Tree - Valued Resampling Dynamics ( Martingale Problems and Applications

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. Ultra-metric spaces extend the class of discrete trees with edge length by allowing behavior such as infinitesimal short edges. We encode genealogies of the population at fixed times as elements in the space of (isometry classes of) ultra-metric measure spaces. The additional probability measure on the ultra-metric space allows to sample from the population. We equip this state space by the Gromov-weak topology and use well-posed martingale problems to construct tree-valued resampling dynamics for both the finite population (tree-valued Moran dynamics) and the infinite population (tree-valued Fleming-Viot dynamics). As an application we study the evolution of the distribution of the lengths of the sub-trees spanned by sequentially sampled " individuals ". We show that ultra-metric measure spaces are uniquely determined by the distribution of the infinite vector of the subsequently evaluated lengths of sub-trees. 1. Introduction. In the present paper we construct and study the evolution of the genealogical structure of the neutral multi-type population model called the Fleming-Viot process ([FV78, FV79, Daw93, EK93, DGV95, Eth01]). This process arises as the large population limit of Moran models. The Moran model (MM) can be described as follows: consider a