Initial measures for the stochastic heat equation

TL;DR

This paper studies nonlinear stochastic heat equations driven by space-time white noise, establishing existence of solutions for arbitrary initial measures and demonstrating boundedness under specific conditions.

Contribution

It proves the existence of solutions for a broad class of nonlinear stochastic heat equations with general Lévy generators and provides bounds on moments, including boundedness for certain initial measures.

Findings

Existence of solutions for all finite initial measures.

Tight bounds on moments of solutions.

Solutions are bounded for compactly supported initial measures when the generator is the Laplacian.

Abstract

We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy process on $\R$, and $\sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $\mathcal{L}f=cf"$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.