Quantitative results for the Fleming-Viot Particle system and quasi-stationary distributions in discrete space

TL;DR

This paper establishes exponential convergence rates to equilibrium and quasi-stationary distributions for discrete Fleming-Viot particle systems, providing explicit quantitative estimates and illustrating with concrete examples.

Contribution

It introduces a method to quantify the convergence rate of Fleming-Viot systems to equilibrium and quasi-stationary distributions using Wasserstein coupling, with explicit bounds.

Findings

Exponential convergence to equilibrium in Wasserstein distance.

Explicit rate of convergence to quasi-stationary distribution.

Uniform convergence of Fleming-Viot process to conditioned process.

Abstract

We show, for a class of discrete Fleming-Viot (or Moran) type particle systems, that the convergence to the equilibrium is exponential for a suitable Wassertein coupling distance. The approach provides an explicit quantitative estimate on the rate of convergence. As a consequence, we show that the conditioned process converges exponentially fast to a unique quasi-stationary distribution. Moreover, by estimating the two-particle correlations, we prove that the Fleming-Viot process converges, uniformly in time, to the conditioned process with an explicit rate of convergence. We illustrate our results on the examples of the complete graph and of N particles jumping on two points.