arXiv:1712.08692v1 [math.DS] 23 Dec 2017 e trco steinterval the is attractor set on trco si eea o nqe hc a ese fr seen equation be differential can scalar which the unique, by not induced general system in Eucli equivalent (or, is often point attractor every attract are point to c assumed again (which only is is attractor it spaces but point global compact A locally manifolds). dimensional on systems for nte eycmo oini hto a of that is attra set notion global common the very of Another Uniqueness set. invariant, bounded strictly every being even set, or compact a being by acterized PE)o utbeHleto aahsae,i hto a of partia that for is spaces, used Banach mainly or Hilbert notion, suitable common on no (PDEs) most the The spaces metric on established. systems dynamical deterministic For Introduction 1 clos set; random closed set; random compact set; limit Omega Keywords. ∗ † ntttfrMteai,M -,Fklä I ehiceUn Technische II, Fakultät 7-5, MA Mathematik, für Institut ntttfrMteai,669Fakut FRG; Frankfurt, 60629 Mathematik, für Institut 02 eln FRG; Berlin, 10623 77 79 01 01 60H25 60H15 60H10 37L99 37C70 h aiyo eswihaeatatdi loe ob comple ve be is to attractor allowed random Classification is Subject attracted a Mathematics are 2010 of which concept sets our of fo that for family hold The out not point need We s property random) this smalle tors. that (possibly of shows of sense which family example the given an (in a minimal attracts which a attr tractor always pullback is general dynamica for there random that a attractors, show We of unique. attractors not random generally that well-known is It iia admattractors random Minimal admatatr ulakatatr ekatatr for attractor; weak attractor; pullback attractor; Random asCrauel Hans ms @ math.tu-berlin.de r´ 1 , 1 s hc,o ore sas on trco.Btalso But attractor. point a also is course, of which, , ∗ rmr 79 eodr 45 55 37B25 35B51 34D55 Secondary 37H99 Primary 1 goa)pitattractor point (global) ihe Scheutzow Michael crauel x @ 9 math.uni-frankfurt.de “ vriä eln taeds1.Jn 136, Juni 17. des Straße Berlin, iversität x y vr nt e) h global The set). finite every ly, n trcigeeycompact, every attracting and lblstattractor set global ´ matadsrcl invariant, strictly and ompact x ino natatri well is attractor an of tion 3 tri immediate. is ctor mtesml dynamical simple the om drno hull random ed eeteuiu global unique the Here . ieeta equations differential l ensae rfinite- or spaces dean cosadweak and actors t.W provide We ets. hc sotnused often is which , t admat- random st) eyarbitrary. tely wr attrac- rward ytmare system l † ygeneral: ry adattractor; ward ti char- is It . 1, 0 1 , and 1 0, 1 , are point attractors. Obviously, there is a minimal point r´attractor,s Y t namelyu t´ 1uYr, 0, 1 . s t´ u In fact, for deterministic systems the following result is well known: Whenever B is an arbitrary family of non-empty subsets of the state space then there exists an attractor for B (i.e., a compact, strictly invariant set attracting every B B) if and only if there exists a compact set such that every B B is attracted by thisP set. Furthermore, there exists a unique minimal attractor for BP, which is given by the closure of the union of the ω-limit sets of all elements of B. This minimal attractor for B is addressed as the B- attractor. If a global set attractor exists then whenever a B-attractor exists it is always a subset of the global set attractor. For random dynamical systems an analogous statement has been established in [6]. However, this result had to assume a separability condition for B. The aim of the present paper is to remove this separability condition, i.e. to establish that for any family B of (possibly even random) sets for which there is a compact random set attracting every element of B, there exists a unique minimal random attractor for B. Furthermore, it is in general not true that this B-attractor is given by the closed random set ΩB ω almost surely, (1) BPB p q ď where ΩB ω denotes the (random) Ω-limit set of B. This is shown using an example of a randomp q dynamical system induced by a stochastic differential equation on the unit circle S1. Here the global set attractor is the whole S1 (which is a strictly invariant compact set). If B is taken to be the family of all deterministic points in S1 it is shown that (1) gives S1, the global set attractor, almost surely. However, the minimal point attractor is a one point set consisting of a random variable, which (pullback) attracts every solution starting in a deterministic point.

