Markov Processes, Brownian Motion, and Time Symmetry
From the reviews of the First Edition: "This excellent book is based on several sets of lecture notes written over a decade and has its origin in a one-semester course given by the author at the ETH, Zürich, in the spring of 1970. The author's aim was to
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Series Editors
M. Berger S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt Editors-in-Chief
A. Chenciner
J. Coates
S.R.S. Varadhan
Kai Lai Chung and John B. Walsh
Markov Processes, Brownian Motion, and Time Symmetry Second Edition
Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo
Kai Lai Chung Department of Mathematics Stanford University Stanford, CA 94305 USA
John B. Walsh Mathematics Department University of British Columbia Vancouver, BC V6T 1Z2 Canada
AMS Subject Classifications (2000): 60Jxx, 60J65
Library of Congress Cataloging in Publication Data Chung, Kai Lai Markov processes, Brownian motion, and time symmetry / Kai Lai Chung, John B. Walsh. p. cm. Includes bibliographical references and index. ISBN 0-387-22026-7 1. Markov processes. 2. Brownian motion processes. I. Walsh, John B. II. Title. QA274.7.C52 2004 519.2 33- -dc22 2004048547 ISBN-10: 0-387-22026-7 ISBN-13: 978-0387-22026-0
Printed on acid-free paper.
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Contents
Preface to the New Edition Preface to the First Edition
ix xi
Chapter 1 Markov Process
1
1.1. 1.2. 1.3. 1.4. 1.5.
Markov Property Transition Function Optional Times Martingale Theorems Progressive Measurability and the Section Theorem Exercises Notes on Chapter 1
Chapter 2 Basic Properties 2.1. 2.2. 2.3. 2.4.
Martingale Connection Feller Process Exercises Strong Markov Property and Right Continuity of Fields Exercises Moderate Markov Property and Quasi Left Continuity Exercises Notes on Chapter 2
Chapter 3 Hunt Process 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.
Defining Properties Exercises Analysis of Excessive Functions Exercises Hitting Times Balayage and Fundamental Structure Exercises Fine Properties Exercises Decreasing Limits Exercises
1 6 12 24 37 43 44
45 45 48 55 56 65 66 73 73
75 75 78 80 87 87 96 105 106 115 116 122
vi 3.7. 3.8.
Contents
Recurrence and Transience Exercises Hypothesis (B) Exercises Notes on Chapter 3
Chapter 4 Brownian Motion 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.
Spat
