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Extinction Probability of Interacting Branching Collision Processes

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2016

Anyue Chen ,
Junping Li ,
Yiqing Chen  and
Dingxuan Zhou
Anyue Chen*
Affiliation:
University of Liverpool and Xian Jiaotong-Liverpool University
Junping Li*
Affiliation:
Central South University
Yiqing Chen*
Affiliation:
University of Liverpool
Dingxuan Zhou*
Affiliation:
City University of Hong Kong
*
∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK.
∗∗∗ Postal address: School of Mathematical Science and Computing Technology, Central South University, Changsha City, Hunan 410075, China. Email address: jpli@mail.csu.edu.cn
∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK.
∗∗∗∗∗ Postal address: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. Email address: mazhou@cityu.edu.hk
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Abstract

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We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

Keywords

Markov branching processcollision branching processinteractionuniquenessregularityextinction probabilityextinction time

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 226 - 259
Copyright
© Applied Probability Trust 

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