Ranked masses in two-parameter Fleming-Viot diffusions

TL;DR

This paper proves that certain measure-valued Fleming-Viot diffusions with Poisson-Dirichlet stationary laws have ranked atom size processes that belong to a known two-parameter family of diffusions, confirming a conjecture and extending classical models.

Contribution

It completes the proof that ranked atom sizes of these diffusions are part of Petrov's two-parameter family, confirming a conjecture and extending Ethier and Kurtz's model.

Findings

Ranked atom size processes are members of Petrov's diffusion family.

Confirms a conjecture by Feng and Sun (2010).

Extends Ethier and Kurtz's model to the case >0.

Abstract

In previous work, we constructed Fleming--Viot-type measure-valued diffusions (and diffusions on a space of interval partitions of the unit interval $[0,1]$) that are stationary with the Poisson--Dirichlet laws with parameters $\alpha\in(0,1)$ and $\theta\geq 0$. In this paper, we complete the proof that these processes resolve a conjecture by Feng and Sun (2010) by showing that the processes of ranked atom sizes (or of ranked interval lengths) of these diffusions are members of a two-parameter family of diffusions introduced by Petrov (2009), extending a model by Ethier and Kurtz (1981) in the case $\alpha=0$. The latter diffusions are continuum limits of up-down Chinese restaurant processes.