Approximating the coefficients in semilinear stochastic partial differential equations

TL;DR

This paper studies how solutions to semilinear stochastic partial differential equations depend continuously on their data, providing approximation results for coefficients in the setting of UMD Banach spaces with applications to parabolic SPDEs.

Contribution

It establishes continuous dependence of solutions on coefficients and initial data for semilinear SPDEs in UMD Banach spaces, including approximation results for coefficients.

Findings

Proves continuous dependence of solutions on data in specific Banach space norms.

Shows convergence of approximate solutions with uniformly sectorial operators and Lipschitz nonlinearities.

Applies results to semilinear parabolic SPDEs with finite-dimensional multiplicative noise.

Abstract

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 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)=X_0, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^{tA}X_0 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.