A Two-parameter Family of Infinite-dimensional Diffusions in the Kingman Simplex

2007/08/17

Functional Analysis and Its Applications 43 (2009), no. 4, 279-296arXiv:0708.1930 [math.PR]

The aim of the paper is to introduce a two-parameter family of infinite-dimensional diffusion processes $X(\alpha,\theta)$ related to Pitman’s two-parameter Poisson-Dirichlet distributions $PD(\alpha,\theta)$. The diffusions $X(\alpha,\theta)$ are obtained in a scaling limit transition from certain finite Markov chains on partitions of natural numbers. The state space of $X(\alpha,\theta)$ is an infinite-dimensional simplex called the Kingman simplex. In the special case when parameter $\alpha$ vanishes, our finite Markov chains are similar to Moran-type model in population genetics, and our diffusion processes reduce to the infinitely-many-neutral-alleles diffusion model studied by Ethier and Kurtz (1981).

Our main results extend those of Ethier and Kurtz to the two-parameter case and are as follows: The Poisson-Dirichlet distribution $PD(\alpha,\theta)$ is a unique stationary distribution for the corresponding process $X(\alpha,\theta)$; the process is ergodic and reversible; the spectrum of its generator is explicitly described. The general two-parameter case seems to fall outside the setting of models of population genetics, and our approach differs in some aspects from that of Ethier and Kurtz. We also consider the case of degenerate series of parameters $\alpha$ and $\theta$ and conclude that the diffusions in finite-dimensional simplexes studied by Ethier and Kurtz (1981) arise as a special case of our two-parameter family of diffusions.

A dynamics of the four most common alleles. The hitting of finite-dimensional subspaces is evident
A dynamics of the four most common alleles. The hitting of finite-dimensional subspaces is evident