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

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(α,θ) distributions, for α∈(0,1) and θ≥0. These processes resolve a conjecture of Feng and Sun (Probab. Theory Related Fields 148 (2010) 501–525). We build on our previous work on (α,0)- and (α,α)-interval partition evolutions. The extension to general θ≥0 is achieved by the construction of a σ-finite excursion measure of a new measure-valued branching diffusion. Our measure-valued processes are Hunt processes on an incomplete subspace of the space of all probability measures and do not possess an extension to a Feller process. In a companion paper, we use generators to show that ranked masses evolve according to a two-parameter family of diffusions introduced by Petrov (Funktsional. Anal. i Prilozhen. 43 (2009) 45–66), extending work of Ethier and Kurtz (Adv. in Appl. Probab. 13 (1981) 429–452).