Stationary Markov processes related to stable Ornstein-Uhlenbeck processes and the additive coalescent

We consider some classes of stationary, counting-measure-valued Markov processes and their companions under time reversal. Examples arise in the Lévy-Itô decomposition of stable Ornstein-Uhlenbeck processes, the large-time asymptotics of the standard additive coalescent, and extreme value theory. These processes share the common feature that points in the support of the evolving counting measure are born or die randomly, but each point follows a deterministic flow during its lifetime.