Quasilinear SPDEs via rough paths

In this talk I will present a new approach to solve singular stochastic PDE which extends directly Gubinelli’s notion of controlled rough paths and is also closely related to Hairer’s theory of regularity structures. The approach is implemented for the variable-coefficient uniformly parabolic PDE

$\large \partial_2 u - a(u)\partial_1^2u - \sigma(u)f =0,$

where $f$ is an irregular random distribution. The assumptions allow, for example, for an $f$ which is white in time and only mildly coloured in space.

The key result is a deterministic stability result (in the spirit of the Lyons-Itô map) for solutions of this equation with respect to $f$ but also the products $vf$ and $\inline v\partial_1v$, with  solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2 v=f$. On the stochastic side it is shown how these (renormalised) products can be constructed for a random $f$.

This talk is based on joint work with F. Otto.

This talk is part of the Probability series.