A TWO-PARAMETER FAMILY OF MEASURE-VALUED DIFFUSIONS WITH POISSON–DIRICHLET STATIONARY DISTRIBUTIONS

Noah Mills Forman, Douglas Rizzolo, Quan Shi

Abstract:

We give a pathwise construction of a two-parameter family of purely-
atomic-measure-valued diffusions in which ranked masses of atoms are sta-
tionary with the Poisson–Dirichlet(α,θ) distributions, for α ∈ (0,1) and
θ ≥ 0. These processes resolve a conjecture of Feng and Sun (Probab. The-
ory 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).