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Fleming-Viot selects the minimal quasi-stationary distribution: the Galton-Watson case. (English. French summary) Zbl 1342.60145

Summary: Consider \(N\) particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits \(0\), at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each \(N\) there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as \(N\) goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
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