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

2009/02/19 by Mueller, Carl, Mytnik, Leonid, Quastel, Jeremy
#35K05 #35R60 #60H15 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.0902.3423

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 ∂_t u = ∂_x² u + u(1-u) + ε√(u(1-u)) W, and ∂_t u = ∂_x² u + u(1-u) + ε√(u) W, where W= W(t,x) is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically 2-π² |log ε²|^-2 up to a factor of order (log|logε|)|logε|^-3.

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