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Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE

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Abstract

We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

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Acknowledgments

The authors wish to thank M. Kovács for fruitful discussions during the preparation of the work [2], which led to improvements of the present paper. We also thank A. Lang and X. Wang for valuable comments on an earlier version of the manuscript and A. Jentzen for making us aware of a reference. The first two authors also acknowledge the kind support by W.-J. Beyn, B. Gentz, and the DFG-funded CRC 701 ’Spectral Structures and Topological Methods in Mathematics’ by making possible an inspiring research stay at Bielefeld University, where part of this work was written.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96, Gothenburg, Sweden

    Adam Andersson & Stig Larsson

  2. Institut für Mathematik, Technische Universität Berlin, Sek. MA 5-3, Straße des 17. Juni 136, 10623, Berlin, Germany

    Raphael Kruse

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  1. Adam Andersson
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  2. Raphael Kruse
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  3. Stig Larsson
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Correspondence to Adam Andersson.

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Andersson, A., Kruse, R. & Larsson, S. Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE. Stoch PDE: Anal Comp 4, 113–149 (2016). https://doi.org/10.1007/s40072-015-0065-7

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  • DOI: https://doi.org/10.1007/s40072-015-0065-7

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