Z  Z 1  .dt / ; .1  e t / .dt / 1 D f ./h./ D a C b C .0;1/ Œ0;1/  C t and letting  ! 0 proves that Z aD

Œ0;1/

 .dt / t

1

2 Œ0; 1/:

Similarly,    Z Z a 1  f ./ t h./ D Cb C  .dt / ; .1e / .dt / 1D    .0;1/ Œ0;1/  C t and letting  ! 1 gives b D  Œ0; 1/1 2 Œ0; 1/; letting  ! 0 we get Z  X 1 1 2 .0; 1: t .dt / D E0 .` / .1/ D b C  ¹0º .0;1/ This proves already our next result. Proposition 15.16. The following relations hold:  1 D E0 .`X /1 .1/ ;  ¹0º 1 D b; m¹0º D  Œ0; 1/  Z 1  .dt / D : rD t a Œ0;1/

mŒ0; 1/ D

284

Chapter 15 Applications to generalized diffusions

If the characteristic function h of the string is known, one can work out the Lévy triplet .a; b; / of f : let 1 h? ./ WD h./ be the Stieltjes function which is conjugate to h in the sense that f ? WD 1= h? is conjugate to f . Write Z c? 1 ? ? h ./ D Cd C  ? .dt /  .0;1/  C t so that Z

  ? .dt / f ./ D h ./ D c C d  C .0;1/  C t Z 1 .1  e t /n? .t / dt; D c? C d ? C ?

?

?

0

Z

where n? .t / D

se st  ? .ds/

.0;1/

is the completely monotone density of . It can be shown, see [224] or [105], that h? is the characteristic function of the so-called dual string m? which is, by definition, the right-continuous inverse of the string m: m? .x/ WD inf¹y > 0 W m.y/ > xº. Finding explicitly the string m 2 MC such that a given f 2 CBF is the Laplace exponent of the inverse local time at zero of a generalized diffusion X corresponding to m is known as the Kre˘ın representation problem. Equivalently, the problem can be stated as follows: given g 2 CM, the density of the Lévy measure  of f , d .d / D g./ d , find the operator A D ddm dx which generates the generalized diffusion X. Surprisingly, there are very few examples of (classes of) completely monotone functions for which this problem has been solved. In Table 15.1 we record those classes with the corresponding operator A. Table 15.1 was established by C. DonatiMartin and M. Yor in [97] and [98], except for the first row which is from [267]. Let Y D .Y t / t0 be a reflected diffusion on Œ0; 1/ with the speed measure m and the scale function s normalized such that s.0/ D 0. The infinitesimal generator of Y is d d . The process s.Y / D .s.Y t // t0 is also a reflected diffusion on Œ0; 1/ having d m ds the same local time at 0 as Y . Therefore, the Laplace exponents of the inverse local times coincide. Let m e D m ı s 1 . The generalized diffusion X corresponding to the string m e and the process s.Y / have the same infinitesimal generator A D d d . This de m dx shows that functions in the left column of Table 15.1 are densities of Lévy measures of the inverse local time at zero of generalized diffusions.

285

Section 15.2 First passage times Table 15.1. Explicit Kre˘ın representations.

AD

g./ c ˛C1

c ˛C1

;

e  ;



1 d2 ı1 d ; C 2 dx 2 2x dx ı D 2.1  ˛/ p  b 0˛ 2x d p K ; A˛ C 2 p  b ˛ 2x dx K A˛ WD

0 0 K c  e ; 

c

d d d m ds

 sinh./

1 d2 C 2 dx 2

>0

p ! b0 p K 2x 1 d C 2 0 p  b 0 2x 2x dx K

˛C1 e .1˛/ ;

A˛  x

d dx

0 < ˛ < 1;  > 0  c

 sinh./

˛C1 ;

b 0 .x 2 =2/ d K ˛=2 A˛ C x b K ˛=2 .x 2 =2/ dx

0 0. Let 0 < u < v < w. Since 0 .v; / D  C

Z .y; / m.dy/; Œ0;v

we conclude that for all  > 0 Z Z 0 C .v; /   .y; / m.dy/   .u;v

.u; / m.dy/ .u;v

D .u; /m.u; v: This gives

Z w 0 .w; / D .v; / C C .y; / dy v Z w 0  C .v; / dy v

(15.44)

0 .v; / D .w  v/C  .w  v/m.u; v .u; /:

We show now that .x;  / grows at least as fast as a polynomial of degree n. Indeed, choose 2n C 1 points 0 < x0 < x1 <    < x2n D x so that every interval .x2k2 ; x2k1 / contains a point of increase of m. Then m.x2k2 ; x2k1  > 0 for all k D 1; 2; : : : ; n. Using repeatedly the inequality (15.44) with x2k2 < x2k1 < x2k ,

288

Chapter 15 Applications to generalized diffusions

we find n

.x; /  

 n

n Y

.x2k  x2k1 /

n Y

kD1 n Y

kD1 n Y

kD1

kD1

.x2k  x2k1 /

m.x2k2 ; x2k1  .x0 ; / m.x2k2 ; x2k1 ;

where we used that .  ; /  1 for every  > 0. This concludes the proof. Remark 15.18. One can adapt the argument used in in Lemma 15.17 to show that  Y .x; z/ z D 1 ; (15.45) x wj;x j

where .wj;x /j 1 is the family of all zeroes of .x;  / arranged in decreasing order. 0 0 .x; z/.x; z/  .x; z/C .x; z/ D 1, it follows The zeroes are also simple. Since C that .x;  / and .x;  / have no common zeroes. PAs a consequence of the convergence of the infinite product (15.43), the series j 1=jzj;x j < 1 converges. Moreover, if 0 < y < x, then .x;  / may have more zeroes than .y;  /. If there are infinitely many points of increase of m in .0; x/, then .x;  / will have an infinite, but countable, number of zeroes. Let us now discuss the relative position of the zeroes of .x;  / and .y;  / for 0 < y < x. For simplicity assume that both .zj;x /j 2N and .zj;y /j 2N are infinite sequences. The argument when these sequences are of different length is similar, the difference being that some of the zeroes do not exist. Lemma 15.19. Assume that 0 < y < x. Then zj;y < zj;x < 0 for all j  1. Proof. We begin with a few remarks that will be useful in the proof. First, since the zeroes of .x;  / are simple, the function  7! .x; / changes sign every time it passes through a zero. Secondly, the function .x; / 7! .x; / is jointly continuous. Thirdly, .0;  /  1, and also .  ; 0/  1. Now fix x > 0. We claim that .y; / > 0 for all .y; / 2 A1;x WD Œ0; x Œz1;x ; 0 such that .y; / ¤ .x; z1;x /. This will prove that z1;y < z1;x for all 0 < y < x. Indeed, as z1;x is the largest zero of .x;  /, we have .x; / > 0 for all  2 .z1;x ; 0. Suppose that there is some .v; / 2 A1;x , .v; / ¤ .x; z1;x /, such that .v; / D 0. Because of the continuity of  the minimum ¯ ® v1 D min v 2 Œ0; x W 9 2 Œz1;x ; 0 such that .v; / D 0 and .v; / 2 A1;x is attained. Since .0;  /  1 and .x; / > 0 for  2 .z1;x ; 0, we have 0 < v1 < x. Choose 1 2 Œz1;x ; 0/ such that .v1 ; 1 / D 0. Assume first that 1 > z1;x . Using

Section 15.2 First passage times

289

the fact that the zeroes of .v1 ;  / are discrete and that .v1 ;  / changes sign every time it passes through a zero, we have, for sufficiently small  > 0, either .v1 ; / < 0 for  2 .1  ; 1 /, or .v1 ; / < 0 for  2 .1 ; 1 C /. Assume that .v1 ; / < 0,  2 .1  ; 1 /. Since ¹ < 0º is open in A1;x , there would exist a point .v; / with v < v1 such that .v; / < 0. This, however, contradicts the minimality of v1 which entails that .v1 ; / > 0 for  2 .1  ; 1 /. The same argument yields .v1 ; / > 0 for  2 .1 ; 1 C / and sufficiently small  > 0. We also get 1  z1;x . Hence, .v; / ¤ 0 for all .v; / 2 Œ0; x .z1;x ; 0. @ We now rule out the possibility that 1 D z1;x . Indeed, since @ .x; z1;x / > 0 and @ 0 0 .v ; z .v1 ; z1;x / > 0, we get from (15.42) that C .x; z1;x / < 0 and C 1 1;x / < 0. @ 0 This implies that there is a v 2 .v1 ; x/ with .v; z1;x / D 0 and C .v; z1;x / > 0. On the other hand, we know already that z1;x is the largest zero of .v;  /, i.e. @ 0 .v; z 0 z1;v D z1;x . By (15.42), C 1;x / D C .v; z1;v / D  @z .v; z1;v / < 0, which is impossible. This concludes the proof that .v; / ¤ 0 for every .v; / 2 A1;x , .v; / ¤ .x; z1;x /. Thus, we have z1;y < z1;x for 0 < y < x. We continue the proof by showing that z2;y < z2;x . Define ¯ ® A2;x WD .v; / W 0  v  x; z2;x    z1;v and note that  < 0 immediately below the curve ¹.v; z1;v / W 0  v  xº. As above, by continuity there exists ¯ ® v2 D min v W 9 2 Œ0; x/ such that .v; / D 0 and .v; / 2 A2;x : Choose 2 such that .v2 ; 2 / D 0 and assume that 2 > z2;x . We argue in the same way as in the first part of the proof. Since the zeroes of .v2 ;  / are discrete and since .v2 ;  / changes sign every time it passes through a zero, we have, for sufficiently small  > 0, either .v2 ; / > 0 for  2 .2  ; 2 /, or .v2 ; / > 0 for  2 .2 ; 2 C /. Suppose that .v2 ; / > 0,  2 .2  ; 2 /. Since ¹ > 0º is open in A2;x , there exists a point .v; / with v < v2 such that .v; / > 0. But this contradicts the minimality of v2 . Therefore, .v2 ; / < 0 for  2 .2  ; 2 /. The same argument yields that .v2 ; / < 0 for  2 .2 ; 2 C / and small enough  > 0. This proves that it cannot happen that 2 > z2;x ; consequently, .v; / ¤ 0 for all .v; / 2 A2;x , with  < z2;x . To rule out the possibility that 2 D z2;x we argue as in the first part of the proof. Thus, .v; / ¤ 0 and, in fact, .v; / < 0 for all .v; / 2 A2;x , .v; / ¤ .x; z2;x /. This proves that we have z2;y < z2;x for 0 < y < x. The inequalities zj;y < zj;x , j  3, can be seen in the same way. Recall that xy is the distribution of the first passage time TyX at level y of the process X started at x. Theorem 15.20. If 0  x < y < r, then xy 2 BO and 0y 2 CE  GGC.

