Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent

Abstract

Regard an element of the set of ranked discrete distributions Δ := {(x 1, x 2,…):x 1≥x 2≥…≥ 0, ∑ i x i = 1} as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the Δ-valued Markov process in which pairs of clusters of masses {x i , x j } merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time −∞ with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991, 1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.