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

Noah Mills Forman, Soumik Pal, Douglas Rizzolo, Matthias Winkel

Abstract:

We introduce diffusions on a space of interval partitions of the unit inter-
val 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 sec-
ond step which requires explicit transition kernels and, what we call, pseudo-
stationarity. This allows us to study processes obtained from the original con-
struction via scaling and time-change. In a sequel paper we establish connec-
tions 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.