Stochastic Flows and Marked Stable Processes

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.