Non-extinction of a Fleming-Viot particle model

TL;DR

This paper proves that in a Fleming-Viot particle model with Brownian motion inside a domain, the particles do not all hit the boundary simultaneously in finite time, ensuring stability of the population distribution.

Contribution

It establishes a non-extinction result for the Fleming-Viot particle system in Lipschitz domains and derives a limit theorem for the empirical distribution.

Findings

Particles do not hit the boundary simultaneously in finite time.

The empirical distribution converges to a limit.

The model remains stable over time.

Abstract

We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.