## Journal of Applied Probability

- J. Appl. Probab.
- Volume 48, Number 2 (2011), 322-332.

### Quasistationary distributions and Fleming-Viot processes in finite spaces

Amine Asselah, Pablo A. Ferrari, and Pablo Groisman

#### Abstract

Consider a continuous-time Markov process with transition rates matrix *Q*
in the state space Λ ⋃ {0}. In the associated Fleming-Viot
process *N* particles evolve independently in Λ with transition
rates matrix *Q* until one of them attempts to jump to state 0. At this
moment the particle jumps to one of the positions of the other particles,
chosen uniformly at random. When Λ is finite, we show that the empirical
distribution of the particles at a fixed time converges as
*N* → ∞ to the distribution of a single particle at the same
time conditioned on not touching {0}. Furthermore, the empirical profile of the
unique invariant measure for the Fleming-Viot process with *N* particles
converges as *N* → ∞ to the unique quasistationary
distribution of the one-particle motion. A key element of the approach is to
show that the two-particle correlations are of order 1 / *N*.

#### Article information

**Source**

J. Appl. Probab. Volume 48, Number 2 (2011), 322-332.

**Dates**

First available in Project Euclid: 21 June 2011

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1308662630

**Digital Object Identifier**

doi:10.1239/jap/1308662630

**Mathematical Reviews number (MathSciNet)**

MR2840302

**Zentralblatt MATH identifier**

05918586

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J25: Continuous-time Markov processes on general state spaces

**Keywords**

Quasistationary distribution Fleming-Viot process

#### Citation

Asselah, Amine; Ferrari, Pablo A.; Groisman, Pablo. Quasistationary distributions and Fleming-Viot processes in finite spaces. J. Appl. Probab. 48 (2011), no. 2, 322--332. doi:10.1239/jap/1308662630. http://projecteuclid.org/euclid.jap/1308662630.