The Annals of Probability

Invariant manifolds for stochastic partial differential equations

Jinqiao Duan,Kening Lu, and Björn Schmalfuss

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Abstract

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

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

Dates
First available: 12 November 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1068646380

Digital Object Identifier
doi:10.1214/aop/1068646380

Mathematical Reviews number (MathSciNet)
MR2016614

Zentralblatt MATH identifier
02077590

Subjects
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

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

Citation

Duan, Jinqiao; Lu, Kening; Schmalfuss, Björn. Invariant manifolds for stochastic partial differential equations. The Annals of Probability 31 (2003), no. 4, 2109--2135. doi:10.1214/aop/1068646380. http://projecteuclid.org/euclid.aop/1068646380.


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References

  • Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
  • Babin, A. B. and Vishik, M. I. (1992). Attractors of Evolution Equations. North-Holland, Amsterdam.
  • Bates, P., Lu, K. and Zeng, C. (1998). Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space. Amer. Math. Soc., Providence, RI.
  • Bensoussan, A. and Flandoli, F. (1995). Stochastic inertial manifold. Stochastics Stochastics Rep. 53 13--39.
  • Caraballo, T., Langa, J. and Robinson, J. C. (2001). A stochastic pitchfork bifurcation in a reaction--diffusion equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 2041--2061.
  • Castaing, C. and Valadier, M. (1977). Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580. Springer, Berlin.
  • Chicone, C. and Latushkin, Y. (1997). Center manifolds for infinite dimensional non-autonomous differential equations. J. Differential Equations 141 356--399.
  • Chow, S.-N., Lu, K. and Lin, X.-B. (1991). Smooth foliations for flows in Banach space. J. Differential Equations 94 266--291.
  • Da Prato, G. and Debussche, A. (1996). Construction of stochastic inertial manifolds using backward integration. Stochastics Stochastics Rep. 59 305--324.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimension. Cambridge Univ. Press.
  • Duan, J., Lu, K. and Schmalfuss, B. (2002). Unstable manifolds for equations with time dependent coefficients. Preprint.
  • Duan, J., Lu, K. and Schmalfuss, B. (2003). Smooth stable and unstable manifolds for stochastic partial differential equations. J. Dynamics Differential Equations. To appear.
  • Girya, T. V. and Chueshov, I. D. (1995). Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems. Sb. Math. 186 29--45.
  • Hadamard, J. (1901). Sur l'iteration et les solutions asymptotiques des equations differentielles. Bull. Soc. Math. France 29 224--228.
  • Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, New York.
  • Koksch, N. and Siegmund, S. (2002). Pullback attracting inertial manifolds for nonautonomous dynamical systems. J. Dynamics Differential Equations 14 889--941.
  • Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
  • Liapunov, A. M. (1947). Problème géneral de la stabilité du mouvement. Princeton Univ. Press.
  • Mohammed, S.-E. A. and Scheutzow, M. K. R. (1999). The stable manifold theorem for stochastic differential equations. Ann. Probab. 27 615--652.
  • Øksendale, B. (1992). Stochastic Differential Equations, 3rd ed. Springer, Berlin.
  • Perron, O. (1928). Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen. Math. Z. 29 129--160.
  • Ruelle, D. (1982). Characteristic exponents and invariant manifolds in Hilbert spaces. Ann. of Math. 115 243--290.
  • Schmalfuss, B. (1997). The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys. 48 951--975.
  • Schmalfuss, B. (1998). A random fixed point theorem and the random graph transformation. J. Math. Anal. Appl. 225 91--113.
  • Schmalfuss, B. (2000). Attractors for the non-autonomous dynamical systems. In Proceedings of the International Conference on Differential Equations (B. Fiedler, K. Gröger and J. Sprekels, eds.) 1 684--690. World Scientific, Singapore.
  • Sell, G. R. (1967). Non-autonomous differential equations and dynamical systems. J. Amer. Math. Soc. 127 241--283.
  • Vishik, M. I. (1992). Asymptotic Behaviour of Solutions of Evolutionary Equations. Cambridge Univ. Press.
  • Wanner, T. (1995). Linearization random dynamical systems. In Expositions in Dynamical Systems (C. K. R. T. Jones, U. Kirchgraber and H. O. Walther, eds.) 203--269. Springer, Berlin.