A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: dU(t)=AU(t)dt+F(t,U(t))dt+G(t,U(t))dWH(t),t>0;U(0)=x0.(SDE)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$\end{document}Here, A is the generator of an analytic C0-semigroup on a UMD Banach space X, H is a Hilbert space, WH is an H-cylindrical Brownian motion, G:[0,T]×X→L(H,XθGA)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$$\end{document} , and F:[0,T]×X→XθFA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$$\end{document} for some θG>-12,θF>-32+1τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$$\end{document} , where τ∈[1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau\in [1, 2]}$$\end{document} denotes the type of the Banach space and XθFA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X_{\theta_F}^{A}}$$\end{document} denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions.