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A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

Published online by Cambridge University Press:  01 July 2016

P. K. Pollett  and
A. J. Roberts
P. K. Pollett*
Affiliation:
The University of Queensland
A. J. Roberts*
Affiliation:
The University of Adelaide
*
∗Postal address: Department of Mathematics, The University of Queensland, St. Lucia, QLD 4067, Australia.
∗∗Postal address: Department of Applied Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, SA 5001, Australia.
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Abstract

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

Keywords

INVARIANT MANIFOLDSQUASISTATIONARY DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 111 - 128
Copyright
Copyright © Applied Probability Trust 1990 

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