Abstract 

Diffusion equations have a smoothing effect on both the initial data and the external forces. Solutions are typically two order of smoothness better than deterministic forcing terms and one order of smoothness better than Ito stochastic forcing terms. In this talk, I'll survey this phenomenon, reaching my latest work on the topic. In this joint work with Pascal Auscher, we show that such smoothing still occurs, even in media that can change in space, time, and the probability variable in an $L^{\infty}$ manner. This is part of quest to understand how noise can regularise fluid mechanics equations. The talk will focus on ideas and directions rather than on techniques (as interesting as they may be to a harmonic analyst).