Dirichlet Forms and Symmetric Markov Processes 9783110889741

Table of contents :
Preface
Notation
Part I. Dirichlet forms
Chapter 1. Basic theory of Dirichlet forms
1.1. Basic notions
1.2. Examples
1.3. Closed forms and semigroups
1.4. Dirichlet forms and Markovian semigroups
1.5. Transience of Dirichlet spaces and extended Dirichlet spaces
1.6. Global properties of Markovian semigroups
Chapter 2. Potential theory for Dirichlet forms
2.1. Capacity and quasi continuity
2.2. Measures of finite energy integrals
2.3. Reduced functions and spectral synthesis
Chapter 3. The scope of Dirichlet forms
3.1. Closability and the smallest closed extensions
3.2. Formulae of Beurling-Deny and LeJan
3.3. Maximum Markovian extensions
Part II. Symmetric Markov processes
Chapter 4. Analysis by symmetric Hunt processes
4.1. Smallness of sets and symmetry
4.2. Identification of potential theoretic notions
4.3. Orthogonal projections and hitting distributions
4.4. Parts of forms and processes
4.5. Continuity, killing and jumps of sample paths
4.6. Quasi notions, fine notions and global properties
Chapter 5. Stochastic analysis by additive functionals
5.1. Positive continuous additive functionals and smooth measures
5.2. Decomposition of additive functionals of finite energy
5.3. Martingale additive functionals and Beurling-Deny formulae
5.4. Continuous additive functionals of zero energy
5.5. Extensions to additive functionals locally of finite energy
5.6. Martingale additive functionals of finite energy and stochastic integrals
5.7. Forward and backward martingale additive functionals
Chapter 6. Transformations of forms and processes
6.1. Perturbed Dirichlet forms and killing by additive functionals
6.2. Traces of Dirichlet forms and time changes by additive functionals
6.3. Transformations by supermartingale multiplicative functionals
Chapter 7. Construction of symmetric Markov processes
7.1. Construction of a Markovian transition function
7.2. Construction of a symmetric Hunt process
7.3. Dirichlet forms and Hunt processes on a Lusin space
A Appendix
A.1 Choquet capacities
A.2 An introduction to Hunt processes
A.3 A summary on martingale additive functionals
A.4 Regular representations of Dirichlet spaces
Notes
Bibliography
Index

Citation preview

de Gruyter Studies in Mathematics 19 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 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, Προ Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev

Masatoshi Fukushima · Yoichi Oshima Masayoshi Takeda

Dirichlet Forms and Symmetric Markov Processes

w

Walter de Gruyter G Berlin · New York 1994 DE

Authors M. Fukushima M. Takeda Department of Mathematical Science Osaka University Toyonaka, Osaka 560 Japan Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstraße D-91054 Erlangen, Germany

Y. Oshima Department of Mathematics Kumamoto University Kumamoto 860 Japan

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: 31-02; 60-02 Keywords: Dirichlet form, symmetric Markov process, potential theory, capacity, additive functional @ 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

Fukushima, Masatoshi, 1935 — Dirichlet forms and symmetric Markov processes / Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda, p. cm. — (De Gruyter studies in mathematics ; 19) Includes bibliographical references and index. ISBN 3-11-011626-X (acid-free) 1. Markov processes. 2. Dirichlet forms. I. Oshima, Yoichi. II. Takeda, Masayoshi, III. Title. IV. Series. QA274.7.F845 1994 519.2'2'33-dc20 94-3434 CIP

Die Deutsche Bibliothek — CIP-Einheitsauf nähme Fukushima, Masatoshi:

Dirichlet forms and symmetric Markov processes / Masatoshi Fukushima ; Yoichi Oshida ; Masayoshi Takeda. - Berlin ; New York : de Gruyter, 1994 (De Gruyter studies in mathematics ; 19) ISB_N3-11-011626-X NE: Oshima, Yoichi:; Takeda, Masayoshi:; GT © Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced 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. Typeset using the authors' LATgX files: Lewis & Leins, Berlin. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Rudolf Hübler, Berlin.

