Nagoya Mathematical Journal

Stochastic calculus over symmetric Markov processes with time reversal

K. Kuwae

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We 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.

Article information

Nagoya Math. J., Volume 220 (2015), 91-148.

Received: 15 November 2012
Revised: 9 November 2013
Accepted: 18 February 2014
First available in Project Euclid: 1 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C25: Dirichlet spaces
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J75: Jump processes

Markov process Dirichlet form Revuz measure martingale additive functionals of finite energy continuous additive functional of zero energy Nakao’s CAF of zero energy Fukushima decomposition semimartingale Dirichlet processes stochastic integral Itô integral Fisk-Stratonovich integral time reversal operator Lyons-Zheng decomposition dual predictable projection progressively additive functional progressively additive functional in the strong sense Feynman-Kac transform Girsanov transform boundary value problem


Kuwae, K. Stochastic calculus over symmetric Markov processes with time reversal. Nagoya Math. J. 220 (2015), 91--148. doi:10.1215/00277630-3335905.

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