cnrs   Université de Toulouse
With cedram.org

Table of contents for this issue | Previous article | Next article
Martin Hairer; Hendrik Weber
Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
Annales de la faculté des sciences de Toulouse Sér. 6, 24 no. 1 (2015), p. 55-92, doi: 10.5802/afst.1442
Article PDF | Reviews MR 3325951

Résumé - Abstract

We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon }$ and correlation length $\delta $ in two and three spatial dimensions. We study diagonal limits $\delta , \varepsilon \rightarrow 0$ and describe fully the large deviation behaviour depending on the relationship between $\delta $ and $\varepsilon $.

The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $\delta $ and fixed noise intensity $\varepsilon $. One key fact is that in order to get non-trivial limits as $\delta \rightarrow 0$, 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 $\delta , \varepsilon $ that guarantees a large deviation principle for diagonal schemes $\varepsilon , \delta \rightarrow 0$ for the equation without renormalisation.

Bibliography

[1] Aida (S.).— Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. II. $P(\varphi )_2$-model on a finite volume. J. Funct. Anal. 256, no. 10, (2009).  MR 2504528 |  Zbl 1179.81075
[2] Aida (S.).— Tunneling for spatially cut-off P(’)2-Hamiltonians. J. Funct. Anal. 263, no. 9, (2012).  MR 2967304 |  Zbl 1257.81027
[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 3240870
[4] Borell (C.).— Tail probabilities in Gauss space. In Vector Space Measures and Applications I, p. 73-82. Springer (1978).  MR 502400 |  Zbl 0397.60015
[5] Borell (C.).— On polynomial chaos and integrability. Probab. Math. Statist 3, no. 2, p. 191-203 (1984).  MR 764146 |  Zbl 0555.60008
[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 2775830 |  Zbl 1232.60049
[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 1941997 |  Zbl 1013.60051
[9] Da Prato (G.) and Debussche (A.).— Strong solutions to the stochastic quantization equations. Ann. Probab. 31, no. 4, p. 1900-1916 (2003).  MR 2016604 |  Zbl 1071.81070
[10] Deuschel (J.-D.) and Stroock (D. W.).— Large deviations, vol. 137 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA (1989).  MR 997938 |  Zbl 0705.60029
[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 2032916 |  Zbl 1050.60068
[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 684578 |  Zbl 0496.60060
[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 3252431 |  Zbl 1172.60306
[14] Hairer (M.).— Introduction to Regularity Structures. ArXiv e-prints (2014). arXiv: 1401.3014. To appear in Braz. J. Prob. Stat.  MR 3274562
[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 3274562
[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 2928722 |  Zbl 1245.60063
[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 1055688 |  Zbl 0703.60095
[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 2284215 |  Zbl 1154.35021
[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 1071528 |  Zbl 0701.60020
[22] Ledoux (M.).— Isoperimetry and Gaussian analysis. In Lectures on probability theory and statistics, p. 165-294. Springer (1996).  MR 1600888 |  Zbl 0874.60005
[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 1935127 |  Zbl 1075.60510
[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 2199801 |  Zbl 1087.60035
[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 1231980 |  Zbl 0782.60026
[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 402915 |  Zbl 0345.60026
[27] Nualart (D.).— The Malliavin calculus and related topics. Springer (2006).  MR 2200233 |  Zbl 1099.60003
Search for an article
Search within the site
top