Seminar on Stochastic Processes, 1981 PDF
Preview Seminar on Stochastic Processes, 1981
Progress in Probability and Statistics Vol. 1 Edited by Peter Huber Murray Rosenblatt Birkhauser Boston •Basel •Stuttgart Seminar on Stochastic Processes, 1981 E. <:inlar, K.L. Chung, R.K. Getoor, editors 1981 Birkhauser Boston • Basel • Stuttgart Editors: E. ~inlar Department of Mathematics Northwestern University Evanston, Illinois 60201 K.L. Chung Department of Mathematics Stanford University Standford, California 94305 R.K. Getoor Department of Mathematics University of California La Jolla, California 92093 LC 81-71608 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, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. ISBN-13: 978-0-8176-3072-0 e-ISBN-13: 978-1-4612-3938-3 DOl: 10.1007/978-1-4612-3938-3 © Birkhauser Boston. 1981 Softcover reprintofthehardcover 1stedition 1981 TABLE OF CONTENTS K.L. CHUNG and K.M. RAO. Feynman-Kac functional and the Schrodinger equation 1 R.K. GETOOR and M.J. SHARPE. Two results on dual excursions 31 F.B. KNIGHT. Characterization of Levy measures of inverse local times of gap diffusion 53 J.W. PITMAN. Levy systems and path decompositions 79 A.O. PITTENGER. Regular birth and death times 111 Z.R. POP-STOJANOVIC and K.M. RAO. Some results on energy 135 J. WALSH and W. WINKLER. Absolute continuity and the fine topology 151 E. ~INLAR and J. JACOD. Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures 159 FOREWORD This volume consists of about half of the papers presented during a three-day seminar on stochastic processes held at Northwestern University in April 1981. The aim of the seminar was to bring together a small group of kindred spirits working on stochastic processes and to provide an informal atmosphere for them to discuss their current work. We plan to hold such a seminar once a year, with slight variations in emphasis to reflect the changing concerns and interests within the field. The invited participants in this year's seminar were J. AZEMA, R.M. BLUMENTHAL, R. CARMONA, K.L. CHUNG, R.K. GETOOR, J. JACOD, F. KNIGHT, S.OREY, A.O. PITTENGER, J. PITMAN, P. PROTTER, M.K. RAO, M. SHARPE, and J. WALSH. We thank them and other participants for the productive liveliness of the seminar. As mentioned above, the present volume is only a fragment of the work discussed at the seminar, the other papers having been already committed to other publications. The seminar was made possible through the enlightened support of the Air Force Office of Scientific Research, Grant No. 80-0252. We are grateful to them as well as the publisher, Birkhauser Boston, for their support and encouragement. E·9· Evanston, 1981 FEYNMAN-KAC FUNCTIONAL AND THE SCHRODINGER EQUATION* by K.L. CHUNG and K.M. RAO The Feynman-Kac formula and its connections with classical analy- sis were inititated in [3J. Recently there has been a revival of interest in the associated probabilistic methods, particularly in applications to quantum physics as treated in [7J. Oddly enough the inherent potential theory has not been developed from this point of view. A search into the literature after this work was under way un- covered only one paper by Khas'minskii [4J which dealt with some rele- vant problems. But there the function q is assumed to be nonnegative and the methods used do not apparently apply to the general case; see the remarks after Corollary 2 to Theorem 2.2 below~* The case of q taking both signs is appealing as it involves oscillatory rather than absolute convergence problems. Intuitively, the Brownian motion must make intricate cancellations along its paths to yield up any determin- able averages. In this respect Theorem 1.2 is a decisive result whose significance has yet to be explored. Next we solve the boundary value problem for the Schrodinger equation (~+2qJP =O. In fact, for a *Research supported in part by a NSF Grant MCS-8001540. **The case q S 0 is "trivial" in the context of this paper. For this case in a more general setting see [9, Chapter 13J. 1 2 K.L. CHUNG and K.M. RAO positive continuous boundary function f, a solution is obtained in the explicit formula given in (2) of §l below, provided that this quantity is finite (at least at one point x in D). Thus the Feynman- Kac formula supplies the natural Green's operator for the problem. For a domain with finite measure, the result is the best possible as it in- cludes the already classical solution of the Dirichlet problem by prob- ability methods. Other results are valid for an arbitrary domain and it seems that some of them are proved here under less stringent condi- tions than usually given in non-probabilistic treatments. For instance, no condition on the smoothness of the boundary is assumed beyond that of regularity in the sense of the Dirichlet problem, and the basic results hold without this regularity. Of course, the Schrodinger equa- tion is a case of elliptic partial differential equations on which there exists a huge literature, but we make no recourse to the latter theory. Comparisons between the methods should prove worthwhile and will be discussed in a separate publication. It is well known that the Schrodinger equation differs essentially from the Laplace equation in that a condition on the size of the domain is necessary to guarantee the uniqueness of solution. In our context it is evident at the outset that the key to this is the quantity uD(x) =EX{ exp(!oTDq(x(t» dt) }, the finiteness of which lies at the base of the probabilistic considerations. As natural as it is from our point of view, this quantity does not lend itself easily to non- probabilistic analysis. The identification in the simplest case (see the remark after Lemma D) as a particular solution of the equation is one of those amusing twists not uncommon in other theories when dealing with an object which has really a simple probabilistic existence. The one-dimensional case of this investigation has appeared in [1] though the orientation is somewhat different there. Asummary of FEYNMAN-KAC FUNCTIONAL 3 the present results has been announced in [2]. 1. Harnack inequality; global bound; boundary limit d Let { X(t), t ~ 0 } be the Brownian motion process in R , d ~ 1; with all paths continuous. The transition semigroup is { Pt , t ~ 0 and Ft is the a-field generated by { Xs' o :$; s :$; t } and augmented in the usual way. The qualifying phrase "almost surely" (a.s.) will be omitted when readily understood. A "set" is always a Borel set and a "function" is always a Borel measurable function. The class of bounded functions will be denoted by bB; if its domain is A this is indi- cated by bB(A). Similarly for other classes of functions to be used later. The sup-norm of f E bB is denoted by IIfll ; restricted to A it is denoted by IIfIIA. pX and EX denote the probability and expec- tation for the process starting at x. For any set B we put inf{ t > 0 I X(t) t B } namely the first exit time from B, with the usual convention that inf ~ = 00. Let q EbB; as an abbreviation we put t f (1) exp{ q(X(s)) ds } o when q is fixed it will be omitted from the notation. A domain in Rd is an open connected set; its boundary is aD =D n DC, where D is the closure and DC the complemen~ of D. For f ~ 0 on aD we put for xED: (2) u(q,f;x)
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