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This paper is concerned with the asymptotic behaviors of the solutions of an impulsive stochastic neural field lattice model driven by nonlinear noise. We first show the existence and uniqueness of weak pullback mean random attractors for the impulsive stochastic systems. Then by the properties of Markov processes, the existence of evolution system of measures for the impulsive stochastic systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô's lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of invariant measures of the highly nonlinear stochastic retarded lattice systems. Although invariant measures of stochastic retarded lattice system has been studied recently, there is so far no result of invariant measure of stochastic retarded lattice systems with highly nonlinear drift or diffusion terms. We first show the existence of invariant measures of the systems. We then prove that any limit point of a tight sequence of invariant measures of the stochastic retarded lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic measures of the stochastic impulsive Hopfield-type lattice systems driven by nonlinear noise. By the properties of periodic Markov processes, the existence of periodic measures for the impulsive stochastic lattice systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems. [ABSTRACT FROM AUTHOR]