Sharp asymptotics of metastable transition times for one dimensional SPDEs
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 129-166.

We consider a class of parabolic semi-linear stochastic partial differential equations driven by space–time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring–Kramers formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.

Nous nous intéressons à une famille d’équations aux dérivées partielles stochastiques paraboliques et semi-linéaires, perturbées par un bruit blanc en espace-temps, définies sur un intervalle réel compact. Nous cherchons à calculer les asymptotiques précises des espérances des temps de transitions entre les états métastables. Nous démontrons dans ce cadre une version en dimension infinie de la formule dite d’Eyring–Kramers. La preuve repose sur l’approximation par un schéma aux différences finies de l’équation aux dérivées partielles stochastique. L’espérance du temps de transition est calculée pour l’approximation puis contrôlée uniformément quelque soit la dimension.

DOI: 10.1214/13-AIHP575
Classification: 82C44,  60H15,  35K57
Keywords: metastability, metastable transition time, parabolic stochastic partial differential equations, reaction-diffusion equations, stochastic Allen–Cahn equations, Eyring–Kramers formula
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Barret, Florent. Sharp asymptotics of metastable transition times for one dimensional SPDEs. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 129-166. doi : 10.1214/13-AIHP575. http://www.numdam.org/articles/10.1214/13-AIHP575/

[1] S. B. Angenent. The Morse–Smale property for a semilinear parabolic equation. J. Differential Equations 62 (3) (1986) 427–442. | MR | Zbl

[2] F. Barret. Temps de transition métastables pour des systèmes dynamiques stochastiques fini et infini-dimensionnels. Ph.D. thesis, École Polytechnique, July 2012.

[3] F. Barret, A. Bovier and S. Méléard. Uniform estimates for metastable transition times in a coupled bistable system. Electron. J. Probab. 15 (12) (2010) 323–345. | MR | Zbl

[4] N. Berglund, B. Fernandez and B. Gentz. Metastability in interacting nonlinear stochastic differential equations. I. From weak coupling to synchronization. Nonlinearity 20 (11) (2007) 2551–2581. | MR | Zbl

[5] N. Berglund, B. Fernandez and B. Gentz. Metastability in interacting nonlinear stochastic differential equations2007) 2583–2614. | MR | Zbl

[6] B. Bianchi, A. Bovier and D. Ioffe. Sharp asymptotics for metastability in the random field Curie–Weiss model. Electron. J. Probab. 14 (2009) 1541–1603. | MR | Zbl

[7] A. Bovier. Metastability. In Methods of Contemporary Statistical Mechanics 177–221. R. Kotecký (Ed.). Lecture Notes in Math. 1970. Springer, Berlin, 2009. | MR | Zbl

[8] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein. Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 (1) (2001) 99–161. | MR | Zbl

[9] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein. Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 (2) (2002) 219–255. | MR | Zbl

[10] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein. Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6 (2) (2004) 399–424. | MR | Zbl

[11] A. Bovier, V. Gayrard and M. Klein. Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. 7 (1) (2005) 69–99. | MR | Zbl

[12] A. Bovier, F. Den Hollander and C. Spitoni. Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Probab. 38 (2) (2010) 661–713. | MR | Zbl

[13] S. Brassesco. Some results on small random perturbations of an infinite-dimensional dynamical system. Stochastic Process. Appl. 38 (1) (1991) 33–53. | MR | Zbl

[14] S. Brassesco and P. Buttà. Interface fluctuations for the D=1 stochastic Ginzburg–Landau equation with nonsymmetric reaction term. J. Statist. Phys. 93 (5–6) (1998) 1111–1142. | MR | Zbl

[15] P. Brunovský and B. Fiedler. Connecting orbits in scalar reaction diffusion equations. II. The complete solution. J. Differential Equations 81 (1) (1989) 106–135. | MR | Zbl

[16] M. Cassandro, A. Galves, E. Olivieri and M. E. Vares. Metastable behavior of stochastic dynamics: A pathwise approach. J. Statist. Phys. 35 (5-6) (1984) 603–634. | MR | Zbl

