Two-Parameter Family of Infinite-Dimensional Diffusions on the Kingman Simplex

L. A. Petrov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 4, pp. 45-66

Résumé

We construct a two-parameter family of diffusion processes $\mathbf{X}_{\alpha,\theta}$ on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers. For $\alpha=0$, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981) in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present paper, we extend Ethier and Kurtz's main results to the two-parameter case. Namely, we show that the (two-parameter) Poisson–Dirichlet distribution $\mathrm{PD}(\alpha,\theta)$ is the unique stationary distribution for the process $\mathbf{X}_{\alpha,\theta}$ and that the process is ergodic and reversible with respect to $\mathrm{PD}(\alpha,\theta)$. We also compute the spectrum of the generator of $\mathbf{X}_{\alpha,\theta}$. The Wright–Fisher diffusions on finite-dimensional simplices turn out to be special cases of $\mathbf{X}_{\alpha,\theta}$ for certain degenerate parameter values.

Détail

Keywords: two-parameter Poisson–Dirichlet measure, Kingman graph
Mots-clés : diffusion process, Ewens–Pitman partition structure.