Genealogical processes for Fleming-Viot models with selection and recombination

Abstract

Infinite population genetic models with general type space incorporating mutation, selection and recombination are considered. The Fleming– Viot measure-valued diffusion is represented in terms of a countably infinite-dimensional process. The complete genealogy of the population at each time can be recovered from the model. Results are given concerning the existence of stationary distributions and ergodicity and absolute continuity of the stationary distribution for a model with selection with respect to the stationary distribution for the corresponding neutral model. 1. Introduction. The Fleming–Viot measure-valued diffusion arises as the large population limit of a wide class of population genetics models. Together with the Dawson–Watanabe process which arises from branching models, it is one of the more well studied measure-valued processes. For a recent review of available results about the Fleming–Viot process, see Ethier and Kurtz (1993) and references therein. Measure-valued diffusions are often motivated by first considering a class of prelimiting finite-population models. The dynamics in such discrete contexts are easily specified in terms of the behavior of the individuals in the population, and the composition of the population is naturally represented as a measure on the set, E, of possible types. Measure-valued diffusions then arise because the associated discrete measure-valued processes behave sensibly (after appropriate rescaling) in the large population limit. On the other hand, the discrete population models which keep track of the fates of individuals make no sense for infinite-population sizes. Thus, while it might be convenient in applications to think of the measure-valued diffusion as describing the evolution of a hypothetically infinite population, it is difficult to make this precise. Donnelly and Kurtz (1996, 1999) have recently given a discrete construction of a class of neutral measure-valued population processes. Loosely speaking, the idea is to “bring back the particles.” First, a (one-dimensional) process P describing the total mass of the measure-valued process is constructed. Conditional on P ,a nE ∞ -valued process, X, is described with the property that