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October  2018, 23(8): 3309-3345. doi: 10.3934/dcdsb.2018282

## Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise

 1 School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China 2 College of Mathematics, Sichuan University, Chengdu 610064, China 3 School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

* Corresponding author: Xueqin Li, lixueqingk@163.com

Received  March 2017 Revised  January 2018 Published  August 2018

Fund Project: The second author is supported by the Fundamental Research Funds for the Central universities (2012017yjsy139)

In this paper, we introduce and study the concepts and properties of Poisson Stepanov-like almost automorphy (or Poisson $S^2$-almost automorphy) for stochastic processes. With appropriate conditions, we apply the results obtained to investigate the asymptotic behavior of the soulutions to SPDEs driven by Lévy noise under $S^2$-almost automorphic coefficients without global Lipschitz conditions. Moreover, the local asymptotic stability of the solutions under local Lipschitz condition is discussed and the attractive domain is also given. Finally, an illustrative example is provided to justify the practical usefulness of the established theoretical results.

Citation: Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
##### References:
 [1] S. Abbas and D. Bahuguna, Almost periodic solutions of neutral functional differential equations, Comput. Math. Appl., 55 (2008), 2593-2601. doi: 10.1016/j.camwa.2007.10.011. Google Scholar [2] D. Applebaum, Lévy Process and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781. Google Scholar [3] P. Bezandry and T. Diagana, Existence of $S^2$-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. Theory Differ. Equ., 35 (2008), 1-19. Google Scholar [4] S. Bochner, A new approach to almost automorphicity, Proc.Natl. Acad. Sci. USA, 48 (1962), 2039-2043. doi: 10.1073/pnas.48.12.2039. Google Scholar [5] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, 52 (1964), 907-910. doi: 10.1073/pnas.52.4.907. Google Scholar [6] Y. K. Chang, Z. H. Zhao and G. M. N'Guérékata, A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations, Nonlinear Anal., 74 (2011), 2210-2219. doi: 10.1016/j.na.2010.11.025. Google Scholar [7] Y. K. Chang, Z. H. Zhao, G. M. N'Guérékata and R. Ma, Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Anal. RWA, 12 (2011), 1130-1139. doi: 10.1016/j.nonrwa.2010.09.007. Google Scholar [8] Y. K. Chang, Z. H. 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 [9] Y. K. Chang, G. M. N'Guérékata and R. Zhang, Stepanov-like weighted pseudo almost automorphic functions via measure theory, Bull. Malays. Math. Sci. Soc., 39 (2016), 1005-1041. doi: 10.1007/s40840-015-0206-1. Google Scholar [10] Z. Chen and W. 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Dudley, Real Analysis and Probability, 2$^{nd}$ edition, Cambridge University Press, 2002. doi: 10.1017/CBO9780511755347. Google Scholar [16] M. M. Fu and Z. X. 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 [17] G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, 2001. doi: 10.1007/978-1-4757-4482-8. Google Scholar [18] G. M. N'Guérékata, Topics in Almost Automorphy, Springer, New York, Boston Dordrecht, London, Moscow, 2005. Google Scholar [19] G. M. N'Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum, 69 (2004), 80-86. doi: 10.1007/s00233-003-0021-0. Google Scholar [20] E. Hernández and H. K. Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522. doi: 10.1006/jmaa.1997.5899. Google Scholar [21] P. Kalamani, D. Baleanu, S. Selvarasu and M. Mallika Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Differ. Equ. , 2016 (2016), Article Number 163, 36pp. doi: 10.1186/s13662-016-0885-4. Google Scholar [22] M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstract and Applied Analysis, 2013 (2013), Article ID 262191, 10 pages. Google Scholar [23] H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal., 69 (2008), 2158-2166. doi: 10.1016/j.na.2007.07.053. Google Scholar [24] K. X. Li, Weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise, J. Math. Anal. Appl., 427 (2015), 686-721. doi: 10.1016/j.jmaa.2015.02.071. Google Scholar [25] Z. X. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149. doi: 10.1016/j.jfa.2013.11.011. Google Scholar [26] O. Mellah and P. Raynaud de Fitte, Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients, Electron. J. Differential Equations, 2013 (2013), 1-7. Google Scholar [27] S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373. Google Scholar [28] K. I. Sato, Lévy Processes and Infinite Divisibility, Cambridge University Press, 1999. Google Scholar [29] C. Tang and Y. K. Chang, Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces, Appl. Anal., 93 (2014), 2625-2646. doi: 10.1080/00036811.2014.880780. Google Scholar [30] Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821. doi: 10.1088/0951-7715/25/10/2803. Google Scholar [31] R. Zhang, Y. K. Chang and G. M. N'Guérékata, New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal. RWA, 13 (2012), 2866-2879. doi: 10.1016/j.nonrwa.2012.04.016. Google Scholar [32] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynam. Syst. Appl., 22 (2013), 269-284. Google Scholar [33] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Almost automorphic solutions to some stochastic functional differential equations with delay, African Diaspora Journal of Mathematics, 15 (2013), 7-25. Google Scholar [34] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear Anal. TMA, 72 (2010), 1886-1894. doi: 10.1016/j.na.2009.09.028. Google Scholar

