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In this paper, a combination of Galerkin’s method and Dafermos’ transformation is first used to prove the existence and uniqueness of solutions for a class of stochastic nonlocal PDEs with long time memory driven by additive noise. Next, the existence of tempered random attractors for such equations is established in an appropriate space for the analysis of problems with delay and memory. Eventually, the convergence of solutions of Wong-Zakai approximations and upper semicontinuity of random attractors of the approximate random system, as the step sizes of approximations approach zero, are analyzed in a detailed way. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball's idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

From both microscopic and macroscopic aspects, we study both discrete and random approximations of the global attractor for a nonlocal lattice dynamical system. First, we establish an abstract Taylor expansion and obtain a global attractor for the continuous-time system. Second, for the discrete-time nonlocal lattice equation via the implicit Euler scheme, we prove the existence of numerical attractors for small time-sizes, and establish the upper semicontinuity of the numerical attractors towards the global attractor as the time-size tends to zero, while the abstract Taylor expansion and error estimation play key roles in the proof. Third, for the random nonlocal lattice equation driven by nonlinear colored noise, we show the existence and upper semicontinuity of random attractors towards the global attractor when the noise-size goes to infinity. Fourth, we establish optimal bounds and several continuities of numerical attractors as well as random attractors under the Hausdorff metric. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown. [ABSTRACT FROM AUTHOR]

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α -model, convective Brinkman-Forchheimer equations, fractional (s , p) -Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others. [ABSTRACT FROM AUTHOR]

In this work, we discuss the large time behaviour of the solutions of two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations on bounded domains. Under the functional setting V ↪ H ↪ V ′ , where H and V are appropriate separable Hilbert spaces and the fact that the embedding V ↪ H is compact, we establish the existence of random attractors in H for the stochastic flow generated by 2D SCBF equations perturbed by additive noise. We prove the upper semicontinuity of the random attractors for 2D SCBF equations in H , when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space V , using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for 2D SCBF equations. Finally, we also comment on the uniqueness of invariant measures. [ABSTRACT FROM AUTHOR]

The large time behavior of deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations ∂ t u - μ Δ u + (u · ∇) u + α u + β | u | r - 1 u + ∇ p = f , ∇ · u = 0 , for r ≥ 3 ( μ , β > 0 for r > 3 and 2 β μ ≥ 1 for r = 3 ), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Furthermore, we establish the upper semicontinuity of random attractors for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero. Finally, we address the existence and uniqueness of invariant measures of 3D stochastic CBF equations. [ABSTRACT FROM AUTHOR]

We study the explicit Euler scheme to approximate the solutions of rough differential equations under a bounded or linear diffusion term, where the drift term satisfies a local Lipschitz continuity and a one-sided linear growth condition. The Euler scheme is then proved to converge for a given solution, where the convergence rate is independent of the initial condition. For a dissipative drift term with a linear growth condition and a bounded diffusion term, the numerical solution under a regular grid generates a random dynamical system (RDS) which admits a random pullback attractor. We also prove that for bounded drift and diffusion terms and under a centered Gaussian noise with stationary increments, the numerical pullback attractor then converges upper semicontinuously to the continuous-time pullback attractor as the time step goes to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior for the three dimensional non-autonomous stochastic Navier-Stokes-Voigt equations on unbounded domains. A continuous non-autonomous random dynamical system for the equations is firstly established. We then obtain pullback asymptotic compactness of solutions and prove that the existence of tempered random attractors for the random dynamical system generated by the equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time. [ABSTRACT FROM AUTHOR]

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.

This paper deals with fractional stochastic nonlocal partial differential equations driven by multiplicative noise. We first prove the existence and uniqueness of solution to this kind of equations with white noise by applying the Galerkin method. Then, the existence and uniqueness of tempered pullback random attractor for the equation are ensured in an appropriate Hilbert space. When the fractional nonlocal partial differential equations are driven by colored noise, which indeed are approximations of the previous ones, we show the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as δ → 0. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations. [ABSTRACT FROM AUTHOR]

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Zähle that the Minkowski content exists in expectation. In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Zähle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors. [ABSTRACT FROM AUTHOR]

This paper reveals dynamical behaviors in the stochastic multimolecule oscillatory reaction model with Poisson jumps. First, this system is proved to have a unique global positive solution via the Lyapunov technique. Second, the existence and uniqueness of general random attractors for its stochastic homeomorphism flow is proved by the comparison theorem, and meanwhile, a criterion for the existence of singleton sets is obtained. Finally, numerical simulations are used to illustrate the predicted random attractors.

The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε→0+\varepsilon \to {0}^{+} on the two-dimensional space.

