Quasi-stationary distributions and Fleming-Viot processes in finite spaces

TL;DR

This paper studies the behavior of Fleming-Viot processes in finite spaces, demonstrating convergence of empirical distributions to quasi-stationary distributions and analyzing particle correlations.

Contribution

It establishes the convergence of empirical distributions and invariant measures of Fleming-Viot processes to quasi-stationary distributions in finite spaces.

Findings

Empirical distribution converges to conditioned single-particle distribution.

Invariant measure converges to the unique quasi-stationary distribution.

Two-particle correlations are of order 1/N.

Abstract

Consider a continuous time Markov chain with rates Q in the state space \Lambda\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in \Lambda with rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to \Lambda instantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When \Lambda is finite, we show that the empirical distribution of the particles at a fixed time converges as N\to\infty to the distribution of a single particle at the same time conditioned on non absorption. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N\to\infty to the unique quasi-stationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations is of order 1/N.