Fleming-Viot particle system driven by a random walk on $\mathbb{N}$

TL;DR

This paper investigates a Fleming-Viot particle system driven by a negative drift random walk on natural numbers, showing convergence of the system's stationary measure to the minimal quasi-stationary distribution and identifying other distributions as metastable states.

Contribution

It demonstrates that the Fleming-Viot system's stationary measure converges to the minimal quasi-stationary distribution as the number of particles grows, linking metastability to other quasi-stationary distributions.

Findings

Convergence of FV stationary measure to minimal qsd as N increases

Identification of other qsd as metastable states

Analysis of FV system dynamics driven by a negative drift random walk

Abstract

Random walk on $\mathbb{N}$ with negative drift and absorption at 0, when conditioned on survival, has uncountably many invariant measures (quasi-stationary distributions, qsd) $\nu_c$. We study a Fleming-Viot(FV) particle system driven by this process and show that mean normalized densities of the FV unique stationary measure converge to the minimal qsd, $\nu_0$, as $N \to \infty$. Furthermore, every other qsd of the random walk ($\nu_c$, $c>0$) corresponds to a metastable state of the FV particle system.