Anthony Sigogne
email: sigogne@univ-mlv.fr

Matthieu Constant
email: mconstan@univ-mlv.fr

Éric Laporte
email: laporte@univ-mlv.fr

Integration of Data from a Syntactic Lexicon into Generative and Discriminative Probabilistic Parsers

.

Introduction

Syntactic lexicons are rich language resources that may contain useful data for parsers like subcategorisation frames, as they provide, for each lexical entry, information about its syntactic behaviors. Most of the time, these lexicons only deal with verbs. Few, like the Lexicon-Grammar [START_REF] Gross | Constructing Lexicon-grammars[END_REF], deal with other categories like nouns, adjectives or adverbs. Many works on symbolic parsing studied the use of a syntactic lexicon, in particular linguistic formalisms like Lexical-Functional Grammars [LFG] [START_REF] Kaplan | Grammar writer's workbench[END_REF][START_REF] Riezler | Parsing the Wall Street Journal using a lexical-functional grammar and discriminative estimation techniques[END_REF][START_REF] Sagot | Analyse automatique du franc ¸ais: lexiques, formalismes, analyseurs[END_REF] or Tree Adjoining Grammars [TAG] [START_REF] Joshi | An introduction to tree adjoining grammars[END_REF][START_REF] Sagot | Intégrer les tables du Lexique-Grammaire à un analyseur syntaxique robuste à grande échelle[END_REF][START_REF] De | Building factorized TAGs with meta-grammars[END_REF]. For probabilistic parsing, we can cite LFG [START_REF] Cahill | Parsing with Automatically Acquired, Wide-Coverage, Robust, Probabilistic LFG Approximations[END_REF][START_REF] O'donovan | Large-scale induction and evaluation of lexical resources from the Penn-II and Penn-III treebanks[END_REF]Schluter and Genabith, 2008), Head-Driven Phrase Structure Grammar [HPSG] [START_REF] Carroll | The automatic acquisition of verb subcategorisations and their impact on the performance of an HPSG parser[END_REF] and Probabilistic Context-Free Grammars [PCFG] [START_REF] Briscoe | Automatic extraction of subcategorization from corpora[END_REF][START_REF] Deoskar | Re-estimation of lexical parameters for treebank PCFGs[END_REF]. The latter has incorporated valence features to PCFGs and lexicons and observes slight improvements on performances. However, lexical resources that contain valence features were obtained automatically from a corpus. Furthermore, valence features are mainly used on verbs. In this paper, we will show how we can exploit information contained in the Lexicon-Grammar in order to improve probabilistic parsers. We will in particular focus on verbs and verbal nouns1 . In section 2, we describe the probabilistic parsers used in our experiments. Section 3 briefly introduces the Lexicon-Grammar. We detail information contained in this lexicon that can be used for parsing. Then, in section 4, we present methods to integrate this information into parsers and, in section 5, we describe our experiments and discuss the obtained results.

Statistical parsers

In our experiments, we used two types of parsers: a generative parser that generates the n-best parses (n most probable parses) for a sentence according to a PCFG; a reranker that reranks the n-best parses generated from the PCFG parser according to a discriminative probabilistic model.

Non-lexicalized PCFG parser

The PCFG parser, used into our experiments, is the Berkeley Parser (called BKY thereafter) [START_REF] Petrov | Learning accurate, compact, and interpretable tree annotation[END_REF] 2 . This parser is based on a nonlexicalized PCFG model. The main problem of non-lexicalized context-free grammars is that preterminal symbols encode too general information which weakly discriminates syntactic ambiguities. BKY tries to handle the problem by generating a grammar containing complex pre-terminals. It follows the principle of latent annotations introduced by [START_REF] Matsuzaki | Probabilistic cfg with latent annotations[END_REF]. It consists in creating iteratively several grammars, which have a tagset increasingly complex. For each iteration, a symbol of the grammar is splitted in several symbols according to the different syntactic behaviors of the symbol that occur in a treebank. Parameters of the latent grammar are estimated with an algorithm based on Expectation-Maximisation (EM). Within the framework of French, [START_REF] Seddah | Adaptation de parsers statistiques lexicalisés pour le franc ¸ais : Une évaluation complète sur corpus arborés[END_REF] have shown that BKY produces state-ofthe-art performances. They have also shown that several parsers, based on the lexicalized paradigm (phrasal nodes are annotated with their headword), achieved lower scores than BKY.

Reranking parser

We have also experimented the integration of a reranker as a post-process of BKY output. For a given sentence s, a reranker selects the best parse y among the set of candidates Y (s) according to a scoring function V θ :

y ⋆ = argmax y∈Y (s) V θ (y) (1)
The set of candidates Y (s) is the n-best parses output of the baseline parser (BKY in our case), Y (s) = {y 1 , y 2 , ..., y n }. The n-best parses correspond to the n most probable parses according to the probability model of the parser. The scoring function V θ is defined by the dot product of a weight vector θ and a feature vector f :

V θ (y) = θ.f (y) = m j=1 θ j .f j (y) (2)
where the feature vector f (y) is a vector of m functions f = (f 1 , f 2 , ..., f m ), and each feature function f j maps a parse y to a real number f j (y).

The first feature f 1 (y) is the probability of the parse given by the n-best parser (cf. [START_REF] Charniak | Coarse-to-fine nbest parsing and MaxEnt discriminative reranking[END_REF]). All remaining features are integer values, and each of them is the number of times that the feature occurs in parse y. Features belong to feature schemas which are abstract schemas from which specific features are instantiated. Feature schemas that we used during our experiments are specified in the table 1. For example, a feature f 10 (y), which is an instance of the feature schema Rule, counts the number of times that a nominal phrase in y is the head of a rule which has a determinant and a noun as children.

The weight vector θ can be estimated by a machine learning algorithm from a treebank corpus which contains the gold parse for each sentence. In our case, we will use the Maximum Entropy estimator, as in [START_REF] Charniak | Coarse-to-fine nbest parsing and MaxEnt discriminative reranking[END_REF]. [START_REF] Collins | Discriminative reranking for natural language parsing[END_REF], and others are from [START_REF] Charniak | Coarse-to-fine nbest parsing and MaxEnt discriminative reranking[END_REF] 3 Lexicon-Grammar

The Lexicon-Grammar [LG] is the richest source of syntactic and lexical information for French3 that focuses not only on verbs but also on verbal nouns, adjectives, adverbs and frozen (or fixed) sentences. Its development started in the 70's by Maurice Gross and his team [START_REF] Gross | Constructing Lexicon-grammars[END_REF]. It is a syntactic lexicon represented in the form of tables. Each table encodes lexical items of a particular category sharing several syntactic properties (e.g. subcategorization information). A lexical item is a lemmatized form that can be present in one or more tables depending on its meaning and its syntactic properties. Each table row corresponds to a lexical item and a column corresponds to a property (e.g. syntactic constructions, argument distribution, and so on). A cell encodes whether a lexical item accepts a given property. Figure 1 shows a sample of verb table 12. In this table, we can see that the verb chérir (to cherish) accepts a human subject (pointed out by a + in the property N0 =: Nhum) but this verb cannot be intransitive (pointed out by ain the property N0 V). Recently, these tables have been made con- sistent and explicit [START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du franc ¸ais[END_REF] in order to be exploitable for NLP. They also have been transformed in a XML-structured format [START_REF] Constant | A generic tool to generate a lexicon for NLP from Lexicon-Grammar tables[END_REF] 4 . Each lexical entry is associated with its table identifier, its possible arguments and its syntactic constructions. For the verbs, we manually constructed a hierarchy of the tables on several levels5 . Each level contains classes which group LG tables which may not share all their defining properties but have a relatively similar syntactic behavior. Many experiments about parsing, within the framework of French [START_REF] Crabbé | Expériences d'analyse syntaxique statistique du franc ¸ais[END_REF][START_REF] Seddah | Adaptation de parsers statistiques lexicalisés pour le franc ¸ais : Une évaluation complète sur corpus arborés[END_REF], have shown that refining the tagset of the training corpus improves performances of the parser. We will follow their works by integrating information from the Lexicon-Grammar to part-of-speech tags. In this article, we will only focus on tables of verbs and verbal nouns.

Table identifiers of the lexical entries are important hints about their syntactic behaviors. For example, the table 31R indicates that all verbs belonging to this table are intransitive. The first experiment, called AnnotTable, consists in augmenting the part-of-speech tag with the table identifier(s) associated with the noun or the verb. For example, the verb chérir (to cherish) belongs to the table 12. Therefore, the induced tag is #tag 12, where #tag is the POS tag associated with the verb. For an ambiguous verb like sanctionner (to punish), belonging to two tables 6 and 12, the induced tag is #tag 6 12.

Then, in the case of verbs, we have done variants of the previous experiment by taking the hierarchy of verb tables into account. This hierarchy provides a tagset with a size which varies according to the level in the hierarchy. Identifiers added to tags depend on the verb and the specific level in the hierarchy. For example, the verb sanctionner, belonging to tables 6 and 12, has a tag #tag QTD2 at level 1. In the case of ambiguous verbs, for a given level in the hierarchy, suffixes contain all classes the verb belongs to. This experiment will be called AnnotVerbs thereafter. In the case of verbal nouns, as such a hierarchy of tables does not exist, we experimented two other methods.

Experimental setup

For our experiments, we used the richest treebank for French, the French Treebank, (later called FTB) [START_REF] Abeillé | Building a treebank for French[END_REF], containing 20,860 sentences and 540,648 words from the newspaper Le Monde (version of 2004). As this corpus is small, we used a cross-validation procedure for the evaluation. This method consists in splitting the corpus into p equal parts, then we compute training on p-1 parts and evaluations on the remaining part. We can iterate this process p times. This allows us to calculate an average score for a sample as large as the initial corpus. In our case, we set the parameter p to 10. We also used the part-of-speech tagset defined in [START_REF] Crabbé | Expériences d'analyse syntaxique statistique du franc ¸ais[END_REF] containing 28 different tags describing some complementary morphological and syntactic features (e.g. verb mood, clitics, ...)8 . Compound words have been merged in order to obtain a single token.

In the following experiments, we will test the impact of modifying the tagset of the training corpus, namely the addition of information from the Lexicon-Grammar described in the section 4. Results on evaluation parts are reported using the standard protocol called PARSEVAL [START_REF] Black | A procedure for quantitatively comparing the syntactic coverage of english grammars[END_REF] for sentences smaller than 40 words. The score f-measure (F1) takes into account the bracketing and categories of nodes (including punctu-ation nodes). For each experiment, we have reported the Baseline results (i.e. the results of BKY trained on the original treebank without annotations from the Lexicon-Grammar). We have also indicated the percentage of distinct annotated verbs and verbal nouns in the entire corpus for each annotation method9 .

Annotation of verb tags

We first conducted experiments on verbs described in section 4, namely AnnotTable and AnnotVerbs. For non-ambiguous verbs, we observe that the experiment AnnotTable highly deteriorates performances. This comes most probably from the grammar which is too fragmented because of the significant size of the part-of-speech tagset (as shown in table 5). However, the effect is reversed as soon as we use levels of the hierarchy of tables (levels 2 and 3 only). The use of the table hierarchy causes the increase of the number of verbs annotated as non-ambiguous and the decrease of the size of the tagset. Considering ambiguous verbs do not improve performances (results are shown only for levels 2 and 3 with maximal ambiguity of 2) because of the large size of the tagset (as for experiment AnnotTable). We can see on Figure 3 absolute gains according to verb annotation methods on evaluation parts.

We have displayed curves for methods AnnotTable and AnnotVerbsX, where X is the level in the hierarchy (without ambiguity). Higher we are in the hierarchy of tables, the more we obtain better performances. Levels 2 and 3 are globally above the baseline for most of their evaluation parts. Therefore, this would mean that table identifiers of verbs and the hierarchy are a real help for parsing and do not produce a random effect. On indicates the average error reduction rate associated with the corresponding average number of error corrected (inside brackets). The NP and PP phrases are those that have the highest number of errors corrected (the low reduction rate can be explained by the fact that these two phrases have the highest number of errors). Furthermore, they are linked to each other because, generally, a PP has a NP kernel. Therefore, if a NP is corrected, the corresponding PP is also corrected (if it is the only error).

Annotation of noun tags

For verbal nouns, we successively conducted several experiments AnnotTable, AnnotNouns and An-notIN, described in section 4. Results are given in table 7. As for verbs, we have reported the results for the experiment AnnotNouns with respect to the parameter Ambiguity (the maximum number of classes being associated with a noun is 3). The various noun annotation methods slightly increase performances of the parser. Unlike verbs, the method AnnotTable does not degrade performances because there are much less nouns in the corpus belonging to the syntactic lexicon (less than 9% as shown in table 8), hence the limited impact of the new tagset. The use of a simple hierarchy of the noun tables, through experiment AnnotNouns, achieves positive gains but, here, insignificant. Moreover, we obtain a slight improvement by annotating some ambiguous nouns. Surprisingly, the method which gives the best result, despite its simplicity, is AnnotIN. We can see in

Combination of annotations

In Table 9: Results from cross-validation evaluation according to combination of annotations Combination of annotations does not increase the gains obtained with the method AnnotVerbs and we even observe a slight decrease.

