Progress in Probability Volume 29 Series Editors Thomas Liggett Charles Newman Loren Pitt Seminar on Stochastic Processes, 1991 E.<;mlar P. Fitzsimmons KL.Chung S.Port M.J. Sharpe T. uggett Editors Mannging Editors Springer Science+Business Media, LLC E.~mlar P. J. Fitzsimmons Dept. of Civil Engineering (Managing Editor) and Operations Research Dept. of Mathematics Princeton University University of Califomia-San Diego Princeton, NJ 08544 La Jolla, CA 92093 K.L.Chung S. Port Dept. of Mathematics T. Liggett Stanford University (Managing Editors) Stanford, CA 94305 Dept. of Mathematics University of Califomia M.J.Sharpe Los Angeles, CA 90024 Dept. of Mathematics University of Califomia-San Diego La Jolla, CA 92093 Library of Congress Cataloglng-ln-PubHcation Data Seminar on stochastic processes, 1991 / edited by E. ~lDlar, K. L. Chung, M. J. Sharpe. p. cm. -- (Progress in probability ; v. 29) ISBN 978-1-4612-6735-5 ISBN 978-1-4612-0381-0 (eBook) DOI 10.1007/978-1-4612-0381-0 1. Stochastic processes--Congresses. I. ~lDlar, E. (Erban), 1941- n. Chung,KaiLai,1917- . m. Sharpe, M. J., 1941- IV. Series: Progress in probability : 29. QA274.A1S443 1992 91-47703 519.2--dc20 CIP Printed on acid-free paper. Cl Springer Science+Business Media New York 1992. Originally published by Birkhäuser Boston in 1992 Softcover reprint ofthe hardcover Ist edition 1992 Copyright is not claimed for works of U.S. Govemment employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, eleclronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress S1reet, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. ISBN 978-1-4612-6735-5 Camera-ready copy prepared by the Authors in TEX. 987654321 This Volume is Dedicated to the Memory of STEVEN OREY LIST OF PARTICIPANTS K. Alexander D. Khoshnevisan J. Rosen R. Banuelos F. Knight T. Salisbury M. Barlow G. Lawler M. Sanz R. Bass T. Liggett R. Schonmann K. Burdzy P. March M. J. Sharpe D. Burkholder M. Marcus C. T. Shih R. Carmona P. McGill H. Sikic E. Qinlar T. Mountford R. Song z. Q. Chen C. Mueller D. Stroock M. Cranston C. Neuhauser G. Swindle R. Dalang X. Pei M. Talagrand R. Darling R. Pemantle L. Taylor S. Evans J. M. Penrose E. Toby N. Falkner Y. Peres Z. Vondracek R. E. Feldman E. Perkins X. Wang P. Fitzsimmons M. Perman J. Watkins R. Getoor J. Picard R. Williams B. Hambly J. Pitman R. Wu T. Harris L. Pitt Z. Zhao H. Hughes S. Port TABLE OF CONTENTS M. CRANSTON In memory of Steven Orey 1 1. BENJAMINI and A correlation inequality for y. PERES tree-indexed Markov chains 7 M. CRANSTON On specifying invariant C7-fields 15 P. J. FITZSIMMONS On the martingale problem for measure-valued Markov branching processes 39 J. GLOVER and M. RAO Potential densities of symmetric Levy processes 53 H. KESTEN An absorption problem for several Brownian motions 59 F. B. KNIGHT Forms of inclusion between processes 73 S. M. KOZLOV, Brownian interpretations J. W. PITMAN and of an elliptic integral M.YOR 83 G. F. LAWLER L-shapes for the logarithmic 1]-model for DLA in three dimensions 97 P. McGILL Remark on the intrinsic local time 123 T. S. MOUNTFORD and Harmonic functions on Denjoy domains S. C. PORT 129 E. A. PERKINS Conditional Dawson-Watanabe processes and Fleming-Viot processes 143 J. ROSEN p-variation of the local times of stable processes and intersection local time 157 M. J. SHARPE Closing values of martingales with finite lifetimes 169 C. T. SHIH Construction of Markov processes from hitting distributions without quasi-Ieft-continuity 187 Z. VONDRACEK A characterization of Brownian motion on Sierpinski spaces 233 FOREWORD The 1991 Seminar on Stochastic Processes was held at the University of California, Los Angeles, from March 23 through March 25, 1991. This was the eleventh in a series of annual meetings which provide researchers with the opportunity to discuss current work on stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Northwestern University, Princeton University, the University of Florida, the University of Virginia, the University of California, San Diego, and the University of British Columbia. Following the successful format of previous years there were five invited lectures. These were given by M. Barlow, G. Lawler, P. March, D. Stroock, M. Talagrand. The enthusiasm and interest of the participants created a lively and stimulating atmosphere for the seminar. Some of the topics discussed are represented