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The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation
Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 931-938.

We study the asymptotics of Allen–Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i) the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii) the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.

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DOI : 10.5802/crmath.117
Classification : 60H15, 35D40
Pierre-Louis Lions 1 ; Panagiotis E. Souganidis 2

1 Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, and CEREMADE, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
2 Department of Mathematics University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Pierre-Louis Lions; Panagiotis E. Souganidis. The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 931-938. doi : 10.5802/crmath.117. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.117/

[1] Matthieu Alfaro; Dimitra Antonopoulou; Georgia Karali; Hiroshi Matano Generation of fine transition layers and their dynamics for the stochastic Allen–Cahn equation (2018) (https://arxiv.org/abs/1812.03804)

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[8] Pierre-Louis Lions; Panagiotis E. Souganidis (in preparation) | DOI | Numdam

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[11] Pierre-Louis Lions; Panagiotis E. Souganidis Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 10, pp. 783-790 | DOI | MR | Zbl

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