The survival probability of critical and subcritical branching processes in finite state space Markovian environment

摘要

Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environmentdefined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite statespace $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d.environment to the Markovian one the classical classification of the branchingprocesses into critical and strongly, intermediate and weakly subcriticalstates. In all these cases, we study the asymptotic behaviour of theprobability that $Z_n>0$ as $n\to+\infty$.