Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 55-92.

Nous étudions l’équation d’Allen-Cahn stochastique conduite par un bruit d’intensité ε et de longueur de corrélation δ en dimensions spatiales deux et trois. Nous considérons la limite δ,ε0 et nous décrivons complètement le comportement des grandes déviations associées, suivant les relations entre δ et ε.

La théorie des structures de régularité récemment développée permet d’analyser le comportement des solutions à intensité de bruit ε fixée dans la limite δ0. Un fait crucial est que, afin d’obtenir des limites non-triviales dans cette limite, il est nécessaire d’introduire des contretermes divergents. La théorie des structures de régularité permet d’analyser rigoureusement de telles procédures de renormalisation pour un nombre d’équations intéressantes.

Notre résultat principal est un principe de grandes déviations pour ces équations renormalisées. Il est alors intéressant de noter que les constantes de renormalisation divergentes disparaissent au niveau de la fonction de taux. Une conséquence de ce résultat est une condition optimale sur le comportement relatif de δ et ε qui garantit l’existence d’un principe de grandes déviations également pour l’équation non-renormalisée dans certains régimes.

We study the stochastic Allen-Cahn equation driven by a noise term with intensity ε and correlation length δ in two and three spatial dimensions. We study diagonal limits δ,ε0 and describe fully the large deviation behaviour depending on the relationship between δ and ε.

The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length δ and fixed noise intensity ε. One key fact is that in order to get non-trivial limits as δ0, it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations.

Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on δ,ε that guarantees a large deviation principle for diagonal schemes ε,δ0 for the equation without renormalisation.

@article{AFST_2015_6_24_1_55_0,
     author = {Hairer, Martin and Weber, Hendrik},
     title = {Large deviations for white-noise driven, nonlinear stochastic {PDEs} in two and three dimensions},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {55--92},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 24},
     number = {1},
     year = {2015},
     doi = {10.5802/afst.1442},
     mrnumber = {3325951},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1442/}
}
TY  - JOUR
AU  - Hairer, Martin
AU  - Weber, Hendrik
TI  - Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2015
SP  - 55
EP  - 92
VL  - 24
IS  - 1
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1442/
DO  - 10.5802/afst.1442
LA  - en
ID  - AFST_2015_6_24_1_55_0
ER  - 
%0 Journal Article
%A Hairer, Martin
%A Weber, Hendrik
%T Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2015
%P 55-92
%V 24
%N 1
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1442/
%R 10.5802/afst.1442
%G en
%F AFST_2015_6_24_1_55_0
Hairer, Martin; Weber, Hendrik. Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 55-92. doi : 10.5802/afst.1442. http://www.numdam.org/articles/10.5802/afst.1442/

[1] Aida (S.).— Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. II. P(ϕ) 2 -model on a finite volume. J. Funct. Anal. 256, no. 10, (2009). | MR | Zbl

[2] Aida (S.).— Tunneling for spatially cut-off P(’)2-Hamiltonians. J. Funct. Anal. 263, no. 9, (2012). | MR | Zbl

[3] Bouchet (F.), Laurie (J.), and Zaboronski (O.).— Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional euler equations. ArXiv e-prints (2014). | MR

[4] Borell (C.).— Tail probabilities in Gauss space. In Vector Space Measures and Applications I, p. 73-82. Springer (1978). | MR | Zbl

[5] Borell (C.).— On polynomial chaos and integrability. Probab. Math. Statist 3, no. 2, p. 191-203 (1984). | MR | Zbl

[6] Borell (C.).— On the Taylor series of a Wiener polynomial. In Seminar Notes on multiple stochastic integration, polynomial chaos and their integration. Case Western Reserve University, Cleveland (1984).

[7] Cerrai (S.) and Freidlin (M.).— Approximation of quasi-potentials and exit problems for multidimensional RDEÕs with noise. Trans. Amer. Math. Soc. 363, no. 7, p. 3853-3892 (2011). | MR | Zbl

[8] Da Prato (G.) and Debussche (A.).— Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196, no. 1, p. 180-210 (2002). | MR | Zbl

[9] Da Prato (G.) and Debussche (A.).— Strong solutions to the stochastic quantization equations. Ann. Probab. 31, no. 4, p. 1900-1916 (2003). | MR | Zbl

[10] Deuschel (J.-D.) and Stroock (D. W.).— Large deviations, vol. 137 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA (1989). | MR | Zbl

[11] E (W.), Ren (W.), and Vanden-Eijnden (E.).— Minimum action method for the study of rare events. Comm. Pure Appl. Math. 57, no. 5, p. 637-656 (2004). | MR | Zbl

[12] Faris (W. G.) and Jona-Lasinio (G.).— Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15, no. 10, p. 3025-3055 (1982). | MR | Zbl

[13] Friz (P.) and Victoir (N.).— Large deviation principle for enhanced Gaussian processes. Ann. Inst. H. Poincaré Probab. Statist. 43, no. 6, p. 775-785 (2007). | MR | Zbl

[14] Hairer (M.).— Introduction to Regularity Structures. ArXiv e-prints (2014). arXiv: 1401.3014. To appear in Braz. J. Prob. Stat. | MR

[15] Hairer (M.).— Singular stochastic PDEs. ArXiv e-prints (2014). arXiv:1403.6353. To appear in Proc. ICM.

[16] Hairer (M.).— A theory of regularity structures. Invent. Math. 198, no. 2, p. 269-504 (2014). | MR

[17] Hohenberg (P. C.) and Halperin (B. I.).— Theory of dynamic critical phenomena. Reviews of Modern Physics 49, no. 3, 435 (1977).

[18] Hairer (M.), Ryser (M. D.) and Weber (H.).— Triviality of the 2D stochastic Allen-Cahn equation. Electron. J. Probab. 17, no. 39, 14 (2012). | MR | Zbl

[19] Jona-Lasinio (G.) and Mitter (P. K.).— Large deviation estimates in the stochastic quantization of ’42. Comm. Math. Phys. 130, no. 1, p. 111-121 (1990). | MR | Zbl

[20] Kohn (A.), Otto (F.), Reznikoff (M. G.) and Vanden-Eijnden (E.).— Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation. Comm. Pure Appl. Math. 60, no. 3, p. 393-438 (2007). | MR | Zbl

[21] Ledoux (M.).— A note on large deviations for Wiener chaos. In Séminaire de Probabilités, XXIV, 1988/89, vol. 1426 of Lecture Notes in Math., 1-14. Springer, Berlin (1990). | Numdam | MR | Zbl

[22] Ledoux (M.).— Isoperimetry and Gaussian analysis. In Lectures on probability theory and statistics, p. 165-294. Springer (1996). | MR | Zbl

[23] Ledoux (M.), Qian (Z.), and Zhang (T.).— Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102, no. 2, p. 265-283 (2002). | MR | Zbl

[24] Millet (A.) and Sanz-Solé (M.).— Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 42, no. 2, p. 245-271 (2006). | Numdam | MR | Zbl

[25] Mayer-Wolf (E.), Nualart (D.), and Pérez-Abreu (V.).— Large deviations for multiple Wiener-Itô integral processes. In Séminaire de Probabilités XXVI, Springer, p. 11-31 (1992). | Numdam | MR | Zbl

[26] Neveu (J.).— Discrete-parameter martingales. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, revised ed., 1975. North-Holland Mathematical Library, Vol. 10. | MR | Zbl

[27] Nualart (D.).— The Malliavin calculus and related topics. Springer (2006). | MR | Zbl

Cité par Sources :