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The fundamental matrix of singularly perturbed Markov chains

Part of: Markov processes General theory of linear operators

Published online by Cambridge University Press:  01 July 2016

Konstantin E. Avrachenkov  and
Jean B. Lasserre
Konstantin E. Avrachenkov*
Affiliation:
University of South Australia
Jean B. Lasserre*
Affiliation:
LAAS-CNRS
*
∗ Postal address: School of Mathematics, University of South Australia, The Levels, SA 5095, Australia.
∗∗ Postal address: 7 Avenue du Colonel Roche, 31 077 Toulouse, Cédex 4, France. Email address: lasserre@laas.fr
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Abstract

We consider a singularly perturbed (finite state) Markov chain and provide a complete characterization of the fundamental matrix. In particular, we obtain a formula for the regular part simpler than a previous formula obtained by Schweitzer, and the singular part is obtained via a reduction process similar to Delebecque's reduction for the stationary distribution. In contrast to previous approaches, one works with aggregate Markov chains of much smaller dimension than the original chain, an essential feature for practical computation. An application to mean first-passage times is also presented.

Keywords

Markov chainsfundamental matrixsingular perturbation

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 47A55: Perturbation theory
Secondary: 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 679 - 697
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Part of this work was done while the second author was visiting the University of South Australia and supported by the Australian Research Council under Grant A49532206.

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