Uniform attractors for non-autonomous random dynamical systems

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Uniform attractors for non-autonomous random dynamical systems Hongyong Cui a,b,∗ , José A. Langa b a School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074,

PR China b Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,

41080 Sevilla, Spain Received 11 July 2016; revised 21 February 2017

Abstract This paper is devoted to establishing a (random) uniform attractor theory for non-autonomous random dynamical systems (NRDS). The uniform attractor is defined as the minimal compact uniformly pullback attracting random set. Nevertheless, the uniform pullback attraction in fact implies a uniform forward attraction in probability, and implies also an almost uniform pullback attraction for discrete time-sequences. Though no invariance is required by definition, the uniform attractor can have a negative semi-invariance under certain conditions. Several existence criteria for uniform attractors are given, and the relationship between uniform and cocycle attractors is carefully studied. To overcome the measurability difficulty, the symbol space is required p to be Polish which is shown fulfilled by the hulls of Lloc (R; Lr ) functions, p, r > 1. Moreover, uniform attractors for continuous NRDS are shown determined by uniformly attracting deterministic compact sets. Finally, the uniform attractor for a stochastic reaction–diffusion equation with translation-bounded external forcing are studied as applications. © 2017 Elsevier Inc. All rights reserved.

Keywords: Random uniform attractor; Cocycle attractor; Non-autonomous random dynamical system

* Corresponding author at: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China. E-mail addresses: [email protected] (H. Cui), [email protected] (J.A. Langa).

http://dx.doi.org/10.1016/j.jde.2017.03.018 0022-0396/© 2017 Elsevier Inc. All rights reserved.

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Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniform attractors for non-autonomous random dynamical systems . . . . . . . . . . . . . . . . . . . . 2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Uniform attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Forward uniform attraction in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Almost uniform attraction for discrete time . . . . . . . . . . . . . . . . . . . . . . . . . 3. Omega-limit sets and the existence of uniform attractors for NRDS . . . . . . . . . . . . . . . . . . . . 3.1. First results based on compact uniformly attracting sets . . . . . . . . . . . . . . . . . . . . . . . 3.2. Alternative dynamical compactnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Comparison on different attractors for NRDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Proper random attractor for skew-product cocycle on extended phase space . . . . . . . . . . 4.1.1. Proper random sets and proper random attractor . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Relationship between uniform attractor for φ and proper random attractor for π 4.2. Cocycle attractors for NRDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Cocycle attractors with autonomous attraction universes . . . . . . . . . . . . . . . . . 4.2.2. Relationship between cocycle and uniform attractors . . . . . . . . . . . . . . . . . . . 4.3. Several corollary results for uniform attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Uniform attractor as pullback attractor of multi-valued RDS . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Uniform attractor for an NRDS is the random attractor for the associated multi-valued RDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Random uniform attractor is determined by uniformly attracting deterministic compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Application: reaction–diffusion equation with translation-bounded forcing and additive white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Translation bounded function and Polish symbol space . . . . . . . . . . . . . . . . . . . . . . . . 6.2. NRDS generated by the reaction–diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Uniform estimates of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Uniform and cocycle attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 6 6 8 10 11 12 13 17 18 18 18 20 22 22 23 25 28 29 30 32 32 35 36 41 42 42

1. Introduction Global attractors play a key role in the study of dynamical systems. For deterministic nonautonomous dynamical systems, there are typically three kinds of global attractors which have drawn much attention in last years: pullback attractors, cocycle attractors and uniform attractors [10,32,38,15]. Each of these three attractors has its own interesting features on one hand, and has close relationship with the others on the other hand. More precisely, the pullback attractor and cocycle attractor for a non-autonomous dynamical system are directly related, while the uniform attractor is exactly the union of elements involved in the pullback/cocycle attractor [5,4,11,24, 22]. When stochastic perturbation is taken into account, it is convenient to go to the nonautonomous random dynamical system (NRDS for short) theory to study the dynamical behavior. An NRDS is a measurable map φ : R+ ×  ×  × X → X with two base flows {ϑt }t∈R and {θt }t∈R acting on  and , respectively, where X denotes the phase space, R+ = [0, ∞) the

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space of elapsed time, (, F , P) a probability space and  the symbol space constructed by time-dependent terms, called the symbol of the system, see [15]. Recently, the concept of cocycle attractors for NRDS was introduced firstly by [21] and then in details by Wang [41,42], where, remarkably, the attraction universes are non-autonomous, i.e., depend not only on stochastic perturbations but also on time or symbols. This makes the cocycle attractor theory of NRDS very close to random dynamical systems (RDS) without non-autonomous forcings, and hence the cocycle attractor there could also be regarded as a nonautonomous generalization of random attractors for (autonomous) RDS. Notice that it is usually popular in deterministic cases to take the collection of bounded or compact sets (in some metric space) as the attraction universe, that is, the attraction universes of cocycle attractors (also of pullback attractors) for deterministic non-autonomous dynamical systems are very often autonomous, see [15,4]. Hence, in [22] the authors studied cocycle attractors for NRDS with autonomous attraction universes, and made a comparison between attractors with different attraction universes. Recall briefly (see Section 4.2) that, given an NRDS φ and an autonomous universe D, the D-cocycle attractor for φ is a non-autonomous random set A = {Aσ (ω)}σ ∈,ω∈ (not belonging to D) such that • A pullback attracts each random set D belonging to D under φ, i.e., lim dist(φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω), Aσ (ω))) = 0,

t→∞

∀ω ∈ , σ ∈ ;

• A is the minimal compact non-autonomous random set satisfying the above condition; • A is invariant under φ, i.e., φ(t, ω, σ, Aσ (ω)) = Aθt σ (ϑt ω),

∀t  0, ω ∈ , σ ∈ .

In this paper, we establish a theory of (random) uniform attractor for NRDS with  compact,  and  invariant under ϑ and θ , respectively (see Section 2). The definition is as follows. Definition 1.1. An (autonomous) random set A ∈ D is said to be the (random) D-uniform attractor for an NRDS φ if (I) A uniformly (pullback) attracts each D ∈ D under φ, namely, lim sup dist(φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω)), A (ω)) = 0,

t→∞ σ ∈

∀ω ∈ ;

(II) A is the minimal compact (autonomous) random set satisfying (I). From the definition it is clear that the random uniform attractor could be regarded as a random generalization of deterministic uniform attractor concept [15,14]. Indeed, when  is a singleton, φ reduces to a deterministic non-autonomous dynamical system, where the uniformly pullback attracting property (I) is equivalent to the uniformly forward attracting property that lim sup dist(φ(t, σ, D), A ) = 0.

t→∞ σ ∈

Because of the random feature involved, the equivalence between pullback and forward uniform attractions fails for random uniform attractors, just like random attractors for autonomous RDS,

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see [1,20,19,26]. However, it is proved that the uniform pullback attraction implies a uniform forward attraction in probability (see Proposition 2.8), which makes a random uniform attractor A still possible to describe the forward dynamical behaviors of an NRDS: (III) A is uniformly forward D-attracting in probability, i.e.,   lim P ω ∈  : sup dist(φ(t, ω, σ, D(ω)), A (ϑt ω)) > ε = 0, t→∞

σ ∈

∀ε > 0, D ∈ D.

Such an interesting connection between pullback attraction and forward attracting in probability property was first introduced in [20,19,37] for random attractors of autonomous RDS. Here we show that such a connection holds for uniform attractors for NRDS. But note that this connection does not hold for random cocycle attractors, as even for deterministic non-autonomous dynamical systems, pullback and forward attractions are not equivalent in general, see, e.g., [33, 31,13]. Compared with D-cocycle attractor A, in addition to the forward attracting in probability property (III), D-uniform attractor A has the following more properties. Firstly, the random uniform attractor is determined by uniformly attracting deterministic compact sets (see Proposition 5.11), that is, (IV) if D is the collection of nonempty compact subsets of X and D ⊂ D, and A is the D-uniform attractor, then P(A = A) = 1, provided that φ is continuous in both  and X. This result is a generalization of the analogous statement for random attractors of (autonomous) RDS established in [17] via Poincaré recurrence theorem. To prove (IV), we make use of multivalued RDS theory which is usually used to deal with dynamical systems without uniqueness, see e.g. [45,6,40,36,29]. It is shown that, given any NRDS φ continuous in both  and X, the mapping  given by (t, ω, x) =



φ(t, ω, σ, x)

σ ∈

is a continuous multi-valued RDS (see Proposition 5.4), called the multi-valued RDS generated by NRDS φ. Moreover, the D-uniform attractor A of φ is exactly the D-random attractor of the multi-valued RDS  (see Theorem 5.5). Then we prove (IV) by showing that the random attractor for the multi-valued RDS  generated by the NRDS φ is determined by attracting compact sets. Secondly, even though by definition we have no invariance property for random uniform attractors (also for deterministic uniform attractors), inspired by [7,24] we have the following negative semi-invariance property (see Proposition 3.5): (V) A is negatively semi-invariant in the sense that A (θt ω) ⊆ (t, ω, A (ω))

for each t  0, ω ∈ ,

provided that φ is continuous in both  and X, where  is the multi-valued RDS generated by φ.

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Thirdly, while cocycle attractors are pullback attracting for each single ω ∈ , the random uniform attractor A can attract almost uniformly (w.r.t. ω ∈ ) for discrete time sequences (see Proposition 2.10): (VI) For each tn → ∞ and any ε > 0, there exists an F ∈ F (depending on {tn }n∈N and ε) with P(F ) < ε such that, for any D ∈ D, n→∞

sup dist(φ(tn , ϑ−tn ω, σ, D(ϑ−tn ω)), A (ω)) −−−−→ 0,

σ ∈

uniformly for all ω ∈  \ F.

This result is an application of Egoroff’s theorem, and clearly holds for random attractors for autonomous RDS (when  is a singleton and thereby autonomous). Even the autonomous result is new in the literature. Fourthly, it is proved that the random uniform attractor A can be composed of states involved in the cocycle attractor A (see Theorem 4.15) which makes it possible to learn uniform attractors via cocycle attractors: (VII) uniform attractor A and cocycle attractor A for an NRDS φ have the relation A (ω) =



Aσ (ω),

∀ω ∈ ,

σ ∈

provided that φ is continuous in both  and X. Note that analogous results for deterministic dynamical systems were established in most recent works [4,11,24]. However, the theory in this paper is established in a rather different way due to the difficulty arising from the measurability problem. We first construct a proper attraction universe DX composed of proper random sets (see Definition 4.1), and then study the DX -random attractor A for the skew-product cocycle generated by the NRDS φ and the base flow {θt }t∈R as a bridge. By developing the relationship between uniform attractor A and the random attractor A and that between A and the cocycle attractor A, we conclude the relationship (VII) between uniform and cocycle attractors, A and A. Existence criteria and characterization of random uniform attractors are established as well. Similar to deterministic cases, random uniform attractor is shown to have a close relationship to compact uniformly attracting random sets (see Theorems 3.5 & 4.16) and can be characterized by omega-limit sets and complete trajectories (see Proposition 4.19). However, in order to prove the measurability, we require the symbol space to be Polish, i.e., a complete metric space with a countable dense subset, which cannot be seen in deterministic attractor theory. The Polish condition is so related to the stochastic features (see [1,18]) that it is also crucial for further analysis of random uniform attractors, but it is shown general enough to cover usual applications, see Section 6.1. In the final section we study a reaction–diffusion equation with both translation-bounded forcing and additive white noise. We first show that the symbol space constructed as the hull of some translation bounded function is compact and Polish, and then study the tempered uniform and cocycle attractors for the equation.

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2. Uniform attractors for non-autonomous random dynamical systems In this section, we first recall some general concepts related to NRDS, and then introduce the definition of random uniform attractor. We also show that, though this uniform attractor is defined uniformly attracting in pullback sense, it is also forward uniformly attracting in probability. Note that this forward attraction in probability property does not hold for random cocycle attractors introduced in [41,22]. Moreover, uniform attractor is pullback attracting almost uniformly (w.r.t. ω ∈ ) for discrete time sequences. 2.1. Preliminaries In this part, we make some basic definitions obeyed throughout this paper expect otherwise stated. Let (X, d) be a Polish metric space, i.e., a separable complete metric space. The Hausdorff semi-metric between non-empty sets in X is defined by dist(A, B) = sup inf d(a, b), a∈A b∈B

A, B ∈ 2X \ ∅.