[17] M. Cassandro, E. Olivieri and P. Picco. Small random perturbations of infinite-dimensional dynamical systems and nucleation theory. Ann. Inst. H. Poincaré Phys. Théor. 44 (4) (1986) 343–396. | Numdam | MR | Zbl

[18] F. Chenal and A. Millet. Uniform large deviations for parabolic SPDEs and applications. Stochastic Process. Appl. 72 (2) (1997) 161–186. | MR | Zbl

[19] R. Courant and D. Hilbert. Methods of Mathematical Physics. Vol. I. Interscience Publishers, Inc., New York, NY, 1953. | MR | Zbl

[20] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge, 1992. | MR | Zbl

[21] H. Eyring. The activated complex in chemical reactions. J. Chem. Phys. 3 (2) (1935) 107.

[22] W. G. Faris and G. Jona-Lasinio. Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15 (1982) 3025–3055. | MR | Zbl

[23] B. Fiedler and C. Rocha. Heteroclinic orbits of semilinear parabolic equations. J. Differential Equations 125 (1) (1996) 239–281. | MR | Zbl

[24] B. Fiedler and C. Rocha. Connectivity and design of planar global attractors of Sturm type. III: Small and platonic examples. J. Dynam. Differential Equations 22 (2) (2010) 121–162. | MR | Zbl

[25] M. I. Freidlin and A. D. Wentzell. Random Perturbations of Dynamical Systems. Springer, New York, 1984. | MR | Zbl

[26] T. Funaki. Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89 (1983) 129–193. | MR | Zbl

[27] A. Galves, E. Olivieri and M. E. Vares. Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 (1987) 1288–1305. | MR | Zbl

[28] C. Geiss and R. Manthey. Comparison theorems for stochastic differential equations finite and infinite dimensions. Stochastic Process. Appl. 53 (1994) 23–35. | MR | Zbl

[29] I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space–time white noise. I. Potential Anal. 9 (1) (1998) 1–25. | MR | Zbl

[30] I. Gyöngy and É. Pardoux. On quasi-linear stochastic partial differential equations. Probab. Theory Related Fields 94 (4) (1993) 413–425. | MR | Zbl

[31] F. R. De Hoog and R. S. Anderssen. Asymptotic formulas for discrete eigenvalue problems in Liouville normal form. Math. Models Methods Appl. Sci. 11 (1) (2001) 43–56. | MR | Zbl

[32] H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7 (4) (1940) 284–304. | MR | Zbl

[33] S. Levit and U. Smilansky. A theorem on infinite products of eigenvalues of Sturm–Liouville type operators. Proc. Amer. Math. Soc. 65 (2) (1977) 299–302. | MR | Zbl

[34] R. Maier and D. Stein. Droplet nucleation and domain wall motion in a bounded interval. Phys. Rev. Lett. 87 (2001) 270601-1–270601-4.

[35] F. Martinelli, E. Olivieri and E. Scoppola. Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times. J. Statist. Phys. 55 (1989) 477–504. | MR | Zbl

[36] F. Martinelli, L. Sbano and E. Scoppola. Small random perturbation of dynamical systems: Recursive multiscale analysis. Stochastics Stochastics Rep. 49 (3–4) (1994) 253–272. | MR | Zbl

[37] E. Olivieri and M. E. Vares. Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications 100. Cambridge Univ. Press, Cambridge, 2005. | MR | Zbl

[38] E. Vanden-Eijnden and M. G. Westdickenberg. Rare events in stochastic partial differential equations on large spatial domains. J. Stat. Phys. 131 (2008) 1023–1038. | MR | Zbl

[39] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’Été de Probabilités de Saint-Flour XIV – 1984 265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986. | MR | Zbl

[40] M. Wolfrum. A sequence of order relations: Encoding heteroclinic connections in scalar parabolic PDE. J. Differential Equations 183 (1) (2002) 56–78. | MR | Zbl

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