Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

TL;DR

This paper proves that the Fleming-Viot process with multiple particles converges to the minimal quasi-stationary distribution of a subcritical Galton-Watson process as the number of particles increases, establishing a link between particle systems and conditioned distributions.

Contribution

It demonstrates the existence and uniqueness of an invariant measure for the Fleming-Viot process and its convergence to the minimal quasi-stationary distribution in the Galton-Watson case.

Findings

Unique invariant measure exists for each N

Empirical distribution converges to minimal quasi-stationary distribution

Convergence holds as number of particles tends to infinity

Abstract

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.