Approximating the coefficients in semilinear stochastic partial differential equations

2010/03/09 by Kunze, Markus, van Neerven, Jan
#47D06 (Secondary) #60H15 (Primary) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1003.1876

Abstract

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X₀ 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), X(0)=X₀, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^tAX₀ 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.

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