Abstract
We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation:
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.
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References
Brzeźniak Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61(3-4), 245–295 (1997)
D.L. Burkholder. Martingales and singular integrals in Banach spaces. In “Handbook of the Geometry of Banach Spaces”, Vol. I, pages 233–269. North-Holland, Amsterdam, 2001.
S.G. Cox. Stochastic Differential Equations in Banach Spaces: Decoupling, Delay equations, and Approximations in Space and Time, 2012. PhD thesis, available online at http://repository.tudelft.nl.
Cox S.G., Hausenblas E.: Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise. Int. J. Comput. Math. 89, 2460–2478 (2012)
Cox S.G., van Neerven J.M.A.M.: Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise. Numer. Math. 125(2), 259–345 (2013)
Da Prato G., Kwapień S., Zabczyk J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23(1), 1–23 (1987)
G. Da Prato and J. Zabczyk. “Stochastic Equations in Infinite Dimensions”, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
Desch W., Schappacher W.: A note on the comparison of C 0-semigroups. Semigroup Forum 35(2), 237–243 (1987)
K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.
M.H.A. Haase. “The Functional Calculus for Sectorial Operators”, volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006.
Jung M.: On the relationship between perturbed semigroups and their generators. Semigroup Forum 61(2), 283–297 (2000)
N.J. Kalton and L. Weis. The H ∞-calculus and square function estimates. In preparation.
Kloeden P.E., Neuenkirch A.: The pathwise convergence of approximation schemes for stochastic differential equations. LMS Journal of Comp. and Math. 10, 235–253 (2007)
M.C. Kunze and J.M.A.M. van Neerven. Approximating the coefficients in semilinear stochastic partial differential equations. J. Evol. Equ., 2011.
M. Ledoux and M. Talagrand. Probability in Banach spaces, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1991.
J.M.A.M. van Neerven. γ-Radonifying operators – a survey. Proceedings of the CMA 44, pages 1–62, 2010.
J.M.A.M. van Neerven and M.C. Veraar. On the stochastic Fubini theorem in infinite dimensions. In Stochastic partial differential equations and applications—VII, volume 245 of Lect. Notes Pure Appl. Math., pages 323–336. Chapman & Hall/CRC, Boca Raton, FL, 2006.
van Neerven J.M.A.M., Veraar M.C., Weis L.: Stochastic integration in UMD Banach spaces. Annals Probab. 35, 1438–1478 (2007)
van Neerven J.M.A.M., Veraar M.C., Weis L.: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255(4), 940–993 (2008)
A. Pazy. “Semigroups of Linear Operators and Applications to Partial Differential Equations”, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983.
D. Revuz and M. Yor. “Continuous Martingales and Brownian Motion”, volume 293 of Grundlehren der Math. Wissenschaften. Springer-Verlag, Berlin, 3rd edition, 1999.
Robinson D.W.: The approximation of flows. J. Functional Analysis 24(3), 280–290 (1977)
Schnaubelt R., Veraar M.: Structurally damped plate and wave equations with random point force in arbitrary space dimensions. Differential Integral Equations 23(9-10), 957–988 (2010)
Weis L.: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann. 319(4), 735–758 (2001)
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Cox, S.G., Hausenblas, E. A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces. J. Evol. Equ. 13, 795–827 (2013). https://doi.org/10.1007/s00028-013-0203-5
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DOI: https://doi.org/10.1007/s00028-013-0203-5