Impact on a reranker

We also experimented the integration of a discriminative reranker (cf. section 2). We practically set to 10 the number of parses generated by BKY for each sentence (therefore, the 10 most probable parses). The following experiment consists in evaluating the impact of the modification of the tagset on a reranker. We called Reranker(Baseline) the experiment using the reranker with BKY trained on the original corpus (without annotations from the Lexicon-Grammar). Reranker(AnnotVerbs) is the experiment based on BKY that is trained on the corpus annotated by the best verb annotation method, An-notVerbs (level 3 of hierarchy without ambiguity First, we can see that Reranker(Baseline) improves performances with an absolute gain of +1,46 as compared with the baseline. These results are comparable to scores obtained for English [START_REF] Charniak | Coarse-to-fine nbest parsing and MaxEnt discriminative reranking[END_REF]. Then, we observe that the experiment Reranker(AnnotVerbs) increases the f-measure by +0,2 compared with Reranker(Baseline) (and to a lesser extent, the tagging accuracy by +0,07). The power of the discriminative model of the reranker implies that the gap of performances between the two experiments based on the reranker is less than the one obtained from experiments only based on BKY (+0,2 against +0,34). In addition, the oracle fmeasure is improved (+0,27), which means that analyses generated by BKY are slightly better. We can see on Figure 5 absolute gains given by the reranker on all evaluation parts according to the two methods described above. Globally, the method Reranker(AnnotVerbs) has a curve slightly above the one of Reranker(Baseline). Note that the first one outperforms the latter on 8 of 10 evaluation parts. All these observations confirm that the syntactic lexicon through the experiment An-notVerbs is able to improve performances on both

Conclusions

The work described in this paper shows that by adding some information from a syntactic lexicon like the Lexicon-Grammar, we are able to improve performances of several probabilistic parsers. These performances are mainly obtained thanks to a hierarchy of verb tables that can limit ambiguity in terms of number of classes associated with a verb. This has the effect of increasing the coverage of verbs annotated according to the level of granularity used. However, once we include some ambiguity, performances drop. Results obtained on verbal nouns with a simple hierarchy of tables are insignificant but suggest a degree of progress with a more complex hierarchy as the one available for verbs. In the near future, we plan to reproduce these experiments by taking into account of word clustering methods introduced by [START_REF] Koo | Simple semi-supervised dependency parsing[END_REF][START_REF] Candito | Improving generative statistical parsing with semi-supervised word clustering[END_REF][START_REF] Candito | Parsing word clusters[END_REF]. Thanks to a semi-supervized algorithm, these methods can reduce the size of the lexicon of the grammar by grouping words according to their behaviors in a treebank. These methods could be complementary to annotation methods described in this paper. Moreover, we plan to exploit the LFG formalism in order to use a syntactic lexicon more easily than for PCFGs, as many works have reported performance improvements for these models [START_REF] Cahill | Parsing with Automatically Acquired, Wide-Coverage, Robust, Probabilistic LFG Approximations[END_REF][START_REF] Deoskar | Re-estimation of lexical parameters for treebank PCFGs[END_REF].

Figure 1 :

 1 Figure 1: Sample of verb table 12

  Figure 2: Sample of the hierarchy of verb tables

Figure 3 :

 3 Figure 3: Absolute gains (F1) of verb annotation methods on evaluation parts (baseline is the horizontal line at 0 on y axis)

Figure 4 :

 4 Figure 4: Absolute gains (F1) of noun annotation methods on evaluation parts (baseline is the horizontal line at 0 on y axis) Figure 4 absolute gains according to noun annotation methods on all evaluation parts. Unlike verbs, absolute gains are closer to the baseline. The best method AnnotIN is able to improve significantly 4 of 10 evaluation parts (+0,4 to +0,8).

Figure 5 :

 5 Figure 5: Absolute gains (F1) given by the reranker on evaluation parts (BKY(baseline) is the horizontal line at 0 on y axis)

Table 1 :

 1 Features used in this work. Those with a △ are from

	Feature schemas
	Rule	Edges
	Word	WordEdges
	Heavy	Heads
	HeadTree	WProj
	Bigrams △	NgramTree
	Trigrams △

Table 2 :

 2 Characteristics of the hierarchy of verb tables

	The hierarchy of tables have the advantage of re-
	ducing the number of classes associated with each
	4 These	resources	are	freely	available	at
	http://infolingu.univ-mlv.fr>Language Resources> Lexi-
	con Grammar>Download

  The first one, called AnnotIN, consists in adding a suffix IN to the tag of a noun if this noun occurs in the syntactic lexicon, and therefore if it is a verbal noun. The second method, called AnnotNouns, consists in creating a hierarchy of noun tables from the table of classes of verbal nouns. This hierarchy is made accordingly to the maximum number of arguments that a noun of a table can have according to defining properties specified for this table. As a consequence, the hierarchy has a single level. For example, nouns of the table N aa can have at most 2 arguments contrary to those of table N an04 which can have only one. The characteristics of each level are specified in table 3 7 (level 0 represents the set of tables of the Lexicon-Grammar). We can state that there are 8,531 distinct nominal forms for 12,351 resulting entries in tables of nouns.

	Level #classes AVG 1 AVG 2
	0	76	162	1.43
	1	3	3,413	1.2

Table 3 :

 3 Characteristics of the hierarchy of noun tables

  The experimental results are shown in the table 4. In the case of the method AnnotVerbs, we varied two parameters, Lvl (for Level) indicating the level of the hierarchy used and Amb. (for Ambiguity) indicating that a tag of a verb is changed only if this verb belongs to a number of classes less than or equal to the number specified by this parameter.

	Method	Lvl/Amb. F1/Tagging Absolute gains (F1)
	Baseline	-/-	85.05/97.43
	AnnotTable -/1	84.49/97.29
	AnnotVerbs 1/1	85.06/97.46
	AnnotVerbs 2/1	85.35/97.41
	AnnotVerbs 3/1	85.39/97.49
	AnnotVerbs 2/2	84.60/97.35
	AnnotVerbs 3/2	85.20/97.48
			-0.5 0.0	+0.5
	Table 4: Results from cross-validation evaluation
	according to verb annotation methods
	Method	Size of tagset % annotated verbs
	Baseline	28	-
	AnnotTable	228	18,6%
	AnnotVerbs 1/1 89	21,5%
	AnnotVerbs 2/1 76	22,5%
	AnnotVerbs 3/1 47	33,9%
	AnnotVerbs 2/2 246	44,7%
	AnnotVerbs 3/2 75	55,7%

Table 5 :

 5 Size of tagset and percentage of annotated verbs according to verb annotation methods

  table 6, we

	Phrase label Meaning	Error reduction
	Ssub	subordinate clause	5,3% (52)
	Sint	internal clause	3,6% (47)
	PP	prepositional phrase 3,1% (272)
	Srel	relative clause	2,2% (17)
	NP	nominal phrase	2,1% (347)
	VPinf	infinitive phrase	2,1% (34)

Table 6 :

 6 Top most error reductions according to phrase label

can see the top most error reductions according to phrase label, for the best verb annotation method (AnnotVerbs with level 3 of the hierarchy). For each phrase, the column called Error reduction

Table 8 :

 8 Size of tagset and percentage of annotated verbal nouns according to noun annotation methods

Table 10 :

 10 ). Results are shown in table 10. The column named Oracle F1/Tagging indicates oracle scores for fmeasure and tagging accuracy. An oracle score is the best global score that we could obtain whether we choose, for each input sentence, the best parse from the n-best parses. With this score, we can estimate the performance limit of a parser and the global quality of parses generated. AnnotVerbs) 86.71/97.49 91.99/98.08 Results from cross-validation evaluation for reranking process.

	Method	F1/Tagging Oracle F1/Tagging
	BKY(Baseline)	85.05/97.43 -
	BKY(AnnotVerbs)	85.39/97.49 -
	Reranker(Baseline)	86.51/97.42 91,72/98.03
	Reranker(

Verbal nouns are nouns playing the role of a predicate in the sentence.

The Berkeley parser is freely available at http://code.google.com/p/berkeleyparser/downloads/list

We can also cite lexicons like LVF[START_REF] Dubois | Les verbes franc ¸ais[END_REF], Dicovalence[START_REF] Eynde | La valence: l'approche pronominale et son application au lexique verbal[END_REF] and Lefff[START_REF] Sagot | The lefff, a freely available, accurate and large-coverage lexicon for french[END_REF].

The hierarchy of verb tables is available at : http://igm.univ-mlv.fr/∼sigogne/arbre-tables.xlsx

We can also state that 3,121 verb forms(3,195 entries) are unambiguous. This means that all their entries occur in a single table.verb of the tables. We will see that this ambiguity reduction is crucial in our experiments.4 Exploitation of the Lexicon-Grammar data

The number of non-ambiguous nouns is 6126 for 6175 entries.

There are 6 distinct tags for verbs and 2 distinct tags for nouns.

The corpus contains 3058 distinct verbal forms and 17003 distinct nominal forms.


Armen Shirikyan
email: armen.shirikyan@u-cergy.fr

Sergey Zelik
email: s.zelik@surrey.ac.uk

Exponential attractors for random dynamical systems and applications

Keywords: AMS subject classifications: 35B41, 35K57, 35R60, 60H15 Random exponential attractors, stochastic PDE's, reactiondiffusion equation

The paper is devoted to constructing a random exponential attractor for some classes of stochastic PDE's. We first prove the existence of an exponential attractor for abstract random dynamical systems and study its dependence on a parameter and then apply these results to a nonlinear reaction-diffusion system with a random perturbation. We show, in particular, that the attractors can be constructed in such a way that the symmetric distance between the attractors for stochastic and deterministic problems goes to zero with the amplitude of the random perturbation.

Introduction

The theory of attractors for partial differential equations (PDE's) has been developed intensively since late seventies of the last century. It is by now well known that many dissipative PDE possesses a minimal attractor, even if the Cauchy problem is not known to be well posed. Moreover, one can establish explicit upper and lower bounds for the dimension of a minimal attractor. A comprehensive presentation of the theory of attractors can be found in [START_REF] Chepyzhov | Attractors for Equations of Mathematical Physics[END_REF][START_REF] Sell | Dynamics of Evolutionary Equations[END_REF]. Similar results were also proved in the case of random dynamical systems (RDS) generated by stochastic PDE's, such as the Navier-Stokes system or reactiondiffusion equations with random perturbations; see [START_REF] Crauel | Attractors for random dynamical systems[END_REF][START_REF] Crauel | Random attractors[END_REF]. A drawback of the theory of attractors is that, in general, it is impossible to have any estimate for the rate of convergence to the minimal attractor. Furthermore, in the case of RDS, the attraction property holds when the initial time goes to -∞, whereas one is usually interested in the large-time asymptotics of solutions for the Cauchy problem with a fixed initial time. To remedy these shortcomings, a concept of exponential attractors was suggested in [START_REF] Eden | Exponential attractors for dissipative evolution equations[END_REF] for deterministic problems. In contrast to the attractors discussed above, they do not possess any minimality property, but still have a finite fractal dimension and, moreover, attract trajectories exponentially fast. We refer the reader to the review paper [START_REF] Miranville | Attractors for dissipative partial differential equations in bounded and unbounded domains[END_REF] (and the references therein) for a detailed account of the results on exponential attractors obtained so far.

The aim of this article is to construct finite-dimensional exponential attractors for some classes of RDS and then to show that the general results are applicable in the case of reaction-diffusion equations. To be precise, let us consider from the very beginning the following problem in a bounded domain D ⊂ R n with a smooth boundary ∂D:

u -a∆u + f (u) = h(x) + η(t, x),
(1.1)

u ∂D = 0, (1.2)
u(0, x) = u 0 (x).

(1.3)

Here u = (u 1 , . . . , u k ) t is an unknown vector function, a is a k × k matrix such that a + a t > 0, f ∈ C 2 (R k , R k ) is a function satisfying some natural growth and dissipativity conditions, h(x) is a deterministic external force acting on the system, and η is a random process, white in time and regular in the space variables; see Section 2.2 for the exact hypotheses imposed on f and η. The Cauchy problem (1.1)-(1.3) is well posed in the space H := L 2 (D, R k ), and we denote by Φ = {ϕ t : H → H, t ≥ 0} the corresponding RDS defined on a probability space (Ω, F , P) with a group of shift operators {θ t : Ω → Ω, t ∈ R} (see Section 2.2). We have the following result on the existence of an exponential attractor for Φ.

Theorem A. There is a random compact set M ω ⊂ H and an event Ω * ⊂ Ω of full measure such that the following properties hold for ω ∈ Ω * .

Semi-invariance. ϕ ω t (M ω ) ⊂ M θtω for all t ≥ 0. Exponential attraction. There is β > 0 such that for any ball B ⊂ H we have

sup u∈B inf v∈M θ t ω ϕ ω t (u) -v ≤ C(B)e -βt , t ≥ 0,
where C(B) is a constant depending only on B.

Finite-dimensionality. There is a number d > 0 such that dim f (M ω ) ≤ d, where dim f stands for the fractal dimension of M ω .

Note that this type of results are well known for non-autonomous dynamical systems (e.g., see [START_REF] Efendiev | Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems[END_REF][START_REF] Miranville | Attractors for dissipative partial differential equations in bounded and unbounded domains[END_REF]). An essential difference between nonautonomous and stochastic systems is that the latter deal with forces which are, in general, unbounded in time, and some key quantities can be controlled only after taking the time average. This turns out to be sufficient for the construction of an exponential attractor.

Let us now assume that the random force η in Eq. (1.1) is replaced by εη, where ε ∈ [-1, 1] is a parameter. We denote by M ε ω the corresponding exponential attractors. Since in the limit case ε = 0 the equation is no longer stochastic, the corresponding attractor M = M 0 is also independent of ω. A natural question is whether one can construct M ε ω in such a way that the symmetric distance between the attractors of stochastic and deterministic equations goes to zero as ε → 0. The following theorem gives a positive answer to that question.

