Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent

TL;DR

This paper establishes a coupling between two different representations of the additive coalescent, linking the Brownian CRT's fragmentation process with Brownian excursions, enhancing understanding of their relationship.

Contribution

It introduces a novel coupling between Aldous-Pitman's and Bertoin's constructions of the additive coalescent using the cut-tree exploration algorithm.

Findings

Unified framework for two representations of the additive coalescent

New insights into the genealogy of CRT fragmentation

Potential applications in stochastic process analysis

Abstract

We construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous \& Pitman, involves the components obtained by logging the Brownian Continuum Random Tree (CRT) by a Poissonian rain on its skeleton as time increases. The second one, due to Bertoin, involves the excursions above its running infimum of a linear-drifted standard Brownian excursion as its drift decreases. Our main tool is the use of an exploration algorithm of the so-called cut-tree of the Brownian CRT, which is a tree that encodes the genealogy of the fragmentation of the CRT.