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 -space obtained by lower-order perturbations.
"Stochastic calculus over symmetric Markov processes with time reversal." Nagoya Math. J. 220 91 - 148, December 2015. https://doi.org/10.1215/00277630-3335905