Surviving particles for subcritical branching processes in random environment

TL;DR

This paper analyzes the survival probabilities and asymptotic behaviors of subcritical branching processes in random environments, distinguishing between different subcritical regimes and providing new insights into their long-term properties.

Contribution

It introduces new asymptotic results for weakly subcritical BPREs and characterizes environment sequences conditioned on survival, advancing understanding of their quasistationary distributions.

Findings

Survival probability proportional to initial particles in SS+IS cases

Conditional on survival, only one initial particle survives almost surely in SS+IS cases

Different asymptotics and environment characterizations for WS case

Abstract

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). %Descendances of particles for BPRE are not independent. 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 $a.s.$ 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.