2 Notation and Preliminaries

Let E be a Polish space, i.e. a separable topological space whose topology is metrizable with a complete metric. Several assertions in the following are formulated in terms of a metric d on E which is referred to without further mentioning. This metric will always be assumed to generate the topology of E and to be complete, even if some of the assertions hold also if d is not complete. For x E and A E we define d x, A inf d x, a : a A with the convention d x, P . For non-emptyĂ subsets Ap andq“B of Et pwe denoteq P theu Hausdorff semi-metricp by Hq“8d B, A : sup d b, A : b B and define d , A : 0 and d B, : in case B p. q “ t p q P u WepH noteq “ that withp thisHq convention“8 both for‰ H the empty family B as well as for B the empty set A is an attractor, in fact the minimal“ one. H “ tHu “ H

2 We denote the Borel σ-algebra on E (i.e. the smallest σ-algebra on E which contains every open set) by E.

Suppose that Ω, F , P is a probability space, T1 Z, R , and p q P t u ϑ : T1 Ω Ω ˆ Ñ t, ω ϑtω p q ÞÑ is a measurable map, such that ϑt : Ω Ω preserves P, and such that ϑt s ϑt ˝ ϑs Ñ ` “ for all s, t T1 and ϑ0 id. Thus ϑt is a classical measurable dynamical system on Ω, F , P . P “ p q p q Definition 1. Given Ω, F , P and ϑt, t T1 as above, E a Polish space, and T2 p q P either R, 0, , Z, or N0 such that T2 T1, a measurable map r 8q Ď ϕ : T2 E Ω E ˆ ˆ Ñ t, x, ω ϕ t, ω x p q ÞÑ p q (or the pair ϕ, ϑ ) is a random dynamical system (RDS) on E if p q (i) ϕ t, ω : E E is continuous for every t T2, P-almost surely p q Ñ P (ii) ϕ t s,ω ϕ t, ϑsω ˝ ϕ s,ω for all s, t T2, and ϕ 0,ω id, for P-almost allp ω`. q “ p q p q P p q “ Remarks 2. (i) Note that we do not assume continuity in t here.

(ii) An RDS ϕ, ϑ is said to be two-sided if T2 is two-sided. For a two-sided RDS ϕ p q ´1 the maps ϕ t, ω are invertible, and ϕ t, ω ϕ t, ϑtω a. s. p q p q “ p´ q (iii) Proposition 25 shows that one can always change ϕ on a set of measure 0 in such a way that properties (i) and (ii) in Definition 1 hold without exceptional sets. In the following we will tacitly assume that ϕ satisfies these slightly stronger assumptions. The question whether exceptional sets of measure zero in (ii) of Definition 1 which may depend on s and t can be eliminated (without destroying possible (right-)- continuity properties of ϕ in the time variable) has been addressed in [1], [2], [17], and [15]. Definition 3. Let Ω, F , P be a probability space and E a Polish space. A random set C is a measurablep subsetq of E Ω (with respect to the product σ-algebra E F ). ˆ b The ω-section of a set C E Ω is defined by Ă ˆ C ω x : x, ω C , ω Ω. p q “ t p q P u P In the case that a set C E Ω has closed or compact ω-sections it is a random set as soon as the mapping ω Ă d x,ˆ C ω is measurable (from Ω to 0, ) for every x E, see [7, Chapter 2]. ThenÞÑC will bep saidq to be a closed or a compactr , respectively,8q randomP ` ˘ set. For any set C E Ω, we define C : x, ω : x C ω Ă ˆ “ tp q P p qu

3 Remark 4. Note that it does not suffice to define random sets by demanding ω d x, C ω to be measurable for every x E. In this case the associated x, ω : xÞÑ C ω pneedq not be an element of E F ,P see [9, Remark 4]. tp q P `p qu ˘ b Definition 5. If ϕ is an RDS then a set D E Ω is said to be forward invari- Ă ˆ ant or strictly invariant, respectively, with respect to ϕ, if ϕ t, ω D ω D ϑtω or p q p q Ă p q ϕ t, ω D ω D ϑtω P-a. s., respectively, for every t 0. p q p q“ p q ě Definition 6. For B E Ω we define the Ω-limit set of B by Ă ˆ

ΩB ω ϕ t, ϑ´tω B ϑ´tω , ω Ω, p q“ T ě0 těT p qp p qq P č ď as in [5, Definition 3.4].

