Lectures from Markov Processes to Brownian Motion
This book evolved from several stacks of lecture notes written over a decade and given in classes at slightly varying levels. In transforming the over lapping material into a book, I aimed at presenting some of the best features of the subject with a min
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Editors
M. E. W. W.
Artin S. S. ehern J. L. Doob A. Grothendieek Heinz F. Hirzebrueh L. Hörmander S. Mac Lane Magnus C. C. Moore J. K. Moser M. Nagata Sehmidt D. S. Seott J. Tits B. L. van der Waerden
M anaging Editors
B. Eckmann
S. R. S. Varadhan
Kai Lai Chung
Lectures from Markov Processes to Brownian Motion With 3 Figures
Springer Science+Business Media, LLC
Kai Lai Chung Department of Mathematics Stanford University Stanford, CA 94305
AMS Subject Classifications (1980): 60Jxx Library of Congress Cataloging in Publication Data Chung, Kai Lai, 1917Lectures from Markov processes to Brownian motion. (Grundlehren der mathematischen Wissenschaften; 249) Bibliography: p. Includes index. 1. Markov processes. 2. Brownian motion processes. L Title. II. Series. QA274.7.C48 519.2'33 81-14413 AACR2 © 1982 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Tnc. in 1982. Softcover reprint of the hardcover 1st edition 1982 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC 9 8 7 6 543 2 1
ISBN 978-1-4757-1778-5 ISBN 978-1-4757-1776-1 (eBook) DOI 10.1007/978-1-4757-1776-1
Contents
Preface
VII
Chapter I
Markov Process 1.1. Markov Property 1.2. Transition Function 1.3. Optional Times 1.4. Martingale Theorems 1.5. Progressive Measurability and the Projection Theorem Notes
6 12 24 37 44
Chapter 2
Basic Properties 2.1. 2.2. 2.3. 2.4.
Martingale Connection FeUer Process Strong Markov Property and Right Continuity of Fields Moderate Markov Property and Quasi Left Continuity Notes
45 48
56 66 73
Chapter 3
Hunt Process 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.
Defining Properties Analysis of Excessive Functions Hitting Times Balayage and Fundamental Structure Fine Properties Decreasing Limits Recurrence and Transience Hypothesis (B) Notes
75 80 87 96 106
116 122 130 135
Chapter 4
Brownian Motion 4.1. Spatial Homogeneity 4.2. Preliminary Properties of Brownian Motion
137 144
Contents
VI
4.3. 4.4. 4.5. 4.6. 4.7.
Harmonie Funetion Diriehlet Problem Superharmonie Funetion and Supermartingale The Role of the Laplaeian The Feynman-Kae Funetional and the Sehrödinger Equation Notes
154
162 174 189 199
206
Chapter 5
Potential Developments 5.1. Quitting Time and Equilibrium Measure 5.2. Some Prineiples of Potential Theory Notes
Bibliography Index
208 218 232 233 237
Preface
This book evolved from several stacks of lecture notes written over a decade and given in classes at slightly varying levels. In transforming the overlapping material into a book, I aimed at presenting some of the best features of the subject with a minimum of prerequisities and technicalities. (Needless to say, one man's technicality is another's professionalism.) But a text frozen in print does not allow for the latitude of the classroom; and the tendency to expand becomes harder to curb without the constraints of time and audience. The result is that this volume contains more topics an
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