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The collision branching process

  • Anyue Chen (a1), Phil Pollett (a2), Hanjun Zhang (a2) and Junping Li (a1)

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.

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Corresponding author

Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK
∗∗ Email address: a.chen@greenwich.ac.uk
∗∗∗ Postal address: Department of Mathematics, University of Queensland, QLD 4072, Australia.
∗∗∗∗ Email address: pkp@maths.uq.edu.au
∗∗∗∗∗ Email address: hjz@maths.uq.edu.au
∗∗∗∗∗∗ Email address: j.li@greenwich.ac.uk

References

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[1] Anderson, W. (1991). Continuous-Time Markov Chains. An Applications-Oriented Approach. Springer, New York.
[2] Asmussen, S., and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.
[3] Athreya, K., and Jagers, P. (1997). Classical and Modern Branching Processes. Springer, Berlin.
[4] Athreya, K., and Ney, P. (1972). Branching Processes. Springer, Berlin.
[5] Chen, A., and Renshaw, E. (1993). Recurrence of Markov branching processes with immigration. Stoch. Process. Appl. 45, 231242.
[6] Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore.
[7] Cottingham, W., and Greenwood, D. (2001). An Introduction to Nuclear Physics, 2nd edn. Cambridge University Press.
[8] Daley, D. J., and Kendall, D. G. (1965). Stochastic rumours. J. Inst. Math. Appl. 1, 4255.
[9] Ezhov, I. I. (1980). Branching processes with group death. Theory Prob. Appl. 25, 202203.
[10] Harris, T. (1963). The Theory of Branching Processes. Springer, Berlin.
[11] Kalinkin, A. V. (1982). Extinction probability of a branching process with interaction of particles. Theory Prob. Appl. 27, 201205.
[12] Kalinkin, A. V. (2002). Markov branching processes with interaction. Russian Math. Surveys 57, 241304.
[13] Kalinkin, A. V. (2003). On the extinction probability of a branching process with two kinds of interaction of particles. Theory Prob. Appl. 46, 347352.
[14] Maki, D. P., and Thompson, M. (1973). Mathematical Models and Applications. Prentice-Hall, Englewood Cliffs, NJ.
[15] Rogers, D., and Hassell, M. P. (1974). General models for insect parasite and predator searching behaviour-interference. J. Animal Ecology 43, 239253.
[16] Sevast′yanov, B. A. (1949). On certain types of Markov processes. Usp. Mat. Nauk 4, 194 (in Russian).
[17] Wang, Z. K., and Yang, X. Q. (1992). Birth and Death Processes and Markov Chains. Springer, Berlin.
[18] Yang, X. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.

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The collision branching process

  • Anyue Chen (a1), Phil Pollett (a2), Hanjun Zhang (a2) and Junping Li (a1)

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