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On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Brenton Gray ,
Phil Pollett  and
Hanjun Zhang
Brenton Gray*
Affiliation:
University of Queensland
Phil Pollett*
Affiliation:
University of Queensland
Hanjun Zhang*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
∗Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
∗Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
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Abstract

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Let S be a countable set and let Q = (qij, i, j ∈ S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (mj, j ∈ S) such that ∑i∈Smiqij = −μmj, j ≠ b. We prove that there exists a Q-process P(t) = (pij(t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑i∈Smipij(t) = e−μtmj, j∈S. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.

Keywords

Markov chainq-matrixbirth–death processconstruction theory

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 713 - 725
Copyright
© Applied Probability Trust 2005 

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