A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

TL;DR

This paper constructs a family of measure-valued diffusions with Poisson-Dirichlet stationary distributions, resolving a conjecture and linking to known diffusions and Chinese restaurant processes.

Contribution

It provides a pathwise construction of a two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions, extending previous work and resolving a conjecture.

Findings

Constructed measure-valued diffusions with Poisson-Dirichlet stationary distributions.

Connected the diffusions to continuum limits of Chinese restaurant processes.

Extended the class of known measure-valued diffusions with explicit stationary measures.

Abstract

We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet$(\alpha,\theta)$ distributions, for $\alpha\in (0,1)$ and $\theta\ge 0$. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(\alpha,0)$- and $(\alpha,\alpha)$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation to unit mass. In a companion paper, we show that the ranked masses of the measure-valued processes evolve according to a two-parameter family of diffusions introduced by Petrov (2009), extending work of Ethier and Kurtz (1981). These ranked-mass diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes.