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August 2015 Some support properties for a class of ${\varLambda}$-Fleming–Viot processes
Huili Liu, Xiaowen Zhou
Ann. Inst. H. Poincaré Probab. Statist. 51(3): 1076-1101 (August 2015). DOI: 10.1214/13-AIHP598

Abstract

For a class of ${\varLambda}$-Fleming–Viot processes with underlying Brownian motion whose associated ${\varLambda}$-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 ${\varLambda}$-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 $\varLambda$-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.

Pour une classe de processus de ${\varLambda}$-Fleming–Viot avec dynamique brownienne sous-jacente dont les ${\varLambda}$-coalescents associés descendent de l’infini, nous obtenons une borne supérieure sur le module de continuité des processus ancestraux définis par la construction look-down de Donnelly et Kurtz. Comme applications, nous obtenons que le module de continuité du processus ${\varLambda}$-Fleming–Viot est majoré à tout temps positif $t$ par la fonction $C\sqrt{t\log(1/t)}$. Nous montrons aussi que le support est simultanément compact pour tout temps positif, et, en cas de compacité au temps initial, l’image est uniformément compacte sur tout intervalle de temps fini. En plus, sous une condition faible sur les taux de ${\varLambda}$-coalescence, nous obtenons une borne supérieure uniforme sur la dimension de Hausdorff du support et de l’image.

Citation

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Huili Liu. Xiaowen Zhou. "Some support properties for a class of ${\varLambda}$-Fleming–Viot processes." Ann. Inst. H. Poincaré Probab. Statist. 51 (3) 1076 - 1101, August 2015. https://doi.org/10.1214/13-AIHP598

Information

Received: 21 September 2013; Accepted: 25 December 2013; Published: August 2015
First available in Project Euclid: 1 July 2015

zbMATH: 1334.60182
MathSciNet: MR3365973
Digital Object Identifier: 10.1214/13-AIHP598

Subjects:
Primary: 60G57
Secondary: 60G17, 60J80

Rights: Copyright © 2015 Institut Henri Poincaré

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Vol.51 • No. 3 • August 2015
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