Stationary distributions for a class of generalized Fleming-Viot processes

TL;DR

This paper characterizes stationary distributions of a class of generalized Fleming-Viot processes with jump mechanisms, using stable random measures, Dirichlet measures, and measure-valued branching processes with immigration.

Contribution

It provides explicit descriptions of stationary distributions for generalized Fleming-Viot processes with specific jump mechanisms, linking them to stable measures and Dirichlet measures.

Findings

Stationary distributions are derived from normalized stable random measures.

Distributions are obtained via biased transformations and mixing with Dirichlet measures.

The approach leverages measure-valued branching processes with immigration.

Abstract

We identify stationary distributions of generalized Fleming-Viot processes with jump mechanisms specified by certain beta laws together with a parameter measure. Each of these distributions is obtained from normalized stable random measures after a suitable biased transformation followed by mixing by the law of a Dirichlet random measure with the same parameter measure. The calculations are based primarily on the well-known relationship to measure-valued branching processes with immigration.