Continuous paths characterization for tree-valued Λ-resampling dynamics

Let $\mathcal V^\Lambda$ be the tree-valued $\Lambda$-resampling dynamics associated with a finite measure $\Lambda\in\mathcal M_f([0,1])$. Continuous-paths conjecture. The following are equivalent: (i) $\mathcal V^\Lambda$ has continuous paths; (ii) $\Lambda=\gamma\delta_0$ for some $\gamma>0$; and (iii) $\mathcal V^\Lambda$ is the tree-valued Fleming–Viot dynamics. The claim characterizes precisely when multiple-merger resampling disappears and the limiting tree-valued process has continuous paths.