Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE

by A. Andersson and R. Kruse and S. Larsson

(Report number 2013-50)

Abstract
In this paper we introduce a new family of refined Watanabe-Sobolev spaces that capture in a fine way 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 Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the Ito formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.

Keywords: SPDE, finite element method, backward Euler, weak convergence, convergence of moments, Malliavin calculus, duality, spatio-temporal discretization, refined Watanabe-Sobolev spaces, Burkholder's inequality

@Techreport{AKL13_550,
  author = {A. Andersson and R. Kruse and S. Larsson},
  title = {Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-50},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-50.pdf },
  year = {2013}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).