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For a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motionin $\mathbb{R}^d$ whose associated $\Lambda$-coalescents come down frominfinity, we obtain sharp global and local modulus of continuities for theancestry processes recovered from the lookdown representations. Asapplications, we prove both global and local modulus of continuities for the$\Lambda$-Fleming-Viot support processes. In particular, if the$\Lambda$-coalescent is the Beta$(2-\beta,\beta)$ coalescent for$\beta\in(1,2]$ with $\beta=2$ corresponding to Kingman's coalescent, then for$h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for thesupport process with modulus function $\sqrt{2\beta/(\beta-1)}h(t)$, and boththe left and right local modulus of continuities hold for the support processwith modulus function $\sqrt{2/(\beta-1)}h(t)$.

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