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

TL;DR

This paper analyzes how perturbations in the linear operator of quasi-linear stochastic differential equations in UMD Banach spaces affect solutions, providing estimates crucial for convergence analysis of discretization schemes.

Contribution

It establishes perturbation estimates for solutions of quasi-linear stochastic PDEs in UMD Banach spaces, focusing on linear operator perturbations and convergence rates.

Findings

Derived bounds for solution differences under operator perturbations

Provided convergence rates for Yosida approximations

Set groundwork for analyzing discretization scheme convergence

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.