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 $\partial u /\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, 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.