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
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.
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The authors are grateful to Marta Sanz-Solé for pointing out the references [3,24] and Mark Veraar for a helpful discussion.
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The authors are supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).
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Kunze, M., van Neerven, J. Approximating the coefficients in semilinear stochastic partial differential equations. J. Evol. Equ. 11, 577–604 (2011). https://doi.org/10.1007/s00028-011-0102-6
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DOI: https://doi.org/10.1007/s00028-011-0102-6