Journal of Evolution Equations

, Volume 13, Issue 4, pp 795–827 | Cite as

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

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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 C 0-semigroup on a UMD Banach space X, H is a Hilbert space, W H 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.

Let A 0 denote the perturbed operator and U 0 the solution to (SDE) with A substituted by A 0. We provide estimates for \({\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}\) in terms of \({D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}\) . Here, \({\delta\in [0, 1]}\) is assumed to satisfy \({0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}\) . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.

Mathematics Subject Classification (2010)

46N40 35R60 35A30 60H15 

Keywords

Perturbations Stochastic differential equations Stochastic convolutions Stochastic partial differential equations Yosida approximation 
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© Springer Basel 2013

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsETH ZürichZurichSwitzerland
  2. 2.Department of Mathematics and Information TechnologyMontana Universität LeobenLeobenAustria

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