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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 ξ of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form

dX(t)=[AX(t)+F(t,X(t))]dt+G(t,X(t))dW H (t),t[0,T],X(0)=ξ,

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 ξ in the norms L p (Ω;C λ ([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.

35R60PDEs with randomness, stochastic PDE
60H15Stochastic partial differential equations
35K58Semilinear parabolic equations
60J65Brownian motion
46B09Probabilistic methods in Banach space theory
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