Remark 7. It is easy to verify that an Ω-limit set is always forward invariant and also that it is strictly invariant for an RDS with two-sided time. The following Lemma, which generalizes [6, Theorem 3.4], provides another sufficient condition for ΩB to be strictly invariant.

Lemma 8. Suppose that B E Ω, K E Ω, and Ω0 Ω. Assume that for all Ă ˆ Ă ˆ Ă ω Ω0, the set K ω is compact and P p q

lim d ϕ t, ϑ´tω B ϑ´tω ,K ω 0. (2) tÑ8 p q p q p q “ ` ˘ Then, for every t 0 and ω Ω0, ΩB ϑtω ϕ t, ω ΩB ω and ΩB ω K ω . If, ě P p q Ă p q p q p q Ă p q moreover, Ω0 F and P Ω0 1, then ΩB is strictly invariant. P p q“

Proof. Fix ω Ω0 throughout the proof. Compactness of K ω and (2) imply ΩB ω P p q p qĂ K ω . Suppose that y ΩB ϑtω for some t 0. Then y limn ϕ tn, ϑ t ϑtω bn p q P p q ě “ Ñ8 p ´ n p qq for sequences tn and bn B ϑt t ω . Consider the sequence ϕ tn t, ϑ t t ω bn, Ñ8 P p ´ n q p ´ ´p n´ q q defined for n with tn t 0. By (2) we have limnÑ8 d ϕ tn t, ϑ´ptn´tqω bn,K ω 0 for n . Compactness´ ě of K ω implies that this sequencep ´ has a convergentq p q subse-“ Ñ 8 p q ` ˘ quence. Choose one and denote its limit by z ω , then z ω ΩB ω . Using the same notation for the subsequence, continuity of ϕ t,p ωq impliesp q P p q p q

ϕ t, ω z ω lim ϕ tn, ϑ´ptn´tqω bn y. p q p q“ nÑ8 p q “

Thus, for any y ΩB ϑtω there exists z ΩB ω with ϕ t, ω z y, whence ΩB ϑtω P p q P p q p q “ p qĂ ϕ t, ω ΩB ω . p q p q The final statement of Lemma 8 follows since ΩB is forward invariant. Now let B be a non-empty family of sets B E Ω. At this point we make no measurability assumptions on the sets in B andĂ thereforeˆ say that a property depending on ω holds almost surely if there is a measurable set of full measure on which the

4 property holds. Analogously, we will interpret statements like “Y 0 in probability” for a real-valued (possibly non-measurable) function ω Y ω . AsÑ usual, we introduce the universal completion F u of F as the intersectionÞÑ ofp allq completions of F with respect to probability measures on F . Note that one automatically has F u F in the case of a complete probability space Ω, F , P . “ p q We define the concept of a random pullback, forward and weak B-attractor of an RDS ϕ, ϑ as usual (except that we do not impose any measurability assumptions on B). pRandomq pullback attractors were first introduced in [8] while the concept of a weak attractor is due to G. Ochs [16]. Definition 9. Suppose that ϕ is an RDS on a Polish space E and B is a non-empty family of subsets of E Ω. Then a set A E Ω is a random attractor for B if ˆ Ă ˆ (i) A is a compact random set (ii) A is strictly ϕ-invariant, i.e.

ϕ t, ω A ω A ϑtω p q p q“ p q P-almost surely for every t T2 with t 0 P ě (iii) A attracts B, i.e.

lim d ϕ t, ϑ´tω B ϑ´tω , A ω 0 P-a.s. (3) tÑ8 p q p q p q “ for every B B; in this` case A is a random pullback˘ attractor for B, or P lim d ϕ t, ω B ω , A ϑtω 0 P-a.s., tÑ8 p q p q p q “ for every B B; in this case` A is a random forward˘ attractor for B, or P lim d ϕ t, ω B ω , A ϑtω 0 in probability (4) tÑ8 p q p q p q “ for every B B; in this` case A is a weak (random)˘ attractor for B. P Remark 10. Note that the property of being a pullback or weak attractor does not depend on the choice of the metric d metrizing the topology of E. This is not true for forward attactors, not even when E R, see [13, Example 2.4] or the example in Section 5. “ Remarks 11. (i) For a weak random attractor condition (4) is equivalent to (3) with almost sure convergence replaced by convergence in probability. See [10] for sufficient conditions for the existence of weak global set attractors. For monotone RDS, the concept of a weak attractor turns out to be more suitable than that of a pullback attractor, see [4] and [14]. (ii) While for non-autonomous systems it is very simple to find examples of attractors which are pullback but not forward, and vice versa (see, e.g., [9]), this is not so simple

5 for random attractors. One of the authors [18] has constructed examples for this to happen as well as examples where only weak random attractors exist which are neither pullback nor forward attractors.

