Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes

Abstract

We show that for $0<\alpha <1$ and $\theta >-\alpha$, the Poisson-Dirichlet distribution with parameter $(\alpha ,\theta )$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $\alpha =0$.