Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} is established by the squeezing approach and that in H01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0^1$$\end{document} by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.


Introduction
Attractor theory is known as a useful tool in the study of infinite-dimensional dynamical systems, especially in numerical simulations and computations. Roughly, if a system has an attractor, any solution trajectory of the system can be tracked by trajectories within the attractor, while, in the meanwhile, if the attractor has finite dimension, finite degrees of freedom are expected to fully determine the asymptotic behavior of the system, though the phase space of the system is infinite dimensional. This is known as a finite-dimensional reduction of infinite-dimensional dynamical systems (Temam 1997;Robinson 2011). In order to describe the long-time behavior of infinitedimensional dynamical systems, one often studies the attractors associated to them. Depending on the setting a problem is proposed, a number of typical attractors have been introduced and extensively studied: global attractors (Temam 1997;Robinson 2001), exponential attractors (Eden et al. 1994), pullback/cocycle attractors (Kloeden and Rasmussen 2011;, uniform attractors (Chepyzhov and Vishik 2002;Bortolan et al. 2020), etc., describing in their own way the asymptotic dynamics of the system under consideration.
Ever since the work of Crauel and Flandoli (1994), the attractor theory has been extended to random dynamical systems for which stochastic perturbations are taken into account (Arnold 1998;Chueshov 2002). Particularly for a non-autonomous random dynamical system (abbrev. NRDS), i.e., a random dynamical system with in addition time-dependent terms (e.g., with a time-dependent forcing field), pullback random attractors have been extensively studied, see, e.g., Wang (2012Wang ( , 2014,  and also Caraballo and Sonner (2017) for pullback exponential attractors. However, the non-autonomous feature of the system prevents one to learn from these pullback attractors about the forward dynamics of the underlying system, and this is the motivation of our previous work , where a random uniform attractor was developed, which provides a certain description of forward dynamics of the system by the property of uniformly forward attracting in probability.
In this paper, we study the random uniform attractors on the conditions that ensure a random uniform attractor to have finite fractal dimension. Thanks to the pioneering works of Mallet-Paret (1976) and Mañé (1981), it is well understood that estimating the fractal dimension of an attractor provides the information that the attractor can be embedded into an Euclidean space R k for some k ∈ N, and this embedding is shown to be linear with a Hölder continuous inverse, see, e.g., Robinson (2011). Hence, estimating the fractal dimension of an attractor is useful in the finite-dimensional reduction of an infinite-dimensional dynamical system. However, since the study of uniform attractors is usually based on a symbol space which contains auxiliary elements that could not belong to the original system, a uniform attractor is more often infinite dimensional. In fact, it has been an untouched problem for almost twenty years that if a uniform attractor of an infinite-dimensional system could be finite dimensional under acceptable conditions, and how flexible the conditions could be.
In a previous work (Cui et al. 2021), we studied the finite-dimensionality of deterministic uniform attractors. By a smoothing property of the underlying system, we established criteria for a uniform attractor to have finite fractal dimension, and the upper bound consists of two parts: the fractal dimension of the symbol space plus an auxiliary number arising from the smoothing property. This structure of the upper bound agrees with the result (Chepyzhov and Vishik 2002, Theorem IX 2.1) of Chepyzhov and Vishik established by studying the quasi-differentials of the system. In addition, we showed in Cui et al. (2021) that the finite-dimensionality of the symbol space is fully determined by the tails of the non-autonomous term of the original system; in other words, the tails of the non-autonomous term are what is crucial for the finite-dimensionality of a uniform attractor.
In this paper, we change the framework to a random environment, which implies crucial theoretical and technical differences with previous papers in the literature, and give an alternative of the smoothing property by a squeezing property. More precisely, we shall present two general criteria of estimating the fractal dimension of random uniform attractors. One is based on a smoothing property of the system, which allows the phase space to be only Banach, but requires an auxiliary space that compactly embedded into the phase space, see Theorem 3.3; the other is based on a squeezing property of the system, where no auxiliary space is needed, but the phase space, in applications, should be Hilbert, see Theorem 3.6. Neither of the two theorems implies the other. Note that smoothing and squeezing properties have already been used in the literature to deal with various problems in dynamics, see for instance (Málek et al. 1994) and also later papers (Caraballo and Sonner 2017;Carvalho and Sonner 2013;Czaja and Efendiev 2011;Efendiev et al. 2000Efendiev et al. , 2003Efendiev and Zelik 2008;Efendiev et al. 2011;Shirikyan and Zelik 2013;Zhao and Zhou 2016) for smoothing property in estimating the fractal dimensions as well as constructing exponential attractors, Foias and Temam (1979) and later literature (Debussche 1997;Eden et al. 1994;Flandoli and Langa 1999;Kloeden and Langa 2007;Zelati and Kalita 2015;Cui et al. 2018a) for the use of squeezing property. Nevertheless, here we need to carefully overcome the difficulty arising jointly from the three features of problem: (a) The lack of the invariance of the random uniform attractor; (b) The superposition of the base flow on the symbol space; (c) The stochastic nature of the problem.
For the first two problems, our previous work (Cui et al. 2021) provides some inspiration of solutions. We carefully make use of the relationship between the uniform attractor A and the cocycle attractor A of the underlying system, where is the symbol space of the system and ( , F, P) a probability space. This allows us to decompose the uniform attractor into sets of cocycle attractor sections, and then the invariance of the cocycle attractor A is useful. Nevertheless, since the absorption time of the random absorbing set is usually random, the analysis in this paper is much more technical than in Cui et al. (2021). Our solution requires the absorbing set absorbs itself after a deterministic time. This condition is in fact slightly stronger than needed, but facilitates our analysis. It also appeared in Shirikyan and Zelik (2013) for the construction of random exponential attractors; our application to a reaction-diffusion equation in Sect. 4 shows that this condition holds naturally in the additive noise case. Multiplicative noise case can also have this property, which will be shown in our future work.
The third problem is the stochastic nature of the setting. Basically, Birkhoff's ergodic theorem is frequently used and so the coefficients in the conditions are often required to have finite expectation which, however, is sometimes difficult to verify in applications. An example is the coefficient κ(ω) lying in the smoothing condition (H 6 ), for which the finite-expectation is unknown for the application of the reaction-diffusion system (4.1). However, the squeezing condition (S) for system (4.1) is verified. In this sense, the squeezing approach seems more applicable than the smoothing one.
On the other hand, an advantage of the smoothing approach is that, once the finitedimensionality of the uniform attractor has been establishd, by the smoothing property one could easily improve it to more regular spaces, see Theorem 3.8. In addition, the coefficient in the smoothing property does not need to have finite expectation, so it does not have the application problem mentioned above. Hence, both the ideas of smoothing and squeezing are useful in estimating the fractal dimension of random uniform attractors. In Sect. 4, we develop an application of a reaction-diffusion system for which the finite-dimensionality of the random uniform attractor in L 2 is established by the squeezing method and in H 1 0 by the smoothing method. An absorbing set with a deterministic absorption time for the system is also constructed, which is crucial for the analysis.
Finally, we note that in Han and Zhou (2019) recently constructed a random uniform exponential attractor for a stochastic reaction-diffusion equation with quasi-periodic forcings. Their result was derived by taking into account an extended phase space and then studying its respective skew-product semiflow, for which the problem is then reduced to an autonomous problem. In this case, i.e., when the skew-product semiflow method applies, the random uniform attractor can be studied by the random attractor of the skew-product semiflow, and, as a particular case, the finite-dimensionality of random uniform attractors can be derived more directly from the finite-dimensional theory of random attractors for autonomous RDS. Here, however, as we consider more general non-autonomous terms than quasi-periodic forcings such that the symbol space is even no longer a linear space, the skew-product semiflow approach fails. Thus, we have to treat the random uniform attractor in its own way, and our method has an advantage that more general non-autonomous terms are allowed. Notations. Throughout the paper, for any metric space (X , d X ) and real number r > 0 we denote by B X (x, r ) the open ball centered at x ∈ X with radius r , and by B X (A, r ) := ∪ x∈A B X (x, r ) we denote the open r -neighborhood of any non-empty subset A of X . Given a precompact set E ⊂ X , we denote by E the cardinality of E, by N X [E; r ] the minimal number of open balls of radius r in X to cover E, and the fractal dimension of E in X is defined by The Hausdorff semi-distance between non-empty sets in X is defined by We denote by B(X ) the Borel sigma-algebra of X .

