On ergodicity of some Markov processes

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-${}^{\ast }$ 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-${}^{\ast }$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations

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• Mean ergodic theorems on norming dual pairs
• Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations