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Particle representations for measure-valued population models. (English) Zbl 0956.60081
The authors consider two continuous time models for the evolution of a finite population in which a type or location, represented by a point in a metric space $$E$$, is associated with each individual. The chances of an individual replicating or dying do not depend on its type. In one model, the population is ordered by decreasing age and in the other, this order does not play a role. Measure-valued processes [see D. A. Dawson, in: Ecole d’Été de probabilités de Saint Flour XXI-1991. Lect. Notes Math. 1541, 1-260 (1993; 799.60080)] are obtained as infinite population limits for a large class of population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of D. Dawson’s and E. A. Perkins’s historical process [Mem. Am. Math. Soc. 454 (1991; Zbl 0754.60062)]. E. Perkins’ historical stochastic integral [Probab. Theory Relat. Fields 94, No. 2, 189-245 (1992; Zbl 0767.60044)] can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behaviour are described.

##### MSC:
 60J60 Diffusion processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G52 Stable stochastic processes
##### Keywords:
measure-valued diffusion; historical process; conditioning
##### Citations:
Zbl 0799.60080; Zbl 0754.60062; Zbl 0767.60044
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##### References:
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