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Existence, uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

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Abstract

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection. The existence criterion of the process is firstly considered. We prove that if the sum of the resurrection rates is finite, then there does not exist any process. An existence criterion is then established when the sum of the resurrection rates is infinite. Some equivalent criteria, possessing the advantage of being easily checked, are obtained for the latter case. The uniqueness criterion for such process is also investigated. We prove that although there exist infinitely many of them, there always exists a unique honest process for a given q-matrix. This unique honest process is then constructed. The ergodicity property of this honest process is analysed in detail. We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.

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References

  1. Harris T E. The Theory of Branching Processes. Berlin-New York: Springer, 1963

    MATH  Google Scholar 

  2. Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972

    MATH  Google Scholar 

  3. Asmussen S, Hering H. Branching Processes. Boston: Birkhäuser, 1983

    MATH  Google Scholar 

  4. Athreya K B, Jagers P. Classical and Modern Branching Processes. Berlin: Springer, 1996

    Google Scholar 

  5. Sevast’yanov B A. Limit theorems for branching stochastic processes of special form. Theory Probab Appl, 2(3): 321–331 (1957)

    Article  Google Scholar 

  6. Zubkov AM. The life-periods of a branching process with immigration. Theory Probab Appl, 17(1): 174–183 (1972)

    Article  MATH  Google Scholar 

  7. Vatutin V A. The asymptotic probability of the first degeneration for branching processes with immigration. Theory Probab Appl, 19(1): 25–34 (1974)

    Article  MATH  Google Scholar 

  8. Vatutin V A. A conditional limit theorem for a critical branching process with immigration. Math Notes, 21: 405–411 (1977)

    MATH  MathSciNet  Google Scholar 

  9. Aksland M. A birth, death and migration process with immigration. Adv in Appl Probab, 7: 44–60 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  10. Aksland M. On interconnected birth and death with immigration. In: Transactions of the Seventh Prague Conference on Information Theory. Dordrecht: Reidel, 1977, 23–28

    Google Scholar 

  11. Adke S R. A birth, death and migration process. J Appl Probab, 6: 687–691 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen M F. Eigenvalues, Inequalities and Ergodic Theory. New York: Springer, 2005

    MATH  Google Scholar 

  13. Foster F G. A limit theorem for a branching process with state-dependent immigration. Ann Math Statist, 42: 1773–1776 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pakes A G. A branching process with a state-dependent immigration component. Adv in Appl Probab, 3: 301–314 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yamazato M. Some results on continuous time branching processes with state-dependent immigration. J Math Soc Japan, 17: 479–497 (1975)

    Article  MathSciNet  Google Scholar 

  16. Pakes A G, Tavaré S. Comments on the age distribution of Markov processes. Adv in Appl Probab, 13: 681–703 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen A Y, Renshaw E. Markov branching processes with instantaneous immigration. Probab Theory Related Fields, 87: 209–240 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chen A Y, Renshaw E. Recurrence of Markov branching processes with immigration. Stochastic Process Appl, 45: 231–242 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen A Y, Renshaw E. Markov branching processes regulated by emigration and large immigration. Stochastic Process Appl, 57: 339–359 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chen A Y, Renshaw E. Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stochastic Process Appl, 88: 177–193 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pakes A G. Absorbing Markov and branching processes with instantaneous resurrection. Stochastic Process Appl, 48: 85–106 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen R R. An extended class of time-continuous branching processes. J Appl Probab, 34: 14–23 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  23. Chen A Y. Ergodicity and stability of generalized Markov branching processes with resurrection. J Appl Probab, 39: 786–803 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Chen A Y. Uniqueness and extinction properties of generalized Markov branching processes. J Math Anal Appl, 274: 482–494 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Li J P, Chen A Y. Markov branching processes with immigration and resurrection. Markov Processes Rel Fields, 12: 139–168 (2006)

    MATH  Google Scholar 

  26. Reuter G E H. Denumerable Markov processes and the associated contraction semigroups on l. Acta Math, 97: 1–46 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  27. Chung K L. Markov Chains with Stationary Transition Probabilities. New York: Springer, 1967

    MATH  Google Scholar 

  28. Yang X Q. The Construction Theory of Denumerable Markov Processes. New York: John Wiley, 1990

    MATH  Google Scholar 

  29. Reuter G E H. Denumerable Markov processes II. J London Math Soc, 34: 81–91 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hou Z T, Zou J Z, Zhang H J, et al. The Q-matrix Problem for Markov Chains (in Chinese). Changsha: Hunan Science and Technology Press, 1994

    Google Scholar 

  31. Feller W. On the integro-differential equations of purely discontinuous Markov processes. Trans Amer Math Soc, 48: 488–515 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  32. Reuter G E H. Denumerable Markov processes III. J London Math Soc, 37: 63–73 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  33. Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. Singapore: World Scientific, 1992

    MATH  Google Scholar 

  34. Anderson W J. Continuous Time Markov Chains. Berlin: Springer, 1991

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Science and Computing Technology, Central South University, Changsha, 410075, China

    JunPing Li

  2. Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

    AnYue Chen

  3. Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

    AnYue Chen

Authors
  1. JunPing Li
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  2. AnYue Chen
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Correspondence to JunPing Li.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10771216), Research Grants Council of Hong Kong (Grant No. HKU 7010/06P) and Project-sponsored by SRF for ROCS, SEM

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Li, J., Chen, A. Existence, uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection. Sci. China Ser. A-Math. 51, 1266–1286 (2008). https://doi.org/10.1007/s11425-008-0027-x

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