290

Chapter 15 Applications to generalized diffusions

Proof. We begin with the case x D 0, that is 0y . By (15.41) and Lemma 15.17, we have 1 Y 1  L .0y I / D D : 1C .y; / jzj;y j j

Since the factors on the right-hand side are Laplace transforms of exponential distributions, Definition 9.15 shows that 0y 2 CE. Now suppose that x ¤ 0. Since x < Py, Lemma 15.19 shows that jzj;x j > jzj;y j. A consequence of Lemma 15.17 is that j 1=jzj;y j < 1. Hence, L .xy I / D

 1 Y .x; /   D 1C 1C .y; / jzj;x j jzj;y j j

is well defined and strictly positive. Corollary 9.17 tells us that xy 2 BO. In the remaining part of this chapter we will give a more precise description of the first passage distributions of generalized diffusions. All generalized diffusions which we have studied so far live on Em  Œ0; r/ and are, in fact, generalized reflected diffusions. We will now consider generalized diffusions on open intervals of the type .r1 ; r2 /, 0 < r1 ; r2  1. The definition of these processes follows closely the one of generalized reflected diffusions. Let m W R ! Œ1; 1 be a non-decreasing, right-continuous function with m.0/ D 0. The family of all such functions will be denoted by M. We use the same letter m to denote the measure on R corresponding to m 2 M. We associate with m 2 M two functions m1 ; m2 W Œ0; 1/ ! Œ0; 1 defined by ´ ´ m..x//; x  0; m.x/; x  0; and m2 .x/ D m1 .x/ D 0; x < 0; 0; x < 0: Both m1 and m2 are non-decreasing and right-continuous, m1 .0/ D m.0/ D 0 and m2 .0/  0. For k D 1; 2, rk D sup¹x W mk .x/ < 1º  1, let Emk be the support of mk restricted to Œ0; rk /, and note that 0 2 Emk need not hold. Set lk WD sup Emk  1. In view of Remark 15.8, we may regard mk as a string of length rk . Let B D .B t ; Px / t0;r1 0 and y 2 .r1 ; r2 / let

h2 .y; / D

.y; / .y; /

Then E0 Œe Ty  D X

and

h.y; / WD

h.y; / .y; /

and P0 .TyX < 1/ D

1 1 h1 ./

C

1 h2 .y;/

for every  > 0

´ 1; r1 r1 Cy ;

X

where

.r2 ; / WD

h./ .r2 ; /

.r2 ; / < 1.

(15.49)

r1 D 1; r1 < 1:

(ii) Suppose that r2 2 EQ m . Then P0 .TrX2 < 1/ > 0 if, and only if, finite. In this case E0 Œe Tr2  D

:

for every  > 0

R r2 0

m.x/ dx is

293

Section 15.2 First passage times

Proof. (i) Recall that .0; / D 1 and E0 Œe Ty  D X

.0; / D 0. It follows from (15.48) that

1 G .0; y/ D G .y; y/ .y; / C h./1 .y; / 1   D .y; / h2 .y; /1 C h1 ./1 h.y; / : D .y; /

(15.50)

  X For the second statement note that P0 TyX < 1 D lim!0 E0 Œe Ty . It follows from the last formula of Proposition 15.16 that lim!0 h1 ./ D h1 .0/ D r1 , while lim!0 h2 .y; / D .y; 0/=.y; 0/ D y. Hence ´ 1; r1 D 1; h.y; / D lim r1 !0 .y; / r1 Cy ; r1 < 1: R r2 (ii) Under the assumption r2 2 EQ m Lemma 15.21 R r2 proves that 0 m.x/ dx < 1 if, and only if, .r2 ; / < 1. Suppose that 0 m.x/ dx D 1 and let y ! r2

in (15.50). We get E0 Œe Tr2  D 0, hence P0 .TrX2 < 1/ D 0. Assume now that R r2 0 m.x/ dx < 1. Since h./ D limx!r2 .x; /= .x; / and .r2 ; / < 1 by Lemma 15.21 (i), we have that .r2 ; / < 1. Let y ! r2 in (15.49). It follows that X h./ E0 Œe Tr2  D ; .r2 ; / X

and since the right-hand side is strictly positive, P.TrX2 < 1/ > 0. Remark 15.23. Since both h1 and h2 .y;  / are in S, it follows that h.y;  / 2 S as well. For x 2 Em and y 2 EQ m such that x < y let xy be the Px -distribution of TyX . Define ¯ ® H WD xy W x 2 Em ; y 2 EQ m ; x < y; m 2 M to be the family of all such first passage distributions. Translating the function m, if necessary, it is enough to consider x D 0. Theorem 15.24. A sub-probability measure  belongs to H if, and only if, there exist 1 2 CE and 2 2 ME with 2 ¹0º D 0 such that  D 1 ? 2 , the representing sequence .bj /j 1 of 1 is either empty or strictly increasing, and the representing measure of  7! . L .2 I //1 has a point mass at every bj .

294

Chapter 15 Applications to generalized diffusions

Proof. Suppose that  D 0y 2 H. By (15.17) we have that    Y z 1 Y z 1 y 1 1C D ; D .y; z/ wj;y bj j

(15.51)

j

where bj WD jwj;y j and .wj;y /j 1 are the zeroes of .y;  /. Since the zeroes .wj;y /j 1 are simple and decreasing, the sequence .bj /j 1 is increasing. Note that the right-hand side of (15.51) is the Laplace transform of a probability measure from CE, cf. Corollary 9.17, so we may define 1 2 CE by L .1 I / WD y= .y; /,   0. Moreover,  7! h.y; /=y 2 S and h.y; 0/=y  1 by Proposition 15.22. It follows by Definition 9.4 that there exists 2 2 ME such that L .2 I / D h.y; /=y. From Proposition 15.22 we have that L .I / D

h.y; / D .y; /

y h.y; / D L .1 I /L .2 I /; .y; / y

that is,  D 1 ? 2 . Since h2 .y;  / 2 S, the function  7! .h2 .y; //1 D .y; /=. .y; // also belongs to S. By Remark 15.18, .y;  / and .y;  / have no common zeroes, i.e. the set of poles of  7! .y; /=. .y; // is precisely ¹0º [ ¹wj;y W j  1º. Hence, the set ¹0º [ ¹bj W j  1º is the support of the representing measure of  7! .h2 .y; //1 . Since 1 1 1 D C ; h.y; / h1 ./ h2 .y; / the representing measure of  7! .h.y; //1 has a point mass at each point in the set ¹0º [ ¹bj W j  1º. But y 1 D ; L .2 I / h.y; / showing that the representing measure of  7! .L .2 I //1 has a point mass at every bn . It remains to check that 2 ¹0º D 0. Note that 2 ¹0º D lim!1 L .2 I / D lim!1 h.y; /=y. By the definition of h.y; / it suffices to show that at least one of the limits lim!1 h1 ./ and lim!1 h2 .y; / is equal to zero. But this follows from (15.23) and the fact that 0 is a point of increase for at least one of the measures m1 and m2 , cf. Remark 15.8. Sufficiency: By assumption we have that ´

1; L .1 I / D Q  j

1C

  1 ; bj

if ¹bj W j  1º D ;; otherwise:

295

Section 15.2 First passage times

Since 2 2 ME it follows that L .2 I  / 2 S as well asR  ! .L .2 ; //1 2 S. Therefore, there exist c  0 and a measure Q satisfying Œ0;1/ .1 C t /1 Q .dt / < 1 such that Z 1 1 DcC Q .dt /  L .2 I / Œ0;1/  C t and that Q has a point mass at each point in ¹0º [ ¹bj W j  1º. Since  Z L .2 I / D c C

Œ0;1/

1  Q .dt / ; Ct

we conclude that Q ¹0º D L .2 I 0/1  1. As 2 ¹0º D 0 we can let  ! 1 to find c > 0 or Q Œ0; 1/ D 1. Write Q as Q 1 C Q 2 where supp.Q 2 / D ¹0º [ ¹bj W j  1º and Q 2 ¹0º D Q ¹0º  1. Thus Q 1 ¹0º D 0. By (7.3) we see that there exist measures 1 and 2 on Œ0; 1/, and real numbers c1  0, c2  0, such that Z 1 Z 1  1 .dt / D Q 1 .dt / ; (15.52) c1 C Œ0;1/  C t .0;1/  C t  1 Z Z 1  2 .dt / D c C Q 2 .dt / : (15.53) c2 C Œ0;1/  C t Œ0;1/  C t Denote the functions defined in (15.52) and (15.53) by g1 and g2 respectively. Then 1  L .2 I / D g1 ./1 C g2 ./1 : (15.54) We record the following facts that follow from (15.52) and (15.53) by letting  ! 1: (a) c1 D 1=Q 1 .0; 1/; (b) if c D 0, then c2 D 1=Q 2 Œ0; 1/, while if c > 0, then c2 D 0; (c) c1 c2 D 0; indeed, if c D 0, then Q Œ0; 1/ D 1, so that either Q 1 Œ0; 1/ D 1 or Q 2 Œ0; 1/ D 1. Further, g1 .0/ D Q 1 ¹0º D 1 and g2 .0/ D Q 2 ¹0º1 2 .0; 1. Note that we can write X 1 

j ; D c C 0 C g2 ./  C bj j 1

where j is the mass of Q 2 at bj , j  0. This shows that 1=g2 has real poles at bj , j  1, hence also real zeroes aj , j  1, which intertwine with the poles: 0 < a1 < b1 < a2 < b2 <    . Therefore,  Q   j 1 C bj (15.55) g2 ./ D Q  :  j 1 C aj

