# American Institute of Mathematical Sciences

• Previous Article
Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy
• DCDS-B Home
• This Issue
• Next Article
Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms
November  2019, 24(11): 5831-5848. doi: 10.3934/dcdsb.2019108

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

 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.

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 [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

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
 [1] Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022 [2] Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793 [3] Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993 [4] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [5] Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103 [6] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 [7] Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050 [8] David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014 [9] Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291 [10] Christian Pötzsche, Stefan Siegmund, Fabian Wirth. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1223-1241. doi: 10.3934/dcds.2003.9.1223 [11] Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089 [12] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [13] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [14] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [15] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [16] Paul Fife, Joseph Klewicki, Tie Wei. Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 781-807. doi: 10.3934/dcds.2009.24.781 [17] Jean-François Couchouron, Mikhail Kamenskii, Paolo Nistri. An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1845-1859. doi: 10.3934/cpaa.2013.12.1845 [18] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 [19] Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065 [20] Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251

2020 Impact Factor: 1.327