Boundary Theory for Symmetric Markov Processes
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516 Martin L. Silverstein
Boundary Theoryfor Symmetric Markov Processes
Springer-Verlag Berlin. Heidelberg. New York 19?6
Author Martin L. Silverstein Department of Mathematics University of Southern California University Park Los Angeles, California 90007 USA
Library of Congress Ca~alogiag ia pdblieation Data
Silverstein, Martin /. 1939Boundary theory for symmetric Markov processes. (Lecture notes in mathematics ; 516) Bibliography: p. Includes index. i. Markov processes. 2. Semigroups. 3. Symmetry groups. I. Title. If. Series: Lecture notes in mathematics "Berlin) ; 516. QA3.L28 no. 516 [QA274.7] 510'.8s [519.2'33] 76-10683
AMS Subject Classifications (1970): 60J 25, 60J 45, 60J 50
ISBN 3-540-07688-3 Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-07688-3 Springer-Verlag New York Heidelberg 9 Berlin 9 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 w 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. 9by Springer-Verlag Berlin - Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Dedicated to the m e m o r y JOSEPH
of m y father
SILVERSTEIN
Introduction. Let This means
Pt' t > 0 be a submarkovian
that each
Pt
semigroup
maps bounded measurable
on a measurable
space
X.=
functions into bounded meas-
urable functions and that
(0.1)
O O.
A l s o i t is u s u a l l y n e c e s s a r y
to impose
h o p e t o do a n y s e r i o u s w o r k .
(0.3) where the
whenever
O P t f whenever f>_ O.
This volume is concerned with the general problem tent classifying markovian
submarkovian
sernigroup
Pt'
semigroups
case when both the
(0.6)
dx
o n X.
Pt
Pt'
of a n a l y z i n g a n d t o s o m e e x -
t > O, w h i c h d o m i n a t e a f i x e d s u b -
t > O.
There are good technical
measure
the first if
and
Pt
reasons
for restricting
a~e symmetric
with respect
This meansthat
IdxPtf(x)g(x) = Idxf(x)ptg(x)
attention to the special to a g i v e n r e f e r e n c e
VI
at least w h e n
f, g are bounded and integrable.
The restriction will be in effect
throughout the volume. Once s y m m e t r y
is imposed, it is convenient to modify the regularity
condition (0. 3) and (0.4).
In place of (0. 3) w e a s s u m e that each
with respect to bounded almost everywhere convergence. symmetry
condition (0.6), guarantees that each
s y m m e t r i c contraction on the Hilbert space a s s u m e that the extended operators
This, together with the
Pt extends uniquely to a bounded
L 2(X, dx).
In place of (0.4) w e
Pt f o r m a semigroup which is continuous
relative to the strong operator topology on LZ(X,dx).
(0.4')
Pt iS continuous
This m e a n s that
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