American Institute of Mathematical Sciences

November  2016, 21(9): 3163-3174. doi: 10.3934/dcdsb.2016091

Approximation for random stable manifolds under multiplicative correlated noises

 1 School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan 430074, China, China 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  December 2015 Revised  February 2016 Published  October 2016

The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.
Citation: Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091
References:
 [1] P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise,, Stochastic Anal. Appl., 2 (1984), 131. doi: 10.1080/07362998408809031. [2] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [3] V. Bally, A. Millet and M. Sanz-Sole, Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations,, Ann. Probab., 23 (1995), 178. doi: 10.1214/aop/1176988383. [4] P. Boxler, Stochastische Zentrumsmannigfaltigkeiten,, Ph.D.thesis, (1988). [5] Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations,, Stochastic Process. Appl., 55 (1995), 329. doi: 10.1016/0304-4149(94)00037-T. [6] Z. Brzeźniak, M. Capinski and F. Flandoli, A convergence result for stochastic partial differential equations,, Stochastics, 24 (1988), 423. doi: 10.1080/17442508808833526. [7] T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations,, Adv. Nonlinear Stud., 10 (2010), 23. [8] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. [9] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations,, J. Dynam. Differential Equations, 16 (2004), 949. [10] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations,, Elsevier, (2014). [11] I. Gyöngy, On the approximations of stochastic partial differential equations I,, Stochastics, 25 (1988), 59. doi: 10.1080/17442508808833533. [12] I. Gyöngy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations,, Appl. Math. Optim., 54 (2006), 315. doi: 10.1007/s00245-006-0873-2. [13] I. Gyöngy, D. Nualart and M. Sanz-Sole, Approximations and support theorems in modulus spaces,, Probab. Theory Related Fields, 101 (1995), 495. [14] I. Gyöngy and T. Pröhle, On the approximation of stochastic partial differential equations and on Stroock-Varadhan support theorem,, Computers Math and Applic., 19 (1990), 65. [15] M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs,, J. Math. Soc. Japan., 67 (2015), 1551. doi: 10.2969/jmsj/06741551. [16] W. Horsthemke and R. Lefever, Noise Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology,, Springer Berlin Heidelberg, (1984). [17] F. Konecny, On Wong-Zakai approximation of stochastic differential equations,, J. Multivariate Anal., 13 (1983), 605. doi: 10.1016/0047-259X(83)90043-X. [18] H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge University Press, (1990). [19] Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space,, Mem. Amer. Math. Soc., 206 (2010). [20] A. Millet and M. Sanz-Sole, The support of the solution to a hyperbolic SPDE,, Probab. Theory Related Fields, 98 (1994), 361. doi: 10.1007/BF01192259. [21] A. Millet and M. Sanz-Sole, Approximation and support theorem for a wave equation in two space dimensions,, Bernoulli, 6 (2000), 887. doi: 10.2307/3318761. [22] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations,, Ann. Probab., 27 (1999), 615. doi: 10.1214/aop/1022677380. [23] S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations,, Mem. Amer. Math. Soc., 196 (2008). doi: 10.1090/memo/0917. [24] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations,, Proc. of Intern. Symp. SDE Kyoto, (1978), 283. [25] R. Pettersson, Wong-Zakai approximations for reflecting stochastic differential equations,, Stochastic Anal. Appl., 17 (1999), 609. doi: 10.1080/07362999908809624. [26] P. Protter, Approximation of solutions of stochastic differential equations driven by semimartingales,, Ann. Probab., 13 (1985), 716. doi: 10.1214/aop/1176992905. [27] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, Proceedings 6th Berkeley Symposium Math. Statist. Probab., 3 (1972), 333. [28] X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical system,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3371010. [29] G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations,, J. Evol. Equ., 6 (2006), 621. doi: 10.1007/s00028-006-0280-9. [30] E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations,, Internat. J. Engrg. Sci., 3 (1965), 213. doi: 10.1016/0020-7225(65)90045-5. [31] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals,, Ann. Math. Statist., 36 (1965), 1560. doi: 10.1214/aoms/1177699916.

