Branching random walks with random environments in time

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