(iii) Clearly each random pullback attractor A for B must satisfy ΩB ω A ω almost surely for every B B (the exceptional sets may depend on B). p qĂ p q P (iv) Instead of assuming an attractor A to be a compact random set it suffices to as- sume A to be a random set such that A ω is compact P-almost surely, which is slighty weaker (see [7, Proposition 2.4] for details).p q Proposition 19 below, applied to the single- ton A, implies existence of a compact random set satisfying the conditions of Definition 9.

Remark 12. For a general family B there may or may not exist a (pullback, forward or weak) random B-attractor for an RDS ϕ and if such an attractor exists then it need not be unique in general. However, as soon as B contains every compact deterministic set, then whenever a weak random attractor for B exists then it is unique, see [13, Lemma 1.3]. Since every pullback and every forward attractor is also a weak attractor, the same uniqueness statement holds for pullback and forward attractors (for pullback attractors this was already established in [5]). Whenever a B-attractor (in whatever sense) is not unique it is natural to ask whether there is a smallest (or minimal) B-attractor. In [6, Theorem 3.4] a condition on B (formulated only for families of deterministic sets) for the existence of a minimal random pullback attractor has been given. It is one of the aims of this paper to show that a minimal random pullback (respectively weak) attractor exists for arbitrary B, provided existence of at least one pullback (respectively weak) B-attractor.

3 Existence of a minimal pullback attractor for B

Given a random dynamical system ϕ, ϑ on the space Ω, F , P taking values in the Polish space E,d we consider a generalp q family B of subsetsp ofq E Ω for which we assume existencep ofq a pullback attractor or, equivalently, the existenceˆ of an attracting compact random set. Note that no measurability conditions whatsoever are imposed on (the elements of) B. The major result of this section is the existence of a minimal random pullback attractor for B. Of course this is immediate as soon as B contains every compact deterministic subset of E (or, more precisely, every C Ω with C E compact); in this case the minimal attractor is the unique global set attractor.ˆ However,Ă for instance for B consisting of all (deterministic) points (or, equivalently, of all finite sets), there may be several random pullback attractors.

Theorem 13. Let B be an arbitrary family of sets B E Ω and let ϕ, ϑ be an RDS on E. Ă ˆ p q

6 (i) Assume that there exists a random set K E Ω such that K ω is P-a.s. compact and K attracts B, i.e. Ă ˆ p q

lim d ϕ t, ϑ´tω B ϑ´tω , K ω 0 P-a.s. tÑ8 p q p q p q “ ` ˘ (and therefore ΩB ω K ω a.s.) for every B B. Then ϕ, ϑ has at least one pullback attractorp forqĂB. p q P p q (ii) If ϕ, ϑ has at least one pullback attractor for B, then the RDS has a minimal p q pullback B-attractor A. In addition, there is a countable sub-family B0 B such Ă that A is also the minimal pullback B0-attractor. (iii) Under the assumptions of (i) the minimal pullback B-attractor A is almost surely contained in K ω . p q Note that under the assumptions of (ii) any pullback attractor K for B will satisfy the assumptions of (i). Therefore, it suffices to show that under the assumptions of (i) there exists a minimal pullback attractor for B and that it satisfies the conclusions of (ii) and (iii). For the proof of the theorem we are going to prepare several results which will also be used in the next section, where we investigate the same question for weak attractors. For the first few results we just need an arbitrary probability space Ω, F , P and a Polish metric space E,d , but no RDS. We will denote the open ballp with centreq x E and radius r 0 byp B qx, r . By D we always denote a fixed countable dense set inPE. The followingą lemma justp requiresq a separable metric (not necessarily Polish) space E,d . It is a straightforward consequence of classical results in topology. p q Lemma 14. For each fixed r 0 there exists a cover R x, r , x D, of E such that R x, r is a closed (possibly empty)ą subset of B x, r , andp forq eachPy E there exists a neighbourhoodp q of y which intersects only finitelyp manyq of the sets R Px, r , x D, and there exists x D such that y is in the interior of R x, r . p q P P p q Proof. For given r 0, the family B x, r , x D, is an open cover of E. Since every metric space is paracompact,ą Theoremp VIII.4.2q P in Dugundji [12] implies the existence of a cover R x, r , x D, as claimed in the lemma. p q P Remark 15. Note that for each r 0 the closed cover R x, r , x D, given in ą p q P Lemma 14, has the property that for every (possibly infinite) subset D0 D the union R x, r is closed. This will be important in the following. Ă xPD0 p q RemarkŤ 16. Note that for each r 0, the family R˚ x, r , x D, is an open cover of E thanks to the last assertion of Lemmaą 14 (here S˚ denotesp q theP interior of the set S).

