Speed of coming down from infinity for $\Lambda$-Fleming-Viot initial support

TL;DR

This paper investigates how quickly the support of certain $ extLambda$-Fleming-Viot processes becomes finite over time, linking the rate to initial measure tails and coalescent speeds.

Contribution

It provides asymptotic characterizations of the support coming down from infinity for $ extLambda$-Fleming-Viot processes with Brownian motion.

Findings

Support becomes finite immediately after t>0

Rates depend on initial measure tail distribution

Rates are linked to coalescent coming down speed

Abstract

The $\Lambda$-Fleming-Viot process is a probability measure-valued process that is dual to a $\Lambda$-coalescent that allows multiple collisions. In this paper, we consider a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion and with associated $\Lambda$-coalescents that come down from infinity. Notably, these processes have the compact support property: the support of the process becomes finite as soon as $t>0$, even though the initial measure has unbounded support. We obtain asymptotic results characterizing the rates at which the initial supports become finite. The rates of coming down are expressed in terms of the asymptotic inverse function of the tail distribution of the initial measure and the speed function of coming down from infinity for the corresponding $\Lambda$-coalescent.