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

### 2007/08/16

Leonid Petrov

*Functional Analysis and Its Applications 43 (2009), no. 4, 279-296* •

arXiv: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