## Electronic Journal of Probability

### 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.

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 24, 684-702.

Dates
Accepted: 28 May 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818495

Digital Object Identifier
doi:10.1214/EJP.v12-415

Mathematical Reviews number (MathSciNet)
MR2318407

Zentralblatt MATH identifier
1127.60088

Rights

#### Citation

Ferrari, Pablo; Maric, Nevena. Quasi Stationary Distributions and Fleming-Viot Processes in Countable Spaces. Electron. J. Probab. 12 (2007), paper no. 24, 684--702. doi:10.1214/EJP.v12-415. https://projecteuclid.org/euclid.ejp/1464818495

#### References

• Burdzy, K., Holyst, R., March, P. A Fleming-Viot particle representation of the Dirichlet Laplacian. 214 (2000), Comm. Math. Phys.
• Cavender, J. A. Quasi-stationary distributions of birth and death processes. Adv. Appl. Prob. 10 (1978), 570-586.
• Darroch, J.N., Seneta, E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4 (1967), 192–196.
• Fernández, R., Ferrari, P.A., Garcia, N. L. Loss network representation of Peierls contours. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. Ann. Probab. 29 (2001), 902–937.
• Ferrari, P.A, Kesten, H., Martínez,S., Picco, P. Existence of quasi stationary distributions. A renewal dynamical approach. Ann. Probab. 23 (1995), 511–521.
• Ferrari, P.A., Martínez, S., Picco, P. Existence of non-trivial quasi-stationary distributions in the birth-death chain. Adv. Appl. Prob. 24 (1992), 725–813.
• Fleming,.W.H., Viot, M. Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 (1979), 817–843.
• Grigorescu, I., Kang,M. Hydrodynamic limit for a Fleming-Viot type system. Stoch. Proc. Appl. 110 (2004), 111–143.
• Jacka, S.D., Roberts, G.O. Weak convergence of conditioned processes on a countable state space. J. Appl. Prob. 32 (1995), 902–916.
• Kipnis, C., Landim, C. Scaling Limits of Interacting Particle Systems (1999) Springer-Verlag, Berlin.
• Löbus, J.-U. A stationary Fleming-Viot type Brownian particle system. (2006) Preprint.
• Nair, M.G., Pollett, P. K. On the relationship between $mu$-invariant measures and quasi-stationary distributions for continuous-time Markov chains. Adv. Appl. Prob. 25 (1993), 82–102.
• Seneta, E. Non-Negative Matrices and Markov Chains. (1981) Springer-Verlag, Berlin.
• Seneta, E., Vere-Jones, D. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3 (1966), 403–434.
• Vere-Jones, D. Some limit theorems for evanescent processes. Austral. J. Statist. 11 (1969), 67–78.
• Yaglom, A. M. Certain limit theorems of the theory of branching stochastic processes (in Russian). Dokl. Akad. Nauk SSSR 56 (1947), 795–798.