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.
Mots clés : stochastic reaction-diffusion equation, sharp interface limit, randomly perturbed boundary motion
@article{AIHPB_2010__46_4_965_0, author = {Weber, Hendrik}, title = {On the short time asymptotic of the stochastic {Allen-Cahn} equation}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {965--975}, publisher = {Gauthier-Villars}, volume = {46}, number = {4}, year = {2010}, doi = {10.1214/09-AIHP333}, mrnumber = {2744880}, zbl = {1210.35307}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP333/} }
TY - JOUR AU - Weber, Hendrik TI - On the short time asymptotic of the stochastic Allen-Cahn equation JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 965 EP - 975 VL - 46 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP333/ DO - 10.1214/09-AIHP333 LA - en ID - AIHPB_2010__46_4_965_0 ER -
%0 Journal Article %A Weber, Hendrik %T On the short time asymptotic of the stochastic Allen-Cahn equation %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 965-975 %V 46 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP333/ %R 10.1214/09-AIHP333 %G en %F AIHPB_2010__46_4_965_0
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/
[1] A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085-1095.
and .[2] Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991) 749-786. | MR | Zbl
, and .[3] Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term. Nonlinear Anal. 28 (1997) 1283-1298. | MR | Zbl
, and .[4] A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differential Equations 13 (2001) 405-425. | MR | Zbl
, and .[5] Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992) 1097-1123. | MR | Zbl
, and .[6] Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991) 635-681. | MR | Zbl
and .[7] Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321-332. | MR | Zbl
and .[8] The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (1995) 221-288. | MR | Zbl
.[9] Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15 (1999) 407-438. | MR | Zbl
.[10] Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991. | MR | Zbl
and .[11] Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem. Interfaces Free Bound. 9 (2007) 1-30. | MR | Zbl
, and .[12] Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications. C. R. Math. Acad. Sci. Paris Sér. I 327 (1998) 735-741. | MR | Zbl
and .[13] Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Birkhäuser, Basel, 1995. | MR | Zbl
.[14] Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533-1589. | MR | Zbl
and .[15] Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144 (1998) 313-355. | MR | Zbl
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