Theorem B. The exponential attractors M ε ω , ε ∈ [-1, 1], can be constructed in such a way that d s (M ε ω , M) → 0 almost surely as ε → 0, where d s stands for the symmetric distance between two subsets of H.

We refer the reader to Section 4 for more precise statements of the results on the existence of exponential attractors and their dependence on a parameter. Let us note that various results similar to Theorem B were established earlier in the case of deterministic PDE's; e.g., see the papers [START_REF] Fabrie | Uniform exponential attractors for a singularly perturbed damped wave equation[END_REF][START_REF] Efendiev | Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems[END_REF], the first of which is devoted to studying the behaviour of exponential attractors under singular perturbations, while the second deals with non-autonomous dynamical systems and proves Hölder continuous dependence of the exponential attractor on a parameter.

We emphasize that the convergence in Theorem B differs from the one in the case of global attractors, for which, in general, only lower semicontinuity can be established. For instance, let us consider the following one-dimensional ODE perturbed by the time derivative of a standard Brownian motion w: u = u -u 3 + ε ẇ.

(1.4)

When ε = 0, the global attractor A for (1.4) is the interval [-1, 1] and is regular in the sense that it consists of the stationary points and the unstable manifolds around them. It is well known that the regular structure of an attractor is very robust and survives rather general deterministic perturbations, and in many cases it is possible to prove that the symmetric distance between the attractors for the perturbed and unperturbed systems goes to zero; see [START_REF] Babin | Attractors of Evolution Equations[END_REF][START_REF] Chepyzhov | Regular attractors and their non-autonomous perturbations[END_REF]. On the other hand, it is proved in [START_REF]Additive noise destroys a pitchfork bifurcation[END_REF] that the random attractor A ε ω for (1.4) consists of a single trajectory and, hence, the symmetric distance between A and A ε ω does not go to zero as ε → 0. In conclusion, let us mention that some results similar to those described above hold for other stochastic PDE's, including the 2D Navier-Stokes system. They will be considered in a subsequent publication.

The paper is organised as follows. In Section 2, we present some preliminaries on random dynamical systems and a reaction-diffusion equation perturbed by a spatially regular white noise. Section 3 is devoted to some general results on the existence of exponential attractors and their dependence on a parameter. In Section 4, we apply our abstract construction to the stochastic reactiondiffusion system (1.1)-(1.3). Appendix gathers some results on coverings of random compact sets and their image under random mappings, as well as the time-regularity of stochastic processes.

Acknowledgements

Notation

Let J ⊂ R be an interval, let D ⊂ R n be a bounded domain with smooth boundary ∂D, and let X be a Banach space. Given a compact set K ⊂ X, we denote by H ε (K, X) its Kolmogorov ε-entropy; see [START_REF] Lorentz | Approximation of Functions[END_REF]. If Y is another Banach space with compact embedding Y ⋐ X, then we write H ε (Y, X) for the ε-entropy of a unit ball in Y considered as a subset in X. We denote by ḂX (v, r) and B X (v, r) the open and closed balls in X of radius r centred at v and by O r (A) the closed r-neighbourhood of a subset A ⊂ X. The closure of A in X is denoted by [A] X . Given any set C, we write #C for the number of its elements.

We shall use the following function spaces: L p = L p (D) denotes the usual Lebesgue space in D endowed with the standard norm • L p . In the case p = 2, we omit the subscript from the notation of the norm. We shall write L p (D, R k ) if we need to emphasise the range of functions. W s,p = W s,p (D) stands for the standard Sobolev space with a norm • s,p . In the case p = 2, we write H s = H s (D) and • s , respectively. We denote by H s 0 = H s 0 (D) the closure in H s of the space of infinitely smooth functions with compact support. C(J, X) stands for the space of continuous functions f : J → X.

When describing a property involving a random parameter ω, we shall assume that it holds almost surely, unless specified otherwise. Given a random function f ω : D → X, we shall say that it is (almost surely) Hölder-continuous if there is γ ∈ (0, 1) such that, for any bounded ball B ⊂ R n , we have

f ω (t 1 ) -f ω (t 2 ) X ≤ C ω |t 1 -t 2 | γ , t 1 , t 2 ∈ B,
where C ω = C ω (B) is an almost surely finite random variable. If f depends on an additional parameter y ∈ Y (that is, f = f y ω (t)), then we say that f is Hölder-continuous uniformly in y if the above inequality holds for f y ω (t) with a random constant C ω (B) not depending on y.

We denote by c i and C i unessential positive constants not depending on other parameters.

Preliminaries

Random dynamical systems and their attractors

Let (Ω, F , P) be a complete probability space, {θ t , t ∈ R} be a group of measurepreserving transformations of Ω, and X be a separable Banach space. Recall that a continuous random dynamical system in X over {θ t } (or simply an RDS in X) is defined as a family of continuous mappings Φ = {ϕ ω t : X → X, t ≥ 0} that satisfy the following conditions:

Measurability. The mapping (t, ω, u) → ϕ ω t (u) from R + × Ω × X to X is measurable with respect to the σ-algebras B R+ ⊗ F ⊗ B X and B X .
Perfect co-cycle property. For almost every ω ∈ Ω, we have the identity

ϕ ω t+s = ϕ θsω t • ϕ ω s , t, s ≥ 0. (2.1)
Time regularity. For almost every ω ∈ Ω, the function (t, τ ) → ϕ θτ ω t (u), defined on R + × R with range in X, is Hölder-continuous with some deterministic exponent γ > 0, uniformly with respect to u ∈ K for any compact subsets K ⊂ X.

An example of RDS is given in the next subsection, which is devoted to some preliminaries on a reaction-diffusion system with a random perturbation.

Large-time asymptotics of trajectories for RDS is often described in terms of attractors. This paper deals with random exponential attractors, and we now define some basic concepts.

Recall that the distance between a point u ∈ X and a subset F ⊂ X is given by d(u, F ) = inf v∈F u -v . The Hausdorff and symmetric distances between two subsets is defined by

d(F 1 , F 2 ) = sup u∈F1 d(u, F 2 ), d s (F 1 , F 2 ) = max d(F 1 , F 2 ), d(F 2 , F 1 ) .
We shall write d X and d s X to emphasise that the distance is taken in the metric of X. Let {M ω , ω ∈ Ω} be a random compact set in X, that is, a family of compact subsets such that the mapping ω → d(u, M ω ) is measurable for any u ∈ X.

Definition 2.1. A random compact set {M ω } is called a random exponential attractor for the RDS {ϕ t } if there is a set of full measure Ω * ∈ F such that the following properties hold for ω ∈ Ω * .

Semi-invariance. For any t ≥ 0, we have

ϕ ω t (M ω ) ⊂ M θtω .
Exponential attraction. There is a constant β > 0 such that

d ϕ ω t (B), M θtω ≤ C(B)e -βt for t ≥ 0, (2.2)
where B ⊂ H is an arbitrary ball and C(B) is a constant that depends only on B.

Finite-dimensionality. There is random variable d ω ≥ 0 which is finite on Ω * such that dim f M ω ≤ d ω .

(2.3) Time continuity. The function t → d s M θtω , M ω is Hölder-continuous on R with some exponent δ > 0.

We shall also need the concept of a random absorbing set . Recall that a random compact set A ω is said to be absorbing for Φ if for any ball B ⊂ X there is T (B) ≥ 0 such that

ϕ ω t (B) ⊂ A θtω for t ≥ T (B), ω ∈ Ω.
(2.4)

All the above definitions make sense also in the case of discrete time, that is, when the time variable varies on the integer lattice Z. The only difference is that the property of time continuity should be skipped for discrete-time RDS and their attractors. In what follows, we shall deal with both situations.

Reaction-diffusion system perturbed by white noise

Let D ⊂ R n be a bounded domain with a smooth boundary ∂D. We consider the reaction-diffusion system (1.1), (1.2), in which u = (u 1 , . . . , u k ) t is an unknown vector function and a is a k × k matrix such that a + a t > 0.

(2.5)

We assume that f ∈ C 2 (R k , R k ) satisfies the following growth and dissipativity conditions:

f (u), u ≥ -C + c|u| p+1 , (2.6) f ′ (u) + f ′ (u) t ≥ -CI, (2.7) |f ′ (u)| ≤ C(1 + |u|) p-1 , (2.8)
where •, • stands for the scalar product in R k , f ′ (u) is the Jacobi matrix for f , I is the identity matrix, c and C are positive constants, and 0 ≤ p ≤ n+2 n-2 . As for the right-hand side of (1.1), we assume h ∈ L 2 (D, R k ) is a deterministic function and η is a spatially regular white noise. That is,

η(t, x) = ∂ ∂t ζ(t, x), ζ(t, x) = ∞ j=1 b j β j (t)e j (x), (2.9)
where {β j (t), t ∈ R} is a sequence of independent two-sided Brownian motions defined on a complete probability space (Ω, F , P), {e j } is an orthonormal basis in L 2 (D, R k ) formed of the eigenfunctions of the Dirichlet Laplacian, and b j are real numbers satisfying the condition

B := ∞ j=1 b 2 j < ∞. (2.10)
In what follows, we shall assume that (Ω, F , P) is the canonical space; that is, Ω is the space of continuous functions ω : R → H vanishing at zero, P is the law of ζ (see (2.9)), and F is the P-completion of the Borel σ-algebra. In this case, the process ζ can be written in the form ζ ω (t) = ω(t), and a group of shifts θ t acts on Ω by the formula (θ t ω)(s) = ω(t + s) -ω(t). Furthermore, it is well known (e.g., see Chapter VII in [START_REF] Stroock | Probability. An Analytic Viewpoint[END_REF]) the restriction of {θ t , t ∈ R} to any lattice T Z is ergodic.

Let us denote H = L 2 (D, R k ) and V = H 1 0 (D, R k ).
The following result on the well-posedness of problem (1.1)-(1.3) can be established by standard methods used in the theory of stochastic PDE's (e.g., see [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF][START_REF] Flandoli | Dissipativity and invariant measures for stochastic Navier-Stokes equations[END_REF]).

Theorem 2.2. Under the above hypotheses, for any u 0 ∈ H there is a stochastic process {u(t), t ≥ 0} that is adapted to the filtration generated by ζ(t) and possesses the following properties:

Regularity: Almost every trajectory of u(t) belongs to the space

X = C(R + , H) ∩ L 2 loc (R + , V ) ∩ L p+1 loc (R + × D).
Solution: With probability 1, we have the relation

u(t) = u 0 + t 0 a∆u -f (u) + h ds + ζ(t), t ≥ 0,
where the equality holds in the space H -1 (D).

Moreover, the process u(t) is unique in the sense that if v(t) is another process with the same properties, then with probability 1 we have u(t) = v(t) for all t ≥ 0.

The family of solutions for (1.1), (1.2) constructed in Theorem 2.2 form an RDS in the space H. Let us describe in more detail a set of full measure on which the perfect co-cycle property and the Hölder-continuity in time are true.

Let us denote by z = z ω (t) the solution of the linear equation

ż -a∆z = h + η(t), (2.11)
supplemented with the zero initial and boundary conditions. Such a solution exists and belongs to the space Y := C(R + , H)∩L 2 loc (R + , V ) with probability 1. Moreover, one can find a set Ω * ∈ F of full measure such that θ t (Ω * ) = Ω * for all t ∈ R and z ω ∈ Y for ω ∈ Ω * . We now write a solution of (1.1)-(1.3) in the form u = z + v and note that v must satisfy the equation

v -a∆v + f (z + v) = 0.
(2.12)

For any ω ∈ Ω * and u 0 ∈ H, this equation has a unique solution v ∈ X issued from u 0 . The RDS associated with (1.1)-(1.2) can be written as

ϕ ω t (u 0 ) = z ω (t) + v ω (t) for ω ∈ Ω * , 0 for ω / ∈ Ω * .
Then Φ = {ϕ t , t ≥ 0} is an RDS in the sense defined in the beginning of Section 2.1, and the time continuity and perfect co-cycle properties hold on Ω * .

3 Abstract results on exponential attractors

Exponential attractor for discrete-time RDS

Let H be a Hilbert space and let Ψ = {ψ ω k , k ∈ Z + } be a discrete-time RDS in H over a group of measure-preserving transformations {σ k } acting on a complete probability space (Ω, F , P). We shall assume that Ψ satisfies the following condition.

Condition 3.1. There is a Hilbert space V compactly embedded in H, a random compact set {A ω }, and constants m, r > 0 such that the properties below are satisfied.

Absorption. The family {A ω } is a random absorbing set for Ψ .

Stability. With probability 1, we have

ψ ω 1 O r (A ω ) ⊂ A σ1ω . (3.1)
Lipschitz continuity. There is an almost surely finite random variable K ω ≥ 1 such that K m ∈ L 1 (Ω, P) and

ψ ω 1 (u 1 ) -ψ ω 1 (u 2 ) V ≤ K ω u 1 -u 2 H for u 1 , u 2 ∈ O r (A ω ). (3.2)
Kolmogorov ε-entropy. There is a constant C and an almost surely finite random variable

C ω such that C ω K m ω ∈ L 1 (Ω, P), H ε (V, H) ≤ C ε -m , (3.3) H ε (A ω , H) ≤ C ω ε -m . (3.4)
The following theorem is an analogue for RDS of a well-known result on the existence of an exponential attractor for deterministic dynamical systems; e.g., see Section 3 of the paper [START_REF] Miranville | Attractors for dissipative partial differential equations in bounded and unbounded domains[END_REF] and the references therein.

