Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

Abstract

This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation \(\frac{\partial u}{\partial t} - \frac{1}{2}\Delta u = b(u){\dot{W}}\), 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. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function \(\rho \), which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho ({\mathbb {R}}^d)\). In particular, our result covers the parabolic Anderson model (i.e., the case when \(b(u) = \lambda u\)) starting from the Dirac delta measure.