Almost automorphy and various extensions for stochastic processes

Bedouhene, Fazia; Challali, Nouredine; Mellah, Omar; Fitte, Paul Raynaud de; Smaali, Mannal

doi:10.1016/j.jmaa.2015.04.014

Cited by 38 publications

(42 citation statements)

References 39 publications

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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%

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Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

N’Guérékata

Kostić

2018

Abstract and Applied Analysis

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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%

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

N’Guérékata

Kostić

2018

Abstract and Applied Analysis

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“…Following , we will introduce almost automorphic functions depending on a parameter. Definition A continuous

f : R \times X \to X

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

(t_{n}^{'})

double-struckR

, there exists a subsequence

(t_{n})

such that, for every

t \in R

and every

x \in X

, the limit

\tilde{f} false(t, x false) = \underset{n \to \infty}{trueprefixlim} f true(t + s_{n}, x true)

exists and, for every bounded subset B of

double-struckX

, the convergence is uniform with respect to

x \in B

, and if the convergence

\underset{n \to \infty}{trueprefixlim} \tilde{f} true(t - s_{n}, x true) = f false(t, x false)

holds uniformly with respect to

x \in B

.…”

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%

Almost automorphic solutions for stochastic differential equations driven by fractional Brownian motion

Chen¹,

Yang

2018

Mathematische Nachrichten

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This paper concerns a class of stochastic differential equations driven by fractional Brownian motion. The existence and uniqueness of almost automorphic solutions in distribution are established provided the coefficients satisfy some suitable conditions. To illustrate the results obtained in the paper, a stochastic heat equation driven by fractional Brownian motion is considered. 1 K E Y W O R D Salmost automorphic solutions, fractional Brownian motion, stochastic differential equations M S C ( 2 0 1 0 ) 34C27, 60E05, 60H10, 93C30 INTRODUCTIONIt is well known that the recurrence of stochastic processes introduced by Kolmogorov in 1930's [24] is similar to the recurrence of dynamical systems introduced by Birkhoff in 1910's [5]. In dynamical systems, the almost automorphy is a class of basic recurrence introduced by Bochner [6], which is a generalization of almost periodicity [7]. Almost automorphy functions and almost automorphic solutions for differential systems have been studied by many authors; see Veech [34], N'Gu'er'ekata [17], Johnson [22], N'Gu'er'ekata [18], Ding, Liang and Xiao [11][12][13], Shen and Yi [31] and N'Gu'er'ekata [19], for more details about these topics and recent developments. Accordingly, the concept of distributionally almost automorphy for stochastic processes was introduced in [14,15].Many authors investigated almost periodic or pseudo almost periodic solutions for stochastic differential equations; see literature, for example, [1,3,8,20,32]. Mellah and De Fitte [28] gave some counterexamples to mean square almost periodicity of solutions for some stochastic differential equations with almost periodic coefficients. Hence it is a unique way to study some stochastic periodicity in distribution for stochastic differential equations. Almost periodic solutions in distribution and fundamental averaging results for stochastic differential equations were studied in [23]. Almost automorphic stochastic processes and their generalizations have been investigated only since 2010 starting with [16]. There are some papers investigating almost automorphy in distribution [14,15,27]. In these paper, almost automorphic solutions in distribution of stochastic differential equations driven by Gaussian processes or non-Gaussian processes were studied. In [2], 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.Recently, the definition and some properties of almost automorphic random functions in probability has been introduced by Ding, Deng and N'Guérékata [10]. Mathematische Nachrichten. 2019;292:983-995.

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“…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%

A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls

Zhang

Han

2019

Optim Control Appl Methods

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Summary This paper introduces the concept of the piecewise weighted pseudo periodicity for stochastic processes. In addition, it establishes a new composition theorem for weighted pseudo almost periodic functions under the non‐Lipschitz conditions. Using this composition theorem, evolution family together with a fixed‐point theorem for condensing maps, we investigate the existence of p‐mean piecewise weighted pseudo almost periodic mild solutions and optimal mild solutions for a class of impulsive stochastic hyperbolic evolution equations in Hilbert spaces. Then, the existence conditions of optimal pairs of systems governed by the nonlinear impulsive stochastic evolution equations are presented. Finally, an example is provided to illustrate the obtained theory.

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Almost automorphy and various extensions for stochastic processes

Cited by 38 publications

References 39 publications

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

Almost automorphic solutions for stochastic differential equations driven by fractional Brownian motion

A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls

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