Chapter Five. Time Changes of Symmetric Markov Processes

2011 ◽  
pp. 166-239
Keyword(s):  
Markov Processes ◽  
Symmetric Markov Processes ◽  
Time Changes

Time Changes of Symmetric Markov Processes

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Processes ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Energy Functional ◽  
Brownian Motions ◽  
Joint Distributions ◽  
Open Set ◽  
Symmetric Markov Processes ◽  
Time Changes

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 = E∖F. 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.

Flux and Lateral Conditions for Symmetric Markov Processes

2008 ◽  
Vol 29 (3) ◽  
pp. 241-269
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Processes ◽  
Symmetric Markov Processes

Boundary Theory for Symmetric Markov Processes

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Processes ◽  
Boundary Theory ◽  
Symmetric Markov Processes ◽  
The Relationship ◽  
Lateral Condition

This chapter proposes a boundary theory for symmetric Markov processes. It begins by investigating the relationship between the space (Fₑ,E) and the space ((ℱ⁰)ref, ℰ 0,ref). Next, the chapter focuses on the restricted spaces ℱ₀∣F, ℱ∣F and their descriptions in terms of the Feller measures U, V, and U α‎ and the Douglas integrals defined by them. The chapter then introduces the lateral condition for the L² generator and studies the case where the set F consists of countably many points that are located in an invariant way under a quasi-homeomorphism. It then turns to one-point extensions and examples of these, and follows up with many-point extensions and their examples as well.

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