Open Access
2023 Diffusions on a space of interval partitions: the two-parameter model
Noah Forman, Douglas Rizzolo, Quan Shi, Matthias Winkel
Author Affiliations +
Electron. J. Probab. 28: 1-46 (2023). DOI: 10.1214/23-EJP946


We introduce and study interval partition diffusions with Poisson–Dirichlet(α,θ) stationary distribution for parameters α(0,1) and θ0. This extends previous work on the cases (α,0) and (α,α) and builds on our recent work on measure-valued diffusions. Our methods for dealing with general θ0 allow us to strengthen previous work on the special cases to include initial interval partitions with dust. In contrast to the measure-valued setting, we can show that this extended process is a Feller process improving on the Hunt property established in that setting. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. Indeed, by studying their infinitesimal generator on suitable quasi-symmetric functions, we relate them to diffusions obtained as scaling limits of composition-valued up-down chains.

Funding Statement

This research is partially supported by NSF grants DMS-1204840, DMS-1444084, DMS-1855568, UW-RRF grant A112251, EPSRC grant EP/K029797/1, NSERC RGPIN-2020-06907, National Key R&D Program of China (No. 2022YFA1006500), National Natural Science Foundation of China (No. 12288201), NSFC grant 12288201.


We thank Soumik Pal for his contributions in early discussions of this project.


Download Citation

Noah Forman. Douglas Rizzolo. Quan Shi. Matthias Winkel. "Diffusions on a space of interval partitions: the two-parameter model." Electron. J. Probab. 28 1 - 46, 2023.


Received: 31 July 2022; Accepted: 4 April 2023; Published: 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583068
Digital Object Identifier: 10.1214/23-EJP946

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

Keywords: branching processes , Chinese restaurant process , infinitely-many-neutral-alleles model , interval partition , regenerative composition structure , self-similar diffusion

Vol.28 • 2023
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