Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-27T07:20:47.616Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Azéma, J., Kaplan-Duflo, M. and Revuz, D. (1967). Mesure invariante sur les classes récurrentes des processus de Markov. Z. Wahrscheinlichkeitsth. 8, 157–181.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Costa, O. L. V. and Dufour, F. (2005a). A sufficient condition for the existence of an invariant probability measure for Markov processes. J. Appl. Prob. 42, 873–878.Google Scholar
Costa, O. L. V. and Dufour, F. (2005b). On the ergodic decomposition for a class of Markov chains. Stoch. Process. Appl. 115, 401–415.Google Scholar
Costa, O. L. V., Fragoso, M. D. and Marques, R. P. (2004). Discrete-Time Markov Jump Linear Systems. Springer, London.Google Scholar
Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23, 1671–1691.Google Scholar
Fragoso, M. D. and Rocha, N. C. S. (2005). Stationary filter for continuous-time Markovian Jump linear systems. SIAM J. Control Optimization 44, 801–815.CrossRefGoogle Scholar
Hernández-Lerma, O. and Lasserre, J. B. (1998). Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math. 54, 99–119.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Prob. 24, 542–574.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993a). Markov Chains and Stochastic Stability. Springer, Berlin.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993b). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993c). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993d). The Doeblin decomposition. In Doeblin and Modern Probability (Contemp. Math. 149), American Mathematical Society, Providence, RI, pp. 211–225.CrossRefGoogle Scholar
Sharpe, M. (1988). General Theory of Markov Processes. Academic Press, London.Google Scholar
Tuominen, P. and Tweedie, R. L. (1979). The recurrence structure of general Markov processes. Proc. London Math. Soc. 39, 554–576.Google Scholar
Tweedie, R. L. (1979). Topological aspects of Doeblin decompositions for Markov chains. Z. Wahrscheinlichkeitsth. 46, 299–305.Google Scholar
Yosida, K. (1980). Functional Analysis, 6th edn. Springer, New York.Google Scholar