2008/06/13 by Andreas Greven, Greven, Andreas, Peter Pfaffelhuber +3
Mathematics · #60J25 (Primary) #60J70 #60K35 #92D10 (Secondary) #Applied mathematics #Coalescent theory #Combinatorics #Computer science #Discrete mathematics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Martingale (probability theory) #Mathematical Dynamics and Fractals #Mathematics #Measure (data warehouse) #Metric Geometry (math.MG) #Metric space #Population #Probability (math.PR) #Probability measure #Stochastic processes and statistical mechanics #Tree (set theory) #Ultrametric space #math.MG #math.PR #msc:60J25 #msc:60J70 #msc:60K35 #msc:92D10
paper · pdf · doi:10.48550/arxiv.0806.2224
openalex publication_date 2008/06/13 · arxiv created 2011/06/23 · arxiv updated 2011/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming-Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled "individuals". We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming-Viot dynamics.