Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.     

Random attractors for stochastic reaction‐diffusion equations with multiplicative noise in

In this paper, we study the random dynamical system generated by a stochastic reaction‐diffusion equation with multiplicative noise and prove the existence of an ‐random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy some growth of arbitrary order .     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise

We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.     

Random attractors of boussinesq equations with multiplicative noise

We study the random dynamical system （RDS） generated by the Benald flow problem with multiplicative noise and prove the existence of a compact random attractor for such RDS.     

The existence of random $\mathcal{D}$-pullback attractors for random dynamical system and its applications

In this paper, we establish a result on the existence of random $\mathcal{D}$-pullback attractors for norm-to-weak continuous non-autonomous random dynamical system. Then we give a method to prove the existence of random $\mathcal{D}$-pullback attractors. As an application, we prove that the non-autonomous stochastic reaction diffusion equation possesses a random $\mathcal{D}$-pullback attractor in $H_0^1$ with polynomial growth of the nonlinear term.     

Existence of global attractors for the three‐dimensional Brinkman–Forchheimer equation

In this paper, we investigate the asymptotic behavior of solutions of the three‐dimensional Brinkman–Forchheimer equation. We first prove the existence and uniqueness of solutions of the equation in L², and then show that the equation has a global attractor in H² when the external forcing term belongs to L². Copyright © 2008 John Wiley & Sons, Ltd.     

Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

This paper is concerned with pullback attractors of the stochastic p  -Laplace equation defined on the entire space Rⁿ${\mathbb{R}}^n$. We first establish the asymptotic compactness of the equation in L²(Rⁿ)$L^2({\mathbb{R}}^n)$ and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on Rⁿ${\mathbb{R}}^n$ is overcome by the uniform smallness of solutions outside a bounded domain.     

Random attractors of reaction-diffusion equations with multiplicative noise in L

In this paper, we study the random dynamical system (RDS) generated by the reaction-diffusion equation with multiplicative noise and prove the existence of a random attractor for such RDS in L^p(D) for any p?2.     

一类非自治Brinkman-Forchheimer系统的拉回$\mathcal{D}$-吸引子

The asymptotic behavior of solutions of the three-dimensional nonautonomous Brinkman-Forchheimer equation is investigated. And the existence of pullback global attractors in L2 (Ω) and H01(Ω) is proved, respectively.     

Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domains

In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as $\delta\rightarrow0$ is proved.     

Asymptotic expansions of backward equations for two-time-scale Markov chains in continuous time

This work develops asymptotic expansions for solutions of systems of backward equations of time- inhomogeneous Maxkov chains in continuous time. Owing to the rapid progress in technology and the increasing complexity in modeling, the underlying Maxkov chains often have large state spaces, which make the computa- tional tasks ihfeasible. To reduce the complexity, two-time-scale formulations are used. By introducing a small parameter ε〉 0 and using suitable decomposition and aggregation procedures, it is formulated as a singular perturbation problem. Both Markov chains having recurrent states only and Maxkov chains including also tran- sient states are treated. Under certain weak irreducibility and smoothness conditions of the generators, the desired asymptotic expansions axe constructed. Then error bounds are obtained.     

Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domains

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $\alpha\in(0,1)$. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $L^{2}(\mathbf{R}^{3})$. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $L^{2}(\mathbf{R}^{3})$ by the tail-estimates of solutions.     

Random attractor for the Ladyzhenskaya model with additive noise

This paper studies the dynamical behavior of the Ladyzhenskaya model with additive noise. With some conditions, we prove that the generated random dynamical system has a compact random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space.     

Asymptotic dynamics for reaction diffusion equations in unbounded domain

In this paper we study the asymptotic dynamics for reaction diffusion equation defined in $\mathbb{R}^n$. We will prove that the equation possesses a fixed point when the nonlinearity satisfies some restrictive conditions and then we show that the fixed point is an exponential attractor.