Stochastic Lattice Dynamical Systems Research Articles

Invariant measures for stochastic delay nonlocal lattice dynamical systems

This paper deals with the dynamics of the stochastic delay lattice systems with fractional discrete Laplacian driven by nonlinear noise. The existence of a invariant measure for the stochastic delay nonlocal lattice systems is established under certain conditions. The idea of the uniform estimates on the tails of the solutions, the technique of diadic division and the Arzela-Ascoli theorem are employed to show the tightness of a family of probability distributions of the solutions.

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>

Random attractor and exponentially stability for the second order nonautonomous retarded lattice systems with multiplicative white noise

In this paper, we first present some conditions for the existence of a random attractor of continuous non-autonomous random dynamical systems (cocycle) and locally coupled stochastic retarded lattice dynamical systems in a space of continuous functions from finite closed interval into a product space of finite weighted spaces of infinite sequences. Then we consider the existence, singleton and exponentially stability of the random attractor for the second order non-autonomous retarded lattice systems with multiplicative white noise.

Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise

The fBm-driving rough stochastic lattice dynamical system with a general diffusion term is investigated. First, an area element in space of tensor is desired to define the rough path integral using the Chen-equality and fractional calculus. Under certain conditions, the considered equation is proved to possess a unique local mild path-area solution.

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.

A Combined Criterion for Existence and Continuity of Random Attractors for Stochastic Lattice Dynamical Systems

The paper is devoted to establishing a combination of sufficient criterion for the existence and upper semi-continuity of random attractors for stochastic lattice dynamical systems. By relying on a family of random systems itself, we first set up the abstract result when it is convergent, uniformly absorbing and uniformly random when asymptotically null in the phase space. Then we apply the results to the second-order lattice dynamical system driven by multiplicative white noise. It is indicated that the criterion depending on the dynamical system itself seems more applicable than the existing ones to lattice differential models.

Stochastic Lattice Dynamical Systems with Fractional Noise

This article studies stochastic lattice dynamical systems driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. First of all, we investigate the existence and uniqueness of pathwise mild solutions to such systems by the Young integration setting and prove that the solution generates a random dynamical system. Further, we analyze the exponential stability of the trivial solution.

Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations

In this paper, we study the Wong-Zakai approximations given by a smoothed approximation of the white noise and their associated long term pathwise behavior for the stochastic lattice dynamical systems. To be exactly, we first establish the existence of the random attractor for the random lattice dynamical system driven by the smoothed noise and then show the convergence of solutions and random attractors to these of stochastic lattice dynamical systems driven by a multiplicative noise and an additive white noise, respectively, when the perturbation parameters tend to 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 stochastic lattice dynamical systems with infinite multiplicative white noise

In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

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.

Random attractors for partly dissipative stochastic lattice dynamical systems with multiplicative white noises

The present paper is devoted to the existence of the random attractor for partly dissipative stochastic lattice dynamical systems with multiplicative white noises.

Existence of Random Attractors for a Class of Second-Order Lattice Dynamical Systems with Brownian Motions

This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.

Random attractor for stochastic lattice dynamical systems with α-stable Lévy noises

The present paper is devoted to the existence of a random attractor for stochastic lattice dynamical systems with α-stable Lévy noises.

Random attractors for stochastic retarded lattice systems

In this paper, we first present some sufficient conditions for the existence of random attractors for general continuous random dynamical systems (RDSs) defined on the separable Banach space consisting of continuous functions from () into the weighted space of infinite sequences and continuous RDSs associated with stochastic retarded lattice dynamical systems in the state space , where is a positive-valued function from into and is a positive constant. Then we apply the obtained abstract result to the first-order stochastic retarded lattice system with random coupled coefficients and additive white noise.

RANDOM ATTRACTORS OF STOCHASTIC LATTICE DYNAMICAL SYSTEMS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

This paper is devoted to consider stochastic lattice dynamical systems (SLDS) driven by fractional Brownian motions with Hurst parameter bigger than 1/2. Under usual dissipativity conditions these SLDS are shown to generate a random dynamical system for which the existence and uniqueness of a random attractor are established. Furthermore, the random attractor is, in fact, a singleton sets random attractor.

Asymptotic behaviors for second order stochastic lattice dynamical systems on [formula omitted] in weighted spaces

In this article, we study the asymptotic behavior of solutions to the second order stochastic lattice differential equation on Zk with additive white noise in weighted space which includes solutions with bounded components. We first consider the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.

Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities

In this paper we study the asymptotic behavior of solutions of a first-order stochastic lattice dynamical system with a multiplicative noise.We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor.

Random Attractors for Stochastic Retarded Lattice Dynamical Systems

This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise. We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove the existence of a compact global random attractor within the set of tempered random bounded sets.

Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

In this article, we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems, we prove the existence of a random compact global attractor.