Preliminaries on Random Uniform Attractors
Given a separable metric space ( , d ), suppose that ⊂ is a compact submetric space endowed with a continuous group {θ t } t∈R acting on it, satisfying that θ 0 σ = σ and θ t (θ s σ ) = θ t+s σ for all σ ∈ , t, s ∈ R, and that the map (t, σ ) → θ t σ is (R × , )-continuous. Moreover, we assume that is invariant under {θ t } t∈R , i.e., θ t = for all t ∈ R. For a set A, let B(A) be the Borel sigma-algebra of A. Denote by ( , F, P) a probability space, which need not be P-complete, endowed also with a flow {ϑ t } t∈R satisfying the following conditions • ϑ 0 = identity operator on ; The two groups {θ t } t∈R and {ϑ t } t∈R acting on and , respectively, are called base flows. As we did not assume the probability space to be complete, we shall not distinguish a full measure subspace˜ from , that is, by saying that a statement holds for all ω ∈ we mean that it holds on˜ almost surely. We denote by E(L) = L(ω)P(dω) the expectation of a random variable L.
Suppose that (X , · X ) is a separable Banach space. The definition of nonautonomous random dynamical systems in X is given as the following.
In applications, an NRDS φ is typically generated by an evolution equation with both a non-autonomous forcing (from the space ) and random perturbations, while the space is formulated via all the time translations of the forcing. In this case, the forcing is called the (non-autonomous) symbol of the equation, and the space is called the symbol space of the NRDS φ.
Definition 2.2 A set-valued map D : → 2 X taking values in the closed subsets of a Polish space X is said to be measurable if for each x ∈ X the map ω → dist X (x, D(ω)) is (F, B(R))-measurable. In this case, D is called a closed random set. If each section D(ω) of D is in addition compact, then D is called a compact random set. D is said to be an open random set if its complement D c is a closed random set.
Due to the attracting property of a random attractor, the distance between trajectories and the attractor is expected to be measurable. This is why we follow the measurability defined above via the distance. Compared with the alternative definition of a closed random set D as a measurable set in × X with all (or almost all) of its sections D(ω) being closed, Definition 2.2 requires more: a measurable set in × X with closed sections is not necessarily a closed random set in the sense of Definition 2.2, and the two definitions coincide only with respect to the universal sigma algebra of F, see for instance (Crauel 2002, Proposition 2.4). On the other hand, for an open set-valued map ω → U (ω) it does not suffice to conclude that U is an open random set from the fact of U being a closed random set, i.e., with dist X (x, U (·)) being measurable we cannot say that U is an open random set. A counterexample is given by Crauel (2002, Remark 2.11).
In the following, a (closed or open) random set D is always mentioned in the sense of Definition 2.2. D is often identified with its image {D(ω)} ω∈ . Given two random sets D 1 and D 2 , D 1 is said to be inside of D 2 if D 1 (ω) ⊆ D 2 (ω) for all ω ∈ .

