scholarly journals Stochastic calculus over symmetric Markov processes with time reversal

2015 ◽  
Vol 220 ◽  
pp. 91-148
Author(s):  
K. Kuwae
Keyword(s):  
Markov Processes ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Stochastic Calculus ◽  
Additive Functional ◽  
Zero Energy ◽  
Additive Functionals ◽  
Stochastic Characterization ◽  
Symmetric Markov Processes

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

scholarly journals Stochastic calculus over symmetric Markov processes with time reversal

2015 ◽  
Vol 220 ◽  
pp. 91-148 ◽  
Author(s):  
K. Kuwae
Keyword(s):  
Markov Processes ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Stochastic Calculus ◽  
Additive Functional ◽  
Zero Energy ◽  
Additive Functionals ◽  
Stochastic Characterization ◽  
Symmetric Markov Processes

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

scholarly journals Errata for Stochastic calculus for symmetric Markov processes

2012 ◽  
Vol 40 (3) ◽  
pp. 1375-1376
Author(s):  
Zhen-Qing Chen ◽  
Patrick J. Fitzsimmons ◽  
Kazuhiro Kuwae ◽  
Tu-Sheng Zhang
Keyword(s):  
Markov Processes ◽  
Stochastic Calculus ◽  
Symmetric Markov Processes

TIME-DEPENDENT DIRICHLET FORMS AND RELATED STOCHASTIC CALCULUS

2004 ◽  
Vol 07 (02) ◽  
pp. 281-316 ◽  
Author(s):  
YOICHI OSHIMA
Keyword(s):  
Continuous Function ◽  
Dirichlet Forms ◽  
Stochastic Calculus ◽  
Time Dependent ◽  
Excessive Function ◽  
Additive Functionals ◽  
Expository Article

Analytic and probabilistic properties of symmetric or non-symmetric Dirichlet forms are well studied. But the processes with parabolic generators are out of the framework of symmetric Dirichlet forms. To cover these cases, we have introduced the time-dependent Dirichlet forms and studied their properties so far. In this expository article, we intend to explain in detail the analytic and probabilistic properties for time-dependent Dirichlet forms parallel to the symmetric Dirichlet forms. New results on a characterization of the minimal α-excessive function dominating a quasi-continuous function as well as the correspondence between additive functionals and smooth mesures are given. In particular, we emphasized the existence of the nontrivial semipolar sets under our settings.

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.

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