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Extinction and explosion of nonlinear Markov branching processes

Part of: Markov processes

Published online by Cambridge University Press:  09 April 2009

Anthony G. Pakes
Anthony G. Pakes
Affiliation:
School of Mathematics and StatisticsUniversity of Western Australia35 Stirling HighwayCrawley WA [email protected]
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Abstract

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This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.

Keywords

Markov branching processregular Markov processpopulation processexplosioninvariant measure and distribution.

MSC classification

Secondary: 60J75: Jump processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 403 - 428
Copyright
Copyright © Australian Mathematical Society 2007

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