The infinitesimal generator of the stochastic Burgers equation

TL;DR

This paper introduces a martingale approach for singular stochastic PDEs of Burgers type, constructing a new domain for their infinitesimal generators to prove existence and uniqueness results, extending previous work.

Contribution

It develops a novel domain construction for the generators of singular stochastic Burgers equations using paracontrolled distributions, enabling new existence and uniqueness proofs.

Findings

Proved existence and uniqueness for the Kolmogorov backward equation.

Extended the uniqueness of energy solutions to a broader class of equations.

Constructed a domain of controlled functions for the generators.

Abstract

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for "energy solutions" of the stochastic Burgers equation of [GP18a] to a wider class of equations.