On the short time asymptotic of the stochastic Allen-Cahn equation
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 965-975.

On étudie le comportement de la solution de l'équation de Allen-Cahn perturbée par un bruit stochastique additif et régularisé. Il est démontré que, dans la limite d'un interface singulière, les solutions évoluent selon la courbure moyenne avec un renforcement stochastique additionnel. Ceci généralise un résultat de Funaki [Acta Math. Sin (Engl. Ser.) 15 (1999) 407-438] pour la dimension spatial d=2 à une dimension quelconque.

A description of the short time behavior of solutions of the Allen-Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki [Acta Math. Sin (Engl. Ser.) 15 (1999) 407-438] in spatial dimension n=2 to arbitrary dimensions.

DOI : 10.1214/09-AIHP333
Classification : 35R60, 53C44
Mots-clés : stochastic reaction-diffusion equation, sharp interface limit, randomly perturbed boundary motion 
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Weber, Hendrik. On the short time asymptotic of the stochastic Allen-Cahn equation. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 965-975. doi : 10.1214/09-AIHP333. http://www.numdam.org/articles/10.1214/09-AIHP333/

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