Hairer, Martin, Ryser, Marc D. and Weber, Hendrik. (2012) Triviality of the 2D stochastic Allen-Cahn equation. Electronic Journal of Probability, Vol.17 . pp. 1-14. ISSN 1083-6489

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Abstract

We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Stochastic analysis
Journal or Publication Title:	Electronic Journal of Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	1083-6489
Date:	2012
Volume:	Vol.17
Page Range:	pp. 1-14
Identification Number:	10.1214/EJP.v17-1731
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), Royal Society (Great Britain), Leverhulme Trust (LT), Hydro-Québec
Grant number:	EP/D071593/1 (EPSRC)
References:	[1] S. Allen and J. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, no. 6, (1979), 1085–1095. [2] S. Albeverio and M. Röckner. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Related Fields 89, no. 3, (1991), 347–386. MR-1113223 [3] F. Barret, A. Bovier, and S. Méléard. Uniform estimates for metastable transition times in a coupled bistable system. Electron. J. Probab. 15, no. 12, (2010), 323–345. MR-2609590 [4] H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations, vol. 343 of Grundlehren der mathematischen Wissenschaften Series. Springer Verlag, 2010. [5] S. Cerrai, and M. Freidlin. Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise. Trans. Amer. Math. Soc. 363, no. 7, (2011), 3853–3892. MR-2775830 [6] G. Da Prato and A. Debussche. Strong solutions to the stochastic quantization equations. Ann. Probab. 31, no. 4, (2003), 1900–1916. MR-2016604 [7] G. Da Prato and L. Tubaro. Wick powers in stochastic PDEs: an introduction. Technical Report UTM 711, University of Trento (2007). [8] G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, vol. 45 of Encyclopedia of mathematics and its applications. Cambridge University Press, 1992. MR-1207136 [9] L.C. EVANS, H.M Soner, and P.E. Souganidis. Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, no. 9, (1992), 1097–1123. MR-1177477 [10] J. Glimm and A. Jaffe. Quantum Physics: A Functional Integral Point of View. Second ed., Springer-Verlag, New York, 1987. MR-0887102 [11] M. Hairer. An introduction to stochastic PDEs. http://www.hairer.org/Teaching.html, 2009. Unpublished lecture notes. [12] T. Ilmanen. Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differential. Geom. 38, no. 2, (1993), 417–461. MR-1237490 [13] R. Kohn, F. Otto, M. Reznikoff, and E. Vanden-Eijnden. Action minimization and sharpinterface limits for the stochastic Allen-Cahn equation. Commun. Pure Appl. Math. 60, no. 3, (2007), 393–438. MR-2284215 [14] L. Landau, and L. Ginzburg. On the theory of superconductivity. J. Expt. Theor. Phys. 20, (1950), 1064–1082. [15] G. Parisi, and Y.S. Wu. Perturbation theory without gauge fixing. Sci. Sinica 24, no. 4, (1981), 483–496. MR-0626795 [16] M.D. Ryser, N. Nigam, and P.F. Tupper. On the well-posedness of the stochastic Allen-Cahn equation in two dimensions. J. Comp. Phys. 231, no. 6, (2012), 2537–2550. [17] J. Walsh. An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour XIV-1984 265–439. MR-0876085
URI:	http://wrap.warwick.ac.uk/id/eprint/48501

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