For any metric space M, we denote by B(M) the Borel sigma-algebra of M. Let (, d ) be a compact Polish metric space which is invariant in the sense that θt  = ,

∀t ∈ R,

where θ is a smooth translation operator such that • θ0 = identity operator on ; • θs ◦ θt = θt+s , ∀t, s ∈ R; • (t, σ ) → θt σ is continuous. A typical example of translation operator is the time shift θt σ (·) := σ (· + t) in some compact subspace  of C(R, R). Finally, we denote by (, F, P) a probability space, which is unnecessarily P-complete, endowed also with a flow {ϑt }t∈R satisfying • • • • •

ϑ0 = identity operator on ; ϑt  = , ∀t ∈ R; ϑs ◦ ϑt = ϑt+s , ∀t, s ∈ R; (t, ω) → ϑt ω is (B(R) × F, F)-measurable; P-preserving: P(ϑt F ) = P(F ), ∀t  0, F ∈ F .

The two groups {θt }t∈R and {ϑt }t∈R acting on  and , respectively, are called base flows. For the ease of notations, we often use θ (or ϑ ), instead of θt (or ϑt ), when describing universal properties valid for every t ∈ R. ˜ instead of . Note that the completion It is convenient to work on a full measure subspace , of the probability space could have impact on the proof of the measurability of the attractor; one can compare, e.g., [20] and [42]. As we did not assume the probability space to be complete, we ˜ from , that is, by saying that a statement holds for all ω ∈  we mean shall not distinguish  ˜ that it holds on  almost surely.

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Definition 2.1. A mapping φ(t, ω, σ, x) : R+ ×  ×  × X → X is said to be a non-autonomous random dynamical system (NRDS) on X (with base flows {ϑt }t∈R on  and {θt }t∈R on ) if (1) φ is (B(R+ ) × F × B() × B(X), B(X))-measurable; (2) φ(0, ω, σ, ·) is the identity on X for each σ ∈  and ω ∈ ; (3) it holds the cocycle property for each fixed σ ∈ , x ∈ X and ω ∈  that φ(t + s, ω, σ, x) = φ(t, ϑs ω, θs σ ) ◦ φ(s, ω, σ, x),

∀t, s ∈ R+ .

An NRDS φ is said to be continuous in  if the mapping σ → φ(t, ω, σ, x) is continuous for each t ∈ R+ , ω ∈  and x ∈ X fixed. In the same way we define the continuity in X. For simplicity, we often speak of an NRDS without mentioning the base flows and even not the phase space. Sometimes we roughly say an NRDS jointly continuous if it is continuous in both  and X. In applications, an NRDS is usually generated by a stochastic differential equation with timedependent (non-autonomous) terms, represented by σ in the above definition and called the (non-autonomous) symbol of the system. One could see Section 6.2 for a particular NRDS generated by a reaction–diffusion equation. To study the uniform attractor for an NRDS φ and for simplicity, we often write φ(t, ω, , B) :=

 

φ(t, ω, σ, x)

σ ∈ x∈B

(2.1)

for each t ∈ R+ , ω ∈ , ∈ 2 \ ∅ and B ∈ 2X \ ∅. In fact, for the case = , the mapping  given by (t, ω, x) := φ(t, ω, , x),

∀t ∈ R+ , ω ∈ , x ∈ X,

is a continuous multi-valued RDS, provided φ is continuous in both  and X. This will be shown in Section 5. Definition 2.2. A random set D(·) in X is defined as a random mapping D:  → 2X \ ∅, ω → D(ω), which is measurable, namely, the mapping ω → distX (x, D(ω)) is (F, B(R))-measurable for each x ∈ X. If each image D(ω) is closed (resp. bounded/compact) in X, then D is called a closed (resp. bounded/compact) random set in X. ¯ a random set D(·) is Note that since for any B ⊂ X nonempty distX (x, B) = distX (x, B), ¯ measurable if and only if its closure D(·) is measurable. We often write D(·) as D or {D(ω)}ω∈ . Given two random sets D1 , D2 , we write D1 ⊂ D2 if D1 (ω) ⊂ D2 (ω) for each ω ∈ , and we then say D1 is smaller than D2 . Note that for each ε > 0, the closed ε-neighborhood Nε (D) of a random set D defined by Nε (D(ω)) = {x ∈ X : dist(x, D(ω))  ε}, ∀ω ∈ , is measurable. For the measurability of random sets we have the following lemma, see [12, Chapter III], [28, Chapter 2.2] and [6, Lemma 2.2].

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Lemma 2.3. (I) Let {Dn }n∈N be a family of closed random sets in a Polish space X. Then ω→



Dn (ω)

n∈N

is a closed random set in X. If in addition {Dn }n∈N is decreasing and every sequence {xn }n∈N with xn ∈ Dn (ω) is precompact, then 

Dn (ω)

n∈N

is non-empty and measurable. (II) For any closed random set D in X there exist countable many random variables fn , n ∈ N, such that fn (ω) ∈ D(ω) for all ω ∈  and D(ω) =



fn (ω).

n∈N

Throughout this paper, we denote by D some class of random sets in X satisfying • D is neighborhood-closed, i.e. for each D ∈ D there exits an ε > 0 such that the closed ε-neighborhood Nε (D) belongs to D; • D is inclusion-closed, i.e., if D ∈ D then each random set smaller than D belongs to D. In applications, D serves as the collection of elements expected to be attracted by an attractor, called the (attraction) universe of the attractor. An example of D is the collection of all the bounded random sets in X. It is important to notice that the universe D is composed of time-independent random sets, and that D reduces to a class of nonempty subsets of X when  is a singleton. To emphasize the time-independence, we often call D and its elements autonomous. Though D can be a particular case of a non-autonomous universe considered by [41], not all the analysis in [41] can be borrowed here. See [22] for a general discussion on cocycle attractors with autonomous attraction universes, where autonomous and non-autonomous attraction universes are compared. 2.2. Uniform attractors In this section, we suppose that φ is an NRDS on X. Definition 2.4. Given two random sets D and B, it is said that D uniformly (pullback) attracts B under the NRDS φ if lim dist(φ(t, ϑ−t ω, , B(ϑ−t ω)), D(ω)) = 0,

t→∞

∀ω ∈ .

(2.2)

If D uniformly attracts every element in D, then D is said to be uniformly D-(pullback) attracting.

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Definition 2.5. A random set A is called the D-(random) uniform attractor of φ, if A belongs to D and is the minimal compact uniformly D-(pullback) attracting set. Definition 2.6. A random set D in X is said to be uniformly D-(pullback) absorbing if for each ω ∈  and B ∈ D there exists a time T = (ω, B) > 0 such that φ(t, ϑ−t ω, , B(ϑ−t ω)) ⊂ D(ω),

for each t  T .

(2.3)

In this paper, the attraction universe D is often omitted when no confusion occurs. Note that by the invariance of  under θ , (2.2) is equivalent to each of the following t→∞

sup dist(φ(t, ϑ−t ω, σ, B(ϑ−t ω)), D(ω)) −−−→ 0;

(2.4)

  t→∞ sup dist φ(t, ϑ−t ω, θ−t σ, B(ϑ−t ω)), D(ω) −−−→ 0.

(2.5)

σ ∈

σ ∈

By (2.5) we see that the uniform attracting property is in fact defined in the pullback sense, and, unlike deterministic uniform attractors, it is not equivalent to the forward uniform attraction t→∞

sup dist(φ(t, ω, σ, B(ϑ−t ω)), D(ϑt ω)) −−−→ 0.

σ ∈

This non-equivalence is the usual case for attractors of RDS, see [1,20,19,26]. But we shall show in next subsection that this pullback uniform attraction implies a forward uniform attraction in probability. Even though random uniform attractor is defined to uniformly attract random sets in D, we shall prove that it is uniquely determined (with full probability) by attracting deterministic compact sets (of course the attraction universe D should include all the deterministic compact sets when talking about this property), see Proposition 5.11. In applications, if g is the non-autonomous symbol of the system, the symbol space  is ˚ often defined as the closure H(g) of the space H(g) of translations of the symbol. Next proposition indicates that, when the NRDS is continuous in , the uniform attractor is actually fully ˚ ˚ determined by H(g) has an equivalence without closure. Since it is known that the space H(g) relationship with the space R of initial time (see [22] for a discussion), this result shows that the uniform attractor gives and is determined by pullback attraction uniformly in initial time, only provided that φ is continuous in . Proposition 2.7. Let φ be an NRDS which is continuous in , and be any a dense subset of . Then a random set K is D-pullback attracting uniformly in  if and only if it is D-pullback attracting uniformly in . Proof. We need to prove that, for each random set B ∈ D, lim dist(φ(t, ϑ−t ω, , B(ϑ−t ω)), K(ω)) = 0

t→∞

holds if and only if

(2.6)

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lim dist(φ(t, ϑ−t ω, , B(ϑ−t ω)), K(ω)) = 0.

(2.7)

t→∞

Only the sufficient condition needs a proof. Suppose that (2.6) does not hold under (2.7), then there exist tn → ∞, σn ∈  and xn ∈ B(ϑ−tn ω) such that dist(φ(tn , ϑ−tn ω, σn , xn ), K(ω))  δ

(2.8)

for some δ > 0. On the other hand, since is dense in  and φ is continuous in , for each n ∈ N there is a σn ∈ such that dist(φ(tn , ϑ−tn ω, σn , xn ), φ(tn , ϑ−tn ω, σn , xn )) < 1/n, which, thanks to (2.7), implies that dist(φ(tn , ϑ−tn ω, σn , xn ), K(ω))  1/n + dist(φ(tn , ϑ−tn ω, σn , xn ), K(ω)) → 0. This contradicts (2.8). The proof is complete.

2

2.2.1. Forward uniform attraction in probability As we have previously stated, the uniform attracting property of random uniform attractors is only in a pullback sense, while that of deterministic uniform attractors is in both forward and pullback senses (indeed, forward and pullback uniform attractions are equivalent in deterministic cases). The next proposition indicates that, the pullback uniform attraction implies a forward uniform attraction in probability. Proposition 2.8. Suppose that a random set A is uniformly D-pullback attracting under an NRDS φ, then it is forward uniformly attracting in probability in the sense that   lim P ω ∈  : sup dist(φ(t, ω, σ, B(ω)), A (ϑt ω)) > ε = 0,

t→∞

σ ∈

∀ε > 0, B ∈ D.

Proof. Given any ε > 0 and B ∈ D, by the uniform D-pullback attraction of A we have   lim P ω ∈  : sup dist(φ(t, ϑ−t ω, σ, B(ϑ−t ω)), A (ω)) > ε = 0.

t→∞

σ ∈

Since  is invariant under ϑ and ϑ is P-preserving, for each t > 0 it holds 



P ω ∈  : sup dist(φ(t, ω, σ, B(ω)), A (ϑt ω)) > ε 

σ ∈



= P ϑ−t ω ∈  : sup dist(φ(t, ω, σ, B(ω)), A (ϑt ω)) > ε 

σ ∈

 = P ω ∈  : sup dist(φ(t, ϑ−t ω, σ, B(ϑ−t ω)), A (ω)) > ε . σ ∈

Hence, we have the result. 2

(2.9)

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2.2.2. Almost uniform attraction for discrete time Notice that the attracting property of a random uniform attractor is expected for each (almost every) fixed sample ω. This is the usual case in the study of random attractors [1]. Now we show that the usual pullback attraction implies an almost uniform (w.r.t. ω ∈ ) pullback attracting property. The following proposition is a generalization of Egoroff’s theorem (see [27, p. 88]). Lemma 2.9. Suppose {Dn }∞ n=0 is a sequence of random sets such that lim dist(Dn , D0 ) = 0,

n→∞

P-a.s.

(2.10)

Then for each ε > 0, there exists an F ∈ F with P(F ) < ε such that lim dist(Dn (ω), D0 (ω)) = 0,

n→∞

uniformly for all ω ∈  \ F.

Proof. Up to a full measure subset, we let (2.10) hold for all ω ∈ . Define m n =

∞ 

{ω ∈  : dist(Di (ω), D0 (ω)) < 1/m}.

i=n

Then it is clear that m n is non-decreasing in n and, by (2.10),  ⊂ lim m n, n→∞

∀m ∈ N.

