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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

Jun Shen 1,, , Kening Lu 2, and  Bixiang Wang 3,

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.

Keywords: Ornstein-Uhlenbeck process, colored noise, approximations, random dynamical systems, center manifolds.
Mathematics Subject Classification: Primary: 60H10; Secondary: 37D10, 37H10.
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.  Google Scholar

[2]

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

[3]

P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38.   Google Scholar

[4]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[5]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
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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.  Google Scholar

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A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, submitted. Google Scholar

[13]

J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228.   Google Scholar

[14]

J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.  Google Scholar

[15]

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

[16]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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[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.  Google Scholar

[22]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380.   Google Scholar

[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.  Google Scholar

[33]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[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). Google Scholar

[35]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[39]

A. Shmatkov, The Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs, PhD thesis, University of Edinburgh, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Rep., 2 (1989), 1-38.   Google Scholar

[4]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[5]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[12]

A. Gu, B. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, submitted. Google Scholar

[13]

J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228.   Google Scholar

[14]

J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.  Google Scholar

[15]

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

[16]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

N. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981.  Google Scholar

[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.  Google Scholar

[22]

D. Kelley and I. Melbourne, Smooth approximation of stochastic differential equations, Ann. Probab., 44 (2016), 479-520.  doi: 10.1214/14-AOP979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton Univ. Press, Princeton, N. J., 1947.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Nakao, On weak convergence of sequences of continuous local martingale, Annales De L'I. H. P., Section B, 22 (1986), 371-380.   Google Scholar

[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.  Google Scholar

[33]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1928), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[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). Google Scholar

[35]

P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab., 13 (1985), 716-743.  doi: 10.1214/aop/1176992905.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[39]

A. Shmatkov, The Rate of Convergence of Wong-Zakai Approximations for SDEs and SPDEs, PhD thesis, University of Edinburgh, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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