Quasi stationary distributions and fleming-viot processes in countable spaces

Abstract

We consider an irreducible pure jump Markov process with rates Q = (q(x, y)) on Λ ⋃ (0) with Λ countable and 0 an absorbing state. A quasi stationary distribution (qsd) is a probability measure ν on Λ that satisfies: starting with ν, the conditional distribution at time t, given that at time t the process has not been absorbed, is still ν. That is, ν(x) = νPt(x)/(Py∈Λ νPt(y)), with Pt the transition probabilities for the process with rates Q. A Fleming-Viot (fv) process is a system of N particles moving in Λ. Each particle moves independently with rates Q until it hits the absorbing state 0; but then instantaneously chooses one of the N−1 particles remaining in Λ and jumps to its position. Between absorptions each particle moves with rates Q independently. Under the condition α := Σx∈Λ inf Q(·, x) > 0 the FV process is ergodic for each N. Under α > C the mean normalized densities of the FV unique stationary measure converge to the qsd of Q, as N → ∞; in this limit the variances vanish. © 2007 Applied Probability Trust.