Hence, limn→∞ P( \ m n ) = 0, and thereby there exists a positive n0 (m) ∈ N such that P( \ m n0 (m) ) <

ε . 2m

Set F=

∞  

 \ m n0 (m) .

m=1

Then F is measurable and P(F ) 

∞

P( \ m n0 (m) ) < ε.

m=1 m m Since  \ F =  ∩ (∩∞ m=1 n0 (m) ), for each m ∈ N and all ω ∈  \ F (then ω ∈ n0 (m) ) it holds

dist(Dn (ω), D0 (ω)) < 1/m, i.e., the limit holds uniformly in  \ F .

2

∀n  n0 (m),

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Proposition 2.10. Suppose that φ is an NRDS and A is the D-uniform attractor. Then for each tn → ∞, and any ε > 0 there exists an F ∈ F (depending on {tn }n∈N and ε) with P(F ) < ε such that, for any D ∈ D, n→∞

sup dist(φ(tn , ϑ−tn ω, σ, D(ϑ−tn ω)), A (ω)) −−−−→ 0,

σ ∈

uniformly for all ω ∈  \ F.

Proof. Since D is neighborhood-closed, there exists a δ > 0 such that Nδ (A ) ∈ D. Hence, since A uniformly attracts Nδ (A ) for each ω ∈ , by Lemma 2.9 we know that for each tn → ∞ and any ε > 0 there exists an F ∈ F with P(F ) < ε such that n→∞

sup dist(φ(tn , ϑ−tn ω, σ, Nδ (A (ϑ−tn ω))), A (ω)) −−−−→ 0,

σ ∈

uniformly for all ω ∈  \ F.

Noticing that Nδ (A ) is in fact a D-uniformly absorbing set, we have completed the proof. 2 3. Omega-limit sets and the existence of uniform attractors for NRDS This section is aimed to establish some criteria for the existence of uniform attractors for NRDS. To ensure the measurability of the random uniform attractor, which is considerably the most important feature of random attractors compared with deterministic ones, the symbol space  is required to be Polish. Omega-limit sets are always important in the study of attractors. For any ⊂  and ω ∈ , the omega limit set of each B ∈ D under NRDS φ is defined by W(ω, , B) =



φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)).

(3.1)

s0 ts

It is important to note that W(ω, , B) = ∪σ ∈ W(ω, σ, B) generally. In fact, we have 

W(ω, σ, B)

σ ∈

=

 

φ(t, ϑ−t ω, θ−t σ, B) ⊆

σ ∈ s∈N ts

⊆

 

s∈N ts

φ(t, ϑ−t ω, θ−t σ, B)

s∈N σ ∈ ts

φ(t, ϑ−t ω, θ−t σ, B) =

s∈N σ ∈ ts

=



φ(t, ϑ−t ω, θ−t , B) =



φ(t, ϑ−t ω, θ−t , B)

s∈N ts



φ(t, ϑ−t ω, θ−t , B) = W(ω, , B).

(3.2)

s0 ts

It is standard to have the following characterization of omega-limit sets. Proposition 3.1. For any ⊂  and ω ∈ , y ∈ W(ω, , B) if and only if there exist sequences tn → ∞, σn ∈ and xn ∈ B(ϑ−tn ω) such that φ(tn , ϑ−tn ω, θ−tn σn , xn ) → y. The following lemma is crucial to prove the measurability of a uniform attractor.

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Lemma 3.2. Suppose that φ is an NRDS continuous in . If ⊂  densely, then for each ω ∈ , W(ω, , B) = W(ω, , B),

∀B ∈ D.

Proof. For fixed ω and B ∈ D, note that W(ω, ℵ, B) =



φ(t, ϑ−t ω, θ−t ℵ, B(ϑ−t ω)),

∀ℵ ⊂ .

s0 ts

Hence, to verify this lemma it suffices to prove ∀t ∈ R+ .

φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)) = φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)),

(3.3)

Only the ⊂ inclusion needs a proof as the inverse is trivial. Let y ∈ φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)). Then there are sequences σn ∈  and xn ∈ B(ϑ−t ω) such that φ(t, ϑ−t ω, θ−t σn , xn ) → y,

as n → ∞.

Since σ → φ(t, ϑ−t ω, θ−t σ, x) is continuous, by the density of in  there is a sequence σn ∈ such that  dist φ(t, ϑ−t ω, θ−t σn , xn ), φ(t, ϑ−t ω, θ−t σn , xn )  1/n,

∀n ∈ N,

which implies that φ(t, ϑ−t ω, θ−t σn , xn ) → y. Hence, y ∈ φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)) as desired. 2 3.1. First results based on compact uniformly attracting sets Lemma 3.3. Suppose that φ is an NRDS continuous in both  and X, with a compact uniformly D-attracting random set K. Then, for any closed random set B ∈ D, W(·, , B) is non-empty, compact and negatively semi-invariant in the sense that W(ϑt ω, , B) ⊂ φ(t, ω, , W(ω, , B)),

∀t  0, ω ∈ .

(3.4)

Moreover, for any closed random set D which uniformly attracts B it holds W(ω, , B) ⊂ D(ω),

∀ω ∈ .

(3.5)

Proof. Non-empty. Take a sequence xn ∈ φ(tn , ϑ−tn ω, θ−tn σn , B(ϑ−tn ω)) with tn → ∞ and σn ∈ . Then, since K uniformly attracts B, dist(xn , K(ω))  dist(φ(tn , ϑ−tn ω, θ−tn σn , B(ϑ−tn ω)), K(ω)) → 0,

as n → ∞.

Since K(ω) is compact, there exists a y ∈ K(ω) such that xn → y in a subsequence sense. Therefore, by Proposition 3.1 we have y ∈ W(ω, , B) and thereby the nonempty is clear.

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Compactness. Notice that, as an intersection of closed sets, W(ω, , B) is closed in X. Now take arbitrarily a y ∈ W(ω, , B). Proposition 3.1 indicates that there exist sequences tn → ∞, σn ∈  and zn ∈ B(ϑ−tn ω) such that φ(tn , ϑ−tn ω, θ−tn σn , zn ) → y.

(3.6)

On the other hand, since B is attracted by K and K is compact, y ∈ K(ω). Hence, W(ω, , B) ⊂ K(ω) is compact, as a closed subset of a compact set is compact. Negative semi-invariance. Let t ∈ R+ be fixed, and take y ∈ W(ϑt ω, , B). Then by Proposition 3.1 there exist sequences t < tn → ∞, σn ∈  and zn ∈ B(ϑ−tn ϑt ω) such that φ(tn , ϑ−tn ϑt ω, θ−tn θt σn , zn ) → y.

(3.7)

Hence, by the invariance of φ we have φ(tn , ϑ−tn ϑt ω, θ−tn θt σn , zn ) = φ(t, ω, σn ) ◦ φ(tn − t, ϑt−tn ω, θt−tn σn , zn ) → y.

(3.8)

On the other hand, since K is compact and uniformly pullback attracts B, there exists a k ∈ K(ω) such that φ(tn − t, ϑt−tn ω, θt−tn σn , zn ) → k

(3.9)

in a subsequence sense. This implies k ∈ W(ω, , B) by Proposition 3.1. Moreover, by the compactness of , σn converges to some σ up to a subsequence. Hence, by the joint continuity of φ we have φ(t, ω, σn ) ◦ φ(tn − t, ϑt−tn ω, θt−tn σn , zn ) → φ(t, ω, σ, k) which along with (3.8) implies y = φ(t, ω, σ, k) ∈ φ(t, ω, σ, W(ω, , B)); negative semiinvariance is clear. To prove (3.5), suppose we are given another closed random set D uniformly attracting B. Then for any y ∈ W(ω, , B), by Proposition 3.1 we have a sequence xn ∈ φ(tn , ϑ−tn ω, θ−tn , B(ϑ−tn ω)) with tn → ∞ such that xn → y. By the uniformly attraction and the closedness of D we know y ∈ D(ω). Therefore, W(ω, , B) ⊂ D(ω) and (3.5) follows. 2 Before the existence theorem we now prove a crucial lemma. The Polish condition of the symbol space  will play a key role to solve the measurability problem. Lemma 3.4. Suppose that φ is an NRDS continuous in both  and X, with a compact uniformly D-attracting random set K. If a closed random set B ∈ D uniformly attracts itself, then W(·, , B) is a compact and negatively semi-invariant random set, and is the minimal closed random set uniformly attracting B. Proof. By Lemma 3.3 we know that for each ω ∈ , W(ω, , B) is non-empty and compact. Now we prove the measurability. First, let us show that

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W(ω, , B) =

∞  

φ(m, ϑ−m ω, , B(ϑ−m ω)),

∀ω ∈ .

15

(3.10)

n∈N m=n

Since B uniformly attracts itself, by (3.5) we have W(·, , B) ⊂ B(·). Hence, by the negative invariance of W(ω, , B), we have W(ω, , B) ⊂ φ(m, ϑ−m ω, , W(ϑ−m ω, , B)) ∀m ∈ N.

⊂ φ(m, ϑ−m ω, , B(ϑ−m ω)), Therefore, W(ω, , B) ⊂

∞  

φ(m, ϑ−m ω, , B(ϑ−m ω)),

∀ω ∈ ,

n∈N m=n

and thereby (3.10) holds, as the inverse inclusion is straightforward. Since  is Polish, suppose = {σi }i∈N is a dense subset of . Denote by Dn (ω) =

∞  

φ(m, ϑ−m ω, θ−m σi , B(ϑ−m ω)),

∀n ∈ N, ω ∈ .

m=n i∈N

Then W(ω, , B) = ∩n∈N Dn (ω) in view of (3.10) and (3.3). Now we first prove that each Dn is measurable. Since B is a non-empty closed random set, by Lemma 2.3 (II) there exists a sequence {fj }j ∈N of measurable functions such that B(ϑ−m ω) = ∪j ∈N fj (ϑ−m ω), which makes φ(m, ϑ−m ω, θ−m σi , B(ϑ−m ω)) =



φ(m, ϑ−m ω, θ−m σi , fj (ω))

j ∈N

as x → φ(m, ϑ−m ω, θ−m σi , x) is continuous. Since φ(m, ϑ−m ω, θ−m σi , x) is (F, B(X))-measurable in ω, it is measurable in the sense of Definition 2.2 as well since it is single-valued. Hence the right-hand side term of the above identity is measurable and then so is the left-hand side term. Therefore, by Lemma 2.3 (I) we know Dn is measurable. On the other hand, it is clear that Dn is decreasing and every sequence {xn } inside W(ω, , B) is precompact since W(ω, , B) is compact itself, by Lemma 2.3 (I) we conclude that W(ω, , B) = ∩n∈N Dn (ω) is measurable. Now we prove by contradiction that W(·, , B) uniformly attracts B. Suppose it is not true, then there exist a δ > 0 and a sequence xn ∈ φ(tn , ϑ−tn ω, θ−tn σn , B(ϑ−tn ω)) with tn → ∞ and σn ∈  such that dist(xn , W(ω, , B)) > δ,

∀n ∈ N.

(3.11)

However, by the uniformly attracting property and the compactness of K again, there is a y ∈ W(ω, , B) such that xn → y, which contradicts (3.11). The minimal property follows from Lemma 3.3. The proof is complete. 2

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Now we give a sufficient condition for the existence of random uniform attractors. Note that, since the attraction universe is inclusion-closed, the necessary statement of the follow result holds true as well. Theorem 3.5. Suppose that φ is an NRDS continuous in both  and X, and is any a dense subset of . If φ has a compact uniformly D-attracting set K and a closed uniformly D-absorbing set B ∈ D, then it has a unique random uniform attractor A ∈ D given by A (ω) = W(ω, , B) = W(ω, , B),

∀ω ∈ .

Moreover, the uniform attractor A is negatively semi-invariant A (ϑt ω) ⊆ φ(t, ω, , A (ω))

for each t  0, ω ∈ .

Proof. The non-empty, compactness and measurability properties, along with minimal and negative semi-invariant properties, are proved by Lemma 3.4. We now prove the uniformly D-attracting property. Since A uniformly attracts B by Lemma 3.4, for each ε > 0 and σ ∈ , ω ∈  fixed, there is a time T > 0 such that  dist φ(t, ϑ−t ω, θ−t , B(ϑ−t ω)), A (ω) < ε,

∀t  T .

