Scaling Limit of the Fleming-Viot Multi-Colour Process

TL;DR

This paper studies the scaling limit of a multi-colour Fleming-Viot process with genetic information, showing convergence to a population genetics Fleming-Viot process as the number of particles grows large.

Contribution

It establishes the convergence of the scaled multi-colour Fleming-Viot process with genetic information to a classical population genetics Fleming-Viot process in the large particle limit.

Findings

Convergence of the genetic information process after rescaling time by N

Extension to Brownian motion with boundary catalyst killing

Different limiting process from classical population genetics

Abstract

We consider the $N$-particle Fleming-Viot process associated to a normally reflected diffusion with soft catalyst killing. The Fleming-Viot multi-colour process is obtained by attaching genetic information to the particles in the Fleming-Viot process. We establish that, after rescaling time by $t\mapsto Nt$, this genetic information converges to the (very different) Fleming-Viot process from population genetics, as $N\rightarrow\infty$. An extension is provided to dynamics given by Brownian motion with hard catalyst killing at the boundary of its domain.