Preface

Part I of this book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms—an axiomatic extension of the classical Dirichlet integrals in the direction of Markovian semigroups. In Part II, this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on the additive functionals. We intend to organize it as a self-contained text book. Part I requires only a first course of functional analysis, while Part II can be read through with the help of "An introduction to Hunt processes" and "A summary on martingale additive functionals" being provided in Appendix. A brief summary at the beginning of each chapter, simple examples presented in many sections and the bibliographic notes stated at the end of the volume will serve to facilitate your use of the text. This is an outgrowth of Fukushima's book "Dirichlet forms and Markov processes" published from Kodansha and North Holland in 1980, partly combined with Oshima's lecture note "Lectures on Dirichlet spaces" delivered at Universität Erlangen-Nürnberg in 1988. Most ingredients in Fukushima's book are maintained as the skeleton of the present volume. But they are reorganized and integrated with many new basic materials developed in the last decade. Some more of basic examples in finite dimensions are included. In the main text except for the first chapter and the last section, the underlying space is assumed to be locally compact. However the infinite dimensional non-locally compact situations can be handled in the present framework as well by making use of "Regular representations of Dirichlet spaces" being provided in the Appendix. We would like to express our hearty thanks to Professor H. Bauer for his warm encouragement in our joint writing of the book. We are grateful to Professors H. Okura and M. Tomisaki for their valuable comments on our preliminary drafts. We also thank Mrs. M. Tsukamoto for her great help in our preparation of the Tex file manuscript. Thanks are due to Dr. M. Karbe of Walter de Gruyter & Co. for his constant and truly generous cooperation. Osaka and Kumamoto, December 1993 Masatoshi Fukushima Yoichi Oshima Masayoshi Takeda

Contents

Preface Notation

v ix

Part I. Dirichlet forms Chapter L Basic theory of Dirichlet forms . 1. .2. .3. .4. .5. .6.

Basic notions Examples Closed forms and semigroups Dirichlet forms and Markovian semigroups Transience of Dirichlet spaces and extended Dirichlet spaces Global properties of Markovian semigroups

3 3 6 15 23 32 46

Chapter 2. Potential theory for Dirichlet forms

64

2.1. Capacity and quasi continuity 2.2. Measures of finite energy integrals 2.3. Reduced functions and spectral synthesis

64 74 92

Chapter 3. The scope of Dirichlet forms

98

3.1. Closability and the smallest closed extensions 3.2. Formulae of Beurling-Deny and LeJan 3.3. Maximum Markovian extensions

98 108 ..117

Part II. Symmetric Markov processes Chapter 4. Analysis by symmetric Hunt processes

133

4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

134 141 148 152 158 167

Smallness of sets and symmetry Identification of potential theoretic notions Orthogonal projections and hitting distributions Parts of forms and processes Continuity, killing and jumps of sample paths Quasi notions, fine notions and global properties

viii

Contents

Chapter 5. Stochastic analysis by additive functionals

180

5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Positive continuous additive functionals and smooth measures Decomposition of additive functionals of finite energy Martingale additive functionals and Beurling-Deny formulae Continuous additive functionals of zero energy Extensions to additive functionals locally of finite energy Martingale additive functionals of finite energy and stochastic integrals 5.7. Forward and backward martingale additive functionals

180 199 212 217 225

Chapter 6. Transformations of forms and processes

258

6.1. Perturbed Dirichlet forms and killing by additive functionals 6.2. Traces of Dirichlet forms and time changes by additive functionals 6.3. Transformations by supermartingale multiplicative functionals

258 265 280

Chapter 7. Construction of symmetric Markov processes

292

7.1. Construction of a Markovian transition function 7.2. Construction of a symmetric Hunt process 7.3. Dirichlet forms and Hunt processes on a Lusin space

292 296 303

A Appendix

308

A. 1 A.2 A.3 A.4

308 310 331 344

Choquet capacities An introduction to Hunt processes A summary on martingale additive functionals Regular representations of Dirichlet spaces

