American Institute of Mathematical Sciences

Advanced
April  2018, 38(4): 1935-1953. doi: 10.3934/dcds.2018078

Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity

Hailong Zhu 1, , Jifeng Chu 2,, and  Weinian Zhang 3,

1. 

School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China
2. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
3. 

College of Mathematics, Sichuan University, Chengdu, China

* Corresponding author

Received  November 2016 Revised  November 2017 Published  January 2018

Fund Project: Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11671118). Hailong Zhu was supported by the National NSF of China (NO. 11301001), China Postdoctoral Science Foundation funded project (NO. 2016M591697), NSF of Anhui Province of China(NO. KJ2017A432, NO. 1708085MA17).

In the setting of mean-square exponential dichotomies, we study the existence and uniqueness of mean-square almost automorphic solutions of non-autonomous linear and nonlinear stochastic differential equations.

Keywords: Mean-square almost automorphic solutions, mean-square exponential dichotomy, stochastic differential equations.
Mathematics Subject Classification: Primary: 60H10, 34C27, 34D09.
Citation: Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078
References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, New York, 1974.

[2]

S. Bochner, A new approach to almost periodicity, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1073/pnas.48.12.2039.

[3]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1016/j.jde.2013.10.018.

[4]

T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.  doi: 10.1016/j.jde.2008.04.001.

[5]

T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.  doi: 10.1016/j.jde.2008.07.025.

[6]

K. Chang, Z. Zhao and G. M. N'Guérékata, Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61 (2011), 384-391.  doi: 10.1016/j.camwa.2010.11.014.

[7]

Z. Chen and W. Lin, Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261 (2011), 69-89.  doi: 10.1016/j.jfa.2011.03.005.

[8]

Z. Chen and W. Lin, Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100 (2013), 476-504.  doi: 10.1016/j.matpur.2013.01.010.

[9]

W. A. Coppel, Dichotomy in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York/Berlin, 1978. doi: 10.1007/BFb0067780.

[10]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013. doi: 10.1007/978-3-319-00849-3.

[11]

H. Ding, W. Long and G. M. N'Guérékata, Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal., 70 (2009), 4158-4164.  doi: 10.1016/j.na.2008.09.005.

[12]

J. D. Dollard and C. N. Friedman, Product Integration with Applications to Differential Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1979. doi: 10.1017/CBO9781107340701.005.

[13]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082.

[14]

M. Fu and Z. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc., 138 (2010), 3689-3701.  doi: 10.1090/S0002-9939-10-10377-3.

[15]

R. D. Gill and S. Johansen, A survey of product integration with a view toward application in survival analysis, Ann. Stat., 18 (1990), 1501-1555.  doi: 10.1214/aos/1176347865.

[16]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numerical Anal., 38 (2000), 753-769.  doi: 10.1137/S003614299834736X.

[17]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205.  doi: 10.1090/S0002-9939-1981-0609651-0.

[18]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[19]

A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods, and Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8384.

[20]

Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.

[21]

C. Lizama and J. G. Mesquita, Almost automorphic solutions of non-autonomous difference equations, J. Math. Anal. Appl., 407 (2013), 339-349.  doi: 10.1016/j.jmaa.2013.05.032.

[22]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[23]

P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc., 61 (1947), 147-192.  doi: 10.1090/S0002-9947-1947-0018719-6.

[24]

J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, in: Pure and Applied Mathematics, vol. 21, Academic Press, 1966.

[25]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005.

[26]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[27]

L. Schlesinger, Neue Grundlagen für einen infinitesimalkalkul der Matrizen, Math. Zeit., 33 (1931), 33-61.  doi: 10.1007/BF01174342.

[28]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows Mem. Amer. Math. Soc. , 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[29]

A. Slavík, Product Integration, its History and Applications, Matfyzpress, Prague, 2007.

[30]

O. M. Stanzhyts'kyi, Investigation of exponential dichotomy of Itô stochastic systems by using quadratic forms, Ukr. Mat. Zh., 53 (2001), 1545-1555. 

[31]

D. Stoica, Uniform exponential dichotomy of stochastic cocycles, Stochastic Process. Appl., 120 (2010), 1920-1928.  doi: 10.1016/j.spa.2010.05.016.

[32]

W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.

[33]

V. Volterra, Sulle equazioni differenziali lineari, Rendiconti Accademia dei Lincei, 4 (1887), 393-396. 

show all references

References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, New York, 1974.

[2]

S. Bochner, A new approach to almost periodicity, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1073/pnas.48.12.2039.

