Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T21:21:58.943Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.Google Scholar
Athreya, K. B. and Jagers, P. (eds) (1972). Classical and Modern Branching Processes. Springer, Berlin.Google Scholar
Athreya, K. B. and Ney, P. E. (1983). Branching Processes. Springer, New York.Google Scholar
Chen, A., Pollett, P., Li, J. and Zhang, H. (2010). Uniqueness, extinction and explosivity of generalised Markov branching processes with pairwise interaction. Methodology Comput. Appl. Prob. 12, 511–531.Google Scholar
Chen, A., Pollett, P., Zhang, H. and Li, J. (2004). The collision branching process. J. Appl. Prob. 41, 1033–1048.Google Scholar
Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, River Edge, NJ.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Kalinkin, A. V. (2002). Markov branching processes with interaction. Russian Math. Surveys 57, 241–304.Google Scholar
Kalinkin, A. V. (2003). On the extinction probability of a branching process with two kinds of interaction of particles. Theory Prob. Appl. 46, 347–352.Google Scholar
Lange, A. M. (2007). On the distribution of the number of final particles in a branching processe with transformations and pairwise interactions. Theory Prob. Appl. 51, 704–714.Google Scholar
Li, J. and Chen, A. (2006). Markov branching processes with immigration and resurrection. Markov Process. Relat. Fields 12, 139–168.Google Scholar
Sevastyanov, B. A. and Kalinkin, A. V. (1982). Random branching processes with interaction of particles. Sov. Math. Dokl. 25, 644–648.Google Scholar