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) = νP t (x)/( y∈Λ 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 Λ and jumps to its position. Between absorptions each particle moves with rates Q independently.Under the condition α := x inf Q(·, x) > sup Q(·, 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.

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

“…Under further assumptions (see (1.8)), there exists a unique invariant measure for the fv process and its profile converges as N → ∞ to the unique qsd. Theorem 1.1 (Ferrari and Maric [3], Theorem 1.2 and 1.4). Let µ be any probability measure on Λ, and µ ⊗N the product probability on …”

“…A natural approach to show that T t µ is close to E N ξ [m(ξ t )] is by establishing that the occupation numbers of two distinct sites, at time t, become independent when N tends to infinity (the so-called propagation of chaos). For this purpose, Ferrari and Maric [3], estimate the correlation of two ξ-particles, when Λ is only assumed countable. Proposition 1.1 (Ferrari and Maric [3], Proposition 3.1).…”

“…For this purpose, Ferrari and Maric [3], estimate the correlation of two ξ-particles, when Λ is only assumed countable. Proposition 1.1 (Ferrari and Maric [3], Proposition 3.1). Let µ be any probability measure on Λ, and µ ⊗N the product probability on Λ N .…”

“…We prove the asymptotic independence of two particles but, in contrast to [3], we consider deterministic initial configurations and obtain bounds on the correlations that hold uniformly on the initial distribution of the particles. This result also holds for countable Λ.…”

“…In Section 2, we construct the processà la Harris following [3]. We use this construction to estimate the correlations and prove Proposition 1.2 in Section 3.…”