Rough Burgers-like equations with multiplicative noise

TL;DR

This paper develops a framework for solving one-dimensional Burgers equations perturbed by multiplicative space-time white noise using rough path theory, establishing existence, uniqueness, and stability of solutions despite the noise's irregularity.

Contribution

It introduces a novel application of controlled rough paths to define and analyze solutions to stochastic Burgers equations with multiplicative noise.

Findings

Existence and uniqueness of solutions under the rough path framework

Solutions are stable under smooth noise approximations

The approach handles the irregularity of space-time white noise

Abstract

We construct solutions to Burgers type equations perturbed by a multiplicative space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.