On criteria of disconnectedness of $\Lambda$-Fleming-Viot support

Xiaowen Zhou

doi:10.1214/ecp.v19-3208

Abstract

The totally disconnectedness of support for super Brownian motion in high dimensions is well known. In this paper, we prove that similar results also hold for $\Lambda$-Fleming-Viot process with Brownian spatial motion provided that the associated $\Lambda$-coalescent does not come down from infinity fast enough. Our proof is another application of the lookdown particle representation for $\Lambda$-Fleming-Viot process. We also discuss the disjointness of independent $\Lambda$-Fleming-Viot supports and ranges in high dimensions. The disconnectedness of the $\Lambda$-Fleming-Viot support remains open in certain low dimensions.

Highlights

It is well known that in dimension one superBrownian motion is absolutely continuous with respect to Lebesgue measure, and in dimensions two and above it is a singular random measure with its support of Hausdorff dimension two
We prove that similar results hold for ΛFleming-Viot processes with Brownian spatial motion provided that the associated Λ-coalescent comes down from infinity fast enough
For a d-dimensional superBrownian motion with binary branching, it is shown by Perkins [18] that at any fixed positive time its support is totally disconnected, i.e. the support contains no nontrivial connected component, in dimension four or above and its support is totally disconnected, uniformly for all positive times, in dimension six or above; see Section III.6 of Perkins [19]

Summary

Introduction

It is well known that in dimension one superBrownian motion is absolutely continuous with respect to Lebesgue measure, and in dimensions two and above it is a singular random measure with its support of Hausdorff dimension two. We prove that similar results hold for ΛFleming-Viot processes with Brownian spatial motion provided that the associated Λ-coalescent comes down from infinity fast enough.

Results

Conclusion