DIFFUSIONS ON A SPACE OF INTERVAL PARTITIONS: POISSON-DIRICHLET STATIONARY DISTRIBUTIONS

• 成果类型: Article
• 署名作者: Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias
• 署名单位: McMaster University; University of Washington; University of Washington Seattle; University of Delaware; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1460
• 发表日期: 2021
• 页码: 793-831
• 关键词: infinite-dimensional diffusions; FAMILY; time

摘要

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters (a, 0) and (a, a). The construction has two steps. The first is a general construction of interval partition processes obtained previously by decorating the jumps of a Levy process with independent excursions. Here, we focus on the second step which requires explicit transition kernels and, what we call, pseudostationarity. 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 continuumtree-valued diffusion.