Stochastic calculus over symmetric Markov processes without time reversal

TL;DR

This paper advances stochastic calculus for symmetric Markov processes by removing the need for time reversal, enabling generalized Fukushima decompositions and Itô formulas under broader conditions.

Contribution

It introduces a refined stochastic calculus framework for symmetric Markov processes without time reversal, extending key functional decompositions and stochastic integrals.

Findings

Extended Nakao's divergence-like functionals without time reversal

Established generalized Fukushima decomposition for new class of functions

Derived a generalized Itô formula for symmetric Markov processes

Abstract

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 It\^{o} formula. (With Errata.)