May  2018, 23(3): 1219-1242. doi: 10.3934/dcdsb.2018149

On the Oseledets-splitting for infinite-dimensional random dynamical systems

1. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

2. 

Friedrich Schiller University, Institute of Mathematics, Enst-Abbe-Platz 2,07743, Jena, Germany

* Corresponding author: Björn Schmalfuss

Dedicated to our friend and colleague Prof. Dr. Igor Dmitrievich Chueshov

Received  March 2017 Revised  July 2017 Published  May 2018 Early access  February 2018

We investigate the Oseledets splitting for Banach space-valued random dynamical systems based on the theory of center manifolds. This technique gives us random one-dimensional invariant spaces which turn out to be the Oseledets subspaces under suitable assumptions. We apply these results to a stochastic parabolic evolution equation driven by a fractional Brownian motion.

Citation: Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149
References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar

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H. Amann, Linear and Quasilinear Parabolic Problems, Basel; Boston; Berlin: Birkhäuser Verlag, 1995.  Google Scholar

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L. Arnold, Random Dynamical Systems, Springer-Verlag Berlin Heidelberg New York, 1991. Google Scholar

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A. T. Bharucha-Reid, Random Integral Equations, Academic Press New York and London, 1972.  Google Scholar

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T. CaraballoJ. DuanK. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.   Google Scholar

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

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X. ChenA. Roberts and J. Duan, Center manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889.  Google Scholar

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C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differ. Equations, 141 (1997), 356-399.  doi: 10.1006/jdeq.1997.3343.  Google Scholar

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S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differ. Equations, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.  Google Scholar

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S.-N. ChowK. Lu and J. Mallet-Paret, Floquet theory for parabolic differential equations, J. Differ. Equations, 109 (1994), 147-200.  doi: 10.1006/jdeq.1994.1047.  Google Scholar

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S.-N. ChowK. Lu and J. Mallet-Paret, Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal, 129 (1995), 245-304.  doi: 10.1007/BF00383675.  Google Scholar

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T. S. Doan and S. Siegmund, Differential equations with random delay, Fields Communication Series, 64, in press, (2013), 279-303.  Google Scholar

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J. DuanK. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.  Google Scholar

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K. -J. Engel and R. Nagel, A Short Course on Operator Semigroups, Springer Verlag, 2006.  Google Scholar

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M. Fabian, P. Habala, P. Hájek. V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, Springer, 2011.  Google Scholar

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M. J. Garrido-AtienzaB. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete. Contin. Dyn. Syst. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.  Google Scholar

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C. Gonzàlez-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn. , 9 (2015), 237-255, arXiv: 1406.1955. doi: 10.3934/jmd.2015.9.237.  Google Scholar

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C. Heil, A Basis Theory Primer, Expanded Edition, Birkäuser, 2011.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.  Google Scholar

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W. Li, K. Lu and B. Schmalfuß, A Hartman-Grobman theorem for scalar stochastic partial differential equations, Preperint. Google Scholar

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Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space, Mem. Amer. Math. Soc. , 206(2010), ⅵ+106 pp.  Google Scholar

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K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differ. Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar

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M. B. Marcus, Hölder conditions for continuous Gaussian processes, Osaka. J. Math., 7 (1970), 483-493.   Google Scholar

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J. Mierczyński and W. Shen, Principal lyapunov exponents and principal floquet spaces of positive random dynamical systems. Ⅰ. general theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365.  doi: 10.1090/S0002-9947-2013-05814-X.  Google Scholar

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S.-E. A. MohammedT. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Memoirs of the American Mathematical Society, 196 (2008), 1-105.   Google Scholar

[27]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, The Annals of Probability, 27 (1999), 615-652.  doi: 10.1214/aop/1022677380.  Google Scholar

[28]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.   Google Scholar

[29]

A. Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer-Verlag New York, 1983.  Google Scholar

[30]

A. Pietsch, History of Banach spaces and linear operators, Birkäuser, 2007.  Google Scholar

[31]

M. Pronk and M. C. Veraar, A new approach to stochastic evolution equations with adapted drift, J. Differ. Equations, 256 (2014), 3634-3683.  doi: 10.1016/j.jde.2014.02.014.  Google Scholar

[32]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert spaces, Annals of math., 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[33]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993.  Google Scholar

[34]

R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, Functional Analytic Methods for Evolution Equations, Series Lecture Notes in Mathematics, 1885 (2014), 401-472.   Google Scholar

[35]

A. V. Skorochod, Random Linear Operators, "Naukova Dumka", Kiev, 1978. 200 pp.  Google Scholar

[36]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag New York, 1998.  Google Scholar

[37]

