Path-properties of the tree-valued Fleming-Viot process
Depperschmidt A, Greven A, Pfaffelhuber P (2013)
Publication Status: Published
Publication Type: Journal article, Original article
Publication year: 2013
Journal
Publisher: Institute of Mathematical Statistics (IMS): OAJ / Institute of Mathematical Statistics
Book Volume: 18
Pages Range: 1-47
Journal Issue: 84
URI: https://projecteuclid.org/euclid.ejp/1465064309
DOI: 10.1214/EJP.v18-2514
Abstract
We consider the tree-valued Fleming-Viot process, (X), with mutation and selection as studied in Depperschmidt, Greven and Pfaffelhuber (2012). 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 neutral 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 apart in the limit ∈ → 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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APA:
Depperschmidt, A., Greven, A., & Pfaffelhuber, P. (2013). Path-properties of the tree-valued Fleming-Viot process. Electronic Journal of Probability, 18(84), 1-47. https://dx.doi.org/10.1214/EJP.v18-2514
MLA:
Depperschmidt, Andrej, Andreas Greven, and Peter Pfaffelhuber. "Path-properties of the tree-valued Fleming-Viot process." Electronic Journal of Probability 18.84 (2013): 1-47.
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