Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ ₀, ξ ₁,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ _n ) on ℝ₊, and reproduce independently new particles according to a probability law p(ξ _n ) on ℕ. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E _ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments. This work was supported by the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012) and Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)