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

We give a pathwise construction of a two-parameter family of purelyatomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet(alpha,theta) distributions, for alpha is an element of(0, 1) and theta >= 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 (alpha, 0)- and (alpha, alpha)-interval partition evolutions. The extension to general theta >= 0 is achieved by the construction of a sigma-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).