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Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces

   
@article{EJP415,
	author = {Pablo Ferrari and Nevena Maric},
	title = {Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {12},
	year = {2007},
	keywords = {Quasi stationary distributions; Fleming-Viot process},
	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.
},
	pages = {no. 24, 684-702},
	issn = {1083-6489},
	doi = {10.1214/EJP.v12-415},    
        url = {http://ejp.ejpecp.org/article/view/415}}