@article{lambert_quasistationary_2007,
Author = {Lambert, Amaury},
Title = {Quasistationary distributions and the continuous-state
branching process conditioned to be never extinct},
Journal = {Electronic Journal of Probability},
Volume = {12},
Pages = {420--446},
abstract = {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 LlogL
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.},
doi = {10.1214/EJP.v12-402},
year = 2007
}