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The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations
Salah-Eldin A. Mohammed, Southern Illinois University at Carbondale, IL, Tusheng Zhang, University of Manchester, England, and Huaizhong Zhao, Loughborough University, Leicestershire, England
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Memoirs of the American Mathematical Society
2008; 105 pp; softcover
Volume: 196
ISBN-10: 0-8218-4250-1
ISBN-13: 978-0-8218-4250-8
List Price: US$65
Individual Members: US$39
Institutional Members: US$52
Order Code: MEMO/196/917
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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 long-term 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 non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. In Part 2, stationary solutions are viewed as cocycle-invariant random points in the infinite-dimensional 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

  • Introduction
Part 1. The stochastic semiflow
  • Basic concepts
  • Flows and cocycles of semilinear see's
  • Semilinear spde's: Lipschitz nonlinearity
  • Semilinear spde's: Non-Lipschitz nonlinearity
Part 2. Existence of stable and unstable manifolds
  • Hyperbolicity of a stationary trajectory
  • The nonlinear ergodic theorem
  • Proof of the local stable manifold theorem
  • The local stable manifold theorem for see's and spde's
  • Acknowledgments
  • Bibliography
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