Notes Bibliography Index

239 248

361 369 389

Notation

We use the following notations for a given measurable space (X, B): f e B ; / is an extended real valued function on X which is -measurable / € + ; / € B and / is non-negative / e Bh ; / € B and / is bounded Given a function space A, we use the following: A+ or ,4+ ; the space of non-negative functions in A Ah ; the space of bounded functions in A When X is a topological space, B(X) (resp. *(X)) denotes the family of all Borel measurable (resp. universally measurable, namely, measurable with respect to every probability measure on X) subsets of X. C(X) denotes the family of all real valued continuous functions on X. In this case, we write C+(X), C/ ; (X),/ e £+(*),/ € Bb(X) for C(X) + , C(X)b,f € B(X)+,f e B(X)h, respectively. Further notations: D(w, v) ; Dirichlet integral f ; Dirichlet space, i.e., the domain £>[£] of a Dirichlet form S fe ; extended Dirichlet space {•^'J/eio.oc] ! minimum completed admissible filtration Ρ(Λ|Σ) ; conditional probability with respect to a σ-field Σ S ; smooth measures S] ; smooth measures in the strict sense SO ; positive Radon measures of finite energy integrals = (M e S0 : μ(Χ) < oo, ||(/ιμ||οο < oo} ; positive continuous additive functionals ; positive continuous additive functionals in the strict sense

Parti Dirichlet Forms

Chapter 1

Basic theory of Dirichlet forms

The Dirichlet form on an L2-space is defined as a Markovian closed symmetric form (§1.1). The link connecting the theory of Dirichlet forms with Markov processes is in that the Markovian nature of a closed symmetric form is equivalent to the Markovian properties of the associated semigroup and resolvent on L2 (§1.4). The domain of a Dirichlet form can be enlarged to the extended Dirichlet space. The recurrence and transience of a Markovian semigroup can be characterized in terms of the associated Dirichlet form and extended Dirichlet space (§1.5, §1.6). Thus the Dirichlet space in the original sense of Beurling and Deny is a specific transient extended Dirichlet space in the present context. Translation invariant Dirichlet forms, Sobolev spaces of order 1 and their extended Dirichlet spaces are studied in examples of §1.2, §1.4, §1.5 and §1.6. Tests of global properties are applied to a local Dirichlet form on R'! associated with the second order differential operator of divergence form at the end of § 1.6.

1.1. Basic notions Let H be a real Hubert space with inner product ( , ). A non-negative definite symmetric bilinear form densely defined on H is henceforth called simply a symmetric form on H. To be precise, E is called a symmetric form on H if the following conditions are satisfied: (£.1) S is defined on £>[£] χ D[£] with values in R 1 , P[£] being a dense linear subspace of H, (£.2) £(«, υ) = £(t», u), E(u + v, w) — £(«, w) + £(u, w), a£(w, υ) = S (au, v), £(w, w) > 0, u, v, w e £>[£], a e R 1 .

We call £>[£] the domain of £. The inner product ( , ) on H is a specific symmetric form defined on the whole space //. Given a symmetric form £ on //, 8a(u,v) £>[£„]

= S(u,v) + a(u,v), u, v e T>[£} = £>[£]

(

·

l)

4

Basic theory of Dirichlet forms

defines a new symmetric form on H for each a. > 0. Note that the space T)[S] is then a pre-Hilbert space with inner product £a. Furthermore Sa and 8 β determine equivalent metrics on T>[S ] for different α, β > 0. If T>[£] is complete with respect to this metric, then S is said to be closed. In other words, a symmetric form 8 is said to be closed if (£".3) u„ [£], B\ (M„ - um, u„ - Um) -»· 0, n, m -> oo =>· 3w e D[£], £1 (M„ — M, «„ — M) —»· 0, n —>· oo. Clearly £>[£] is then a real Hubert space with inner product €a for each a > 0. We say that a symmetric form S is dosable if the following condition is fulfilled: u„ € T>[£], £(un - um, u,, - um) -> 0, n, m -> oo, (M„,M„) -» 0, « —»· oo =>· £(HH, κ,,) -> Ο, « —> oo.

2

Given two symmetric forms is said to be an extension of £ (1) if X>[£ (l) ] c £>[£(2)] and