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2013 Path-properties of the tree-valued Fleming–Viot process
Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber
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Electron. J. Probab. 18: 1-47 (2013). DOI: 10.1214/EJP.v18-2514

Abstract

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.

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Andrej Depperschmidt. Andreas Greven. Peter Pfaffelhuber. "Path-properties of the tree-valued Fleming–Viot process." Electron. J. Probab. 18 1 - 47, 2013. https://doi.org/10.1214/EJP.v18-2514

Information

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

zbMATH: 1284.60166
MathSciNet: MR3109623
Digital Object Identifier: 10.1214/EJP.v18-2514

Subjects:
Primary: 60J25, 60K35
Secondary: 60J68, 92D10

JOURNAL ARTICLE
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