A Two-Parameter Family of Infinite-Dimensional Diffusions in the Kingman Simplex

Understand

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).