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2020 Diffusions on a space of interval partitions: construction from marked Lévy processes
Noah Forman, Soumik Pal, Douglas Rizzolo, Matthias Winkel
Electron. J. Probab. 25(none): 1-46 (2020). DOI: 10.1214/20-EJP521

Abstract

Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.

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Noah Forman. Soumik Pal. Douglas Rizzolo. Matthias Winkel. "Diffusions on a space of interval partitions: construction from marked Lévy processes." Electron. J. Probab. 25 1 - 46, 2020. https://doi.org/10.1214/20-EJP521

Information

Received: 2 December 2019; Accepted: 6 September 2020; Published: 2020
First available in Project Euclid: 29 October 2020

MathSciNet: MR4169174
Digital Object Identifier: 10.1214/20-EJP521

Subjects:
Primary: 60J25, 60J60, 60J80
Secondary: 60G18, 60G52, 60G55

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