Theorem 3.2. Assume that the discrete-time RDS Ψ satisfies Condition 3.1. Then Ψ possesses an exponential attractor M ω . Moreover, the attraction property holds for the norm of V :

d V ψ ω k (B), M σ k ω ≤ C(B)e -βk for k ≥ 0, (3.5)
where B ⊂ H is an arbitrary ball and C(B) and β > 0 are some constants not depending on k.

The proof given below will imply that (3.5) holds for B = A ω with C(B) = r, and that in inequality (3.5) the constant in front of e -βk has the form

C(B) = 2 T (B) r, (3.6)
where T (B) is a time after which the image of the ball B under the mapping ψ ω k belongs to the absorbing set A σtω . Furthermore, as is explained in Remark 3.4 below, under an additional assumption, the fractal dimension dim f (M ω ) can be bounded by a deterministic constant.

Proof. We repeat the scheme used in the case of deterministic dynamical systems. However, an essential difference is that we have a random parameter and need to follow the dependence on it. In addition, the constants entering various inequalities are now (unbounded) random variables, and we shall need to apply the Birkhoff ergodic theorem to bound some key quantities.

Step 1: An auxiliary construction. Let us define a sequence of random finite sets V k (ω) in the following way. Applying Lemma 5.1 with δ ω = (2K ω ) -1 r to the random compact set A ω , we construct a random finite set U 0 (ω) such that

d s A ω , U 0 (ω) ≤ δ ω , (3.7) ln #U 0 (ω) ≤ 2 m C ω δ -m ω ≤ (4/r) m C ω K m ω . (3.8) Since K ω ≥ 1, we have δ ω ≤ r/2, whence it follows that U 0 (ω) ⊂ O r (A ω ). Setting V 1 (σ 1 ω) = ψ ω 1 (U 0 (ω))
, in view of (3.1), (3.2), and (3.7), we obtain

ψ ω 1 (A ω ) ⊂ u∈V1(σ1ω) B V (u, r/2) =: C 1 (ω), V 1 (σ 1 ω) ⊂ O r/2 ψ ω 1 (A ω ) ∩ A σ1ω . Now note that C 1 (ω) is a random compact set in H. Moreover, it follows from (3.3) and (3.8) that H ε (C 1 (ω), H) ≤ ln #V 1 (σ 1 ω) + H 2ε/r (V, H) ≤ (4/r) m C ω K m ω + (r/2) m Cε -m . (3.9) Applying Lemma 5.1 with δ ω = (4K σ1ω ) -1 r to C 1 (ω), we construct a random finite set U 1 (ω) such that d s C 1 (ω), U 1 (ω) ≤ δ ω , ln #U 1 (ω) ≤ H δω /2 (C 1 (ω), H) ≤ (4/r) m C ω K m ω + 2 m CK m σ1ω .
Repeating the above argument and setting

V 2 (σ 2 ω) = ψ σ1ω 1 (U 1 (ω))
, we obtain

ψ σ1ω 1 C 1 (ω) ⊂ u∈V2(σ2ω) B V (u, r/4) =: C 2 (ω), V 2 (σ 2 ω) ⊂ O r/4 ψ σ1ω 1 (C 1 (ω) ∩ A σ2ω .
Moreover, C 2 (ω) is a random compact set H whose ε-entropy satisfies the inequality (cf. (3.9))

H ε (C 2 (ω), H) ≤ ln #V 2 (σ 2 ω) + H 4ε/r (V, H) ≤ (4/r) m C ω K m ω + 2 m CK m σ1ω + (r/4) m Cε -m .
Iterating this procedure and recalling that σ k : Ω → Ω is a one-to-one transformation, we construct random finite sets V k (ω), k ≥ 1, and unions of balls

C k (ω) := u∈V k (σ k ω) B V (u, 2 -k r)
such that the following properties hold for any integer k ≥ 1:

ψ ω k (A ω ) ⊂ C k (ω), (3.10) V k (ω) ⊂ O 2 1-k r ψ σ-1ω 1 (C k-1 (σ 1-k ω)) ∩ A ω , (3.11) ln #V k (ω) ≤ (4/r) m C σ -k ω K m σ -k ω + 2 m C k-1 j=1 K m σ j-k ω .
(3.12)

Step 2: Description of an attractor. Let us define a sequence of random finite sets by the rule

E 1 (ω) = V 1 (ω), E k (ω) = V k (ω) ∪ ψ σ-1ω 1 E k-1 (σ -1 ω) , k ≥ 2.
The very definition of E k implies that

ψ ω 1 (E k (ω)) ⊂ E k+1 (σ 1 ω). (3.13) and since #V k (ω) ≤ #V k+1 (σ 1 ω), it follows from (3.12) that ln #E k (ω) ≤ ln k + ln #V k (ω) ≤ ln k + (4/r) m C σ -k ω K m σ -k ω + 2 m C k-1 j=1 K m σ j-k ω . (3.14)
Furthermore, it follows from (3.10) that

d V ψ ω k (A ω ), V k (σ k ω) ≤ 2 -k r, k ≥ 0. (3.15)
We now define a random compact set M ω by the formulas

M ω = M ′ ω V , M ′ ω = ∞ k=1 E k (ω). (3.16)
We claim that M ω is a random exponential attractor for Ψ . Indeed, the semiinvariance follows immediately from (3.13). Furthermore, inequality (3.15) implies that

d V ψ ω k (A ω ), M σ k ω ≤ 2 -k r for any k ≥ 0.
Recalling that A ω is an absorbing set and using inclusion (2.4), together with the co-cycle property, we obtain

d V ψ ω k (B), M σ k ω ≤ d V ψ σT ω k-T (A σT ω ), M σ k-T (σT ω) ≤ 2 T -k r,
where T = T (B) is the constant entering (2.4). This implies the exponential attraction inequality (3.5) with β = ln 2 and C(B) = 2 T (B) r. It remains to prove that M ω has a finite fractal dimension. This is done in the next step.

Step 3: Estimation of the fractal dimension. We shall need the following lemma, whose proof is given at the end of this subsection.

Lemma 3.3. Under the hypotheses of Theorem 3.2, for any integers l ≥ 0, k ∈ Z, and m ∈ [0, l], we have

d V E k (σ k ω), ψ σ k-m ω m (A σ k-m ω ) ≤ 2 2-(k-l) r l j=1 K σ k-j ω .
(3.17) Inequality (3.17) with m = l and ω replaced by σ -k ω implies that

d V E k (ω), ψ σ -l ω l (A σ -l ω ) ≤ 2 2-(k-l) r l j=1 K σ-j ω , (3.18)
where k ≥ 1 is arbitrary. On the other hand, in view of (3.15) with k = l and ω replaced by σ -l ω, we have

d V ψ σ -l ω l (A σ -l ω ), V l (ω) ≤ 2 -l r.
Combining this with (3.18), we obtain

d V k≥n E k (ω), V l (ω) ≤ r 2 -l + 2 2-(n-l) l j=1 K σ-j ω , (3.19)
where n ≥ 1 and l ∈ [1, n] are arbitrary integers. Since {E k (ω)} is an increasing sequence and

V l (ω) ⊂ E l (ω) ⊂ M ω for any l ≥ 1, inequality (3.19) implies that d s V M ω , n l=1 V l (ω) ≤ r inf l∈[1,n] 2 -l + 2 2-(n-l) l j=1 K σ-j ω =: ε n (ω), (3.20)
where n ≥ 1 is arbitrary. If we denote by N ε (ω) the minimal number of balls of radius ε > 0 that are needed to cover M ω , then inequality (3.20) implies that

N εn(ω) (ω) ≤ n k=1 #V k (ω). Since V k ⊂ E k and #V k (ω) ≥ #V k-1 (σ -1 ω), it follows from (3.12) that ln N εn(ω) (ω) ≤ ln n #E n (ω) ≤ ln n 2 #V n (ω) ≤ 2 ln n + (4/r) m C σ-nω K m σ-nω + 2 m C n-1 k=1 K m σ -k ω . (3.21)
Since K m ∈ L 1 (Ω, P), by the Birkhoff ergodic theorem (see Section 1.6 in [START_REF] Walters | An Introduction to Ergodic Theory[END_REF]), we have

lim n→∞ 1 n n k=1 K m σ -k ω = ξ ω , (3.22)
where ξ ω is an integrable random variable. This implies, in particular, that

n -1 K m σ-nω → 0 as n → ∞. By a similar argument, n -1 C σ-nω K m σ-nω → 0 as n → ∞. Combining this with (3.22) and (3.21), we derive ln N εn(ω) (ω) ≤ 2 m Cξ ω n + o ω (n), (3.23)
where, given α ∈ R, we denote by o ω (n α ) any sequences of positive random variables such that n -α o ω (n α ) → 0 a. s. as n → ∞. On the other hand, since the function log 2 x is concave, it follows from (3.22) that

m l l j=1 log 2 K σ-j ω ≤ log 2 1 l l j=1 K m σ-j ω = log 2 ξ ω + o ω (1) , (3.24)
whence we conclude that the random variable ε n defined in (3.20) satisfies the inequality

ε n (ω) ≤ r inf l∈[1,n] 2 -l + 4 • 2 -(n-l) (ξ ω + o ω (1)) l/m .
Taking l = mn 2m + log 2 (ξ ω + o ω (1))

-1 , we obtain

ε n (ω) ≤ 5 exp - mn ln 2 2m + log 2 (ξ ω + o ω (1)) . (3.25)
Combining this inequality with (3.23), we derive

lim n→∞ ln N εn(ω) (ω) ln ε -1 n (ω) ≤ 2 m Cξ ω (ln ξ ω + 2m) m ln 2 =: d ω .
It is now straightforward to see

dim f (M ω ) = lim sup ε→0 + ln N ε ln ε -1 ≤ d ω . (3.26)
The proof of the theorem is complete.

Remark 3.4. It follows from (3.26) that if the random variable ξ ω entering the Birkhoff theorem is bounded (see (3.22)), then the fractal dimension of M ω can be bounded by a deterministic constant. For instance, if the group of shift operators {σ k } is ergodic, then ξ ω is constant, and the conclusion holds. This observation will be important in applications of Theorem 3.2.

Proof of Lemma 3.3. The co-cycle property (2.1) and inclusion (3.1) imply that ψ

σ k-m ω m (A σ k-m ω ) ⊃ ψ σ k-l ω l (A σ k-l ω ) for m ≤ l.
Hence, it suffices to establish (3.17) for m = l.

We first note that inequality (3.2), inclusion (3.11), and the definition of

C k (ω) imply that 1 d V V n (ω), ψ σ-1ω 1 (V n-1 (σ -1 ω) ≤ 2 2-n rK σ-1ω ,
where n ≥ 1 is an arbitrary integer, and we set V 0 (ω) = U 0 (ω). Combining this with (3.2) and the co-cycle property, for any integers n ≥ 1 and q ≥ 0 we derive

d V ψ ω q (V n (ω)), ψ σ-1ω q+1 (V n-1 (σ -1 ω) ≤ 2 2-n r q j=0 K σj-1ω . (3.27)
Applying (3.27) to the pairs (n, q) = (k -i, i), i = 0, . . . , l -1, with ω replaced by σ k-i ω, using the triangle inequality, and recalling that K ω ≥ 1, we obtain

d V V k (σ k ω), ψ σ k-l ω l (V k-l (σ k-l ω)) ≤ r l-1 i=0 2 2-k+i i j=0 K σ k-i+j-1 ω ≤ 2 2-(k-l) r l j=1 K σ k-j ω , 1
In the case n = 1, the left-hand side of this inequality is zero.

where k ≥ 1 and p ∈ [1, k] are arbitrary integers. A similar argument based on the application of (3.27) to the pairs (n, q) = (k -s -i, s + i), i = 0, . . . , l -s -1, with ω replaced by σ k-s ω, enables one to prove that for any integer n ∈ [1, k] we have

d V ψ σ k-s ω s (V k-s (σ k-s ω)), ψ σ k-l ω l (V k-l (σ k-l ω)) ≤ 2 2-(k-l) r l j=1 K σ k-j ω ,
(3.28) where s ∈ [0, l -1] is an arbitrary integer. Recalling that V n (ω) ⊂ A ω for any n ≥ 1 (see (3.11)), we deduce from (3.28) that

d V ψ σ k-s ω s (V k-s (σ k-s ω)), ψ σ k-l ω l (A σ k-l ω ) ≤ 2 2-(k-l) r l j=1 K σ k-j ω (3.29) for any integer s ∈ [0, k -1]. Since E k (σ k ω) = k-1 s=0 ψ σ k-s ω s (V k-s (σ k-s ω)),
inequality (3.29) immediately implies (3.17) with m = l.

Dependence of attractors on a parameter

We now turn to the case in which the RDS in question depends on a parameter. Namely, let Y ⊂ R and T ⊂ R be bounded closed intervals. We consider a discrete-time RDS Ψ y = {ψ y,ω k : H → H, k ≥ 0} depending on the parameter y ∈ Y and a family of measurable isomorphisms {θ τ : Ω → Ω, τ ∈ T }. We assume that θ τ commutes with σ 1 for any τ ∈ T , and the following uniform version of Condition 3.1 is satisfied. Condition 3.5. There is a Hilbert space V compactly embedded in H, almost surely finite random variables R y ω , R ω ≥ 0, and positive constants m, r, and α ≤ 1 such that R y ω ≤ R ω for all y ∈ Y , and the following properties hold.

