Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman’s (1978) representation theorem (J. Lond. Math. Soc. (2) 18 374–380) states that any exchangeable partition of ℕ can be represented as a paintbox based on random mass-partition. Similarly, composition (i.e., ordered ℕ) an interval-partition (Gnedin (1997) Ann. Probab. 25 1437–1450). Our first main result is coalescent process (not necessarily Markovian) nondecreasing valued in interval-partitions, called nested interval-partition, generalizing the notion comb metric space introduced Lambert and Uribe Bravo (2017) (p-Adic Numbers Ultrametric Anal. Appl. 9 22–38) to represent compact ultrametric spaces. As special case, we show Λ-coalescent obtained from unique interval Λ-comb, which Markovian with explicit transitions. This directly relates flow bridges Bertoin Le Gall (2003) (Probab. Theory Related Fields 126 261–288). We also display particularly simple description so-called evolving (Pfaffelhuber Wakolbinger (2006) Stochastic Process. 116 1836–1859) by comb-valued Markov process. Next, prove measure U, under mild measure-theoretic assumptions leaf set tree composed separable subtree backbone, are grafted additional subtrees, act star-trees standpoint sampling. Displaying this weak isometry requires us extend Gromov-weak topology Greven, Pfaffelhuber Winter (2009) 145 285–322), was initially designed for spaces, nonseparable It allows such there (1) indistinguishable U topology; weakly isometric if has complete backbone; (3) separable.