Dirichlet forms of diffusion processes on Thoma simplex

TL;DR

This paper analyzes a family of diffusion processes on the infinite-dimensional Thoma simplex, revealing their boundary behavior and providing a new description of their Dirichlet forms, advancing understanding of infinite-dimensional stochastic processes.

Contribution

It introduces a new description of the Dirichlet forms for diffusions on the Thoma simplex and characterizes their boundary behavior, extending prior models in infinite-dimensional diffusion theory.

Findings

Diffusions jump into a dense face of the Thoma simplex immediately after start.

The face acts as a natural state space for the diffusions.

A new description of the Dirichlet forms for the processes is provided.

Abstract

We study a prominent two-parametric family of diffusion processes $X_{z,z'}$ on an infinite-dimensional Thoma simplex. The family was constructed by Borodin and Olshanski in 2007 and it closely resembles Ethier-Kurtz's infinitely-many-neutral-allels diffusion model on Kingman simplex (1981) and Petrov's extension of Ethier-Kurtz's model (2007). The processes $X_{z,z'}$ have unique symmetrizing measures, namely, the boundary $z$-measures, which play the role of Poisson-Dirichlet measures in our context. We establish the following behavior of diffusions $X_{z,z'}$: immediately after the initial moment they jump into a dense face of Thoma simplex and then always stay there. In other words, the face acts as a natural state space for the diffusions, while other points of the simplex act like an entrance boundary for our process. As a key intermediate step we study the Dirichlet forms of the diffusions $X_{z,z}$ and find a new description for them.