Absorption and continuity. For any ball B ⊂ H there is a time T (B) ≥ 0 such that

ψ y,θτ ω k (B) ⊂ A y ω for k ≥ T (B), y ∈ Y , τ ∈ T , ω ∈ Ω, (3.30)
where we set A y ω = B V (R y ω ). Moreover, there is an integrable random variable L ω ≥ 1 such that

|R y1 θτ 1 ω -R y2 θτ 2 ω | ≤ L ω |y 1 -y 2 | α + |τ 1 -τ 2 | α (3.31)
for y 1 , y 2 ∈ Y , τ 1 , τ 2 ∈ T , and ω ∈ Ω.

Stability. With probability 1, we have

ψ y,ω 1 O r (A y ω ) ⊂ A y σ1ω
for y ∈ Y .

(3.32)

Hölder continuity. There are almost surely finite random variables K y ω , K ω ≥ 1 such that K y ω ≤ K ω for all y ∈ Y , (RK) m ∈ L 1 (Ω, P), and

ψ y1,θτ 1 ω 1 (u 1 )-ψ y2,θτ 2 ω 1 (u 2 ) V ≤ K y1,y2 ω |y 1 -y 2 | α +|τ 1 -τ 2 | α + u 1 -u 2 H (3.33) for y 1 , y 2 ∈ Y , τ 1 , τ 2 ∈ T , u 1 , u 2 ∈ O r (A y1 ω ∪ A y2 ω )
, and ω ∈ Ω, where we set

K y1,y2 ω = max(K y1 ω , K y2 ω ).
Kolmogorov ε-entropy. Inequalities (3.3) holds with some C not depending on ε.

In particular, for any fixed y ∈ Y , the RDS Ψ y satisfies Condition 3.1 and, hence, possesses an exponential attractor M y ω . The following result is a refinement of Theorem 3.2.

Theorem 3.6. Let Ψ y be a family of RDS satisfying Condition 3.5. Then there is a random compact set (y, ω) → M y ω with the underlying space Y × Ω and a set of full measure Ω * ∈ F such that the following properties hold.

Attraction. For any y ∈ Y , the family {M y ω } is a random exponential attractor for Ψ y . Moreover, the attraction property holds uniformly in y and ω in the following sense: for any ball B ⊂ H there is C(B) > 0 such that

sup y∈Y d V ψ y,ω k (B), M y σ k ω ≤ C(B)e -βk for k ≥ 0, ω ∈ Ω * , (3.34)
where β > 0 is a constant not depending on B, k, y, and ω.

Hölder continuity. There are finite random variables P ω and γ ω ∈ (0, 1] such that

d s V (M y1 ω , M y2 ω ) ≤ P ω |y 1 -y 2 | γω for y 1 , y 2 ∈ Y , ω ∈ Ω * . (3.35)
If, in addition, the random variable ξ ω entering (3.22) is bounded, then γ ω can be chosen to be constant, and we have the inequality

d s V (M y θτ 1 ω , M y θτ 2 ω ) ≤ Q ω |τ 1 -τ 2 | γ for y ∈ Y , τ 1 , τ 2 ∈ T , ω ∈ Ω * , (3.36)
where γ ∈ (0, 1], and Q ω is a finite random constant.

In addition, it can be shown that all the moments of the random variables P ω and Q ω are finite. The proof of this property requires some estimates for the rate of convergence in the Birkhoff ergodic theorem. Those estimates can be derived from exponential bounds for the time averages of some norms of solutions. Since the corresponding argument is technically rather complicated, we shall confine ourselves to the proof of the result stated above.

Proof of Theorem 3.6. To establish the first assertion, we repeat the scheme used in the proof of Theorem 3.2, applying Corollary 5.3 and Lemma 5.5 instead of Lemma 5.1 to construct coverings of random compact sets. Namely, let us denote by U y k (ω), V y k (ω), and C y k (ω) with y ∈ Y the random sets described in the proof of Theorem 3.2 for the RDS Ψ y . In particular, U y k (ω) is a random finite set such that

d s U y k (ω), C y k (ω) ≤ 2 k+1 K σ k ω -1 r, k ≥ 0, (3.37)
where C y 0 (ω) = A y ω and

C y k (ω) = u∈V y k (σ k ω) B V (u, 2 -k r), V y k (σ k ω) = ψ y,σ k-1 ω 1 U y k-1 (ω) (3.38)
for k ≥ 1. We apply Corollary 5.3 to construct a random finite set R → U 0,R satisfying (5.14)-(5.16) with δ = r 2K y ω and then define U y 0 (ω) := U 0,R y ω . The subsequent sets U y k (ω), k ≥ 1, are constructed with the help of Lemma 5.5. What has been said implies the following bound for the number of elements of U y k (ω) (cf. (3.12)):

ln #U y k (ω) ≤ 4 32 r m C y ω K m ω + 4 m C k j=1 K m σj ω ,
where C y ω = C(R y ω ) m . This enables one to repeat the argument of the proof of Theorem 3.2 and to conclude that the random compact set defined by relations (3.16) is an exponential attractor for Ψ y (with a uniform rate of attraction).

We now turn to the property of Hölder continuity for M y ω . Inequalities (3.35) and (3.36) are proved by similar arguments, and therefore we give a detailed proof for the first of them and confine ourselves to the scheme of the proof for the other. Inequality (3.35) is established in four steps.

Step 1. We first show that

d s V V y1 k (ω), V y2 k (ω) ≤ |y 1 -y 2 | α k j=1 j i=1 K σ-iω , (3.39)
where |y 1 -y 2 | ≤ 1. The proof is by induction on k. For k = 1, the random finite set U y 0 (ω) does not depend on ω. Recalling that V y 1 (ω) = ψ y,σ-1ω 1 (U y 0 (σ σ-1 ω)) and using (3.33), for |y 1 -y 2 | ≤ 1 we get the inequality

d s V V y1 1 (ω), V y2 1 (ω) ≤ K σ-1ω |y 1 -y 2 | α ,
which coincides with (3.39) for k = 1. Assuming that inequality (3.39) is established for 1 ≤ k ≤ m, let us prove it for k = m + 1. In view of Lemma 5.5, the random finite set U y m (ω) satisfying (3.37) can be constructed in such a way that

d s U y1 m (ω), U y2 m (ω) ≤ d s V y1 m (σ m ω), V y2 m (σ m ω) .
Combining this with (3.33), we see that

d s V V y1 m+1 (ω), V y2 m+1 (ω) ≤ K σ-1ω |y 1 -y 2 | α + d s V V y1 m (σ -1 ω), V y2 m (σ -1 ω) .
Using inequality (3.39) with k = m and ω replaced by σ -1 ω to estimate the second term on the right-hand side, we arrive at (3.39) with k = m + 1.

Step 2. We now prove that

d s V (M y1 ω , M y2 ω ) ≤ 2ε n (ω) + |y 1 -y 2 | α n k=1 k k i=1 K σ-iω , (3.40)
where n ≥ 1 is an arbitrary integer and ε n (ω) is defined in (3.20). Indeed, inequality (3.20), which was proved in the case of a single RDS, remains true in the present parameter-dependent setting:

d s V M yp ω , n k=1 V yp k (ω) ≤ ε n (ω), p = 1, 2.
Combining this with (3.40) and the obvious inequality

d s V (A 1 ∪ A 2 , B 1 ∪ B 2 ) ≤ d s V (A 1 , B 1 ) + d s V (A 2 , B 2 ),
we derive

d s V (M y1 ω , M y2 ω ) ≤ 2ε n (ω) + n k=1 d s V (V y1 k (ω), V y2 k (ω)).
Using (3.39) to estimate each term of the sum on the right-hand side, we arrive at (3.40).

Step 3. Suppose now we have shown that

n k=1 k k i=1 K σ-iω ≤ exp(ζ n ω n), n ≥ 1, ( 3
d s V (M y1 ω , M y2 ω ) ≤ 10 exp(-η n ω n) + exp(ζ n ω n)|y 1 -y 2 | α , (3.42)
where we set

η n ω = m ln 2 2m + log 2 (ξ ω + o ω (1))
.

We wish to optimize the choice of n in (3.42). To this end, first note that

lim n→∞ η n ω =: η ω = m ln 2 2m + log 2 ξ ω > 0.
Let n 1 (ω) ≥ 1 be the smallest integer such that

ζ n ω ≤ 2ζ ω , η n ω ≥ η ω 2 for n ≥ n 1 (ω), (3.43)
and let n 2 (ω, r) be the smallest integer greater than 2η -1 ω γ ω ln r -1 , where the small random constant γ ω > 0 will be chosen later. Note that if r ≤ τ ω := exp -n1(ω)ηω 2γω , then n 2 (ω, r) ≥ n 1 (ω). Combining (3.42) and (3.43), for |y 1 -y 2 | ≤ τ ω and n = n 2 (ω, |y 1 -y 2 |), we obtain

d s V (M y1 ω , M y2 ω ) ≤ 10 exp(-η ω n/2) + exp(2ζ ω n)|y 1 -y 2 | α ≤ 10 |y 1 -y 2 | γω + |y 1 -y 2 | α-8ζω γω/ηω . Choosing γ ω = αη ω η ω + 8ζ ω , (3.44)
we obtain

d s V (M y1 ω , M y2 ω ) ≤ 11 |y 1 -y 2 | γω for |y 1 -y 2 | ≤ τ ω .
This obviously implies the required inequality (3.35) with an almost surely finite random constant P ω .

Step 4. It remains to prove (3.41). In view of (3.24), we have

k i=1 K σ-iω ≤ ξ ω + o ω (1) n/m for 1 ≤ k ≤ n. (3.45) It follows that n k=1 k k i=1 K σ-iω ≤ n(n + 1) 2 ξ ω + o ω (1) n/m ,
whence we obtain (3.41) with

ζ n ω = ζ ω + o ω (1), ζ ω = 1 m ln ξ ω . (3.46)
This completes the proof of (3.35). It is straightforward to see from (3.44) and the explicit formulas for ζ ω and η ω that if ξ ω ≥ 1 is bounded, then γ ω can be chosen to be independent of ω.

We now turn to the scheme of the proof of (3.36). Suppose we have shown that (cf. (3.39))

d s V V y k (θ τ1 ω), V y k (θ τ1 ω) ≤ D k (ω)|τ 1 -τ 2 | α for y ∈ Y , |τ 1 -τ 2 | ≤ 1, (3.47)
where we set

D k (ω) = c L σ -k ω k i=1 K σ-iω + k j=1 j i=1 K σ-iω , (3.48)
and c ≥ 1 is the constant in (5.12). In this case, repeating the argument used in Step 2, we derive (cf. (3.40))

d s V (M y θτ 1 ω , M y θτ 2 ω ) ≤ ε n (θ τ1 ω) + ε n (θ τ2 ω) + D n (ω) |τ 1 -τ 2 | α , (3.49)
where n ≥ 1 is an arbitrary integer and ε n (ω) is defined in (3.20). If we prove that (cf. (3.41))

D n (ω) ≤ exp(ζ n ω n), lim n→∞ ζ n ω = ζ ∈ R + a. s., (3.50)
then the argument of Step 3 combined with the boundedness of ξ ω implies the required inequality (3.36). To prove (3.50), note that, by the Birkhoff theorem, there is an integrable random variable λ ω ≥ 1 such that

n k=1 L σ -k ω = nλ ω + o ω (n), n ≥ 1.
Combining this with (3.41), we obtain inequality (3.50) (with larger random variables ζ n ω ). Thus, it remains to establish inequality (3.47). Its proof is by induction on k. It follows from (5.16) and (3.31) that

d s U 0 (θ τ1 ω), U 0 (θ τ2 ω) ≤ c R θτ 1 ω -R θτ 2 ω ≤ c L ω |τ 1 -τ 2 | α . Since V y 1 (ω) = ψ y,σ-1ω 1
(U 0 (σ -1 ω)), using (3.33) we derive the inequality

d s V V y 1 (θ τ1 ω), V y 1 (θ τ2 ω) ≤ K σ-1ω 1 + c L σ-1ω |τ 1 -τ 2 | α ,
which coincides with (3.47) for k = 1. Let us assume that (3.47) is true for k = m and prove it for k = m + 1. In view of (5.23), the random finite set U y m (ω) satisfies the inequality

d s U y m (ω 1 ), U y m (ω 2 ) ≤ d s V y m (σ m ω 1 ), V y m (σ m ω 2 ) ,
where ω i = θ τi ω for i = 1, 2. It follows that

d s V V y m+1 (ω 1 ), V y m+1 (ω 2 ) ≤ K σ-1ω |τ 1 -τ 2 | α + d s V y m (σ m ω 1 ), V y m (σ m ω 2 ) .
The induction hypothesis now implies inequality (3.47) with k = m + 1. The proof of Theorem 3.6 is complete.

As in the case of Theorem 3.2, inequality (3.34) holds for B = A ω with C(B) = r. Furthermore, if the group of shift operators {σ k } is ergodic, then the Hölder exponent in (3.35) is a deterministic constant (cf. Remark 3.4).

Finally, if ψ y,ω k , R y ω , and K y ω do not depend on ω for some y = y 0 ∈ Y , then the exponential attractor M y0 ω constructed in the above theorem is also independent of ω. Indeed, M y ω was defined in terms of ψ y,ω k , R y ω , K y ω and the random finite sets U y k (ω) that form δ-nets for the random compact sets C y k (ω). As is mentioned after the proof of Lemma 5.5, these δ-nets are independent of ω if so are the random compact sets to be covered. Using this observation, it is easy to prove by recurrence that C y0 k (ω) and U y0 k (ω) do not depend on ω, and therefore the same property is true for the attractor M y0 ω .

