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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 (X 1 ,X 2 ,): Z(t)=lim n 1 n i=1 n δ X k (t) . The interaction among the particles X 1 ,X 2 , 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 ,) 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:
60J70Applications of Brownian motions and diffusion theory
60J25Continuous-time Markov processes on general state spaces
60K35Interacting random processes; statistical mechanics type models; percolation theory
60J80Branching processes
92D10Genetics
60G57Random measures