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doi: 10.3934/dcds.2020324

Mean-square random invariant manifolds for stochastic differential equations

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.

Citation: Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 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. CaraballoI. 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. CaraballoJ. DuanK. 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. ChenA. 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. 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

[16]

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

[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]

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. CaraballoI. 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. CaraballoJ. DuanK. 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. ChenA. 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. 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

[16]

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

[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]

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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