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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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Time Changes of Symmetric Markov Processes

Time Changes of Symmetric Markov Processes

(p.166) Chapter Five Time Changes of Symmetric Markov Processes
Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Zhen-Qing Chen

Masatoshi Fukushima

Princeton University Press

This chapter discusses the time change. It first relates the perturbation of the Dirichlet form to a Feynman-Kac transform of X and deals with characterization of the Dirichlet form (Ĕ,̆‎F) of a time-changed process. The chapter next introduces the concept of the energy functional of a general symmetric transient right process, as well Feller measures on F relative to the part process X⁰ of X on the quasi open set E₀ = EF. It derives the Beurling-Deny decomposition of the extended Dirichlet space (̆Fₑ,Ĕ) living on F in terms of the due restriction of E to F with additional contributions by Feller measures. Finally, Feller measures are described probabilistically as the joint distributions of starting and end points of the excursions of the process X away from the set F using an associated exit system. Examples related to Brownian motions and reflecting Brownian motions are also provided.

Keywords:   time changes, time-changed process, energy functional, Feller measures, Beurling-Deny decomposition, Brownian motions, reflecting Brownian motions, perturbed Dirichlet forms

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