Local Operators and Markov Processes
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Mathematisches Institut der Universit~it Erlangen-NLirnberg Advisers: H. Bauer and K. Jakobs
816 Lucretiu Stoica
Local Operators and Markov Processes
Springer-Verlag Berlin Heidelberg New York 1980
Author Lucretiu Stoica Department of Mathematics, INCREST Bdul. Pacii 220 77538 Bucharest/Romania
AMS Su bject Classifications (1980): 31 D 05, 60 J 25, 60 J 35, 60 J 40, 60J45 ISBN 3-540-10028-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10028-8 Springer-Verlag NewYork Heidelberg Berlin Libraryof Congress Cataloging in PublicationData.Stoic& Lucretiu, 1949- Local operators and Markov processes.(Lecture notes in mathematics;816) Bibliography:p. Includes index. 1. Markov processes.2. Operatortheory.3. Potential,Theory of. I. Title. II. Series: Lecture notes in mathematics(Berlin); 816. QA3.L28 no. 816 [QA274.7] 510s [519.2'33] 80-20217 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 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Introduction The p r e s e n t local
character
subject
has
such that
Meyer
been
treated
functions
and c o m p l e t e d
[19],
I.Cuculescu
A.Cornea
[13].
The c o n v e r s e
process,
was
potential
theoretic
T h e n J.C. each
first
Taylor
continuous
Feller,
yields
and A . C o r n e a type
locally
a harmonic
classical
introduced
and the k e r n e l s
by G . M o k o b o d z k i [23],
from the a x i o m a t i c
and D . S i b o n y Roth
table base. open
Similar
is a s s u m e d
sets
potential
to e l l i p t i c
(in [23]
) was
In C h a p t e r
space X such
transition
function
is unique.
was
a local o p e r a t o r
between
this n o t i o n
boundary
object
space,
value
out
by G . L u m e r
studied here
X, with
operators
our
a coun-
local ope-
and to have a base of
differs
a continuous
that its c h a r a c t e r i s t i c
On a
t h e o r y was n o i n t e d
the first to c o n s i d e r
I we c o n s t r u c t
of
to
operators.
for the P o i s s o n - D i r i c h l e t approach
theory
of a local o p e r a t o r
theoretic
principle
that
for the d i f f e r e n t i a l
considered
differential
to obey a m a x i m u m
but the spirit of the p r e s e n t state
were
[14].
is strong
potential
21. F u r t h e r
L, on a locally c o m p a c t
that are r e g u l a r
that G . L u m e r
onerators
[34].
axiomatic
[5] p r o v e d
He a s s o c i a t e d
potential
Markov
of C . C o n s t a n t i n e s c u
The r e l a t i o n
[31] Th.
to local
~24] and J.P. The m a i n
process.
continuous
ker
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