Condensation in Fleming--Viot particle systems with fast selection mechanism

TL;DR

This paper analyzes the behavior of Fleming--Viot particle systems under rapid selection, revealing condensate formation and its evolution as the killing rate becomes infinitely large, with convergence proofs and dynamic descriptions.

Contribution

It provides a rigorous analysis of condensate formation and evolution in Fleming--Viot systems with fast selection, including convergence results and dynamic characterizations.

Findings

Condensate forms due to fast selection mechanism.

The condensate evolves according to a Markov chain influenced by mutation and killing rates.

Convergence of the particle system is established under specific asymptotic regimes.

Abstract

We study the Fleming--Viot particle system in a discrete state space, in the regime of a fast selection mechanism, namely with killing rates which grow to infinity. This asymptotics creates a time scale separation which results in the formation of a condensate of all particles, which then evolves according to a continuous-time Markov chain with jump rates depending in a nontrivial way on both the underlying mutation dynamics and the relative speed of growth to infinity of the killing rate between neighbouring sites. We prove the convergence of the particle system and completely describe the dynamics of the condensate in the case where the number of particles is kept fixed, and partially in the case when the number of particles goes to infinity together with (but slower than) the minimal killing rate.