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

  • Adam AnderssonEmail author
  • Raphael Kruse
  • Stig Larsson
Article

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.

Keywords

SPDE Finite element method Backward Euler Weak convergence Convergence of moments Malliavin calculus  Duality Spatio-temporal discretization 

Mathematics Subject Classification

60H15 60H07 65C30 65M60 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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