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

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.

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.

DOI : https://doi.org/10.1214/09-AIHP333
Classification:  35R60,  53C44
Keywords: 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},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {965-975},
     doi = {10.1214/09-AIHP333},
     zbl = {1210.35307},
     mrnumber = {2744880},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 46 (2010) no. 4, pp. 965-975. doi : 10.1214/09-AIHP333. http://www.numdam.org/item/AIHPB_2010__46_4_965_0/

[1] S. M. Allen and J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085-1095.

[2] Y. Chen, Y. Giga and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991) 749-786. | MR 1100211 | Zbl 0696.35087

[3] X. Chen, D. Hilhorst and E. Logak. Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term. Nonlinear Anal. 28 (1997) 1283-1298. | MR 1422816 | Zbl 0883.35013

[4] N. Dirr, S. Luckhaus and M. Novaga. A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differential Equations 13 (2001) 405-425. | MR 1867935 | Zbl 1015.60070

[5] L. Evans, H. Soner and P. Souganidis. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992) 1097-1123. | MR 1177477 | Zbl 0801.35045

[6] L. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991) 635-681. | MR 1100206 | Zbl 0726.53029

[7] L. Evans and J. Spruck. Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321-332. | MR 1068927 | Zbl 0776.53005

[8] T. Funaki. The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (1995) 221-288. | MR 1337253 | Zbl 0834.60066

[9] T. Funaki. Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15 (1999) 407-438. | MR 1736690 | Zbl 0943.60060

[10] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991. | MR 1121940 | Zbl 0734.60060

[11] M. Katsoulakis, G. Kossioris and O. Lakkis. Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem. Interfaces Free Bound. 9 (2007) 1-30. | MR 2317297 | Zbl 1129.35088

[12] P. Lions and P. Souganidis. Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications. C. R. Math. Acad. Sci. Paris Sér. I 327 (1998) 735-741. | MR 1659958 | Zbl 0924.35203

[13] A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Birkhäuser, Basel, 1995. | MR 1329547 | Zbl 0816.35001

[14] P. De Mottoni and M. Schatzman. Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533-1589. | MR 1672406 | Zbl 0840.35010

[15] N. Yip. Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144 (1998) 313-355. | MR 1656479 | Zbl 0930.60047