Dirichlet Forms and Analysis on Wiener Space [Reprint 2010 ed.] 9783110858389, 9783110129199

Table of contents :
I General Dirichlet forms
1 Closed forms
2 Sub-Markovian symmetric operators
3 Dirichlet forms and Dirichlet operators
4 The carré du champ operator
5 Locality
6 Functional calculus
7 Absolute continuity of image measures
8 Capacity
9 Distributions of finite energy
II Dirichlet forms on vector spaces
1 Standard Dirichlet structure on RN*
2 Standard structure on the Wiener space
3 Abstract Wiener spaces
4 Dirichlet forms and directional derivatives
5 An absolute continuity criterion
6 Operators D and δ
7 Sobolev spaces
III Analysis on Wiener space
1 Operations on chaos decompositions
2 Derivation operator
3 Calculus on stochastic integrals
4 Representation of positive distributions
IV Stochastic differential equations
1 Solution for a fixed initial condition
2 Existence of densities
3 Regularity of the flow
4 Accurate versions of the flow
V The algebra of Dirichlet structures
1 Image structures
2 Tensor products and projective limits
3 Other constructions of Dirichlet structures
4 Dirichlet-independence
5 Substructures and conditioning
VI An extension of Girsanov’s theorem
1 Distribution-measures
2 Extension of Girsanov’s theorem
3 Examples
VII Quasi-everywhere convergence
1 Derivation operator
2 Ergodic theorems
3 Convergence of martingales
4 Stochastic differential equations
Notes
Bibliography
Index

Citation preview

de Gruyter Studies in Mathematics 14 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, Wilhelm Klingenberg 2 Semimartingales, Michel Metivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 1 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann

Nicolas Bouleau · Francis Hirsch

Dirichlet Forms and Analysis on Wiener Space

w

Walter de Gruyter G Berlin · New York 1991 DE

Authors Nicolas Bouleau CERMA Ecole Nationale des Fonts et Chaussees La Courtine F-93167 Noisy-le-Grand Cedex France

Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstrasse 1 Vz D-8520 Erlangen, FRG

Francis Hirsch University of Evry Boulevard des Coquibus F-91025 Evry Cedex France

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistrasse 101 CH-8092 Zürich Switzerland

1991 Mathematics Subject Classification: 60-02; 60Fxx, 60Gxx, 60Hxx, 60Jxx. 31-02; 31Cxx. 46-02; 46Cxx, 46Exx, 46Fxx, 46Gxx, 46Nxx. 47-02; 47Axx, 47Dxx. © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Bouleau, Nicholas. Dirichlet forms and analysis on Wiener space / Nicholas Bouleau, Francis Hirsch. p. cm. — (De Gruyter studies in mathematics ; 14) Includes bibliographical references and index. ISBN 3-11-012919-1 (alk. paper) 1. Dirichlet forms. 2. Malliavin calculus. I. Hirsch, F. (Francis) II. Title. III. Series. QA274.2.B68 1991 519.2-dc20 91-32819 CIP Die Deutsche Bibliothek — Cataloging-in-Publication Data Bouleau, Nicolas: Dirichlet forms and analysis on Wiener space / Nicolas Bouleau ; Francis Hirsch. — Berlin ; New York : de Gruyter, 1991 (De Gruyter studies in mathematics ; 14) ISBN 3-11-012919-1 NE: Hirsch, Francis:; GT © Copyright 1991 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Gerike GmbH, Berlin. Binding: Dieter Mikolai, Berlin. Cover design: Rudolf Hübler, Berlin.

