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Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. (English) Zbl 1127.60082

Summary: We consider continuous-state branching (CB) processes which become extinct (i.e., hit 0) with positive probability. We characterize all the quasi-stationary distributions (QSD) for the CB-process as a stochastically monotone family indexed by a real number. We prove that the minimal element of this family is the so-called Yaglom quasi-stationary distribution, that is, the limit of one-dimensional marginals conditioned on being nonzero.
Next, we consider the branching process conditioned on not being extinct in the distant future, or \(Q\)-process, defined by means of Doob \(h\)-transforms. We show that the \(Q\)-process is distributed as the initial CB-process with independent immigration, and that under the \(L \log L\) condition, it has a limiting law which is the size-biased Yaglom distribution (of the CB-process). More generally, we prove that for a wide class of nonnegative Markov processes absorbed at 0 with probability 1, the Yaglom distribution is always stochastically dominated by the stationary probability of the \(Q\)-process, assuming that both exist. Finally, in the diffusion case and in the stable case, the \(Q\)-process solves an SDE with a drift term that can be seen as the instantaneous immigration.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K05 Renewal theory
60F05 Central limit and other weak theorems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G18 Self-similar stochastic processes