We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao’s divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized Itô formula.
"Stochastic calculus over symmetric Markov processes without time reversal." Ann. Probab. 38 (4) 1532 - 1569, July 2010. https://doi.org/10.1214/09-AOP516