  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.
