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

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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 βₘ (if the sampled parts are distinct) or splitting the part with probability βₛ according to a law σ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if σ 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.