Non-autonomous Stochastic Lattice Research Articles

Non-autonomous stochastic lattice systems with Markovian switching

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic lattice systems with Markovian switching. We first show existence of an evolution system of measures of the stochastic system. We then study the pullback (or forward) asymptotic stability in distribution of the evolution system of measures. We finally prove that any limit point of a tight sequence of an evolution system of measures of the stochastic lattice systems must be an evolution system of measures of the corresponding limiting system as the intensity of noise converges zero. In particular, when the coefficients are periodic with respect to time, we show every limit point of a sequence of periodic measures of the stochastic system must be a periodic measure of the limiting system as the noise intensity goes to zero.

Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations

<p style='text-indent:20px;'>A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.</p>

Wong-Zakai approximations and long term behavior of second order non-autonomous stochastic lattice dynamical systems with additive noise

<abstract><p>In this article, we investigate the Wong-Zakai approximations of a class of second order non-autonomous stochastic lattice systems with additive white noise. We first prove the existence and uniqueness of tempered pullback random attractors for the original stochastic system and its Wong-Zakai approximation. Then, we establish the upper semicontinuity of these attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.</p></abstract>

Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems

In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic lattice systems. We first prove that the approximate system has a unique tempered pullback attractor under much weaker conditions than the original system. When the stochastic system is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero.

Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays

<p style='text-indent:20px;'>In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.</p>

Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise

We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system defined on the product space of [Formula: see text]-weighted spaces of infinite sequences. Then, based on this criterion, we prove the existence of random uniform exponential attractors for stochastic lattice systems and stochastic FitzHugh–Nagumo lattice systems that are both with quasi-periodic forces.

Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise

This paper is concerned with the global existence and random dynamics of non-autonomous stochastic second-order lattice systems driven by infinite-dimensional nonlinear noise defined on higher-dimensional integer sets. We first show the existence and uniqueness of mean square solutions to the equations when the nonlinear drift term has a polynomial growth of arbitrary order and the diffusion term is locally Lipschitz continuous. We then prove that the mean random dynamical system associated with the solution operator possesses a unique tempered weak pullback mean random attractor in a Bochner space under certain conditions. We finally establish the existence of invariant measures for the stochastic systems in \begin{document}$ \ell^2\times\ell^2 $\end{document} by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions.

Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.

Regularity, forward-compactness and measurability of attractors for non-autonomous stochastic lattice systems

This paper is concerned with the regularity, forward-compactness and measurability of pullback random attractors of non-autonomous stochastic lattice systems with multiplicative white noise as well as random coefficients in weighted spaces. The existence, uniqueness and forward-compactness of pullback random attractors in weighted space ℓσ2 are proved for the systems when the random coefficient in the front of the discrete Laplacian and the time-dependent functions satisfy certain conditions. In addition, this forward-compact (ℓσ2,ℓσ2)-attractor is proved to be a forward-compact bi-spatial (ℓσ2,ℓσ2∩ℓσp)-attractor that is forward-compact and measurable in ℓσ2∩ℓpσ, and attracts all random subset of ℓσ2 under the topology of ℓσ2∩ℓσp. The difficulty of establishing the measurability of the attractors in ℓσ2∩ℓσp is overcome by proving the identity of the attractors on two different universes.

Random exponential attractor for second order non-autonomous stochastic lattice dynamical systems with multiplicative white noise in weighted spaces

This paper is concerned with the random exponential attractor for second order non-autonomous stochastic lattice system with multiplicative white noise and unbounded nonlinearity. Firstly, we transfer the stochastic lattice system into a random lattice system without noise term whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Then we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle in the product weighted space of sequences, which improved the existing conditions. Finally, we prove the existence of a random exponential attractor for the considered system in weighted space of sequences.

Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise

We first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable Banach space and weighted spaces of infinite sequences. Then we apply our abstract result to study the existence of random exponential attractors for non-autonomous first order dissipative lattice dynamical systems with multiplicative white noise.

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems in weighted spaces

In this paper, we first consider the existence of random attractors for the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise in a weighted space $l_{\rho }^{2} \times l_{\rho }^{2} $ ; then we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients

In this paper, we consider the existence of random attractors in a weighted space $ l_\rho ^2 $ for first-order non-autonomous stochastic lattice system with random coupled coefficients and multiplicative/additive white noise, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space [formula omitted

In this article, some sufficient conditions on the regularity of random attractors are first provided for general random dynamical systems in the weighted space ℓρp(p>2) of infinite sequences. They are then used to study the asymptotic dynamics of a class of non-autonomous stochastic lattice differential equations with spatially valued additive noises. The existences of tempered random attractors for this in both spaces ℓρ2 and ℓρp are proved respectively, which implies that the obtained ℓρ2-random attractor is compact and attracting in the topology of ℓρp space. To solve this, a common embedding space of ℓρ2 and ℓρp is constructed and some new estimates are also developed here.

Random Attractors for Non-autonomous Stochastic Lattice FitzHugh-Nagumo Systems with Random Coupled Coefficients

In this paper, we study the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and multiplicative white noise. We consider the existence of random attractors in a weighted space $l_\rho^2 \times l_\rho^2$ for this system, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.

Random attractor of non-autonomous stochastic Boussinesq lattice system

In this paper, we first consider the existence of tempered random attractor for second-order non-autonomous stochastic lattice dynamical system of nonlinear Boussinesq equations effected by time-dependent coupled coefficients and deterministic forces and multiplicative white noise. Then, we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise

We study asymptotic stability of a class of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise. Under certain conditions, we prove such systems have a unique tempered complete quasi-solution which exponentially pullback attracts all solutions starting from a tempered random set. When the non-autonomous deterministic forcings are time-periodic, we obtain the existence, uniqueness and exponential stability of pathwise random periodic solutions for the stochastic lattice systems with delay. The convergence of the tempered complete quasi-solution (periodic solution) is also established when time delay approaches zero.

Attractors of non-autonomous stochastic lattice systems in weighted spaces

We study the asymptotic behavior of solutions to a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. We also construct maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external forcing terms are periodic in time, we show the random attractors are pathwise periodic. In addition, we exhibit a non-autonomous stochastic lattice system which possesses an infinite-dimensional tempered random attractor.