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 distributions, for and . These processes resolve a conjecture of Feng and Sun (Probab. Theory Related Fields 148 (2010) 501–525). We build on our previous work on - and -interval partition evolutions. The extension to general 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).
Funding Statement
This research is partially supported by NSF Grants DMS-1204840, DMS-1308340, DMS-1612483, DMS-1855568, UW-RRF grant A112251, EPSRC Grant EP/K029797/1.
Citation
Noah Forman. Douglas Rizzolo. Quan Shi. Matthias Winkel. "A two-parameter family of measure-valued diffusions with Poisson–Dirichlet stationary distributions." Ann. Appl. Probab. 32 (3) 2211 - 2253, June 2022. https://doi.org/10.1214/21-AAP1732
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