Dirichlet forms of diffusion processes on Thoma simplex
Abstract
We study a prominent two-parametric family of diffusion processes Xz,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 Xz,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 Xz,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 Xz,z and find a new description for them.
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