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

Regard an element of the set of ranked discrete distributions Delta := {(x( 1), x(2),...) : x(1) greater than or equal to x(2) greater than or equal to ... greater than or equal to 0, Sigma (i) x(i) = 1} as a fragmentation of unit mass into clusters of masses xi. The additive coalescent is the Delta -valued Markov process in which pairs of clusters of masses {x(i), x(j)} me rge into a cluster of mass x(i) + x(j) at rate x(i) + x(j). Aldous and Pitm an (1998) showed that a version of this process starting from time -infinit y 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 proc ess may be constructed analogously from a new family of inhomogeneous conti nuum random trees.