296

Chapter 15 Applications to generalized diffusions

PThis shows that the representing measure of g2 is supported in .aj /j 1 and j 1 1=aj < 1. Let m1 and m2 be strings having g1 and g2 as their characteristic functions. Since g2 .0/ 2 .0; 1, it follows that the length r2 of the string m2 satisfies r2 D g2 .0/ 2 .0; 1. Moreover, R r by Lemma 15.21 (ii) we conclude that either m.r2 / < 1, or m.r2 / D 1 and 0 2 m.x/ dx < 1. In the second case it holds that r2 2 EQ m . In the first case one can extend m2 without changing its characteristic function g2 such that the length of the extended m2 is strictly larger than r2 – e.g. extend m2 to be the constant m.r2 / on an interval to the right of r2 . We use the same letter for the extended string. Next 1 Œ0; 1/ D lim g1 ./ D !1

lim!1

R

1 1 .0;1/ Ct

Q 1 .dt /

D 1;

hence by the second line in Proposition 15.16 we get that m1 ¹0º D 0. Define m by ´ m1 ..x//; x < 0; m.x/ D x  0: m2 .x/; Then m 2 M. Let .  ; / and .  ; / be the solutions of equations (15.46) and (15.47) with this m. Then clearly g1 and g2 can be expressed in terms of  and as .x; / ; x!r1 .x; / .r2 ; / .x; / g2 ./ D lim D : x!r2 .x; / .r2 ; /

g1 ./ D  lim

Comparing the last formula with (15.55) we see that the family of all the zeroes of .r2 ;  / is equal to ¹bj W j  1º. Hence,  Y  1C ; .r2 ; / D bj j

implying that L .1 I / D

1 : .r2 ; /

Together with (15.54) this gives that L .I / D L .1 I / L .2 I / D

g./ .r2 ; /

where g./ WD .g1 ./1 C g2 ./1 /1 . Since c1 D 0 or c2 D 0, we see that 0 is in the support of m. Hence, it follows from Proposition 15.22 that  is the hitting time distribution of r2 starting at 0 for the generalized diffusion corresponding to m.

Section 15.2 First passage times

297

Comments 15.25. Kre˘ın’s theory of strings was developed by M. G. Kre˘ın in the period from 1951 to 1954 in a series of papers [220, 221, 222, 223, 224]. An excellent exposition of these results is given in [196] which also contains historical and bibliographical remarks on the subject. An equally good source for the theory of strings are Chapters 5 and 6 of [105]. Our presentation is a combination of those two references. A complete proof of Theorem 15.7 can be found in either of these references. Definition 15.10 is from [105] as well as Lemma 15.11 which says that R W L2 ..1; r/; m/ ! D.A/ is one-to-one and onto. The image of L2 ..1; r/; m/ under R is dense in L2 ..1; r/; m/ and both R and A are selfadjoint. Applications of Kre˘ın’s string theory to Markov processes are studied in the 1970’s. It seems that Langer, Partzsch and Schütze are the first to use this connection in [238] where it is shown that the operator A (acting on an appropriate domain) is the infinitesimal generator of a C0 -semigroup. Path properties of the corresponding Feller process are studied by Groh in [137]. S. Watanabe, [365] gives the first construction of the process as a time-change of (reflected) Brownian motion and introduces the name quasi-diffusion. This name is adopted by Küchler in [228, 229] and [230] who looks at the properties of transition densities of the process. The name generalized diffusion is probably introduced by N. Ikeda, S. Kotani and S. Watanabe in [172], and is used again by Tomisaki, [349]. Related works are [198, 216] and [199]. The term gap diffusion is used by Knight in [209]. The question of characterizing Lévy measures of inverse local times of diffusions was posed in [178, p. 217]. In its original form, which requires that m has full support, the problem is very difficult and still unsolved. In the context of generalized diffusions it was independently solved by Knight [209] and Kotani and Watanabe [217]. Bertoin [46] revisited the problem and gave another proof. The proof we give follows essentially the approach from [209] and [217]. One of the key steps in the proof is the identification of the probabilistically defined Green function G .x; y/ with the analytically defined R .x; y/ D .x; / .y; /. This seems to be part of the folklore; as we were unable to give a precise reference, we decided to include a proof of Proposition 15.15. This proof follows [178, pp. 126–7]. The relations between m and  , given in Proposition 15.16, appear in [224] and can be derived using the spectral theory of strings, see [105, p. 192]. Note that a consequence of the last formula is that r D lim!0 h./ D h.0C/. The problem of the characterization of the hitting time distributions in the case of nonsingular diffusion on an interval .r1 ; r2 / was studied by Kent in [204], see also [203] for complete proofs, and [206]. He showed that xy 2 BO if the boundary r1 is not natural. The case presented in Theorem 15.20 covers an instantaneously reflecting, non-killing left boundary. The complete characterization of hitting time distributions for generalized diffusions, Theorem 15.24, was obtained by Yamazato in [377]. A slight drawback of this result is that the condition on 2 is

298

Chapter 15 Applications to generalized diffusions

given in terms of the representing measure of . L .2 I //1 . In [378] Yamazato expressed the condition on 2 in terms of the function 2 appearing in the representation (9.4).

Chapter 16

Examples of complete Bernstein functions

Below we provide a list of examples of complete Bernstein functions. We restrict ourselves to complete Bernstein functions of the form Z Z  t  .dt /: .1  e / m.t / dt D f ./ D .0;1/ .0;1/  C t Whenever possible, we provide expressions for the representing density m.t / and measure  .dt /, respectively; for simplicity, we do not distinguish between a density and the measure given in terms of the density. We also state whether f is a Thorin– Bernstein function, i.e. whether f is of the form   Z  f ./ D  .dt /: log 1 C t .0;1/ By ‘unknown if in TBF’ we indicate that we were not able to determine whether f 2 TBF or not. The third column of the tables contains references and remarks how to derive the corresponding assertions. Some more lengthy comments can be found in Sections 16.12.1–16.12.3 below. We record here again the relations between the Lévy density m and the Stieltjes measure  and the Thorin measure  , cf. Remark 6.10 (iii) and Remark 8.3 (ii): Z m.t / D e ts s  .ds/I .0;1/ Z e ts  .0; s/ dsI m.t / D .0;1/ Z e ts  .ds/I t  m.t / D .0;1/

t   .dt / D  .0; t / dt: Only a few publications contain extensive lists of (complete) Bernstein functions. We used the papers by Jacob and Schilling [185] (29 entries) and Berg [35] (11 entries) as a basis for the tables in this chapter.

300

Chapter 16 Examples of complete Bernstein functions

16.1 Special functions used in the tables Our standard references for this section are Abramowitz and Stegun [1], Erdélyi et al. [106, 107] and Gradshteyn and Ryzhik [134]. Trigonometric functions sec.x/ D

1 ; cos.x/

csc.x/ D

1 sin.x/

Hyperbolic functions sinh.x/ D

e x  e x ; 2

cosh.x/ D

e x C e x ; 2

tanh.x/ D

Gamma and related functions Gamma function Z .x/ D

1

e t t x1 dt

0

Digamma function ‰0 .x/ D ‰.x/ D

 0 .x/ d log .x/ D dx .x/

Trigamma function ‰1 .x/ D

1 X 1 d2 log .x/ D 2 dx .x C m/2 mD0

Tetragamma function ‰2 .x/ D

1 X 1 d3 log .x/ D  dx 3 .x C m/3 mD0

Incomplete Gamma functions Z 1 .˛; x/ D t ˛1 e t dt D x .˛1/=2 e x=2 W.˛1/=2;˛=2 .x/ x Z x t ˛1 e t dt D ˛ 1 x ˛ 1 F1 .˛; ˛ C 1; x/

.˛; x/ D 0

Beta and related functions Beta function Z

1

t x1 .1  t /y1 dt D

B.x; y/ D 0

.x/ .y/ .x C y/

sinh.x/ cosh.x/

Section 16.1 Special functions used in the tables Incomplete Beta functions Z x t 1 .1  t /1 dt Bx .; / D 0

Bx .; / Ix .; / D B1 .; / Exponential integrals Z 1 t e Ei.x/ D  dt (Cauchy principal value integral if x > 0) t x Z 1 t e dt D Ei.x/ D .0; x/; x > 0 E1 .x/ D t x Z 1 xt e dt; m  2; x > 0 Em .x/ D tm 1 Sine and cosine integrals Z 1 sin t si.x/ D  dt; t Zx 1 cos t ci.x/ D  dt; t x

x>0 x>0

Error functions Z x 2 2 e t dt Erf.x/ D p  0 Erfc.x/ D 1  Erf.x/ Bessel and modified Bessel functions Bessel functions J .x/ D

1 X

.1/n

nD0

Y .x/ D

.x=2/C2n nŠ . C n C 1/

J cos./  J .x/ sin./

Modified Bessel functions I .x/ D K .x/ D

1 X

.x=2/C2n nŠ . C n C 1/ nD0  I .x/  I .x/ 2 sin./

Bessel function zeroes j;n ;

n D 1; 2; : : :

zeroes of J

301

302

Chapter 16 Examples of complete Bernstein functions

Mittag-Leffler function E˛;ˇ .x/ D

1 X

xn .n˛ C ˇ/ nD0

Hypergeometric functions Confluent hypergeometric functions 1 F1 .a; b; x/ D

1 X .a/k x k .b/k kŠ

kD0

Hypergeometric functions 2 F1 .a;

b; c; x/ D

1 X .a/k .b/k x k .c/k kŠ

kD0

Whittaker functions Whittaker functions of the first kind   1 1 M; .x/ D x C 2 e x=2 1 F1    C ; 2 C 1; x 2 Whittaker functions of the second kind W; .x/ D

.2/ .2/ M; .x/ C M; .x/ 1 1 . 2    / . 2   C /

modified Whittaker functions ˆ˛;ˇ .x/ D x 1 e x=2 W; .x/; where ˛ D   ; ˇ D   ;  D

.˛ C ˇ/ ˛ˇ ; D 2 2

Parabolic cylinder functions D .x/ D 2=2C1=2 x 1=2 W=2C1=4; 1=4 .x 2 =2/

Tricomi functions .1  b/ 1 F1 .a; b; x/ .a  b C 1/ .b  1/ 1b x C 1 F1 .a  b C 1; 2  b; x/ .a/    1 F1 .a; b; x/ 1 F1 .a  b C 1; 2  b; x/ D  x 1b sin.b/ .a  b C 1/.b/ .a/.2  b/

‰.a; b; x/ D

Section 16.1 Special functions used in the tables Lommel’s functions of two variables U .y; x/ D

1 X nD0

.1/n 

V .y; x/ D cos

 y C2n x

y x2  C C 2 2y 2

JC2n .x/  C U2 .y; x/

Partial sums of the exponential function en .x/ D

n X xk ; kŠ

n D 0; 1; 2; : : :

kD0

Hermite polynomials Hn .x/ D .1/n e x

2

d n x 2 e ; dx n

n D 0; 1; 2; : : :

Generalized Laguerre polynomials / L. n .x/

! n j X j nC x D .1/ ; n  j jŠ

n D 0; 1; 2; : : :

j D0

Lambert function The Lambert W function is defined as the unique solution of the equation W .x/ e W .x/ D x such that W W Œe 1 ; 1/ ! Œ1; 1/, W .0/ D 0 and, W j.0;1/ > 0.

