Wave propagation for 1-dimensional reaction-diffusion equations with nonzero random drift

TL;DR

This paper analyzes wave propagation in 1D reaction-diffusion equations with random drift, revealing that positive average drift causes wave fronts to move toward negative infinity, using large deviations and probabilistic methods.

Contribution

It introduces a novel probabilistic approach to understand wave front behavior in reaction-diffusion systems with random drift, highlighting the drift's impact on wave direction.

Findings

Wave fronts can propagate toward negative infinity when average drift is positive.

Probabilistic methods reveal the physical mechanism behind wave front formation.

The drift shifts the free-energy reference level without changing fluctuation structure.

Abstract

We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the asymptotic wave fronts propagating to the positive and negative directions are both pushed in the negative direction, leading to the possibility that both wave fronts propagate toward negative infinity. Our proof is based on the Large Deviations Principle for diffusion processes in random environments, as well as an analysis of the Feynman-Kac formula. Such probabilistic arguments also reveal the underlying physical mechanism of the wave fronts formation: the drift acts as an external field that shifts the (quenched) free-energy reference level without altering the intrinsic fluctuation structure of the system.