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Symmetric Markov Processes PDF

295 Pages·1974·3.82 MB·English
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Lecture Notes ni Mathematics Edited yb .A Dold dna .B Eckmann 624 Martin .L Silverstein Symmetric Markov Processes Springer-Verlag Berlin-Heidelberg - New York 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 Silverstein, Martin L 1939 ~ S2Tmnetric Markov processes. (Lecture notes in mathematics ; 426) Bibliography: p. i. Markov processes. 2. Potential, Theory of. .I Title. II. Series. Lecture notes in mathematics (Berlin) ; 426. QA3.LT8 no. 426 [0~274.7] 510'.8s [519.2'33] 74-22376 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 photo- copying machine or similar means, dna 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. DEDICATED TO W. FELLER Introduction This monograph si concerned with symmetric Markov processes and especially with Dirichlet spaces as tool a rof analyzing them. The volume as a whole focuses on the problem of classiftyhien g symmetric submarkovian semigroups which dominate a given one. The main results are contained in Chapter llI and especially in Section 20. A modified reflected space si determined by a boundary A together with an intensity rof jumping to A rather than to the dead point. Every dominating semigroup which si actually an extension si subordinate to least at one modified reflected space. The extensions subordinate to a given modified reflected space are classified by certain Dirichlet spaces which live on the appropriate .~/ When the intensity rof jumping to vanishes identically, the subordinate extensions lla have the same local generator as the given one. The most general dominating semigroup which si not an extension is obtained by first suppressing jumps to the dead point and/or replacing them byj umps within the state space and then taking an extension. Some genertahle ory si developed in Chapter .I A decomposition of the Dirichlet form into "killlng", "jumping" and "diffusion" is accomplished In Chapter .II Examples are discussed in Chapter .VI Each chapter is prefaced by a short summary. The main prerequisite si familiarity with the theory of martingales as developed by P. A. Meyer and hls school. Little si needed from the theory of Markov processes as such, except from the point of vlew of motivation. Vl For a treatment classification of theory in the context of diffusions we refer to [20] and [30]. nI fact ti si M. Fukushlma's paper [20] that inspired our own research this in area and his influence si apparent throughout the volume. For more information on "sample space constructions" rof extensions of a given process we to refer Freedman's book [52] where current references thteo literature can be found. The expert typing was done by Elsie E. Walker at the University of Southern California. Notations Throughout the volume X si a separable locally compact = Hausdorff space and dx si a Radon measure on X which charges every nonempty open set. The indicator of a set will be denoted both by 1A and I(A). The integral of a function ~ over the set determined by a condition such as "X E .,'/ will be denoted both by ~ X e 1 ~: ~ t t and 4 X(I e )'I ~ . The measure which sA absolutely continuous with t respect to a given measure p and has density ~ will often be represented 0% • /~ . The subcollection of bounded functions ni F__ wall be denoted by F= b. llA functions are real valued. In particular L 2 (dx) or L 2 (X,dx) si the real Hilbert space of square integrable functions on the measure space (X,dx) and Ccom(=X), C0(X_~ are the collections of real valued continuous functions on ~ respectively with compact support and "vanishing at .ytinifni " Questions of measurability are generally taken rof granted-thus functions are usually understood to be measurable with respect to the obvious sigma algebra. Table of Contents .I General Theory_ 1 .i Transience and Recurrence 3 2. Regular Dirichlet Spaces_ 02 3 Some Potential Theory_ 42 4 Construction of Processes 93 5 An Approximate Markov Process 16 6 Additive Functionals_ 96 7 Balayage 87 8 Random Time Change 48 .II Decomposition of the Dirichlet Form 79 9. Potentials ni the Wide Sense 98 .0I The L~vy Kernel 201 .II The Diffusion Form_ 211 .21 Characterization of a and 621 Ill. Structure Theory_ 031 13. Preliminary Formula 331 14. The Reflected Dirichlet Space 341 15. First Structure Theorem 251 16. The Recurrent Case 851 17. Scope of First Structure Theorem 165 18. The Enveloping Dirichlet Space 371 .91 Equivalent Regular Representations 871 20. Second Structure Theorem 381 .12 Third Structure Theorem 216 .VI Examples 022 22. Diffusions with Bounded Scale; No gnilliK 222 23. Diffusions with Bounded Scale; Nontrivial Killing_ 522 24. Unbounded Scale 732 25. Infinitely Divisible Processes 842 26. Stable Markov Chains 452 27. General Markov Chains 258 Chapter I. General Theory This chapter unifies and extends some of the results ni [44] and [46]. In Section I we establish the connection between the submarkov property rof symmetric semigroups P and the contractivity property t rof Dirichlet spaces .)E,F_( This was tsrif discovered by A. Beurling and .; Deny ]i[ but apparently ti was M. Fukushima who tsrif appreciated sti significance rof Markov processes. Also in Section I we introduce what seems to be the appropriate notion of "irreducibility" and we distinguish thter ansient and recurrent cases. We define the extended Dirichlet space F(e ) by completing F relative to the E form alone taht( is, without adding a piece of the standard inner product) and we show that )E,)e(F=( si an honest Hilbert space when (F,E) si transient. nI Section 2 we show how a given Dirichlet space can be transformed a into regular one by introducing an appropriate modification of the state space. Our construction differs only slightly from Fukushima's ni .]12[ Some potential theory rof regular Dirichlet spaces si developed in Section 3 and used ni Section 4 to construct a "decent" Markov process. The main result was tsrif established by Fukushima [22]. Our approach differs from his ni that we avoid Ray resolvents and quasl-homeomorphisms. nI Section 5 we adapt G. A. Hunt's construction of "approximate Markov chains" to our situation. nI Section 6 we introduce various additive functionals, some of which are used to develop a theory of balayage in Section 7. nI Section 8 we study random time change. We show that the time 1.2 changed process si symmetric relative to the "time changing measure" and we identify the time changed Dirlchlet space. One immediate application si that fi (F,E) si recurrent then the constant function 1 si nI the extended space F(e ) and the norm E(I,I) -- . 0 nI particular -(e) si not a Hllbert space which complements the result in Section 1 rof the transient case.

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