Abstract

Regularity structures, paracontrolled distributions and all that provide pathwise, deterministic tools to solve and study singular stochastic PDEs over finite-dimensional spaces. From a probabilistic point of view we would also like to understand the associated Kolmogorov backward equations, which can be interpreted as infinite-dimensional singular SPD Es. I will discuss on the example of the conservative stochastic Burgers equation how to construct a space of (para-) paracontrolled distributions in which the backward equation is well posed. As an application we obtain a martingale formulation and an alternative proof for the well-posedness of “energy solutions”, without using the Cole-Hopf transform. The approach extends to some other singular SPD Es with Gaussian invariant measures and quadratic nonlinearities. This is joint work with Massimiliano Gubinelli.