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 {\em quasi stationary distribution \rm} (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 $\alpha:=\sum_{x\in\Lambda}\inf Q(\cdot,x) > \sup Q(\cdot,0):=C$ 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.