Exact Modulus of Continuities for \(\Lambda \)-Fleming–Viot Processes with Brownian Spatial Motion

For a class of \(\Lambda \) $\mathrm{\Lambda }$ -Fleming–Viot processes with Brownian spatial motion in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ whose associated \(\Lambda \) $\mathrm{\Lambda }$ -coalescents come down from infinity, we obtain sharp global and local moduli of continuity for the ancestral processes recovered from the associated lookdown representations. As applications, we establish both global and local moduli of continuity for the \(\Lambda \) $\mathrm{\Lambda }$ -Fleming–Viot support processes. In particular, if the \(\Lambda \) $\mathrm{\Lambda }$ -coalescent is the Beta \((2-\beta ,\beta )\) $(2-\beta ,\beta )$ coalescent for \(\beta \in (1,2]\) $\beta \in (1,2]$ with \(\beta =2\) $\beta =2$ corresponding to Kingman’s coalescent, then for \(h(t)=\sqrt{t\log (1/t)}\) $h(t)=\sqrt{tlog(1/t)}$ , the global modulus of continuity holds for the support process with modulus function \(\sqrt{2\beta /(\beta -1)}h(t)\) $\sqrt{2\beta /(\beta -1)}h(t)$ , and both the left and right local moduli of continuity hold for the support process with modulus function \(\sqrt{2/(\beta -1)}h(t)\) $\sqrt{2/(\beta -1)}h(t)$ .