Ennio Fedrizzi

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 consider a few easy examples (ODEs) to introduce some of these regularizing mechanisms, then detail two cases where we have regularization for a PDE: the (stochastic) linear transport equation $$\partial_t u(t,x) + b(t,x) \cdot \nabla u(t,x) \Big(+ \nabla u(t,x) \circ \frac{dW_t}{dt} \Big) = 0$$ and a (stochastic) kinetic equation with force term $$\partial_t f(t,x,v) + v \cdot \nabla_x f(t,x,v) + F(x) \cdot \nabla_v f(t,x,v) \Big(+ \nabla_v f(t,x,v) \circ \frac{dW_t}{dt} \Big) = 0.$$ I will present some classical results for these two equations, related to well-posedness and regularity of solutions, that can be obtained under weaker hypothesis in the stochastic setting. For both equations, results are obtained through the analysis of the regularity properties of characteristics: they solve a stochastic differential equation (SDE), which is degenerate for the kinetic equation. We'll see that characteristics are more regular than one could expect: this can be shown using the regularizing effects of an associated parabolic or elliptic (degenerate, for the kinetic equation) PDE. If time allows, I will conclude by discussing some ongoing work on regularization by noise for a nonlinear PDE, the Burgers equation: $$\partial_t u(t,x) + b(t,x) \cdot \nabla u^2(t,x) \Big(+ \nabla u(t,x) \circ \frac{dW_t}{dt} \Big) = 0.$$ These results are from joint works with Franco Flandoli, Benjamin Gess, Enrico Priola and Julien Vovelle.