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  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.

[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.

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