Effect of Noise on Front Propagation in Reaction-Diffusion equations of KPP type

TL;DR

This paper investigates how space-time white noise affects the speed of traveling fronts in reaction-diffusion equations of KPP type, confirming a conjecture about the asymptotic behavior of the front speed as noise intensity diminishes.

Contribution

It proves the Brunet-Derrida conjecture that noise causes a specific logarithmic correction to the front speed in stochastic KPP equations.

Findings

Confirmed the Brunet-Derrida conjecture on front speed correction

Derived asymptotic formula for front speed with noise

Quantified the impact of white noise on wave propagation

Abstract

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot W, $ and $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u}\dot W, $ where $\dot W= \dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically $ 2-\pi^2 |\log \epsilon^2|^{-2} $ up to a factor of order $ (\log|\log\epsilon|)|\log\epsilon|^{-3}$.