On the other hand, for each D ∈ D and ω ∈ , there is a time TD (ω) > 0 such that 

φ(t, ϑ−t ω, θ−t , D(ϑ−t ω)) ⊂ B(ω)

tTD (ω)

as B is a uniformly D-absorbing set. Hence,  dist φ(t + T , ϑ−t−T ω, θ−t−T , D(ϑ−t−T ω)), A (ω)  = dist φ(T , ϑ−T ω, θ−T , φ(t, ϑ−t ϑ−T ω, θ−t θ−T , D(ϑ−t ϑ−T ω)), A (ω)   dist φ(T , ϑ−T ω, θ−T , B(ϑ−T ω)), A (ω) < ε,

∀t  TD (ϑ−T ω),

which indicates that A uniformly pullback attracts D. Since B ∈ D, A belongs to D as well since A ⊂ B and D is inclusion-closed. The proof is complete. 2 Remark 3.6. Note that, in Theorem 3.5, K unnecessarily belongs to D. In Theorem 4.16 we will prove that a compact uniformly D-attracting set K ∈ D alone is a sufficient (and also necessary) condition for the existence of the uniform attractor. Remark 3.7. In applications, the symbol space  is often defined as the closed hull of the nonautonomous forcing. Theorem 3.5 (and also Theorem 4.16) indicate that, under the required conditions, the uniform attractor is actually determined by the hull without the closure. See also the discussion before Proposition 2.7.

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3.2. Alternative dynamical compactnesses Theorem 3.5 implies a direct relationship between uniform attracting sets and uniform attractors. However, the existence of a compact uniformly attracting set K is often nontrivial to establish. Therefore, several dynamical notations of compactness were introduced in attractor theory, such as asymptotic compactness, pullback omega-limit compactness, flattening and squeezing properties [35,30,23,16], etc. Now we introduce analogous concepts in the context of uniform attractors, and show that these compactnesses of an NRDS will ensure the omega-limit set of a uniformly D-absorbing set to be a compact uniformly D-attracting set. Definition 3.8. An NRDS φ on a Banach space (X,  · ) is called uniformly D-(pullback) flattening if for each D ∈ D, ε > 0, ω ∈  there exist a T0 = T0 (D, ε, ω) > 0 and a finite-dimensional subspace Xε of X such that (i) ∪tT0 Pε φ(t, ϑ−t ω, , D(ϑ−t ω)) is bounded, and (ii) (I − Pε ) ∪tT0 φ(t, ϑ−t ω, , D(ϑ−t ω)) < ε, where Pε : X → Xε is a bounded projection. Definition 3.9. An NRDS φ on a Banach space (X,  ·) is called uniformly D-(pullback) omegalimit compact if for each D ∈ D, ε > 0, ω ∈  there exists a T1 = T1 (D, ε, ω) > 0 such that ⎛ κ⎝



⎞ φ(t, ϑ−t ω, , D(ϑ−t ω))⎠ < ε,

tT1

where κ denotes the Kuratowski measure [35] of noncompactness of sets defined as   κ(B) = inf δ : B has a finite cover by balls of X of diameter less than δ ,

∀B ⊂ X.

Definition 3.10. An NRDS φ on a Banach space (X,  · ) is called uniformly D-(pullback) asymptotically compact if for each D ∈ D, ω ∈  and any sequences 0 < tn → ∞ and xk ∈ D(ϑ−tn ω) the set {φ(tn , ϑ−tn ω, , xk )} is precompact in X. Theorem 3.11. Suppose that X is a uniformly convex Banach space (particularly, a Hilbert space). The following dynamical compactness properties of an NRDS φ on X are equivalent: (i) uniformly D-(pullback) flattening; (ii) uniform D-(pullback) omega-limit compactness; (iii) uniform D-(pullback) asymptotically compactness, where the uniformly convex property of X is only for the relation (iii)⇒(i). Moreover, each of these dynamical compactnesses implies that the omega-limit set W(·, , B) of a D-uniformly absorbing set B ∈ D is a compact D-uniformly attracting random set. Proof. Similar to, e.g., [30, Theorems 4.5 & 4.6], or [16, Section 2].

2

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The above theorem implies that these dynamical compactnesses could replace the requirement of a compact uniformly D-attracting set in Theorem 3.5, since they are stronger under suitable conditions. 4. Comparison on different attractors for NRDS In this section, we study the relationship between uniform and cocycle attractors for NRDS. To this end, we first introduce a dynamical system named skew-product cocycle generated by an NRDS and define a proper random attractor with proper attraction universe for the skew-product cocycle as a bridge. 4.1. Proper random attractor for skew-product cocycle on extended phase space Denote by (X, dX ) the extended phase space  × X, that is, χ ∈ X if and only if it has the form χ = {σ } × {x} for some σ ∈  and x ∈ X, endowed with the skew-product metric given by dX (χ1 , χ2 ) = d (σ1 , σ2 ) + dX (x1 , x2 ),

∀χj = {σj } × {xj } ∈ X.

Clearly, any set B in X has the form B = ∪σ ∈ {σ } × B(σ ), where each B(σ ) is a (possibly empty) subset of X called the σ -section of B. Denote by Pσ the mapping from each B ⊂ X to its σ -section, i.e., Pσ B := B(σ ), and let PX B =



  Pσ B = x ∈ X : there is some σ ∈  such that {σ } × {x} ∈ B .

σ ∈

Then PX is the projection from X to X. Denote by P the projection from X to . Given an NRDS φ, define a mapping π : R+ ×  × X → X by   π t, ω, {σ } × {x} = {θt σ } × {φ(t, ω, σ, x)}.

(4.1)

Then the mapping π is a (random) cocycle, namely, satisfying • π is (B(R+ ) × F × B(X), B(X))-measurable; • π(0, ω, χ) = χ , ∀ω ∈ , χ ∈ X; • the cocycle property π(t + s, ω, χ) = π(t, ϑs ω, π(s, ω, χ)) for each t, s ∈ R+ , ω ∈ , χ ∈ X. The cocycle π is called the skew-product cocycle generated by φ (and θ ). Note that π is continuous, namely, the mapping χ → π(·, ·, χ) is continuous in X, if and only if φ is continuous in both  and X. Very often, we write π(t, ω, χ) as π(t, ω)χ for convenience. 4.1.1. Proper random sets and proper random attractor For the cocycle π , a particular (autonomous) RDS, it is sensible to study its long-time behavior in terms of random attractors. In order to serve uniform attractors, we define proper random sets in X and then define a proper random attractor for π .

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Definition 4.1. A set-valued mapping B(·) :  → 2X \ ∅ is called a random set in X if for each χ ∈ X the mapping ω → distX (χ, B(ω)) is (F , B(R+ ))-measurable. If, moreover, B satisfies that Pσ (B(ω)) = ∅

for each σ ∈ , ω ∈ ,

(4.2)

and that PX (B) ∈ D,

(4.3)

then it is said to be proper. Note that condition (4.2) is equivalent to that P (B(ω)) ≡ 

for all ω ∈ ,

(4.4)

which implies that the stochastic perturbation happens only to the X-component. Let   DX = B : B is a proper random set in X . Example elements of DX are random sets in the form  × D = { × D(ω)}ω∈ with D ∈ D. Now we define the proper random attractor, which pullback attracts proper random sets in X. Definition 4.2. A random set A ∈ DX is called a DX -random attractor of the skew product cocycle π if (i) A is compact; (ii) A is DX -pullback attracting, that is, lim distX (π(t, ϑ−t ω, D(ϑ−t ω)), A(ω)) = 0,

t→∞

∀ω ∈ , D ∈ DX ;

(iii) A is invariant under π , that is, π(t, ω, A(ω)) = A(ϑt ω),

∀t ∈ R+ , ω ∈ .

Note that, since A ∈ DX , A attracts itself by definition. Moreover, by the invariance property, it is the minimal compact random set in X satisfying (ii). Random attractors for (autonomous) cocycles have been relatively much studied during recent years, see for instance [20,26,3]. But due to the special setting of the attraction universe DX , no results in the literature could be directly applied, since the collection DX is even not inclusionclosed due to the requirement (4.2). Hence, we must be careful when proving the existence of a DX -random attractor, see Proposition 4.3.

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4.1.2. Relationship between uniform attractor for φ and proper random attractor for π Now we are interested in the relationship between the uniform attractor A of an NRDS φ and the DX -random attractor A of the corresponding skew-product cocycle π . This helps understand the random uniform attractor as implied by the following results. Proposition 4.3. If the NRDS φ is continuous in both  and X and has a D-uniform attractor A , then the (continuous) skew product cocycle π generated by φ has a DX -random attractor (given by (4.6) below). Proof. We first claim that the extended set  × A is a compact DX -pullback attracting set of π belonging to DX . Indeed, it is clearly a random set in X and  × A ∈ DX as Pσ ( × A ) ≡ A ∈ D for each σ ∈ ; the compactness follows from that of A directly; for any D ∈ DX , by the uniform attraction of A we have  distX π(t, ϑ−t ω)D(ϑ−t ω),  × A (ω)   = distX π(t, ϑ−t ω) {σ } × Pσ D(ϑ−t ω),  × A (ω) = distX



σ ∈

{θt σ } × φ(t, ϑ−t ω, σ, Pσ D(ϑ−t ω)),  × A (ω)

(4.5)

σ ∈

  distX  × (∪σ ∈ φ(t, ϑ−t ω, σ, PX D(ϑ−t ω)),  × A (ω)  = sup distX φ(t, ϑ−t ω, σ, PX D(ϑ−t ω)), A (ω) → 0, as t → ∞, σ ∈

and thereby  × A is DX -pullback attracting. Now we construct the DX -random attractor. Since D is neighborhood closed, there exists an ε > 0 such that the closed ε-neighborhood Nε (A ) of A belongs to D. Set K =  × Nε (A ). Then K ∈ DX is a closed DX -absorbing set of π . Consider the omega-limit set of K (under π ) given by K (ω) =



π(t, ϑ−t ω, K(ϑ−t ω)),

∀ω ∈ .

(4.6)

s0 ts

Following the proof of [41, Lemmas 2.17 & 2.21] and [42, Theorem 2.14] we know that K is a (non-empty) compact random set in X which is invariant and DX -pullback attracting (under π ). In order to show that K is the DX -random attractor of π , we need to prove further that (4.2) and (4.3) hold for K . For each σ ∈  and ω ∈ , since the uniform attractor A pullback attracts PX (K), we have dist(φ(n, ϑ−n ω, θ−n σ, PX (K(ϑ−n ω))), A (ω)) → 0,

as n → ∞.

Hence, take any sequence xn ∈ Pθ−n σ K(ϑ−n ω), by the compactness of A (ω) there exists a y ∈ A (ω) such that n→∞

φ(n, ϑ−n ω, θ−n σ, xn ) −−−−→ y

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holds in a subsequence sense. This means that π(n, ϑ−n ω, {θ−n σ } × {xn }) → {σ } × {y},

as n → ∞,

and thereby {σ } × {y} ∈ K (ω). Hence, y ∈ Pσ (K (ω)) and (4.2) holds. Take arbitrarily a y ∈ X, and recall that K =  × Nε (A ). Then for each ω fixed, y ∈ PX (K (ω)), i.e. {σ } × {y} ∈ K (ω) for some σ ∈ , if and only if there exist sequences 0 < tn → ∞ and σn ∈ , xn ∈ Nε (A (ϑ−tn ω)) such that π(tn , ϑ−tn ω, {θ−tn σn } × {xn })) → {σ } × {y}, or in other words by (4.1), that σn → σ and φ(tn , ϑ−tn ω, θ−tn σn , xn )) → y, which is equivalent to that y ∈ W(ω, , Nε (A )) by Proposition 3.1. Hence, we conclude that PX (K (ω)) = W(ω, , Nε (A )),

∀ω ∈ .

