On the short time asymptotic of the stochastic Allen–Cahn equation

Hendrik Weber

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 965-975
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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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 46.4 (2010): 965-975. <http://eudml.org/doc/242183>.

@article{Weber2010,
abstract = {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.},
author = {Weber, Hendrik},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic reaction–diffusion equation; sharp interface limit; randomly perturbed boundary motion; stochastic reaction-diffusion equation},
language = {eng},
number = {4},
pages = {965-975},
publisher = {Gauthier-Villars},
title = {On the short time asymptotic of the stochastic Allen–Cahn equation},
url = {http://eudml.org/doc/242183},
volume = {46},
year = {2010},
}

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
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 965
EP - 975
AB - 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.
LA - eng
KW - stochastic reaction–diffusion equation; sharp interface limit; randomly perturbed boundary motion; stochastic reaction-diffusion equation
UR - http://eudml.org/doc/242183
ER -

References

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  1. [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. [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. Zbl0696.35087MR1100211
  3. [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. Zbl0883.35013MR1422816
  4. [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. Zbl1015.60070MR1867935
  5. [5] L. Evans, H. Soner and P. Souganidis. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992) 1097–1123. Zbl0801.35045MR1177477
  6. [6] L. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991) 635–681. Zbl0726.53029MR1100206
  7. [7] L. Evans and J. Spruck. Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321–332. Zbl0776.53005MR1068927
  8. [8] T. Funaki. The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (1995) 221–288. Zbl0834.60066MR1337253
  9. [9] T. Funaki. Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15 (1999) 407–438. Zbl0943.60060MR1736690
  10. [10] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991. Zbl0734.60060MR1121940
  11. [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. Zbl1129.35088MR2317297
  12. [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. Zbl0924.35203MR1659958
  13. [13] A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Birkhäuser, Basel, 1995. Zbl0816.35001MR1329547
  14. [14] P. de Mottoni and M. Schatzman. Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347 (1995) 1533–1589. Zbl0840.35010MR1672406
  15. [15] N. Yip. Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144 (1998) 313–355. Zbl0930.60047MR1656479

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