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

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 $$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial z^2}+b(u(t,z))+\dot{W}(t,z),$$ 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.

46 pages

Cited by in corpus (1)

• Well-posedness by noise for scalar conservation laws