Approximation of Stochastic Invariant Manifolds

1. Front Matter

   Pages i-xv

   PDF

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

   PDF

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.

• 37L65,37D10,37L25,35B42,37L10,37L55.
• Leading-Order Taylor Approximations
• Lyapunov-Perron Integrals
• Stochastic Invariant Manifolds
• Stochastic Partial Differential Equations
• Weak Non-Resonance Conditions
• partial differential equations
• ordinary differential equations

“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)

• University of California, Los Angeles, USA

  Mickaël D. Chekroun, Honghu Liu

• Indiana University, Bloomington, USA

  Shouhong Wang

Policies and ethics