Multiple Poisson-Dirichlet diffusions on generalized Kingman simplices

TL;DR

This paper introduces a new class of infinite-dimensional diffusions on generalized Kingman simplices, modeling the evolution of frequencies of multiple types with labels, extending classical distributions and connecting to Poisson-Dirichlet measures.

Contribution

It constructs a novel infinite-dimensional diffusion process on generalized Kingman simplices with a new stationary distribution extending Poisson-Dirichlet distributions.

Findings

The diffusion process converges to a limit described by a specific infinitesimal operator.

The stationary measure is a new multiple Poisson-Dirichlet distribution.

Special cases recover known models like the two-mark diffusion and the infinitely-many-neutral-alleles model.

Abstract

We construct a new class of infinite-dimensional diffusions taking values in a generalized Kingman simplex. Our model describes the temporal evolution of the relative frequencies of infinitely-many types which are "labeled" by an arbitrary finite number of marks or colors, but "unlabeled" within each mark. We start with a finite-dimensional construction which extends to Wright-Fisher diffusions a self-similarity property known for Dirichlet distributions, and corresponds to a multiple skew-product representation of the Wright-Fisher diffusion relative to the marks in the population. After ranking decreasingly the frequencies within each mark, we identify the limit in distribution of the resulting diffusion when the number of types for each mark goes to infinity, and describe its infinitesimal operator. The limiting process reduces to a diffusion in the Thoma simplex in the special case of only two marks, whereas the infinitely-many-neutral-alleles model is recovered when all frequencies have the same mark. The stationary measure of the limit diffusion is shown to be the recently introduced multiple Poisson-Dirichlet distribution, which extends Kingman's Poisson-Dirichlet distribution and is the de Finetti representing measure for a family of random partitions whose elements are marked.