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 and 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. 

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[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. 

[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.

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[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. 

[35]

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

[36]

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

[37]

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

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[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. 

[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.

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[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. 

[35]

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

[36]

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

[37]

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

[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.

[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.

[1]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

[2]

Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377

[3]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[4]

Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091

[5]

Jana Rodriguez Hertz. Some advances on generic properties of the Oseledets splitting. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4323-4339. doi: 10.3934/dcds.2013.33.4323

[6]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835

[7]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51

[8]

Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042

[9]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55

[10]

Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677

[11]

Martin Golubitsky, Claire Postlethwaite. Feed-forward networks, center manifolds, and forcing. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2913-2935. doi: 10.3934/dcds.2012.32.2913

[12]

Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure and Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643

[13]

Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008

[14]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555

[15]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237

[16]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[17]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098

[18]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[19]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113

[20]

Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (379)
  • HTML views (339)
  • Cited by (1)

Other articles
by authors

[Back to Top]