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March 2021 Diffusions on a space of interval partitions: Poisson–Dirichlet stationary distributions
Noah Forman, Soumik Pal, Douglas Rizzolo, Matthias Winkel
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Ann. Probab. 49(2): 793-831 (March 2021). DOI: 10.1214/20-AOP1460

Abstract

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson–Dirichlet laws with parameters α,0 and α,α. The construction has two steps. The first is a general construction of interval partition processes obtained previously by decorating the jumps of a Lévy process with independent excursions. Here, we focus on the second step which requires explicit transition kernels and, what we call, pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (Adv. in Appl. Probab. 13 (1981) 429–452) and Petrov (Funktsional. Anal. i Prilozhen. 43 (2009) 45–66). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step toward resolving longstanding conjectures by Feng and Sun on measure-valued Poisson–Dirichlet diffusions and by Aldous on a continuum-tree-valued diffusion.

Acknowledgments

This research is partially supported by NSF Grants DMS-1204840, DMS-1308340, DMS-1612483, DMS-1855568, UW-RRF Grant A112251, EPSRC Grant EP/K029797/1, NSERC RGPIN-2020-06907.

Citation

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Noah Forman. Soumik Pal. Douglas Rizzolo. Matthias Winkel. "Diffusions on a space of interval partitions: Poisson–Dirichlet stationary distributions." Ann. Probab. 49 (2) 793 - 831, March 2021. https://doi.org/10.1214/20-AOP1460

Information

Received: 1 October 2019; Revised: 1 July 2020; Published: March 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.1214/20-AOP1460

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

Keywords: Aldous diffusion , Excursion theory , infinitely-many-neutral-alleles model , interval partition , Poisson–Dirichlet distribution

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 2 • March 2021
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