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Effect of noise on front propagation in reaction-diffusion equations of KPP type

C. Mueller,L. Mytnik,J. Quastel

Published 2009 in Inventiones mathematicae

ABSTRACT

AbstractWe 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 $$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u(1-u)}\dot{W},$$ and $$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u}\dot{W},$$ 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 for small ε is $$2-\pi^2|{\log}\,\epsilon^2|^{-2}+O((\log|{\log}\,\epsilon|)|{\log}\,\epsilon|^{-3}).$$

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