Stochastic Flows and Marked Stable Processes

TL;DR

This paper constructs a space-time partition using coupled stochastic squared Bessel flows, revealing connections to stable processes and excursion theory, and unifying several recent models in stochastic analysis.

Contribution

It introduces a novel construction linking squared Bessel flows, stable processes, and excursion theory, unifying various models of interval partitions and shredded disks.

Findings

Cells correspond to squared Bessel excursions with negative parameter

Partition relates to jumps of a spectrally positive stable process

Unifies models like interval partition evolutions and shredded disks

Abstract

We construct a random partition of the space-time plane $\mathbb{R}_+\times \mathbb{R}$ using two coupled stochastic squared Bessel flows, whose parameters differ by $\delta\in (0,2)$. We show that the cells of this partition correspond to squared Bessel excursions with a negative parameter $-\delta$ which are embedded within the jumps of a spectrally positive $(1+\frac\delta 2)$ stable process. In particular, we demonstrate that interval partition evolutions [Forman et. al. 2020] and stable shredded disks [Bj\"ornberg, Curien and Stef\'ansson 2022] arise naturally in this framework.