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

Abstract

For a class of Λ-Fleming-Viot processes with underlying Brownian motion whose associated Λ-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 Λ-Fleming-Viot support process has one-sided modulus of continuity (with modulus function C 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 Λ-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.

Instantaneous support propagation for $Λ$-Fleming-Viot processes

“…The lookdown representation encodes a genealogy of the Fleming-Viot process, and often plays the role of the historical process for superprocesses. Using the lookdown representation, more recent work of Liu and Zhou [29,30] has established, among other properties, the compact support property, the one-sided modulus of continuity and Hausdorff dimension of the support process for the Λ-Fleming-Viot process with Brownian mutation with the associated Λ-coalescent coming down from infinity. For a measure-valued process, instantaneous support propagation occurs if the (closed) support of the random measure can reach arbitrarily far away within arbitrarily small time.…”

Existence, uniqueness and ergodicity for the centered Fleming-Viot process

“…As the support of the original Fleming-Viot process is compact at all positive times P F V ν −a.s. [36],…”

“…Proof. It is proved in [36] that the support of the Λ−Fleming-Viot process associated to a Λ−coalescent which comes down from infinity, is compact at all positive times. Our case corresponds to Kingman's coalescent.…”

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

“…Birkner and Blath [5] pointed out that the Λ-Fleming-Viot process does not have a compact support if the associated Λ-coalescent does not come down from infinity. Applying the lookdown construction, for a class of Λ-Fleming-Viot processes that come down from infinity Liu and Zhou [20,21] showed the compact support property and a one-sided modulus of continuity for the support at fixed time, and found the bounds of Hausdorff dimensions on the support and range. Note that the compact support property fails if the spatial motion allows jumps.…”

“…In this paper we are interested in improving the existing modulus of continuity in [21] for Λ-Fleming-Viot process. Given h(t) = t log (1/t) for t > 0, it is well known that √ 2h(x) is the exact modulus of continuity function for almost all Brownian paths.…”