Mickaël D. Chekroun, Honghu Liu, Shouhong Wang

Approximation of Stochastic Invariant Manifolds (eBook, PDF)

Stochastic Manifolds for Nonlinear SPDEs I

• Format: PDF

Produktbeschreibung

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.

Produktdetails

• Verlag: Springer International Publishing
• Erscheinungstermin: 20. Dezember 2014
• Englisch
• ISBN-13: 9783319124964
• Artikelnr.: 43784419

Inhaltsangabe

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.

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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.

Rezensionen

"The book under review is the first in a two-volume series and deals with approximation of stochastic manifolds that are invariant for dynamics of a parabolic Stratonovich SPDE driven by a one-dimensional Wiener process. ... The book is aimed at readers interested in stochastic partial differential equations and random dynamical systems." (Martin Ondreját, zbMATH 1319.60002, 2015)