In the following R x, r , x D, will always denote a closed cover as in Lemma 14 (which, of course, is not unique).p q TheP following result asserts that every subset of E Ω has a closed random hull. ˆ

7 Proposition 17. Let K E Ω. There exists a (unique) smallest closed random set Kˆ which contains K in theĂ followingˆ sense: K Kˆ and for every random set S for which K ω S ω for almost all ω Ω and for whichĂ S ω is closed for almost all ω Ω, we havep qĂKˆ ω p q S ω for almostP all ω Ω. p q P p qĂ p q P Proof. For G E define P M G : ω : K ω G . “ t p qX ‰ Hu ˚ G This set will not be measurable in general. Let βG : P M : inf P M : M G “ p G q “ t p q P F , M M and let M1 M2 ... be sets in F with M Mi, i N, such that Ą u Ą˜ G Ą 8 Ă P limi P Mi βG. Define M : 1 Mi and Ñ8 p q“ “ i“ 8 Ş 1 1 ˆ ˜ Rpx, n q K : R x, n M . (5) “ n“1 xPD p qˆ č ď ` ˘ Kˆ is a random set. Using Lemma 14 and Remark 15, we see that Kˆ ω is closed for 1 1 Rpx, q p q all ω Ω. Since ω d y, R x, M˜ n ω is measurable for each x, y and n, P ÞÑ p p n qˆ qp q the same is true for the map ω d y, Kˆ ω , so that Kˆ is a closed random set. By ` ˘ construction, we have K ω Kˆ ÞÑω forp allpωqq Ω. Indeed, p qĂ p q P 8 1 1 Rpx, n q K R x, n M . “ n“1 xPD p qˆ č ď ` ˘ It remains to show the minimality property of Kˆ . Let S be a set as in the statement of the proposition and let G E. Then, almost surely, P K ω G S ω G p qX Ă p qX and therefore S ω G for almost every ω M G. Since E is Polish, the projection theorem (see, e.g.,p qX Dellacherie‰ H and Meyer [11,P III.44]) implies that the set ω Ω: S ω G is in F u and therefore S ω G for almost all ω Mt˜ GP. Let p qX ‰ Hu p qX ‰ H P Ω0 F be a set of full measure such that S ω is closed for all ω Ω0 and S ω G P G 1 p q P p qX ‰ H for all ω M˜ Ω0 and all G R x, , x D, n N. Let ω Ω0 and y Kˆ ω . P X “ p n q P P P P p q By definition of Kˆ this means that for every n N there exists some x D such that 1 1 ˜ Rpx, n q P P y R x, n and ω M , so d y,S ω 2 n. Since S ω is closed this implies y P S ωp , soq the proofP of the propositionp p isqq complete. ď { p q P p q Remark 18. The assertion of Proposition 17 is wrong without the word closed (at both places where it appears). As an example consider the deterministic case in which Ω is a singleton and let K be a set which is not in E. There exists no smallest measurable subset of E which contains K.

8 u Proposition 19. Assume that Kα, α I, is a family of sets in E F . Then there P b exists a closed random set A such that for every α I we have Kα ω A ω almost surely and A is the minimal set with this property: AP ω S ω almostp q surely Ă p forq every p qĂ p q S E Ω for which S ω is almost surely closed and for which Kα ω S ω almost surelyĂ forˆ every α I. p q p qĂ p q P Furthermore, there exists a countable subset I0 I such that Ă

A ω Kα ω (6) p q“ p q αPI0 ď up to a set of measure zero.

