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

Kening Lu; Alexandra Neamţu; Björn Schmalfuss

doi:10.3934/dcdsb.2018149

Article Contents

2018, Volume 23, Issue 3: 1219-1242. Doi: 10.3934/dcdsb.2018149

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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
* Corresponding author: Björn Schmalfuss

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

Received: February 28, 2017

Revised: June 30, 2017

Early access: February 2018

Published: June 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 37H15, 37L55; Secondary: 60H15, 35R60, 60J65, 34D45.

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References

	P. Acquistapace , Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988) , 433-457.
	P. Acquistapace and B. Terreni , A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987) , 47-107.
	H. Amann, Linear and Quasilinear Parabolic Problems, Basel; Boston; Berlin: Birkhäuser Verlag, 1995.
	L. Arnold, Random Dynamical Systems, Springer-Verlag Berlin Heidelberg New York, 1991.
	A. T. Bharucha-Reid, Random Integral Equations, Academic Press New York and London, 1972.
	T. Caraballo , J. Duan , K. Lu and B. Schmalfuß , Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010) , 23-52.
	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
	X. Chen , A. Roberts and J. Duan , Center manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015) , 606-632. doi: 10.1080/10236198.2015.1045889.
	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.
	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.
	S.-N. Chow , K. Lu and J. Mallet-Paret , Floquet theory for parabolic differential equations, J. Differ. Equations, 109 (1994) , 147-200. doi: 10.1006/jdeq.1994.1047.
	S.-N. Chow , K. Lu and J. Mallet-Paret , Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal, 129 (1995) , 245-304. doi: 10.1007/BF00383675.
	T. S. Doan and S. Siegmund, Differential equations with random delay, Fields Communication Series, 64, in press, (2013), 279-303.
	J. Duan , K. 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.
	K. -J. Engel and R. Nagel, A Short Course on Operator Semigroups, Springer Verlag, 2006.
	M. Fabian, P. Habala, P. Hájek. V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, Springer, 2011.
	M. J. Garrido-Atienza , B. 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.
	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.
	C. Heil, A Basis Theory Primer, Expanded Edition, Birkäuser, 2011.
	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.
	W. Li, K. Lu and B. Schmalfuß, A Hartman-Grobman theorem for scalar stochastic partial differential equations, Preperint.
	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.
	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.
	M. B. Marcus , Hölder conditions for continuous Gaussian processes, Osaka. J. Math., 7 (1970) , 483-493.
	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.
	S.-E. A. Mohammed , T. 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.
	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.
	V. I. Oseledets , A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968) , 197-231.
	A. Pazy, Semigroups of Linear Operators and Applications to PDEs, Springer-Verlag New York, 1983.
	A. Pietsch, History of Banach spaces and linear operators, Birkäuser, 2007.
	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.
	D. Ruelle , Characteristic exponents and invariant manifolds in Hilbert spaces, Annals of math., 115 (1982) , 243-290. doi: 10.2307/1971392.
	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993.
	R. Schnaubelt , Asymptotic behaviour of parabolic nonautonomous evolution equations, Functional Analytic Methods for Evolution Equations, Series Lecture Notes in Mathematics, 1885 (2014) , 401-472.
	A. V. Skorochod, Random Linear Operators, "Naukova Dumka", Kiev, 1978. 200 pp.
	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag New York, 1998.
	J. M. A. M. van Neerven, Stochastic Evolution Equations, ISEM Lecture Notes 2007/08.
	M. Zähle , Integration with respect to fractal functions and stochastic calculus Ⅰ, Probab. Theory Relat. Fields, 111 (1998) , 333-374. doi: 10.1007/s004400050171.
	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.