303

304

Chapter 16 Examples of complete Bernstein functions

16.2 Algebraic functions No Function f ./ 1

˛ ;

2

. C 1/˛  1;

3

1  .1 C /˛1 ;

4

 ; Ca

5

p

Comment

0 0; b  1

p 1 2 .a C a a2 C 2/ 2 ! r a2 a 1C Cp  a2 ; C  log 2 2

14.2(27) in [107]

Theorem 2.2 of [210]

a>0

39

  log 2.1  a/ p    log 1 C   .1 C /2  4a.1  a/ ;

Example 3.2.3 of [67]

1 0

51

    1 1  log log 1 C ; log log 1 C a Ca a>0

Theorem 3.6 of [187], Theorem 1.2 of [50]. See §16.12.1

325

Section 16.4 Logarithmic functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L, S:

closed expression unknown — T:

unknown if in TBF

L:

closed expression unknown

S:

1 1 1.a;1Ca/ .t / 2 .t  a/.1 C a  t /  C .log 1Cat /2 t a

T:

f … TBF

L:

closed expression unknown

S:

1 1 1.1=.1Ca/; 1=a/ .t / .1  at /..1 C a/t  1/  2 C .log .1Ca/t 1 /2 1at

T:

f … TBF

L:

closed expression unknown

S:

8 ˆ ˆ0; ˆ  ! ˆ C1 ˆ < 1 1 arctan 1 log at t a  ; t 2  ˆ ˆ ˆ ˆ 1 ˆ : ; t

T:

00

53



54

  en .log /

  1   en log 1   ; lognC1 1 ;

n2N

 lognC1  ;   en .log /

n2N

log 55

56

57

n2N

nC1



   lognC1 1  ; 1  en log 1

n2N

p   log 1 C 2 C 2 .1 C /

[124, 125, 126]

[124, 125, 126]

[124, 125, 126]

[124, 125, 126]

Theorem 3.1 of [187]. See §16.12.1

327

Section 16.4 Logarithmic functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

closed expression unknown

S:

8 ˆ 0; ˆ  1 ˆ 0 ˆ 1 ˆ ˆ arctan 1 log .1Ca/t 1=a

1

1  2 C log .1Ca/t 1at

2 1.1=.1Ca/;1=a/ .t /

L, S, T:

closed expression unknown

L, S:

closed expression unknown — T:

unknown if in TBF

L, S:

closed expression unknown — T:

unknown if in TBF

L, S:

closed expression unknown — T:

unknown if in TBF

1 t =2

e

  t I0 2

L:

t

S:

8 p 2 ˆ < arcsin. t /; 0 < t < 1 t ˆ :1; t 1 t

T:

1 1 1.0;1/ .t / p  t .1  t /

328

Chapter 16 Examples of complete Bernstein functions

No Function f ./ p 58 1C 1C 2 log 2

59

log.1 C ˛ /;

60

log

1 ; .1 C /˛  ˛

61

log

˛ ; .1 C /˛  1

62

Comment Theorem 3.1* of [187]. See §16.12.1

0 0

Comment 1.431(2) of [134], 5.5(1) of [107], Corollary 6.3

1.431(4) of [134], 5.5(1) of [107], Corollary 6.3

Entry 70, Theorem 8.2 (v)

14.2(71) of [107], Theorem 8.2 (v)

Section 16.6 Hyperbolic functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

1 X 1  2 n2 t =2 e t nD1

S:

1 X 1 .t / 1 2 2 t . n =2; 1/ nD1

T:

1 X

ı 2 n2 =2 .dt /

nD1

L:

1 X 1  2 .n 1 /2 t =2 2 e t nD1

S:

1 X 1 .t / 1 2 2 t . .n1=2/ =2; 1/ nD1

T:

1 X

ı 2 .n1=2/2 =2 .dt /

nD1

L:

1 1 X 1  2 .n 1 /2 t =2 X 1  2 n2 t =2 2  e e t t nD1 nD1

S:

1 X 1 .t / dt 1 2 2 2 2 t . .n1=2/ =2;  n =2/ nD1

T:

f … TBF

L:

closed expression unknown

S:

p   1 p log 1 C b 2 tan2 .a t / 2 t

T:

f … TBF

335

336

Chapter 16 Examples of complete Bernstein functions

No Function f ./ p  p  74  log 1 C b coth.a / ;

Comment 14.2(72b) of [107], Theorem 8.2 (v)

a; b > 0

75

p p   log b sinh.a / p  C c cosh.a /  a;

14.2(68) in [107], Theorem 8.2 (v)

a > 0; b; c  1

16.7 Inverse hyperbolic functions No Function f ./ 76

sinh1 ./ p p D log. 1 C  C /

Comment

Section 16.7 Inverse hyperbolic functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

closed expression unknown

S:

p   1 p log 1 C b 2 cot2 .a t / 2 t

T:

f … TBF

L:

closed expression unknown

S:

p p   1 p log b 2 sin2 .a t / C c 2 cos2 .a t / 2 t

T:

f … TBF;

b¤c

Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

closed expression unknown

S:

8 p ˆ arcsin. t / ˆ < ; t ˆ 1 ˆ : ; 2t

T:

00

16.9 Bessel functions No Function f ./ 110

Comment

 .  1/ log 2  log ./ p p C  C log K . /;  > 1=2

p ! 111 p K1 . / p  1 ; K . /

Formula below (2.3) of [175]

Entry 110 and Remark 8.3  > 1=2

351

Section 16.9 Bessel functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L, S:

closed expression unknown — T:

L:

1 2.t C a/3=2

S:

e at p t

T:

f … TBF

L:



S: T:

unknown if in TBF

  5 .t C a/5=2 2

p

t e at

f … TBF

Lévy (L), Stieltjes (S) and Thorin (T) representation measures L, S:

closed expression unknown

T:

1 p 2 t

L:

S:

T:

2 1  p  p p   t J2 . t / C Y2 . t /

closed expression unknown 1 p

 t

2 1   p  p p   t J2 . t / C Y2 . t /

f … TBF

!

!

352

Chapter 16 Examples of complete Bernstein functions

No Function f ./

Comment

112 .  1/ log 2 C log ./ p p    log K . /;

Proof of Theorem 1 in [175]

0 <  < 1=2 ! p 113 p K1 . /  p 1 ; K . /

Entry 112 and Remark 8.3 (iii)

0 <  < 1=2

114 .  /.log

p

  log 2/ p p C log K . /  log K . /;

(4.2) of [176], Theorem 8.2 (ii)

0<0 IC1 . / C I . /

Theorem 9 of [266], Theorem 8.2 (v)

120 p 

p IC1 . / p p ; >0 IC1 . / C I . /

Theorem 9 of [266], Theorem 8.2 (v)

121 

1=2

p K . / p ; KC . /

(2.1) of [174], (67) on p. 97 of [106]

0 <  < 1;   

p p 122 p a1 I . / C a2 I1 . /  p p ; a3 I . / C a4 I1 . /  > 0; a1 ; a2 ; a3 ; a4 > 0

Theorem 3.1 of [177]

Section 16.9 Bessel functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

2

1 X

C3 j;n

J .j;n / j;n 2 t e J0 .j;n /

C1 j;n

J .j;n / ı 2 .dt / J0 .j;n / j;n

nD1

S:

2

1 X nD1

T:

f … TBF

L:

closed expression unknown

S:

p J2 . t / p p 2  t JC1 . t / C J2 . t /

T:

f … TBF

L:

closed expression unknown

S:

p 2 JC1 . t/ p p 2  t JC1 . t / C J2 . t /

T:

f … TBF

L:

closed expression unknown

S:

p p p p 1 JC . t /Y . t /  J . t /YC . t / p p 2 2 t =2 JC . t / C YC . t/

T:

unknown if in TBF

L:

closed expression unknown

S:

p p 2 . t/ 1 a1 a3 J2 . t / C a2 a4 J1 p p p 2  t a32 J2 . t / C a42 J1 . t/

T:

unknown if in TBF

1 p

1 p

355

356

Chapter 16 Examples of complete Bernstein functions

No Function f ./ p p 123 p a1 K . / C a2 K1 . / p p ;  a3 K . / C a4 K1 . /

Comment Theorem 3.3 of [177]

  0; a1 ; a2 ; a3 ; a4 > 0

    124 2 I ap K ap ;  

14.3(22) in [107], Theorem 8.2 (v)

a > 0;  2 R

125 1C e a K .a/;

14.3(36) in [107]

a > 0; 1=2 <  < 1=2

126

p  e a K .a/; a > 0; 1=2 <  < 1=2

14.3(39) in [107]

357

Section 16.9 Bessel functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

S:

closed expression unknown  2 p  p p p 2.a2 a4 C a1 a3 / 2 . t / C Y1 . t / C a2 a4 J2 . t / C Y2 . t / C p a1 a3 J1  t   p p p p 2 2 2 2  a4 J1 . t /  a3 Y2 . t / C a3 J2 . t / C a4 Y1 . t/

T:

unknown if in TBF

L:

2a2 .3=2 C / 2.1C/ t p 2 F1  .1 C /

S:

J2 .at /

T:

f … TBF

L:

21C a .3=2 C / 3=2 .t C a/.t C 2a/3=2 t p  sec./

S:

1 t  e at I .at / sec./

T:

unknown if in TBF

L:



1 3 4a2 C ; C ; 1 C 2;  2 2 2 t



cos./

21C a .t C a/3=2C ´     3 3  2 5  2 a2

4 .t C a/2  C  ./ 2 F1 ; ; 1  ; 2 4 4 .t C a/2    μ 3 3 C 2 5 C 2 a2 2 Ca  C  ./ 2 F1 ; ; 1 C ; 2 4 4 .t C a/2

