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

AMINE ASSELAH; PABLO A. FERRARI; PABLO GROISMAN

JOURNAL OF APPLIED PROBABILITY

APPLIED PROBABILITY TRUST

Lugar: Londres; Año: 2011 vol. 48 p. 322 - 332

0021-9002

Consider a continuous time Markov chain with rates Q in the state space Lambdacup{0} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in Lambda with rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to Lambda instantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When Lambda is finite, we show that the empirical distribution of the particles at a fixed time converges as N oinfty 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 oinfty 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.