Particle Representations for Measure-Valued Population Models

Donnelly, Peter ; Kurtz, Thomas G.

Ann. Probab., Tome 27 (1999) no. 1, p. 166-205 / Harvested from Project Euclid

Résumé

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. Measure-valued 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 $E^{\infty}$ -valued particle model$X=(X_1, X_2\ldots)$ such that, for each $t \geq 0,(X_1(t),X_2(t),\ldots)$ is exchangeable. 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.

Détail

Publié le : 1999-01-14
Classification:  Fleming-Viot process,  Dawson-Watanabe process,  superprocess,  measure-valued diffusion,  exchangeability,  genealogical processes,  coalescent,  historical process,  conditioning,  60J25,  60K35,  60J70,  60J80,  92D10