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Birth-death processes with disaster and instantaneous resurrection

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anyue Chen ,
Hanjun Zhang ,
Kai Liu  and
Keith Rennolls
Anyue Chen*
Affiliation:
University of Greenwich
Hanjun Zhang*
Affiliation:
University of Queensland
Kai Liu*
Affiliation:
University of Liverpool
Keith Rennolls*
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, School of Physical Sciences, University of Queensland, QLD 4072, Australia.
∗∗∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK.
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK.
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Abstract

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Keywords

Birth-death processdisasterinstantaneous resurrectionunstable continuous-time Markov chainexistenceuniquenessrecurrenceergodicityequilibrium distributionsymmetryreversibility

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 267 - 292
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Chen, A. Y. and Liu, K. (2003). Birth–death processes with an instantaneous reflection barrier. J. Appl. Prob. 40, 163–179.Google Scholar
Chen, A. Y. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209–240.Google Scholar
Chen, A. Y. and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427–456.Google Scholar
Chen, A. Y. and Renshaw, E. (1995). Markov branching processes regulated by emigration and large immigration. Stoch. Process. Appl. 57, 339–359.Google Scholar
Chen, A. Y. and Renshaw, E. (2000). Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stoch. Process. Appl. 88, 177–193.Google Scholar
Chen, M. F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.Google Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pure Appl. 9, 301–345.Google Scholar
Freedman, D. (1983). Approximating Countable Markov Chains, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Hou, Z. T. and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.Google Scholar
Kendall, D. G. and Reuter, G. E. (1954). Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators on l. In Proc. Internat. Cong. Math. (Amsterdam), Vol. III, North-Holland, Amsterdam, pp. 377–415.Google Scholar
Kingman, J. F. C. (1972). Regenerative Phenomena. John Wiley, New York.Google Scholar
Kolmogorov, A. N. (1951). On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov. Gos. Univ. Učenye Zapiski Mat. 148, 53–59 (in Russian).Google Scholar
Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on l. Acta. Math. 97, 1–46.Google Scholar
Reuter, G. E. H. (1959). Denumerable Markov processes. II. J. London Math. Soc. 34, 81–91.CrossRefGoogle Scholar
Reuter, G. E. H. (1962). Denumerable Markov processes. III. J. London Math. Soc. 37, 63–73.Google Scholar
Reuter, G. E. H. (1969). Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitsth. 13, 315–320.Google Scholar
Rogers, L. C. G. and Williams, D. (1986). Construction and approximation of transition matrix functions. Adv. Appl. Prob. Suppl., 133–160.Google Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales, Vol. 2. John Wiley, New York.Google Scholar
Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 1, 2nd edn. John Wiley, Chichester.Google Scholar
Williams, D. (1967). A note on the Q-matrices of Markov chains. Z. Wahrscheinlichkeitsth. 7, 116–121.Google Scholar
Williams, D. (1976). The Q-matrix problems. In Séminaire de Probabilités X (Lecture Notes Math. 511), ed. Meyer, P. A., Springer, Berlin, pp. 216–234.Google Scholar
Yang, X. Q. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.Google Scholar