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 $$\frac{\partial u}{\partial t} - \frac{1}{2}\Delta u = b(u){\dot{W}}$$ ∂ u ∂ t - 1 2 Δ u = b ( u ) W ˙ , where b is assumed to be a globally Lipschitz continuous function and the noise $${\dot{W}}$$ 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 $$\rho $$ ρ , which together guarantee the existence of an invariant measure in the weighted space $$L^2_\rho ({\mathbb {R}}^d)$$ L ρ 2 ( R d ) . In particular, our result covers the parabolic Anderson model (i.e., the case when $$b(u) = \lambda u$$ b ( u ) = λ u ) starting from the Dirac delta measure.