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

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

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.

Citation: Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078
References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

K. ChangZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

H. DingW. 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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

K. ChangZ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

H. DingW. 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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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 & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525

[12]

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

[13]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113

[14]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[15]

S. Kanagawa, K. Inoue, A. Arimoto, Y. Saisho. Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method. Conference Publications, 2005, 2005 (Special) : 463-475. doi: 10.3934/proc.2005.2005.463

[16]

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

[17]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[18]

Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287

[19]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501

[20]

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, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019213

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (124)
  • HTML views (318)
  • Cited by (0)

Other articles
by authors

[Back to Top]