Annals of Probability

On ergodicity of some Markov processes

Tomasz Komorowski, Szymon Peszat, and Tomasz Szarek

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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.

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Ann. Probab., Volume 38, Number 4 (2010), 1401-1443.

First available in Project Euclid: 8 July 2010

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]

Ergodicity of Markov families invariant measures stochastic evolution equations passive tracer dynamics


Komorowski, Tomasz; Peszat, Szymon; Szarek, Tomasz. On ergodicity of some Markov processes. Ann. Probab. 38 (2010), no. 4, 1401--1443. doi:10.1214/09-AOP513.

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