by the articles in this volume. P. J. Fitzsimmons T. M. Liggett S. C. Port Los Angeles, 1991 In Memory of Steven Orey M. CRANSTON The mathematical community has lost a cherished colleague with the passing of Steven Orey. This unique and thoughtful man has left those who knew him with many pleasant memories. He has also left us with important contributions in the development of the theory of Markov processes. As a friend and former student, I wish to take this chance to recall to those who know and introduce to * those who do not a portion of his lifework. Steven was born in Berlin and at an early age fled with his family first to Lisbon then to New York. His university studies, both undergraduate and graduate, were completed at Cornell University. It was there that he met his wife Delores who was a student in philosophy. Upon graduation in 1953, Steven took a position at the University of Minnesota. Here he remained throughout his career. In addition to his research contributions, he has authored two books [32] and [46] and directed· several Ph.D. students. Among these are Robert Anderson, Carol Bezuidenhout, Tzuu-Shuh Chiang, Dean Isaacson, Peter March, Timo Seppalainen, Ananda Weerasinghe, Albert Wang and myself. A common theme runs through much of the early work of Steven Orey, namely the ergodic behavior of Markov chains. His work grew to include many related topics in the theory of Markov processes including central limit theorems, renewal theory, tail O"-fields, and large deviations. He also wrote about Gaussian processes as well as control theory and optimization. Still this does not give a complete account of his work as his first interest was logic, a subject on which he * I apologize to those whose work has been overlooked in this account. Omis sions are due solely to my limited knowledge. Finally, Peter March has provided me with valuable assistance. 2 M.CRANSTON has written several papers. However, I would like to confine myself to a summary of what I have referred to as the common theme, the ergodic theory of Markov chains, to related topics and to some of his other work which has been influential. Steven's Ph.D. thesis was written at Cornell in logic under J. B. Rosser. Before graduating he attended lectures by William Feller on one-dimensional diffusions. A year spent at Berkeley early in his career further propelled him in the direction of probability. Evidently, he was inspired by the ideas of Doeblin and this influence appears often in Steven's work. His first paper on probability theory [4] established the central limit theorem for a sequence of m-dependent random variables. The condition put forward there reduces to the usual Lindeberg condition in the case m = 0, a feature that had been lacking in previous theorems of this type. What might fairly be called the principal interest of his career appeared in his next work [5]. Here he is interested in examining the relation between Harris recurrence and Doeblin's condition for Markov chains. Among the main results are a ratio ergodic theorem and a very pretty central limit theorem. The latter concerns the additive functional E~':~ !(Xk) , of values of a given function! of the first n positions of the chain. Doeblin had handled this functional in the case of a recurrent chain by noting that excursions from a given state are independent and identically distributed. Steven observed that with Harris recurrent chains the excursions from a set form a Markov chain satisfying Doeblin's condition and with sufficient independence for a central limit result to hold. Other highlights from his work of the early 1960's are [12], [7] and [11]. The first of these contains (in approximately two pages) perhaps his most well-known result. Here he considers a recurrent, aperiodic, irreducible Markov chain on a countable state space. Using a coupling or cancellation idea due to Doeblin, he proves that the total variation of the difference between the distributions of two copies of the chain at time n, starting from any two probability distributions, goes to zero as n tends to infinity. The clever idea used here is to partition the integers into two classes: those times when the mass from the first distribution exceeds the mass from the second at a fixed state and the complementary set of times. The chain must visit the fixed state infinitely often on one of these sets of