Invariance principles and Log-distance of F-KPP fronts in a random medium

TL;DR

This paper investigates the behavior of F-KPP fronts in a random medium, showing they lag logarithmically behind linearized solutions and establishing central limit theorems for their fluctuations.

Contribution

It generalizes Bramson's results to random environments and proves new invariance principles for the fronts of F-KPP and parabolic Anderson models.

Findings

Front lag is at most logarithmic in time.

Functional central limit theorems are established for the fronts.

Results extend classical homogeneous case to random media.

Abstract

We study the front of the solution to the F-KPP equation with randomized non-linearity. Under suitable assumptions on the randomness involving spatial mixing behavior and boundedness, we show that the front of the solution lags at most logarithmically in time behind the front of the solution of the corresponding linearized equation, i.e. the parabolic Anderson model. This can be interpreted as a partial generalization of Bramson's findings for the homogeneous setting. Building on this result, we establish functional central limit theorems for the fronts of the solutions to both equations.