Generalized couplings and ergodic rates for SPDEs and other Markov models

TL;DR

This paper develops verifiable conditions for exponential or subexponential ergodicity of Markov processes, including nonlinear SPDEs like 2D stochastic Navier-Stokes, using a new generalized coupling method.

Contribution

It introduces a new generalized coupling approach and provides verifiable criteria for ergodicity of Markov processes lacking the strong Feller property.

Findings

Established conditions for ergodicity of nonlinear SPDEs.

Proved exponential ergodicity for 2D stochastic Navier-Stokes equations.

Developed a new version of the generalized coupling method.

Abstract

We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.