Ranked masses in two-parameter Fleming–Viot diffusions

Previous work 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 respect to the Poisson–Dirichlet random measures with parameters $\alpha \in (0,1)$ and $\theta >-\alpha$ . In this paper, we complete the proof that these processes resolve a conjecture by Feng and Sun [Probab. Theory Related Fields 148 (2010), pp. 501–525] 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 [Funct. Anal. Appl. 43 (2009), pp. 279–296], extending a model by Ethier and Kurtz [Adv. in Appl. Probab. 13 (1981), pp. 429–452] in the case $\alpha =0$ .