Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains

TL;DR

This paper studies the existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, expanding classes of nonlinearities and operators for which ergodic behavior can be established.

Contribution

It extends the classes of nonlinear functions and elliptic operators for which invariant measures exist, especially in unbounded domains, and proves uniqueness in specific cases.

Findings

Invariant measures exist for broader classes of nonlinearities and operators.

Existence of invariant measures implies ergodic behavior in the studied systems.

Uniqueness of invariant measure shown for equations with Schrödinger-type operators on half space.

Abstract

In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + \sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $\sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions $f$ and $\sigma$ and elliptic operators $A$ for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $A$ is the Shr\"{o}dinger-type operator $A = \frac{1}{\rho}(\text{div} \rho \nabla u)$ where $\rho = e^{-|x|^2}$ is the Gaussian weight.