Tree-valued resampling dynamics Martingale problems and applications

• 成果类型: Article
• 署名作者: Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita
• 署名单位: University of Erlangen Nuremberg; University of Freiburg; University of Duisburg Essen
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0413-8
• 发表日期: 2013
• 页码: 789-838
• 关键词: bessel-bridges; REPRESENTATION; models; integration; parts; time

摘要

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.