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

Abstract

We consider the effect of perturbations to a quasi-linear parabolic stochastic differential equation set in a UMD Banach space $X$. To be precise, we consider perturbations of the linear part, i.e. the term concerning a linear operator $A$ generating an analytic semigroup. We provide estimates for the difference between the solution to the original equation $U$ and the solution to the perturbed equation $U_0$ in the $L^p(\Omega;C([0,T];X))$-norm. In particular, this difference can be estimated $|| R(\lambda:A)-R(\lambda:A_0) ||$ for sufficiently smooth non-linear terms. The work is inspired by the desire to prove convergence of space discretization schemes for such equations. In this article we prove convergence rates for the case that $A$ is approximated by its Yosida approximation, and in a forthcoming publication we consider convergence of Galerkin and finite-element schemes in the case that $X$ is a Hilbert space.