show all references

##### References:
 [1] S. Abbas and D. Bahuguna, Almost periodic solutions of neutral functional differential equations, Comput. Math. Appl., 55 (2008), 2593-2601. doi: 10.1016/j.camwa.2007.10.011. Google Scholar [2] D. Applebaum, Lévy Process and Stochastic Calculus, 2$^{nd}$ edition, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781. Google Scholar [3] P. Bezandry and T. Diagana, Existence of $S^2$-almost periodic solutions to a class of nonautonomous stochastic evolution equations, Electron. J. Qual. Theory Differ. Equ., 35 (2008), 1-19. Google Scholar [4] S. Bochner, A new approach to almost automorphicity, Proc.Natl. Acad. Sci. USA, 48 (1962), 2039-2043. doi: 10.1073/pnas.48.12.2039. Google Scholar [5] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, 52 (1964), 907-910. doi: 10.1073/pnas.52.4.907. Google Scholar [6] Y. K. Chang, Z. H. Zhao and G. M. N'Guérékata, A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations, Nonlinear Anal., 74 (2011), 2210-2219. doi: 10.1016/j.na.2010.11.025. Google Scholar [7] Y. K. Chang, Z. H. Zhao, G. M. N'Guérékata and R. Ma, Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations, Nonlinear Anal. RWA, 12 (2011), 1130-1139. doi: 10.1016/j.nonrwa.2010.09.007. Google Scholar [8] Y. K. Chang, Z. H. 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 [9] Y. K. Chang, G. M. N'Guérékata and R. Zhang, Stepanov-like weighted pseudo almost automorphic functions via measure theory, Bull. Malays. Math. Sci. Soc., 39 (2016), 1005-1041. doi: 10.1007/s40840-015-0206-1. Google Scholar [10] 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 [11] 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 [12] T. Diagana and G. M. N'Guérékata, Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett., 20 (2007), 462-466. doi: 10.1016/j.aml.2006.05.015. Google Scholar [13] T. Diagana and G. M. N'Guérékata, Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., 86 (2007), 723-733. doi: 10.1080/00036810701355018. Google Scholar [14] T. Diagana, Evolution equations in generalized Stepanov-like pseudo almost automorphic spaces, Electron. J. Diff. Equ., 2012 (2012), 1-19. Google Scholar [15] R. M. Dudley, Real Analysis and Probability, 2$^{nd}$ edition, Cambridge University Press, 2002. doi: 10.1017/CBO9780511755347. Google Scholar [16] M. M. Fu and Z. X. 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 [17] G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, 2001. doi: 10.1007/978-1-4757-4482-8. Google Scholar [18] G. M. N'Guérékata, Topics in Almost Automorphy, Springer, New York, Boston Dordrecht, London, Moscow, 2005. Google Scholar [19] G. M. N'Guérékata, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum, 69 (2004), 80-86. doi: 10.1007/s00233-003-0021-0. Google Scholar [20] E. Hernández and H. K. Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522. doi: 10.1006/jmaa.1997.5899. Google Scholar [21] P. Kalamani, D. Baleanu, S. Selvarasu and M. Mallika Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Differ. Equ. , 2016 (2016), Article Number 163, 36pp. doi: 10.1186/s13662-016-0885-4. Google Scholar [22] M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstract and Applied Analysis, 2013 (2013), Article ID 262191, 10 pages. Google Scholar [23] H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal., 69 (2008), 2158-2166. doi: 10.1016/j.na.2007.07.053. Google Scholar [24] K. X. Li, Weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise, J. Math. Anal. Appl., 427 (2015), 686-721. doi: 10.1016/j.jmaa.2015.02.071. Google Scholar [25] Z. X. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149. doi: 10.1016/j.jfa.2013.11.011. Google Scholar [26] O. Mellah and P. Raynaud de Fitte, Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients, Electron. J. Differential Equations, 2013 (2013), 1-7. Google Scholar [27] S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373. Google Scholar [28] K. I. Sato, Lévy Processes and Infinite Divisibility, Cambridge University Press, 1999. Google Scholar [29] C. Tang and Y. K. Chang, Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces, Appl. Anal., 93 (2014), 2625-2646. doi: 10.1080/00036811.2014.880780. Google Scholar [30] Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821. doi: 10.1088/0951-7715/25/10/2803. Google Scholar [31] R. Zhang, Y. K. Chang and G. M. N'Guérékata, New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal. RWA, 13 (2012), 2866-2879. doi: 10.1016/j.nonrwa.2012.04.016. Google Scholar [32] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynam. Syst. Appl., 22 (2013), 269-284. Google Scholar [33] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Almost automorphic solutions to some stochastic functional differential equations with delay, African Diaspora Journal of Mathematics, 15 (2013), 7-25. Google Scholar [34] Z. H. Zhao, Y. K. Chang and J. J. Nieto, Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear Anal. TMA, 72 (2010), 1886-1894. doi: 10.1016/j.na.2009.09.028. Google Scholar
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