Cocycle and Uniform Attractors of Non-autonomous Random Dynamical Systems
Before the definition of attractors, let us introduce first the attraction universe D which is a collection of some random sets that are expected to be attracted by an attractor. In this paper, we consider the universe D of all tempered random sets in X , i.e., where a closed random set D in X is said to be tempered if D(ω) X := sup x∈D(ω) x X ≤ R(ω) for some random variable R(·) : → R which is tempered, i.e., lim t→±∞ ln R(ϑ t ω) |t| = 0 ln · = log e · , ∀ω ∈ .
(2.1) Definition 2.3 ) A family A = {A σ (·)} σ ∈ of compact random sets is said to be the D-cocycle attractor of an NRDS φ, if • A is D-pullback attracting, i.e., for each D ∈ D, • (Minimality) if A = {A σ (·)} σ ∈ is another family of compact random sets satisfying the above condition, then It is observed in applications that every section A σ (·) of a cocycle attractor A can often belong to the attraction universe, i.e., A σ (·) ∈ D for every σ ∈ . This leads to the possibility of replacing the minimality condition in Definition 2.3 by the condition that A σ (·) ∈ D for every σ ∈ . We above follow the definition of , where a detailed analysis for the existence criteria, characterization and robustness of cocycle attractors was given.

Definition 2.4 (Cui and Langa 2017) A compact random set
(ii) (Minimality) A is inside of any compact random set satisfying (i).
The random uniform attractor can be regarded as a random generalization of the deterministic uniform attractor (Haraux 1988;Chepyzhov and Vishik 2002) or a nonautonomous generalization of the autonomous random attractor (Crauel et al. 1997;Crauel and Flandoli 1994;Flandoli and Schmalfuss 1996). Since a uniform attractor describes the dynamics in a uniform way w.r.t. symbols in the symbol space , the attraction universe D in consideration consists of some σ -independent autonomous random sets. This is also the case in the deterministic uniform attractor theory, where the attraction universe is usually the collection of (autonomous) bounded sets in the phase space.
Due to the nature of random perturbations in general applications, the uniform attraction of the attractor is defined in the pullback sense, but it implies the forward attraction in probability as given below; so a random uniform attractor describes also a forward dynamics of the NRDS φ.
Recall that a closed random set B = {B(ω)} ω∈ is said to be a uniformly Dpullback absorbing set, if for each D ∈ D and ω ∈ there is an absorption time Note that the absorption time T D (ω) in applications is usually a random variable in ω, and is even a deterministic number for some particular random set D. The latter observation is crucial for our analysis later, see condition (H 3 ) in Theorems 3.3 and 3.6. Then, we have the following existence conditions for a D-uniform attractor and also some properties the attractor possesses.
Theorem 2.6 (Cui and Langa 2017) Suppose that φ is a ( × X , X )-continuous NRDS. If φ has a compact uniformly D-pullback attracting set K and a closed uniformly Dabsorbing set B ∈ D, then it has a unique random D-uniform attractor A ∈ D given by Moreover, the following properties hold: (i) The NRDS φ has also a D-cocycle attractor A = {A σ (·)} σ ∈ which satisfies the relation

