Regularization by noise and flows of solutions for a stochastic heat equation

TL;DR

This paper proves the existence and uniqueness of solution flows for a stochastic heat equation with non-Lipschitz drift and space-time white noise, extending regularization by noise results to SPDEs.

Contribution

It establishes the existence, uniqueness, and path-by-path uniqueness of solutions for a stochastic heat equation with irregular drift, extending prior SDE results to SPDEs.

Findings

Existence and uniqueness of solution flows for the SPDE.

Path-by-path uniqueness for solutions with certain initial conditions.

Extension of regularization by noise phenomena to stochastic partial differential equations.

Abstract

Motivated by the regularization by noise phenomenon for SDEs we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation $$\frac{\partial u}{\partial t}=\frac12\frac{\partial^2 u}{\partial z^2} + b(u(t,z)) + \dot{W}(t,z), $$ where $\dot W$ is a space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof we also establish the so-called path--by--path uniqueness for any initial condition in a certain class on the same set of probability one. This extends recent results of Davie (2007) to the context of stochastic partial differential equations.