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

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.