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 \Lambda\cup\{0\} with \Lambda countable and 0 an absorbing state. A quasi-stationary distribution (qsd) is a probability measure \nu on \Lambda that satisfies: starting with \nu, the conditional distribution at time t, given that at time t the process has not been absorbed, is still \nu. That is, \nu(x) = \nu P_t(x)/(\sum_{y\in\Lambda}\nu P_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 \Lambda. 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 \Lambda and jumps to its position. Between absorptions each particle moves with rates Q independently. Under the condition \alpha:=\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 \alpha>0 the {\fv} process is ergodic for each N. Under \alpha>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.