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Stochastic flows associated to coalescent processes
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  • Published: June 2003

Stochastic flows associated to coalescent processes

  • Jean Bertoin1 &
  • Jean-Françcois Le Gall2 

Probability Theory and Related Fields volume 126, pages 261–288 (2003)Cite this article

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Abstract.

 We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Möhle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size. We define a bridge to be a right-continuous process (B(r),r[0,1]) with nondecreasing paths and exchangeable increments, such that B(0)=0 and B(1)=1. We show that flows of bridges are in one-to-one correspondence with the so-called exchangeable coalescents. This yields an infinite-dimensional version of the classical Kingman representation for exchangeable partitions of ℕ. We then propose a Poissonian construction of a general class of flows of bridges and identify the associated coalescents. We also discuss an important auxiliary measure-valued process, which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized Fleming-Viot process.

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires and Institut universitaire de France, Université Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France. e-mail: jbe@ccr.jussieu.fr, , , , , , FR

    Jean Bertoin

  2. DMA, Ecole Normale Supérieure, 45, rue d'Ulm, F-75005 Paris, France. e-mail: jean-francois.le.gall@ens.fr, , , , , , FR

    Jean-Françcois Le Gall

Authors
  1. Jean Bertoin
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  2. Jean-Françcois Le Gall
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Additional information

Received: 26 November 2002 / Revised version: 10 February 2003 / Published online: 15 April 2003

Mathematics Subject Classification (2000): 60G09, 60J25, 92D30

Key words or phrases: Flow – Coalescence – Exchangeability – Bridge

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Bertoin, J., Gall, JF. Stochastic flows associated to coalescent processes. Probab. Theory Relat. Fields 126, 261–288 (2003). https://doi.org/10.1007/s00440-003-0264-4

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  • Issue Date: June 2003

  • DOI: https://doi.org/10.1007/s00440-003-0264-4

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Keywords

  • General Class
  • Fixed Size
  • Asymptotic Model
  • Stochastic Flow
  • Coalescent Process
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