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

This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation \(\frac{\partial u}{\partial t} - \frac{1}{2}\Delta u = b(u){\dot{W}}\) $\frac{\partial u}{\partial t}-\frac{1}{2}\mathrm{\Delta }u=b(u)\dot{W}$ , where b is assumed to be a globally Lipschitz continuous function and the noise \({\dot{W}}\) $\dot{W}$ 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 \(\rho \) $\rho$ , which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho ({\mathbb {R}}^d)\) $L_{\rho }^2({\mathbb{R}}^d)$ . In particular, our result covers the parabolic Anderson model (i.e., the case when \(b(u) = \lambda u\) $b(u)=\lambda u$ ) starting from the Dirac delta measure.