Proof. We can and will assume without loss of generality that the family Kα, α I, is closed under finite unions. For an arbitrary set G E and α I we define P P P G M : ω Ω: Kα ω G . α “ t P p qX ‰ Hu G F u Since E is Polish, Mα by the projection theorem [11, III.44]. We can extend the probability measurePP to F u in a unique way (we will use the same notation for G the extension). Note that on the set Mα each S as in the lemma necessarily satisfies S ω G almost surely and therefore the same must be true for A (which we yet need p qX ‰ H G to define). Let βα,G : P Mα . Then, there exists a sequence αj αj G , j 1, 2,... G“ “ p q “ in I such that P M βG : sup βα,G. Define αj Õ` “˘ αPI ` ˘ 8 G G M : Mαj . “ j“1 ď G The set M depends on the choice of the sequence αj , but different choices give sets which only differ by a set of measure 0. Changing the sets M G on a set of P-measure 0, ` ˘ α we can and will in fact assume that all these sets (and also the sets M G) are even in F . G G Note that for each α I we have Mα M up to a set of measure 0. Recall that D is a countable and denseP subset of E. DefiningĂ

Bpx,rq Cr : B x, r M , C : C1{n, “ xPD p qˆ “ nPN ď ` ˘ č one may hope to be able to show that A : C has the required properties. While the measurability property clearly holds it is not“ clear that A ω is closed. One might there- p q fore take the closure of the right hand side in the definition of Cr, but then measurability of A is not clear. We will therefore change the definition of Cr in such a way that it becomes a closed random set for each r 0. Then C will be a closed random set as well. ą Define R x, r as in Lemma 15 and put p q Rpx,rq Ar : R x, r M , A : A1{n. “ xPD p qˆ “ nPN ď ` ˘ č

9 Clearly, Ar E F and Ar ω is closed for each ω Ω and each r 0. As in the proof P b p q P ą of Proposition 17 it follows that Ar is even a closed random set and hence the same is true for A.

For S ω as in the proposition and α I we have to show that Kα ω A ω S ω for P-almostp q all ω Ω. P p q Ă p q Ă p q P Fix α I. For each r 0 we have P ą Rpx,rq Rpx,rq Kα ω R x, r Mα ω R x, r M ω Ar ω p qĂ xPD p qˆ p qĂ xPD p qˆ p q“ p q ´ ď ` ˘¯ ´ ď ` ˘¯ up to a set of measure 0 (which may depend on α). Hence, up to a set of measure 0,

Kα ω A ω . (7) p qĂ p q To finish the proof it suffices to show that

A ω Kαj pRpx,1{nqq ω (8) p q“ xPD,nPN,jPN p q ď up to a null set (observe that the right hand side is in S ω almost surely). Note that the inclusion “ ” follows from (7) and the fact that A ω isp closed,q so it remains to show the inclusion “Ą”. p q Ă For every n N and almost every ω Ω we have P P 1 2{n A ω R x, K 1 ω , p qĂ p n qĂ αj pRpx, {mqqp q Rpx,1{nq xPD,mPN,jPN xPD,Mď Qω ď where the upper index 2 n denotes the closed 2 n-neighbourhood of a set. Taking intersections over all n N{, (8) follows and the proof{ of the proposition is complete. P Remark 20. The assertion of Proposition 19 becomes wrong if the word closed is deleted. As an example take Ω a singleton, E R, I a non-measurable subset of R, and “ Kα α for α I. There is no minimal measurable subset of R which contains I. “ t u P Our next goal is to clarify whether forward and strict invariance, respectively, of a set K are inherited by Kˆ given by (the proof of) Proposition 17. Lemma 21. (i) If K E Ω is forward invariant then so is Kˆ . Ă ˆ (ii) If K E Ω is strictly invariant, and if Kˆ ω is compact for almost every ω Ω, then KĂˆ isˆ strictly invariant. p q P

Proof. Let K be forward invariant, so for fixed t 0 we have ϕ t, ω K ω K ϑtω . Here and in the following equalities and inclusionsě are meant upp to setsq p ofqĂ measurep 0.q Hence ´1 ´1 ˆ K ω ϕ t, ω K ϑtω ϕ t, ω K ϑtω . p qĂp p qq p p qqĂp p qq p p qq

