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In this paper, we prove the existence and regularity of a tempered pullback attractor for a non-autonomous stochastic fractional equation involving an abstract non-local operator defined as L K u (x) = lim ε ↘ 0 ∫ R N ∖ B ε (x) (u (x) − u (y)) K (x − y) d y , x ∈ R N , where K : R N ∖ { 0 } → (0 , + ∞) is the kernel of L K which satisfies the general fractional-type condition of order s , 0 < s < 1 and B ε (x) is a ball in R N centered at x with radius ε. The asymptotical compactness of solutions in X 0 s is demonstrated by the spectral decomposition and truncation technique, where X 0 s is a Hilbert space with s ∈ (0 , 1) and N > 2 s. As a special example, we obtain the existence and regularity of tempered pullback attractor of stochastic fractional reaction-diffusion equation. [ABSTRACT FROM AUTHOR]

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on RN with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space L2(RN) × L2(RN) is actually a compact, measurable and attracting set in H1(RN) × L2(RN). A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in H1(RN) × L2(RN), despite of the loss of the high-order integrability of the difference of solutions for this system. [ABSTRACT FROM AUTHOR]

In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi-spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non-autonomous stochastic fractional power dissipative equation on RN with additive white noise and a polynomial-like growth nonlinearity of order p,p ≥ 2. We prove that this equation generates a bi-spatial (L2(RN),Hs(RN) ∩ Lp(RN))-continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in Hs(RN) ∩ Lp(RN), where Hs(RN) is a fractional Sobolev space with s ∈ (0,1). Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in Hs(RN) and Lp(RN) with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in Hs(RN) ∩ Lp(RN),s ∈ (0,1), N ≥ 1. [ABSTRACT FROM AUTHOR]

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in L 2 ( R N ) × L 2 ( R N ) by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most p -times integrability where p is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in ( L p ( R N ) ∩ H 1 ( R N ) ) × L 2 ( R N ) with respect to the initial values belonging to L 2 ( R N ) × L 2 ( R N ) . As an application of the result on the continuity, the existence of a pullback attractor in ( L p ( R N ) ∩ H 1 ( R N ) ) × L 2 ( R N ) is proved for arbitrary N ≥ 1 and p > 2 . [ABSTRACT FROM AUTHOR]

In this article, we study the long-time random dynamics of non-autonomous stochastic parabolic p -Laplacian equations driven by a Wiener-type multiplicative noise. By a new asymptotic a priori estimate method, not involving the plus or minus sign of the nonlinearity, the omega-limit compact of solutions in L p ( R N ) ∩ L q ( R N ) is proved for any p , q > 2 and N ≥ 1 , where q is the polynomial growth exponent of the nonlinearity. If 2 < p < N , by a boot strap technique, we obtain the higher-order integrability of difference of solutions near the initial time. As an application, we show that the obtained L 2 -random attractor is attracting in the topology of L 2 + δ ( R N ) for any δ > 0 . [ABSTRACT FROM AUTHOR]

In this article, some sufficient conditions on the regularity of random attractors are first provided for general random dynamical systems in the weighted space ℓ ρ p ( p > 2 ) of infinite sequences. They are then used to study the asymptotic dynamics of a class of non-autonomous stochastic lattice differential equations with spatially valued additive noises. The existences of tempered random attractors for this in both spaces ℓ ρ 2 and ℓ ρ p are proved respectively, which implies that the obtained ℓ ρ 2 -random attractor is compact and attracting in the topology of ℓ ρ p space. To solve this, a common embedding space of ℓ ρ 2 and ℓ ρ p is constructed and some new estimates are also developed here. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the existence of -random attractor is established for a stochastic reaction–diffusion equation on the whole space . This random attractor is a compact and invariant tempered set which attracts every tempered random subset of in the topology of . The nonlinearity is supposed to satisfy some growth of arbitrary order , where . The -asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions. [Copyright &y& Elsevier]

This paper is devoted to the investigation of the asymptotic behavior of solutions of the stochastic magneto-hydrodynamics equations driven by random exterior forced terms both in the velocity and in the magnetic field. The nonlinear term is supposed to be time-dependent and satisfies certain dissipative conditions. The existence of a random attractor for this random dynamical system is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the higher-order Wong–Zakai approximations of the non-autonomous stochastic reaction–diffusion equation driven by additive/multiplicative white noises. The solutions between the approximation equation and stochastic reaction–diffusion equation are compared in higher-order spaces, in terms of the initial data. Based on these results and the known L 2 -upper semi-continuity, we prove that the random attractor of the approximation random system converges to that of the non-autonomous stochastic reaction–diffusion equation with additive/multiplicative white noises in L p (R N) ∩ H 1 (R N) when the size of the approximation shrinks to zero. • Approximation is obtained in L p ∩ H 1 for equations with additive noise. • Approximation is obtained in L p ∩ H 1 for equations with multiplicative noises. • Upper semi-continuity w.r.t. the size of approximations is derived in L p ∩ H 1 . [ABSTRACT FROM AUTHOR]