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

Let (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 law generated by the trajectories of (X-n)(n >= 0) starting at X-0 = i is an element of X. We study the asymptotic behaviour of the joint survival probability P-i (Z(n) > 0, X-n = j), j is an element of X as n -> +infinity in the critical and strongly, intermediate and weakly subcritical cases. (C) 2018 Elsevier B.V. All rights reserved.