Note that W(·, , Nε (A )) is a compact random set in X proved by Lemma 3.4, and that W(·, , Nε (A )) ⊂ A (since it is the minimal closed random set uniformly attracting Nε (A ) by Lemma 3.4). Therefore, we conclude that PX (K (·)) = W(·, , Nε (A )) ∈ D by the inclusionclosed property of D. Hence, (4.3) holds. The proof is complete. 2 Proposition 4.4. Let φ be an NRDS. If the random cocycle π generated by φ has a DX -random attractor A ∈ DX , then the random set A := PX A is the D-random uniform attractor for φ. Proof. The compactness and measurability of A follows from A directly. Let us prove uniformly attracting property of A . Notice that, for each x ∈ X, σ ∈ , ω ∈  and t  0, we have distX (x, A (ω)) = inf distX (x, Pσ A(ω)) σ ∈  A(ω)) + d (σ, σ )  inf (x, P dist X σ σ ∈  = distX {σ } × {x}, ∪σ ∈ {σ } × Pσ A(ω) . Hence, for any D ∈ D, since D with D(ω) :=  × D(ω) belongs to DX and thereby attracted by A, we have  sup distX φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω)), A (ω) σ ∈

  sup distX {σ } × φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω)), ∪σ ∈ {σ } × Pσ A(ω) σ ∈

   = sup distX π t, ϑ−t ω, {θ−t σ } × D(ϑ−t ω) , A(ω) σ ∈

    = distX π t, ϑ−t ω, D(ϑ−t ω) , A(ω) → 0,

as t → ∞,

where the uniform attraction of A follows. To see the minimal property, we assume A is another closed uniformly attracting set of φ. Then, in view of (4.5), it is clear that  × A is a closed pullback attracting set of π . On the other hand, the pullback attractor A is the minimal closed pullback attracting set of π since A ∈ DX is invariant and pullback attracts itself. Therefore, it holds A ⊂  × A , and thereby A = PX A ⊂ A ; the minimal property follows. 2

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Corollary 4.5. Suppose that φ is an NRDS continuous in both  and X. Then the D-random uniform attractor A of the NRDS exists if and only if so does the DX -random attractor A of the associated skew product cocycle. Moreover, it holds that A (ω) = PX A(ω),

∀ω ∈ .

4.2. Cocycle attractors for NRDS In this section, we first study the relationship between cocycle attractor for any NRDS and proper random attractor for the generated skew-product cocycle, and then we conclude the relationship between uniform and cocycle attractors. 4.2.1. Cocycle attractors with autonomous attraction universes Now we introduce several basic concepts and known results related to cocycle attractors with autonomous attraction universes. Definition 4.6. A two parameterized mapping A:  ×  → 2X \ ∅, (σ, ω) → Aσ (ω) is called a non-autonomous random set in X if, for each σ ∈ , Aσ (·) is measurable in the sense of Definition 2.2. If for each σ ∈ , Aσ is a compact/bounded random set, then A is called compact/bounded. Definition 4.7. A non-autonomous random set B = {Bσ (·)}σ ∈ is called a D-pullback attracting set of an NRDS φ if  lim dist φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω)), Bσ (ω) = 0, ∀D ∈ D, σ ∈ , ω ∈ . t→∞

Definition 4.8. A non-autonomous random set A = {Aσ (·)}σ ∈ is called a D-(random) cocycle attractor of an NRDS φ if (1) every Aσ (·) is a compact random set in X; (2) A is D-pullback attracting; (3) A is invariant under φ, that is, φ(t, ω, σ, Aσ (ω)) = Aθt σ (ϑt ω),

∀t ∈ R+ ;

(4) A is the minimal among all the compact non-autonomous random sets in X satisfying (2). It is worth emphasizing that random cocycle attractors without minimal property (4) are not unique due to the fact that the attractor does not belong to D in general. This is a general difference from these random attractors attracting non-autonomous random sets, which have drawn much attention in recent years, see [41,42,34,25], etc. The relationship between cocycle attractors with autonomous and non-autonomous attraction universes was studied in [22]. Lemma 4.9. [22] Suppose that φ is an NRDS continuous in X. If there exists a compact uniformly D-pullback attracting set K ∈ D, then φ has a unique D-cocycle attractor A = {Aσ (·)}σ ∈ given by Aσ (·) = W(·, σ, K),

∀σ ∈ .

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Lemma 4.10. [22] Suppose that A = {Aσ (ω)} is the D-cocycle attractor of an NRDS φ. Then (i) if φ is continuous in both  and X, and there is a compact random set D ∈ D such that ∪σ ∈ Aσ (·) ⊆ D(·), then A is upper semi-continuous in , namely, dist(Aσ (ω), Aσ0 (ω)) → 0,

whenever σ → σ0 in ;

(ii) conversely, if A is upper semi-continuous in , then ∪σ ∈ Aσ (ω) is compact for each ω ∈ . In Lemma 4.10 (ii), it is unclear whether the mapping ω → ∪σ ∈ Aσ (ω) is measurable or not. However, it will be shown that, if the NRDS φ is continuous in both  and X and has a D-random uniform attractor, then ∪σ ∈ Aσ (·) is measurable since it is actually the uniform attractor itself. Definition 4.11. A mapping ξ :  × R → X is called a (σ -driven) complete trajectory of an NRDS φ if ξ(ϑt ω, t) = φ(t − s, ϑs ω, θs σ, ξ(ϑs ω, s)) for each t  s and ω ∈ . If there exists a random set B ∈ D such that ∪t∈R ξ(·, t) ⊂ B(·), then ξ is called a D-complete trajectory. Lemma 4.12. [22] Suppose that A = {Aσ (ω)} is the D-random cocycle attractor of φ and, moreover, there is a random set B ∈ D such that ∪σ ∈ Aσ (·) ⊂ B(·). Then   Aθt σ (ϑt ω) = ξ(ϑt ω, t) : ξ is a σ -driven D-complete trajectory of φ , ∀t ∈ R, σ ∈ , ω ∈ . 4.2.2. Relationship between cocycle and uniform attractors To see the relationship between D-random cocycle attractors and D-random uniform attractors, let us show first the relationship between D-random cocycle attractors and DX -random attractors of the corresponding skew product cocycle π . For any proper random set A in X, i.e., A ∈ DX , we write {Pσ (A(ω))}σ ∈,ω∈ as {Pσ (A)}σ ∈ . Note that it is unclear whether each ω → Pσ (A(ω)) is a random mapping or not, but Proposition 4.14 will indicate that it is if A is the DX -random attractor of the skew-product cocycle π , provided φ is continuous in X. Proposition 4.13. Given an NRDS φ, suppose that the skew-product cocycle π generated by φ has the DX -random attractor A. Then the set {Pσ (A)}σ ∈ is invariant under the NRDS φ, i.e. Pθt σ (A(ϑt ω)) = φ(t, ω, σ, Pσ (A(ω))),

for all t  0, σ ∈ , ω ∈ ,

and is D-pullback attracting under φ in the sense that lim dist(φ(t, ϑ−t ω, θ−t σ, D(ϑ−t ω)), Pσ (A(ω))) = 0,

t→∞

∀ω ∈ , σ ∈ , D ∈ D.

(4.7)

Proof. We first prove the invariance. Let σ ∈ , ω ∈  and t ∈ R+ be arbitrarily fixed, and y ∈ Pθt σ (A(ϑt ω)), which implies that {θt σ } × {y} ∈ A(ϑt ω). Then by the invariance of A that A(ϑt ω) = π(t, ω)A(ω), there exists a {σ } × {x} ∈ A(ω) such that {θt σ } × {y} = π(t, ω){σ } × {x} = {θt σ } × {φ(t, ω, σ , x)}, which indicates that σ = σ and y = φ(t, ω, σ, x) with x ∈ Pσ (A(ω)). Hence,

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Pθt σ (A(ϑt ω)) ⊂ φ(t, ω, σ, Pσ (A(ω))). On the other hand, since {θt σ } × φ(t, ω, σ, Pσ (A(ω))) = π(t, ω, {σ } × Pσ A(ω))



⊂ π(t, ω)A(ω) = A(ϑt ω) =

{σ } × Pσ (A(ϑt ω)),

σ ∈

we have φ(t, ω, σ, Pσ (A(ω))) ⊂ Pθt σ (A(ϑt ω)). Hence, the invariance is clear. Next, we prove (4.7) by contradiction. If it is not the case, then there are δ > 0, ω ∈ , D ∈ D and sequences tn → ∞ and xn ∈ φ(tn , ϑ−tn ω, θ−tn σ, D(ϑ−tn ω)) such that dist(xn , Pσ (A(ω)))  δ,

∀n ∈ N.

Denote by χn := {σ } × {xn } and D :=  × D ∈ DX . Then by the pullback attraction of A under π we have   distX (χn , A(ω))  distX {σ } × φ(tn , ϑ−tn ω, θ−tn σ, D(ϑ−tn ω)), A(ω)   = distX π(tn , ϑ−tn ω){θ−tn σ } × D(ϑ−tn ω), A(ω)    distX π(tn , ϑ−tn ω)D(ϑ−tn ω), A(ω) → 0, as n → ∞. Therefore, as A(ω) = ∪σ ∈ {σ } × Pσ (A(ω)) is compact, there is a χ¯ ∈ {σ } × Pσ (A(ω)) for some σ ∈  such that, up to a subsequence, χn = {σ } × {xn } → χ¯ , which implies that σ = σ and xn → Pσ χ¯ ∈ Pσ (A(ω)); a contradiction.

2

Proposition 4.14. Suppose that φ is an NRDS continuous in X. If the cocycle π generated by φ has a DX -random attractor A, then φ has a D-cocycle attractor A = {Aσ (·)}σ ∈ . Moreover, they have the relation Aσ (ω) = Pσ A(ω),

∀ω ∈ , σ ∈ .

(4.8)

Proof. Proposition 4.4 indicates that the NRDS φ has a D-uniform attractor A = PX A, and also a D-cocycle attractor A by Lemma 4.9. Now we prove the relation (4.8). By the invariance of {Pσ (A)}σ ∈ established in Proposition 4.13 we have  dist(Pσ (A(ω)), Aσ (ω)) = dist φ(t, ϑ−t ω, θ−t σ, Pθ−t σ (A(ϑ−t ω))), Aσ (ω)   dist φ(t, ϑ−t ω, θ−t σ, A (ϑ−t ω)), Aσ (ω) → 0,

as t → ∞, ∀ω ∈ ,

since A pullback attracts A . Hence, Pσ (A) ⊂ Aσ for each σ ∈ . Let us prove the converse inclusion. Since A is a D-cocycle attractor, for each σ ∈ , Aσ is a compact random set in X. Hence, the mapping ω → ∪j ∈N Aθ−j σ (ω) is a closed random set in

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X by Lemma 2.3. Moreover, by the minimal property of A, ∪j ∈N Aθ−j σ is smaller than A and thereby belongs to D. Hence,  dist(Aσ (ω), Pσ (A(ω))) = dist φ(n, ϑ−n ω, θ−n σ, Aθ−n σ (ϑ−n ω)), Pσ (A(ω))     dist φ n, ϑ−n ω, θ−n σ, ∪j ∈N Aθ−j σ (ϑ−n ω) , Pσ (A(ω)) → 0,

as n → ∞, ∀ω ∈ ,

where we have used (4.7). Thus, it follows that Aσ ⊂ Pσ (A) for each σ ∈ . The proof is complete. 2 Since Proposition 4.14 shows the relationship between the DX -random attractor A of π and the cocycle attractor A of φ, while Corollary 4.5 indicates the relationship between the DX -random attractor A of π and the uniform attractor A of φ, we are able to give the relationship between uniform and cocycle attractors of φ as follows. Theorem 4.15. Suppose that φ is an NRDS continuous in both  and X. Then if φ has a D-uniform attractor A , then it also has a D-cocycle attractor A = {Aσ (·)}σ ∈ . Moreover, the two attractors admit the relation A (ω) =



Aσ (ω),

∀ω ∈ .

(4.9)

σ ∈

Proof. The existence of the D-uniform attractor A implies that of the D-cocycle attractor A by Lemma 4.9, and implies that of the DX -random attractor A of the skew-product cocycle generated by φ by Proposition 4.3. Then from Corollary 4.5 and Proposition 4.14 it follows that A (ω) = PX A(ω) =



Pσ A(ω) =

σ ∈

The proof is complete.



Aσ (ω),

∀ω ∈ .

σ ∈

2

4.3. Several corollary results for uniform attractors Now we present several important properties of uniform attractors, making use of the relation (4.9) given by Theorem 4.15. We first strengthen the existence Theorem 3.5 to the following one, in which we characterize the uniform attractor by the omega limit set of arbitrarily a compact uniformly attracting set K instead of that of uniformly absorbing sets. Theorem 4.16. Let be any a dense subset of  and φ an NRDS continuous in both  and X. If K ∈ D is a compact uniformly D-attracting set, then φ has a unique D-uniform attractor A ∈ D given by A (ω) = W(ω, , K) = W(ω, , K),

∀ω ∈ .

