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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 is concerned with periodic measures of fractional stochastic reaction-diffusion equations with variable time delay defined on unbounded domains. We first prove the existence of periodic measures of the Markov process associated with the equation by showing the weak compactness of distribution laws of a family of solutions based on the uniform estimates and the equicontinuity of solutions in probability. We then establish the regularity of periodic measures and prove the tightness of the set of all periodic measures of the equation for small noise intensity. As a result, any sequence of periodic measures possesses at least one limit point, and we further prove every such limit must be a periodic measure of the corresponding limiting system. Finally, under further assumptions on the nonlinear terms, we establish the uniqueness and the exponential mixing property of periodic measures in the sense of Wasserstein metric. All the results of the paper are also valid for invariant measures of the corresponding autonomous version of the stochastic equation with constant time delay. The idea of uniform tail estimates of solutions in probability is employed to overcome the difficulty introduced by the non-compactness of Sobolev embeddings on unbounded domains. [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 devoted to the existence of invariant measures and their limiting behavior of the stochastic Schrödinger lattice systems with respect to noise intensity. We prove the set of all invariant measures of the stochastic systems is weakly compact when the noise intensity varies in a bounded interval. We further show any limit of a sequence of invariant measures of the perturbed systems must be an invariant measure of the limiting system. [ABSTRACT FROM AUTHOR]

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures. [ABSTRACT FROM AUTHOR]

The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Z k with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations are established in L 2 (Ω , ℓ 2). The existence of invariant measures for the autonomous equations is established in ℓ 2 by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions. It seems that this is the first time to study the random attractors and invariant measures of stochastic equations with superlinear noise. [ABSTRACT FROM AUTHOR]

In this paper, existence of invariant measure is mainly investigated for a fractional stochastic delay reaction–diffusion equation defined on unbounded domains. We first establish the mean-square uniform smallness of the tails of the solutions in order to overcome the non-compactness of standard Sobolev embeddings on unbounded domains. We then show the weak compactness of a family of probability distributions of the solutions by combining the Ascoli–Arzelŕ theorem, the uniform tail-estimates as well as the technique of dyadic division. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of invariant measures in the Wasserstein sense for fractional stochastic reaction–diffusion equations defined on unbounded domains as the noise intensity approaches zero. Based on uniform estimates of solutions, we prove the family of invariant measures of the stochastic equations converges to the invariant measure of the corresponding deterministic equations in terms of the Wasserstein metric. We also provide the rate of such convergence. [ABSTRACT FROM AUTHOR]

This article is concerned with the global existence and random dynamics of the non-autonomous p -Laplacian lattice system defined on the entire integer set driven by infinite-dimensional nonlinear noise. The existence and uniqueness of mean square solutions to the equations are proved when the nonlinear drift and diffusion terms are locally Lipschitz continuous. It is shown that the mean random dynamical system generated by the solution operators has a unique tempered weak pullback random attractor in a Bochner space. The existence of invariant measures for the stochastic equations in the space of square summable sequences is also established. The idea of uniform tail-estimates of solutions is employed to show the tightness of a family of distribution laws of the solutions. [ABSTRACT FROM AUTHOR]

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 by showing the tightness of a family of distribution laws of solutions via the idea of uniform tail-estimates on the solutions. [ABSTRACT FROM AUTHOR]

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on R n. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]