Almost Periodically Distributed Solutions for Diffusion Equations in Duals of Nuclear Spaces

Tudor, Constantin

doi:10.1007/s002459900089

Cited by 3 publications

(1 citation statement)

References 13 publications

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“…The study of almost periodicity for SDEs seems to start with the Romanian school, which studied Bohr almost periodicity in one-dimensional distribution (APOD) of solutions to almost periodic SDEs: first Halanay [23], then mostly Constantin Tudor and his collaborators in many papers, among them [4,30,33,34,36]. It was Tudor [35] who proposed the notions of almost periodicity in finite-dimensional distribution (APFD) and almost periodicity in (path) distribution, that we call here APPD.…”

Section: Introductionmentioning

confidence: 99%

Almost periodicity and periodicity for nonautonomous random dynamical systems

Fitte

2020

Stoch. Dyn.

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We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.

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Section: Introductionmentioning

confidence: 99%

Almost periodicity and periodicity for nonautonomous random dynamical systems

Fitte

2020

Stoch. Dyn.

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Some periodic type solutions for stochastic reaction–diffusion equation with cubic nonlinearities

Gao¹

2017

Computers & Mathematics with Applications

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The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations

Gao

2019

Stoch. Dyn.

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Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. In this paper, we will establish an averaging principle for multiscale stochastic linearly coupled complex cubic-quintic Ginzburg-Landau equations with slow and fast time scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved, and as a consequence, the system can be reduced to a single stochastic complex cubic-quintic Ginzburg-Landau equation with a modified coefficient.where T > 0, I = (0, 1), Q = I × (0, T ), the stochastic perturbations are of additive type, W 1 and W 2 are mutually independent Wiener processes on a complete stochastic basis (Ω, F, F t , P), which will be specified later, we denote by E the expectation with respect to P. The noise coefficients σ 1 and σ 2 are positive constants and the parameter ε is small and positive, which describes the ratio of time scale between the process A ε and B ε . With this time scale the variable A ε is referred as slow component and B ε as the fast component.Many problems in the natural sciences give rise to singularly perturbed systems of stochastic partial differential equations. In the past four decades, singularly perturbed systems have been the focus of extensive research within the framework of averaging methods. The separation of scales is then taken to advantage to derive a reduced equation, which approximates the slow components. Conditions under which the averaging principle can be applied to this kind of system are well known in the classical literature.Multiscale stochastic partial differential equations(SPDEs) arise as models for various complex systems, such model arises from describing multiscale phenomena in, for example, nonlinear oscillations, material sciences, automatic control, fluids dynamics, chemical kinetics and in other areas leading to mathematical description involving "slow" and "fast" phase variables. The study of the asymptotic behavior of such systems is of great interest. In this respect, the question of how the physical effects at large time scales influence the dynamics of the system is arisen. We focus on this question and show that, under some dissipative conditions on fast variable equation, the complexities effects at large time scales to the asymptotic behavior of the slow component can be omitted or neglected in some sense.The theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time-scales and is relatively mature for stochastic dynamical systems. The theory of averaging principle serves as a tool in study of the qualitative behaviors for complex systems with multiscales, it is essential for describing and understanding the asymptotic behavior of dynamical systems with fast and slow variables. Its basic idea is to approximate the original system by a reduced system. The theory of averaging for deterministic dynamica...

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Almost Periodically Distributed Solutions for Diffusion Equations in Duals of Nuclear Spaces

Cited by 3 publications

References 13 publications

Almost periodicity and periodicity for nonautonomous random dynamical systems

Almost periodicity and periodicity for nonautonomous random dynamical systems

Some periodic type solutions for stochastic reaction–diffusion equation with cubic nonlinearities

The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations

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