S:

cos./ at p e K .at /  t

T:

f … TBF

358

Chapter 16 Examples of complete Bernstein functions

16.10 Miscellaneous functions No Function f ./ 127

Comment

1 I  .; /; .a  / 1 a

14.2(10) in [107], Lemma 2.1 in [79]

a > 0;  <  < 1

  128 F 1; 1  ; 2  ;  a ; 2 1  a > 0; 0 <  < 1

129

    2 F1 1; 1 C ; 2 C ;  ; a

a > 0; 0 <  < 1

130

    2 F1   1; ; ; 1  ; a

a > 0; 0 <  < 

131

 .2a/ .1  /

. C .1 C

a > 0; 0 <  < 1

 / 2a 2 F1  / 2a

    ; 1C ;1 ; 1;  C 2a 2a

14.2(9) in [107], Lemma 2.1 in [79]

[98]

Section 16.10 Miscellaneous functions

Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

3 at sin./ 1 a .2  / .at / 2 e 2 W 1 ; 2 .at / 2 2 

S:

sin./  t .a C t / 

T:

unknown if in TBF

L:

closed expression unknown

S:

1    t 1.0;a/ .t / a1

T:

f … TBF

L:

closed expression unknown

S:

a t  1.a;1/ .t /

T:

a1 ıa .dt / C .1  / t  1.a;1/ .t / dt

L:

 ./ C1  C1 at 2 a 2 t e 2 W 1 ; 2C .at / 2 2 .  /

S:

./ a1 t 1 .a C t /1 ./ .  /

T:

unknown if in TBF 

L:

a sinh.at /

1C e .1/ at

S:

closed expression unknown

T:

unknown if in TBF

359

360

Chapter 16 Examples of complete Bernstein functions

No Function f ./ 132

Comment   [98]  ./  2 .2a/ .1  / ; ;1 2 F1 1;  .2  / a a    !    C 2a   ; 2C ;1    2 F1 1;  C  2a 2a  2   C 2a a > 0; 0 <  < 1;  D

133

 ˆ˛;ˇ ./; or:

134

1C 2

1 1 ˛> ;   ;  0;  > 0

135

C1=2 e a=2 W; .a/;

14.3(50) of [107], (9.221) of [134]

a > 0; 1=2 <  <  C 1=2

136

 e a=2 W; .a/; a > 0; jj  1=2 <  < jj C 1=2

14.3(53) of [107], (9.222) of [134]

Section 16.10 Miscellaneous functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures  1C a L: sinh.at / S:

closed expression unknown

T:

unknown if in TBF

L:

closed expression unknown

S:

t ˛Cˇ e t ˆ˛;ˇ .t /      ˛ C 12  ˇ C 12

T:

unknown if in TBF

L:

closed expression unknown

S:

a.C1/=2  at t e  ./

T:

f … TBF

L:

closed expression unknown

S:

t 1=2 e at =2 M; .at /   .2 C 1/     C 12

T:

unknown if in TBF

L:

closed expression unknown

S:

t 1 e at =2 W; .at /      C  C 12     C 12

T:

unknown if in TBF



361

362

Chapter 16 Examples of complete Bernstein functions

No Function f ./ 137

138



Comment

‰.a C 1; b C 1; / ; ‰.a; b; /

1  1=2

(1.4) of [176]

a > 0; b < 1

D1 .1=2 / ; D .1=2 /

(1.3) of [176]

>0

p 2   139 r    1 D=1=2  a 2 p ;  C 2  2 2 

p. 254 of [245]

 > 0; a 2 R

140 

   . 2 / 1 F1 . 2 ; ˇ; a/ ‰. 2 ; ˇ; a/

.ˇ/ a1ˇ

p. 257 of [245] ;

a D 2= 2 ; ˇ D  C 1  > 0;   0;  > 0 141 e a

ı 2 .  C 1/M 1   .a/ W 1   .a/

D

2



2



2 . C 1/ 1  ; D 2jˇj; a D 2 2jˇj ı jˇj

ˇ < 0; ı > 0;   0

pp. 260–261 of [245] ;

Section 16.10 Miscellaneous functions Lévy (L), Stieltjes (S) and Thorin (T) representation measures L:

closed expression unknown

S:

t b e t j‰.a; b; t e i  /j2 .a C 1/ .a  b C 1/

T:

unknown if in TBF

L:

closed expression unknown

S:

p

T:

unknown if in TBF

L:

closed expression unknown

S:

jD .it 1=2 /j2 t 3=2 2 . C 1/

1 X nD0

1

1   1=2 a2   p 2 Hn a  e ı.nC1=2/ .dt / nŠ 

2n

T:

unknown if in TBF

L:

closed expression unknown 1 X

S:

nŠ  1/ .a/ ı n .dt / aˇ 1 L.ˇ n .ˇ C n/ nD0

T:

unknown if in TBF

L:

closed expression unknown

S:

1¹ 0: (6.21)  1 C t2  C t 0 Conversely, any function of the form (6.21) is in CBF. Thus there is a one-to-one correspondence between CBF and the set ± ° measurable (6.22)  D . ; / W 2 R and  W .0; 1/ ! Œ0; 1 : In order to find new complete Bernstein functions it is enough to identify ‘simple’ functions  W .0; 1/ ! Œ0; 1 such that the integral  Z 1 1 t .t / dt  1 C t2  C t 0 can be explicitly calculated. Then we can combine the Tables 16.2–16.10 of complete Bernstein functions with Theorems 6.2 and 6.17. Recall that every complete Bernstein function f admits a Stieltjes representation Z   .dt /;  > 0; (6.7) f ./ D a C b C .0;1/  C t R (a; b  0 and  satisfies .0;1/ .1 C t /1  .dt / < 1) as well as a Nevanlinna–Pick representation   Z 1 t f ./ D ˛ C ˇ C .1 C t 2 / .dt /  (6.17) 2 1 C t  C t .0;1/ R1 ( isRa finite measure on .0; 1/ with 0 t 1 .dt / < 1 and ˛; ˇ  0 such that ˛  .0;1/ t 1 .dt /). The measure  in (6.7) is called the Stieltjes measure of f and the measure  in (6.17) is called the Nevanlinna–Pick measure of f . The quantities appearing in the formulae (6.7) and (6.17) are related as follows: Z 1 1 C t2 1 .dt /; b D ˇ; and  .dt / D .dt /: aD˛ t t 0 Now we can inspect the Tables 16.2–16.10 of complete Bernstein functions to see which entry f has a representation measure .1 C t 2 / .dt / D t  .dt / which is of the form .t / dt with some  W .0; 1/ ! Œ0; 1. If so, g./ WD exp.f .// is another complete Bernstein function.

0

1 e

2 e

1 2e

a>

a>

a>0

a

a

1 ; e b>0

a

1 ; e b>0

Parameters

p t

.t =/1=2 e at

p apt te

t e a

1  e at

1 e t t n .n  1/Š

t e at 1.b;1/ .t /

t e at 1.0;b/ .t /

Density .t/

Entry 108

Entry 106

Entry 105

Entry 104

Entry 103

Entry 102

Entry 101

Comments

Section 16.11 CBFs given by exponential representations

369





1 1C2

30

29



.1 C /1C1=

12a

1C

1C 2

 1C

a



a.1  a/.  1/2 .a  1/.1a  1/



28

 2C22

e 1=2 e



e =4  1

1=

27

26

2

4e 1= .1 C /1=

25

a aC

e arctan.1=a/



Function g./

r  p     3=2 e a Erfc a exp a

24

23

No 3 2e

a 2 Œ0; 1

a 2 .0; 1/

a>1

a>

Parameters

1t

.t

t

/ cos a

 .t a Ct 1a / sin a arctan 1.0;1/ .t / 1t.t a t 1a / cos a

 .t a Ct 1Ca / sin a 1 arctan 1.1;1/ .t / C 1 a 1Ca

a1Œ0;1 .t / C .1  a/1.1;1/ .t /

.1  t /1Œ0;1 .t / C .1  t 1 /1.1;1/ .t /

1



1 1 C t2

t 1Ct

1 1Œ1;1/ .t / t

1 1Œa;1/ .t / t

t 3=2 e at

Density .t/

[142]

[141]

[15]

Entry 109

Comments

370 Chapter 16 Examples of complete Bernstein functions

371

Section 16.12 Additional comments

16.12 Additional comments 16.12.1 Entries 7–10, 12, 29–31, 40, 49–52, 57, 58, 60, 61 The entries 7–10, 12, 29–31, 40, 49–52, 57, 58, 60 and 61 are taken from [50] and [187]. Since these papers are written from a probabilistic point of view, we give a brief sketch how the representation measures can be derived from the information given there. Let G be a strictly positive random variable with distribution function F and assume that Z 1 dt < 1: .1 ^ t / EŒe tG  t 0 Define ² Z 1 ³ dt g./ WD exp  .1  e t / EŒe tG  : t 0 An application of the Frullani integral identity, cf. [50, (1.64), p. 323] yields ´ Z μ ´   μ 

   log 1 C D exp  F .dt / : g./ D exp  E log 1 C G t .0;1/ Thus,

Z

f ./ WD  log g./ D

1 0

.1  e

t

/ EŒe

tG

dt D  t

Z

  F .dt / log 1 C t .0;1/ 

is contained in TBF with the Thorin measure  .dt / D F .dt /. By Remark 8.3 (iii), f 0 ./ is in CBF with Stieltjes measure equal to the Thorin measure of f , namely F .dt /. By [50, Thm. 1.7, Point 3] we have   1 0 ; f ./ D E CG hence,



 f ./ D E CG 0



Z D

.0;1/

 F .dt /: Ct

We illustrate the method for the family of random variables G˛ , 0  ˛  1. Their densities are given in formulae (3.3), (3.4), (3.5) and (3.6) of [187]: ˛ sin. ˛/ t ˛1 .1  t /˛1 1Œ0;1 .t /; .1  ˛/ .1  t /2˛  2.1  t /˛ t ˛ cos. ˛/ C t 2˛ 0 < ˛ < 1; G1 is uniformly distributed on Œ0; 1 ; 1 ; C is a standard Cauchy random variable. G0 D 1 C exp. C /

fG˛ .t / D

372

Chapter 16 Examples of complete Bernstein functions

The basic formula is [187, (3.2), p. 41] saying that for 0  ˛  1 ³ ² Z 1 1 .G˛ / t tG˛ dt : E.e / D exp  .1  e / EŒe  t 0 In [187] explicit formulae for the Laplace exponent of 1 .G˛ / are provided, i.e. explicit formulae for the function g defined above with G standing for any member of the family G˛ . The distribution F .dt / can now be computed from the density fG˛ .t /.

