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Stability of Markovian processes II: continuous-time processes and sampled chains

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Sean P. Meyn  and
R. L. Tweedie
Sean P. Meyn*
Affiliation:
University of Illinois
R. L. Tweedie*
Affiliation:
Colorado State University
*
∗ Postal address: University of Illinois, Coordinated Science Laboratory, 1101 W. Springfield Ave., Urbana, IL 61801, USA.
∗∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.
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Abstract

In this paper we extend the results of Meyn and Tweedie (1992b) from discrete-time parameter to continuous-parameter Markovian processes Φ evolving on a topological space.

We consider a number of stability concepts for such processes in terms of the topology of the space, and prove connections between these and standard probabilistic recurrence concepts. We show that these structural results hold for a major class of processes (processes with continuous components) in a manner analogous to discrete-time results, and that complex operations research models such as storage models with state-dependent release rules, or diffusion models such as those with hypoelliptic generators, have this property. Also analogous to discrete time, ‘petite sets', which are known to provide test sets for stability, are here also shown to provide conditions for continuous components to exist.

New ergodic theorems for processes with irreducible and countably reducible skeleton chains are derived, and we show that when these conditions do not hold, then the process may be decomposed into an uncountable orbit of skeleton chains.

Keywords

IRREDUCIBLE MARKOV PROCESSESERGODICITYRECURRENCERESOLVENTSDIFFUSIONS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 487 - 517
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

This work was commenced at Bond University, and developed there and at the Australian National University, the University of Illinois, and Colorado State University.

Work supported in part by NSF initiation grant #ECS 8910088.

References

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