Superlinear stochastic heat equation on $\mathbb{R}^d$

TL;DR

This paper investigates the stochastic heat equation on Euclidean space with Gaussian noise, establishing existence and uniqueness of solutions even with superlinear growth in coefficients, extending prior one-dimensional results.

Contribution

It extends the theory of stochastic heat equations to higher dimensions with superlinear coefficients, providing new existence and uniqueness results.

Findings

Proved existence of solutions in higher dimensions.

Established uniqueness under superlinear growth conditions.

Extended prior one-dimensional results to $\

Abstract

In this paper, we study the stochastic heat equation (SHE) on $\mathbb{R}^d$ subject to a centered Gaussian noise that is white in time and colored in space. We establish the existence and uniqueness of the random field solution in the presence of locally Lipschitz drift and diffusion coefficients, which can have certain superlinear growth. This is a nontrivial extension of the recent work by Dalang, Khoshnevisan and Zhang (2019), where the one-dimensional SHE on $[0,1]$ subject to space-time white noise has been studied.