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\cup\{0\} with Λcountable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure νon Λthat 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) = νP_t(x)/(\sum_{y\inΛ}νP_t(y)), with P_t 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 Λand jumps to its position. Between absorptions each particle moves with rates Q independently. Under the condition α:=\sum_x\inf Q(\cdot,x) > \sup Q(\cdot,0):=C we prove existence of qsd for Q; uniqueness has been proven by Jacka and Roberts. When α>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 \to \infty; in this limit the variances vanish.