On the short time asymptotic of the stochastic Allen-Cahn equation
Weber, Hendrik
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010), p. 965-975 / Harvested from Numdam
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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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1214/09-AIHP333
Classification:  35R60,  53C44 
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     title = {On the short time asymptotic of the stochastic Allen-Cahn equation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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     doi = {10.1214/09-AIHP333},
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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) pp. 965-975. doi : 10.1214/09-AIHP333. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_4_965_0/

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