Some support properties for a class of Lambda-Fleming-Viot processes

TL;DR

This paper establishes continuity properties and bounds for the support of Lambda-Fleming-Viot processes with Brownian motion, showing support compactness, uniform bounds, and Hausdorff dimension estimates under certain conditions.

Contribution

It proves a one-sided modulus of continuity for ancestry processes and derives support compactness and Hausdorff dimension bounds for a class of Lambda-Fleming-Viot processes.

Findings

Support process has one-sided modulus of continuity with modulus C√(t log(1/t)).

Support is compact at all positive times and uniformly over finite intervals.

Upper bounds on Hausdorff dimension of support and range under mild conditions.

Abstract

For a class of $\Lambda$-Fleming-Viot processes with underlying Brownian motion whose associated $\Lambda$-coalescents come down from infinity, we prove a one-sided modulus of continuity result for their ancestry processes recovered from the lookdown construction of Donnelly and Kurtz. As applications, we first show that such a $\Lambda$-Fleming-Viot support process has one-sided modulus of continuity (with modulus function $C\sqrt{t\log(1/t)}$) at any fixed time. We also show that the support is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over any finite time interval. In addition, under a mild condition on the $\Lambda$-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.