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Zbl 1231.35324
Kunze, Markus; van Neerven, Jan
Approximating the coefficients in semilinear stochastic partial differential equations.
(English)
[J] J. Evol. Equ. 11, No. 3, 577-604 (2011). ISSN 1424-3199; ISSN 1424-3202/e

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 $$\cases{\mathrm d}X(t) = [AX(t) + F(t, X(t))] \, {\mathrm d}t + G(t, X(t)) \, {\mathrm d}W_H(t),\quad t \in [0,T],\\ X(0) = \xi, \endcases$$ 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 2000:
*35R60 PDE with randomness
60H15 Stochastic partial differential equations
35K58
60J65 Brownian motion
46B09 Probabilistic methods in Banach space theory

Keywords: stochastic integration; Banach spaces; multiplicative noise

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