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August  2019, 39(8): 4797-4840. doi: 10.3934/dcds.2019196

## Convergence and center manifolds for 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: Jun Shen, junshen85@163.com

Received  November 2018 Published  May 2019

Fund Project: This work was supported by NSFC (11501549, 11331007, 11831012), 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 study the convergence and pathwise dynamics of random differential equations driven by colored noise. We first show that the solutions of the random differential equations driven by colored noise with a nonlinear diffusion term uniformly converge in mean square to the solutions of the corresponding Stratonovich stochastic differential equation as the correlation time of colored noise approaches zero. Then, we construct random center manifolds for such random differential equations and prove that these manifolds converge to the random center manifolds of the corresponding Stratonovich equation when the noise is linear and multiplicative as the correlation time approaches zero.

Citation: Jun Shen, Kening Lu, Bixiang Wang. Convergence and center manifolds for differential equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4797-4840. doi: 10.3934/dcds.2019196
##### 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] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38. [4] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977. [6] 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. [7] S.-N. Chow and K. Lu, Ck center unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320. doi: 10.1017/S0308210500014682. [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] B. 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. [10] W. Gerstner, W. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014. [11] 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. [12] A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, submitted. [13] J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228. [14] J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981. [16] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. [17] N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. RIMS, Kyoto Univ., 13 (1977), 285-300. doi: 10.2977/prims/1195190109. [18] T. Jiang, X. 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. [19] 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. [20] N. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981. [21] 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. [22] D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979. [23] M. Klosek-Dygas, B. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441. doi: 10.1137/0148023. [24] 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. [25] T. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. doi: 10.1214/aop/1176990334. [26] T. Kurtz and P. Protter, Wong-zakai corrections, random evolutions, and simulation schemes for SDEs, Stochastic Analysis, 331-346, Academic Press, Boston, MA, 1991. [27] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947. [28] E. J. McShane, Stochastic differential equations and models of random processes, in: Proc. 6th Berkeley Sympos., Math. Statist. Probab., vol. 3,263-294, Univ. California Press, Berkeley, Calif., 1972. [29] 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. [30] S. 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. [31] S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380. [32] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, in: Proceedings of the International Symposium on Stochastic Differential Equations, 283-296, Wiley, New York-Chichester-Brisbane, 1978. [33] O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160. doi: 10.1007/BF01180524. [34] 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). [35] P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. doi: 10.1214/aop/1176992905. [36] L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511984730. [37] 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. [38] 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. [39] A. Shmatkov, The Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs, PhD thesis, University of Edinburgh, 2005. [40] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in: Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., vol. 3,333-359, Univ. California Press, Berkeley, Calif., 1972. [41] H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298. doi: 10.1090/S0002-9904-1977-14312-7. [42] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608. [43] G. Uhlenbeck and L. Ornstein, On the theory of the Brownian motion, Phys. Rev., 36 (1930), 823-841. doi: 10.1103/PhysRev.36.823. [44] 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. [45] M. Wang and G. Uhlenbeck, On the theory of Brownian motion. â…¡, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. [46] T. Wanner, Linearization of random dynamical systems, in: C.K.R.T. Jonesm, U. Kirchgraber, H.O. Walther (Eds.), Dynamics Reported, vol. 4,203-269, Springer, Berlin/Heidelberg/New York, 1995. [47] E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Engng Sci., 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5. [48] 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.

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), 131-186. doi: 10.1080/07362998408809031. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [3] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38. [4] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977. [6] 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. [7] S.-N. Chow and K. Lu, Ck center unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 303-320. doi: 10.1017/S0308210500014682. [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] B. 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. [10] W. Gerstner, W. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Cambridge University Press, Cambridge, 2014. [11] 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. [12] A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, submitted. [13] J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228. [14] J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1981. [16] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. [17] N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. RIMS, Kyoto Univ., 13 (1977), 285-300. doi: 10.2977/prims/1195190109. [18] T. Jiang, X. 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. [19] 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. [20] N. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981. [21] 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. [22] D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520. doi: 10.1214/14-AOP979. [23] M. Klosek-Dygas, B. Matkowsky and Z. Schuss, Colored noise in dynamical systems, SIAM J. Appl. Math., 48 (1988), 425-441. doi: 10.1137/0148023. [24] 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. [25] T. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070. doi: 10.1214/aop/1176990334. [26] T. Kurtz and P. Protter, Wong-zakai corrections, random evolutions, and simulation schemes for SDEs, Stochastic Analysis, 331-346, Academic Press, Boston, MA, 1991. [27] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947. [28] E. J. McShane, Stochastic differential equations and models of random processes, in: Proc. 6th Berkeley Sympos., Math. Statist. Probab., vol. 3,263-294, Univ. California Press, Berkeley, Calif., 1972. [29] 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. [30] S. 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. [31] S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380. [32] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, in: Proceedings of the International Symposium on Stochastic Differential Equations, 283-296, Wiley, New York-Chichester-Brisbane, 1978. [33] O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160. doi: 10.1007/BF01180524. [34] 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). [35] P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743. doi: 10.1214/aop/1176992905. [36] L. Ridolfi, P. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511984730. [37] 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. [38] 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. [39] A. Shmatkov, The Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs, PhD thesis, University of Edinburgh, 2005. [40] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in: Proc. 6-th Berkeley Symp. on Math. Stat. and Prob., vol. 3,333-359, Univ. California Press, Berkeley, Calif., 1972. [41] H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point, Bull. Amer. Math. Soc., 83 (1977), 296-298. doi: 10.1090/S0002-9904-1977-14312-7. [42] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6 (1978), 19-41. doi: 10.1214/aop/1176995608. [43] G. Uhlenbeck and L. Ornstein, On the theory of the Brownian motion, Phys. Rev., 36 (1930), 823-841. doi: 10.1103/PhysRev.36.823. [44] 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. [45] M. Wang and G. Uhlenbeck, On the theory of Brownian motion. â…¡, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. [46] T. Wanner, Linearization of random dynamical systems, in: C.K.R.T. Jonesm, U. Kirchgraber, H.O. Walther (Eds.), Dynamics Reported, vol. 4,203-269, Springer, Berlin/Heidelberg/New York, 1995. [47] E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Engng Sci., 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5. [48] 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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