Exponential attractor for continuous-time RDS

We now turn to a construction of an exponential attractor for continuous-time RDS. Let us fix a bounded closed interval Y ⊂ R and consider a family of RDS Φ y = {ϕ y,ω t : H → H, t ≥ 0}, y ∈ Y . We shall always assume that the associated group of shift operators θ t : Ω → Ω satisfies the following condition.

Condition 3.7. The discrete-time dynamical system {θ kτ0 : Ω → Ω, k ∈ Z} is ergodic for any τ 0 > 0.

Given τ 0 > 0, consider a family of discrete-time RDS

Ψ y = {ψ y,ω k , k ∈ Z + } defined by ψ y,ω k (u) = ϕ y,ω kτ0 (u), u ∈ H, k ≥ 0, ω ∈ Ω.
with the group {σ k = θ kτ0 , k ∈ Z} as the associated family of shift operators. The following theorem is the main result of this section.

Theorem 3.8. Suppose there is τ 0 > 0 such that the family {Ψ y , y ∈ Y } satisfies Condition 3.5, in which T = [-τ 0 , τ 0 ] and the measurable isomorphism θ τ coincides with the shift operator. Furthermore, suppose that Condition 3.7 is also satisfied, A y ω = B V (R y ω ) is a random absorbing set for Φ y , and the mapping (t, τ, y, u) → ϕ y,θτ ω t

(u), R + × [-τ 0 , τ 0 ] × Y × V → H, (3.51)
is uniformly Hölder continuous on compact subsets with a universal deterministic exponent. Then there is a random compact set (y, ω) → M y ω in H with the underlying space Y × Ω such that the following properties hold.

Attraction. For any y ∈ Y , the random compact set M y ω is an exponential attractor for Φ y . Moreover, the fractal dimension of M y ω is bounded by a universal deterministic constant, and the attraction property holds for the norm of V uniformly with respect to y ∈ Y :

d V ϕ y,ω t (B), M y θtω ≤ C(B)e -βt , t ≥ 0, y ∈ Y. (3.52)
Here B ⊂ H is an arbitrary ball, C(B) and β are positive deterministic constants, and the inequality holds with probability 1.

Hölder continuity. The function (t, y) → M y θtω is Hölder-continuous from Y × R to the space of random compact sets in H with the metric d s H . More precisely, there is γ ∈ (0, 1] such that for any T > 0 and an almost surely finite random variable P ω,T we have

d s H M y1 θt 1 ω , M y2 θt 2 ω ≤ P ω,T |t 1 -t 2 | γ + |y 1 -y 2 | γ (3.53) for y 1 , y 2 ∈ Y , t 1 , t 2 ∈ [-T, T ],
and ω ∈ Ω.

Proof. By rescaling the time, we can assume that τ 0 = 1. Let us denote by { M y ω } the random compact set constructed in Theorem 3.6 for the family of discrete-time RDS Ψ y and define

M y ω = τ ∈[0,1] ϕ y,θ-τ ω τ M y θ-τ ω . (3.54)
We shall prove that {M y ω } possesses all the required properties.

Step 1: Measurability. Let us show that (y, ω) → M y ω is a random compact set. We need to prove that, for any u ∈ H, the function

(y, ω) → inf v∈M y ω u -v
is measurable. To this end, we shall apply Proposition 5.6 to the family of compact sets

(y, ω) → K (y,ω) = {(τ, u) ∈ [0, 1] × H : u ∈ M y θ-τ ω }
and the random mapping

ψ (y,ω) : [0, 1] × H → H, (τ, u) → ϕ y,θ-τ ω τ (u).
It is straightforward to see that M y ω = ψ (y,ω) (K (y,ω) ). If we prove that ψ (y,ω) and K (y,ω) satisfy the hypotheses of Proposition 5.6, then we can conclude that M y ω is a random compact set in H. For any fixed (y, ω), the mapping (τ, u) → ψ (y,ω) (τ, u) is continuous. On the other hand, the measurability in ω and the continuity in y of the mapping ϕ y,θ-τ ω τ (u) imply that, for any fixed (τ, u), the mapping ψ (y,ω) (τ, u) is measurable. Furthermore, for any (τ, u)

∈ [0, 1] × H, the mapping (y, ω) → inf (τ ′ ,u ′ )∈K (y,ω) |τ -τ ′ | + u -u ′ = inf τ ′ ∈Q∩[0,1] inf u ′ ∈ M y θ -τ ω u -u ′
is measurable, so that K (y,ω) is a random compact set. Thus, the application of Proposition 5.6 is justified.

Step 2: Semi-invariance. Since { M y ω } is an exponential attractor for the discrete-time RDS Ψ y , for any y ∈ Y with probability 1 we have

ϕ y,ω k ( M y ω ) ⊂ M y θ k ω , k ≥ 0.
It follows that, for any rational s ∈ R and y ∈ Y , the inequality

ϕ y,θsω k ( M y θsω ) ⊂ M y θ k+s ω , k ≥ 0, (3.55)
takes place almost surely. The continuity in (s, y) of all the objects entering inequality (3.55) implies that it holds, with probability 1, for all s ∈ R, y ∈ Y , and k ≥ 0. The semi-invariance can now be established by a standard argument. Namely, for any τ ∈ [0, 1] and t ≥ 0, we choose an integer k ≥ 0 so that σ = t + τ -k ∈ [0, 1) and write

ϕ y,ω t ϕ y,θ-τ ω τ M y θ-τ ω = ϕ y,θ-τ ω k+σ M y θ-τ ω = ϕ y,θ k-τ ω σ ϕ y,θ-τ ω k M y θ-τ ω ⊂ ϕ y,θ-σ(θtω) σ M y θ k-τ ω = ϕ y,θ-σ(θtω) σ M y θ-σ(θtω) ⊂ M y θtω ,
where we used (3.55) to derive the first inclusion. Since the above relation is true for any τ ∈ [0, 1], we conclude that M y ω is semi-invariant under ϕ ω t .

Step 3: Exponential attraction. We first note that, with probability 1, where the inequality holds a.s. for all rational numbers s ∈ R. The continuity in s of all the objects entering inequality (3.56) implies that, with probability 1, it remains true for all s ∈ R. We now fix an arbitrary ball B ⊂ H and denote by T (B) ≥ 0 the instant of time after which the trajectories starting from B are in A θtω . For any t ≥ T (B) + 1, we choose s ∈ [T (B), T (B) + 1) such that k := t -s is an integer and use the cocycle property to write

sup y∈Y d V ϕ y,ω k (A ω ), M y θ k ω ≤ r e -βk , k ≥ 0.
d V ϕ y,ω t (B), M y θtω = d V ϕ y,θsω k (ϕ y,ω s (B)), M y θtω) ≤ d V ϕ y,θsω k (A θsω ), M y θ k+s ω) .
Taking the supremum in y ∈ Y and using (3.56), we obtain

sup y∈Y d V ϕ y,ω t (B), M y θtω ≤ r e -βk ≤ r e T (B)+1 e -βt .
This proves inequality (3.52) with C(B) = r e T (B)+1 .

Step 4: Fractal dimension. As was established in the proof of Theorem 3.2, the fractal dimension of M y ω admits the explicit bound (see (3.26))

dim f ( M y ω ) ≤ 2 m Cξ ω (ln ξ ω + 2m) m ln 2 ,
where ξ ω is the random variable defined in (3.22). Since the group {σ k } is ergodic, ξ ω is constant, and dim f ( M y ω ) can be estimated, with probability 1, by a constant not depending on y and ω. Since the function τ → ϕ y,θ-τ ω τ (u) and τ → M y θ-τ ω are Hölder continuous with a deterministic exponent, it is easy to prove that the fractal dimension of M y ω is bounded by a universal constant.

Step 5: Time continuity. Since mapping (3.51) is Hölder continuous, the required inequality (3.53) will be established if we prove that (3.53) is true for M y ω . However, this is an immediate consequence inequalities (3.35), (3.36) and the ergodicity of the group of shift operators {σ k }. The proof of the theorem is complete.

As in the case of discrete-time RDS, if ϕ y,ω t , R y ω , and the random objects entering Condition 3.5 do not depend on ω for some y 0 , then the exponential attractor M y0 ω is also independent of ω. This fact follows immediately from representation (3.54), because ϕ y0,θ-τ ω τ and M y0 θ-τ ω do not depend on ω.

4 Application to a reaction-diffusion system

Formulation of the main result

In this section, we apply Theorem 3.8 to the reaction-diffusion (1.1) in which the amplitude of the random force depends on a parameter. Namely, we consider the equation u

-a∆u + f (u) = h(x) + ε η(t, x), x ∈ D, (4.1)
where D ⊂ R n is a bounded domain with smooth boundary and ε ∈ [-1, 1] is a parameter. Concerning the matrix a, the nonlinear term f , and the external forces h and η, we assume that they satisfy the hypotheses described in Section 2.2, with the stronger condition p ≤ n n-2 for n ≥ 3. Moreover, we impose a higher regularity on the external force, assuming that

h ∈ H 1 0 (D, R k ) ∩ H 2 (D, R k ), B 3 := ∞ j=1 λ 3 j b 2 j < ∞, (4.2)
where λ j denotes the j th eigenvalue of the Dirichlet Laplacian. This condition ensures that almost every trajectory of a solution for Eq. (4.1) with f ≡ 0 is a continuous function of time with range in H 3 . The following theorem is the main result of this section.

Theorem 4.1. Under the above hypotheses, for any ε ∈ [-1, 1] problem (4.1), (1.2) possesses an exponential attractor M ε ω . Moreover, the sets M ε ω can be constructed in such a way that M 0 ω does not depend on ω, the fractal dimension of M ε ω is bounded by a universal deterministic constant, the attraction property holds uniformly with respect to ε, and

d s H M ε1 ω , M ε2 ω ≤ P ω |ε 1 -ε 2 | γ for ε 1 , ε 2 ∈ [-1, 1], (4.3)
where γ ∈ (0, 1] is a constant and P ω is an almost surely finite random variable.

To prove this result, we shall apply Theorem 3.8. For the reader's convenience, let us describe briefly the conditions we need to check, postponing their verification to the next subsection.

Recall that H = L 2 , V = H 1 0 , and the probability space (Ω, F , P) and the corresponding group of shits operators θ t were defined in Section 2.2. The ergodicity of the restriction of {θ t } to any lattice τ 0 Z is well known (see Condition 3.7), and the Kolmogorov ε-entropy of a unit ball in V regarded as a subset in H can be estimated by Cε -n , where n is the space dimension (see the fourth item of Condition 3.5). We shall prove that the following properties are true for a sufficiently large τ 0 > 0.

Absorbing set. There are random variables

R ε ω , R ω ≥ 0 such that R ε ω ≤ R ω for all ε ∈ [-1, 1], R ∈ L q (
Ω, P) for any q ≥ 1, and for any ball B ⊂ H and a sufficiently large T (B) > 0 we have

u ε,θτ ω (t; u 0 ) ∈ B V (R ε θtω ) for t ≥ T (B), |τ | ≤ τ 0 , |ε| ≤ 1, u 0 ∈ B, (4.4)
where u ε,ω (t; u 0 ) denotes the solution of (4.1), (1.2), (1.3). Moreover, R ε ω satisfies inequality (3.31) with y i = ε i ∈ [-1, 1] for an integrable random variable L ω and a deterministic constant α ∈ (0, 1].

Stability. There is r > 0 such that u ε,ω (τ 0 ; u 0 ) ∈ B V (R ε θτ 0 ω ) for |ε| ≤ 1, u 0 ∈ O r B V (R ε ω ) . (4.5)
Hölder continuity. There is α > 0 such that, for any T > 0 and any random variable r ω > 0 all of whose moments are finite, one can construct a family of random variables K ε ω ≥ 1 satisfying the inequalites u ε1,θτ 1 ω (t 1 ; u 01 ) -u ε2,θτ 2 ω (t 2 ; u 02 )

≤ K ε1,ε2 ω |ε 1 -ε 2 | + |τ 1 -τ 2 | α + u 01 -u 02 + |t 1 -t 2 | α , (4.6) u ε1,θτ 1 ω (τ 0 ; u 01 ) -u ε2,θτ 2 ω (τ 0 ; u 02 ) 1 ≤ K ε1,ε2 ω |ε 1 -ε 2 | + |τ 1 -τ 2 | α + ũ01 -ũ02 , (4.7)
where

|ε i | ≤ 1, |τ i | ≤ τ 0 , 0 ≤ t i ≤ T , u 0i ∈ B V (r ω ), ũ0i ∈ B H (r ω ), and we set K ε1,ε2 ω = max{K ε1 ω , K ε1 ω }.
Moreover, there is a random variable K ω belonging to L q (Ω, P) for any q ≥ 1 such that K ε ω ≤ K ω for all ε ∈ [-1, 1]. We shall also prove that the random variables R 0 ω and K 0 ω are constants. If these properties are established, then all the hypotheses of Theorem 3.8 are fulfilled, and its application to the RDS associated with problem (4.1), (1.2) gives the conclusions of Theorem 4.1.