We consider a system of two coupled stochastic lattice equations driven by additive white noise processes, where the strength of the coupling is given by a parameter κ ≥ 0. We show that these equations generate a random dynamical system which has a random pullback attractor. This attractor naturally depends on the parameter κ. When the intensity of the coupling becomes large, we observe that the components of the given system synchronize. To describe this phenomenon, we prove the upper semicontinuity of the family of attractors with respect to the attractor of a specific limiting system. [ABSTRACT FROM AUTHOR]

We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset of the parameterized space such that the binary map is continuous at all points of with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space and the infinity of noise means that the equation is deterministic. [ABSTRACT FROM AUTHOR]

We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity. [ABSTRACT FROM AUTHOR]

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small. [ABSTRACT FROM AUTHOR]

Based on the existence of pullback attractors for stochastic suspension bridge in [7], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem. [ABSTRACT FROM AUTHOR]

In this paper we study asymptotic behavior of a class of stochastic plate equations with memory and additive noise. First we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions. Second we consider the existence and upper semicontinuity of random attractors for the equation. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with additive white noise, for which the nonlinear damping has a critical cubic growth rate. By showing the pullback asymptotic compactness of the stochastic dynamic systems, we prove the existence of a random attractor in H 0 1 × L 2 $H_{0}^{1}\times L^{2}$ .

In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.

In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generate a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of a Feller Markov Invariant Measure. [ABSTRACT FROM AUTHOR]

The Wong–Zakai approximations given by a stationary process and attractors for stochastic degenerate parabolic equations are considered in this paper. We first establish the existence and uniqueness of tempered pullback attractors for the Wong–Zakai approximations of stochastic degenerate parabolic equations. We then prove that the attractors of Wong–Zakai approximations converge to the attractor of stochastic degenerate parabolic equations driven by multiplicative white noise. [ABSTRACT FROM AUTHOR]

The dynamical behaviour of the stochastic non-autonomous p-Laplace lattice equation driven by variable delays, random viscosity, multiplicative noise and non-Lipschitz nonlinearity is concerned. The existence of global solutions for the equation is proved, while the solutions are possibly multi-valued and generate a multi-valued dynamical system. The existence of a pullback attractor is shown. Moreover, the measurability of the pullback attractor as well as the multi-valued cocycle is proved by using the weak continuity of the discrete p-Laplace operator and a countable decomposition of the Wiener probability space. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space Hα(Rn) with α ∈ (0,1) as well as their time derivatives in L2(Rn). Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively. [ABSTRACT FROM AUTHOR]

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to the dynamical behavior of stochastic coupled suspension bridge equations of Kirchhoff type. For the deterministic cases, there are many classical results such as existence and uniqueness of a solution and long-term behavior of solutions. To the best of our knowledge, the existence of random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type is not yet considered. We intend to investigate these problems. We first obtain the dissipativeness of a solution in higher-energy spaces H3(U)×H01(U)×(H2(U)∩H01(U))×H01(U) $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0}^{1}(U))\times H_{0}^{1}(U)$. This implies that the random dynamical system generated by the equation has a random attractor in (H2(U)∩H01(U))×L2(U)×H01(U)×L2(U) $(H^{2}(U)\cap H_{0}^{1}(U))\times L^{2}(U) \times H_{0}^{1}(U)\times L^{2}(U)$, which is a tempered random set in the space in H3(U)×H01(U)×(H2(U)∩H01(U))×H01(U) $H^{3}(U)\times H_{0}^{1}(U)\times (H^{2}(U)\cap H_{0} ^{1}(U))\times H_{0}^{1}(U)$.

Abstract We consider a Wong–Zakai process, which is the difference of a Wiener-like process. We then prove that there are random attractors for non-autonomous Ginzburg–Landau equations driven by linear multiplicative noise in terms of Wong–Zakai process and Wiener-like process, respectively. Moreover, we establish the upper semi-continuity of random attractors as the size of difference noise tends to zero.

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray- α model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic generalized porous media equations and stochastic generalized p -Laplace equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with additive white noise, for which the nonlinear damping has a critical cubic growth rate. By showing the pullback asymptotic compactness of the stochastic dynamic systems, we prove the existence of a random attractor in H 0 1 × L 2 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is devoted to the well-posedness and long-time behavior of a stochastic suspension bridge equation with memory effect. The existence of the random attractor for the stochastic suspension bridge equation with memory is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. [ABSTRACT FROM AUTHOR]

This paper is devoted to the convergence of bi-spatial random attractors as a family of bounded domains is extended to be unbounded. Some criteria in terms of expansion and restriction are provided to ensure that the unbounded-domain attractor is approximated by the family of bounded-domain attractors in both upper and lower semi-continuity senses. The theoretical results are applied to show that the stochastic FitzHugh–Nagumo coupled equations have an attractor in p -times Lebesgue space irrespective of whether the domain is bounded or unbounded. Furthermore, we prove that the family of bounded-domain attractors continuously converges to the unbounded-domain attractor, and the latter can be constructed by the metric-limit set of all bounded-domain attractors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic model of incompressible non-Newtonian fluids of second grade on a bounded domain of with multiplicative noise. We first show that the solutions to the stochastic equations of second grade fluids generate a continuous random dynamical system. Second, we investigate the Fréchet differentiability of the random dynamical system. Finally, we establish the asymptotic compactness of the random dynamical system, and the existence of random attractors for the random dynamical system, we also obtain the upper semi-continuity of the perturbed random attractors when the noise intensity approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long-time behavior of the solutions for stochastic plate equations with memory and additive noise. First we establish a continuous cocycle for the equation and the pullback asymptotic compactness of solutions. Second we prove the existence of random attractors for the equation. [ABSTRACT FROM AUTHOR]