Spatial ergodicity for SPDEs via a Poincaré-type inequality

Abstract

Consider a parabolic stochastic PDE of the form ${\partial }_{t}u=\frac{1}{2}\Delta u+\sigma (u)\eta$, where $u=u(t,x)$ for $t\ge 0$ and $x\in {\mathbb{R}}^d$, $\sigma :\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous and non random, and $\eta$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)\equiv 1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs \cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.

References in corpus (1)

• On strongly Petrovskii's parabolic SPDEs in arbitrary dimension and the stochastic Cahn-Hilliard equation

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