Skip to main content

Large deviation principle for a stochastic Allen–Cahn equation

Abstract

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. 2.

    Bonic, R., Frampton, J., Tromba, A.: \(\lambda \)-manifolds. J. Funct. Anal. 3, 310–320 (1969)

    Article  MATH  Google Scholar 

  3. 3.

    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. Stat. 31(1), 81–118 (1995)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Braides, A.: \(\Gamma \)-Convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002)

    Book  MATH  Google Scholar 

  5. 5.

    Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab. 36(4), 1390–1420 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for stochastic flows of diffeomorphisms. Bernoulli 16(1), 234–257 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cerrai, S., Freidlin, M.: Approximation of quasi-potentials and exit problems for multidimensional RDE’s with noise. Trans. Am. Math. Soc. 363(7), 3853–3892 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Dal Maso, G.: An introduction to \(\Gamma \)-convergence. In: Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Inc., Boston, MA (1993)

  10. 10.

    De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    de Mottoni, P., Schatzman, M.: Development of interfaces in \({ R}^N\). Proc. R. Soc. Edinb. Sect. A 116(3–4), 207–220 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  13. 13.

    Evans, L.C.: Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI (2010)

  14. 14.

    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45(9), 1097–1123 (1992)

  15. 15.

    Faris, W.G., Jona-Lasinio, G.: Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15(10), 3025–3055 (1982)

  16. 16.

    Feng, J.: Large deviation for diffusions and Hamilton–Jacobi equation in Hilbert spaces. Ann. Probab. 34(1), 321–385 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102(2), 221–288 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Funaki, T.: Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15(3), 407–438 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Garsia, A.M., Rodemich, E., Rumsey Jr., H.: A real variable lemma and the continuity of paths of some gaussian processes. Indiana Univ. Math. J 20(565–578), 1971 (1970)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Giaquinta, M., Martinazzi, L.: An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, vol. 11 of Appunti, 2nd edn. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa (2012)

  21. 21.

    Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Hairer, M., Ryse, M.D., Weber, H.: Triviality of the 2D stochastic Allen–Cahn equation. Electron. J. Probab. 17(39), 14 (2012)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Hairer, M., Weber, H.: Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions. Ann. Fac. Sci. Toulouse Math. (6) 24(1), 55–92 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Hofmanová, M., Röger, M., von Renesse, M.: Weak solutions for a stochastic mean curvature flow of two-dimensional graphs. Probab. Theory Relat. Fields (2016). doi:10.1007/s00440-016-0713-5

  25. 25.

    Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38(2), 417–461 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Kohn, R., Otto, F., Reznikoff, M.G., Vanden-Eijnden, E.: Action minimization and sharp-interface limits for the stochastic Allen–Cahn equation. Commun. Pure Appl. Math. 60(3), 393–438 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Kohn, R.V., Reznikoff, M.G., Tonegawa, Y.: Sharp-interface limit of the Allen–Cahn action functional in one space dimension. Calc. Var. Partial Differ. Equ. 25(4), 503–534 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence, RI (1996)

    MATH  Google Scholar 

  29. 29.

    Kunita, H.: Stochastic Flows and Stochastic Differential Equations, volume 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997) (Reprint of the 1990 original)

  30. 30.

    Ladyzenskaya, OA, Solonnikov, VV, Ural’tseva, NN: Linear–Quasilinear Equations of Parabolic Type. Translation AMS, vol. 23. American Mathematical Society Providence, RI (1968)

  31. 31.

    Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Sér. I Math. 327(8), 735–741 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Magni, A., Röger, M.: Variational analysis of a mean curvature flow action functional. Calc. Var. Partial Dif. 52(3), 609–639 (2015)

  33. 33.

    Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 357–383 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Modica, L., Mortola, S.: Un esempio di \(\Gamma \)-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Mugnai, L., Röger, M.: The Allen–Cahn action functional in higher dimensions. Interfaces Free Bound. 10(1), 45–78 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Mugnai, L., Röger, M.: Convergence of perturbed Allen–Cahn equations to forced mean curvature flow. Indiana Univ. Math. J. 60(1), 41–75 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Texts in Applied Mathematics. Springer, New York (2004)

    MATH  Google Scholar 

  38. 38.

    Röger, M., Weber, H.: Tightness for a stochastic Allen–Cahn equation. Stoch. Partial Differ. Equ. Anal. Comput. 1(1), 175–203 (2013)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Roubíček, T.: Nonlinear Partial Differential Equations with Applications, volume 153 of International Series of Numerical Mathematics, 2nd edn. Birkhäuser/Springer, Basel AG, Basel (2013)

  40. 40.

    Weber, H.: On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. H. Poincaré Probab. Statist. 46(4), 965–975 (2010)

  41. 41.

    Westdickenberg, M.G., Tonegawa, Y.: Higher multiplicity in the one-dimensional Allen–Cahn action functional. Indiana Univ. Math. J. 56(6), 2935–2989 (2007)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially funded by the DFG-Forschergruppe 718 Analysis and Stochastics in Complex Physical Systems. We thank Hendrik Weber for helpful discussions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Matthias Röger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Heida, M., Röger, M. Large deviation principle for a stochastic Allen–Cahn equation. J Theor Probab 31, 364–401 (2018). https://doi.org/10.1007/s10959-016-0711-7

Download citation

Keywords

  • Large deviations
  • Stochastic partial differential equations
  • Stochastic flows
  • Allen–Cahn equation

Mathematics Subject Classification (2010)

  • 60F10
  • 60H15
  • 35R60
  • 49J45