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Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)$
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Zhen-Qing Chen and Masatoshi Fukushima

Print publication date: 2011

Print ISBN-13: 9780691136059

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691136059.001.0001

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Basic Properties and Examples of Dirichlet Forms

Basic Properties and Examples of Dirichlet Forms

Chapter:
(p.37) Chapter Two Basic Properties and Examples of Dirichlet Forms
Source:
Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)
Author(s):

Zhen-Qing Chen

Masatoshi Fukushima

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691136059.003.0002

This chapter introduces the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and presents their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. The chapter then presents some basic examples of Dirichlet forms, with special attention paid to their basic properties as well as explicit expressions of the corresponding extended Dirichlet spaces. Hereafter the chapter discusses the analytic potential theory for regular Dirichlet forms, and presents some conditions for the demonstrated Dirichlet form (E,F) to be local.

Keywords:   Dirichlet forms, transience, recurrence, irreducibility, Markovian symmetric operators, Dirichlet spaces, extended Dirichlet spaces, analytic potential theory, regular Dirichlet forms, local properties

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