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Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces


 
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1. Title Title of document Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces
 
2. Creator Author's name, affiliation, country Pablo A. Ferrari; Universidade de Sao Paulo
 
2. Creator Author's name, affiliation, country Nevena Maric; Syracuse University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Quasi stationary distributions; Fleming-Viot process
 
3. Subject Subject classification 60F ; 60K35
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) FAPESP; CNPq; IM-AGIMB
 
7. Date (YYYY-MM-DD) 2007-05-28
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/415
 
10. Identifier Digital Object Identifier 10.1214/EJP.v12-415
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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