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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  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
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