3)
and, for each ω fixed, the set-valued map σ → A σ (ω) is upper semi-continuous: where a D-complete trajectory of the NRDS φ is a map ξ : × R → X for which there exists σ ∈ such that ξ(ϑ t ω, t) = φ(t − s, ϑ s ω, θ s σ, ξ(ϑ s ω, s)) for all t ≥ s and ω ∈ , and that there exists D ∈ D such that ∪ t∈R ξ(·, t) ⊂ D(·); (iv) A is fully determined by uniformly attracting deterministic compact sets: if we denote by A the random D-uniform attractor of φ with D the collection of all non-empty compact sets in X , then

Remark 2.7
The random D-uniform attractor A of a ( × X , X )-continuous NRDS is inside of any uniformly D-pullback absorbing set B, since from the negative semiinvariance (2.4) of A and the uniform pullback absorption of B it follows , for t large enough.

Conjugate Attractors and Their Structural Relationship
The idea of conjugate dynamical systems has been widely used in, e.g., transforming a stochastic PDE to a deterministic PDE with random parameters, see, e.g., Chueshov (2002), Flandoli and Lisei (2004) and Cui et al. (2016). Now we study in a more abstract framework the attractors under this transformation.
Suppose that X andX are two Banach spaces (where X =X is allowed), and that φ andφ are two NRDS with the same base flows (θ, ) and (ϑ, ) in phase spaces X andX , respectively. Definition 2.8 φ andφ are said to be conjugate NRDS if there is a map T : × X →X , which is called a cohomology of φ andφ, with properties (2.5) Let D andD be collections of tempered closed random sets in X and inX , respectively. In the following, we will need the cohomology T be a bijection between D and D, i.e., for each D ∈ D there is a uniqueD ∈D and for eachD ∈D there is a unique D ∈ D such thatD(ω) = T(ω, D(ω)) for all ω ∈ . A particular example of such a cohomology T is given later in the application section, see (4.16).
Theorem 2.9 Suppose that φ andφ are conjugate NRDS with cohomology T satisfying (2.5), and that the cohomology T is a bijection between D andD. If φ has a D-uniform attractor A in X , thenφ has aD-uniform attractorÃ inX , and vice versa. Moreover, the two attractors have the relatioñ (2.6) Proof Without loss of generality, we suppose that φ has a D-uniform attractor A , and then we prove by definition thatÃ (ω) := T(ω, A (ω)) defines theD-uniform attractor ofφ. Clearly,Ã is compact and measurable, since so is A and T is a homeomorphism.
Hence,Ã is uniformlyD-pullback attracting underφ. In the same way, the minimality ofÃ follows from that of A .
In the same way, we have the corresponding theorem for conjugate cocycle attractors.
Theorem 2.10 Suppose that φ andφ are conjugate NRDS with cohomology T satisfying (2.5), and that the cohomology T is a bijection between D andD. If φ has a D-cocycle attractor A in X , thenφ has aD-cocycle attractorÃ inX , and vice versa. Moreover, the two attractors have the relatioñ (2.7)

Remark 2.11
The structural relationships (2.6) and (2.7) allow one to learn the structure of an attractor from that of its conjugate attractor. For instance, conjugate attractors could share the same fractal dimension, e.g., when the cohomology T(ω, x) is linear in x.

Finite-Dimensionality of Random Uniform Attractors
In this section, we estimate the fractal dimension of random uniform attractors. Two approaches will be presented, one by a smoothing property and the other by a squeezing property of the system. Then in Sect. 3.3, we will improve the finite-dimensionality to more regular Banach spaces. First recall that, given a precompact subset E of a Banach space X , the fractal dimension (also called box-counting dimension or capacity dimension) of E in X is defined as where N X [E; r ] denotes the minimal number of open balls of radius r in X that are necessary to cover E. Note that dim F (E; X ) = dim F (Ē; X ), whereĒ denotes the closure of E.