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Proof. Since D is neighborhood-closed, there exists an ε > 0 such that the closed ε-neighborhood Nε (K) of K belongs to D. Then since K is a uniformly attracting set, Nε (K) is a closed uniformly absorbing set. By Theorem 3.5 we have A (ω) = W(ω, , Nε (K)). Clearly, A (ω) ⊇ W(ω, , K). Now we prove A (ω) ⊂ W(ω, , K). Notice that W(ω, , A ) = A (ω),

∀ω ∈ .

Indeed, since, by Lemma 3.4, W(·, , A ) is the minimal closed random set which uniformly pullback attracts A , it holds W(ω, , A ) ⊂ A (ω) as A uniformly attracts itself as well; the inverse inclusion follows from W(ω, , A ) =

 

φ(t, ϑ−t ω, θ−t σ, A (ϑ−t ω))

s0 ts σ ∈

=

    φ t, ϑ−t ω, θ−t σ, ∪σ ∈ Aσ (ϑ−t ω)

s0 ts σ ∈

⊇

    φ t, ϑ−t ω, θ−t σ, Aθ−t σ (ϑ−t ω)

s0 ts σ ∈

=

 

Aσ (ω) =

s0 ts σ ∈



A (ω) = A (ω),

s0 ts

where the relation (4.9) and the invariance of the cocycle attractor are employed. Therefore, for K a compact random uniformly attracting set, A (ω) = W(ω, , A ) ⊂ W(ω, , K) as A (ω) ⊂ K(ω) by the minimality of A . The proof is complete. 2 Proposition 4.17. Let φ be an NRDS continuous in both  and X and have a compact uniformly D-pullback attracting set K ∈ DX (hence φ has a D-cocycle attractor A by Lemma 4.9). Suppose that y ∈ X is such that φ(tn , ϑ−tn ω, θ−tn σn , xn ) → y for some sequences tn → ∞, σn ∈  and xn ∈ D(ϑ−tn ω). Then y ∈ Aσ (ω), k→∞

where σ is such that there exists a subsequence σnk −−−→ σ . Proof. By Theorem 4.16 and Proposition 4.3 we know that the existence of a compact uniformly D-pullback attracting set implies the existence of the D-uniform attractor A for φ and that of the DX -random attractor A for the generated skew-product cocycle π . Denote by Dˆ :=  × D. Then Dˆ belongs to DX and is attracted by A under π . Hence, by (4.1) and Proposition 4.14 we have

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   distX {σnk } × φ tnk , ϑ−tnk ω, θ−tnk σnk , xnk , ∪σ ∈ {σ } × Aσ (ω)    = distX π tnk , ϑ−tnk ω, {θ−tnk σnk } × {xnk } , A(ω)  ˆ −tn ω)), A(ω) −k→∞  distX π(tnk , ϑ−tnk ω, D(ϑ −−→ 0. k k→∞

Therefore, from σnk −−−→ σ and φ(tn , ϑ−tn ω, θ−tn σn , xn ) → y it follows that y ∈ Aσ (ω).

2

Proposition 4.18. Suppose that φ is an NRDS continuous in both  and X. If the NRDS φ has a uniform attractor A , then it also has a cocycle attractor A = {Aσ (·)}σ ∈ which is upper semi-continuous in symbols dist(Aσ (ω), Aσ0 (ω)) → 0,

whenever σ → σ0 in ,

and satisfies A (ω) = ∪σ ∈ Aσ (ω) for each ω ∈ . Proof. The proof is concluded by Theorem 4.15 and Lemma 4.10.

2

Proposition 4.19. Suppose that φ is an NRDS continuous in both  and X. If φ has a D-random uniform attractor A , then   A (ϑt ω) = ξ(ϑt ω, t) : ξ is a D-complete trajectory of φ , Proof. The proof is concluded by (4.9) and Lemma 4.12.

∀t ∈ R, ω ∈ .

2

With (4.9) we are able to prove neatly the negative semi-invariance, Proposition 3.5, of a uniform attractor. Proposition 4.20. Suppose that φ is an NRDS continuous in both  and X. Then the uniform attractor A of φ is negatively semi-invariant, that is, A (ϑt ω) ⊆ (t, ω, A (ω)),

∀t ∈ R+ , ω ∈ ,

where (t, ω, x) := ∪σ ∈ φ(t, ω, σ, x). (Note that  is actually a multi-valued RDS, see Proposition 5.4.) Proof. By the relation (4.9) and the invariance of the cocycle attractor, A (ϑt ω) =

 σ ∈

⊆



σ ∈

Aθt σ (ϑt ω) =



φ(t, ω, σ, Aσ (ω))

σ ∈

φ(t, ω, σ, ∪σ ∈ Aσ (ω)) = (t, ω, A (ω)).

2

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5. Uniform attractor as pullback attractor of multi-valued RDS It is known that the theory of multi-valued dynamical systems is often used to deal with differential equations without uniqueness of solutions, see [2,36,40,24] and references therein. In this section we shall show that it is also powerful for the study of uniform attractors, as a uniform attractor for an NRDS could be regarded as the random attractor of a corresponding multi-valued RDS. Let C(X) be the collection of non-empty closed sets in X. Definition 5.1. A set-valued mapping  : R+ ×  × X → C(X) is called a multi-valued RDS if (1)  is (B(R+ ) × F × B(X), B(X))-measurable; (2) (0, ω, x) = x, ∀ω ∈ , x ∈ X; (3) it holds the negative invariance property (t + s, ω, x) ⊂ (t, ϑs ω, (s, ω, x)),

∀t, s ∈ R+ , ω ∈ , x ∈ X.

If it holds in (3) the identity, then  is called a strict multi-valued RDS. Moreover, the multivalued RDS  is said to be upper (or lower) semi-continuous if the mapping (t, ω, ·) is upper (or lower) semi-continuous for each fixed t, ω. If it is both upper and lower semi-continuous, then it is said continuous. Definition 5.2. A random set A (·) ∈ D is called a D-random attractor for a multi-valued RDS  if (1) A is compact; (2) A is D-(pullback) attracting under , namely, for any B ∈ D it holds that dist((t, ϑ−t ω, B(ϑ−t ω)), A (ω)) → 0,

as t → ∞;

(3) A is negatively invariant in the sense that A (ϑt ω) ⊂ (t, ω, A (ω)),

∀t ∈ R+ , ω ∈ .

Remark 5.3. Note that the negative invariance of a D-random attractor implies the minimal property that for any closed random set A which is D-pullback attracting under the multi-valued RDS  it holds A ⊂ A . This can be seen from dist(A (ω), A (ω))  dist((t, ϑ−t ω, A (ϑ−t ω)), A (ω)) → 0,

as t → ∞.

Hence, random attractors for multi-valued RDS must be unique. Several papers can be found on the study of attractors for multi-valued RDS, e.g., [9,8,44,43]. Here we will not go further on the general multi-valued attractor theory, but show a relationship between uniform attractor and multi-valued attractor theories.

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5.1. Uniform attractor for an NRDS is the random attractor for the associated multi-valued RDS In this part, we show that uniform attractors could be studied using multi-valued (autonomous) RDS theory. First we show that any continuous NRDS generates a continuous multi-valued RDS. Proposition 5.4. Suppose that φ is an NRDS on X continuous in both  and X. Then the mapping  on R+ ×  × X defined by (t, ω, x) =



φ(t, ω, σ, x)

(5.1)

σ ∈

is a continuous multi-valued RDS, which is said generated by (single-valued) NRDS φ. Proof. Since the NRDS φ is continuous in  and  is compact,  takes values in C(X), and hence (t, ω, x) =





φ(t, ω, σ, x) =

σ ∈

φ(t, ω, σ, x),

(5.2)

σ ∈

where is an arbitrary countable dense set of . Thus, in view of Lemma 2.3, the measurability of  is clear. The negative invariance of  follows from that of φ and the invariance of . Indeed, (t + s, ω, x) =



φ(t + s, ω, σ, x) ⊂

σ ∈

⊂



φ(t, ϑs ω, θs σ, φ(s, ω, σ, x))

σ ∈

   φ t, ϑs ω, θs σ, ∪σ ∈ φ(s, ω, σ , x)

σ ∈

= (t, ϑs ω, (s, ω, x)),

∀t, s ∈ R+ , ω ∈ , x ∈ X.

Now we prove the continuity by proving the upper and lower semi-continuity, respectively. Let xn → x. First we prove upper semi-continuity by contradiction. Suppose for some ε > 0 it holds that dist((t, ω, xn ), (t, ω, x)) > ε,

∀n ∈ N.

Then by (5.1) we have a sequence σn ∈  such that dist(φ(t, ω, σn , xn ), (t, ω, x)) > ε,

∀n ∈ N.

Noticing that σn → σ for some σ ∈  by the compactness of , up to a subsequence we have φ(t, ω, σn , xn ) → φ(t, ω, σ, x) ∈ (t, ω, x) as φ is continuous in both  and X; a contradiction. To see the lower semi-continuity, it suffices to notice that for each y ∈ (t, ω, x), there is a σ ∈  such that y = φ(t, ω, σ, x) which is approximated by φ(t, ω, σ, xn ) which belongs to (t, ω, xn ). The proof is complete. 2

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Theorem 5.5. Suppose that φ is an NRDS on X which is continuous in both  and X, and that  is the continuous multi-valued RDS generated by φ. Then the random set A is the D-uniform attractor of φ if and only if it is the D-random attractor of . Proof. First it is trivial to observe that, for any B ∈ D the pullback attraction lim dist((t, ϑ−t ω, B(ϑ−t ω)), A (ω)) = 0

t→∞

is equivalent to lim dist(φ(t, ϑ−t ω, , B(ϑ−t ω)), A (ω)) = 0.

t→∞

Hence, A is D-pullback attracting under  if and only it is uniformly D-attracting under φ. If A is the random uniform attractor of φ, then Theorem 3.5 implies the negative invariance of A under , and hence it is the random attractor of . Conversely, if A is the random attractor of , since by Remark 5.3 we have shown the minimal property of A , it is clearly the random uniform attractor of φ. 2 5.2. Random uniform attractor is determined by uniformly attracting deterministic compact sets In this section, we show that uniform attractor for any jointly continuous NRDS φ is determined by uniformly attracting deterministic compact sets. This could be regarded as a nonautonomous generalization of analogous results in [17] for (autonomous) RDS. Let  be the continuous multi-valued RDS generated by φ. The omega-limit set of a compact set D ⊂ X is defined as D (ω) =



(t, ϑ−t ω, D),

∀ω ∈ .

s0 ts

Note that the omega-limit set D of a compact set D for a general multi-valued RDS is not measurable in general. But for the continuous multi-valued RDS  generated by NRDS φ, in view of [20, Theorem 3.11] it is clear that the omega-limit set is measurable at least with respect to the P-completion of F , since  and X are Polish. In this part we shall work in this P-completion sense. Hereafter, for any random sets I and D, we write P{ω ∈  : I (ω) ⊂ D(ω)} simply as P(I ⊂ D). For the ease of description, let   D = K : K is a (deterministic and non-empty) compact subset of X ⊂ D. Now we define the random attractor pullback attracting compact sets. Definition 5.6. A random set A is said to be a D-random attractor of  if (1) A is compact; (2) A is D-(pullback) attracting under , namely, for any K ∈ D it holds that dist((t, ϑ−t ω, K), A(ω)) → 0,

as t → ∞;

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(3) A is negatively invariant under , i.e., A(ϑt ω) ⊂ (t, ω, A(ω)),

∀t ∈ R+ , ω ∈ .

The following lemma is a multi-valued generalization of [17, Proposition 5.2]. Lemma 5.7. Suppose that I is a random set which is negatively invariant under . Then for each compact set D ⊂ X we have P(I ⊂ D)  P(I ⊂ D ). Proof. The proof is achieved similarly to [17, Proposition 5.2] using Poincaré recurrence theorem. 2 Lemma 5.8. Suppose that I is a compact random set which is negatively invariant under . Then for each ε > 0 there exists a compact deterministic set K ⊂ X such that P(I ⊂ K )  1 − ε. Proof. Similar to [17, Corollary 5.4].

