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


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.


First passage time Exponential decay Quasi stationary distribution 

Mathematics Subject Classification (2010)

92D25 60J28 



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


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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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