Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces

TL;DR

This paper establishes precise criteria for ergodicity and stability of Feller semigroups on Polish spaces, with applications to the 2D Navier-Stokes equations under stochastic forcing.

Contribution

It introduces sharp criteria for ergodicity and stability of Feller semigroups, extending their analysis to complex systems like the stochastic 2D Navier-Stokes equations.

Findings

Sharp criteria for ergodicity and asymptotic stability established.

Application to 2D Navier-Stokes equations with degenerate stochastic forcing.

Framework applicable to general Feller semigroups on Polish spaces.

Abstract

We provide some sharp criteria for studying the ergodicity and asymptotic stability of general Feller semigroups on Polish metric spaces. As application, the 2D Navier-Stokes equations with degenerate stochastic forcing will be simply revisited.