Invariance Principles and Log-Distance of F-KPP Fronts in a Random Medium

We study the front of the solution to the F-KPP equation with randomized non-linearity. Under suitable assumptions on the randomness including 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 (Bramson in Commun Pure Appl Math 31(5):531-581, 1978) for the homogeneous setting. Partially building on this result and its derivation, we establish functional central limit theorems for the fronts of the solutions to both equations.

Automated summary

In this study, we investigated the front of the solution to the F-KPP equation with randomized non-linearity and analyzed its evolution process. By making appropriate assumptions on the randomness, including spatial mixing behavior and boundedness, we obtained a relationship between the front of the solution and the front of the corresponding linearized equation, which can be interpreted as a partial generalization of previous findings in the homogeneous setting. Additionally, we established functional central limit theorems for the fronts of the solutions to both equations.