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

Abstract

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(\alpha ,0)$ and $(\alpha ,\alpha )$. 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 (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards resolving longstanding conjectures by Feng and Sun on measure-valued Poisson-Dirichlet diffusions, and by Aldous on a continuum-tree-valued diffusion.

50 pages, 8 figures. Following arXiv:1909.02584 [math.PR], this is the second in a sequence of three papers that will collectively supersede arXiv:1609.06706 [math.PR]

References in corpus (2)

• Alpha-diversity processes and normalized inverse-Gaussian diffusions
• On the Aldous diffusion on Continuum Trees. I