Approximation of stochastic invariant manifolds [electronic resource] : stochastic manifolds for nonlinear SPDEs I /
Chekroun, Mickaël D.author.
Liu, Honghu.author.
Wang, Shouhong.author.
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.
Description based upon print version of record.
Includes bibliographical references and index.
General 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.
Dynamics.
Ergodic theory.
Partial differential equations.
Probabilities.
Differential equations.
Dynamical Systems and Ergodic Theory.
Partial Differential Equations.
Probability Theory and Stochastic Processes.
Ordinary Differential Equations.