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\iteman{ZMATH 06136721}
\itemau{Lindner, Felix; Schilling, Ren\'e L.}
\itemti{Weak order for the discretization of the stochastic heat equation driven by impulsive noise.}
\itemso{Potential Anal. 38, No. 2, 345-379 (2013).}
\itemab
Summary: We study the approximation of the distribution of $X _{T }$, where $(X _{t })_{t \in [0, T]}$ 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 $(Z _{t })_{t \in [0, T]}$ is an impulsive cylindrical process and the operator $Q$ describes the spatial covariance structure of the noise; we assume that $A ^{ - \alpha }$ has finite trace for some $\alpha > 0$ and that $A ^{\beta } Q$ is bounded for some $\beta \in (\alpha - 1, \alpha $]. 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 $\theta $-method in time (parameter $\Delta t = T/N$). 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 $\gamma < 1 - \alpha + \beta $. 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.
\itemrv{~}
\itemcc{60H15 65M60 60H35 60G51 60G52 65C30}
\itemut{weak order; stochastic heat equation; impulsive cylindrical process; infinite dimensional L\'evy process; finite element; Euler scheme}
\itemli{doi:10.1007/s11118-012-9276-y}
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