Proof of Theorem 4.1

Step 1: Absorbing set. Let U ε,ω (t) be the unique stationary solution of the equation u

-a∆u = ε η(t, x), t ∈ R, (4.8)
supplemented with the Dirichlet boundary condition (1.2). It is straightforward to see that, with probability 1,

U ε,θτ ω (t) = ε U ω (t + τ ), t, τ ∈ R, (4.9)
where U ω (t) = U 1,ω (t). Using the Itô formula and the regularity assumption (4.2), one can prove that2

E e δ sup t Mt < ∞, M t (ω) := U ω (t) 2 3 + t 0 U ω (s) 2 4 ds -C(1 + |t|), (4.10)
where δ > 0 and C > 0 are deterministic constant, and the supremum is taken over t ∈ R. Moreover, by Proposition 5.8, inequality (5.31) holds for U . Solutions of (4.1), (1.2) can be written as

u ε,θτ ω (t, x) = ε U ω (t + τ, x) + v ε,τ,ω (t, x), (4.11)
where v = v ε,τ,ω is the solution of the problem

v -a∆v + f (v + ε U ω (t + τ )) = h(x), (4.12
)

v ∂D = 0, (4.13) v(0, x) = v 0 (x), (4.14)
where

v 0 (x) = u 0 (x) -ε U ω (τ, x).
In what follows, we shall often omit the subscripts ε and ω to simplify notation. We wish to derive some a priori estimates for v. Since the corresponding argument is rather standard, we only sketch it.

Taking the scalar product of (4.12) in L 2 and carrying out some transformations, we derive

∂ t v 2 + c 1 v 2 + v 2 1 + v p+1 L p+1 ≤ C 1 1 + h 2 -1 + ε U p+1 L p+1
, (4.15) where U = U ω (• + τ ), and we used inequalities (2.5), (2.6), and (2.8). Let us fix any δ ∈ (0, c 1 ). Applying the Gronwall inequality, using the continuity of the embedding H 1 ⊂ L p+1 , and recalling that |τ | ≤ τ 0 , we obtain

v(t) 2 + c 1 t 0 e -c1(t-σ) v 2 1 + v p+1 L p+1 dσ ≤ e -c1t v 0 2 + R ε,1 θtω , (4.16)
where we set

R ε,1 ω = C 1 0 -∞ e δσ 1 + h 2 -1 + ε U ω (σ + τ 0 ) p+1 1 dσ.
We now derive a similar estimate for the H 1 norm of v. Taking the scalar product of (4.12) with -2(t -s)∆v in L 2 , after some transformations we derive

∂ t (t -s) ∇v 2 + c 2 (t -s) v 2 2 ≤ ∇v 2 + C 2 (t -s) 1 + h 2 + v p+1 L p+1 + ε U p+1 2 .
Integrating in t ∈ (s, s + 1), we obtain

∇v(s + 1) 2 + c 2 s+1 s (σ -s) ∆v 2 dσ ≤ C 3 + s+1 s ∇v 2 + C 2 v p+1 L p+1 dσ + C 2 s+1 s ε U p+1 2 dσ,
where

C 3 = C 2 (1 + h 2 )
. Taking s = t -1 and using (4.16) to estimate the second term on the right-hand side, we obtain

v(t) 2 1 ≤ C 3 + C 4 e -c1t v 0 2 + R ε,1 θtω + C 2 R ε,2 θtω , t ≥ 1, (4.17)
where we set 3

R ε,2 ω = 0 -∞ e δ(σ+3) ε U ω (σ + τ 0 ) p+1 2 dσ.
Let us define R ε ω by the relation

(R ε ω ) 2 = 8 1 + C 3 + C 4 R ε,1 ω + C 2 R ε,2 ω + sup σ≤0 e δ(σ+2τ0) ε U ω (σ + τ 0 ) 2 1 (4.18) and set R ω = R 1 ω . It is clear that R ε ω ≤ R ω for all ε ∈ [-1, 1]
. Relations (4.11) and (4.17) imply that

u ε,θτ ω (t) 2 1 ≤ 2C 4 e -c1t u 0 2 + ε U ω (τ ) 2 + 4 -1 (R ε θtω ) 2 -1, t ≥ 1, (4.19)
whence we see (4.4) holds for any ball B ⊂ H and a sufficiently large T (B) > 0. Moreover, it follows from (4.10) and (4.18) that all the moments of R ω are finite. Finally, Proposition 5.8 and the stationarity of U imply that R ε ω satisfies (3.31) with a constant α ∈ (0, 1/2) and an integrable random variable L ω .

Step 2: Stability. It follows from (4.18) that the stability property (4.5) with parameters r > 0 and τ 0 > 0 will certainly be satisfied if

2C 4 e -c1τ0 (R ε ω + r) 2 + ε U ω (τ ) 2 + 4 -1 (R ε θτ 0 ω ) 2 -1 ≤ (R ε θτ 0 ω ) 2 . (4.20) Let us note that (R ε θtω ) 2 ≥ e -δt (R ε ω ) 2 , t ≥ 0. (4.21)
We now take an arbitrary r > 0 and choose τ 0 > 0 so large that 4C 4 e -c1τ0 r 2 ≤ 1, 16C 4 e -(c1-δ)τ0 ≤ 1.

In this case, inequality (4.20) holds, so that the stability condition is fulfilled.

Step 3: Hölder continuity. Representation (4.11) implies that it suffices to establish analogues of (4.6) and (4.7) for solutions of problem (4.12)-(4.14).

3 To have an absorbing set, one could take for R ε,2 ω the integral of ε U ω (σ + τ 0 ) p+1 2 in σ ∈ [-3, 0]. However, in this case the stability condition may not hold, and therefore we define R ε,2 ω in a different way. Our choice ensures that (4.21) holds for the radius of the absorbing ball.

Namely, we first prove that for any random variable r ω > 0 with finite moments there is a family of almost surely finite random variables K ε ω such that

v 1 (t) -v 2 (t) ≤ K ε1,ε2 ω ε 1 U ω1 -ε 2 U ω2 L ∞ (0,T ;H 1 ) + v 01 -v 02 , (4.22) v 1 (τ 0 ) -v 2 (τ 0 ) 1 ≤ K ε1,ε2 ω ε 1 U ω1 -ε 2 U ω2 L ∞ (0,τ0;H 2 ) + v 01 -v 02 , (4.23)
where

|ε i | ≤ 1, |τ i | ≤ τ 0 , 0 ≤ t ≤ T , v 0i ∈ B H (r ω
), and we set ω i = θ τi ω, v i (t) = v εi,τi,ω , and K ε1,ε2 ω = max{ K ε1 ω , K ε2 ω }. Moreover, our proof will imply that K ε ω ≤ K ω for all ε ∈ [-1, 1], where the random constant K belongs to L q (Ω, P) for any q ≥ 1. Once these properties are established, the Hölder continuity of U ω (t) and relations (4.10) and (4.11) will prove inequalities (4.6) and (4.7) with t 1 = t 2 . We shall next show that the solutions of (4.12)-(4.14) with v 0 ∈ B V (r ω ) satisfy the inequality

v ε,τ,ω (t 1 ; v 0 ) -v ε,τ,ω (t 2 ; v 0 ) ≤ K ε ω |t 1 -t 2 | α , t 1 , t 2 ∈ [0, T ], (4.24)
with possibly a larger random constant K ε ω with the same property. This will complete the proof of the property of Hölder continuity and that of Theorem 4.1.

We begin with (4.22). To simplify the presentation, we shall assume that n ≥ 3. In what follows, we denote by {K ε,i ω , ε ∈ [-1, 1]} (where i ≥ 1) families of random variables that can be bounded by a random constant belonging to L q (Ω, P) for any q ≥ 1. The difference v = v 1 -v 2 satisfies the equation v -a∆v + f (u 1 ) -f (u 2 ) = 0 (4.25)

and the boundary and initial conditions (4.13) and (4.14), where u i = v i +ε i U ω i and v 0 = v 01 -v 02 . Taking the scalar product of (4.25) with 2v in L 2 and using the "monotonicity" assumption (2.7), we derive

∂ t v 2 + 2c 3 ∇v 2 ≤ -f (u 1 ) -f (u 2 ), v ≤ C v 2 + C 5 ξ L q + |u 1 | p-1 ξ L q + |u 2 | p-1 ξ L q v 1 ≤ C v 2 + c 3 ∇v 2 + C 6 1 + u 1 2(p-1) L p+1 + u 2 2(p-1) L p+1 ξ 2 1 ,
where q = 2n n+2 and ξ = ε 1 U ω1 -ε 2 U ω2 . Applying the Gronwall inequality, we obtain

v(t) 2 +c 3 t 0 e C(t-σ) ∇v 2 dσ ≤ e Ct v 0 2 +C 7 max{K ε,1 ω , K ε,2 ω } ξ 2 L ∞ (0,T ;H 1 ) , ( 4
.26) where 0 ≤ t ≤ T , C 7 = C 7 (T ), and we set

K ε,1 ω = T 0 1 + u ε,ω (σ) p+1 L p+1 dσ.
Lemma 5.1. Let {A ω } be a random compact set in a Hilbert space H. Then for any measurable function δ = δ ω satisfying the inequality 0 < δ ≤ 1 one can construct a random finite set U δ (ω) ⊂ H such that for

d s A ω , U δ (ω) ≤ δ ω , (5.1) ln #U δ (ω) ≤ H δω /2 (A ω , H). (5.2)
Moreover, if δ ω ≡ δ is constant, then one can replace δ ω /2 in the right-hand side (5.2) by δ.

Note that inequality (5.1) is equivalent to the inclusions

A ω ⊂ u∈U δ (ω) B H (u, δ ω ), U δ (ω) ⊂ O δω (A ω ).
(5.3)

Proof. We first assume that δ ω ≡ δ. Let {u k } ⊂ H be a dense sequence. For any k = {k 1 , . . . , k n } ⊂ N, define the random varable

Z ω (k ) =    1, A ω ⊂ n i=1 B(u ki , δ), 0, otherwise.
Since A ω is a (random) compact set, for any ω there is a finite subset k ⊂ N such that Z ω (k ) = 1. Let Ω n be the set of those ω ∈ Ω for which there is an n-tuple k ⊂ N such that Z ω (k ) = 1 and Z ω (k ′ ) = 0 for any subset k ′ ⊂ N containing less than n elements. Then we have Ω = ∪ n≥1 Ω n . Furthermore, since Ω n is the intersection of the measurable sets

#k =n-1 {Z ω (k ) = 0} and #k =n {Z ω (k ) = 1},
we have Ω n ∈ F for any n ≥ 1. Thus, it suffices to construct U δ on each subset Ω n . Indexing the set of all n-tuples k ⊂ N in an arbitrary way, it is easy to construct measurable functions I k : Ω n → {0, 1} such that, for any ω ∈ Ω n , we have

#k (ω) = n, A ω ⊂ k∈k (ω) B H (u k , δ), B H (u k , δ) ∩ A ω = ∅, (5.4)
where k (ω) = {k ∈ N : I k (ω) = 1} and k ∈ k (ω) in the third relation. We claim that U δ (ω) = {u k , k ∈ k (ω)} satisfies the required properties. Indeed, for any u ∈ H, we have

d(u, U δ (ω)) = min{ u -u k : I k (ω) = 1}, ω ∈ Ω n ,
whence it follows easily that U δ (ω) is a random finite set. Furthermore, inclusions (5.3) (which are equivalent to inequality (5.1)) are consequences of the second and third relations in (5.4). Let us prove that inequality (5.2) holds with δ ω /2 replaced by δ ω ; that is, ln n ≤ H δ (A ω , H) for ω ∈ Ω n .

(5.5)

To see this, note that the set A ω admits a covering by balls

{B j } such that ln #{B j } ≤ H δ (A ω , H), diam(B j ) ≤ δ.
Choosing arbitrary points u kj in every ball B j , we see that one can cover A ω by the balls {B H (u kj , δ)}. The choice of n now implies that n ≤ #{B j }, whence it follows that (5.5) holds.

We now turn to the case of an arbitrary function δ ω such that 0 k) . In view of what has been proved above, on each Ω (k) one can construct a random finite set U k (ω) such that, for ω ∈ Ω (k) , we have

< δ ω ≤ 1. Let us define Ω (k) = {ω ∈ Ω : 2 -k < δ ω ≤ 2 1-k }, so that Ω = ∪ k≥1 Ω (
d s A ω , U k (ω) ≤ 2 -k , ln #U k (ω) ≤ H 2 -k (A ω , H). Setting U δ (ω) = U k (ω) for ω ∈ Ω (k)
, we obtain the required covering. The proof of the lemma is complete.

The second result shows that, if a random compact set depends on a parameter in a Lipschitz manner, then the random finite set constructed above can be chosen to have a similar dependence on the parameter. To prove it, we shall need the following auxiliary construction.

Let us denote by ∆ n ⊂ R n the set of vectors θ = (θ 1 , . . . , θ n ) such that θ i ≥ 0 and i θ i = 1. Given subsets W i ⊂ H, 1 ≤ i ≤ n, a vector θ ∈ ∆ n , and a number α > 0, we define

[W 1 , . . . , W n ] α θ = n i=1 θ i u i : u i ∈ W i , u i -u j H ≤ α for 1 ≤ i, j ≤ n . It is straightforward to check that ln #[W 1 , . . . , W n ] α θ ≤ ln(#W 1 ) + • • • + ln(#W n ), (5.6) d s [W 1 , . . . , W n ] α θ 1 , [W 1 , . . . , W n ] α θ 2 ≤ α|θ 1 -θ 2 |, (5.7)
where

θ j = (θ j 1 , . . . , θ j n ) and |θ 1 -θ 2 | = max i |θ 1 i -θ 2 i |. Moreover, if A ⊂ H and r i ≥ 0 are such that A ⊂ u∈Wi B H (u, r i ), 1 ≤ i ≤ n, then for any θ ∈ ∆ n we have A ⊂ u∈[W1,...,Wn] r θ B H (u, max{r i , 1 ≤ i ≤ n}), (5.8)
where r = max{r i + r j , 1 ≤ i, j ≤ n}.

