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We consider branching random walk in spatial random branching environment(BRWRE) in dimension one, as well as related differential equations: theFisher-KPP equation with random branching and its linearized version, theparabolic Anderson model (PAM). When the random environment is bounded, we showthat after recentering and scaling, the position of the maximal particle of theBRWRE, the front of the solution of the PAM, as well as the front of thesolution of the randomized Fisher-KPP equation fulfill quenched invarianceprinciples. In addition, we prove that at time t the distance between themedian of the maximal particle of the BRWRE and the front of the solution ofthe PAM is in O(ln t). This partially transfers results from Bramson [Comm.Pure Appl. Math. 31 (1978), no. 5, 531--581] to the setting of BRWRE.

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