Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size

Felix Otto (MPI for Mathematics in the Sciences Leipzig)
Hendrik Weber (University of Warwick)
Maria G. Westdickenberg (RWTH Aachen)


We study the invariant measure of the one-dimensional stochastic Allen Cahn equation for a small noise strength and a large but finite system with so-called Dobrushin boundary conditions, i.e., inhomogeneous $\pm 1$ Dirichlet boundary conditions that enforce at least one transition layer from $-1$ to $1$. (Our methods can be applied to other boundary conditions as well.) We are interested in the competition between the "energy'' that should be minimized due to the small noise strength and the "entropy'' that is induced by the large system size.

Specifically, in the context of system sizes that are exponential with respect to the inverse noise strength---up to the ``critical'' exponential size predicted by the heuristics---we study the extremely strained large deviation event of seeing \emph{more than the one transition layer} between $\pm 1$ that is forced by the boundary conditions. We capture the competition between energy and entropy through upper and lower bounds on the  probability of these unlikely extra transition layers. Our bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from $-1$ to $+1$ is exponentially close to one. Our second result then studies the distribution of the transition layer. In particular, we establish that, on a super-logarithmic scale, the position of the transition layer is approximately uniformly distributed.

In our arguments we use local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

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Pages: 1-76

Publication Date: February 24, 2014

DOI: 10.1214/EJP.v19-2813


  • Bertini, Lorenzo; Brassesco, Stella; Buttà, Paolo. Soft and hard wall in a stochastic reaction diffusion equation. Arch. Ration. Mech. Anal. 190 (2008), no. 2, 307--345. MR2448321
  • Bertini, Lorenzo; Brassesco, Stella; Buttà, Paolo. Dobrushin states in the $\phi^ 4_ 1$ model. Arch. Ration. Mech. Anal. 190 (2008), no. 3, 477--516. MR2448325
  • Betz, Volker; Lőrinczi, József. Uniqueness of Gibbs measures relative to Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 5, 877--889. MR1997216
  • Bogachev, Vladimir I. Gaussian measures. Mathematical Surveys and Monographs, 62. American Mathematical Society, Providence, RI, 1998. xii+433 pp. ISBN: 0-8218-1054-5 MR1642391
  • Brassesco, S.; De Masi, A.; Presutti, E. Brownian fluctuations of the interface in the $D=1$ Ginzburg-Landau equation with noise. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 1, 81--118. MR1340032
  • Cassandro, Marzio; Orlandi, Enza; Presutti, Errico. Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Probab. Theory Related Fields 96 (1993), no. 1, 57--96. MR1222365
  • den Hollander, Frank. Large deviations. Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. x+143 pp. ISBN: 0-8218-1989-5 MR1739680
  • Faris, William G.; Jona-Lasinio, Giovanni. Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15 (1982), no. 10, 3025--3055. MR0684578
  • Feynman, R. P. Space-time approach to non-relativistic quantum mechanics. Rev. Modern Physics 20, (1948). 367--387. MR0026940
  • Freidlin, Mark I. Random perturbations of reaction-diffusion equations: the quasideterministic approximation. Trans. Amer. Math. Soc. 305 (1988), no. 2, 665--697. MR0924775
  • Freidlin, M. I.; Wentzell, A. D. Random perturbations of dynamical systems. Translated from the 1979 Russian original by Joseph Szücs. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer-Verlag, New York, 1998. xii+430 pp. ISBN: 0-387-98362-7 MR1652127
  • Funaki, T. The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (1995), no. 2, 221--288. MR1337253
  • M. Hairer. An Introduction to Stochastic PDEs. phArXiv e-prints .
  • Jona-Lasinio, G.; Martinelli, F.; Scoppola, E. New approach to the semiclassical limit of quantum mechanics. I. Multiple tunnelings in one dimension. Comm. Math. Phys. 80 (1981), no. 2, 223--254. MR0623159
  • Kifer, Ju. I. Some results concerning small random perturbations of dynamical systems. (Russian) Teor. Verojatnost. i Primenen. 19 (1974), 514--532. MR0353451
  • Modica, Luciano; Mortola, Stefano. Un esempio di $\Gamma ^{-}$-convergenza. (Italian) Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285--299. MR0445362
  • R. Peierls. On Ising's model of ferromagnetism. phMathematical Proceedings of the Cambridge Philosophical Society, 32, (1936), 477--481.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Reznikoff, Maria G.; Vanden-Eijnden, Eric. Invariant measures of stochastic partial differential equations and conditioned diffusions. C. R. Math. Acad. Sci. Paris 340 (2005), no. 4, 305--308. MR2121896
  • Simon, Barry. Functional integration and quantum physics. Pure and Applied Mathematics, 86. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. ix+296 pp. ISBN: 0-12-644250-9 MR0544188
  • Vanden-Eijnden, Eric; Westdickenberg, Maria G. Rare events in stochastic partial differential equations on large spatial domains. J. Stat. Phys. 131 (2008), no. 6, 1023--1038. MR2407378
  • Varadhan, S. R. S. Large deviations and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. v+75 pp. ISBN: 0-89871-189-4 MR0758258
  • Weber, Hendrik. Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation. Comm. Pure Appl. Math. 63 (2010), no. 8, 1071--1109. MR2642385
  • A. D. Wentzell and M. I. Freidlin. O malykh sluchainykh vozmuschcheniyakh dinamicheskikh sistem. phUspekhi Mat. Nauk 25, no. 1, (1970), 3-55. English translation of title: On small random perturbations of dynamical systems.
  • Zabczyk, J. Symmetric solutions of semilinear stochastic equations. Stochastic partial differential equations and applications, II (Trento, 1988), 237--256, Lecture Notes in Math., 1390, Springer, Berlin, 1989. MR1019609

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