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
(Ω;C([0,T];X))
-norm. In particular, this difference can be estimated
||R(λ:A)−R(λ:
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.