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Approximating the coefficients in semilinear stochastic partial differential equations. (English) Zbl 1231.35324

Summary: We investigate, in the setting of UMD Banach spaces $E$, the continuous dependence on the data $A,F,G$ and $\xi$ of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form

$\left\{\begin{array}{c}\mathrm{d}X\left(t\right)=\left[AX\left(t\right)+F\left(t,X\left(t\right)\right)\right]\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t+G\left(t,X\left(t\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{W}_{H}\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],\hfill \\ X\left(0\right)=\xi ,\hfill \end{array}\right\$

where ${W}_{H}$ is a cylindrical Brownian motion in a Hilbert space $H$. We prove continuous dependence of the compensated solutions $X\left(t\right)-{e}^{tA}\xi$ in the norms ${L}^{p}\left({\Omega };\phantom{\rule{0.166667em}{0ex}}{C}^{\lambda }\left(\left[0,T\right];\phantom{\rule{0.166667em}{0ex}}E\right)\right)$ assuming that the approximating operators ${A}_{n}$ are uniformly sectorial and converge to $A$ in the strong resolvent sense and that the approximating nonlinearities ${F}_{n}$ and ${G}_{n}$ are uniformly Lipschitz continuous in suitable norms and converge to $F$ and $G$ pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite dimensional multiplicative noise.

MSC:
 35R60 PDEs with randomness, stochastic PDE 60H15 Stochastic partial differential equations 35K58 Semilinear parabolic equations 60J65 Brownian motion 46B09 Probabilistic methods in Banach space theory
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