Approximation of Stochastic Invariant Manifolds

Stochastic Manifolds for Nonlinear SPDEs I

Mickaël D. Chekroun, Honghu Liu, Shouhong Wang (Autoren)

Buch | Softcover

XV, 127 Seiten

2015 | 2015
Springer International Publishing (Verlag)
978-3-319-12495-7 (ISBN)

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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.

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.

"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)

Erscheint lt. Verlag	13.1.2015
Reihe/Serie	SpringerBriefs in Mathematics
Zusatzinfo	XV, 127 p. 1 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	231 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
Themenwelt	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
Schlagworte	37L65,37D10,37L25,35B42,37L10,37L55. • Leading-Order Taylor Approximations • Lyapunov-Perron Integrals • Ordinary differential equations • Partial differential equations • Stochastic Invariant Manifolds • stochastic partial differential equations • Weak Non-Resonance Conditions
ISBN-10	3-319-12495-1 / 3319124951
ISBN-13	978-3-319-12495-7 / 9783319124957
Zustand	Neuware