Branching random walk with a random environment in time

Abstract: We consider a branching random walk on $\mathbb{R}$ with a stationary and ergodic environment $\xi=(\xi_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. For the case where the corresponding branching process ${Z_n(\mathbb{R})}$ $ (n\in\mathbb{N})$ is supercritical, we establish large deviation principles, central limit theorems and a local limit theorem for the sequence of counting measures ${Z_n}$, and prove that the position $R_n$ (resp. $L_n$) of rightmost (resp. leftmost) particles of generation $n$ satisfies a law of large numbers.