Symmetric Markov Processes
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426 Martin L. Silverstein
Symmetrie Markov Processes
Springer-Verlag Berlin · Heidelberg · NewYork 1974
Prof. Martin L. Silverstein University of Southern California Dept. of Mathematics University Park Los Angeles, CA 90007/USA
Library of Congress Cataloging in Publication Data
1939Silverstein, Martin L Symmetric Markov processes. (Lecture notes in mathematics ; 426)
Bibliography: p.
1. Markov processes. 2. Potential, Theory of. I. Title. II. Series. Lecture notes in mathematics (Berlin) ; 426. 510'.8s [519.2'33]74-22376 QA3.L28 no. 426 [QA274.7]
AMS Subject Classifications (1970): 60J25, 60J45, 60J50 ISBN 3-540-07012-5 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07012-5 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin · Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
DEDICA TED
TO
W. FELLER
Introduction This monograph is concerned with symmetric Markov processes and especially with Dirichlet spaces as a tool for analyzing them. The volume as a whole focuses on the problern of classifying the symmetric submarkovian semigroups which dominate a given one. The main results are contained in Chapter III and especially in Section 20. A modified reflected space is determined by a boundary together with an intensity for jumping to
t:,
t:,
rather than to the dead point.
Every dominating semigroup which is actually an extension is subordinate to at least one modified reflected space.
The extensions subordinate to a
given modified reflected space are classified by certain Dirichlet spaces which live on the appropriate
t:,.
When the intensity for jumping to
t:,
vanishes identically, the subordinate extensions all have the same local generator as the given one.
The most general dominating semigroup
which is not an extension is obtained by first suppressing jumps to the dead point and/or replacing them by jumps within the state space and then taking an extension. Some general theory is developed in Chapter I. A decomposition of the Dirichlet form into killing 11 , 11
11
jumping
11
and "diffusion" is
accomplished in Chapter II. Examples are discussed in Chapter N. Each chapter is prefaced by a short summary. The main prerequisite is familiarity with the theory of martingales as developed by P. A. Meyer and his school. Little is needed from the theory of Markov processes as such, except from the point of view of motivation.
VI
For a treatment of classification theory in the context of diffusions we refer to [20] and [30]. In fact it is M. Fukushima 's
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