The Annals of Probability

Particle Representations for Measure-Valued Population Models

Peter Donnelly and Thomas G. Kurtz

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Abstract

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.

Article information

Source
Ann. Probab. Volume 27, Number 1 (1999), 166-205.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677258

Digital Object Identifier
doi:10.1214/aop/1022677258

Mathematical Reviews number (MathSciNet)
MR1681126

Zentralblatt MATH identifier
0956.60081

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Fleming-Viot process Dawson-Watanabe process superprocess measure-valued diffusion exchangeability genealogical processes coalescent historical process conditioning

Citation

Donnelly, Peter; Kurtz, Thomas G. Particle Representations for Measure-Valued Population Models. Ann. Probab. 27 (1999), no. 1, 166--205. doi:10.1214/aop/1022677258. https://projecteuclid.org/euclid.aop/1022677258


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  • UNITED KINGDOM MADISON, WISCONSIN 53706-1388 E-MAIL: donnelly@stats.ox.ac.uk E-MAIL: kurtz@math.wisc.edu