16.12.2 Entries 20, 65 and 66 The entries 20, 65 and 66 are from [68]. We give an explanation for 65, which comes from [68, Appendix 1.5, pp. 121–122]. Take a D 0, b D 2, x D y D 1, replace ˛ in [68] by . Then the formula for the -Green function reads p Z 1 cosh2 . 2/ p D G .1; 1/ D p e t p.t; 1; 1/ dt; 2 sinh. 2/ 0 where

! Z 1  n  1 X n2  2 t=8 C D e cos2 e ts Q .ds/; 2 2 Œ0;1/

1 p.t; 1; 1/ D 2

nD1

and

 n  X1 1 cos2 ı 2 2 : Q D ı0 C 4 2 2 n  =8 1

nD1

Thus, Z G .1; 1/ D Hence,

1

e

t

Z dt

e Œ0;1/

0

ts

Z Q .ds/ D

Œ0;1/

1 Q .ds/: Cs

p Z 1 cosh2 . 2/ 1  .ds/ p  D p 2 sinh. 2/ 4 Œ0;1/  C s

with D

1  n  X 1 cos2 ı 2 2 : 2 2 n  =8

nD1

16.12.3 Entries 80 and 81 Entry 80 (and, similarly, 81) can be obtained in the following way. Entry 70 shows that p  p  f1 ./ D log sinh. 2/  log. 2/

373

Section 16.12 Additional comments

is in CBF. From entry 78 we know that f2 ./ D

2 1 cosh1 .1 C / 2

is also in CBF. Therefore, f1 ı f2 2 CBF. A straightforward computation gives that f1 ı f2 D f . Proposition 6.2 of [262] gives the Lévy density m.t /.

Appendix

In this appendix we collect a few concepts and results which are not used in a uniform way or which are particular to one branch of mathematics.

A.1 Vague and weak convergence of measures Throughout this section E will be a locally compact separable metric space equipped with its Borel  -algebra B.E/. We write M C .E/ for the set of all locally finite, inner regular Borel measures on E, i.e. the Radon measures on E. Most of the time we will use Greek letters ; ; : : : to denote elements in M C .E/; MbC .E/ are all finite measures, i.e. all  2 M C .E/ such that .E/ < 1. Full proofs for the topics in this section can be found in Bauer [26, Sections 30, 31] or Schwartz [329, vol. 3, Chapter V.13]. Definition A.1. A sequence .n /n2N in M C .E/ converges vaguely to  2 M C .E/, if Z Z lim u.x/ n .dx/ D u.x/ .dx/ for all u 2 Cc .E/: (A.1) n!1 E

E

A sequence of finite measures .n /n2N in MbC .E/ converges weakly to  2 MbC .E/, if Z Z lim u.x/ n .dx/ D u.x/ .dx/ for all u 2 Cb .E/: (A.2) n!1 E

E

We denote the closure of Cc .E/ with respect to the uniform norm kk1 by C1 .E/; these are the continuous functions vanishing at infinity. Note that .C1 .E/; k  k1 / is a Banach space. It is not hard to see that we can replace (A.1) by Z Z lim u.x/ n .dx/ D u.x/ .dx/ for all u 2 C1 .E/: (A.3) n!1 E

E

Remark A.2. If E is a compact topological space, M C .E/ D MbC .E/, and vague convergence and weak convergence coincide. Weak convergence is, in general, not the weak convergence used in functional analysis. Topologically, the space of signed finite Radon measures Mb .E/ WD ¹   W ;  2 MbC .E/º can be identified with the topological dual of C1 .E/. Thus, vague convergence is actually the topological weak* convergence  .Mb .E/; C1 .E//.

Section A.1 Vague and weak convergence of measures

375

C Remark R A.3. Let . t / t0 be a family of measures in M .E/ such that the function t 7! E u.x/  t .dx/ is measurable for every u 2 Cc .E/, and let  be a measure on Œ0; 1/ such that Z Z ƒ.u/ WD u.x/  t .dx/ .dt / < 1 (A.4) Œ0;1/

E

for all u 2 Cc .E/. Then ƒ defines a positive linear functional on Cc .E/ which we may identify with a measure  2 M C .E/. Formally, Z D  t .dt / Œ0;1/

and we call this a vague integral, provided that (A.4) holds for all u 2 Cc .E/. Using standard approximation techniques it is easy to show that Z .B/ D  t .B/ .dt /; B 2 B.E/: Œ0;1/

We have the following relation between vague convergence and weak convergence of measures. Theorem A.4. A sequence of measures .n /n2N , n 2 MbC .E/, converges weakly to  if, and only if, it converges vaguely to  and if the total masses converge to the total mass of the limit lim n .E/ D .E/: n!1

A vaguely convergent sequence of measures .n /n2N , n 2 M C .E/, is necessarily vaguely bounded, i.e. for each u 2 Cc .E/ or for each compact set K  E we have Z sup u dn < 1 or sup n .K/ < 1; n2N

n2N

but some kind of converse is also true. Theorem A.5. A subset F  M C .E/ is vaguely relatively compact if, and only if, F is vaguely bounded. This means, in particular, that every vaguely bounded sequence of Radon measures has at least one vaguely convergent subsequence. The Banach–Alaoglu theorem for the dual pair .C1 .E/; M C .E// can thus be written as Corollary A.6 (Banach–Alaoglu). Every ball ¹ 2 MbC .E/ W .E/  rº of radius r > 0 is vaguely, or weak*, compact. Weak convergence can also be characterized via the portmanteau theorem.

376

Appendix

Theorem A.7. A sequence of measures .n /n2N , n 2 MbC .E/, converges weakly to a measure  2 MbC .E/ if one, hence all, of the following equivalent conditions are satisfied: (i) lim supn!1 n .F /  .F / for all closed sets F  E; (ii) lim inf n!1 n .G/  .G/ for all open sets G  E; (iii) limn!1 n .B/ D .B/ for all Borel sets B  E with .@B/ D 0. If n ;  are finite measures on the real line R, we can add a further equivalence in terms of their distribution functions: (iv) limn!1 n .1; x D .1; x for all x 2 R such that ¹xº D 0. Recall that F .x/ WD .1; x is a distribution function and that we have a one-toone relation between finite measures on the real line and all bounded, non-decreasing and right-continuous functions F W R ! R. In the context of vague convergence and weak convergence the classical Helly’s selection theorem becomes: Corollary A.8 (Helly). If .Fn /n2N is a sequence of distribution functions which is uniformly bounded, i.e. supn2N;x2R jFn .x/j < 1, then there exists a distribution function F .x/ and a subsequence .Fnk /k2N such that lim Fnk .x/ D F .x/

k!1

at all continuity points x of F .

The corresponding sequence of finite measures, .nk /k2N converges vaguely to the finite measure  induced by F . It converges weakly if, and only if, limR!1 supk ŒFnk .R/  Fnk .R/ D 0, or equivalently, if limk!1 nk .R/ D .R/. Lemma A.9. Let .n /n2N be a sequence of finite measures on Œ0; 1/. Then the vague limit  D limn!1 n exists if, and only if, limn!1 L .n I x/ D g.x/ exists for all x > 0. If this is the case, g.x/ D L .I x/. The measures converge weakly if, and only if, limn!1 L .n I x/ D g.x/ for all x  0. If this is the case, g.x/ D L .I x/. Proof. The assertion on vague convergence is a consequence of the closedness of the set CM, see the proof of Theorem 1.6. If, in addition, n!1

L .n ; 0/ D n Œ0; 1/ ! Œ0; 1/ D L .I 0/ holds, we can use Theorem A.4 to get weak convergence.

Section A.2 Hunt processes and Dirichlet forms

377

A.2 Hunt processes and Dirichlet forms In this section we collect some basic definitions and results pertinent to Hunt processes and the theory of Dirichlet forms. Let E be a locally compact separable metric space and let E@ denote its one-point compactification. If E is already compact, then @ is added as an isolated point. The Borel  -algebras on E and E@ will be denoted by B D B.E/ and B@ WD B.E@ /, respectively. Let .; F / be a measurable space. A stochastic process with time-parameter set Œ0; 1/ and state space E is a family X D .X t / t0 of measurable maps X t W  ! E. A filtration on .; F / is a non-decreasing family .G t / t0 of sub- -algebras of F . T The filtration .G t / t0 is said to be right-continuous if G t D G tC WD s>t Gs for every t  0. The stochastic process X D .X t / t0 is adapted to the filtration .G t / t0 if for every t  0, X t is G t measurable. A stopping time with respect to the filtration .G t / t0 is a mapping T W  ! Œ0; 1 such that ¹T  t º 2 G t for each t  0. The  -algebra GT is defined as GT D ¹F 2 F W F \ ¹T  t º 2 G t for all t  0º. Let 0 WD  .Xs W s  0/ and for t  0, F t0 WD  .Xs W 0  s  t /. F1 Definition A.10. Let X D .X t / t0 be a stochastic process on .; F / with state space .E@ ; B@ /, and  let .Px /x2E@ be a family of probability measures on .; F /. The family X D .X t / t0 ; .Px /x2E@ is called a Markov process on .E; B/ if the following conditions hold true (M1) x 7! Px .X t 2 B/ is B.E/ measurable for every B 2 B.E/. (M2) There exists a filtration .G t / t0 on .; F / such that X is adapted to .G t / t0 and Px .X tCs 2 B j G t / D PX t .Xs 2 B/; Px -a.s. (A.5) for every x 2 E, t; s  0 and B 2 B.E/. (M3) P@ .X t D @/ D 1 for all t  0. The Markov process X is called normal if, in addition, (M4) Px .X0 D x/ D 1 for all x 2 E. The condition (M2) is called the Markov property of X with respect to the filtration .G t / t0 . Let M1 .E@ / denote the family of Rall probability measures on E@ . For  2 M1 .E@ / and ƒ 2 F we define P .ƒ/ WD E@ Px .ƒ/ .dx/. For  2 M1 .E@ / we denote   0 by F1 the completion of F1 with respect to P and F t is the P -completion T   of F t0 within F1 ; finally, set F t WD 2M1 .E@ / F t where t 2 Œ0; 1. Then .F t / t0 is called the minimal completed admissible filtration. If X has the Markov 0 / property with respect to .F tC t0 , then the filtration .F t / t0 is right-continuous, see e.g. [123, Lemma A.2.2].