Preface

to Jacques Deny

Two important research trends, the theory of Dirichlet forms and the Malliavin calculus, are so closely connected as to merit a book detailing their interplay which, it is hoped, should contribute to their mutual enrichment. The first, and elder, of the two trends is the branch of potential theory which concerns Markov semigroups symmetric with respect to a σ-finite measure. The pivotal contributions of Beurling and Deny showed that this setting presents particularities suited for specific Hubert space techniques, with the remarkable property that contractions operate, which is to say that the domains of Dirichlet forms are function spaces stable by composition with Lipschitz functions. Subsequently developed notably by Silverstein, Ancona, Le Jan and Fukushima, this theory has provided a simple powerful framework allowing one to treat, within a functional calculus of class C1, functions which are not, as Fukushima has shown, semimartingales on trajectories of the associated Markov process. This is exposited in a locally compact setting in Fukushima [2], now a standard reference text. The second trend forms, from its origins in the work of Malliavin, a mathematical framework extending to infinite dimensions the reasoning of integration by parts, which gives rise to regularity of the density function of a random variable X thanks to estimates on Έ[ψ^(Χ)], where φ is a test function. The first success of this approach was a probabilistic proof of the hypoellipticity theorem of H rmander. With the works of Stroock, Kusuoka and Stroock, Bismut, Ikeda and Watanabe, and Shigekawa, this approach has developed into a penetrating tool for the study of stochastic differential equations and Wiener functionals. There now exist many treatments of the "stochastic calculus of variations", by and large well developed, of which we cite: Stroock [1], Watanabe [1], Ikeda and Watanabe [2], Zakai[l], Bell [1], and particularly Ocone [1] which is an excellent introduction to this subject. This field is by now quite rich and diversified, leaning on the one hand towards different theories of distributions on Wiener space and on infinite dimensional spaces (e.g., Kree, Lascar, Hida, Kuo, Watanabe, and Korezlioglu and Ustunel), and leaning on the other hand towards a precise analysis of functionals by exploiting capacities associated to Sobolev spaces on Wiener space and the geometric study of submanifolds of Wiener space obtained by lifting manifolds of Kn (e.g., Malliavin, Airault, and Feyel and La Pradelle). In order to extend the techniques of Dirichlet forms to a setting which encompasses the case of the Ornstein-Uhlenbeck semi-group in Wiener space, the first

vi

Preface

step consisted of discarding hypotheses of local compactness used hitherto. This is pursued in Chapter I, which shows that the ideas of Beurling and Deny, as well as the functional calculus and the theory of capacity in Dirichlet spaces, can overcome this restriction. This chapter also introduces the concept of a carre du champ operator which plays an important role throughout the text. In Chapter II, the measurable structure of the space is enriched with a vector space structure, with which can be formulated a more intimate connection with the Dirichlet form. From this, with rather general hypotheses, we establish that the almost-everywhere strict positivity of the determinant of the Gram-Malliavin matrix of a functional by itself implies the existence of a density for the law of the functional. The following chapter assembles the requisite tools to exploit this result on Wiener space, with special attention given to the derivation operator of Feyel and La Pradelle. This operator is, in many respects, a more powerful tool than the conventional gradient operator normally used in the stochastic calculus of variations. As Lipschitz functions operate in Dirichlet spaces, the principle behind Ito's proof, showing the existence of solutions to stochastic differential equations with Lipschitz coefficients using Picard's iteration, remarkably, carries over to our context by replacing the L2 norm with the Dirichlet norm. This leads with relative ease to numerous results in Chapter IV concerning the existence of a density for the solution, the regularity of the associated flow, as well as the existence, for this flow, of refined versions up to a set of zero capacity. The originality of this chapter stems from the very mild hypotheses on the coefficients of the equation (Lipschitz hypotheses). Here the power and flexibility of Dirichlet form methods are most strikingly revealed. Chapter V shows that the theory of general Dirichlet forms is, in a way, to Malliavin calculus what the theory of abstract measures is to the theory of Radon measures and Schwartz distributions. The notion of a Dirichlet structure — a probability space equipped with a Dirichlet form with no Gaussian assumption — is shown to be a robust tool, facilitating constructions analogous to those used in the calculus of probabilities for defining canonical spaces of processes, which themselves may be naturally accompanied by Dirichlet forms. We show for example that the local structure on Poisson space, introduced by Carlen and Pardoux, is readily interpreted as a product structure. Moreover, the conditional calculus developed by Nualart and Zakai extends to these general situations. A particularly interesting property of the structure on Wiener space induced by the Ornstein-Uhlenbeck semi-group is the fact that some distributions which are positive measures, possibly singular with respect to the Wiener measure, may be interpreted as laws of processes. This leads to a generalization of Girsanov's theorem, which is developed in Chapter VI. The final chapter extends classical convergence theorems for martingales and ergodic theory to convergence theorems outside a set of zero capacity. Although numerous examples are treated, including Birkoff's theorem, Strassen's theorem, and quadratic Brownian variation, the emphasis is on the methods themselves. These methods derive from the extension of the existence of a quasi-continuous version to elements in Dirichlet space with values in a Banach space, an exten-

