Some support properties for a class of ${\varLambda}$-Fleming–Viot processes

Abstract

Using Donnelly and Kurtz's lookdown construction, we prove that the Lambda-Fleming-Viot process with underlying Brownian motion has a compact support at any fixed time provided that the associated Lambda-coalescent comes down from infinity not too slowly. We also find both upper and lower bounds on Hausdorff dimension for the support at any fixed time. When the associated Lambda-coalescent has a nontrivial Kingman component, the Hausdorff dimension for the support is exactly two at any fixed time. For such a Lambda-Fleming-Viot process, we further prove a one-sided modulus of continuity result for the ancestry process recovered from Donnelly and Kurtz's lookdown construction. As an application, we can prove that its support process also has the one-sided modulus of continuity (with modulus function C\sqrt{tlog(1/t)}) at any fixed time. In addition, we obtain that the support process is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over time interval [0,t) for all t>0. Under a mild condition on the Lambda-coalescence rates, we also find a uniform upper bound on Hausdorff dimension for the support and an upper bound on Hausdorff dimension for the range.