The compact support property for the $\Lambda$-Fleming-Viot process with underlying Brownian motion

TL;DR

This paper proves that the $ ext{Lambda}$-Fleming-Viot process with Brownian motion has a compact support at fixed times under certain coalescent conditions, and estimates the Hausdorff dimension of this support.

Contribution

It establishes conditions for compact support of the process and provides bounds on the Hausdorff dimension, advancing understanding of its geometric properties.

Findings

Support is compact if the coalescent comes down from infinity sufficiently quickly.

Bounds on the Hausdorff dimension of the support are derived.

The lookdown construction is used to analyze the process.

Abstract

Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the $\Lambda$-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding $\Lambda$-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support.