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Acta Applicandae Mathematicae

, Volume 131, Issue 1, pp 197–212 | Cite as

Exponentiality of First Passage Times of Continuous Time Markov Chains

  • Romain Bourget
  • Loïc Chaumont
  • Natalia Sapoukhina
Article
  • 227 Downloads

Abstract

Let \((X,\mathbb{P}_{x})\) be a continuous time Markov chain with finite or countable state space S and let T be its first passage time in a subset D of S. It is well known that if μ is a quasi-stationary distribution relative to T, then this time is exponentially distributed under \(\mathbb {P}_{\mu}\). However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution μ for T to be exponentially distributed under \(\mathbb{P}_{\mu}\). We show in addition how quasi-stationary distributions can be expressed in terms of any initial law which makes the distribution of T exponential. We also study two examples in branching processes where exponentiality does imply quasi-stationarity.

Keywords

First passage time Exponential decay Quasi stationary distribution 

Mathematics Subject Classification (2010)

92D25 60J28 

Notes

Acknowledgement

We are very grateful to Professor Servet Martínez to have pointed out the reference [8] and given access to its preliminary version.

References

  1. 1.
    Aguilée, R., Claessen, D., Lambert, A.: Allele fixation in a dynamic metapopulation: founder effects vs refuge effects. Theor. Popul. Biol. 76(2), 105–117 (2009) CrossRefzbMATHGoogle Scholar
  2. 2.
    Athreya, K.B., Ney, P.: Renewal approach to the Perron-Frobenius theory of non-negative kernels on general state spaces. Math. Z. 179, 507–529 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bertoin, J., Doney, R.A.: Some asymptotic results for transient random walks. Adv. Appl. Probab. 28(1), 207–226 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bourget, R.: Modélisation stochastique des processus d’adaptation d’une population de pathogènes aux résistances génétiques des hôtes. Thèse de doctorat, Université d’Angers (2013) Google Scholar
  5. 5.
    Bourget, s.R., Chaumont, L., Sapoukhina, N.: Timing of pathogen adaptation to a multicomponent treatment. PLoS ONE 8(8), e71926 (2013). doi: 10.1371/journal.pone.0071926 CrossRefGoogle Scholar
  6. 6.
    Cattiaux, P., Méléard, S.: Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction. J. Math. Biol. 60(6), 797–829 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Champagnat, N., Lambert, A.: Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann. Appl. Probab. 17(1), 102–155 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Collet, P., Martínez, S., San Martín, J.: Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems. Probability and Its Applications (New York). Springer, Heidelberg (2013) CrossRefzbMATHGoogle Scholar
  9. 9.
    Durrett, R., Moseley, S.: Evolution of resistance and progression to disease during clonal expansion of cancer. Theor. Popul. Biol. 77, 42–48 (2010) CrossRefGoogle Scholar
  10. 10.
    Durrett, R., Schmidt, D., Schweinsberg, J.: A waiting time problem arising from the study of multi-stage carcinogenesis. Ann. Appl. Probab. 19(2), 676–718 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Haas, B., Rivero, V.: Quasi-stationary distributions and Yaglom limits of self-similar Markov processes (2011). Preprint arXiv:1110.4795
  12. 12.
    Handel, A., Longini, I.M., Antia, R.: Antiviral resistance and the control of pandemic influenza: the roles of stochasticity, evolution and model details. J. Theor. Biol. 256, 117–125 (2009) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Iwasa, Y., Michor, F., Nowak, M.A.: Evolutionary dynamics of invasion and escape. J. Theor. Biol. 226, 205–214 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Jacka, S.D., Roberts, G.O.: Weak convergence of conditioned processes on a countable state space. J. Appl. Probab. 32(4), 902–916 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Komarova, N.: Stochastic modeling of drug resistance in cancer. J. Theor. Biol. 239, 351–366 (2006) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kyprianou, A.E., Palmowski, Z.: Quasi-stationary distributions for Lévy processes. Bernoulli 12(4), 571–581 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lambert, A.: Probability of fixation under weak selection: a branching process unifying approach. Theor. Popul. Biol. 69(4), 419–441 (2006) CrossRefzbMATHGoogle Scholar
  18. 18.
    Lambert, A.: Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12, 420–446 (2007). Paper no. 14 CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lambert, A.: Population dynamics and random genealogies. Stoch. Models 24(suppl. 1), 45–163 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Méléard, S.: Quasi-stationary distributions for population processes. In: VI Escuela De Probabilidad y Procesos Estocásticos (2009) Google Scholar
  21. 21.
    Nair, M.G., Pollett, P.K.: On the relationship between μ-invariant measures and quasi-stationary distributions for continuous-time Markov chains. Adv. Appl. Probab. 25(1), 82–102 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Nåsell, I.: Extinction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol. 211, 11–27 (2001) CrossRefGoogle Scholar
  23. 23.
    Pollett, P.K., Vere Jones, D.: A note on evanescent processes. Aust. J. Stat. 34(3), 531–536 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Reuter, G.E.H.: Competition processes. In: Proc. 4th Berkeley. Sympos. Math. Statist. and Prob., vol. II, pp. 421–430. Univ. California Press, Berkeley (1961) Google Scholar
  25. 25.
    Schweinsberg, J.: The waiting time for m mutations. Electron. J. Probab. 13, 1442–1478 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Serra, M.C.: On the waiting time to escape. J. Appl. Probab. 43(1), 296–302 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Serra, M.C., Haccou, P.: Dynamics of escape mutants. Theor. Popul. Biol. 72, 167–178 (2007) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Romain Bourget
    • 1
    • 2
    • 3
    • 4
  • Loïc Chaumont
    • 4
  • Natalia Sapoukhina
    • 1
    • 2
    • 3
  1. 1.INRAUMR1345 Institut de Recherche en Horticulture et Semences—IRHSBeaucouzé CedexFrance
  2. 2.AgroCampus-OuestUMR1345 Institut de Recherche en Horticulture et Semences—IRHSAngersFrance
  3. 3.Université d’AngersUMR1345 Institut de Recherche en Horticulture et Semences—IRHSAngersFrance
  4. 4.LAREMA UMR CNRS 6093Université d’AngersAngers Cedex 01France

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