## Annals of Probability

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

### Particle Representations for Measure-Valued Population Models

Peter Donnelly and Thomas G. Kurtz

#### 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