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 $\left(\mathit{\alpha },0\right)$ and $\left(\mathit{\alpha },\mathit{\alpha }\right)$. 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.