378

Appendix

Let X be a Markov process with respect to a filtration .G t / t0 . Then X is called a strong Markov process with respect to .G t / t0 if the filtration .G t / t0 is rightcontinuous and P .XT Cs 2 B j GT / D PXT .Xs 2 B/;

P -a.s.;

(A.6)

for every .G t / t0 -stopping time T such that P .T < 1/ D 1, and for all probability measures  2 M1 .E@ /, B 2 B@ , s  0. A Markov process X is said to be quasi left-continuous if for any sequence .Tn /n2N of .G t / t0 -stopping times increasing to a stopping time T it holds that   P lim XTn D XT ; T < 1 D P .T < 1/ for all  2 M1 .E@ /: (A.7) n!1

Definition A.11. Let X be a normal strong Markov process on .E; B.E// with respect to the filtration .G t / t0 satisfying the following condition (M5)

(i) X t .!/ D @ for every t  .!/, where .!/ D inf¹t  0 W X t D @º is the lifetime of X; (ii) For each t  0, there exists a map  t W  !  such that X t ıs D X tCs , s  0; (iii) For each ! 2 , the sample path t 7! X t .!/ is right-continuous on Œ0; 1/ and has left limits on .0; 1/.

If X is also quasi left-continuous, then X is called a Hunt process. The process X is a Hunt process if, and only if, X is a strong Markov process and quasi left-continuous with respect to the minimal completed admissible filtration .F t / t0 , see e.g. [123, Theorem A.2.1]. Let X be a Markov process with respect to the filtration .G t / t0 . For x 2 E, t  0 and B 2 B.E/ define the transition function of the Markov process X by P t .x; B/ WD Px .X t 2 B/:

(A.8)

Then P t is a transition kernel on .E; B.E//, i.e. x 7! P t .x; B/ is B.E/ measurable for all B 2 B.E/, and B 7! P t .x; B/ is a sub-probability measure R on B.E/ for all x 2 E. For a measurable function  W E ! R we set P t .x/ WD E .y/P t .x; dy/ whenever the integral makes sense. The Markov property of X implies that .P t / t0 is a Markov transition function, i.e. for every B.E/ measurable non-negative function  W E ! E, (A.9) P t Ps  D P tCs ; t; s  0: Note that P t .x/ D Ex ..X t /1¹t< º /, where Ex denotes the expectation with respect to the probability measure Px . If we agree that every function  on E is extended to E@ by letting .@/ D 0, then we can write P t .x/ D Ex ..X t //. Note

379

Section A.2 Hunt processes and Dirichlet forms

that the right-continuity of t 7! X t .!/ for all ! implies that .t; !/ 7! X t .!/ is also measurable, hence .t; !/ 7! 1B .X t .!// is measurable for every B 2 B. By Fubini’s theorem it follows that t 7! P t .x; B/ D Ex 1B .X t / is also measurable. The Laplace transform of the transition function Z 1 e t P t .x; B/ dt; x 2 E; B 2 B.E/;  > 0; (A.10) G .x; B/ D 0

is called the resolvent kernel of the Markov process X. The transition function P t .x; B/ is said to be a Feller transition function if every P t maps C1 .E/ into C1 .E/ and if lim t!0 P t .x/ D .x/ for all x 2 E and for all  2 C1 .E/. If .P t / t0 is a Feller transition function on E, then there exists a Hunt process on E having .P t / t0 as its transition function, compare [123, Theorem A.2.2.] or [60, Theorem I.9.4.]). From now on we assume that m is a positive Radon measure on .E; B.E// with full support, i.e. supp.m/ D E. The transition function .P t / t0 is called m-symmetric if for all non-negative measurable functions  and and for all t  0, Z Z .x/P t .x/ m.dx/ D .x/P t .x/ m.dx/: (A.11) E

E

A Hunt process X with an m-symmetric transition function P t is called an m-symmetric Hunt process. For m-symmetric transition functions we see, using Cauchy’s inequality, Z Z  2 P t .x/ m.dx/  .x/2 m.dx/; for all  2 Bb .E/ \ L2 .E; m/: E

E

This implies that P t can for all t  0 be uniquely extended to a contraction operator on L2 .E; m/ which we denote by T t . Thus, T t W L2 .E; m/ ! L2 .E; m/. It is easy to see that .T t / t0 is a semigroup on L2 .E; m/, i.e. T tCs D T t Ts for all t; s  0. In the same way we can extend the resolvent kernels Gˇ , ˇ > 0, to bounded operators on L2 .E; m/ which we denote by Rˇ . The operators .Rˇ /ˇ >0 satisfy the resolvent equation R˛   Rˇ  D .ˇ  ˛/R˛ Rˇ ; ˛; ˇ > 0;

 2 L2 .E; m/:

(A.12)

Remark A.12. The families .T t / t0 and .Rˇ /ˇ >0 are a priori defined on the underlying Banach space, while .P t / t0 and .Gˇ /ˇ >0 are given by the process and have a pointwise meaning. It is possible to show, cf. [123, Theorem 4.2.3], that we have T t u D P t u and Rˇ u D Gˇ u m-almost everywhere on E. Assume that X is an m-symmetric Hunt process with transition function .P t / t0 . Then it holds that lim P t .x/ D lim Ex .X t / D .x/

t!0

t!0

for all x 2 E and all  2 Cc .E/:

380

Appendix

This implies, see e.g. [123, Lemma 1.4.3], that the semigroup .T t / t0 is strongly continuous: L2 - lim t!0 T t  D  for every  2 L2 .E; m/. Similarly, the resolvent .R />0 is strongly continuous in the sense that L2 - lim!1 R  D  for every  2 L2 .E; m/. The (infinitesimal) generator A of a strongly continuous semigroup .T t / t0 on L2 .E; m/ is defined by ´ A WD lim t!0 T t  t ® ¯ (A.13) D.A/ WD  2 L2 .E; m/ W A exists as a strong limit : Let .G />0 be a strongly continuous resolvent on L2 .E; m/. It follows from the resolvent equation (A.12) and the strong continuity that G is invertible. The generator A of .G />0 on L2 .E; m/ is defined by ´ A WD   G1  (A.14) D.A/ WD G .L2 .E; m//: Note that the resolvent equation shows that A and D.A/ are independent of  > 0. Lemma A.13. The generator of a strongly continuous resolvent is a negative semidefinite self-adjoint operator. Moreover, the generator of a strongly continuous semigroup coincides with the generator of its resolvent. Since the operator A is positive semi-definite and self-adjoint we can use the spectral calculus to define its non-negative square root .A/1=2 with the domain D..A/1=2 /. Definition A.14. Let X be an m-symmetric Hunt process on E, .T t / t0 the corresponding strongly continuous semigroup on L2 .E; m/, and A the generator of the semigroup. The Dirichlet form of X is the symmetric bilinear form E on L2 .E; m/ defined by ´ E.; / D h.A/1=2 ; .A/1=2 iL2 (A.15) D.E/ D D..A/1=2 /: Here h; iL2 denotes the inner product in L2 .E; m/. The form E is a closed symmetric form on L2 .E; m/. For ˇ > 0 let Eˇ .; / WD E.; / C ˇh; iL2 ;

;

2 D.E/:

Then .D.E/; Eˇ / is a Hilbert space. Since the semigroup .T t / t0 is Markovian, the form E is also Markovian, cf. [123, Theorem 1.4.1], which means that all normal contractions operate on E, see [123, p. 5]. The Dirichlet form E of X is called regular if there exists a subset C of D.E/ \ Cc .E/ such that C is dense in D.E/ with respect to E1 and dense in Cc .E/ with respect to the uniform norm.

381

Section A.2 Hunt processes and Dirichlet forms

Clearly, R .L2 .E; m// D G .L2 .E; m// D D.A/  D.E/. Therefore the connection between the resolvent .R />0 and the Dirichlet form E can be expressed by E .R ; / D h; iL2 ;  2 L2 .E; m/; 2 D.E/: (A.16) For an open set B  E let ¯ ® Cap.B/ WD inf E1 .; / W   1 m-a.e. on B ;

(A.17)

and for an arbitrary A  E, ® ¯ Cap.A/ WD inf Cap.B/ W B open, B  A :

(A.18)

Then Cap.A/ is called the 1-capacity or simply the capacity of the set A. Let A  E. A statement depending on points in A is said to hold quasi everywhere (q.e. for short) if there exists a set N  A of zero capacity such that the statement holds for all points in A n N . For any open subset D of E, let D D inf¹t > 0 W X t … Dº be the first exit time of X from D. The process X D obtained by killing the process X upon exiting from D is defined by ´ X t ; t < D ; X tD WD @; t  D : Then X D is again an m-symmetric Hunt process on D. The Dirichlet form associated with X D is .ED ; D.ED // where ED D E and D.ED / D ¹u 2 F W u D 0 E-q.e. on E n Dº;

(A.19)

cf. [123, pp. 153–154]. Let .P tD / t0 and .GD />0 denote the transition semigroup and resolvent of X D . For  > 0 let   HD .x/ WD Ex e D .XD / : Then for every  2 L2 .E; m/ such that  D 0 q.e. on E n D, it holds that G .x/ D GD .x/ C HD G .x/

for x 2 D;

(A.20)

and E .GD ; HD G / D 0. This is the E -orthogonal decomposition of G  into GD  2 D.ED / and its complement, cf. [123, Sections 4.3 and 4.4]. Definition A.15. A measure  on .E; B.E// is called smooth if it charges no set of zero capacity and if there exists an increasing sequence .Fn /n2N of closed sets such that .Fn / < 1 for all n 2 N and lim Cap.K n Fn / D 0

n!1

for every compact set K.