Preface

vii

sion rendered simple using the derivation operator mentioned above. We also study approximations to solutions of stochastic differential equations obtained by discretization. Numerous important themes are still under active study and, although some are alluded to in this work, many lie beyond the scope of the present book. Of notable interest is the theory of capacity associated to different Sobolev spaces, leading to the notion of a slim set and the theory of redefinition, as developed by Malliavin, and further pursued by Fukushima, Sugita, Airault, Ren, Feyel and La Pradelle,...Another important theme is the construction and the study of processes associated with Dirichlet forms, treated in the locally compact case, by Silverstein, Le Jan, Fukushima,..., and , in the general case, by Kusuoka, Albeverio, Ma, Röckner,... The book does not suppose any requisite background knowledge aside from classical results of probability theory and analysis. Exercises are included at the end of each section to develop examples, improve comprehension, and indicate extensions and current research topics. In view of the vast recent literature on this subject (roughly 5000 references), the bibliography is restricted to those works having a clear relevance to our treatment. Finally, we wish to express our gratitude to Heinz Bauer for his warm encouragement for this enterprise. Paris, July 1991

Nicolas Bouleau Francis Hirsch

Contents

I

General Dirichlet forms 1 Closed forms 2 Sub-Markovian symmetric operators 3 Dirichlet forms and Dirichlet operators 4 The carre du champ operator 5 Locality 6 Functional calculus 7 Absolute continuity of image measures 8 Capacity . . . " 9 Distributions of finite energy

1 1 6 12 16 28 35 43 52 61

II

Dirichlet forms on vector spaces 1 Standard Dirichlet structure on R N 2 Standard structure on the Wiener space 3 Abstract Wiener spaces 4 Dirichlet forms and directional derivatives 5 An absolute continuity criterion 6 Operators D and 8 7 Sobolev spaces

69 69 78 90 96 105 110 115

III Analysis on Wiener space 1 Operations on chaos decompositions 2 Derivation operator 3 Calculus on stochastic integrals 4 Representation of positive distributions

119 119 134 143 147

IV Stochastic differential equations 1 Solution for a fixed initial condition 2 Existence of densities 3 Regularity of the 4 Accurate versions of the

151 151 160 167 174

V

flow flow

The algebra of Dirichlet structures 1 Image structures 2 Tensor products and projective limits 3 Other constructions of Dirichlet structures 4 Dirichlet-independence 5 Substructures and conditioning

185 186 200 213 217 223

χ

Contents

VI An extension of Girsanov's theorem

1 2 3

Distribution-measures Extension of Girsanov's theorem Examples

VII Quasi-everywhere convergence

1 2 3 4

Derivation operator Ergodic theorems Convergence of martingales Stochastic differential equations

243

243 248 254 265

265 272 287 293

Notes

299

Bibliography

309

Index

321

Chapter I

General Dirichlet forms

We study, in this chapter, the general properties of Dirichlet forms on a measure space. In section 1, we consider any real Hubert space E, equipped with its scalar product ( , )E and with the associated norm || \\B- Beginning with section 2, a measure space (Ω, J-, m) is fixed where m is a positive σ-finite measure, and E is taken to be L2 (m), the space of classes of real square m-integrable functions. (In the notation, m will be omitted most of the time). Throughout the book, the terms positive, increasing, and decreasing are understood in the wide sense. Otherwise, we use strictly positive, strictly increasing, and strictly decreasing.

1 Closed forms 1.1 This first subsection is devoted to defining the basic notions of the theory.

1.1.1 A closed form is a quadratic form £ defined on a subspace D (E) = E) dense in E, which is positive (i.e., V/ € E), £(/) > 0) and such that E) equipped with the norm

H/||D = Ill/Ill + £(/)]1/2 is a Hubert space. The bilinear symmetric form on Ε) χ E) associated with £ will be denoted by £(/,0) (W £ DI £(/,/) = £(/)) and ( , )D will denote the scalar product associated with || ||D. We shall denote by F the set of closed forms. Hereafter "operator of E" shall always mean a linear map from a subspace of E (the domain of the operator) into E.

