Nonnegative solutions to stochastic heat equation with nonlinear drift

TL;DR

This paper constructs nonnegative solutions for a one-dimensional stochastic heat equation with nonlinear drift and multiplicative noise, expanding understanding of solution existence under nonlinear conditions.

Contribution

It introduces a method to establish nonnegative solutions for SPDEs with nonlinear drift and noise, which was previously less understood.

Findings

Successfully constructed nonnegative solutions

Extended solution existence theory to nonlinear drift cases

Provided conditions for solution positivity

Abstract

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and $\sigma:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function with suitable property. We will construct nonnegative solutions to such SPDEs.