Branching random walks with random environments in time

Abstract

We consider a branching random walk on ℝ with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let \(\tilde Z_n (t)\) be its Laplace transform. We show the convergence of the free energy n−1log \(\tilde Z_n (t)\), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale \(\tilde Z_n (t)/\mathbb{E}[\tilde Z_n (t)|\xi ]\).