Dirichlet Forms of Diffusion Processes on the Thoma Simplex

We study a prominent two-parameter family of diffusion processes Xz,z' X z , z ′ on the infinite-dimensional Thoma simplex. This family was constructed by Borodin and Olshanski in 2007 and it closely resembles both the Ethier–Kurtz infinitely-many-neutral-alleles diffusion model on the Kingman simplex (1981) and Petrov’s subsequent extension (2007). The processes Xz,z' X z , z ′ have unique symmetrizing measures, namely, the boundary z -measures, which play the role of the Poisson–Dirichlet measures in this context. We establish the following behavior for Xz,z' X z , z ′ : Immediately after the initial moment, the processes jump into a dense face of the Thoma simplex and then remain there forever. In other words, this face acts as the natural state space for the diffusions, while the remaining points of the simplex serve as an entrance boundary. As a key intermediate step, we analyze the associated Dirichlet forms and provide a new description of them.