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Potential Analysis

, Volume 38, Issue 2, pp 345–379 | Cite as

Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

  • Felix LindnerEmail author
  • René L. Schilling
Article

Abstract

We study the approximation of the distribution of X T , where (X t ) t ∈ [0, T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise,
$$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$
Here (Z t ) t ∈ [0, T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A − α has finite trace for some α > 0 and that A β Q is bounded for some β ∈ (α − 1, α]. A discretized solution \((X_h^n)_{n\in\{0,1,\ldots,N\}}\) is defined via the finite element method in space (parameter h > 0) and a θ-method in time (parameter Δt = T/N). For \(\varphi \in C^2_b(H;{\mathbb R})\) we show an integral representation for the error \(|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|\) and prove that
$$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$
where γ < 1 − α + β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.

Keywords

Weak order Stochastic heat equation Impulsive cylindrical process Infinite dimensional Lévy process Finite element Euler scheme 

Mathematics Subject Classifications (2000)

Primary 60H15 65M60; Secondary 60H35 60G51 60G52 65C30 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institut für Mathematische StochastikTechnische Universität DresdenDresdenGermany

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