J. M. A. M. van Neerven, Stochastic Evolution Equations, ISEM Lecture Notes 2007/08. Google Scholar

[38]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅰ, Probab. Theory Relat. Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

[39]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅱ, Math. Nachr., 225 (2001), 145-183.  doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.  Google Scholar

show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar

[3]

H. Amann, Linear and Quasilinear Parabolic Problems, Basel; Boston; Berlin: Birkhäuser Verlag, 1995.  Google Scholar

[4]

L. Arnold, Random Dynamical Systems, Springer-Verlag Berlin Heidelberg New York, 1991. Google Scholar

[5]

A. T. Bharucha-Reid, Random Integral Equations, Academic Press New York and London, 1972.  Google Scholar

[6]

T. CaraballoJ. DuanK. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.   Google Scholar

[7]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

[8]

X. ChenA. Roberts and J. Duan, Center manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889.  Google Scholar

[9]

C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differ. Equations, 141 (1997), 356-399.  doi: 10.1006/jdeq.1997.3343.  Google Scholar

[10]

S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differ. Equations, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.  Google Scholar

[11]

S.-N. ChowK. Lu and J. Mallet-Paret, Floquet theory for parabolic differential equations, J. Differ. Equations, 109 (1994), 147-200.  doi: 10.1006/jdeq.1994.1047.  Google Scholar

[12]

S.-N. ChowK. Lu and J. Mallet-Paret, Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal, 129 (1995), 245-304.  doi: 10.1007/BF00383675.  Google Scholar

[13]

T. S. Doan and S. Siegmund, Differential equations with random delay, Fields Communication Series, 64, in press, (2013), 279-303.  Google Scholar

[14]

J. DuanK. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dyn. Differ. Equ., 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.  Google Scholar

[15]

K. -J. Engel and R. Nagel, A Short Course on Operator Semigroups, Springer Verlag, 2006.  Google Scholar

[16]

M. Fabian, P. Habala, P. Hájek. V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, Springer, 2011.  Google Scholar

[17]

M. J. Garrido-AtienzaB. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete. Contin. Dyn. Syst. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.  Google Scholar

[18]

C. Gonzàlez-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn. , 9 (2015), 237-255, arXiv: 1406.1955. doi: 10.3934/jmd.2015.9.237.  Google Scholar

[19]

C. Heil, A Basis Theory Primer, Expanded Edition, Birkäuser, 2011.  Google Scholar

[20]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.  Google Scholar

[21]

W. Li, K. Lu and B. Schmalfuß, A Hartman-Grobman theorem for scalar stochastic partial differential equations, Preperint. Google Scholar

[22]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space, Mem. Amer. Math. Soc. , 206(2010), ⅵ+106 pp.  Google Scholar

[23]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differ. Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.  Google Scholar

[24]

M. B. Marcus, Hölder conditions for continuous Gaussian processes, Osaka. J. Math., 7 (1970), 483-493.   Google Scholar

[25]

J. Mierczyński and W. Shen, Principal lyapunov exponents and principal floquet spaces of positive random dynamical systems. Ⅰ. general theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365.  doi: 10.1090/S0002-9947-2013-05814-X.  Google Scholar

[26]

S.-E. A. MohammedT. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Memoirs of the American Mathematical Society, 196 (2008), 1-105.   Google Scholar

[27]

S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, The Annals of Probability, 27 (1999), 615-652.  doi: 10.1214/aop/1022677380.  Google Scholar

[28]

V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.   Google Scholar

[29]

A. Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer-Verlag New York, 1983.  Google Scholar

[30]

A. Pietsch, History of Banach spaces and linear operators, Birkäuser, 2007.  Google Scholar

[31]

M. Pronk and M. C. Veraar, A new approach to stochastic evolution equations with adapted drift, J. Differ. Equations, 256 (2014), 3634-3683.  doi: 10.1016/j.jde.2014.02.014.  Google Scholar

[32]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert spaces, Annals of math., 115 (1982), 243-290.  doi: 10.2307/1971392.  Google Scholar

[33]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993.  Google Scholar

[34]

R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, Functional Analytic Methods for Evolution Equations, Series Lecture Notes in Mathematics, 1885 (2014), 401-472.   Google Scholar

[35]

A. V. Skorochod, Random Linear Operators, "Naukova Dumka", Kiev, 1978. 200 pp.  Google Scholar

[36]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag New York, 1998.  Google Scholar

[37]

J. M. A. M. van Neerven, Stochastic Evolution Equations, ISEM Lecture Notes 2007/08. Google Scholar

[38]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅰ, Probab. Theory Relat. Fields, 111 (1998), 333-374.  doi: 10.1007/s004400050171.  Google Scholar

[39]

M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅱ, Math. Nachr., 225 (2001), 145-183.  doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.  Google Scholar

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