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abstract:
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 written in abstract form as
\[dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T],\]
driven by an impulsive space time noise whose covariance operator $Q$ is given. $A^{-\alpha}$ is assumed to have finite trace for some $\alpha>0$ and $seitenumbruch unterbindenA^\beta
Q$ is assumed to be bounded for some $\beta\geq0$. \\
A discretized solution $(X_h^n)_{n\in\{0,1,\ldots,N\}}$ is defined via finite element methods in space (parameter $h>0$) and implicit Euler schemes in time
(parameter $\Delta t=T/N$). For suitable functions $\varphi$ defined on $H$, it is shown that
\[|\mathds{E}\varphi(X^N_h)-\mathds{E}\varphi(X_T)|=O(h^\gamma+\Delta t^{\gamma/2})\]
where $\gamma |
