Periodic and almost periodic solutions for semilinear stochastic equations

Prato, Giuseppe Da; Tudor, Constantin

doi:10.1080/07362999508809380

Cited by 108 publications

(64 citation statements)

References 16 publications

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“…Bochner-almost periodicity in distribution as defined here is proved for some solutions of stochastic differential equations in [13,19,1,9]. Remark 2.2 If X is almost periodic in distribution, the family ( X(r)) r∈I is tight in C k .…”

Section: Almost Periodicity and Asymptotic Almost Periodicity In Distmentioning

confidence: 93%

“…For example, in the works of C. Tudor and his collaborators (see in particular [13,19,1,9]), almost periodicity in distribution of the solutions is proved. On the other hand, in a series of papers (in particular [2,3,4]) Bezandry and Diagana prove almost periodicity in quadratic mean, which amounts to almost periodicity in probability plus uniform integrability of the square of the norms.…”

Section: Introductionmentioning

confidence: 99%

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Bochner-Almost Periodicity for Stochastic Processes

Bedouhene

Mellah

Fitte

2012

Stochastic Analysis and Applications

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We compare several notions of almost periodicity for continuous processes defined on the time interval I = R or I = [0, +∞) with values in a separable Banach space E (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when I = R, and to asymptotic Bohr-almost periodicity when I = [0, +∞).

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Section: Almost Periodicity and Asymptotic Almost Periodicity In Distmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Bochner-Almost Periodicity for Stochastic Processes

Bedouhene

Mellah

Fitte

2012

Stochastic Analysis and Applications

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“…Recently, the existence of almost periodic or pseudo almost periodic solutions to some stochastic differential equations have been considered in many publications, such as [16][17][18][19][20][21][22] and references therein. In a very recent article [23], the authors introduced a new concept of square-mean almost automorphic stochastic process.…”

Section: D[x(t) − F T B 1 X(t) ] = [Ax(t) + G(t B 2 X(t))]dt + H(tmentioning

confidence: 99%

Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space

2011

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This article deals primarily with the existence and uniqueness of square-mean almost automorphic mild solutions for a class of stochastic differential equations in a real separable Hilbert space. We study also some properties of square-mean almost automorphic functions including a compostion theorem. To establish our main results, we use the Banach contraction mapping principle and the techniques of fractional powers of an operator. Mathematics Subject Classification (2000) 34K14, 60H10, 35B15, 34F05.

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“…Almost periodicity for stochastic differential equations in infinite dimensional spaces has been studied by many authors, compare for instance [9,17]. Let H denote a real separable Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

Bonaccorsi

Ziglio

2014

Random Operators and Stochastic Equations

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Abstract. Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with ascillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the mean square norm for all times.

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Periodic and almost periodic solutions for semilinear stochastic equations

Cited by 108 publications

References 16 publications

Bochner-Almost Periodicity for Stochastic Processes

Bochner-Almost Periodicity for Stochastic Processes

Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

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