RANDOM PULLBACK EXPONENTIAL ATTRACTORS: GENERAL EXISTENCE RESULTS FOR RANDOM DYNAMICAL SYSTEMS IN BANACH SPACES

. We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.


1.
Introduction. Studying the longtime behavior of infinite dimensional dynamical systems can often be reduced to analyzing the dynamics on the global attractor. Global attractors are compact subsets of the phase space that are strictly invariant under the time evolution of the system and attract all bounded subsets as time tends to infinity. If the global attractor exists, it is unique, and in most cases of finite fractal dimension. The rate of convergence to the attractor, however, is typically unknown. It can be arbitrarily slow and hence, global attractors are generally not stable under perturbations. To overcome these drawbacks the notion of an exponential attractor was introduced in [7]. Exponential attractors are compact, semi-invariant sets of finite fractal dimension that contain the global attractor and attract all bounded subsets at an exponential rate. Due to the exponential rate of convergence they are more stable under perturbations. However, since exponential attractors are only semi-invariant under the time evolution of the system, they are not unique.
Different methods have been developed to show the existence of exponential attractors for infinite dimensional dynamical systems. The first existence proof was established for semigroups acting in Hilbert spaces, cf. [7]. It is non-constructive, based on the so-called squeezing property of the semigroup and essentially uses the

TOMÁS CARABALLO AND STEFANIE SONNER
Hilbert space structure of the phase space. In [9] an alternative method and explicit construction of exponential attractors for discrete time semigroups in Banach spaces was proposed. The approach relies on the compact embedding of the phase space into an auxiliary normed space, the existence of a bounded absorbing set and the so-called smoothing property of the semigroup. This property implies that the semigroup is eventually compact and is mainly satisfied by parabolic problems.
More recently, in [6,11] and [2,3] the construction of exponential attractors [9] has been extended to non-autonomous dynamical systems using the notion of pullback attraction. In [6] and [11] evolution processes satisfying the smoothing property were considered and the existence of a fixed bounded pullback absorbing set was assumed. Moreover, rather strong regularity assumptions were imposed that mainly restrict applications to parabolic problems. In [2,3] the existence of pullback exponential attractors was shown under significantly weaker hypotheses and the construction was generalized for asymptotically compact evolution processes. In particular, the Hölder continuity of the process, as hypothesized in [6,11], could be omitted and the fixed bounded pullback absorbing set was replaced by a timedependent family of absorbing sets. As a consequence, the results allow for more general non-autonomous terms in the equations and are also applicable, e.g. to hyperbolic problems. Moreover, unlike in [6,11] the pullback attractors are allowed to be unbounded in the past, a property that is inherent to random pullback attractors in most applications.
The aim of our paper is to extend the construction [2,3] to the setting of random dynamical systems. We formulate general existence results for random pullback exponential attractors and derive explicit estimates for their fractal dimension. The generalizations developed in [2,3] are hereby essential, since the absorbing sets as well as the constants in the estimates depend on the random parameter, and hence, are time-dependent. Moreover, random attractors of PDEs perturbed by additive or multiplicative noise are typically unbounded in the past.
Exponential attractors for random dynamical systems have previously been considered in [12], however, under restrictive assumptions that are difficult to verify in applications. The construction was carried out in the setting of Hilbert spaces and the attraction universe was the family of deterministic sets. The random dynamical system was assumed to satisfy the smoothing property and to be uniformly Hölder continuous in time. Moreover, certain stability assumptions and the compactness of the absorbing set were imposed. We improve this result in various directions and show that several of these hypotheses are not required. We consider asymptotically compact random dynamical systems in Banach spaces, i.e. the cocycle can be represented as a sum of operators satisfying the smoothing property and a family of contractions, and the attraction universe is the family of tempered random sets. Our proof yields the measurability of the random exponential attractor without the technical auxiliary results needed and established in [12], and does not require a stability assumption or the compactness of the absorbing set. Moreover, we derive explicit estimates for the fractal dimension of the attractors. For continuous time random dynamical systems we propose to weaken the notion of positive invariance. This allows to simplify the construction of random pullback exponential attractors such that the assumption of Hölder continuity in time of the cocycle can be omitted.
In a forthcoming paper [1] we will apply the theoretical results to a stochastic semilinear damped wave equation with multiplicative noise, where D ⊂ R n , n ≥ 3, is a bounded domain and W : Ω → C 0 (R) a standard scalar Wiener process. Assuming that the nonlinearity f is subcritical and dissipative and the noise |σ| is small w.r.t. β we prove the existence of a random pullback exponential attractor and derive estimates for its fractal dimension. The previous existence result for random exponential attractors [12] is not applicable to problem (1), since it is based on the smoothing property and Hölder continuity in time of the generated random dynamical system, i.e. on properties that are not satisfied in this situation. The outline of our paper is as follows: In Section 2 we collect several notions from the theory of random dynamical systems and recall results about entropy properties of embeddings that we will need in the sequel. General existence results for random pullback exponential attractors are derived in Section 3, where the construction is first carried out for discrete time random dynamical systems and subsequently extended to the continuous time setting.

