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

TL;DR

This paper constructs and analyzes two related diffusions on interval partitions with Poisson-Dirichlet stationary distributions, linking them to Levy processes, Bessel excursions, and continuum limits of Chinese restaurant processes, advancing understanding of diffusions on real trees.

Contribution

It introduces a novel construction of diffusions on interval partitions with specific Poisson-Dirichlet stationary laws, connecting Levy processes, Bessel excursions, and continuum limits.

Findings

Constructed diffusions are stationary with Poisson-Dirichlet laws.

Linked diffusions to spectrally positive Levy processes and Bessel excursions.

Progress towards diffusion models on the space of real trees.

Abstract

We construct a pair of related diffusions on a space of interval partitions of the unit interval $[0,1]$ that are stationary with the Poisson-Dirichlet laws with parameters (1/2,0) and (1/2,1/2) respectively. These are two particular cases of a general construction of such processes obtained by decorating the jumps of a spectrally positive L\'evy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partitions are members of a two parameter family of diffusions 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 describing a diffusion on the space of real trees whose existence has been conjectured by Aldous.