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Ergodic properties and ergodic decompositions of continuous-time Markov processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

O. L. V. Costa  and
F. Dufour
O. L. V. Costa*
Affiliation:
Escola Politécnica da Universidade de São Paulo
F. Dufour*
Affiliation:
Université Bordeaux I
*
∗Postal address: Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica da Universidade de São Paulo, CEP 05508 900, São Paulo, Brazil. Email address: oswaldo@lac.usp.br
∗∗Postal address: Mathématiques Appliqées de Bordeaux, Université Bordeaux I, 351 cours de la Liberation, 33405 Talence Cedex, France. Email address: dufour@math.u-bordeaux1.fr
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Abstract

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In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named the T'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

Keywords

Markov processcontinuous timeinvariant probability measurelimit kernelFoster-Lyapunov criterionergodic decompositionDoeblin decompositionYosida decomposition

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 767 - 781
Copyright
© Applied Probability Trust 2006 

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