2

Corollary 5.9. Suppose that I is a compact random set which is negatively invariant under . If A is a D-random attractor of , then P(I ⊂ A) = 1. Proof. By Lemma 5.8 we see that for each ε > 0 there exists a compact set K ∈ D such that P(I ⊂ K )  1 − ε. On the other hand, since A is the D-random attractor, K ⊂ A for any K ∈ D. Hence, P(I ⊂ A)  P(I ⊂ K )  1 − ε, which completes the proof.

∀ε > 0,

2

The above corollary implies that D-random attractor is P-almost surely unique, as any D-random attractor is negatively invariant. Noticing also that any D-random attractor is also a D-random attractor, we have the following result. Proposition 5.10. Suppose that A is the D-random attractor of  and A is the D-random attractor of , then P(A = A) = 1. Proof. Firstly, by Corollary 5.9 we have P(A ⊂ A) = 1. On the other hand, by Lemma 5.8 we know that for any ε > 0 there exists a compact set K ∈ D such that P(A ⊂ K )  1 − ε. Note that, since D ⊂ D, A is D-pullback attracting as well, which indicates that K ⊂ A for any K ∈ D. Hence, P(A ⊂ A )  1 − ε for each ε > 0 and thereby P(A ⊂ A ) = 1. The proof is complete. 2

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Now we rewrite Proposition 5.10 to indicate that uniform attractors for jointly continuous NRDS are determined by uniformly attracting deterministic compact sets. Proposition 5.11. Suppose φ is an NRDS which is continuous in both  and X with a D-random uniform attractor A , and A is the random uniform attractor which uniformly attracts deterministic compact sets. Then P(A = A) = 1. 6. Application: reaction–diffusion equation with translation-bounded forcing and additive white noise In this section we take a reaction–diffusion equation with both translation-bounded nonautonomous forcing and additive white noise as an example to better illustrate our theoretical analysis. We first prove that the translation bounded condition ensures that the time-translation operators are continuous in time and the symbol space constructed as the hull of the nonautonomous forcing is compact and Polish, and then we show that the NRDS generated by this reaction–diffusion equation has a tempered random uniform attractor. 6.1. Translation bounded function and Polish symbol space   For any domain O ⊂ RN (N ∈ N), let H = L2 (O),  ·  . Suppose that the space L2loc (R; H ) consists of all functions g which maps from R to H and is two-power integrable in Bochner sense on any bounded segment, i.e., t2 g(s)2 ds < ∞ for any bounded [t1 , t2 ] ⊂ R. t1

The space L2loc (R; H ) is endowed with the two-power mean convergence topology on any bounded segment of R, i.e., gn → g in L2loc (R; H ) means that t2 gn (s) − g(s)2 ds → 0,

for any bounded [t1 , t2 ] ⊂ R.

t1

A function g ∈ L2loc (R; H ) is said to be translation compact in L2,w loc (R; H ) if its hull H(g) := 2,w {θt g(·) : t ∈ R} is compact in Lloc (R; H ), where θt g(·) := g(· + t),

∀t ∈ R, g ∈ L2loc (R; H ),

(6.1)

and the closure is in the local weak convergence topology sense, i.e., σn → σ in H(g) if and only if t2 t1



 v(s), σn (s) − σ (s) ds → 0

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for each bounded (t1 , t2 ) ⊂ R and v ∈ L2loc (R; H ). Note that g is translation compact in 2 L2,w loc (R; H ) if and only if it is translation bounded in Lloc (R; H ), as indicated by the following result. Proposition 6.1. [15, Proposition 4.2] Let g ∈ L2loc (R; H ) be translation compact in L2,w loc (R; H ). Then (1) (2) (3) (4)

the translation operator θt defined by (6.1), ∀t ∈ R, is continuous on H(g) in L2,w loc (R; H ); the hull of g is translation invariant H(g) = θt H(g), ∀t ∈ R; any function σ ∈ H(g) is translation compact in L2,w loc (R; H ) and H(σ ) ⊆ H(g); equivalently, g is translation bounded in L2loc (R; H ), i.e., τ η(g) := sup

τ ∈R τ −1

g(s)2 ds < ∞;

(6.2)

(5) for any σ ∈ H(g), η(σ )  η(g). We will need the following bound. Proposition 6.2. Let g ∈ L2loc (R; H ) be translation compact in L2,w loc (R; H ). Then 0 sup

σ ∈H(g) −∞

eλs σ (s)2 ds 

η(g) , 1 − e−λ

(6.3)

where η(g) is the constant given by (6.2). Proof. For each σ ∈ H(g), by Proposition 6.1 (5) and (6.2) we have 0

−(n−1)

 e σ (s) ds = eλs σ (s)2 ds λs

2

n∈N −n

−∞





e

−λ(n−1)

e

−λ(n−1)

n∈N





−(n−1) 

σ (s)2 ds −n

η(σ )

n∈N



n∈N

e−λ(n−1) η(g) =

η(g) . 2 1 − e−λ p

Next proposition indicates that the mapping t → θt g is (R, Lloc (R; Lr (O)))-continuous for p every g ∈ Lloc (R; Lr (O)) with p, r > 1, which ensures the hull of g to be Polish.

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p

Proposition 6.3. Let g ∈ Lloc (R; Lr (O)) with p, r > 1. Then for any bounded interval (T1 , T2 ) ⊂ R we have T2

p

g(· + ε) − g(·)Lr (O) ds → 0,

as ε → 0.

T1

Proof. Without loss of generality, let ε ∈ (−1, 1). Denote by X = Lp (T1 , T2 ; Lr (O)). Then we need to prove that g(· + ε) converges to g strongly in X. Let (p, q) and (r, s) be conjugate indices. Firstly, for any v ∈ C ∞ (T1 − 1, T2 + 1; Ls (O)) we have   T2     (g(s + ε) − g(s))v(s) dxds   T1 O

  T2    T2        g(s + ε) v(s) − v(s + ε) dxds  +  g(s + ε)v(s + ε) − g(s)v(s) dxds    T1 O

T1 O

  T2    T2         g(s + ε)v (ξs ) dxds  +  g(s + ε)v(s + ε) − g(s)v(s) dxds  = |ε| T1 O

T1 O

   k|ε|g(s)Lp (R;Lr ) +  loc

T2 +ε

T2  g(s)v(s) dxds −

T1 +ε O

  ε→0 g(s)v(s) dxds  −−−→ 0,

T1 O

where the last inequality is due to Hölder’s inequality and k is such that

sup

ξ ∈(T1 −1,T2 +1)

v (ξ )Ls (O)

 k. Therefore, g(· + ε) converges weakly to g in X since C ∞ (T1 , T2 ; Ls (O)) is dense in X ∗ = Lq (T1 , T2 ; Ls (O)) (the dual of X). Hence, from  T2 g(· + ε)X =

g(s T1

p + ε)Lr (O) ds

1/p

 T2 +ε 1/p ε→0 p = g(s)Lr (O) ds −−−→ gX T1 +ε

we conclude that g(· + ε) indeed converges to g(·) strongly in X, as weak convergence along with norm convergence implies the strong convergence for uniformly convex X. The proof is complete. 2 In the following, let O ⊂ RN be a bounded domain with smooth boundary, and let  = H(g), the hull of a given translation bounded function g ∈ L2loc (R; H ), endowed with the local weak convergence topology and a group of translation operator {θt }t∈R given by (6.1) acting on . Then  is a compact metric space [15, p. 105]. Moreover, since each translation operator θt defined by (6.1) is clearly continuous on  and the mapping t → θt g is (R, )-continuous

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by Proposition 6.3 (taking p = r = 2), the symbol space  is Polish since the set := {θr g : r is a rational number} is a countable dense subset of . The group {θt }t∈R is a base flow on . 6.2. NRDS generated by the reaction–diffusion equation Consider the following stochastic reaction–diffusion equation with additive scalar white noise du + (λu − u)dt = f (x, u)dt + σ (x, t)dt + h(x)dω,

x ∈ O, t  0,

(6.4)

endowed with the initial and boundary conditions u(x, t)|t=0 = u0 (x),

(6.5)

u(x, t)|∂ O = 0,

where λ > 0 is a constant, σ ∈  and h(x) ∈ W 2,p (O) for some p  2 is the perturbation intensity. The nonlinear term f (x, u) is assumed to satisfy the following standard conditions f (x, s)s  −α1 |s|p + ψ1 (x),

(6.6)

|f (x, s)|  α2 |s|p−1 + ψ2 (x), ∂f (x, s)  α3 , ∂s   ∂f    (x, s)  ψ3 (x), ∂x

(6.7) (6.8) (6.9)

where αj are positive constants, ψ1 ∈ L1 (O) and ψ2 , ψ3 ∈ H (= L2 (O)). Let us define the probability space . Let  = {ω ∈ C(R; R) : ω(0) = 0}, and F the Borel sigma-algebra induced by the compact-open topology of , P the two-sided Wiener measure on (, F). Define the translation operator ϑ on  by ϑt ω = ω(· + t) − ω(t),

∀t ∈ R, ω ∈ .

Then P is ergodic and invariant under ϑ [26]. Denote by 0 z(ω) = −λ

eλτ ω(τ ) dτ,

∀ω ∈ .

(6.10)

−∞

Then z(ω) is a stationary solution of the one-dimensional Ornstein–Uhlenbeck equation dz(ϑt ω) + λz(ϑt ω)dt = dω. ˜ ⊂  of full measure such that z(ϑt ω) is continuous Moreover, there exists a ϑ-invariant subset  ˜ in t for every ω ∈  and the random variable |z(·)| is tempered satisfying

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lim e−εt |z(ϑ−t ω)| = 0,

t→∞

∀ω ∈ ,

(6.11)

˜ and . for each ε > 0 [1]. Hereafter, we will not distinguish  Consider the following conjugate deterministic problem with random coefficients ⎧ dv ⎪ ⎨ dt + λv − v = f (x, v + hz(ϑt ω)) + σ (x, t) + z(ϑt ω)h(x), v(x, t)|t=0 = v0 (x), ⎪ ⎩ v(x, t)|∂ O = 0.

(6.12)

Following a standard method of [39,15] we know, for each initial data v0 ∈ H , (6.12) has a unique solution v(·, ω, σ, v0 ) ∈ C([0, ∞); H ) ∩ L2loc ((0, ∞); V ) with v(0, ω, σ, v0 ) = v0 . Moreover, v is (F, B(H ))-measurable in ω and continuous in σ and v0 . For each t  0, ω ∈ , σ ∈  and u0 ∈ H , set φ(t, ω, σ, u0 ) = v(t, ω, σ, u0 − hz(ω)) + hz(ϑt ω).

(6.13)

Then φ(t, ω, σ, u0 ) is the solution of (6.4) at time t with initial data u0 (at time t = 0) satisfying Definition 2.1. Hence, (4.1) defines an NRDS continuous both in initial data and symbols. For the equation (6.4) we study the tempered uniform and cocycle attractors. Take the universe of tempered random sets in H as the attraction universe D, i.e.,   D = D : D is a bounded random set in H satisfying lim e−λt D(ϑ−t ω)2 = 0, ∀ω ∈  . t→∞

Clearly, the (autonomous) universe D is both inclusion-closed and neighborhood-closed. 6.3. Uniform estimates of solutions Now we establish uniform estimates which are always essential in particular studies. The calculation techniques are standard in spirit. But noticing that most, if not all, recent publications on NRDS took the real line as the symbol space, we prove in details since our symbol space is the hull H(g). Lemma 6.4. For each D ∈ D and ω ∈ , there exists a time T = T (D, ω) > 1 such that, for every σ ∈ , t v(t, ϑ−t ω, θ−t σ, v0 ) +

eλ(s−t) v(s, ϑ−t ω, θ−t σ, v0 )2 ds

2

0

t +

t e

λ(s−t)

∇v(s) ds +

0

0

0 c −∞

p

eλ(s−t) vp ds

2

0 eλs σ (s)2 ds + c

eλs |z(ϑs ω)|p ds + c

−∞

holds uniformly in v0 ∈ D and t  T , where c is an absolute positive constant.

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Proof. Take the inner product of (6.12) with v in H to obtain 1 d v(t, ω, σ, v0 )2 + λv2 + ∇v2 2 dt   = v f (x, v + hz(ϑt ω)) + σ (t) + z(ϑt ω)h dx.

