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

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 $(\log |\log \epsilon |)|\log \epsilon {|}^{-3}$.

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