Stochastic regularization effects of semi-martingales on random functions

TL;DR

This paper extends the Itô-Tanaka trick to random functions by developing new results on Fokker-Planck SPDEs and BSPDEs, addressing an open question in stochastic regularization effects.

Contribution

It introduces a generalized Itô-Tanaka trick for random functions and provides new theoretical results on Fokker-Planck SPDEs and BSPDEs.

Findings

Extended the Itô-Tanaka trick to random mappings

Derived new results on Fokker-Planck SPDEs and BSPDEs

Addressed an open question in stochastic regularization

Abstract

In this paper we address an open question formulated in [17]. That is, we extend the It{\^o}-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker-Planck equation associated to X, to random mappings f. To this end we provide new results on a class of adpated and non-adapted Fokker-Planck SPDEs and BSPDEs.