Regularization by noise for transport and kinetic equations

Luogo: Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"
Ore: 11.00 

• Relatore: Ennio Fedrizzi (Applied Mathematical Department, ENSTA ParisTech)

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

I will first mention a few easy examples (ODEs) to introduce some of these regularising mechanisms, then detail two cases where we have regularisation 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 strategy of proof based on stochastic exponentials and an associated parabolic PDE, which allows to obtain, under even weaker hypothesis, well-posedness for stochastic PDEs in a class of solutions which are only regular in mean.
This will be illustrated by an application to the transport equation.

Referente: Stefano Bonaccorsi