Invariant measures for the nonlinear stochastic heat equation with no drift term

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear stochastic heat equation with Gaussian noise, establishing conditions for the existence of invariant measures in weighted spaces, including the parabolic Anderson model.

Contribution

It identifies specific conditions on initial data and noise correlation that guarantee invariant measures for the nonlinear stochastic heat equation, extending to the parabolic Anderson model.

Findings

Existence of invariant measures under certain conditions.

Applicable to the parabolic Anderson model from Dirac initial data.

Provides moment formulas for analysis.

Abstract

This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function $\rho$, which will together guarantee the existence of an invariant measure in the weighted space $L^2_\rho(\mathbb{R}^d)$. In particular, our result includes the parabolic Anderson model (i.e., the case when $b(u) = \lambda u$) starting from the Dirac delta measure.