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July 2010 On ergodicity of some Markov processes
Tomasz Komorowski, Szymon Peszat, Tomasz Szarek
Ann. Probab. 38(4): 1401-1443 (July 2010). DOI: 10.1214/09-AOP513


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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Tomasz Komorowski. Szymon Peszat. Tomasz Szarek. "On ergodicity of some Markov processes." Ann. Probab. 38 (4) 1401 - 1443, July 2010.


Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1214.60035
MathSciNet: MR2663632
Digital Object Identifier: 10.1214/09-AOP513

Primary: 60H15 , 60J25
Secondary: 76N10

Keywords: Ergodicity of Markov families , Invariant measures , passive tracer dynamics , stochastic evolution equations

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 4 • July 2010
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