On ergodic properties of stochastic PDEs

TL;DR

This paper reviews ergodic properties of stochastic PDEs across various settings, including dissipative, singular, degenerate, and models with degenerate coefficients, highlighting recent results and phase transitions related to noise intensity.

Contribution

It provides a comprehensive overview of ergodic behavior in stochastic PDEs, including new results on the parabolic Anderson model with degenerate noise in higher dimensions.

Findings

Ergodic properties are established under basic dissipative conditions.

Degenerate noise in stochastic Navier-Stokes equations can still lead to ergodicity.

A phase transition in ergodic behavior is observed in the parabolic Anderson model depending on noise intensity.

Abstract

In this note we review several situations in which stochastic PDEs exhibit ergodic properties. We begin with the basic dissipative conditions, as stated by Da Prato and Zabczyk in their classical monograph. Then we describe the singular case of SPDEs with reflection. Next we move to some degenerate (and thus more demanding) settings. Namely we recall some results obtained around 2006, concerning stochastic Navier-Stokes equations with a very degenerate noise. We finish the article by handling some cases with degenerate coefficients. This includes a new result about the parabolic Anderson model in dimension $d\ge 3$, driven by a general class of noises and fairly general initial conditions. In this context, a phase transition is observed, expressed in terms of the noise intensity.