Exact modulus of continuities for $\Lambda$-Fleming-Viot processes with Brownian spatial motion

TL;DR

This paper establishes precise global and local modulus of continuity results for $ ext{Lambda}$-Fleming-Viot processes with Brownian motion, especially for Beta coalescents, revealing detailed regularity properties of their support and ancestry processes.

Contribution

It provides the first sharp global and local modulus of continuity results for $ ext{Lambda}$-Fleming-Viot processes with Brownian motion, including explicit formulas for Beta coalescents.

Findings

Global modulus of continuity with specific bounds for support processes.

Local modulus of continuity results for support processes.

Explicit formulas for Beta coalescents with parameter $eta$.

Abstract

For a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $\Lambda$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $\Lambda$-Fleming-Viot support processes. In particular, if the $\Lambda$-coalescent is the Beta$(2-\beta,\beta)$ coalescent for $\beta\in(1,2]$ with $\beta=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the support process with modulus function $\sqrt{2\beta/(\beta-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(\beta-1)}h(t)$.