Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct

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 Llog 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 a SDE with a drift term that can be seen as the instantaneous immigration.