Preliminaries.
2.1. Random dynamical systems. We recall basic notions from the theory of random dynamical systems that we will need in the subsequent sections and introduce the concept of random exponential attractors. Here and in the sequel, we assume (Ω, F, P) is a probability space and (V, · V ) a Banach space. Moreover, let T denote R or Z, and T + be the non-negative real numbers, or integers respectively. Definition 2.1. A random dynamical system (θ, ϕ) on V consists of a measurable and measure-preserving dynamical system {θ t } t∈T , θ t : Ω → Ω, on (Ω, F, P), i.e.
where Id denotes the identity operator in Ω, and a cocycle mapping ϕ : Definition 2.2. A random set B is a subset of Ω × V that is measurable with respect to the product σ-algebra F ⊗ B V , where B V denotes the Borel σ-algebra of V . Moreover, the ω-section of a random set B is defined by A random set B is called tempered, if there exists a random variable r B (ω) ≥ 0 such that B(ω) is contained in a ball with center zero and radius r B (ω) and 6386 TOMÁS CARABALLO AND STEFANIE SONNER lim t→±∞ 1 |t| log + r B (θ t (ω)) = 0.
We will denote a general family of random sets by D. It is usually called universe and can represent, for instance, the family of bounded deterministic sets, or the family of tempered random sets.
In the remainder of this subsection, when stating properties involving a random parameter we assume that they hold a.s., unless otherwise specified (i.e., there exists a subsetΩ ⊂ Ω of full measure such that the property is satisfied for all ω ∈Ω).
and it pullback attracts the family D, i.e.
There exist several criteria ensuring the existence of random pullback attractors. The simplest one states that if a random dynamical system in V possesses a compact random pullback D-attracting set, then a random pullback D-attractor exists (see Theorem 4 in [5]).
Theorem 2.4. Let (θ, ϕ) be a random dynamical system on a separable Banach space V . There exists a random pullback D-attractor for (θ, ϕ) if and only if there exists a compact random pullback D-attracting set K, i.e., the sections K(ω) are compact and Remark 1. If a random dynamical system possesses a random pullback D-attractor A and the universe D contains the family of compact deterministic sets, then A is unique a.s. (see Corollary 1 in [5]).
We now use the concept of random pullback attractors to introduce exponential attractors for random dynamical systems.
Moreover, the fractal dimension of M(ω) is finite, i.e. there exists a random variable and M is pullback D-attracting at an exponential rate, i.e. there exists α > 0 such that lim Here, dim f (·) denotes the fractal dimension, i.e. if A ⊂ V is precompact, then where N V ε (A) denotes the minimal number of ε-balls in V with centers in A needed to cover A.

(Kolmogorov)
ε-entropy and entropy numbers. Our construction of random pullback exponential attractors is based on the embedding of the phase space into an auxiliary normed space, and the entropy properties of this embedding will play a crucial role. In this subsection we recall the corresponding notions and results that we will need in the sequel.
The (Kolmogorov) ε-entropy of a precompact subset A of a Banach space V is defined as denotes the minimal number of ε-balls in V with centers in A needed to cover A. It was first introduced by Kolmogorov and Tihomirov in [10]. The order of growth of H V ε as ε tends to zero is a measure for the massiveness of the set A in V , even if its fractal dimension is infinite.
If V and U are Banach spaces such that the embedding V → → U is compact we use the notation Remark 2. The ε-entropy is related to the entropy numbers e k for the embedding V → U, which are defined by If the embedding is compact, then e k is finite for all k ∈ N. For certain function spaces the entropy numbers can explicitly be estimated (see [8]). For instance, if D ⊂ R n is a smooth bounded domain, then the embedding of the Sobolev spaces . Moreover, the entropy numbers grow polynomially, namely, e k k − l 1 −l 2 n (see Theorem 2, Section 3.3.3 in [8]), and consequently, for some constant c > 0.
Here and in the sequel, we write f g, if there exist positive constants c 1 and c 2 such that

