Abstract
Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rates of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.
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References
Ciarlet, P.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North Holland, Amsterdam (1978)
Cohen, A.: Numerical Analysis of Wavelet Methods. Studies in Mathematics and its Applications, vol. 32. Elsevier, Amsterdam (2003)
Gyöngy, I., Krylov, N.V.: On stochastic equations with respect to semi-martingales II, Ito formula in Banach spaces. Stochastics 6, 153–173 (1982)
Gyöngy, I.: On stochastic equations with respect to semimartingales III. Stochastics 7, 231–254 (1982)
Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal. 11, 1–37 (1999)
Gyöngy, I., Martinez, T.: Solutions of partial differential equations as extremals of convex functionals. Acta Math. Hungar. 109, 127–145 (2005)
Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23, 99–134 (2005)
Gyöngy, I., Millet, A.: Rate of convergence of implicit approximations for stochastic evolution equations. In: Baxendale, P., Lototsky, S. (eds.) Stochastic Differential Equations: Theory and Applications (A volume in honor of Boris L. Rosovskii), vol. 2, pp. 281–310. World Scientific Interdisciplinary Mathematical Sciences. World Scientific, Singapore (2007)
Krylov, N.V., Rosovskii, B.L.: On Cauchy problem for linear stochastic partial differential equations. Math. USSR Izvestija 11(4), 1267–1284 (1977)
Krylov, N.V., Rosovskii, B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Études Mathématiques. Dunod Gauthiers-Villars (1969)
Mueller-Gronbach, T., Ritter, K.: An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise (2006). arXiv math.PR/0604600
Pardoux, E.: Équations aux dérivées partielles stochastiques nonlinéares monotones. Étude de solutions fortes de type Itô, Thèse Doct. Sci. Math. Univ. Paris Sud (1975)
Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3–2, 127–167 (1979)
Pardoux, E.: Filtrage non linéaire et équations aux derivées partielles stochastiques associées. In: École d’été de Probabilités de Saint-Flour 1989. Lecture Notes in Math., vol. 1464, pp. 67–163. Springer, New York (1981)
Rozovskii, B.: Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht (1990)
Yan, Y.B.: Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. Numer. Math. 44, 829–847 (2004)
Yoo, H.: An analytic approach to stochastic partial differential equations and its applications. Thesis, University of Minnesota (1998)
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This paper was written while the first named author was visiting the University of Paris 1.
The research of this author is partially supported by EU Network HARP.
The research of the second named author is partially supported by the research project BMF2003-01345.
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Gyöngy, I., Millet, A. Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations. Potential Anal 30, 29–64 (2009). https://doi.org/10.1007/s11118-008-9105-5
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DOI: https://doi.org/10.1007/s11118-008-9105-5
Keywords
- Stochastic evolution equations
- Monotone operators
- Coercivity
- Space time approximations
- Galerkin method
- Wavelets
- Finite elements