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

Abstract

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ 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, \({G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}\) , and \({F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}\) for some \({\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}\) , where \({\tau\in [1, 2]}\) denotes the type of the Banach space and \({X_{\theta_F}^{A}}\) 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.