Curvature Motion Perturbed by a Direction-Dependent Colored Noise

  • Clément Denis
  • Tadahisa FunakiEmail author
  • Satoshi Yokoyama
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)


The aim of this paper is twofold. First we give a brief overview of several results on the deterministic and stochastic motions by mean curvature and their derivation under the so-called sharp interface limit. Then, we study the motions by mean curvature perturbed by a direction-dependent Gaussian colored noise described by \(V=\kappa + \dot{W}(t,\mathbf n )\). This part is a generalization of (Funaki, Acta Math Sin (Engl Ser), 15:407–438, 1999) [10] where the noise is independent from space. We derive a uniform moment estimate on solutions of approximating equations and prove a Wong–Zakai type convergence theorem (in law) for the SPDEs for the curvature of a convex curve in two-dimensional space before the time the curve exhibits a singularity.


Stochastic partial differential equation Motion by mean curvature Wong–Zakai theorem Colored noise 

MSC 2010

60H15 35K93 



We thank the referee who suggested the proof of Theorem 1, in particular, the SPDE (26) and the reference [5]. We acknowledge the support from the training course at ENS Cachan, under which C. Denis could visit Tokyo and stayed for four months in 2016.


  1. 1.
    Alfaro, M., Antonopoulou, D., Karali, G., Matano, H.: Generation and propagation of fine transition layers for the stochastic Allen-Cahn equation (preprint, 2016)Google Scholar
  2. 2.
    Allen, S.M., Cahn, J.W.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  3. 3.
    Bellettini, G.: Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations. Lecture Notes, vol. 12. Scuola Normale Superiore di Pisa (2013)Google Scholar
  4. 4.
    Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  5. 5.
    Debussche, A., de Moor, S., Hofmanova, M.: A regularity result for quasilinear stochastic partial differential equations of parabolic type. SIAM J. Math. Anal. 47, 1590–1614 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dirr, N., Luckhaus, S., Novaga, M.: A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differ. Equ. 13, 405–425 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Es-Sarhir, A., von Renesse, M.: Ergodicity of stochastic curve shortening flow in the plane. SIAM J. Math. Anal. 44, 224–244 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Func. Anal. 109, 257–288 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102, 221–288 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Funaki, T.: Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15, 407–438 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Funaki, T.: Lectures on Random Interfaces. SpringerBriefs in Probability and Mathematical Statistics. Springer, Berlin (2016)CrossRefGoogle Scholar
  12. 12.
    Funaki, T., Yokoyama, S.: Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation. arXiv:1610.01263
  13. 13.
    Funaki, T., Nakada, S., Yokoyama, S.: A stochastically perturbed volume preserving mean curvature flow (preprint, 2017)Google Scholar
  14. 14.
    Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
  15. 15.
    Giga, Y., Mizoguchi, N.: Existence of periodic solutions for equations of evolving curves. SIAM J. Math. Anal. 27, 5–39 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gurtin, M.E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford (1993)zbMATHGoogle Scholar
  17. 17.
    Hofmanová, M., Röger, M., von Renesse, M.: Weak solutions for a stochastic mean curvature flow of two-dimensional graphs. Probab. Theory Relat. Fields 168, 373–408 (2017)Google Scholar
  18. 18.
    Kawasaki, K., Ohta, T.: Kinetic drumhead model of interface I. Prog. Theoret. Phys. 67, 147–163 (1982)CrossRefGoogle Scholar
  19. 19.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  20. 20.
    Lee, K.: Generation and motion of interfaces in one-dimensional stochastic Allen-Cahn equation. J. Theor. Probab., published online (2016)Google Scholar
  21. 21.
    Lee, K.: Generation of interfaces for multi-dimensional stochastic Allen-Cahn equation with a noise smooth in space, arXiv:1604.06535
  22. 22.
    Lions, P.L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Ser. I Math. 326, 1085–1092 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lions, P.L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Ser. I Math. 327, 735–741 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Otto, F., Weber, H.: Quasilinear SPDEs via rough paths. arXiv:1605.09744
  25. 25.
    Röger, M., Weber, H.: Tightness for a stochastic Allen-Cahn equation. Stoch. Partial Differ. Equ. Anal. Comput. 1, 175–203 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Weber, H.: On the short time asymptotic of the stochastic Allen-Cahn equation. Ann. Inst. H. Poincaré Probab. Statist. 46, 965–975 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Clément Denis
    • 1
  • Tadahisa Funaki
    • 2
    • 3
    Email author
  • Satoshi Yokoyama
    • 2
    • 3
  1. 1.Département de mathématiquesEcole Normale Supérieure (ENS) de Paris -SaclayCachanFrance
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  3. 3.Department of Mathematics, School of Fundamental Science and EngineeringWaseda UniversityTokyoJapan

Personalised recommendations