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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 & Continuous Dynamical Systems - A, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078
References:
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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

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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

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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

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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.  Google Scholar

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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

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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

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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

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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

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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.  Google Scholar

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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

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

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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

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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

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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

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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

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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. 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.  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. 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.  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

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