## Electronic Journal of Probability

### An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two

#### Abstract

We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon }$. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring–Kramers formula for the limiting renormalised stochastic PDE. The effect of the “infinite renormalisation” is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring–Kramers law by a renormalised Carleman–Fredholm determinant.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 41, 27 pp.

Dates
Accepted: 23 April 2017
First available in Project Euclid: 28 April 2017

https://projecteuclid.org/euclid.ejp/1493345028

Digital Object Identifier
doi:10.1214/17-EJP60

Mathematical Reviews number (MathSciNet)
MR3646067

Zentralblatt MATH identifier
1362.60059

#### Citation

Berglund, Nils; Di Gesù, Giacomo; Weber, Hendrik. An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two. Electron. J. Probab. 22 (2017), paper no. 41, 27 pp. doi:10.1214/17-EJP60. https://projecteuclid.org/euclid.ejp/1493345028

#### References

• [1] Shigeki Aida, Semi-classical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. II. $P(\phi )_2$-model on a finite volume, J. Funct. Anal. 256 (2009), no. 10, 3342–3367.
• [2] Florent Barret, Sharp asymptotics of metastable transition times for one dimensional SPDEs, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 1, 129–166.
• [3] Florent Barret, Anton Bovier, and Sylvie Méléard, Uniform estimates for metastable transition times in a coupled bistable system, Electron. J. Probab. 15 (2010), no. 12, 323–345.
• [4] Nils Berglund, Kramers’ law: validity, derivations and generalisations, Markov Process. Related Fields 19 (2013), no. 3, 459–490., arXiv:1106.5799.
• [5] Nils Berglund, Bastien Fernandez, and Barbara Gentz, Metastability in interacting nonlinear stochastic differential equations. I. From weak coupling to synchronization, Nonlinearity 20 (2007), no. 11, 2551–2581.
• [6] Nils Berglund and Barbara Gentz, Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond, Electron. J. Probab. 18 (2013), no. 24, 58.
• [7] Anton Bovier and Frank den Hollander, Metastability. a potential-theoretic approach, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 351, Springer, Cham, 2015.
• [8] Anton Bovier, Michael Eckhoff, Véronique Gayrard, and Markus Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 4, 399–424.
• [9] Giuseppe Da Prato and Arnaud Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916.
• [10] Giuseppe Da Prato and Luciano Tubaro, Wick powers in stochastic PDEs: an introduction, Tech. Report UTM 711, University of Trento, 2007.
• [11] Giacomo Di Gesù and Dorian Le Peutrec, Small noise spectral gap asymptotics for a large system of nonlinear diffusions, Journal of Spectral Theory, to appear (2017), 1–46, arXiv:1506.04434.
• [12] Henry Eyring, The activated complex in chemical reactions, The Journal of Chemical Physics 3 (1935), no. 2, 107–115.
• [13] William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15 (1982), no. 10, 3025–3055.
• [14] Robin Forman, Functional determinants and geometry, Invent. Math. 88 (1987), no. 3, 447–493.
• [15] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1998, Translated from the 1979 Russian original by Joseph Szücs.
• [16] James Glimm and Arthur Jaffe, Quantum physics, Springer-Verlag, New York-Berlin, 1981, A functional integral point of view.
• [17] Martin Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504.
• [18] Martin Hairer, Marc D. Ryser, and Hendrik Weber, Triviality of the 2D stochastic Allen-Cahn equation, Electron. J. Probab. 17 (2012), no. 39, 14.
• [19] Martin Hairer and Hendrik Weber, Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 1, 55–92.
• [20] Bernard Helffer, Markus Klein, and Francis Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26 (2004), 41–85.
• [21] G. Jona-Lasinio and P. K. Mitter, Large deviation estimates in the stochastic quantization of $\phi ^4_2$, Comm. Math. Phys. 130 (1990), no. 1, 111–121.
• [22] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940), 284–304.
• [23] Robert S. Maier and Daniel L. Stein, Droplet nucleation and domain wall motion in a bounded interval, Phys. Rev. Lett. 87 (2001), 270601–1.
• [24] Fabio Martinelli, Enzo Olivieri, and Elisabetta Scoppola, Small random perturbations of finite- and infinite-dimensional dynamical systems: unpredictability of exit times, J. Statist. Phys. 55 (1989), no. 3-4, 477–504.
• [25] Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227.
• [26] David Nualart, The Malliavin calculus and related topics, second ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006.
• [27] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974, Computer Science and Applied Mathematics.
• [28] Joran Rolland, Freddy Bouchet, and Eric Simonnet, Computing Transition Rates for the 1-D Stochastic Ginzburg–Landau–Allen–Cahn Equation for Finite-Amplitude Noise with a Rare Event Algorithm, J. Stat. Phys. 162 (2016), no. 2, 277–311.
• [29] Barry Simon, Trace ideals and their applications, second ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005.
• [30] Makoto Sugiura, Metastable behaviors of diffusion processes with small parameter, J. Math. Soc. Japan 47 (1995), no. 4, 755–788.
• [31] Pavlos Tsatsoulis and Hendrik Weber, Spectral Gap for the Stochastic Quantization Equation on the 2-dimensional Torus, Annales de l’Institut Henri Poincaré, to appear (2017), 1–46, arXiv:1609.08447.
• [32] Rongchan Zhu and Xiangchan Zhu, Three-dimensional Navier-Stokes equations driven by space-time white noise, J. Differential Equations 259 (2015), no. 9, 4443–4508.