Skip to main content
×
×
Home

Birth-death processes with disaster and instantaneous resurrection

  • Anyue Chen (a1), Hanjun Zhang (a2), Kai Liu (a3) and Keith Rennolls (a1)
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.

Copyright
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@gre.ac.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.
References
Hide All
Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.
Chen, A. Y. and Liu, K. (2003). Birth–death processes with an instantaneous reflection barrier. J. Appl. Prob. 40, 163179.
Chen, A. Y. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209240.
Chen, A. Y. and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427456.
Chen, A. Y. and Renshaw, E. (1995). Markov branching processes regulated by emigration and large immigration. Stoch. Process. Appl. 57, 339359.
Chen, A. Y. and Renshaw, E. (2000). Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stoch. Process. Appl. 88, 177193.
Chen, M. F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.
Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pure Appl. 9, 301345.
Freedman, D. (1983). Approximating Countable Markov Chains, 2nd edn. Springer, New York.
Hou, Z. T. and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.
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. 377415.
Kingman, J. F. C. (1972). Regenerative Phenomena. John Wiley, New York.
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, 5359 (in Russian).
Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on l. Acta. Math. 97, 146.
Reuter, G. E. H. (1959). Denumerable Markov processes. II. J. London Math. Soc. 34, 8191.
Reuter, G. E. H. (1962). Denumerable Markov processes. III. J. London Math. Soc. 37, 6373.
Reuter, G. E. H. (1969). Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitsth. 13, 315320.
Rogers, L. C. G. and Williams, D. (1986). Construction and approximation of transition matrix functions. Adv. Appl. Prob. Suppl., 133160.
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales, Vol. 2. John Wiley, New York.
Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 1, 2nd edn. John Wiley, Chichester.
Williams, D. (1967). A note on the Q-matrices of Markov chains. Z. Wahrscheinlichkeitsth. 7, 116121.
Williams, D. (1976). The Q-matrix problems. In Séminaire de Probabilités X (Lecture Notes Math. 511), ed. Meyer, P. A., Springer, Berlin, pp. 216234.
Yang, X. Q. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 114 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th September 2018. This data will be updated every 24 hours.