Electronic ISBN:  9781470405236 
Product Code:  MEMO/196/917.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 196; 2008; 105 ppMSC: Primary 60;
The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the longterm behavior of the solution field near stationary points. The analysis is in two parts. In Part 1, the authors prove general existence and compactness theorems for \(C^k\)cocycles of semilinear see's and spde's. The results cover a large class of semilinear see's as well as certain semilinear spde's with Lipschitz and nonLipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycleinvariant random points in the infinitedimensional state space. The pathwise local structure of solutions of semilinear see's and spde's near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution.

Table of Contents

Chapters

Introduction

1. The stochastic semiflow

1.1 Basic concepts

1.2 Flows and cocycles of semilinear see’s

1.3 Semilinear spde’s: Lipschitz nonlinearity

1.4 Semilinear spde’s: NonLipschitz nonlinearity

2. Existence of stable and unstable manifolds

2.1 Hyperbolicity of a stationary trajectory

2.2 The nonlinear ergodic theorem

2.3 Proof of the local stable manifold theorem

2.4 The local stable manifold theorem for see’s and spde’s


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The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Such characterization is realized through the longterm behavior of the solution field near stationary points. The analysis is in two parts. In Part 1, the authors prove general existence and compactness theorems for \(C^k\)cocycles of semilinear see's and spde's. The results cover a large class of semilinear see's as well as certain semilinear spde's with Lipschitz and nonLipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycleinvariant random points in the infinitedimensional state space. The pathwise local structure of solutions of semilinear see's and spde's near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution.

Chapters

Introduction

1. The stochastic semiflow

1.1 Basic concepts

1.2 Flows and cocycles of semilinear see’s

1.3 Semilinear spde’s: Lipschitz nonlinearity

1.4 Semilinear spde’s: NonLipschitz nonlinearity

2. Existence of stable and unstable manifolds

2.1 Hyperbolicity of a stationary trajectory

2.2 The nonlinear ergodic theorem

2.3 Proof of the local stable manifold theorem

2.4 The local stable manifold theorem for see’s and spde’s