Tree-valued Fleming-Viot dynamics with mutation and selection
Depperschmidt A, Greven A, Pfaffelhuber P (2012)
Publication Status: Published
Publication Type: Journal article, Original article
Publication year: 2012
Journal
Publisher: Institute of Mathematical Statistics (IMS)
Book Volume: 22
Pages Range: 2560-2615
Journal Issue: 6
URI: https://projecteuclid.org/euclid.aoap/1353695962
DOI: 10.1214/11-AAP831
Abstract
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 treevalued 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 Girsanovtype 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from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic.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. © Institute of Mathematical Statistics, 2012.
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APA:
Depperschmidt, A., Greven, A., & Pfaffelhuber, P. (2012). Tree-valued Fleming-Viot dynamics with mutation and selection. Annals of Applied Probability, 22(6), 2560-2615. https://doi.org/10.1214/11-AAP831
MLA:
Depperschmidt, Andrej, Andreas Greven, and Peter Pfaffelhuber. "Tree-valued Fleming-Viot dynamics with mutation and selection." Annals of Applied Probability 22.6 (2012): 2560-2615.
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