On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

TL;DR

This paper introduces a flexible framework using asymptotic coupling to prove the uniqueness of invariant measures in nonlinear stochastic PDEs, including parabolic and hyperbolic types, and offers a new method for establishing invariant measures.

Contribution

It presents a novel approach based on asymptotic coupling for proving uniqueness of invariant measures and a simple framework for existence in challenging cases.

Findings

Asymptotic coupling effectively proves uniqueness of invariant measures.

The framework applies to both parabolic and hyperbolic stochastic PDEs.

A new method for establishing invariant measures when traditional approaches fail.

Abstract

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.