Asymptotics of certain coagulation-fragmentation processes and invariant Poisson-Dirichlet measures

We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the part with probability $\beta_s$ according to a law $\sigma$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $\sigma$ is the uniform measure then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invariant measures. We also derive transience and recurrence criteria for these chains.