The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

TL;DR

This paper analyzes the fragmentation process of the Brownian excursion and stable tree, providing a detailed description of how fragments disintegrate and characterizing the sizes of fragments near total disintegration.

Contribution

It introduces a novel representation of tagged fragment masses using Doob transformations and describes the asymptotic behavior of fragment sizes in the Brownian fragmentation.

Findings

Representation of tagged fragment mass via Doob transformation

Description of typical fragment disintegration process

Limit theorem for fragment sizes near dust

Abstract

We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.