10 The right hand side is a random subset with closed ω-sections since ϕ t, ω is continuous and therefore contains Kˆ , so Kˆ is invariant. p q Next suppose that K is strictly invariant. Changing Kˆ on a set of measure 0 if necessary we can and will assume that Kˆ is a compact random set. For t 0 fixed we have ě ˆ ϕ t, ω K ω ϕ t, ω K ω K ϑtω . p q p qĄ p q p qĄ p q ˆ If the left hand side is a random set with closed ω-sections, then it must contain K ϑtω and strict invariance of Kˆ follows. Using the representation (5) of the set Kˆ , wep seeq that if ϕ t, S , where S C F with C a deterministic compact subset of E and F F ,p is a¨qp randomq set, then“ theˆ same is true for ϕ t, Kˆ . Observe that the map y P d y,ϕ t, ω S ω is measurable for each y Ep since¨qp qd y,ϕ t, ω S ω forÞÑω p F , pd y,ϕqp t,p ωqqqS ω d y,ϕ t, ω C forPω F andp d y,ϕp t,qp ω pC qqq “r 8iff R p p qp p qqq “ p p q q P p p q q ă d y,ϕ t, ω xi r for all i where xi is a countable dense set in C and r 0. This showsp p thatq ϕqt, ă ω Kˆ ω is a randomp set.q To see that it has closed ω-sections weą use the fact that Kˆ ωp isq compactp q and ϕ t, ω is continuous. This completes the proof of the lemma. p q p q ˆ ˆ Proof [of Theorem 13]. We apply Proposition 19 to KB : ΩB, B B, where ΩB is “ P constructed from ΩB as in Proposition 17, and obtain a smallest closed random set A ˆ containing ΩB almost surely for every B B. Clearly A is also the minimal closed P random set A containing ΩB almost surely for each B B. We claim that (a slight modification of) A is the minimal pullback B-attractor.P By assumption, the set K ω ˆ p q from the assertion of Theorem 13 contains ΩB ω and, by Proposition 17, also ΩB ω almost surely for every B B. By minimality,p weq have A ω K ω almost surely.p q Since K ω is almost surelyP compact so is A ω . Changingp Aqon Ă ap setq N of measure zero containingp q those ω for which A ω is notp compactq (e.g. by redefining A ω e for ω N with some fixed e E)p weq can assume that A is a compact randomp q “ set. t u PropositionP 19 further impliesP that A can be represented as the closure of a countable ˆ union of sets ΩB , B B, almost surely. Since ΩB is strictly invariant by Lemma 8 and ˆ P ˆ ΩB ω is almost surely compact, ΩB is strictly invariant by Lemma 21, and so is A and thep proofq of Theorem 13 is complete. Remark 22. It is natural to ask if Theorem 13 remains true if the set K is not required to be E F -measurable (but all other assumptions hold). This is not the case in general. As an example,b take Ω 0, 1 equipped with Lebesgue measure P on the Borel sets, take ϑ id, E R, and“ rϕ sid (with discrete or continuous time). Let f : Ω R be a function“ whose“ graph B“ E Ω is non-measurable and let B : B . ThenÑ Ă ˆ “ t u ΩB ω B ω f ω for all ω, hence ΩB B and the assumptions of Theorem 13 holdp withq “ Kp q “B texceptp qu that K is not a random“ set. If a pullback-B-attractor exists then it cannot“ possibly be contained in K, so (iii) of Theorem 13 does not hold. One might hope that Theorem 13 remains true if one replaces K by Kˆ in part (iii). However, also this fails to hold true since K E Ω with K ω compact for almost all ω does not guarantee that Kˆ ω is almostĂ surelyˆ compactp andq if we choose B K then no B-pullback attractor exists.p q “

11 If we impose additional measurability assumptions on B then measurability of K is not required for Theorem 13 to hold. Indeed, if ΩB is a random set for every B B (or u P even just ΩB E F ), then define A as in Proposition 19. Due to formula (6) we see that A ω KP ωbalmost surely, so all assumptions of Theorem 13 hold with K replaced by A. p qĂ p q

