2011/01/31 by Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber
Economics, Econometrics and Finance · Mathematics · #Advanced Topology and Set Theory #Ergodic theory #Ergodicity #Markov chain #Mathematics #Measure (data warehouse) #Metric space #Population #Pure mathematics #Statistics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Ultrametric space #math.PR
paper · pdf · doi:10.1214/11-aap831
published as Annals of Applied Probability 2012, Vol. 22, No. 6, 2560-2615 · Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2012/11/23 · arxiv created 2012/11/29 · arxiv updated 2012/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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from Dawson and Greven [On the effects of migration in spatial Fleming–Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming–Viot measure-valued diffusion is ergodic.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.