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The collision branching process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Anyue Chen ,
Phil Pollett ,
Hanjun Zhang  and
Junping Li
Anyue Chen*
Affiliation:
University of Greenwich
Phil Pollett*
Affiliation:
University of Queensland
Hanjun Zhang*
Affiliation:
University of Queensland
Junping Li*
Affiliation:
University of Greenwich
*
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK
∗∗∗ Postal address: Department of Mathematics, University of Queensland, QLD 4072, Australia.
∗∗∗ Postal address: Department of Mathematics, University of Queensland, QLD 4072, Australia.
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK
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Abstract

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Keywords

Regularityextinction probabilityhitting time

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1033 - 1048
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anderson, W. (1991). Continuous-Time Markov Chains. An Applications-Oriented Approach. Springer, New York.Google Scholar
Asmussen, S., and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.Google Scholar
Athreya, K., and Jagers, P. (1997). Classical and Modern Branching Processes. Springer, Berlin.Google Scholar
Athreya, K., and Ney, P. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Chen, A., and Renshaw, E. (1993). Recurrence of Markov branching processes with immigration. Stoch. Process. Appl. 45, 231–242.Google Scholar
Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore.CrossRefGoogle Scholar
Cottingham, W., and Greenwood, D. (2001). An Introduction to Nuclear Physics, 2nd edn. Cambridge University Press.Google Scholar
Daley, D. J., and Kendall, D. G. (1965). Stochastic rumours. J. Inst. Math. Appl. 1, 42–55.CrossRefGoogle Scholar
Ezhov, I. I. (1980). Branching processes with group death. Theory Prob. Appl. 25, 202–203.Google Scholar
Harris, T. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Kalinkin, A. V. (1982). Extinction probability of a branching process with interaction of particles. Theory Prob. Appl. 27, 201–205.CrossRefGoogle 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.CrossRefGoogle Scholar
Maki, D. P., and Thompson, M. (1973). Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Rogers, D., and Hassell, M. P. (1974). General models for insect parasite and predator searching behaviour-interference. J. Animal Ecology 43, 239–253.CrossRefGoogle Scholar
Sevast′yanov, B. A. (1949). On certain types of Markov processes. Usp. Mat. Nauk 4, 194 (in Russian).Google Scholar
Wang, Z. K., and Yang, X. Q. (1992). Birth and Death Processes and Markov Chains. Springer, Berlin.Google Scholar
Yang, X. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.Google Scholar