Smoothing Approach
Now we present the first approach by a smoothing property of the system. This approach allows the phase space X to be only Banach, but technically requires an auxiliary Banach space Y ⊂ X with compact embedding I : Y → X . The following lemma of Sobolev compactness embedding gives examples of such Banach spaces.
Lemma 3.1 (Temam 1997 We will need the Kolmogorov ε-entropy of the compact embedding I : Y → X , also called the Kolmogorov ε-entropy of Y in X , ε > 0, which is defined as (3.1) For this Kolmogorov ε-entropy, the following estimate is useful.
As a particular case, for some α > 0, Now we give our main criterion for a random uniform attractor to have finite fractal dimension. Suppose that (H 3 ) φ has a tempered uniformly D-pullback absorbing set B = {B(ω)} ω∈ which pullback absorbs itself after a deterministic period of time, i.e., there exists a deterministic time Lipschitz continuous in symbols within the absorbing set B: for a random variable L(·) : → R + with finite expectation E(L) < ∞; (H 5 ) Y is a separable Banach space densely and compactly embedded into X , and for any ε > 0 the Kolmogorov ε-entropy of Y in X satisfies (3.7) Under these hypotheses, the random uniform attractor A = {A (ω)} ω∈ of φ has finite fractal dimension that can be bounded by a deterministic number. More precisely, Theorem 3.3 Suppose that φ is an NRDS in X with D-uniform attractor A . If conditions (H 1 )-(H 6 ) hold, then the uniform attractor A has finite fractal dimension in X : for any ν ∈ (0, 1), (3.8) In particular, taking ν = 1/2, (3.9) Remark 3.4 Note that (i) The upper bound given above is deterministic and uniform w.r.t. ω ∈ ; (ii) The entropy condition (H 5 ) depends only on the spaces X and Y , and is independent of the system. Lemma 3.2 is useful in order to obtain such a property; (iii) In condition (H 3 ), we required a deterministic absorbing time T B , which seems unusual in the literature. However, this demand can be naturally satisfied by a broad class of applications. In Sect. 4.3, we will show by a reaction-diffusion model that in additive noise cases the closed random absorbing set B constructed in the usual way will be satisfactory, see Proposition 4.5. Multiplicative noises need some slight modification in constructing the absorbing set, which will be shown in our future work.

Proof of Theorem 3.3
Let ν ∈ (0, 1) be given and fixed, and suppose without loss of generality thatt = T B = 1 in hypotheses (H 3 ) and (H 6 ). Since the absorbing random set B is tempered, we have for points x ω ∈ B(ω) and some tempered random variable R(·) satisfying (2.1).
Since Y is compactly embedded into X the unit ball B Y (0, 1) in Y is covered by a finite number of ν 2κ(ω) -balls in X , and we denote by N (ω) the minimum number of such balls that are necessary for this, i.e., Next, for each ω ∈ and σ ∈ we construct sets U n (ω, σ ) ⊆ B(ω) by induction on n ∈ N such that Note that the bound (3.12) of cardinality is independent of σ . For n = 1, by the smoothing property (3.7) in hypothesis (H 6 ) we have , and with this we have Assuming that the sets U k (ω, σ ) have been constructed for all 1 ≤ k ≤ n, ω ∈ and σ ∈ , we now construct the sets U n+1 (ω, σ ). Given ω ∈ and σ ∈ , by the cocycle property of φ we have and by the induction hypothesis the smoothing property (H 6 ) we obtain Now, to find a finite cover of the random uniform attractor A let us make a decomposition of it using the structure (2.3). By the compactness of the symbol space , for any positive number η > 0 there exists a finite cover of by at least M η := N [ ; η] balls of radius η, i.e., there are centers σ l ∈ , l = 1, 2, . . . , M η , such that (3.14) For each l = 1, . . . , M η , denote by where A is the D-cocycle attractor of φ. Then by (2.3), the random uniform attractor A is decomposed as In the following, we shall find finite covers for each A l (ω). Note that the constant M η is independent of ω ∈ , and depends only on the symbol space and the corresponding given number η.

Squeezing Approach
Theorem 3.3 gives a criterion on the finite-dimensionality of random uniform attractors where, however, the finiteness of the expectation of the coefficient κ(ω) in the smoothing condition (H 6 ) is usually not easy to obtain in real applications. To overcome this, we next propose an alternative method using a squeezing condition instead. The squeezing, in applications, applies mainly to a Hilbert phase space X , but allows the coefficients to be an exponential with only the order having finite expectation (the expectation of the entire exponential does not need to be finite, see (S)). We first recall the following lemma of finite-coverings of balls in Euclidian spaces.
Lemma 3.5 (Debussche 1997, Lemma 1.2) Let E be an Euclidean space with algebraic dimension equals to m ∈ N and R ≥ r > 0 be positive numbers. Then for any x ∈ E, it holds In other words, any ball in E with radius R > 0 can be covered by k(R, r ) balls of radius r > 0.
Let φ be an NRDS, T B > 0 be as in (H 3 ) and suppose in addition the following squeezing property: (S) φ satisfies a random uniformly squeezing property on B, i.e., there exist t ≥ T B , δ ∈ (0, 1/4), an m-dimensional orthogonal projection P : X → P X (dim(P X) = m) and a random variable ζ(·) : We have then the following criterion for the finite-dimensionality of random uniform attractors.
Theorem 3.6 Suppose that φ is an NRDS in X with D-uniform attractor A . If conditions (H 1 )-(H 4 ) and (S) hold, then A has finite fractal dimension in X : for any (3.31) Proof Suppose without loss of generality thatt = T B = 1 in hypotheses (H 3 ) and (S).

