Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher.KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher.KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in O(lnt). This partially transfers classical results of Bramson (Comm. Pure Appl. Math. 31 (1978) 531.581) to the setting of BRWRE.