On the Stationary Measure for Markov Branching Processes

A previous study determined criteria ensuring that a probability distribution supported in positive integers is the limiting conditional law of a subcritical Markov branching process. It is known that there is an close connection between the limiting conditional law and the stationary measure of the transition semigroup. This paper revisits that theme of by seeking tractable criteria ensuring that a sequence on positive integers is the stationary measure of a subcritical or critical Markov branching process. These criteria are illustrated with several examples. The subcritical case motivates consideration of the Sibuya distribution, leading to the demonstration that members of a certain family of complete Bernstein functions, in fact, are Thorin–Bernstein. The critical case involves deriving a notion of the limiting law of population size given that extinction occurs at a precise future time. Examples are given, and some show an interesting relation between stationary measures and Hausdorff moment sequences.