Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Open access
Date
2021-12Type
- Journal Article
Abstract
Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000515653Publication status
publishedExternal links
Journal / series
Applied Mathematics & OptimizationVolume
Pages / Article No.
Publisher
SpringerSubject
weak convergence; stochastic wave equations; multiplicative noise; hyperbolic Anderson model; spatial approximationOrganisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
Funding
ETH-47 15-2 - Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with Levy noise (ETHZ)
156603 - Numerical approximations of nonlinear stochastic ordinary and partial differential equations (SNF)
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