Invariance principle for the maximal position process of branching Brownian motion in random environment

Abstract: In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $\xi = \left(\xi(x)\right)_{x\in\mathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time $t$ satisfies a quenched strong law of large numbers and an annealed invariance principle.