Exponential ergodicity of Stochastic Evolution Equations with reflection

TL;DR

This work establishes exponential ergodicity for stochastic evolution equations with reflection in an infinite-dimensional ball by developing a coupling framework with a distance-like metric and $d$-small sets, yielding a unique invariant measure and exponential convergence in a Wasserstein-type distance $W_d$. The method hinges on a Lyapunov function and contraction properties over suitable time windows to satisfy a set of hypotheses that ensure ergodicity. The authors prove the Feller property and existence of invariant measures for the reflected SEE, and then apply the framework to the reflected stochastic Navier–Stokes equations in two dimensions, obtaining explicit exponential convergence rates to the invariant measure. The results provide a rigorous approach to ergodicity in SPDEs with reflection and deliver concrete implications for stochastic fluid dynamics models with hard-wall constraints.

Abstract

In this paper, we establish an exponential ergodicity for stochastic evolution equations with reflection in an infinite dimensional ball. As an application, we obtain the exponential ergodicity of stochastic Navier-Stokes equations with reflection. A coupling method plays an important role.

Theorems & Definitions (23)

• Definition 2.1
• Proposition 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 4.1
• Remark 4.1
• Definition 4.1
• Definition 4.2