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A countable representation of the Fleming-Viot measure-valued diffusion. (English) Zbl 0869.60074
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 $\left({X}_{1},{X}_{2},\cdots \right)$: $Z\left(t\right)={lim}_{n\to \infty }\frac{1}{n}{\sum }_{i=1}^{n}{\delta }_{{X}_{k}\left(t\right)}$. The interaction among the particles ${X}_{1},{X}_{2},\cdots$ 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 $\left({X}_{1},{X}_{2},\cdots \right)$ 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.
##### MSC:
 60J70 Applications of Brownian motions and diffusion theory 60J25 Continuous-time Markov processes on general state spaces 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J80 Branching processes 92D10 Genetics 60G57 Random measures