[3]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1016/j.jde.2013.10.018.

[4]

T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.  doi: 10.1016/j.jde.2008.04.001.

[5]

T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.  doi: 10.1016/j.jde.2008.07.025.

[6]

K. Chang, Z. Zhao and G. M. N'Guérékata, Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61 (2011), 384-391.  doi: 10.1016/j.camwa.2010.11.014.

[7]

Z. Chen and W. Lin, Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261 (2011), 69-89.  doi: 10.1016/j.jfa.2011.03.005.

[8]

Z. Chen and W. Lin, Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100 (2013), 476-504.  doi: 10.1016/j.matpur.2013.01.010.

[9]

W. A. Coppel, Dichotomy in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York/Berlin, 1978. doi: 10.1007/BFb0067780.

[10]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013. doi: 10.1007/978-3-319-00849-3.

[11]

H. Ding, W. Long and G. M. N'Guérékata, Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal., 70 (2009), 4158-4164.  doi: 10.1016/j.na.2008.09.005.

[12]

J. D. Dollard and C. N. Friedman, Product Integration with Applications to Differential Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1979. doi: 10.1017/CBO9781107340701.005.

[13]

L. C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082.

[14]

M. Fu and Z. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc., 138 (2010), 3689-3701.  doi: 10.1090/S0002-9939-10-10377-3.

[15]

R. D. Gill and S. Johansen, A survey of product integration with a view toward application in survival analysis, Ann. Stat., 18 (1990), 1501-1555.  doi: 10.1214/aos/1176347865.

[16]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numerical Anal., 38 (2000), 753-769.  doi: 10.1137/S003614299834736X.

[17]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205.  doi: 10.1090/S0002-9939-1981-0609651-0.

[18]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[19]

A. G. Ladde and G. S. Ladde, An Introduction to Differential Equations: Stochastic Modeling, Methods, and Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8384.

[20]

Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.

[21]

C. Lizama and J. G. Mesquita, Almost automorphic solutions of non-autonomous difference equations, J. Math. Anal. Appl., 407 (2013), 339-349.  doi: 10.1016/j.jmaa.2013.05.032.

[22]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[23]

P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc., 61 (1947), 147-192.  doi: 10.1090/S0002-9947-1947-0018719-6.

[24]

J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, in: Pure and Applied Mathematics, vol. 21, Academic Press, 1966.

[25]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005.

[26]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[27]

L. Schlesinger, Neue Grundlagen für einen infinitesimalkalkul der Matrizen, Math. Zeit., 33 (1931), 33-61.  doi: 10.1007/BF01174342.

[28]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows Mem. Amer. Math. Soc. , 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[29]

A. Slavík, Product Integration, its History and Applications, Matfyzpress, Prague, 2007.

[30]

O. M. Stanzhyts'kyi, Investigation of exponential dichotomy of Itô stochastic systems by using quadratic forms, Ukr. Mat. Zh., 53 (2001), 1545-1555. 

[31]

D. Stoica, Uniform exponential dichotomy of stochastic cocycles, Stochastic Process. Appl., 120 (2010), 1920-1928.  doi: 10.1016/j.spa.2010.05.016.

[32]

W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.

[33]

V. Volterra, Sulle equazioni differenziali lineari, Rendiconti Accademia dei Lincei, 4 (1887), 393-396. 

[1]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
[2]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[3]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051
[4]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[5]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715
[6]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221
[7]

Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154
[8]

Pablo Pedregal. Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. Electronic Research Announcements, 2001, 7: 72-78.

[9]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021040
[10]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[11]

Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105
[12]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
[13]

Gaston Mandata N ' Guerekata. Remarks on almost automorphic differential equations. Conference Publications, 2001, 2001 (Special) : 276-279. doi: 10.3934/proc.2001.2001.276
[14]

Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 409-425. doi: 10.3934/dcdss.2021082
[15]

Octavian G. Mustafa, Yuri V. Rogovchenko. Existence of square integrable solutions of perturbed nonlinear differential equations. Conference Publications, 2003, 2003 (Special) : 647-655. doi: 10.3934/proc.2003.2003.647
[16]

Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003
[17]

Rui Zhang, Yong-Kui Chang, G. M. N'Guérékata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525
[18]

Aníbal Coronel, Christopher Maulén, Manuel Pinto, Daniel Sepúlveda. Almost automorphic delayed differential equations and Lasota-Wazewska model. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1959-1977. doi: 10.3934/dcds.2017083
[19]

Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022012
[20]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (412)
  • HTML views (370)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy