Particle Representations for Measure - Valued Population Models

Models of populations in which a type or location, represented by a point in a metric space E, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measurevalued processes are obtained as infinite population limits for a large class of neutral population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population and the de Finetti measures associated with an E -valued particle model Ž . Ž Ž . Ž . . X X , X , . . . such that, for each t 0, X t , X t , . . . is exchange1 2 1 2 able. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both neutral Fleming Viot and Dawson Watanabe processes. The particle model gives a simple representation of the Dawson Perkins historical process and Perkins’s historical stochastic integral can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behavior are described.