382

Appendix

Every Radon measure on .E; B.E// that charges no set of zero capacity is smooth, see [123, p. 81]. For a Borel set B  E, let TB D inf¹t > 0 W X t 2 Bº be the hitting time of X to B. A set N  E is called exceptional for X if there exists a Borel set NQ  N such that Pm .TNQ < 1/ D 0. Definition A.16. Let C D .C t / t0 be a stochastic process with values in Œ0; 1. Then C is called a positive continuous additive functional (PCAF) of X if (a) C is adapted to the minimal completed admissible filtration .F t / t0 and if there exists a set ƒ 2 F1 and an exceptional set N  E such that (b) Px .ƒ/ D 1 for all x 2 E n N , (c)  t ƒ  ƒ for all t > 0, (d) t 7! C t .!/ is continuous on Œ0; .!// for all ! 2 ƒ, (e) for all ! 2 ƒ we have

8 ˆ 0 and for quasi every x 2 E we have Px .C t D C t / D 1. Let C D .C t / t0 be a positive continuous additive functional of X. For a Borel function  W E ! Œ0; 1/ and  > 0 define Z   t UC .x/ WD Ex e .X t / dC t : Œ0;1/

The PCAF C and the smooth measure  are said to be in Revuz correspondence if for every  > 0 and all non-negative Borel functions  and it holds that Z (A.21) h; UC iL2 D .G / d: In this case, the measure  is called the Revuz measure of the PCAF C . The condition (A.21) is one of several equivalent conditions describing the Revuz correspondence; for the others we refer to [123, Theorem 5.1.3]. Theorem A.17. The family of all equivalence classes of positive continuous additive functionals of X and the family of all smooth measures are in one-to-one correspondence under the Revuz correspondence.

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Index

additive functional 382 analytic function of bounded type 85 canonical factorization 85 angular limit 76, 90 Balakrishnan’s formula 217 Bernstein function 21, 202 characterization 27, 212 complete see complete Bernstein function cone structure 28 convergence of 29 definition 21 extended 52, 87, 127 extremal representation 33 !  holomorphic on H 25 is negative definite 38, 41 Lévy measure 22 Lévy triplet 22 Lévy–Khintchine representation 22, 51, 58, 65, 69, 184, 205, 269 preserves sectors 25 relation with CM 27, 49, 53 relative compactness 7 special see special Bernstein function stability 28 subclass PBF 61, 139 tail behaviour 24 transformation see transformation of BF Bernstein’s theorem 3, 21, 40, 70 Bochner’s theorem 46 Bochner–Schoenberg theorem 212 Bondesson class 117 and first passage time 289 as vague closure 120 definition 117 stability 117

BO  ID 117 BO 6 SD 128 capacity 243, 381 Choquet representation for BF 33 for CBF 95 for CM 8 for S 18 for TBF 114 for positive definite fns 40–41 complete Bernstein function 69 CBF  S 80 and inverse local time 270 characterization 69–70, 92, 93, 96, 99, 181 composition with H 176 cone structure 94 conjugate pair 92 convergence of 95 definition 69 exponential of 87 exponential representation 86, 99, 100, 119, 124, 366 extended see extended CBF extremal representation 95 is Nevanlinna–Pick function 70 is operator monotone 190 is Pick function 70 kernel representation 108 Lévy vs. Stieltjes measure 80, 246 log-convexity of CBF 99, 100, 102 logarithm of 87 multiplicative representation 87 Nevanlinna representation 78, 102 potential density 162 relation with S 93, 96, 99 stability 93, 94, 98, 118 Stieltjes measure 75

407

Index Stieltjes representation 75, 92, 93, 95, 101, 117, 181, 185 symmetric 103, 105 CBF  SBF 159 completely monotone function 2 compl. self-decomposable 60 cone structure 5 convergence of 6 definition 2 extremal representation 8 infinitely divisible 51 is positive definite 38, 41 iterated differences 40 logarithmically see log-completely monotone on R 13 p-times self-decomposable 59 representation 3 self-decomposable 55 stability 5, 28, 29, 58, 376 stable 55 contractivity hypercontractivity 241 supercontractivity 241 ultracontractivity 233, 241 convolution of exp. distributions 125 and first passage time 289, 293 CE  GGC 125 convolution semigroup 48, 202, 205 characterization 49 definition 48 potential measure of 63 subordinate 65 Dirichlet form 243, 380 dissipative operator 180, 201 spectrum 180 distribution S-self-decomposable 123 infinitely divisible 51, 54 Laplace exponent of 128 overview 127 relations among 128 self-decomposable see self-decomposable distribution stable 55

excessive function 261 extended CBF 80 integral representation 80 extended Bernstein function see Bernstein function Feller transition function 379 first passage time and BO 289 and CE 289, 293 and GGC 289 and ME 293 generalized diffusion 285 fractional power 129 Balakrishnan’s formula 217 in BF vii, 21, 33 in CBF viii, 95 in CM vii, 56 in S viii, 19 in TBF viii of operators 216 functional calculus 222 functional inequality Nash-type 233, 235, 236 Sobolev-type 233 super-Poincaré 239 weak Poincaré 240 generalized diffusion 269 first passage time 285 inverse local time 270 local time 269 generalized Gamma convolution 121 and first passage time 289 as vague closure 123 definition 121 stability 122 GGC  BO 122 GGC  SD 122 GGC 6 ME 129 GGC 6 SD1 129 harmonic function 262 Heinz–Kato inequality 191 Hellinger–Stone formula 182 Helly’s selection theorem 376

408 Hirsch function 172 characterization 173 composition with CBF 176 cone structure 173 definition 172 relation with P 175 relation with SBF 175 relation with S 175 stability 173 Hunt process 242, 269, 378 inconsistency unavoidable 1–411 infinitely divisible see distribution, random variable inverse local time and CBF 270 generalized diffusion 270 killed Brownian motion 260 killed process 243 subordinate 246 Kre˘ın correspondence 284 examples 284–285 Kre˘ın representation problem 284 Kre˘ın–Milman representation see Choquet representation Laplace exponent 49 of distribution 127 relations among 128 Laplace transform 1 uniqueness 2 Lévy process 187, 212, 252 Lévy triplet 22 Lévy–Khintchine formula 46 local time generalized diffusion 269 log-CM 127 log-S 127 log-completely monotone function characterization 53 definition 53 relation with P 64 log-convex function 164 log-convex sequence 164, 173

Index Markov process 377 definition 377 generator 380 normal 377 quasi left-continuous 378 resolvent kernel 379 semigroup 379 strong 378 transition function 378 Markov property 377 strong 378 matrix monotone function 188 measure Lévy 22, 299 Nevanlinna–Pick 79 Pick 79 Stieltjes 75, 299 Thorin 111, 299 mixture of exponential distributions and first passage time 293 characterization 119 definition 118 stability 120 ME  BO 120 ME 6 GGC 129 ME 6 SD 128 monotone matrix function 188 monotone of order n see n-monotone function multiply monotone function see n-monotone function n-monotone function 8, 14 vs. n-monotone sequence 14 integral representation 8–9 iterated differences 42 on R 13 stability 12 negative definite function 36, 212 in the sense of Schoenberg 46, 187, 212 Nevanlinna factorization 85 Nevanlinna–Pick function 77 integral representation 77 relation with CBF 83 relation with S 82–83 Nevanlinna–Pick measure 79 non-tangential limit 76, 90

Index operator monotone function definition 188 is CBF 195 p-monotone function see n-monotone function Phillips’ theorem 204 Pick function 77 integral representation 77 relation with CBF 83 relation with S 82–83 Pick matrix 108 portmanteau theorem 375 positive definite function 35 conditionally 36 in the sense of Bochner 45 potential 64, 97, 257 relation with H 175 structure 97 potential measure 63 -potential measure 64 potential operator 257 -potential operator 270 quasi everywhere 243, 381 random variable infinitely divisible 51, 54 stable 55 representing measure for BF 22 for CBF 75 for CM 4 for PBF 61 for S 17 for TBF 111 for n-monotone functions 8–9 for Nevanlinna–Pick fns 77, 79 resolvent equation 179, 201, 379 Revuz measure 382 S-self decomposable distribution characterization 123 definition 123 is GGC \ ME 124 Schoenberg’s theorem 36 Schoenberg–Bochner theorem 212

409 self-adjoint operator dissipative 180, 201 order 188 resolvent 179, 201 resolvent equation 179 resolvent estimate 201 spectral theorem 183 spectrum 179, 201 self-decomposable distribution 55 completely (SD1 ) 60–61, 121 Lévy measure of 57 p-times (SDp ) 60 GGC 6 SD1 129 SD1 \ CE 125 SD1 \ ME 121 SD 6 BO 127 SD 6 ME 127 semigroup see also convolution semigroup, see also subordinate semigroup algebraic 35 C0 -semigroup 200 contraction semigroup 200, 202 generator 200, 380 hypercontractive 241 intrinsically ultracontractive 257, 261 not hypercontractive 255 strongly continuous 200, 380 supercontractive 241 ultracontractive 233, 241, 247 smooth measure 381 special subordinator 257 special Bernstein function 159, 164 cone structure 171 conjugate Lévy triplet 161–162 conjugate pair 159 counterexample 169–170 definition 159 potential density 161 relation with H 175 stability 171 SBF 6 CBF 169–170 special functions 300–303 special subordinator 159, 164 characterization 160 factorization of potential density 163, 258, 259 potential density 161

410 spectral theorem 183, 189 Stieltjes function 16 CBF  S 80 and string 273 cone structure 17 definition 16 extremal representation 18 log-convexity of S 99 negative of 82–83 primitive 97 relation with CBF 93, 96, 99 relation with H 175 stability 17, 94, 120 Stieltjes inversion formula 75 Stieltjes transform see Stieltjes function generalized 20, 116 string 268 and Stieltjes function 273 characteristic function 274 definition 268 dual 284 length 268 subordinate generator 203 CBF-subordinator 220 domain 208, 219, 227–230 functional calculus 222 limits 230 moment inequality 209 operator core 204 Phillips’ formula 205 spectrum 211 subordinate process 65, 211, 244 killed 245 subordinate Brownian motion 91, 135, 216 subordinate semigroup 65, 202–203 definition 203 in Hilbert space 203 of measures 65

Index subordinator 49, 211 integral w.r.t. 131 killed 50 transition probabilities 49 symmetric ˛-stable process 252 table of distributions 127–128 of Kre˘ın correspondences 284 of Laplace exponents 127–128 of transformations of BF 134, 136 tangent functional 233 Thorin–Bernstein function 109, 114 TBF  114 characterization 109, 112, 124 cone structure 113 definition 109, 114 extremal representation 114, 123 is in CBF 109 relation with S 113 Thorin measure 111 Thorin representation 111, 114, 122 transformation of BF compositions 142, 143 continuity 140, 146, 149, 152 definition 132 domain 133 Lévy triplet of 136 not continuous 141 not one-to-one 142 overview 134, 136 parametrized families 133 range 133, 149, 152, 155, 156 PBF 155, 156 transition function 49, 378 vague convergence 374 vague integral 375 weak convergence 374