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Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion. (English) Zbl 1186.37094
Summary: We consider a class of stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. The existence of local random unstable manifolds is shown if the linear parts of these SPDEs are hyperbolic. For this purpose we introduce a modified Lyapunov-Perron transform, which contains stochastic integrals. By the singularities inside these integrals we obtain a special Lyapunov-Perron’s approach by treating a segment of the solution over time interval [0,1] as a starting point and setting up an infinite series equation involving these segments as time evolves. Using this approach, we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium.

37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
35R60 PDEs with randomness, stochastic partial differential equations
60J65 Brownian motion
Full Text: DOI
[1] Arnold, L., Random dynamical systems, Springer monogr. math., (1998), Springer-Verlag Berlin
[2] Bates, P.; Lu, K.; Zeng, C., Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. amer. math. soc., vol. 135, (1998) · Zbl 1023.37013
[3] Bensoussan, A.; Flandoli, F., Stochastic inertial manifold, Stochastics rep., 53, 1-2, 13-39, (1995) · Zbl 0854.60059
[4] Caraballo, T.; Chueshov, I.; Langa, J., Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18, 2, 747-767, (2005) · Zbl 1071.35130
[5] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuss, J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, working paper · Zbl 1201.60063
[6] Caraballo, T.; Kloeden, P.; Schmalfuß, B., Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. math. optim., 50, 3, 183-207, (2004) · Zbl 1066.60058
[7] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, Lecture notes in math., vol. 580, (1977), Springer-Verlag Berlin · Zbl 0346.46038
[8] Chueshov, I.D.; Girya, T.V., Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sb. math., 186, 1, 29-45, (1995) · Zbl 0851.60036
[9] Da Prato, G.; Debussche, A., Construction of stochastic inertial manifolds using backward integration, Stochastics rep., 59, 3-4, 305-324, (1996) · Zbl 0876.60040
[10] Decreusefond, L.; Üstünel, A.S., Stochastic analysis of the fractional Brownian motion, Potential anal., 10, 2, 177-214, (1999) · Zbl 0924.60034
[11] Duan, J.; Lu, K.; Schmalfuß, B., Invariant manifolds for stochastic partial differential equations, Ann. probab., 31, 4, 2109-2135, (2003) · Zbl 1052.60048
[12] Duan, J.; Lu, K.; Schmalfuß, B., Smooth stable and unstable manifolds for stochastic evolutionary equations, J. dynam. differential equations, 16, 4, 949-972, (2004) · Zbl 1065.60077
[13] Flandoli, F.; Lisei, H., Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications, Stoch. anal. appl., 22, 6, 1385-1420, (2004) · Zbl 1063.60089
[14] M.J. Garrido-Atienza, K. Lu, B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, 2009, submitted for publication · Zbl 1200.37075
[15] M.J. Garrido-Atienza, B. Maslowski, B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Internat. J. Bifur. Chaos, in press · Zbl 1202.37073
[16] Gubinelli, M.; Lejay, A.; Tindel, S., Young integrals and SPDEs, Potential anal., 25, 4, 307-326, (2006) · Zbl 1103.60062
[17] Hadamard, J., Sur l’iteration et LES solutions asymptotiques des equations differentielles, Bull. soc. math. France, 29, 224-228, (1901) · JFM 32.0314.01
[18] P. Hornok, Rauhe Pfade Theorie und zufällige dynamische Systeme getrieben durch frakktionale Brownsche bewegung, 2006, Diplomarbeit
[19] Jost, C., A note on ergodic transformations of self-similar Volterra Gaussian processes, Electron. comm. probab., 12, 259-266, (2007), (electronic) · Zbl 1129.60037
[20] Kunita, H., Stochastic flows and stochastic differential equations, (1990), Cambridge University Press · Zbl 0743.60052
[21] Lu, K.; Schmalfuß, B., Invariant manifolds for stochastic wave equations, J. differential equations, 236, 460-492, (2007) · Zbl 1113.37056
[22] Lyapunov, A.M., Problème géneral de la stabilité du mouvement, Ann. of math. stud., vol. 17, (1947), (originally published in Russian, 1892)
[23] Lyons, T.J., Differential equations driven by rough signals, Rev. mat. iberoamericana, 14, 2, 215-310, (1998) · Zbl 0923.34056
[24] Maslowski, B.; Nualart, D., Evolution equations driven by a fractional Brownian motion, J. funct. anal., 202, 1, 277-305, (2003) · Zbl 1027.60060
[25] Maslowski, B.; Schmalfuß, B., Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stoch. anal. appl., 22, 6, 1577-1607, (2004) · Zbl 1062.60060
[26] Mohammed, S.-E.A.; Scheutzow, M.K.R., The stable manifold theorem for stochastic differential equations, Ann. probab., 27, 2, 615-652, (1999) · Zbl 0940.60084
[27] Mohammed, S.; Zhang, T.; Zhao, H., The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. amer. math. soc., 196, (2008) · Zbl 1169.60014
[28] Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. math., 53, 1, 55-81, (2002) · Zbl 1018.60057
[29] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Springer appl. math. ser., (1983), Springer-Verlag Berlin · Zbl 0516.47023
[30] Perron, O., Über stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen, Math. Z., 29, 129-160, (1928) · JFM 54.0456.04
[31] Perron, O., Über stabilität und asymptotisches verhalten der Lösungen eines systems endlicher differenzengleichungen, J. reine angew. math., 161, 41-64, (1929) · JFM 55.0869.02
[32] Schmalfuß, B., A random fixed point theorem and the random graph transformation, J. math. anal. appl., 225, 1, 91-113, (1998) · Zbl 0931.37019
[33] Sell, G.R.; You, Y., Dynamics of evolutionary equations, Appl. math. sci., vol. 143, (2002), Springer-Verlag Berlin · Zbl 1254.37002
[34] Tindel, S.; Tudor, C.; Viens, F., Stochastic evolution equations with fractional Brownian motion, Probab. theory related fields, 127, 2, 186-204, (2003) · Zbl 1036.60056
[35] Wanner, T., Linearization random dynamical systems, (), 203-269 · Zbl 0824.34069
[36] Zähle, M., Integration with respect to fractal functions and stochastic calculus, I, Probab. theory related fields, 111, 3, 333-374, (1998) · Zbl 0918.60037
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