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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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Symmetric Markovian Semigroups and Dirichlet Forms

Symmetric Markovian Semigroups and Dirichlet Forms

(p.1) Chapter One Symmetric Markovian Semigroups and Dirichlet Forms
Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Zhen-Qing Chen

Masatoshi Fukushima

Princeton University Press

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.

Keywords:   symmetric Markovian semigroups, Dirichlet forms, Hausdorff topological space, quasi-regular Dirichlet forms, quasi-homeomorphism, Dirichlet spaces, m-tight special Borel

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