On a high-dimensional nonlinear stochastic partial differential equation

TL;DR

This paper studies a complex high-dimensional nonlinear stochastic PDE driven by space-correlated Gaussian noise, establishing the existence of weak solutions, and extends known results from the one-dimensional stochastic Burgers equation to higher dimensions.

Contribution

It introduces an approximation method to prove the existence of weak solutions for a high-dimensional nonlinear SPDE with non-Lipschitz coefficients, generalizing the stochastic Burgers equation.

Findings

Existence of weak solutions for the high-dimensional nonlinear SPDE.

Extension of stochastic Burgers equation results to arbitrary dimensions.

Handling of non-Lipschitz noisy coefficients in the analysis.

Abstract

In this paper we investigate a nonlinear stochastic partial differential equation (spde in short) perturbed by a space-correlated Gaussian noise in arbitrary dimension $d\geq1$, with a non-Lipschitz coefficient noisy term. The equation studied coincides in one dimension with the stochastic Burgers equation. Existence of a weak solution is established through an approximation procedure.