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(α, θ) distributions, for α ∈ (0, 1) and θ ≥ 0. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on (α, 0)-and (α, α)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.

Ranked masses in two-parameter Fleming-Viot diffusions

“…The construction of Fleming-Viot processes corresponding to EKP(α, θ) diffusions was one of the longest-standing open problems in the field. As noted in [16], identifying the evolution of ranked atom masses of the process constructed there with the EKP(α, θ) diffusion completes the argument that those processes are the associated Fleming-Viot processes. Since the mechanism by which new types are created by the Fleming-Viot process is explicit in the construction, this also explains how types are created in the EKP(α, θ) diffusion.…”

“…Introduction. Fleming-Viot processes corresponding to Petrov's [25] two-parameter extension of Ethier and Kurtz infinitely-many-neutral-alleles diffusion model [9] were recently constructed in [16,38]. The existence of these Fleming-Viot processes was conjectured in [11].…”

“…Efforts to construct them through analytic approaches such as Dirichlet forms encountered significant challenges that have not yet been overcome, though progress has been made [11,12,43]. In contrast to the analytic approaches, these processes are constructed path-wise in [16,38] using Poisson random measures. While these constructions give easy access to a number of interesting properties of the Fleming-Viot processes, particularly sample path properties, deriving an analytic characterization of the processes remains challenging.…”

“…In this paper, we initiate the study of the analytic properties of the processes constructed in [16,38]. Our main result is that we identify the evolution of the sequence of ranked atom masses in a Fleming-Viot process with parameters 0 < α < 1, θ > −α as the corresponding diffusion constructed by Petrov.…”

Dual process in the Two-parameter Poisson-Dirichlet diffusion

“…It is reversible with a twoparameter Poisson-Dirichlet stationary distribution PD(α, θ). A measure-valued diffusion version is constructed in Griffiths et al (2021) and another by Forman et al (2020) which uses a completely different construction from stable processes. The two-parameter Poisson-Dirichlet diffusion is an extension of the infinitely-many-alleles diffusion studied, in the case with α = 0, by Ethier and Kurtz (1981) who derive it as the continuous limit of a sequence of discrete time, finite-allele Wright-Fisher models.…”