Transition functions of diffusion processes with the Jack parameter on the Thoma simplex

TL;DR

This paper studies a family of infinite-dimensional diffusion processes on a simplex, showing their transition functions have continuous densities, extending previous results and using symmetric functions related to Laguerre polynomials.

Contribution

It proves the transition functions of these diffusion processes have continuous densities with respect to their symmetrising measures, generalizing earlier findings.

Findings

Transition functions have continuous densities with respect to symmetrising measures.

Generalization of earlier results by Ethier and Feng et al.

Uses a basis in symmetric functions related to Laguerre polynomials.

Abstract

The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski (arXiv:0706.1034; arXiv:0902.3395), and they include, as limit objects, the Ethier-Kurtz's infinitely-many-neutral-allels diffusion model (1981) and its extension found by Petrov (arXiv:0708.1930). Each process X from our family possesses a symmetrising measure M. Our main result is that the transition function of X has continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in the algebra of symmetric functions related to Laguerre polynomials.