In 2009, Petrov constructed a two-parameter extension of Ethier and Kurtz’s infinitely-many-neutral-alleles diffusions. These processes have the Poisson-Dirichlet distributions as unique stationary measures and are limits of some Markov chains driven by Chinese Restaurant-type dynamics. Recent works of Forman, Pal, Rizzolo, Shi, and Winkel then constructed ordered analogues of Petrov’s diffusions. These possess the analogous stationary distributions, but lacked the connection to the Chinese Restaurant Process. This connection, which appears as a conjecture of Rogers and Winkel, has since been established through joint efforts.

In this talk, I will discuss my contribution to this conjecture – namely, a construction of the ordered diffusions based on Chinese-restaurant dynamics. This construction is simpler than the original one, which takes a pathwise approach using marked Lévy processes, and allows for a detailed analysis of the diffusions. For example, we can compute explicitly the separation distance between the diffusion and its stationary distribution, under the worst initial condition.

Based on joint works with Douglas Rizzolo and Valentin Féray.

Ponente

Kelvin Rivera, Investigador Postdoctoral, IIMAS