1.1.2 A negative self-adjoint operator is an operator A of E with domain D(A) dense in E, which is self-adjoint (i.e., A* = A where A* denotes the adjoint of ^4)

2

/. General Dirichlet forms

and satisfying VfeD(A)

(Af,f)E0 of everywhere defined symmetric operators of E satisfying i) PO = I

Vt,s>0

Pt+s = PtPs (semi-group)

ii) V/ 6 E

limt->oPtf = f (strong continuity)

iii) V/ 6 E

Vi > 0

||Pt/||B < \\f\\E (contraction).

The set of symmetric semi-groups will be denoted by Π.

1.2 The following propositions give the relationship between the three notions introduced above. For the proofs, see for example Fukushima [2], §1.3. Proposition 1.2.1 If Tl — (Pt)t>0 is a symmetric semi-group, A = Θ(Π) is defined as the generator of Π, in other words

D(A) = {/; lim

~

exists}

and V/ 6 D(A)

Af = lim — 0 of every where-defined operators of L2, satisfying properties i) and ii) of 1.1.3 and iii)' Vi > 0, Pt is a sub-Markovian symmetric operator. (In particular, by 2.2.4, iii) of 1.1.3 is also satisfied.) The set of these semi-groups will be denoted by ΠοWe now consider a semi-group (it) t>0 ε Πο . For each ί > 0 , Pt contraction of L1 associated with Pt (see 2.2.2).

is the

Proposition 2.4.2 The family (P( )t>o is a strongly continuous contraction semi-group in L1, whose generator A^1' is the smallest closed extension of the restriction of the generator A of (Pt)t>0 to {/ £ D(A) Π L1; Af e L1}. Proof. 1) To show the first point, it suffices, by 2.2.2, to prove V/ G L1 n L00

lim Ptf = f in L1.

Then let f € L1 nL°°, / > 0, and let An be an increasing sequence in T such that Vn m(An) < +00 and Ω = UnAn.

lim [(Ptf)dm > lim / (Ptf)dm = tlim(U n , Ptf)L, = f t

t—07

t^OJAn

~°

J An

fdm.

10

/. General Dirichlet forms

Hence lim l (Ptf)dm> l fdm t->oj J and therefore lim i(Ptf)dm=

f fdm.

(1.1)

Let now (tn) be a vanishing sequence. As Ptnf —> / in I/2, there exists such that lim Ptnfc f = f m-a.e. k—>oo

By dominated convergence we have lim(P t

fc—>oo

k

/A/) = /

inL 1 .

This yields lim(P t / Λ /) = /

in L1,

(1.2)

and the equality II^J - /||LI = j(Ptf + f- 2(Ptf Λ /))dm gives, by (I.I) and (1.2), limP t / = /

inL 1 .

2) To show the second point, suppose f £D(A)nLl and Af e Ζλ Then j

7

t

Pf

-

f

tj

t

^ -

1

/*

1

' P s (A/)d S = 7 ι- Jo '-Jo

hence / € D(A^) and Finally let / e ^»(ylt 1 )). Setting R = (I-A)~l, it is easy to see (by the formula R = J0°° e tPtdt) that R is also a symmetric sub-Markovian operator and

On the other hand 3gn € L1 Π L2 such that gn —» (/ - A (1) )/ in L1. Now, A5n 6 D(A) Π L1 and A(Rgn) = -gn + Rgn € L1. But Rgn = RWgn —» / in L1 and ^( ff„) —>· -(/ - ^ (1) )/ + / = A^ 1 )/ in L1 , which completes the proof. G 2.4.2.1 Remark. In the same way, it can be seen that A is the smallest closed extension in L2 of the restriction of A^ to {/ 6 D(A^) nL 2 ; A^ f e Proposition 2.4.3 If f £ D (A) and f2 e £>(A) ί/ien /2 £

2. Sub-Markovian symmetric operators

11

Proof. It suffices, by 2.4.2, to show A ( f 2 ) e L1. By 2.2.3 P t ( f 2 ) + f - 2/PJ > (Ptf - /) 2 > 0. Hence the inequality J[(Pt(f2)

- / 2 ) - 2/(Pt/ - f ) ] d m < -ij f ( P t f - f)dm

yields by Fatou's lemma

A(/ 2 ) - 2fAf e L1 and therefore Λ(/ 2 ) e L1.

