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Approximation of Stochastic Invariant Manifolds

Stochastic Manifolds for Nonlinear SPDEs I

  • Book
  • © 2015

Overview

Authors:
  1. Mickaël D. Chekroun

    1. University of California, Los Angeles, USA

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  2. Honghu Liu

    1. University of California, Los Angeles, USA

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  3. Shouhong Wang

    1. Indiana University, Bloomington, USA

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Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xv
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  2. General Introduction

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 1-7

  3. Stochastic Invariant Manifolds: Background and Main Contributions

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 9-12

  4. Preliminaries

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 13-27

  5. Existence and Attraction Properties of Global Stochastic Invariant Manifolds

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 29-47

  6. Local Stochastic Invariant Manifolds: Preparation to Critical Manifolds

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 49-59

  7. Local Stochastic Critical Manifolds: Existence and Approximation Formulas

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 61-92

  8. Approximation of Stochastic Hyperbolic Invariant Manifolds

    • Mickaël D. Chekroun, Honghu Liu, Shouhong Wang
    Pages 93-98

  9. Back Matter

    Pages 99-127
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • University of California, Los Angeles, USA

    Mickaël D. Chekroun, Honghu Liu

  • Indiana University, Bloomington, USA

    Shouhong Wang

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