Brownian Continuum Random Tree Research Articles

The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree

We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal Holder exponents of the height functions.

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

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.