2015
DOI: 10.1016/j.jmaa.2015.04.014
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Almost automorphy and various extensions for stochastic processes

Abstract: We compare different modes of pseudo almost automorphy and variants for stochastic processes: in probability, in quadratic mean, or in distribution in various senses. We show by a counterexample that squaremean (pseudo) almost automorphy is a property which is too strong for stochastic differential equations (SDEs). Finally, we consider two semilinear SDEs, one with almost automorphic coefficients and the second one with pseudo almost automorphic coefficients, and we prove the existence and uniqueness of a mil… Show more

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Cited by 39 publications
(42 citation statements)
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References 39 publications
(65 reference statements)
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“…The concepts of Weylalmost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by Abbas [37] in 2012. Besides the concepts of Stepanovlike almost automorphic functions, our results apply also to the classes of Weyl-almost automorphic functions and Besicovitch almost automorphic functions, introduced in [38] (cf. [7,39] for more details).…”
Section: Stepanov and Weyl Generalizations Of (Asymptotically) Almostmentioning
confidence: 99%
See 1 more Smart Citation
“…The concepts of Weylalmost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by Abbas [37] in 2012. Besides the concepts of Stepanovlike almost automorphic functions, our results apply also to the classes of Weyl-almost automorphic functions and Besicovitch almost automorphic functions, introduced in [38] (cf. [7,39] for more details).…”
Section: Stepanov and Weyl Generalizations Of (Asymptotically) Almostmentioning
confidence: 99%
“…. , ( −1 ( )) ≥0 ) is a global -regularized -resolvent propagation family for (1) and (30) holds, then is injective for every ∈ C with R > and̃( ) ̸ = 0 and equalities (37)- (38) are fulfilled.…”
Section: Is a Global -Regularized 2 -Uniqueness Propagation Family Fomentioning
confidence: 99%
“…Following , we will introduce almost automorphic functions depending on a parameter. Definition A continuous f:R×XX is almost automorphic with respect to the first variable, uniformly with respect to the second variable in bounded subsets of double-struckX if, for every sequence (tn) in double-struckR, there exists a subsequence (tn) such that, for every tR and every xX, the limit ffalse(t,xfalse)=trueprefixlimnftrue(t+sn,xtrue)exists and, for every bounded subset B of double-struckX, the convergence is uniform with respect to xB, and if the convergence trueprefixlimnftrue(tsn,xtrue)=ffalse(t,xfalse)holds uniformly with respect to xB.…”
Section: Preliminariesmentioning
confidence: 99%
“…In these paper, almost automorphic solutions in distribution of stochastic differential equations driven by Gaussian processes or non‐Gaussian processes were studied. In , the history of almost automorphic stochastic processes and their generalizations were given, and the existence and uniqueness of pseudo almost automorphic mild solutions for stochastic differential equations were also obtained.…”
Section: Introductionmentioning
confidence: 99%
“…Therefore, we must import the stochastic effects into the investigation of differential systems. Recently, many authors studied the existence of pseudo almost periodic solutions and weighted pseudo almost periodic solutions for stochastic differential equations in Hilbert spaces.…”
Section: Introductionmentioning
confidence: 99%