Diffusions on a space of interval partitions: construction from Bertoin's ${\tt BES}_0(d)$, $d\in(0,1)$

TL;DR

This paper represents Bertoin's measure-valued process within a space of interval partitions, connecting it to recent diffusion models constructed via spectrally positive stable Lévy processes and Bessel excursions.

Contribution

It introduces a new representation of Bertoin's ${ t BES}_0(d)$ process in the space of interval partitions, linking it to recent diffusion constructions from stable Lévy processes.

Findings

Bertoin's process can be represented as an interval partition diffusion.

The process is connected to spectrally positive stable Lévy processes with jumps marked by Bessel excursions.

This representation unifies different constructions of similar diffusions.

Abstract

In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable L\'evy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.