Fractal Dimension in more Regular Spaces
By Theorems 3.3 and 3.6, we have established the finite-dimensionality of the random uniform attractor A = {A (ω)} ω∈ in the phase space X . Now we are interested to improve the finite-dimensionality to a more regular space Y ⊂ X . But to be more general, in the following we study the problem in a Banach space Z for which Z = Y is a particular case.
Let (Z , · Z ) be a Banach space and suppose the NRDS φ takes values in Z , i.e., for each u ∈ X , ω ∈ and σ ∈ we have φ (t, ω, σ, u) ∈ Z for t > 0. Suppose also that the random uniform attractor is such that A (ω) ⊆ X ∩ Z for all ω ∈ . In the following, we shall show that under a ( × X , Z )-smoothing property the fractal dimension in Z of the random uniform attractor can be bounded by the dimension of it in X plus the dimension of the symbol space in .
The ( × X , Z )-smoothing condition we need is as follows: There is at > 0 such that for some positive constants δ 1 , δ 2 > 0 and a random variableL(ω) > 0 it holds is often more applicable in applications since powers δ 1 and δ 2 are allowed but no powers are allowed in (H 6 ). In fact, it is open whether or not Theorem 3.3 can be established using a weaker version of (H 6 ) with condition (3.7) weakened to: there exists some power δ ∈ (0, 1], (3.7 ) Theorem 3.8 Let φ be an NRDS which is ( × X , X )-continuous and has a Duniform attractor A ⊆ X ∩ Z . Suppose that A has finite fractal dimension in X , i.e., dim F (A (ω); X ) < c(ω) < ∞. Then if (H 1 ) and (H 7 ) are satisfied, A has finite fractal dimension in Z as well: Remark 3.9 Notice that (i) Z does not have to be a subset of X and no embedding from Z into X was required, so the theorem applies to the case of, e.g., X = L 2 (R) and Z = L p (R) with p > 2; (ii) If the fractal dimension of A in X is uniformly (w.r.t. ω ∈ ) bounded, i.e., dim F A (ω); X ≤ d for all ω ∈ , where d is a deterministic constant, then the fractal dimension of A in Z is also uniformly bounded by a deterministic number: Let ω ∈ . Since A (ω) is compact in X we have and by the negative semi-invariance of A (see (2.4)) we obtain (3.52) for all l = 1, . . . , M ε 1/δ 1 and i = 1, . . . , N X A (ϑ −t ω); ε 1/δ 2 , so by (3.51) we obtain Since r (t, ω, δ 1 , δ 2 ) is independent of ε and A (ϑ −t ω) is finitely dimensional in X then in a standard way of taking the limit as ε → 0 + we conclude

