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March  2021, 41(3): 1449-1468. doi: 10.3934/dcds.2020324

Mean-square random invariant manifolds for stochastic differential equations

Bixiang Wang

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

Received  February 2020 Published  September 2020

We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

Keywords: Mean-square random invariant manifold, random stable manifold, random unstable manifold, stochastic equation, backward equation.
Mathematics Subject Classification: Primary: 37D10, 60H15; Secondary: 35B42, 35L25, 37L55.
Citation: Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.  Google Scholar

[3]

P. W. Bates, K. Lu and C. Zeng, Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space, Memoirs of the AMS, Vol. 135, American Mathematical Society, Providence, 1998. doi: 10.1090/memo/0645.  Google Scholar

[4]

A. Bensoussan and F. Flandoli, Stochastic inertial manifold, Stochastics and Stochastic Rep., 53 (1995), 13-39.  doi: 10.1080/17442509508833981.  Google Scholar

[5]

P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.  Google Scholar

[6]

P. Brune and B. Schmalfuss, Inertial manifolds for stochastic PDE with dynamical boundary conditions, Communications on Pure and Applied Analysis, 10 (2011), 831-846.  doi: 10.3934/cpaa.2011.10.831.  Google Scholar

[7]

T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.  doi: 10.1088/0951-7715/18/2/015.  Google Scholar

[8]

T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Advanced Nonlinear Studies, 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.  Google Scholar

[9]

X. Chen, A. J. Roberts and J. Duan, Center manifolds for infinite dimensional random dynamical systems, Dynamical Systems, 34 (2019), 334-355.  doi: 10.1080/14689367.2018.1531972.  Google Scholar

[10]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynamics and Differential Equations, 13 (2001), 355-380.  doi: 10.1023/A:1016684108862.  Google Scholar

[11]

I. Chueshov, M. Scheutzow and B. Schmalfuss, Continuity properties of inertial manifolds for stochastic retarded semilinear parabolic equations, 353-375, in Interacting Stochastic Systems by J. Deuschel and A. Greven, 2005, Springer, Berlin. doi: 10.1007/3-540-27110-4_16.  Google Scholar

[12]

I. D. Chueshov and T. V. Girya, Inertial manifolds for stochastic dissipative dynamical systems, Doklady Acad. Sci. Ukraine, 7 (1994), 42-45.   Google Scholar

[13]

I. D. Chueshov and T. V. Girya, Inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise, Lett. Math. Phys., 34 (1995), 69-76.  doi: 10.1007/BF00739376.  Google Scholar

[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Second Edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[15]

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

[16]

J. Duan, K. 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

[17]

M. J. Garrido-Atienza, K. 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]

T. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sbornik: Mathematics, 186 (1995), 29-46.  doi: 10.1070/SM1995v186n01ABEH000002.  Google Scholar

[19]

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

[20]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Analysis and Applications, 9 (1991), 445-459.  doi: 10.1080/07362999108809250.  Google Scholar

[21]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[22] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990.   Google Scholar
[23]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988.  doi: 10.1002/cpa.20083.  Google Scholar

[24]

Z. Lian and K. Lu, Lyapunov Exponents and Invariant Manifolds for Infinite-Dimensional Random Dynamical Systems in a Banach Space, Mem. Amer. Math. Soc., 206 2010, No. 967,106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, Journal of Dynamics and Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[29]

B. Wang, Periodic and almost periodic random inertial manifolds for non-autonomous stochastic equations, Continuous and Distributed Systems II, 189–208, Studies in Systems, Decision and Control, Vol 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_11.  Google Scholar

[30]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported, 4, Springer, Berlin, 1995., 203–269. 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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.  Google Scholar

[3]

P. W. Bates, K. Lu and C. Zeng, Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space, Memoirs of the AMS, Vol. 135, American Mathematical Society, Providence, 1998. doi: 10.1090/memo/0645.  Google Scholar

[4]

A. Bensoussan and F. Flandoli, Stochastic inertial manifold, Stochastics and Stochastic Rep., 53 (1995), 13-39.  doi: 10.1080/17442509508833981.  Google Scholar

[5]

P. Boxler, A stochastic version of center manifold theory, Probab. Theory Related Fields, 83 (1989), 509-545.  doi: 10.1007/BF01845701.  Google Scholar

[6]

P. Brune and B. Schmalfuss, Inertial manifolds for stochastic PDE with dynamical boundary conditions, Communications on Pure and Applied Analysis, 10 (2011), 831-846.  doi: 10.3934/cpaa.2011.10.831.  Google Scholar

[7]

T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767.  doi: 10.1088/0951-7715/18/2/015.  Google Scholar

[8]

T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations, Advanced Nonlinear Studies, 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102.  Google Scholar

[9]

X. Chen, A. J. Roberts and J. Duan, Center manifolds for infinite dimensional random dynamical systems, Dynamical Systems, 34 (2019), 334-355.  doi: 10.1080/14689367.2018.1531972.  Google Scholar

[10]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynamics and Differential Equations, 13 (2001), 355-380.  doi: 10.1023/A:1016684108862.  Google Scholar

[11]

I. Chueshov, M. Scheutzow and B. Schmalfuss, Continuity properties of inertial manifolds for stochastic retarded semilinear parabolic equations, 353-375, in Interacting Stochastic Systems by J. Deuschel and A. Greven, 2005, Springer, Berlin. doi: 10.1007/3-540-27110-4_16.  Google Scholar

[12]

I. D. Chueshov and T. V. Girya, Inertial manifolds for stochastic dissipative dynamical systems, Doklady Acad. Sci. Ukraine, 7 (1994), 42-45.   Google Scholar

[13]

I. D. Chueshov and T. V. Girya, Inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise, Lett. Math. Phys., 34 (1995), 69-76.  doi: 10.1007/BF00739376.  Google Scholar

[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Second Edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[15]

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

[16]

J. Duan, K. 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

[17]

M. J. Garrido-Atienza, K. 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]

T. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Sbornik: Mathematics, 186 (1995), 29-46.  doi: 10.1070/SM1995v186n01ABEH000002.  Google Scholar

[19]

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

[20]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Analysis and Applications, 9 (1991), 445-459.  doi: 10.1080/07362999108809250.  Google Scholar

[21]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[22] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990.   Google Scholar
[23]

W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988.  doi: 10.1002/cpa.20083.  Google Scholar

[24]

Z. Lian and K. Lu, Lyapunov Exponents and Invariant Manifolds for Infinite-Dimensional Random Dynamical Systems in a Banach Space, Mem. Amer. Math. Soc., 206 2010, No. 967,106 pp. doi: 10.1090/S0065-9266-10-00574-0.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, Journal of Dynamics and Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[29]

B. Wang, Periodic and almost periodic random inertial manifolds for non-autonomous stochastic equations, Continuous and Distributed Systems II, 189–208, Studies in Systems, Decision and Control, Vol 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_11.  Google Scholar

[30]

T. Wanner, Linearization of random dynamical systems, Dynamics Reported, 4, Springer, Berlin, 1995., 203–269. doi: 10.1007/978-3-642-61215-2_4.  Google Scholar

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