Instantaneous support propagation for $\Lambda$-Fleming-Viot processes

TL;DR

This paper proves that the support of a $ ext{Lambda}$-Fleming-Viot process with L\'evy mutation propagates instantaneously, ensuring the support at any positive time is contained within the initial support, using a lookdown particle representation.

Contribution

It establishes the instantaneous support propagation property for $ ext{Lambda}$-Fleming-Viot processes with L\'evy mutation, a novel result in measure-valued population models.

Findings

Support propagates instantaneously at any positive time.

Support at time t is contained within the initial support.

Uses Donnelly-Kurtz's lookdown particle representation.

Abstract

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with L\'evy mutation and resampling mechanism associated to a general $\Lambda$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(\nu * Z_t) \subseteq S(Z_t)$, where $\nu$ is the L\'evy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process.