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

International audienceLet (Z n) n 0 be a branching process in a random environment defined by a Markov chain (X n) n 0 with values in a finite state space X. Let P i be the probability generated by the trajectories of (X n) n 0 starting at X 0 = i ∈ X. We study the asymptotic behaviour of the joint survival probability P i (Z n > 0 , X n = j), j ∈ X as n → +∞ in the critical and strongly, intermediate and weakly subcritical cases