Regularization by noise for transport and kinetic equations

For some differential equations the addition of a carefully chosen, random noise term can produce a regularizing effect (e.g. solutions are more regular, or restored uniqueness).

I will first mention a few easy examples (ODEs) to introduce some of these regularizing mechanisms, then detail two cases where we have regularization for a PDE: the linear transport equation and a kinetic equation with force term. I will present some classical results for these two equations, related to well-posedness and regularity of solutions, that in the stochastic setting can be obtained under weaker hypothesis. These results are based on a careful analysis of the stochastic characteristics and the regularising properties of some associated parabolic/elliptic PDE.

If time allows, I will conclude by introducing a new different strategy of proof based on stochastic exponentials and an associated parabolic PDE, which allows to obtain wellposedness for stochastic PDEs in a class of solutions which are only regular in mean. This will illustrated by the application to the transport equation.

This work is supported by LABEX MILYON / ANR-10-LABX-0070.