TOMÁS CARABALLO AND STEFANIE SONNER
3. Construction of random pullback exponential attractors. Let (V, · V ) be a separable Banach space, (θ, ϕ) be a random dynamical system on V and D denote the universe of tempered random sets. Our construction of random exponential attractors is based on the compact embedding of the phase space into an auxiliary normed space, the decomposition of the cocycle as a sum of operators satisfying the smoothing property and a family of contractions, and the existence of a tempered pullback D-absorbing random set. We assume that the following properties are satisfied on a subset of full measure Ω ⊂ Ω that, for simplicity, we will denote again by Ω.
for some positive constants c and γ. (H 1 ) There exists a random closed set B ∈ D that is pullback D-absorbing, i.e. for every D ∈ D and ω ∈ Ω there exists T D,ω ≥ 0 such that and we assume that T D,θ−τ (ω) ≤ T D,ω for all τ ∈ T + . Moreover, the cocycle ϕ can be represented as sum ϕ = φ + ψ, where φ : T + × Ω × V → V, and ψ : T + × Ω × V → V, are families of operators satisfying the following hypotheses: (H 2 ) There exists a positivet ≥ T B,ω such that the family φ satisfies the smoothing property within B, i.e. there exists a random variable κ(ω) such that where γ is the growth exponent of the ε-entropy in (H 0 ).
Our main result is the following existence result for discrete time random dynamical systems that we later extend for continuous time random dynamical systems.
Theorem 3.1. Let (θ, ϕ) be a discrete time random dynamical system on V , i.e. T = Z and the assumptions (H 0 )-(H 3 ) be satisfied. Then, for any ν ∈ (0, 1 2 − λ) there exists a random pullback exponential attractor M ν for (θ, ϕ) in V , and the fractal dimension of its sections is bounded by and c and γ are the constants determined by the entropy properties in (H 0 ).
In the particular case that the constant κ in (H 2 ) can be chosen uniformly w.r.t. ω we recover the bound for the fractal dimension of deterministic pullback exponential attractors in [2,3], namely, + λ)) .