Finite-Dimensional Symbol Space 6 of Continuous Functions
In condition (H 1 ), the symbol space is required to be finite dimensional, which is technical itself in real applications. In fact, even in the deterministic uniform attractor theory the uniform attractor A and the compact symbol space have a close relationship, which can be seen from, for instance, the presentation where {A(σ )} σ ∈ forms the cocycle attractor of the underlying system (Bortolan et al. 2014). This structure provides a view of the uniform attractor A as the image set of the map π : → A , σ → A(σ ). In a simple case that π is single-valued, i.e., each A(σ ) is a single point in the phase space, the fractal dimension of A can equal that of the symbol space when, e.g., π is Lipschitz continuous. For this reason, we do not expect a general finite-dimensional result of uniform attractors for infinite-dimensional symbol spaces. Now from the application point of view we present some conditions that ensure a symbol space to be finite-dimensional. Note that for a non-autonomous evolution equation with time-dependent term g (called the (non-autonomous) symbol of the equation), the symbol space is often formulated as the hull H(g) of g with all the time translations of g being included. More precisely, for g ∈ with being a complete metric space, (3.53) and θ s : → are translation operators on : In our previous work (Cui et al. 2021), taking as the Fréchet space of continuous functions we gave conditions on a function g ∈ that ensure the hull of g to have finite fractal dimension, which weakened the known condition of quasi-periodicity needed by Chepyzhov and Vishik (2002) in applications. Now we recall briefly the main results, since they give us insights about what applications our theorem can apply to. We begin with two preliminary concepts of almost periodic functions and quasiperiodic functions. The readers are referred to, e.g., Amerio and Prouse (1971) and Chepyzhov and Vishik (2002).
Let (X , d X ) be a complete metric space and ξ(·) : R → X a continuous map. For any ε > 0, a number τ ∈ R is said to be an ε-period of ξ if sup s∈R d X ξ(s + τ ), ξ(s) ≤ ε.
If for any ε > 0 the ε-periods of function ξ form a relatively dense set in R, i.e., there is a number l = l(ε) > 0 such that for any α ∈ R the interval [α, α + l] contains an ε-period τ of ξ , then ξ is said to be an almost periodic function. Note that for an almost periodic function ξ , the set of values {ξ(t) : t ∈ R} is precompact in X . Also, ξ is uniformly continuous on R, and the sum of almost periodic functions is an almost periodic function. Clearly, periodic functions are almost periodic.
A particular class of almost periodic functions is the quasi-periodic functions. For k ∈ N, let T k = [R mod 2π ] k be the k-dimensional torus and denote by C(T k ; X ) the set of continuous functions ϕ ∈ C(R k ; X ) which are 2π -periodic in each argument, i.e., for each i = 1, . . . , k k} is a set of rationally independent real numbers, i.e., if n 1 , . . . , n k ∈ Z are integers such that n 1 α 1 + · · · + n k α k = 0 then n 1 = · · · = n k = 0. For ϕ ∈ C(T k ; X ), a function ξ : R → X with the form is said to be quasi-periodic (with k frequences) with values in X . Note that periodic functions are particular quasi-periodic functions, and quasi-periodic functions are almost periodic.
For the space b := C b (R; X ) of bounded continuous functions with the supremum metric, Chepyzhov and Vishik (2002) showed that the hull of Lipschitz continuous quasi-periodic functions has finite fractal dimension. More precisely, Lemma 3.10 (Chepyzhov and Vishik 2002, Proposition IX.2 is given as in (3.53) with the closure taken over the supremum metric.
Moreover, the following lemma indicates that a necessary condition for H b (ξ ) to be finite dimensional in b is that ξ is almost periodic.
Lemma 3.11 (Chepyzhov and Vishik 2002, Theorem V.1 In order to study evolution equations with more general non-autonomous terms than quasi-periodic ones, in our previous work (Cui et al. 2021), we considered the space = C(R; X ) of continuous functions with the Fréchet metric [−n,n] d X ξ 1 (s), ξ 2 (s) , n ∈ N.
In this space, the translation operators θ t are Lipschitz but with t-dependent Lipschitz constants, as indicated by the following lemma.
Lemma 3.12 (Cui et al. 2021, Proposition 4.3) For any t ∈ R the translation operator θ t on = C(R; X ) is Lipschitz: In addition, the following theorem shows that the finite-dimensionality of the hull H(ξ ) of a function ξ in is fully determined by the tails of the function.
Theorem 3.13 (Cui et al. 2021, Theorem 4.12) Suppose that g + , g − ∈ = C(R; X ) are two functions with finite-dimensional hulls H(g + ) and H(g − ) in , respectively. If g ∈ is a function such that (G1) g is Lipschitz continuous from R to X ; (G2) g converges forward to g + and backwards to g − exponentially, i.e., there exist a time T * ≥ 0 and constants C, β > 0 such that Then, the hull H(g) of g is finite dimensional in with Note that, by Lemma 3.10, quasi-periodic functions are examples of g + and g − . Theorem 3.13 allows us to consider in applications some non-autonomous terms that are not almost periodic, for instance, the smoothly switching forcing g ∈ C(R; R) such that for which by Theorem 3.13 (with g + (t) ≡ 1 and g − (t) ≡ −1) we have dim F H(g); C(R; R) ≤ 1. More examples and comments were given in Cui et al. (2021). Finally we recall a useful lemma.
Lemma 3.14 (Cui et al. 2021, Lemma 5.1) Let X , · X be a Banach space. If g ∈ := C(R; X ) and the hull H(g) of g is compact in , then there is a constant c = c(g) > 0 such that for any σ 1 , σ 2 ∈ H(g) , ∀t ≥ τ and q ≥ 1.

Applications to a Stochastic Reaction-Diffusion Equation
In this section, we study a stochastic reaction-diffusion equation as an application of our theoretical analysis. Note that, under certain conditions, the existence and some preliminary results of the random uniform attractor have been established recently by . Now, with the non-autonomous term strengthened such that the symbol space is finite dimensional, we show the finite-dimensionality of the random uniform attractor.

