Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Huili Liu and Xiaowen Zhou

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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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 1076-1101.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759240

Digital Object Identifier
doi:10.1214/13-AIHP598

Mathematical Reviews number (MathSciNet)
MR3365973

Zentralblatt MATH identifier
1334.60182

Subjects
Primary: 60G57: Random measures
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G17: Sample path properties

Keywords
${\varLambda}$-Fleming–Viot process Measure-valued process ${\varLambda}$-coalescent Lookdown construction Ancestry process Compact support Modulus of continuity Hausdorff dimension

Citation

Liu, Huili; Zhou, Xiaowen. Some support properties for a class of ${\varLambda}$-Fleming–Viot processes. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 1076--1101. doi:10.1214/13-AIHP598. https://projecteuclid.org/euclid.aihp/1435759240


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