Π

Exercise 2.1 Show the corresponding properties to 2.2.2 and 2.4.2 for the spaces Lp (1 < ρ < oo). Exercise 2.2 Let g (resp. /) be a function from Ω into IR (resp. Rp) such that

i

and

wvuj, (*j ' t,- ir. ι \ — g(uj ι >\\ ^ V^ t ιpictM )\ ^~ y ι

(g is said to be a normal contraction of /). Show that there exists F 6 71° such that

Show a similar result using contractions in place of normal contractions. Exercise 2.3 Suppose m is a probability measure and Γι a sub-a-field of T . The conditional expectation is denoted E^1 . Show that if / e (L 2 ) p and F e Tp \\Fof-

Exercise 2.4 Show that, if F is a function from Rp into R satisfying p

Vx, y € R p \F(x) - F(y)\2 < V^ \Xi - yt t=l

2

and F(0) = 0,

12

/. General Dirichlet forms

then

v/e(L 2 ) p (where £p is the form defined in 2.3 )

3 Dirichlet forms and Dirichlet operators 3.1 Let us give first the definitions and notation.

3.1.1

A closed form S with domain E) satisfying

/ e D => (/ Λ 1) e D and £(/ Λ 1) < £"(/) is called a Dirichlet form. Let FO denote the subset of Dirichlet forms in the set F of closed forms.

3.1.2

A self-adjoint operator A satisfying

VfeD(A) is called a Dirichlet operator. Such an operator belongs to A, and AQ will denote the subset of A of the Dirichlet operators.

3.2 The following proposition links the three notions of a Dirichlet form, a Dirichlet operator and a sub-Markovian symmetric operator to each other. Proposition 3.2.1 Let Π = (Pt)t>0 £ Π, A £ A, £ G F associated in the sense of 1.2, i.e., A = Φ(5) = Θ(Π) ε = Φ(Π). Then

(5 e F0).

(Π e Π0)

Proof. 1) Suppose first that Π = (Pt)t>0 belongs to Πο and let

ε = Φ(π).

3. Dirichlet forms and Dirichlet operators

13

The function z £R—>ζΛΐ €R belongs to T®. Hence by proposition 2.3.2 V/eL2

Vi>0

((/Al)-Pt(/Al),/Al)ia 0 be the semi-group generated by A (A = Θ(Π)). To show that (i > t) i > 0 is sub-Markovian, it suffices to show that VA > 0

\R\ = \(XI -A)"1

is sub-Markovian

(indeed Ptf = lim^^ e~tx Σ"=0 ^λ η (λΑ λ ) η / )- Then let / £ L 2 , / < 1 and g = \Rxf

.

We therefore have

g £ D(A)

and λ^ - Ag = Xf.

Hence

and so

(g-f)(g-i)dm 1}) = 0 and therefore g < 1 . So

/ < 1 =Φ A A/ < 1 and, by homogeneity,

/ > 0 => ΧΗχ/ > 0. D

14

/. General Dirichlet forms

3.3 From the preceding proposition and the results of section 2 we shall deduce the main property that the normal contractions operate on the domain of a Dirichlet form. From now on a Dirichlet form £ with domain E) is fixed. Proposition 3.3.1 Suppose f e Dp and F e T® .Then

Fof gD

and

Proof. Let (-Pt)i>0 be the sub-Markovian symmetric semi-group associated to E (cf. proposition 3.2.1). Then E> = {/; lim -£Pt ( / ) < + « > }

and

(where £pt is defined in 2.3). Then the result follows immediately from proposition 2.3.2. Π Corollary 3.3.2 The space E) Π Z/°° zs on algebra and

Proof: By homogeneity, we can suppose that ||/||L°° = 1 and ||