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

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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

$$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u(1-u)}\dot{W},$$

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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Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Carl Mueller

  2. Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Leonid Mytnik

  3. Departments of Mathematics and Statistics, University of Toronto, Toronto, Ontario, Canada

    Jeremy Quastel

Authors
  1. Carl Mueller
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  2. Leonid Mytnik
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  3. Jeremy Quastel
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Corresponding author

Correspondence to Jeremy Quastel.

Additional information

C. Mueller was supported by an NSF grant.

L. Mytnik was supported in part by the Israel Science Foundation (grant No. 1162/06).

J. Quastel was supported by NSERC, Canada.

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Mueller, C., Mytnik, L. & Quastel, J. Effect of noise on front propagation in reaction-diffusion equations of KPP type. Invent. math. 184, 405–453 (2011). https://doi.org/10.1007/s00222-010-0292-5

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  • DOI: https://doi.org/10.1007/s00222-010-0292-5

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