Stochastic regularization effects of semi-martingales on random functions

Since the late 70’s it is well-known that the addition of a random force in an ill-posed ODE may bring back the system well-posed. This kind of phenomenon is known as "stochastic regularization effect" or "regularization by noise". A breakthrough in this domain is contained in the paradigm known under the name of "The Itô-Tanaka trick" which links the time average of a non-smooth map $f$ along the solution of an SDE with the solution of a Fokker-Planck PDE. However, this approach is restricted to deterministic mappings $f$. The aim of this talk is to go beyond this limitation. More precisely, we propose an extended version of the Itô-Tanaka trick to random mappings $f$. This talk is based on a joint work with Romain Duboscq.