# 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 and 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), 131-186. doi: 10.1080/07362998408809031. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 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), 178-222. doi: 10.1214/aop/1176988383. [4] P. Boxler, Stochastische Zentrumsmannigfaltigkeiten, Ph.D.thesis, Institut fiir Dynamische Systeme, Universitat Bremen, 1988. [5] Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55 (1995), 329-358. 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), 423-445. 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), 23-52. [8] 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. [9] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. [10] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [11] I. Gyöngy, On the approximations of stochastic partial differential equations I, Stochastics, 25 (1988), 59-85. 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), 315-341. 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), 495-509. [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), 65-70. [15] M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan., 67 (2015), 1551-1604. 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), 605-611. doi: 10.1016/0047-259X(83)90043-X. [18] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 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), vi+106 pp. [20] A. Millet and M. Sanz-Sole, The support of the solution to a hyperbolic SPDE, Probab. Theory Related Fields, 98 (1994), 361-387. 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), 887-915. 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), 615-652. 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), vi+105 pp. doi: 10.1090/memo/0917. [24] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, Proc. of Intern. Symp. SDE Kyoto, (1978), 283-296. [25] R. Pettersson, Wong-Zakai approximations for reflecting stochastic differential equations, Stochastic Anal. Appl., 17 (1999), 609-617. doi: 10.1080/07362999908809624. [26] P. Protter, Approximation of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. 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., University of California Press, Berkeley , 3 (1972), 333-359. [28] X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical system, J. Math. Phys., 51 (2010), 042702, 12pp. doi: 10.1063/1.3371010. [29] G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655. 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), 213-229. 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), 1560-1564. doi: 10.1214/aoms/1177699916.

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##### 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), 131-186. doi: 10.1080/07362998408809031. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 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), 178-222. doi: 10.1214/aop/1176988383. [4] P. Boxler, Stochastische Zentrumsmannigfaltigkeiten, Ph.D.thesis, Institut fiir Dynamische Systeme, Universitat Bremen, 1988. [5] Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55 (1995), 329-358. 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), 423-445. 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), 23-52. [8] 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. [9] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. [10] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, Amsterdam, 2014. [11] I. Gyöngy, On the approximations of stochastic partial differential equations I, Stochastics, 25 (1988), 59-85. 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), 315-341. 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), 495-509. [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), 65-70. [15] M. Hairer and E. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan., 67 (2015), 1551-1604. 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), 605-611. doi: 10.1016/0047-259X(83)90043-X. [18] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 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), vi+106 pp. [20] A. Millet and M. Sanz-Sole, The support of the solution to a hyperbolic SPDE, Probab. Theory Related Fields, 98 (1994), 361-387. 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), 887-915. 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), 615-652. 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), vi+105 pp. doi: 10.1090/memo/0917. [24] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, Proc. of Intern. Symp. SDE Kyoto, (1978), 283-296. [25] R. Pettersson, Wong-Zakai approximations for reflecting stochastic differential equations, Stochastic Anal. Appl., 17 (1999), 609-617. doi: 10.1080/07362999908809624. [26] P. Protter, Approximation of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. 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., University of California Press, Berkeley , 3 (1972), 333-359. [28] X. Sun, J. Duan and X. Li, An impact of noise on invariant manifolds in nonlinear dynamical system, J. Math. Phys., 51 (2010), 042702, 12pp. doi: 10.1063/1.3371010. [29] G. Tessitore and J. Zabczyk, Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655. 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), 213-229. 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), 1560-1564. doi: 10.1214/aoms/1177699916.
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