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The stable manifold theorum for semilinear stochastic evolution equations and stochastic partial differential equations. II: Existence of stable and unstable manifolds.

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preprint
posted on 01.08.2005, 13:08 by Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao
This article is a sequel, aimed at completing the characterization of the pathwise local structure of solutions of semi-linear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. The characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semi-linear see’s and spde’s (Theorems 4.1-4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments (Theorem 2.1).

History

School

  • Science

Department

  • Mathematical Sciences

Pages

546478 bytes

Publication date

2003

Notes

This pre-print has been submitted, and accepted, to the journal, Journal of Functional Analysis [© Elsevier]. The definitive version: MOHAMMED, S. A., ZHANG, T. and ZHAO, H., 2004. The stable manifold theorum for semilinear stochastic evolution equations and stochastic partial differential equations. Journal of Functional Analysis, 206(2), pp. 253-306, is available at: http://www.sciencedirect.com/science/journal/00221236.

Language

en