Open Access
December 2012 Tree-valued Fleming–Viot dynamics with mutation and selection
Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber
Ann. Appl. Probab. 22(6): 2560-2615 (December 2012). DOI: 10.1214/11-AAP831


The Fleming–Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming–Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions.

The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces.

To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming–Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming–Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.


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Andrej Depperschmidt. Andreas Greven. Peter Pfaffelhuber. "Tree-valued Fleming–Viot dynamics with mutation and selection." Ann. Appl. Probab. 22 (6) 2560 - 2615, December 2012.


Published: December 2012
First available in Project Euclid: 23 November 2012

zbMATH: 1316.92048
MathSciNet: MR3024977
Digital Object Identifier: 10.1214/11-AAP831

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

Keywords: ancestral selection graph , Coalescent , Duality , Fleming–Viot process , Genealogical tree , Girsanov theorem , measure-valued diffusion , metric measure space , Resampling , tree-valued Fleming–Viot dynamics

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.22 • No. 6 • December 2012
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