The Annals of Probability

Invariant manifolds for stochastic partial differential equations

Jinqiao Duan, Kening Lu, and Björn Schmalfuss

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Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite- and infinite-dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite-dimensional random dynamical systems generated by stochastic partial differential equations. We first introduce a random graph transform and a fixed point theorem for nonautonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

Article information

Ann. Probab., Volume 31, Number 4 (2003), 2109-2135.

First available in Project Euclid: 12 November 2003

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 37L25: Inertial manifolds and other invariant attracting sets 37D10: Invariant manifold theory

Invariant manifolds cocycles nonautonomous dynamical systems stochastic partial differential equations generalized fixed points


Duan, Jinqiao; Lu, Kening; Schmalfuss, Björn. Invariant manifolds for stochastic partial differential equations. Ann. Probab. 31 (2003), no. 4, 2109--2135. doi:10.1214/aop/1068646380.

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