Fleming-Viot processes : two explicit examples

TL;DR

This paper explores two explicit examples of Fleming-Viot processes, providing detailed formulas and analysis for a complete graph random walk and a two-state Markov chain, advancing understanding of these stochastic systems.

Contribution

It extends previous work by deriving explicit formulas and spectral properties for Fleming-Viot processes in specific discrete settings, including complete graphs and two-state chains.

Findings

Explicit formulas for invariant distribution and correlations on complete graphs

Spectral gap estimates for the Fleming-Viot process on complete graphs

Reduction of the two-state case to birth-death process analysis

Abstract

The purpose of this paper is to extend the investigation of the Fleming-Viot process in discrete space started in a previous work to two specific examples. The first one corresponds to a random walk on the complete graph. Due to its geometry, we establish several explicit and optimal formulas for the Fleming-Viot process (invariant distribution, correlations, spectral gap). The second example corresponds to a Markov chain in a two state space. In this case, the study of the Fleming-Viot particle system is reduced to the study of birth and death process with quadratic rates.