Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates sum to 1 is the natural state space for the process. In fact, the complementary set acts like an entrance boundary.
"A property of Petrov's diffusion." Electron. Commun. Probab. 19 1 - 4, 2014. https://doi.org/10.1214/ECP.v19-3684