Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

TL;DR

This paper establishes existence and uniqueness of solutions for a stochastic heat equation with distributional drift, including skew equations, using a novel approach based on the stochastic sewing lemma.

Contribution

It introduces a framework for solving stochastic heat equations with distributional drifts, extending previous results to more singular cases and including skew equations.

Findings

Existence and uniqueness of strong solutions under specified conditions.

Existence of weak solutions for more singular drifts.

Extension of classical results to distributional and measure-valued drifts.

Abstract

We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=\delta_0$ which corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gy\"ongy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in L\^e~(2020).