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In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of invariant measures of the stochastic delay lattice systems. Under certain conditions, we first show the existence of invariant measures of the systems and then establish the stability in distribution of the solutions. We finally prove that any limit point of a tight sequence of invariant measures of the stochastic delay lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero. In particular, when the stochastic delay lattice systems are stable in distribution, we show the invariant measures of the perturbed systems converge to that of the limiting system. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time. [ABSTRACT FROM AUTHOR]

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 partial differential equations driven by a white noise. We prove that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. When the stochastic partial differential equation is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previously studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous as the radius of the channel tends to zero. [ABSTRACT FROM AUTHOR]