Approximation of Stochastic Invariant Manifolds electronic resource Stochastic Manifolds for Nonlinear SPDEs I / by Mickaël D. Chekroun, Honghu Liu, Shouhong Wang.
Material type: TextSeries: SpringerBriefs in MathematicsPublication details: Cham : Springer International Publishing : Imprint: Springer, 2015Description: XV, 127 p. 1 illus. in color. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783319124964Subject(s): mathematics | Dynamics | Ergodic theory | Differential Equations | Partial Differential Equations | Probabilities | Mathematics | Dynamical Systems and Ergodic Theory | Partial Differential Equations | Probability Theory and Stochastic Processes | Ordinary Differential EquationsDDC classification: 515.39 | 515.48 LOC classification: QA313Online resources: Click here to access onlineGeneral Introduction -- Stochastic Invariant Manifolds: Background and Main Contributions -- Preliminaries -- Stochastic Evolution Equations -- Random Dynamical Systems -- Cohomologous Cocycles and Random Evolution Equations -- Linearized Stochastic Flow and Related Estimates -- Existence and Attraction Properties of Global Stochastic Invariant Manifolds -- Existence and Smoothness of Global Stochastic Invariant Manifolds -- Asymptotic Completeness of Stochastic Invariant Manifolds -- Local Stochastic Invariant Manifolds: Preparation to Critical Manifolds -- Local Stochastic Critical Manifolds: Existence and Approximation Formulas -- Standing Hypotheses -- Existence of Local Stochastic Critical Manifolds -- Approximation of Local Stochastic Critical Manifolds -- Proofs of Theorem 6.1 and Corollary 6.1 -- Approximation of Stochastic Hyperbolic Invariant Manifolds -- A Classical and Mild Solutions of the Transformed RPDE -- B Proof of Theorem 4.1 -- References.
This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
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