Proposition 5.2. Let Y ⊂ R be a closed interval and let {A y ω , y ∈ Y } be a family of random compact sets in a Hilbert space H such that

d s (A y1 ω , A y2 ω ) ≤ C |y 1 -y 2 | for y 1 , y 2 ∈ Y , (5.9)
where C ≥ 1 is a finite random constant. Then there exists a random finite set (δ, y, ω) → U δ,y (ω) with the underlying space (0, 1] × Y × Ω such that

d s A y ω , U δ,y (ω) ≤ δ, (5.10) ln #U δ,y (ω) ≤ 4 H 2 -4 δ (A y ω , H),
(5.11)

d s U δ1,y1 (ω), U δ2,y2 (ω) ≤ c |δ 1 -δ 2 | + C |y 1 -y 2 | , (5.12)
where y, y 1 , y 2 ∈ Y , δ, δ 1 , δ 2 ∈ (0, 1], and c ≥ 1 is an absolute constant.

In particular, taking a measurable function δ = δ ω with range in (0, 1], we can construct a random finite set (y, ω) → U δ,y (ω) such that

d s U δ,y1 (ω), U δ,y2 (ω) ≤ c C |y 1 -y 2 |,
(5.13) and inequalities (5.10) and (5.11) hold with δ = δ ω in the right-hand side.

The proof given below will imply that if A y ω does not depend on ω for some y = y 0 , then the random set U δ,y (ω) satisfying (5.10)-(5.12) can be chosen in such a way that U δ,y0 (ω) is also independent of ω. Furthermore, if A y ω does not depend on ω for all y ∈ Y , then U δ,y is also independent of ω. The latter observation implies the following corollary used in the main text.

Corollary 5.3. Let V ⊂ H be two Hilbert spaces with compact embedding. Then there is a random finite set (δ, R) → U δ,R with the underlying space (0, 1] × R + such that

d s H B V (R), U δ,R ≤ δ,
(5.14) ln #U δ,R ≤ 4 H δ/16R (V, H), (5.15)

d s H (U δ,R1 , U δ,R2 ) ≤ c |R 1 -R 2 |, (5.16)
where R, R 1 , R 2 ≥ 0 and δ ∈ (0, 1] are arbitrary, and c > 0 is an absolute constant.

To prove this result, it suffices to apply Proposition 5.2 to the non-random compact set B V (R) depending on the parameter R ∈ R + .

Proof of Proposition 5.2. Without loss of generality, we assume that the random variable C is constant, since one can represent Ω as the union of the subsets Ω l = {ω ∈ Ω : l ≤ C < l + 1} and construct required random finite sets on each Ω l .

Let us fix an integer k ≥ 1 and denote by ν k < C -1 2 -k-4 the largest number such that N k := ν -1 k is an integer. We now set y k j = jν k for j ∈ Z + . In view of Lemma 5.1, there are random finite sets U k j (ω) ⊂ H such that d s A yj ω , U k j (ω) ≤ 2 -k-3 , (5.17) ln #U k j (ω) ≤ H 2 -k-3 (A yj ω , H), (5.18) Lemma 5.5. Let Y be an arbitrary metric space, let K ⊂ H be a compact subset, let (y, ω) → V y (ω) be a random finite set, and let

q A 1 A 2 A 3 A 4 A Π 1 Π 4 Π 3 Π 2
A y ω = v∈V y (ω) (v + K).
Then there is a random finite set (δ, y, ω) → U δ,y (ω) with the underlying space (0, 1] × Y × H such that (5.10) holds, and ln #U δ,y (ω) ≤ ln(#V y (ω) + H δ/2 (K, H), (5.22) d s U δ,y1 (ω 1 ), U δ,y2 (ω 2 ) ≤ d s V y1 (ω 1 ), V y2 (ω 2 ) .

(5.23)

Proof. Applying Lemma 5.1 to the random compact set δ → δK with the underlying space (0, 1], we construct a random finite set δ → U δ such that d s (δK, U δ ) ≤ δ 2 , ln(#U δ ) ≤ H δ 2 /2 (δK, H) = H δ/2 (K, H).

It is straightforward to see that the random set U δ,y (ω) = δ -1 U δ + V y (ω) = {δ -1 u + v : u ∈ U δ , v ∈ V y (ω)} possesses all required properties.

As is clear from the proof, if V y (ω) does not depend on ω for some y = y 0 , then the random set U δ,y0 (ω) constructed in Lemma 5.5 is also independent of ω.

Proof of Lemma 5.4. Given a point A ∈ Π, we divide the rectangle Π into four smaller rectangles Π i (see Figure 1). It is easy to prove that the functions θ i (A) = Area(Π i ) Area(Π) , 1 ≤ i ≤ 4, (5.24) possess the required properties.

Image of random compact sets

Proposition 5.6. Let X and Y be Polish spaces, let (Ω, F ) be a measurable space, let {K ω , ω ∈ Ω} be a random compact set in X, and let ψ ω : X → Y be a family of continuous mappings such that, for any u ∈ X, the mapping ω → ψ ω (u) is measurable from Ω to Y . Then {ψ ω (K ω ), ω ∈ Ω} is a random compact set in Y .

Proof. Let us set fix u ∈ Y and define a function F u : Ω → R by

F u (ω) = d Y u, ψ ω (K ω ) = inf v∈Kω d Y (u, ψ ω (v)).
We need to prove that this function is measurable. Let {u k } ⊂ X be a dense sequence. Repeating the argument used in the proof of Lemma 5.1, we can construct a sequence of random finite sets K n ω ⊂ {u k } such that ) . Thus, it suffices to establish the measurability of F n u . To this end, note that, in view of (5.26), we have .

d s X (K ω , K n ω ) ≤ 1 n , ( 5
F n u (ω) = inf
This relation readily implies the required property.

Kolmogorov-Čentsov theorem

The Kolmogorov-Čentsov theorem provides a sufficient condition for Höldercontinuity of trajectories of a random process. We shall need the following qualitative version of that result.

Theorem 5.7. Let X be a Banach space and let {ξ t , 0 ≤ t ≤ T } be an X-valued random process with almost surely continuous trajectories that is defined on a probability space (Ω, F , P) and satisfies the inequality where C p > 0 is a constant not depending on t and s. Then for any γ ∈ (0, 1/2) there is a constant K γ > 0 and an almost surely positive random variable t γ such that ξ t (ω) -ξ s (ω) X ≤ K γ |t -s| γ for |t -s| ≤ t γ (ω), (5.28) E t -q γ < ∞ for any q ≥ 1.

E ξ t -ξ s
(5.29) To this end, we note that ζ 2 2 = j c 2 j β 2 j (t), where c j = b j λ j . The monotone convergence theorem and the Burkholder inequality (see Theorem 2.10 in [START_REF] Hall | Martingale Limit Theory and Its Application[END_REF]) imply that where we used the fact that j c j β j (t) is a zero-mean Gaussian random variable with variance t j c 2 j . This proves (5.32) and completes the proof of the proposition.

.

  This work was supported by the Royal Society-CNRS grant Long time behavior of solutions for stochastic Navier-Stokes equations (No. YFDRN93583). The first author was supported by the ANR grant STOSYMAP (No. ANR 2011 BS01 015 01).

  .41) where ζ n ω ≥ 1 is a sequence of almost surely finite random variables such that lim n→∞ ζ n ω =: ζ ω < ∞ with probability 1. In this case, combining (3.40) with (3.25) and (3.41), we derive

  cf. discussion following Theorem 3.2. It follows that sup y∈Y d V ϕ y,θsω k (A θsω ), M y θ k+s ω ≤ r e -βk , k ≥ 0, (3.56)

Figure 1 :

 1 Figure 1: Division of Π into four rectangles

  .25)K n ω = {u k , I n k (ω) = 1},(5.26)whereI n k : Ω → {0, 1}, k, n ≥ 1, are measurable functions. It follows from (5.25) that F u (ω) = lim inf n→∞ F n u (ω), F n u (ω) = d Y u, ψ ω (K n ω

  u, ψ ω (v)) = inf k≥1 d Y (u, ψ ω (u k )) I n k (ω)

2pX≤

  C p |t -s| p for any t, s ∈ [0, T ], p ≥ 1, (5.27)

≤

  C 3 (p) |t| p ,

  X = H 1 . Since U is stationary, we can assume that s = 0. Equation (4.8) implies that Using (4.10), we see that the mean value of first term on the right-hand side can be estimated by C|t| p . Thus, the required inequality will be established if we show that E ζ 2p 2 ≤ C p |t| p .(5.32)

	t
	U (t) -U (0) =	a∆U (r) dr + ζ(t),
	0
	whence, applying the Hölder inequality, it follows that
	U (t) -U (0) 2p 2 ≤ 2 2p-1 |t| p	0	t	a∆u 2 2 dr	p	+ ζ 2p 2	.

For instance, see Proposition 2.4.10 in[START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF] for the more complicated case of the Navier-Stokes system.

We thank A. Iftimovici for the simple geometric argument proving Lemma

5.4.

Inequality (4.26) immediately implies (4.22).

To prove (4.23), we first note that, in view of (4.26), there is a measurable function s : Ω → R such that, with probability 1, we have s ω ∈ [ τ0 4 , 3τ0 4 ] and ∇v(s ω ) 2 ≤ C 8 v 0 2 + K ε,1 ω ξ 2 L ∞ (0,τ0;H 1 ) . (4.27)

Let us take the scalar product of (4.25) with -2∆v in L 2 . After some transformations, we obtain

v+ξ L q ∆v , (4.28)

where q = 2n n-2 . Since H 1 ⊂ L q and H 1 ⊂ L n(p-1) , applying the interpolation and Cauchy-Schwartz inequalities, from (4.28) we derive

Integrating in t ∈ [s ω , τ 0 ] and using (4.26), (4.27) and (4.19) (we can assume that τ 0 ≥ 4), we obtain (4.23).

It remains to establish inequality (4.24). We shall only outline its proof. Taking the scalar product of (4.12) with -2∆v and using some standard arguments (cf. derivation of (4.17)), we obtain

Combining this with (4.10) and (4.12), we see that

It follows that v is Hölder continuous with the exponent 1/2. Since U ω is also Hölder continuous in time, in view of (4.11) we arrive at the required result. Finally, it is not diffucult to see that the random variables R 0 ω and K 0 ω are constant. The proof of the theorem is complete.

Appendix

Coverings for random compact sets

In this section, we have gathered three auxiliary results on coverings of random compact sets by balls centred at the points of random finite sets. The first of them establishes the existence of a "minimal" covering with an explicit bound of the number of balls in terms of the Kolmogorov ε-entropy of the random compact set in question.

where we write y j instead of y k j to simplify the notation. We now need the following lemma, whose proof 4 is given at end of this section.

Lemma 5.4. Let A 1 , . . . , A 4 be the vertices of a rectangle Π ⊂ R 2 . Then there are Lipschitz functions

and y j ≤ y ≤ y j+1 , denote by A δ,y ∈ Π the point with the coordinates (δ, y). Let us define

, where θ(δ, y) = (θ i (A δ,y ), 1 ≤ i ≤ 4) ∈ ∆ 4 . We claim that U δ,y (ω) satisfies the required properties.

Indeed, it follows from the choice of y j that

Combining this with (5.17), we see that

(5.20) Inclusion (5.8) now implies that

(5.21)

On the other hand, the definition of U δ,y (ω) and inequality (5.20) imply that

Combining this with (5.21), we obtain (5.10). Inequality (5.19) implies that an ε-covering for A y ω with y j ≤ y ≤ y j+1 is an (ε + 2 -k-4 )-covering for A yj ω . Taking ε = 2 -k-4 , we see that

Combining this with (5.18) and (5.6), we obtain (5.11):

Finally, inequality (5.12) follows from (5.7) and the explicit form of the functions θ i (A) (see (5.24)):

The proof of the proposition is complete.

Sketch of the proof. We repeat the argument used in Section 2.2.B of [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]. Without loss of generality, we can assume that T = 1. Let us fix any γ ∈ (0, 1/2) and introduce the events

where n ≥ 1 and 1 ≤ k ≤ 2 n . It follows from (5.27) and the Chebyshev inequality that

Summing up over k = 1, . . . , 2 n , we derive

Choosing p ≥ 1 so large that α p > 0 and applying the Borel-Cantelli lemma, we construct an almost surely finite random integer n 0 ≥ 1 such that ω / ∈ Ω n for n ≥ n 0 (ω) and ω ∈ Ω n0-1 if n 0 (ω) ≥ 2. In particular, we have

(5.30)

As is shown in the proof of Theorem 2.8 of [KS91, Chapter 2], inequality (5.30) implies (5.28) with K γ = 2/(1 -2 -γ ) and t 0 = 2 -n0 . Thus, the theorem will be proved if we show that E 2 qn0 < ∞ for any q ≥ 1. To this end, note that {n 0 = m} ⊂ Ω m-1 for any m ≥ 2. It follows that

2 qm-αp(m-1) .

Choosing p ≥ 1 so large that α p > q, we see that the series on the right-hand side of the above inequality converges.

Note that one can rewrite (5.28) and (5.29) in the form

where C γ is a random variable with finite moments. We now apply the above result to establish a time-regularity property for the process U ω defined in the beginning of Section 4.2.

Proposition 5.8. For any γ ∈ (0, 1/2) and any T > 0 there is a random variable C γ,T > 0 all of whose moments are finite such that U (t) -U (s) 2 ≤ C γ,T |t -s| γ for t, s ∈ [-T, T ].

(5.31)

Proof. In view of the remark following the proof of Theorem 5.7, it suffices to check that U satisfies inequality (5.27) with [0, T ] replaced by [-T, T ] and