Electronic Journal of Probability

Path-properties of the tree-valued Fleming–Viot process

Andrej Depperschmidt, Andreas Greven, and Peter Pfaffelhuber

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We consider the tree-valued Fleming–Viot process, (Xt )t≥0 , with mutation and selection. This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming–Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent.

In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the Fleming–Viot genealogies at fixed time t arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than ε apart in the limit ε → 0. Second, we answer two open questions about the sample paths of the tree-valued Fleming–Viot process. We show that for all t > 0 almost surely the marked metric measure space Xt has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming–Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov’s formula giving absolute continuity.

Article information

Electron. J. Probab. Volume 18 (2013), paper no. 84, 47 pp.

Accepted: 19 September 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J68: Superprocesses 92D10: Genetics {For genetic algebras, see 17D92}

Marked tree-valued Fleming–Viot process path properties selection mutation Kingman coalescent

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Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Path-properties of the tree-valued Fleming–Viot process. Electron. J. Probab. 18 (2013), paper no. 84, 47 pp. doi:10.1214/EJP.v18-2514. http://projecteuclid.org/euclid.ejp/1465064309.

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