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

Abstract

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.

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Acknowledgements

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.

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Correspondence to Charles-Edouard Bréhier.

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Bréhier, CE., Goudenège, L. Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation. Bit Numer Math 60, 543–582 (2020). https://doi.org/10.1007/s10543-019-00788-x

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Keywords

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

Mathematics Subject Classification

  • 60H15
  • 65C30
  • 60H35