November  2020, 40(11): 6201-6246. doi: 10.3934/dcds.2020276

Invariant manifolds and foliations for random differential equations driven by colored noise

1. 

School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2. 

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

3. 

Department of Mathematics New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Kening Lu, klu@math.byu.edu

Received  October 2019 Revised  May 2020 Published  July 2020

Fund Project: This work was supported by NSFC (11501549, 11831012, 11331007, 11971330), NSF (1413603), the Fundamental Research Funds for the Central Universities (YJ201646) and International Visiting Program for Excellent Young Scholars of SCU

In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches zero.

Citation: Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276
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P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38.   Google Scholar

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J. DuanK. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

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M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667.  doi: 10.1016/j.jde.2009.11.006.  Google Scholar

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M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in(1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.  Google Scholar

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T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[24]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701, 19pp. doi: 10.1063/1.5017932.  Google Scholar

[25]

A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  doi: 10.1016/0022-0396(67)90016-2.  Google Scholar

[26]

A. Liapounoff, ProblÜme Gén´eral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947.  Google Scholar

[27]

K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518.  doi: 10.1142/S0219493708002421.  Google Scholar

[28]

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

[29]

S.-E. A. Mohammed, T. Zhang and H. Zhao, The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations, Memories AMS Vol. 196, Amer. Math. Soc., Providence, R.I., 2008. doi: 10.1090/memo/0917.  Google Scholar

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[31]

O. Perron, $\ddot{\text{U}}$ber Stabilit$\ddot{\text{a}}$t und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[32]

V. A. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1297–1324 (in Russian).  Google Scholar

[33]

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

[34]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113.  doi: 10.1006/jmaa.1998.6008.  Google Scholar

[35]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.  Google Scholar

[36]

J. Shen, K. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 4797-4840. doi: 10.3934/dcds.2019196.  Google Scholar

[37]

J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623. doi: 10.1016/j.jde.2018.10.008.  Google Scholar

[38]

A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224.  doi: 10.1016/0022-1236(87)90086-3.  Google Scholar

[39]

T. Wanner, Linearization of random dynamical systems, in: C. K. R. T. Jonesm, U. Kirchgraber, H. O. Walther (Eds.), Dynamics Rep., vol. 4,203–269, Springer, Berlin/Heidelberg/New York, 1995. doi: 10.1007/978-3-642-61215-2_4.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38.   Google Scholar

[3]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in banach space, Mem. Amer. Math. Soc., 135 (1998), 129 pp. doi: 10.1090/memo/0645.  Google Scholar

[4]

P. W. BatesK. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc., 352 (2000), 4641-4676.  doi: 10.1090/S0002-9947-00-02503-4.  Google Scholar

[5]

P. W. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[6]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[7]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

S-N. ChowX.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations, 94 (1991), 266-291.  doi: 10.1016/0022-0396(91)90093-O.  Google Scholar

[10]

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

[11]

S.-N. Chow and K. Lu, $C^k$ center unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320.  doi: 10.1017/S0308210500014682.  Google Scholar

[12] G. Da Prato and J. Zabczyk., Stochastic Equations in Infinite Dimension., Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781107295513.  Google Scholar
[13]

J. DuanK. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.  Google Scholar

[14]

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

[15]

F. Flandoli, Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs Vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995.  Google Scholar

[16]

H. GaoM. J. Garrido-Atienza and B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.  Google Scholar

[17]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion, J. Differential Equations, 248 (2010), 1637-1667.  doi: 10.1016/j.jde.2009.11.006.  Google Scholar

[18]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in(1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar

[19]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Disc. Cont. Dyn. Sys.-Series B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[20]

J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228.   Google Scholar

[21]

J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.  Google Scholar

[22]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.  Google Scholar

[23]

T. JiangX. Liu and J. Duan, Approximation for random stable manifolds under multiplicative correlated noises, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3163-3174.  doi: 10.3934/dcdsb.2016091.  Google Scholar

[24]

T. Jiang, X. Liu and J. Duan, A Wong-Zakai approximation for random invariant manifolds, J. Math. Phys., 58 (2017), 122701, 19pp. doi: 10.1063/1.5017932.  Google Scholar

[25]

A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  doi: 10.1016/0022-0396(67)90016-2.  Google Scholar

[26]

A. Liapounoff, ProblÜme Gén´eral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947.  Google Scholar

[27]

K. Lu and B. Schmalfuß, Invariant foliations for stochastic partial differential equations, Stoch. Dyn., 8 (2008), 505-518.  doi: 10.1142/S0219493708002421.  Google Scholar

[28]

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

[29]

S.-E. A. Mohammed, T. Zhang and H. Zhao, The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations, Memories AMS Vol. 196, Amer. Math. Soc., Providence, R.I., 2008. doi: 10.1090/memo/0917.  Google Scholar

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[31]

O. Perron, $\ddot{\text{U}}$ber Stabilit$\ddot{\text{a}}$t und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[32]

V. A. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1297–1324 (in Russian).  Google Scholar

[33]

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

[34]

B. Schmalfuß, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113.  doi: 10.1006/jmaa.1998.6008.  Google Scholar

[35]

J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differential Equations, 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.  Google Scholar

[36]

J. Shen, K. Lu and B. Wang, Convergence and center manifolds for differential equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 4797-4840. doi: 10.3934/dcds.2019196.  Google Scholar

[37]

J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623. doi: 10.1016/j.jde.2018.10.008.  Google Scholar

[38]

A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224.  doi: 10.1016/0022-1236(87)90086-3.  Google Scholar

[39]

T. Wanner, Linearization of random dynamical systems, in: C. K. R. T. Jonesm, U. Kirchgraber, H. O. Walther (Eds.), Dynamics Rep., vol. 4,203–269, Springer, Berlin/Heidelberg/New York, 1995. doi: 10.1007/978-3-642-61215-2_4.  Google Scholar

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