Decomposition of Dirichlet Processes and its Application

Abstract

We extend the forward-backward martingale approach to Stratonovich integrals developed by Zheng and Lyons to the general context of Dirichlet spaces. From this perspective, it is clear that the Stratonovich integral of an $L^2$ 1-form against a Dirichlet process is well defined, coordinate invariant, and obeys appropriate chain rules. The paper continues by examining the tightness and continuity of the mapping from Dirichlet forms to probability measures on path space. Some positive results are obtained for a class of infinite-dimensional diffusions.