Diffusions on a space of interval partitions: Poisson-Dirichlet stationary distributions

Journal article

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.

Authors: Forman, N; Pal, S; Rizzolo, D; Winkel, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Annals of Probability
• Volume: 49
• Issue: 2
• Pages: 793 - 831
• Publication date: 2021-03-17
• Acceptance date: 2020-07-07
• DOI: 10.1214/20-AOP1460
• ISSN: 0091-1798
• Language: English
• Keywords: 60G18; math.PR; FFR; 60J60; 60G55; 60J80; 60G52; 60J25