Some support properties for a class of $𝛬$-Fleming–Viot processes

Liu, Huili; Zhou, Xiaowen

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1076-1101.

Pour une classe de processus de $𝛬$-Fleming–Viot avec dynamique brownienne sous-jacente dont les $𝛬$-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 $𝛬$-Fleming–Viot est majoré à tout temps positif $t$ par la fonction $C\sqrt{tlog(1/t)}$. 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 $𝛬$-coalescence, nous obtenons une borne supérieure uniforme sur la dimension de Hausdorff du support et de l’image.

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 $C\sqrt{tlog(1/t)}$) 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.

Mots clés : ${\varLambda }$-Fleming–Viot process, measure-valued process, ${\varLambda }$-coalescent, lookdown construction, ancestry process, compact support, modulus of continuity, Hausdorff dimension

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