Approximating the coefficients in semilinear stochastic partial differential equations

Abstract

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 $$ \left\{ \begin{array}{l} {\rm d}X(t) = [AX(t) + F(t, X(t))] \, {\rm d}t + G(t, X(t)) \, {\rm d}W_H(t),\quad t \in [0,T], X(0) = \xi, \end{array} \right. $$ where WH is a cylindrical Brownian motion in a Hilbert space H. We prove continuous dependence of the compensated solutions X(t) − etAξ in the norms Lp(Ω;Cλ([0, T]; E)) assuming that the approximating operators An are uniformly sectorial and converge to A in the strong resolvent sense and that the approximating nonlinearities Fn and Gn 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.