Abstract
In this paper, we introduce and study the concepts and properties of Poisson Stepanov-like almost automorphy (or Poisson \begin{document}$S^2$\end{document} -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 Levy noise under \begin{document}$S^2$\end{document} -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.
Highlights
Since it was introduced in 1960s by Bochner [4, 5], the theory of almost automorphy, which is an important generalization of the classical almost periodicity, has had a long lasting impact on the development of discrete and continuous dynamic systems, for more details about this topics we refer the reader to [17, 18]
Noise or stochastic perturbation is unavoidable in the real world as well as in man-made systems, it is of great significance to import the stochastic effects into the differential equations and the almost automorphic functions
In 2011, Chang et al [7] investigated the existence of S2-almost automorphic mild solutions for a stochastic differential equation, which generalized the concept of square-mean almost automorphy in another way
Summary
Since it was introduced in 1960s by Bochner [4, 5], the theory of almost automorphy, which is an important generalization of the classical almost periodicity, has had a long lasting impact on the development of discrete and continuous dynamic systems, for more details about this topics we refer the reader to [17, 18]. A stochastic process x ∈ L2loc(R, L2(P, H)) is said to be Stepanov-like almost automorphic if its Bochner transform xb : R → L2(0, 1; L2(P, H)) is squaremean almost automorphic in the sense that for sequence of real numbers {sn} there exist a subsequence {sn} and a stochastic process y ∈ L2loc(R, L2(P, H)) such that t+1 t+1. Since F, F1, F2 : R×L2(P, H)×K → L2(P, H) are Poisson Stepanov-like almost automorphic functions, for every sequence of real numbers {sn}, there exists a subsequence {sn} and a function F : R × L2(P, H) × K → L2(P, H) with F1(·, u, x), F2(·, u, x) ∈ L2loc(R, L2(P, H)) such that t+1.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
