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 L logL condition, it has a limiting law which is the size-biased Yaglom distribution (of the