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November  2019, 24(11): 5831-5848. doi: 10.3934/dcdsb.2019108

Strong convergence of neutral stochastic functional differential equations with two time-scales

Junhao Hu 1, and  Chenggui Yuan 2,

1. 

College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
2. 

Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK

Received  July 2018 Revised  December 2018 Published  November 2019 Early access  June 2019

The purpose of this paper is to discuss the strong convergence of neutral stochastic functional differential equations (NSFDEs) with two time-scales. The existence and uniqueness of invariant measure of the fast component is proved by using Wasserstein distance and the stability-in-distribution argument. The strong convergence between the slow component and the averaged component is also obtained by the the averaging principle in the spirit of Khasminskii's approach.

Keywords: Two time-scales, neutral functional differential equations, exponential ergodicity, invariant measure, averaging principle.
Mathematics Subject Classification: 60H10, 60H30.
Citation: Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108
References:
[1]

J. Bao, Q. Song, G. Yin and C. Yuan, Ergodicity and strong limit results for two-time-scale functional stochastic differential equations, Stoc. Anal. Appl., 35 (2017), 1030-1046.  doi: 10.1080/07362994.2017.1349613.  Google Scholar

[2]

J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.  doi: 10.1016/j.na.2013.12.001.  Google Scholar

[3]

J. Bao, G. Yin and C.Yuan, Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity, arXiv: 1308.2018. Google Scholar

[4]

D. Blömker, M. Hairer and G. A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.  doi: 10.1088/0951-7715/20/7/009.  Google Scholar

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C.-E. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007.  Google Scholar

[6]

M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, Second Edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, In: London Mathematical Society. Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

R. D. Drive, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, in Nonlinear Differential Equations and Nonlinear Mechanics, 474-484, Academic Press, New York, 1963.  Google Scholar

[9]

W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.  Google Scholar

[10]

A. Es-Sarhir, M. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200.   Google Scholar

[11]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, second ed., New York, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[12]

H. Fu, L. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279.  doi: 10.1016/j.spa.2015.03.004.  Google Scholar

[13]

D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM Multiscale Model. Simul., 6 (2007), 577-594.  doi: 10.1137/060673345.  Google Scholar

[14]

D. Givon, I. G. Kevrekidis and R. Kupferman, Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Commun. Math. Sci., 4 (2006), 707-729.  doi: 10.4310/CMS.2006.v4.n4.a2.  Google Scholar

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M. Hairer, J. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

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Y. Kabanov and S. Pergamenshchikov, Two-Scale Stochastic Systems, Springer, Berlin, 2003. doi: 10.1007/978-3-662-13242-5.  Google Scholar

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R. Z. Khasminskii, On an averaging principle for Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.   Google Scholar

[18]

M. S. Kinnally and R. J. Williams, On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints, Electron. J. Probab., 15 (2010), 409-451.  doi: 10.1214/EJP.v15-756.  Google Scholar

[19]

S. B. Kuksin and A. L. Piatnitski, Khasminski-Whitman averaging for randonly perturbed KdV equations, J Math Pures Appl., 89 (2008), 400-428.  doi: 10.1016/j.matpur.2007.12.003.  Google Scholar

[20]

H. J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.  doi: 10.1007/s00245-010-9104-y.  Google Scholar

[21]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[22]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.  doi: 10.4310/CMS.2010.v8.n4.a11.  Google Scholar

[23]

Y. Liu and G. Yin, Asymptotic expansions of transition densities for hybrid jump-diffusions, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 1-18.  doi: 10.1007/s10255-004-0143-5.  Google Scholar

[24]

X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[25]

S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984.  Google Scholar

[26]

M. Rei$\beta$, M. Riedle and O. van Gaans, Delay differential equations driven by Lévy processes: Stationarity and Feller properties, Stochastic Process. Appl., 116 (2006), 1409-1432.  doi: 10.1016/j.spa.2006.03.002.  Google Scholar

[27]

G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Ed., Springer, New York, NY, 2013. doi: 10.1007/978-1-4614-4346-9.  Google Scholar

show all references

References:
[1]

J. Bao, Q. Song, G. Yin and C. Yuan, Ergodicity and strong limit results for two-time-scale functional stochastic differential equations, Stoc. Anal. Appl., 35 (2017), 1030-1046.  doi: 10.1080/07362994.2017.1349613.  Google Scholar

[2]

J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.  doi: 10.1016/j.na.2013.12.001.  Google Scholar

[3]

J. Bao, G. Yin and C.Yuan, Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity, arXiv: 1308.2018. Google Scholar

[4]

D. Blömker, M. Hairer and G. A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.  doi: 10.1088/0951-7715/20/7/009.  Google Scholar

[5]

C.-E. Bréhier, Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593.  doi: 10.1016/j.spa.2012.04.007.  Google Scholar

[6]

M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, Second Edition, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, In: London Mathematical Society. Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

R. D. Drive, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, in Nonlinear Differential Equations and Nonlinear Mechanics, 474-484, Academic Press, New York, 1963.  Google Scholar

[9]

W. E, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.  Google Scholar

[10]

A. Es-Sarhir, M. Scheutzow and O. van Gaans, Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200.   Google Scholar

[11]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, second ed., New York, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[12]

H. Fu, L. Wan and J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279.  doi: 10.1016/j.spa.2015.03.004.  Google Scholar

[13]

D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM Multiscale Model. Simul., 6 (2007), 577-594.  doi: 10.1137/060673345.  Google Scholar

[14]

D. Givon, I. G. Kevrekidis and R. Kupferman, Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Commun. Math. Sci., 4 (2006), 707-729.  doi: 10.4310/CMS.2006.v4.n4.a2.  Google Scholar

[15]

M. Hairer, J. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[16]

Y. Kabanov and S. Pergamenshchikov, Two-Scale Stochastic Systems, Springer, Berlin, 2003. doi: 10.1007/978-3-662-13242-5.  Google Scholar

[17]

R. Z. Khasminskii, On an averaging principle for Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.   Google Scholar

[18]

M. S. Kinnally and R. J. Williams, On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints, Electron. J. Probab., 15 (2010), 409-451.  doi: 10.1214/EJP.v15-756.  Google Scholar

[19]

S. B. Kuksin and A. L. Piatnitski, Khasminski-Whitman averaging for randonly perturbed KdV equations, J Math Pures Appl., 89 (2008), 400-428.  doi: 10.1016/j.matpur.2007.12.003.  Google Scholar

[20]

H. J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.  doi: 10.1007/s00245-010-9104-y.  Google Scholar

[21]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.  Google Scholar

[22]

D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.  doi: 10.4310/CMS.2010.v8.n4.a11.  Google Scholar

[23]

Y. Liu and G. Yin, Asymptotic expansions of transition densities for hybrid jump-diffusions, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 1-18.  doi: 10.1007/s10255-004-0143-5.  Google Scholar

[24]

X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[25]

S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984.  Google Scholar

[26]

M. Rei$\beta$, M. Riedle and O. van Gaans, Delay differential equations driven by Lévy processes: Stationarity and Feller properties, Stochastic Process. Appl., 116 (2006), 1409-1432.  doi: 10.1016/j.spa.2006.03.002.  Google Scholar

[27]

G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Ed., Springer, New York, NY, 2013. doi: 10.1007/978-1-4614-4346-9.  Google Scholar

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