show all references

References:
 [1] P. Acquistapace and B. Terreni, An approach to Ito linear equations in Hilbert spaces by approximation of white noise with coloured noise,, Stochastic Anal. Appl., 2 (1984), 131. doi: 10.1080/07362998408809031. [2] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [3] V. Bally, A. Millet and M. Sanz-Sole, Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations,, Ann. Probab., 23 (1995), 178. doi: 10.1214/aop/1176988383. [4] P. Boxler, Stochastische Zentrumsmannigfaltigkeiten,, Ph.D.thesis, (1988). [5] Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations,, Stochastic Process. Appl., 55 (1995), 329. doi: 10.1016/0304-4149(94)00037-T. [6] Z. Brzeźniak, M. Capinski and F. Flandoli, A convergence result for stochastic partial differential equations,, Stochastics, 24 (1988), 423. doi: 10.1080/17442508808833526. [7] T. Caraballo, J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for random and stochastic partial differential equations,, Adv. Nonlinear Stud., 10 (2010), 23. [8] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. [9] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations,, J. Dynam. Differential Equations, 16 (2004), 949. [10] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations,, Elsevier, (2014). [11] I. Gyöngy, On the approximations of stochastic partial differential equations I,, Stochastics, 25 (1988), 59. doi: 10.1080/17442508808833533. [12] I. Gyöngy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations,, Appl. Math. Optim., 54 (2006), 315. doi: 10.1007/s00245-006-0873-2. [13] I. Gyöngy, D. Nualart and M. Sanz-Sole, Approximations and support theorems in modulus spaces,, Probab. Theory Related Fields, 101 (1995), 495. [14] I. Gyöngy and T. Pröhle, On the approximation of stochastic partial differential equations and on Stroock-Varadhan support theorem,, Computers Math and Applic., 19 (1990), 65. [15] M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs,, J. Math. Soc. Japan., 67 (2015), 1551. doi: 10.2969/jmsj/06741551. [16] W. Horsthemke and R. Lefever, Noise Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology,, Springer Berlin Heidelberg, (1984). [17] F. Konecny, On Wong-Zakai approximation of stochastic differential equations,, J. Multivariate Anal., 13 (1983), 605. doi: 10.1016/0047-259X(83)90043-X. [18] H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge University Press, (1990). [19] Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space,, Mem. Amer. Math. Soc., 206 (2010). [20] A. Millet and M. Sanz-Sole, The support of the solution to a hyperbolic SPDE,, Probab. Theory Related Fields, 98 (1994), 361. doi: 10.1007/BF01192259. [21] A. Millet and M. Sanz-Sole, Approximation and support theorem for a wave equation in two space dimensions,, Bernoulli, 6 (2000), 887. doi: 10.2307/3318761. [22] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations,, Ann. Probab., 27 (1999), 615. doi: 10.1214/aop/1022677380. [23] S.-E. A. Mohammed, T. Zhang and H. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations,, Mem. Amer. Math. Soc., 196 (2008). doi: 10.1090/memo/0917. [24] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations,, Proc. of Intern. Symp. SDE Kyoto, (1978), 283. [25] R. Pettersson, Wong-Zakai approximations for reflecting stochastic differential equations,, Stochastic Anal. Appl., 17 (1999), 609. doi: 10.1080/07362999908809624. [26] P. Protter, Approximation of solutions of stochastic differential equations driven by semimartingales,, Ann. Probab., 13 (1985), 716. doi: 10.1214/aop/1176992905. [27] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, Proceedings 6th Berkeley Symposium Math. Statist. Probab., 3 (1972), 333. [28] X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical system,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3371010. [29] G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations,, J. Evol. Equ., 6 (2006), 621. doi: 10.1007/s00028-006-0280-9. [30] E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations,, Internat. J. Engrg. Sci., 3 (1965), 213. doi: 10.1016/0020-7225(65)90045-5. [31] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals,, Ann. Math. Statist., 36 (1965), 1560. doi: 10.1214/aoms/1177699916.
 [1] Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008 [2] Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-31. doi: 10.3934/dcdsb.2019104 [3] Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651 [4] Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013 [5] Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451 [6] Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071 [7] Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 [8] Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142 [9] Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871 [10] Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285 [11] Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525 [12] Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649 [13] Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269 [14] Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038 [15] Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032 [16] Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133 [17] Nikolai Dokuchaev. On strong causal binomial approximation for stochastic processes. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1549-1562. doi: 10.3934/dcdsb.2014.19.1549 [18] Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757 [19] Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049 [20] Giuseppe Da Prato. Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 637-647. doi: 10.3934/dcdss.2013.6.637

2018 Impact Factor: 1.008