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Invariant manifolds for stochastic partial differential equations. (English) Zbl 1052.60048
Stochastic partial differential equations with multiplicative noises are considered and the existence of invariant manifold is studied. Firstly, random graph transforms are introduced and a fixed point theorem for nonautonomous systems is given. Generalized fixed points of this transform give the desired manifold.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37H10 Generation, random and stochastic difference and differential equations
37L55 Infinite-dimensional random dynamical systems; stochastic equations
37D10 Invariant manifold theory for dynamical systems
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References:
[1] Arnold, L. (1998). Random Dynamical Systems . Springer, Berlin. · Zbl 0906.34001
[2] Babin, A. B. and Vishik, M. I. (1992). Attractors of Evolution Equations . North-Holland, Amsterdam. · Zbl 0778.58002
[3] Bates, P., Lu, K. and Zeng, C. (1998). Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space . Amer. Math. Soc., Providence, RI. · Zbl 1023.37013
[4] Bensoussan, A. and Flandoli, F. (1995). Stochastic inertial manifold. Stochastics Stochastics Rep. 53 13–39. · Zbl 0854.60059
[5] 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. · Zbl 0996.60070
[6] Castaing, C. and Valadier, M. (1977). Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580 . Springer, Berlin. · Zbl 0346.46038
[7] Chicone, C. and Latushkin, Y. (1997). Center manifolds for infinite dimensional non-autonomous differential equations. J. Differential Equations 141 356–399. · Zbl 0992.34033
[8] Chow, S.-N., Lu, K. and Lin, X.-B. (1991). Smooth foliations for flows in Banach space. J. Differential Equations 94 266–291. · Zbl 0749.58043
[9] Da Prato, G. and Debussche, A. (1996). Construction of stochastic inertial manifolds using backward integration. Stochastics Stochastics Rep. 59 305–324. · Zbl 0876.60040
[10] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimension . Cambridge Univ. Press. · Zbl 0761.60052
[11] Duan, J., Lu, K. and Schmalfuss, B. (2002). Unstable manifolds for equations with time dependent coefficients.
[12] Duan, J., Lu, K. and Schmalfuss, B. (2003). Smooth stable and unstable manifolds for stochastic partial differential equations. J. Dynamics Differential Equations . · Zbl 1052.60048
[13] Girya, T. V. and Chueshov, I. D. (1995). Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems. Sb. Math. 186 29–45. · Zbl 0851.60036
[14] Hadamard, J. (1901). Sur l’iteration et les solutions asymptotiques des equations differentielles. Bull. Soc. Math. France 29 224–228. · JFM 32.0314.01
[15] Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840 . Springer, New York. · Zbl 0456.35001
[16] Koksch, N. and Siegmund, S. (2002). Pullback attracting inertial manifolds for nonautonomous dynamical systems. J. Dynamics Differential Equations 14 889–941. · Zbl 1025.34042
[17] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations . Cambridge Univ. Press. · Zbl 0743.60052
[18] Liapunov, A. M. (1947). Problème géneral de la stabilité du mouvement . Princeton Univ. Press.
[19] Mohammed, S.-E. A. and Scheutzow, M. K. R. (1999). The stable manifold theorem for stochastic differential equations. Ann. Probab. 27 615–652. · Zbl 0940.60084
[20] Øksendale, B. (1992). Stochastic Differential Equations , 3rd ed. Springer, Berlin.
[21] Perron, O. (1928). Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen. Math. Z. 29 129–160. · JFM 54.0456.04
[22] Ruelle, D. (1982). Characteristic exponents and invariant manifolds in Hilbert spaces. Ann. of Math. 115 243–290. JSTOR: · Zbl 0493.58015
[23] Schmalfuss, B. (1997). The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys. 48 951–975. · Zbl 0887.34057
[24] Schmalfuss, B. (1998). A random fixed point theorem and the random graph transformation. J. Math. Anal. Appl. 225 91–113. · Zbl 0931.37019
[25] 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. · Zbl 0971.37038
[26] Sell, G. R. (1967). Non-autonomous differential equations and dynamical systems. J. Amer. Math. Soc. 127 241–283. JSTOR: · Zbl 0189.39602
[27] Vishik, M. I. (1992). Asymptotic Behaviour of Solutions of Evolutionary Equations . Cambridge Univ. Press. · Zbl 0797.35016
[28] 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. · Zbl 0824.34069
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