A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

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$(\alpha ,\theta )$ distributions, for $\alpha \in (0,1)$ and $\theta \ge 0$. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(\alpha ,0)$- and $(\alpha ,\alpha )$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation to unit mass. In a companion paper, we show that the ranked masses of the measure-valued processes evolve according to a two-parameter family of diffusions introduced by Petrov (2009), extending work of Ethier and Kurtz (1981). These ranked-mass diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes.

57 pages, 5 figures

References in corpus (4)

• Distributions of linear functionals of two parameter Poisson--Dirichlet random measures
• Alpha-diversity processes and normalized inverse-Gaussian diffusions
• Diffusions on a space of interval partitions: Poisson-Dirichlet stationary distributions
• Diffusions on a space of interval partitions: construction from marked Lévy processes

Cited by in corpus (2)

• Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions
• Dual process in the two-parameter Poisson-Dirichlet diffusion