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

TL;DR

This paper establishes quenched invariance principles for the maximal particle in branching random walk in random environments and related models, demonstrating convergence to a limiting process and quantifying median-front distances.

Contribution

It extends invariance principles to BRWRE and related equations, connecting particle maxima with solution fronts under random environments.

Findings

Maximal particle position satisfies quenched invariance principles after scaling.

The front of the PAM solution aligns with the maximal particle front, with bounded median-front distance.

Results generalize Bramson’s classical findings to random environment settings.

Abstract

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(ln t). This partially transfers results from Bramson [Comm. Pure Appl. Math. 31 (1978), no. 5, 531--581] to the setting of BRWRE.