Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Andersson, Adam; Lindner, Felix

doi:10.3934/dcdsb.2019081

Cited by 2 publications

(3 citation statements)

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“…These relations also allow for the treatment of equations with multiplicative noise, cf. the accompanying paper [3]. Finally, we note that in [56] Malliavin calculus methods are used in the spirit of [27] to derive a weak convergence result for spatial approximations of linear equations with square-integrable Lévy noise, which is very similar to an earlier result from [26].…”

Section: Introductionsupporting

confidence: 60%

“…As a first application of our theory, we analyze the weak order of convergence of space-time discretizations for a class of linear SPDE driven by α-stable noise, α ∈ (1, 2). In the accompanying paper [3] we also treat semi-linear equations.…”

Section: Introductionmentioning

confidence: 99%

“…Acknowledgement. We thank Kristin Kirchner, Raphael Kruse, Annika Lang and Stig Larsson for the participation in early discussions regarding this work and [3].…”

mentioning

confidence: 99%

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Poisson Malliavin calculus in Hilbert space with an application to SPDE

Andersson,

Lindner

2017

Preprint

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In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment estimates, and thus allows for a simple treatment of the Malliavin operators. The main part of the theory is developed for general Poisson random measures, defined on a σ-finite measure space, with minimal conditions. The theory is shown to apply to a space-time setting, suitable for studying stochastic partial differential equations. As an application, we analyze the weak order of convergence of space-time approximations for a class of linear equations with α-stable noise, α ∈ (1, 2). For a suitable class of test functions, the weak order of convergence is found to be α times the strong order.

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Section: Introductionsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 99%

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Poisson Malliavin calculus in Hilbert space with an application to SPDE

Andersson,

Lindner

2017

Preprint

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Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Mukam,

Tambue

2024

Computational Methods in Applied Mathematics

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This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

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Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

Cited by 2 publications

References 32 publications

Poisson Malliavin calculus in Hilbert space with an application to SPDE

Poisson Malliavin calculus in Hilbert space with an application to SPDE

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

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