(6.14)

By (6.6), (6.7) and Young’s inequality we have  vf (x, v + hz(ϑt ω))dx 



=

(v + hz(ϑt ω))f (x, v + hz(ϑt ω))dx −

 −α1 v

p + hz(ϑt ω)p



 +

|hz(ϑt ω)||v + hz(ω)|p−1 dx

ψ1 (x)dx + α2

 |hz(ϑt ω)||ψ2 | dx  −

+

hz(ϑt ω)f (x, v + hz(ϑt ω))dx

α1 p vp + c(|z(ϑt ω)|p + 1), 2

(6.15)

where and hereafter c denotes an absolute positive constant which may change its value when necessary. As 

  λ 4 v σ (t) + z(ϑt ω)h dx  v2 + σ 2 + c(|z(ϑt ω)|p + 1), 4 λ

(6.16)

by (6.14)–(6.16) we conclude that d 3 8 p v2 + λv2 + ∇v2 + α1 vp  σ 2 + c(|z(ϑt ω)|p + 1). dt 2 λ

(6.17)

Multiply (6.17) by eλt and then integrate over (0, t) to obtain λ v(t, ω, σ, v0 ) + 2

t eλ(s−t) v(s, ω, σ, v0 )2 ds

2

0

t +

t e

λ(s−t)

∇v(s) ds + α1

0

e

p

eλ(s−t) vp ds

2

(6.18)

0 −λt

8 v0  + λ

t

2

t e

λ(s−t)

σ (s) ds + c

eλ(s−t) (|z(ϑs ω)|p + 1) ds.

2

0

Replacing ω and σ with ϑ−t ω and θ−t σ , we have

0

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λ v(t, ϑ−t ω, θ−t σ, v0 ) + 2

t eλ(s−t) v(s, ϑ−t ω, θ−t σ, v0 )2 ds

2

0

t +

t e

∇v(s) ds + α1

0

p

eλ(s−t) vp ds

2

λ(s−t)

(6.19)

0

e

−λt

8 v0  + λ

t

2

t e

λ(s−t)

σ (s − t) ds + c

eλ(s−t) (|z(ϑs−t ω)|p + 1) ds.

2

0

0

Since v0 ∈ D(ϑ−t ω), by the tempered condition of D there exists a T = T (ω, D) > 1 such that

λ v(t, ϑ−t ω, θ−t σ, v0 ) + 2

t eλ(s−t) v(s, ϑ−t ω, θ−t σ, v0 )2 ds

2

0

t +

t e

λ(s−t)

∇v(s) ds + α1

0

p

eλ(s−t) vp ds

2

0

1+

8 λ

t eλ(s−t) σ (s − t)2 ds + c 0

8  λ

(6.20)

t eλ(s−t) (|z(ϑs−t ω)|p + 1)ds 0

0

0 e σ (s) ds + c λs

eλs |z(ϑs ω)|p ds + c,

2

−∞

∀t  T ,

−∞

by which the proof is completed. 2 Lemma 6.5. For each D ∈ D and ω ∈ , there exists a time T > 1 given by Lemma 6.4 such that, for every σ ∈ , t

t ∇v(t, ϑ−t ω, θ−t σ, v0 ) ds + 2

t−1

p

v(t, ϑ−t ω, θ−t σ, v0 )p ds

t−1

0 c

0 eλs σ (s)2 ds + c

−∞

eλs |z(ϑs ω)|p ds + c

−∞

holds uniformly in v0 ∈ D and t  T , where c is an absolute positive constant. Proof. For any t  T , by Lemma 6.4,

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t

t ∇v(t, ϑ−t ω, θ−t σ, v0 ) ds +

p

v(t, ϑ−t ω, θ−t σ, v0 )p ds

2

t−1

39

t−1

t e

λ

t e

λ(s−t)

∇v(t, ϑ−t ω, θ−t σ, v0 ) ds + e 2

t−1

t−1

t

t eλ(s−t) ∇v(t, ϑ−t ω, θ−t σ, v0 )2 ds + eλ

 eλ 0

p

eλ(s−t) v(t, ϑ−t ω, θ−t σ, v0 )p ds 0

0 c

p

eλ(s−t) v(t, ϑ−t ω, θ−t σ, v0 )p ds

λ

0 eλs σ (s)2 ds + c

−∞

eλs |z(ϑs ω)|p ds + c.

2

−∞

Lemma 6.6. For each D ∈ D and ω ∈ , there exists a time T > 1 (given by Lemma 6.4) such that, for every σ ∈ , 0 ∇v(t, ϑ−t ω, θ−t σ, v0 )  c

0 e σ (s) ds + c

2

λs

2

−∞

eλs |z(ϑs ω)|p ds + c

−∞

holds uniformly in v0 ∈ D and t  T , where c is an absolute positive constant. Proof. Multiply (6.12) by −v and then integrate over O to obtain 1 d ∇v(t, ω, σ, v0 )2 + λ∇v2 + v2 2 dt   = − v f (x, v + hz(ϑt ω)) + σ (t) + z(ϑt ω)h dx.

(6.21)

By (6.7)–(6.9) we have  −

vf (x, v + hz(ϑt ω))dx  =−

 uf (x, u) dx +

hz(ϑt ω)f (x, u) dx

 ∂f ∂f (x, u) + |∇u|2 (x, u) dx + hz(ϑt ω)f (x, u)dx ∂x ∂u    ∇uψ3  + α3 ∇u2 + |hz(ϑt ω)| α2 |u|p−1 + ψ2 dx =

 

∇u

p

 c∇v2 + cvp + c|z(ϑt ω)|p + c, where we have used the notation u = v + hz(ϑt ω). Since

(6.22)

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40



  v σ (t) + z(ϑt ω)h dx  v2 + σ 2 + c|z(ϑt ω)|2 ,

−

(6.23)

from (6.21)–(6.23) it follows that d p ∇v(t, ω, σ, v0 )2  c∇v2 + cvp + c|z(ϑt ω)|p + 2σ 2 + c. dt

(6.24)

For each t  T , let s ∈ (t − 1, t). Integrate (6.24) over (s, t) to obtain ∇v(t, ω, σ, v0 ) − ∇v(s, ω, σ, v0 )  c 2

2

t 

p ∇v(τ )2 + v(τ )p dτ

t−1

+c

t 

(6.25)



|z(ϑτ ω)|p + σ (τ )2 dτ + c.

t−1

Then integrating (6.25) with respect to s over (t − 1, t) and replacing ω and σ by ϑ−t ω and θ−t σ , respectively, we have ∇v(t, ϑ−t ω, θ−t σ, v0 )2 t

t

c

p

∇v(s, ϑ−t ω, θ−t σ, v0 ) ds + c

v(s, ϑ−t ω, θ−t σ, v0 )p ds

2

t−1

0 +c

t−1



|z(ϑτ ω)|p + σ (τ )2 dτ + c.

−1

Hence, by Lemma 6.5 we conclude that 0 ∇v(t, ϑ−t ω, θ−t σ, v0 )  c

0 e σ (s) ds + c

2

2

λs

−∞

where we have used the relation

0

−1 σ (s)

eλs |z(ϑs ω)|p ds + c,

−∞ 2 ds

 eλ

0

−1 e

λs σ (s)2 ds.

2

By (4.1) and Lemma 6.6 we have the following uniform estimate for solutions of (6.4). Corollary 6.7. For each D ∈ D and ω ∈ , there exist a time T = T (D, ω) > 1 and an absolute positive constant L such that, for every σ ∈ , 0 ∇φ(t, ϑ−t ω, θ−t σ, u0 )  L 2

−∞

holds uniformly in u0 ∈ D and t  T .

0 e σ (s) ds + L λs

2

−∞

eλs |z(ϑs ω)|p ds + L|z(ω)| + L

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6.4. Uniform and cocycle attractors For each ω ∈  and σ ∈ , let us define 

Lη(g) E(ω) = u ∈ V : ∇u  +L 1 − e−λ



0

2

e |z(ϑs ω)| ds + L|z(ω)| + L , λs

p

∀ω ∈ .

−∞

(6.26)

where L and η(g) are positive constants determined by Corollary 6.7 and g (6.2), respectively, and |z(·)| is the tempered random variable given by (6.10). Then by Sobolev compactness embeddings, E is a compact random set in H belonging to D, and, furthermore, Corollary 6.7 and (6.3) indicate that it is a uniformly D-pullback absorbing set for the NRDS φ. Now we show the existence of D-cocycle and D-uniform attractors, and also more properties discussed in theoretical parts. First, let us see the cocycle attractor. Theorem 6.8. The NRDS φ generated by the stochastic reaction–diffusion equation (6.4) has a unique D-random cocycle attractor A = {Aσ (·)}σ ∈ given by Aσ (ω) = W(ω, σ, E),

∀σ ∈ , ω ∈ ,

(6.27)

where E is the random set defined by (6.26). Moreover, the cocycle attractor A has the following properties: (I) it is upper semi-continuous in symbols, i.e., for each ω ∈ , distH (Aσ (ω), Aσ0 (ω)) → 0,

whenever σ → σ0 in ;

(6.28)

(II) it is uniformly compact, i.e., for each ω ∈ , the set ∪σ ∈ Aσ (ω) is compact in H ; (III) it is characterized by D-complete trajectories of φ, i.e.,   Aσ (ω) = ξ(ω, 0) : ξ is a σ -driven D-complete trajectory of φ , ∀σ ∈ , ω ∈ . Proof. As E ∈ D is a compact uniformly D-pullback absorbing set for the NRDS φ, Lemma 4.9 indicates that the NRDS φ has a unique D-random cocycle attractor A with characterizations (6.27). Since ∪σ ∈ Aσ (ω) ⊂ E(ω) and E(ω) is compact in H , A is upper semi-continuous and ∪σ ∈ Aσ (ω) is compact in H by Lemma 4.10. Property (III) follows from Lemma 4.12. The proof is complete. 2 Now, by Theorem 3.5, Theorem 4.15, Proposition 2.8 and Proposition 5.11, we are able to strengthen Theorem 6.8 to the following result. Theorem 6.9. The NRDS φ generated by the stochastic reaction–diffusion equation (6.4) has a D-uniform attractor A ∈ D and a D-cocycle attractor A satisfying

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A (ω) = W(ω, , E)  = Aσ (ω)

(6.29)

σ ∈

  = ξ(ω, 0) : ξ is a D-complete trajectory of φ , ∀ω ∈ , where E is the random set defined by (6.26). Moreover, the uniform attractor A is forwardattracting in probability and is determined by uniformly attracting deterministic compact sets in H , and the cocycle attractor A is upper semi-continuous in symbols satisfying (6.28). Remark 6.10. (i) Observe that (6.29) indicates that the mapping ω → ∪σ ∈ Aσ (ω) is a compact random set as it is in fact the uniform attractor, which improves Theorem 6.8 (II) where the measurability is not proved. (ii) From (6.27) and (6.29) it follows that W(ω, , E) =



W(ω, σ, E),

σ ∈

for E given by (6.26). But note that for a general random set, we only have the ⊃ inclusion, see (3.2). Acknowledgments The authors thank the anonymous referee for his/her carefully reading the script and helpful comments. H. Cui was supported by State Scholarship Fund 201506990049 and NSFC Grant 11571283. J.A. Langa was partially supported by Junta de Andalucía under Proyecto de Excelencia FQM-1492, Brazilian–European partnership in Dynamical Systems (BREUDS) from the FP7-IRSES grant of the European Union and FEDER Ministerio de Economía y Competitividad grant MTM2015-63723-P. References [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. [2] J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997) 475–502. [3] P.W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction–diffusion equations on unbounded domains, J. Differential Equations 246 (2009) 845–869. [4] M. Bortolan, A. Carvalho, J. Langa, Structure of attractors for skew product semiflows, J. Differential Equations 257 (2014) 490–522. [5] M.C. Bortolan, T. Caraballo, A.N. Carvalho, J. Langa, Skew product semiflows and Morse decomposition, J. Differential Equations 255 (2013) 2436–2462. [6] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuß, J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst. 21 (2008) 415–443. [7] T. Caraballo, J. Langa, V. Melnik, J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal. 11 (2003) 153–201. [8] T. Caraballo, J. Langa, J. Valero, Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, J. Math. Anal. Appl. 260 (2001) 602–622. [9] T. Caraballo, J. Langa, J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal. 48 (2002) 805–829.

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