Preliminary Settings and the Symbol Space
We consider the following reaction-diffusion equation with additive scalar white noise (4.1) endowed with the initial and boundary conditions where O ⊂ R N , N ∈ N, is a bounded smooth domain and λ > 0 is a constant. The nonlinear term f ∈ C 1 R, R is assumed to satisfy the following standard conditions where all the coefficients are positive constants and the growth order p ≥ 2. Let h(x) ∈ W 2,2 p−2 (O) for simplicity. To establish a smoothing property (H 6 ), we will also need the growth order p to satisfy (4.7) This ensures the continuous embedding for some constant c > 0, where · := · L 2 (O) , see Robinson (2001, Theorem 5.26). The probability space ( , F, P) is defined in a standard way. Let F be the Borel sigma-algebra induced by the compact-open topology of and P the two-sided Wiener measure on ( , F). Define the translation operators ϑ t on by Then P is ergodic and invariant under ϑ (see Flandoli and Schmalfuss 1996). Setting we have that z(ω) is a stationary solution of the one-dimensional Ornstein-Uhlenbeck equation dz(ϑ t ω) + λz(ϑ t ω)dt = dω. (4.10) Moreover, there is a ϑ-invariant subset˜ ⊂ of full measure such that z(ϑ t ω) is continuous in t for every ω ∈˜ and the random variable |z(·)| is tempered (see Fan 2006, Lemma 1). Hereafter, we will not distinguish˜ and .
In order to study the finite-dimensionality of the random uniform attractor, we need the non-autonomous forcing g to have a finite-dimensional hull in some metric space . By the analysis in Sect. 3.4, for the current reaction-diffusion equation we take := C R; L 2 (O) and assume that (G) g ∈ = C R; L 2 (O) and the hull of g H(g) = θ r g : r ∈ R d has finite fractal dimension in , i.e., dim F H(g); < ∞.
Note that, by Lemma 3.10, Lipschitz continuous quasi-periodic functions are examples of such g's with condition (G), and Theorem 3.13 indicates that Lipschitz continuous functions with tails eventually exponentially converging to quasi-periodic functions satisfy condition (G) as well. Some concrete examples were given in Cui et al. (2021). Now for the reaction-diffusion equation (4.1), we define the symbol space as the hull of the forcing g in : Then condition (G) ensures that is a finite-dimensional compact subset of . Moreover, the group {θ t } t∈R of translation operators forms a base flow on .
In the rest, we shall study the finite-dimensionality of A in H by checking the conditions (H 1 )-(H 4 ) and (S) in Theorem 3.6, and in V by checking (H 7 ) in Theorem 3.8. The existence of the random uniform attractor was proved previously by , using Theorem 2.6.

Estimates of Solutions
The estimates of solutions in this section will be achieved in a standard way as in  in spirit, but we need more details that are crucial for us to bound the fractal dimension of the uniform attractor afterwards. Note that the condition (4.7) on p is not needed in this section.
where C > 0 is a positive constant independent of v 0 ∈ H, σ ∈ and ω ∈ .
Proof Take the inner product of (4.12) with v in H to obtain (4.18) By (4.3), (4.4) and Young's inequality, we have, (4.20) by (4.18)-(4.20) we conclude that Replacing ω and σ by ϑ −t ω and θ −t σ , we have and the proof is complete.
Lemma 4.2 (Estimate in V ) Let conditions (4.3)-(4.6) hold. Then for any > 0, there is a constant c such that any solution v of (4.12) with initial value v 0 ∈ H satisfies where c is a positive constant depending only on .
To establish the (H , V )-smoothing property of the system, we need the following estimates.
For later purpose, we state the following corollary.

The Absorbing set B with Deterministic Absorption Time
Now we construct an admissible uniformly D-absorbing set B satisfying (H 3 ). Since = H(g) is compact in = C(R; H ) we know from Chepyzhov and Vishik (2002, Proposition V.2 (4.34) Hence, there exists a uniform bound C b > 0 such that Proof The temperedness of B follows from that of ρ(·). Now we show the uniformly D-pullback absorbing property. By Lemma 4.1, we know that for any tempered set D ∈ D (i.e., there is some tempered random variable R(·) such that D(ω) 2 ≤ R(ω)), the solutions with initial data in D satisfy (4.37) In addition, since the random variable R(·) is tempered, there exists a random variable T D (·) > 0 such that e −λt R(ϑ −t ω) ≤ 1 for all t ≥ T D (ω). Hence, for all t ≥ T D (ω), so B is a uniformly D-pullback absorbing set. Now we show that B uniformly attracts itself after a deterministic period of time T B . By (4.37), Hence, take T B > 0 such that Then, for all t ≥ T B , which by the definition (4.36) indicates that as desired.
The proof is complete.
For n ∈ N, set H n := span{e 1 , e 2 , . . . , e n }, and let P n : H → H n and Q n := I − P n be the orthonormal projectors on H . Then, λ n+1 Q n v 2 ≤ ∇v 2 , ∀v ∈ V , n ∈ N.