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

2012/03/07 by Sonja Cox, Cox, Sonja, Erika Hausenblas +1
Mathematics · #35A30 #35R60 #46N40 #60H15 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:35A30 #msc:35R60 #msc:46N40 #msc:60H15

paper · pdf · doi:10.48550/arxiv.1203.1606

arxiv created 2012/03/07 · arxiv updated 2012/03/08

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 U₀ in the L^p(Ω;C([0,T];X))-norm. In particular, this difference can be estimated || R(λ:A)-R(λ:A₀) || 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.

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