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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

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