Quasi stationary distributions and Fleming-Viot processes in countable spaces

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)/( ∑ y∈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 := ∑ x∈Λ inf Q(·, x) > supQ(·, 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 → ∞; in this limit the variances vanish .