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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 $\begin{cases}{\mathrm d}X(t) = [AX(t) + F(t, X(t))] \, {\mathrm d}t + G(t, X(t)) \, {\mathrm d}W_H(t),\quad t \in [0,T],\\ X(0) = \xi, \end{cases}$ where $$W _{H }$$ is a cylindrical Brownian motion in a Hilbert space $$H$$. We prove continuous dependence of the compensated solutions $$X(t) - e ^{tA } \xi$$ in the norms $$L ^{p }(\Omega ; \, C ^{\lambda }([0, T]; \, E))$$ 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 partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35K58 Semilinear parabolic equations 60J65 Brownian motion 46B09 Probabilistic methods in Banach space theory
##### Keywords:
stochastic integration; Banach spaces; multiplicative noise
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