An Invariance Principle for Random Traveling Waves in One Dimension

Abstract

We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is either the ignition nonlinearity or the bistable nonlinearity, under suitable restrictions on the size of the spatial fluctuations. It is known that the solution develops an interface which propagates with a well-defined speed in the large-time limit. The main result of this article is a functional central limit theorem for the random interface position.

Slowdown for Time Inhomogeneous Branching Brownian Motion

“…Finally, we mention that results for the homogenization of the KPP equation are available, see e.g. [N11,NRRZ12] and references therein. In the terminology we employ here, those results correspond to fast varying, or microscopic, time inhomogenuities.…”

(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation

“…Again, the proof takes advantage of analyzing (PAM) first. Let us note here that in [Nol11b], a corresponding invariance principle has been derived for non-linearities that are either ignition type or bistable; note however, that -as will be explained below -on a logarithmic in time scale these fronts behave quite differently from the fronts to (F-KPP) in our context. For a different and due technical reasons restricted set of initial conditions, Nolen [Nol11a] has derived a central limit theorem for the position of the front of the solution to (F-KPP) by analytic means.…”

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

“…In this lingo, our set of initial conditions corresponds to the critical regime, and Corollary 1.5 (for the critical regime) also suggests that the randomness of the functional central limit theorem is already coming from the environment, and not necessarily due to the random initial condition. Furthermore, in [29] a corresponding invariance principle for the front has been derived in the case where the non-linearity in (F-KPP) is either ignition typ or bistable.…”