Stochastic Flows and Marked Stable Processes

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 [Björnberg, Curien and Stefánsson 2022] arise naturally in this framework.