Regularisation by multiplicative noise for reaction-diffusion equations

TL;DR

This paper proves the existence and uniqueness of solutions for a stochastic reaction-diffusion equation driven by multiplicative white noise with irregular drift, using advanced stochastic analysis techniques.

Contribution

It introduces a novel approach combining stochastic sewing and Malliavin calculus to handle irregular drift in stochastic PDEs.

Findings

Unique solution existence established for the equation.

Applicable to drifts with regularity index greater than -1.

Method extends analysis of stochastic PDEs with distributional coefficients.

Abstract

We consider the stochastic reaction-diffusion equation in $1+1$ dimensions driven by multiplicative space-time white noise, with a distributional drift belonging to a Besov-H\"older space with any regularity index larger than $-1$. We assume that the diffusion coefficient is a regular function which is bounded away from zero. By using a combination of stochastic sewing techniques and Malliavin calculus, we show that the equation admits a unique solution.