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Zbl 0869.60074
Donnelly, Peter; Kurtz, Thomas G.
A countable representation of the Fleming-Viot measure-valued diffusion. (English)
[J] Ann. Probab. 24, No. 2, 698-742 (1996). ISSN 0091-1798

In the first two sections it is shown that the Fleming-Viot process $Z$ can be obtained as limit of the empirical measures of a certain particle system $(X_1,X_2, \dots) $: $Z(t)= \lim_{n\to\infty} {1\over n} \sum^n_{i=1} \delta_{X_k(t)}$. The interaction among the particles $X_1,X_2,\dots$ has a relatively simple description. Thus the above representation turns out to be a useful device, which is applied in the three subsequent sections of this article to derive a variety of properties of the Fleming-Viot process $Z$. First a connection between the genealogical structure of the population model and the particle system $(X_1,X_2,\dots)$ is established. Then a criterion for the strong ergodicity of $Z$ is given, and the speed of convergence to equilibrium is analyzed. The final section is devoted to the derivation of numerous support properties of the sample paths of $Z$. Some of these assertions extend previously known results to the case of more general mutation operators.
[A.Schied (Berlin)]

MSC 2000:: *60J70 Appl. of diffusion theory
60J25 Markov processes with continuous parameter
60K35 Interacting random processes
60J80 Branching processes
92D10 Genetics
60G57 Random measures

Keywords: genealogical process; strong ergodicity; sample path properties; Fleming-Viot process; empirical measures; population model; speed of convergence

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