Two-parameter Family of Diffusion Processes in the Kingman Simplex

The aim of the paper is to introduce a two-parameter family of infinitedimensional diffusion processes X related to Pitman’s two-parameter PoissonDirichlet distributions P. The diffusions X are obtained in a scaling limit transition from certain finite Markov chains on partitions of natural numbers. The state space of X is an infinite-dimensional simplex called the Kingman simplex. In the special case when parameter α 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 P is a unique stationary distribution for the corresponding process X; 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.