Invariant Measures for Stochastic Reaction-Diffusion Equations

Abstract: We study the long-time behavior of systems governed by nonlinear reaction-diffusion type equations $du = (Au + f(u))dt + \sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $\sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. This equation is known to have a uniformly bounded (in time) solution provided $f(u)$ possesses certain dissipative properties. The existence of a bounded solution implies, in turn, the existence of an invariant measure for this equation, which is an important step in establishing the ergodic behavior of the underlying physical system. In my presentation I will talk about expanding the existing class of nonlinearities $f$ and $\sigma$, for which the invariant measure exists. We also show that the equation has a unique invariant measure if $A$ is a Shrodinger-type operator $A = 1/\rho(div \rho \nabla u)$ where $\rho = e^{-|x|^2}$ is the Gaussian weight. In this case the source of dissipation comes from the operator $A$ instead of the nonlinearity $f$. The main idea is to show that the reaction-diffusion equation has a unique bounded solution, defined for all $t \in \R$, i.e. that can be extended backwards in time. This solution is an analog of the trivial solution for the linear heat equation.