Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation

  • Charles-Edouard BréhierEmail author
  • Ludovic Goudenège


This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the one-dimensional stochastic Allen–Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to \(\frac{1}{2}\), like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.


Stochastic partial differential equations Splitting schemes Allen–Cahn equation Weak convergence Kolmogorov equation 

Mathematics Subject Classification

60H15 65C30 60H35 



The authors would like to thank the two anonymous referees for their remarks and suggestions. They also want to thank Arnaud Debussche for discussions, and for suggesting the approach to prove Lemma 4.4. They also wish to thank Jialin Hong and Jianbao Cui for helpful comments and suggestions to improve the presentation of the manuscript.


  1. 1.
    Allen, S., Cahn, J.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 27(6), 1085–1095 (1979)CrossRefGoogle Scholar
  2. 2.
    Andersson, A., Kruse, R., Larsson, S.: Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 4(1), 113–149 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andersson, A., Larsson, S.: Weak convergence for a spatial approximation of the nonlinear stochastic heat equation. Math. Comput. 85(299), 1335–1358 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Becker, S., Gess, B., Jentzen, A., Kloeden, P.E.: Strong convergence rates for explicit space time discrete numerical approximations of stochastic Allen–Cahn equations (2017). arXiv preprint arXiv:1711.02423
  5. 5.
    Bréhier, C.-E.: Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs. J. Complex. 56 (2019) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bréhier, C.-E., Cui, J., Hong, J.: Strong convergence rates of semi-discrete splitting approximations for stochastic Allen-Cahn equation. IMA J. Numer. Anal. 39, 2096–2134 (2018). CrossRefzbMATHGoogle Scholar
  7. 7.
    Bréhier, C.-E., Debussche, A.: Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient. J. Math. Pures Appl. 119, 193–254 (2018) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bréhier, C.-E., Goudenège, L.: Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete Contin. Dyn. Syst. B 24(8), 4169–4190 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cerrai, S.: Second Order PDE’s in Finite and Infinite Dimension. Lecture Notes in Mathematics. A probabilistic approach, vol. 1762. Springer, Berlin (2001)CrossRefGoogle Scholar
  10. 10.
    Conus, D., Jentzen, A., Kurniawan, R.: Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Ann. Appl. Probab. 29(2), 653–716 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Da Prato, G., Debussche, A.: An integral inequality for the invariant measure of a stochastic reaction–diffusion equation. J. Evol. Equ. 17(1), 197–214 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  13. 13.
    Debussche, A.: Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comput. 80(273), 89–117 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Debussche, A., Printems, J.: Weak order for the discretization of the stochastic heat equation. Math. Comput. 78(266), 845–863 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grisvard, P.: Caractérisation de quelques espaces d’interpolation. Arch. Ration. Mech. Anal. 25(1), 40–63 (1967)CrossRefGoogle Scholar
  16. 16.
    Hefter, M., Jentzen, A., Kurniawan, R.: Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces (2016). arXiv preprint arXiv:1612.03209
  17. 17.
    Jentzen, A., Kloeden, P.E.: Taylor Approximations for Stochastic Partial Differential Equations. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 83. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011)CrossRefGoogle Scholar
  18. 18.
    Jentzen, A., Kurniawan, R.: Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients (2015). arXiv:1501.03539
  19. 19.
    Kopec, M.: Quelques contributions à l’analyse numérique d’équations stochastiques. PhD thesis, Ecole normale supérieure de Rennes-ENS Rennes (2014)Google Scholar
  20. 20.
    Kovács, M., Larsson, S., Lindgren, F.: On the backward Euler approximation of the stochastic Allen–Cahn equation. J. Appl. Probab. 52(2), 323–338 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kovács, M., Larsson, S., Lindgren, F.: On the discretisation in time of the stochastic Allen-Cahn equation. Math. Nachrichten 291(5–6), 966–995 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kruse, R.: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Lecture Notes in Mathematics, vol. 2093. Springer, Cham (2014)CrossRefGoogle Scholar
  23. 23.
    Liu, Z., Qiao, Z.: Wong-Zakai approximations of stochastic Allen-Cahn equation. Int. J. Numer. Anal. Mod. 16, 681–694 (2019)MathSciNetGoogle Scholar
  24. 24.
    Liu, Z., Qiao, Z.: Strong approximation of monotone stochastic partial differential equations driven by white noise. IMA J. Numer. Anal. (2019).
  25. 25.
    Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, New York (2014)CrossRefGoogle Scholar
  26. 26.
    Majee, A., Prohl, A.: Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math. 18(2), 297–311 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications (New York), 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  28. 28.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefGoogle Scholar
  29. 29.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  30. 30.
    Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10(4), 545–551 (1959)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wang, X.: Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete Contin. Dyn. Syst. 36(1), 481–497 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, X.: An efficient explicit full discrete scheme for strong approximation of stochastic Allen–Cahn equation (2018). arXiv preprint arXiv:1802.09413
  33. 33.
    Wang, X., Gan, S.: Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise. J. Math. Anal. Appl. 398(1), 151–169 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CNRS, UMR5208, Institut Camille JordanUniv Lyon, Université Claude Bernard Lyon 1VilleurbanneFrance
  2. 2.CNRS - FR3487, Fédération de Mathématiques de CentraleSupélec, CentraleSupélecUniversité Paris-SaclayGif-sur-YvetteFrance

Personalised recommendations