no. 8
Équations aux dérivées partielles, Probabilités
The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation
Lions, Pierre-Louis1  ; Souganidis, Panagiotis E.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
Comptes Rendus. Mathématique, Tome 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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Révisé le :
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DOI : 10.5802/crmath.117
Classification : 60H15, 35D40
Lions, Pierre-Louis 1 ; Souganidis, Panagiotis E. 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 
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Lions, Pierre-Louis; Souganidis, Panagiotis E. The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation. Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 931-938. doi : 10.5802/crmath.117. http://www.numdam.org/articles/10.5802/crmath.117/

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