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 Lévy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.
Highlights
We define interval partitions, following Aldous [3, Section 17] and Pitman [12, Chapter 4].Definition 1.1
An interval partition is a set β of disjoint, open subintervals of some finite real interval [0, M ], that cover [0, M ] up to a Lebesgue-null set
We refer to the elements of an interval partition as its blocks
Summary
We define interval partitions, following Aldous [3, Section 17] and Pitman [12, Chapter 4].Definition 1.1. If N = i∈I δ(si,fi) is a point process of times si ∈ [0, S] and excursions fi of excursion lengths ζi (spindle heights), and X is a real-valued process with jumps ∆X(si) := X(si) − X(si−) = ζi at times si, i ∈ I, we define the interval partition SKEWER(y, N, X) at level y, as follows.
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