Symmetric Markov Processes

295 Pages·1974·3.82 MB·English

by  Prof. Martin L. Silverstein (auth.)

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
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© by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany.
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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.