Quasi-stationary distributions and Fleming-Viot processes in finite spaces

2009/04/20 by Asselah, Amine, Ferrari, Pablo A., Groisman, Pablo · 1 citation
#60J25 #60K35 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0904.3039

Abstract

Consider a continuous time Markov chain with rates Q in the state space Λ∪\0\ with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in Λwith rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to Λinstantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When Λis finite, we show that the empirical distribution of the particles at a fixed time converges as N→∞ to the distribution of a single particle at the same time conditioned on non absorption. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N→∞ to the unique quasi-stationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations is of order 1/N.

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• Fleming-Viot Selection of the Yaglom Limit for Age-Structured Bellman-Harris Processes, with Application to Livestock Epidemic Surveillance

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