Large and moderate deviations for a ${\mathbb{R}}^d$-valued branching random walk with a random environment in time

Abstract

We consider a ${\mathbb{R}}^d$-valued branching random walk 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 $n$. With the help of the uniform convergence of martingale and the multifractal analysis, we establish a large deviation result for the measures $Z_n$ as well as a moderate deviation principle.