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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 University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691136059.001.0001

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date: 13 December 2017

Reflected Dirichlet Spaces

Reflected Dirichlet Spaces

Chapter:
(p.240) Chapter Six Reflected Dirichlet Spaces
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/9780691136059.003.0006

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. The chapter next establishes several equivalent notions of reflected Dirichlet space (ref,ref) for a regular transient Dirichlet form (E,F). One of these equivalent notions is then used to define reflected Dirichlet space for a regular recurrent Dirichlet form. Moreover, the chapter gives yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms. Various concrete examples of reflected Dirichlet spaces are also exhibited for regular Dirichlet forms. Finally, the chapter defines a Silverstein extension of a quasi-regular Dirichlet form (E,F) on L²(E; m) and investigates the equivalence of analytic and probabilistic concepts of harmonicity.

Keywords:   reflected Dirichlet spaces, terminal random variables, harmonic functions, regular recurrent Dirichlet forms, Silverstein extension, quasi-regular Dirichlet forms, harmonicity, transient regular Dirichlet forms

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