Remark 4.
Our results improve the previous existence result for random pullback exponential attractors by Shirikyan & Zelik [12]. The hypotheses are significantly weaker and easier to verify in applications. In particular, we generalize the construction for asymptotically compact random dynamical systems, i.e. for cocycles that can be represented as sum of operators φ satisfying the smoothing property and a family of contractions ψ. Moreover, we formulate the setting in Banach spaces instead of Hilbert spaces and replace the attraction universe of bounded deterministic sets by tempered random sets. The hypotheses on the random pullback absorbing set B are essentially weaker, since we do not suppose its compactness nor impose growth conditions for the ε-entropy of its sections. The measurability of the exponential attractor is achieved by a modified construction that does not require the technical auxiliary results in [12]. In Section 3.2 we extend the construction for continuous time random dynamical systems. In order to apply the approach in [2] developed for non-autonomous deterministic problems, that does not require the Hölder continuity in time of the cocycle, we propose to weaken the notion of positive invariance for continuous time random exponential attractors. It essentially simplifies the construction and leads to better, explicit estimates for the fractal dimension. During the revision of our article S. Zhou published an existence result for exponential attractors for non-autonomous random dynamical systems [13] and applied the results to stochastic lattice systems. The setting is different, since he considers non-autonomous random problems, and the hypotheses are more difficult to verify in applications.
We first prove Theorem 3.1 and subsequently construct random exponential attractors for continuous time random dynamical systems in Subsection 3.2.
Proof of Theorem 3.1. Without loss of generality we assume thatt = 1 in assumptions (H 2 ) and (H 3 ).
We follow and extend the method used in [2] to construct pullback exponential attractors for nonautonomous evolution processes. Different from the deterministic setting, the constants now depend on the random parameter ω, and the construction has to be done in such a way that the random pullback exponential attractor is measurable.
Construction of measurable sets of centres. Let n ∈ N and δ(ω) = β n R(θ −n (ω)). We recall that V(ω) = j∈N v j (ω) is a measurable selection for B(ω), the sets U n (ω) ⊂ ϕ n, θ −n (ω), V(θ −n (ω)) and ϕ n, θ −n (ω), V(θ −n (ω)) ⊂ u∈U n (ω) For an l-tuple k = (k 1 , . . . , k l ) ∈ N l , we define the random variable G k : Ω → {0, 1}, , v k j (θ−n(ω)) , 0 otherwise, and denote by Ω l ⊂ Ω those ω ∈ Ω for which there exists an l-tuple k such that G k (ω) = 1 and Gk(ω) = 0 for anyk containing less than l elements. The sets Ω l are intersections of measurable sets, and hence, are measurable. Since Ω = l∈N Ω l , it suffices to construct the set of measurable centres U n on each subset Ω l . For l ∈ N let σ : N → N l , σ(i) = k = (k 1 , . . . , k l ) be an indexing of all l-tuples in N l . We define random variables and F i (ω) = 0, otherwise. Finally, we set and U n (ω) = ϕ n, θ −n (ω), U n (ω)) . The set U n is a finite random set, since for all v ∈ V , which by the continuity and measurability of the cocycle ϕ implies that also U n is a finite random set. Constructing for all l ∈ N the sets U n on Ω l we obtain the random finite set U n (ω), ω ∈ Ω. If necessary, we now replace for all n ∈ N and ω ∈ Ω the sets U n (ω) by the sets U n (ω), and obtain a family of random finite sets U n , n ∈ N, which by construction, satisfies the properties (U 1 )-(U 3 ).
Then, the family E n (ω), n ∈ N 0 , satisfies These relations can be proved by induction, and are immediate consequences of the definition of the sets E n (ω), the properties (U 1 )-(U 3 ) of the family U n (ω), n ∈ 6394 TOMÁS CARABALLO AND STEFANIE SONNER N 0 , and property (4). Using the sets E n (ω) we define M ν (ω) = n∈N0 E n (ω) and show that its closure is a random pullback exponential attractor for (θ, ϕ).
Positive ϕ-invariance. The set M ν is positively ϕ-invariant: Indeed, for all k ∈ N and ω ∈ Ω property (E 1 ) implies that Since ϕ is continuous, it follows the positive ϕ-invariance of M ν , Compactness and finite fractal dimension. We first prove that the sections M ν (ω) are precompact and of finite fractal dimension in V . For any m ∈ N and n ≥ m the cocycle property implies that where property (4) was used in the last inclusion. Consequently, for all m ∈ N we obtain Let ε m > 0, m ∈ N, be a sequence converging to 0 as m → ∞. Since B is tempered and β ∈ (0, 1), there exists a subsequence m j , j ∈ N, such that m j → ∞ as j → ∞ and holds. Hence, it follows that and we can estimate the number of ε m -balls in V needed to cover M(ω) by where we used properties (U 2 ) and (E 2 ). This proves the precompactness of M ν (ω) in V , and taking the closure M ν (ω) = M ν (ω) , ω ∈ Ω, we obtain compact subsets in V .
For the fractal dimension of M ν (ω) we obtain .
Let now α ∈ (β, 1) be arbitrary and δ = ln( α β ). Since B is tempered there exists n 0 ∈ N such that β n R(θ −n (ω)) < α n ∀n ≥ n 0 . Consequently, it follows that where we used the growth of the ε-entropy H ε (V ; U ) in (H 0 ) and assumption (H 2 ). This estimates holds for all α ∈ (β, 1), which implies the bound stated in the Theorem. Finally, since , the fractal dimension of the sections M ν (ω) is bounded by the same value.
Pullback exponential attraction. It remains to show that M ν pullback attracts all tempered random sets at an exponential rate. By assumption (H 1 ) for any D ∈ D and ω ∈ Ω there exists N D,ω ∈ N such that ϕ m, θ −(m+k) (ω), D(θ −(m+k) (ω)) ⊂ B(θ −k ω) for all m ≥ N D,ω and k ∈ N. If n ≥ N D,ω + 1, i.e. n = N D,ω + n 0 for some n 0 ∈ N, then dist V H ϕ n, θ −n (ω), D(θ −n (ω)) , M ν (ω) for some constants C ≥ 0 and α > 0, where we used that B is tempered in the last inequality. Finally, the sections M ν (ω) = M ν (ω) V are certainly pullback Dattracting at an exponential rate, since Measurability. By Proposition 1.3.1 in [4] the pullback exponential attractor is a random set if and only if M ν is a random set. Moreover, M ν is the countable union of the sets E n , n ∈ N, and hence, it suffices to show that each set E n is a random set. However, E n is the union of U n and images of the sets U n0 , n 0 < n, under the continuous and measurable cocycle ϕ. Since U n = { U n (ω), ω ∈ Ω} is a finite random set for all n ∈ N, it follows the measurability of the sets E n , n ∈ N, and therefore the measurability of M ν .
This shows that M ν is a random pullback exponential attractor for the discrete random dynamical system (θ, ϕ) in V .

3.2.
Continuous time random dynamical systems. We now consider the continuous time setting, i.e. T = R. If (θ, ϕ) is a continuous time random dynamical system satisfying the hypotheses (H 0 )-(H 3 ), we can construct as in the previous subsection a random set satisfying all the properties of a random pullback exponential attractor, except for the positive ϕ-invariance. To obtain a positively ϕ-invariant attractor requires additional assumptions, namely, the Hölder continuity in time of the cocycle.
(H 4 ) The cocycle ϕ is Hölder continuous in time within B, in particular, there exist constants δ ω ∈ (0, 1] and K ω > 0 such that for all s, t ∈ [0,t] and n ∈ N 0 , where E n (ω) are the sets of centers constructed in the proof of Theorem 3.1 and dist V H, symm (·, ·) denotes the symmetric Hausdorff distance in V .