On ergodicity of some Markov processes

TL;DR

This paper establishes criteria for the existence, uniqueness, and ergodic behavior of invariant measures for certain Markov processes in Polish spaces, with applications to stochastic evolution equations and passive tracer models.

Contribution

It introduces a new criterion based on the e-property for invariant measure existence and weak ergodicity, applied to stochastic evolution equations in Hilbert spaces.

Findings

Proves existence and uniqueness of invariant measures under specified conditions.

Establishes weak-* ergodicity for Markov processes with the e-property.

Derives a law of large numbers for passive tracer trajectories.

Abstract

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.