# 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.