Surviving particles for subcritical branching processes in random environment

The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives a.sa.s. These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.