4 Existence of a minimal weak attractor for B

Now we discuss the corresponding question for weak attractors. We continue to use covers R x, r , x D, as in the previous section. p q P Theorem 23. Let B be an arbitrary family of sets B E Ω and let ϕ, ϑ be an RDS on E. Ă ˆ p q (i) Assume that there exists a random set K E Ω such that K ω is P-a.s. compact and K attracts B, i.e. Ă ˆ p q

lim d ϕ t, ϑ´tω B ϑ´tω , K ω 0 in probability, tÑ8 p q p q p q “ ` ˘ for every B B. Then ϕ, ϑ admits at least one weak attractor for B. P p q (ii) If the RDS ϕ, ϑ has a weak attractor for B, then it has a minimal weak B- p q attractor A. In addition, there is a countable sub-family B1 B such that A is Ă also the minimal weak B1-attractor. (iii) Under the assumptions of (i) the minimal weak B-attractor A satisfies A ω K ω almost surely. p qĂ p q

Proof. For G E and B B define P P G F ˚ VB : V : lim P V ω : ϕ t, ϑ´tω B ϑ´tω G 0 “ P tÑ8 X t p qp p qqX ‰ Hu “ ! ) ` G ˘ G and put β : sup P V : V VB . Choose an increasing sequence Vi VB such that “ t p q P 8u G P limiÑ8 P Vi β. Then V : i“1 Vi VB (since outer measures are monotone and subadditive).p q “ Put M G : Ω V“and defineP B “ z Ť

8 1 1 Rpx, n q KB : R x, n MB . “ n“1 xPD p qˆ č ď ´ ¯

Fix B B. Then KB is a closed random set. We first show that KB is a minimal weak P B -attractor and that KB is contained in K. t u Step 1: KB ω K ω almost surely p qĎ p q

12 Let B B, G E and denote St : ω : ϕ t, ϑ´tω B ϑ´tω G for t 0 and P δP “ t p q p ˚ qX ‰ Hu ě Mδ : ω : K ω G for δ 0. Then limtÑ8 P St Mδ 0 by assumption, “ t c pG qX ‰ Hu Gą δ p z q “G whence Mδ VB and therefore MB Mδ, so K G on MB almost surely for each δ Q P 0, . This implies Ď X ‰ H P Xp 8q K G on M G a.s., X ‰ H B provided G is closed (using the fact that K ω is almost surely compact). In particular, p q we have 1 Rpx, q R x, 1 n K3{n ω on M n a.s. p { qĎ p q B and therefore

8 1 8 R x, 3{n p n q 3{n KB ω K ω MB ω K ω K ω a.s., p qĎ n“1 xPD p qˆ p qĎ n“1 p q“ p q č ď´ ¯ č proving Step 1.

Our next goal is to show that KB attracts B in probability. Generally, we say (in this proof) that a random set S E Ω attracts B if Ă ˆ

lim d ϕ t, ϑ´tω B ϑ´tω ,S ω 0 in probability. tÑ8 p q p q p q “ ` ˘ Step 2: The following properties are easy to verify:

a) If S1 and S2 attract B, then so does S1 S2. X b) If Sn, n N, are random sets such that Sn ω is compact for almost all ω and such P 8p q that Sn attract B, n N, then so does 1 Sn. P n“ 1 1 Rpx, n q Step 3: We show that En : R x, ŞM attracts B for each n N. “ xPD p n qˆ B P ´ ¯ Let ε 0 and let K E be compactŤ such that P K ω K 1 ε. Let D0 D be a ą Ă ˚ 1 p p qĎ qě ´ ˚Ă 1 c finite set such that K xPD0 R x, n and let δ : inf d y,K : y xPD0 R x, n . Note that δ 0. We haveĂ p q “ t p q P p q u ą Ť Ť ` ˘ ˚ P d ϕ t, ϑ tω B ϑ tω , En ω ε p ´ q p ´ q p q ě 1 ´ ˚ ¯ 1 1 Rpx, n q P ` d ϕ t, ϑ tω B ϑ tω˘ R x, , R x, M ε ď p ´ q p ´ qX p n q p n qˆ B ě ´ ´xPD´ ¯ xPD0´ ¯¯ ¯ ˚ ď δ ď P ϕ t, ϑ´tω B ϑ´tω K ď p q p qĆ 1 ˚ 1 1 Rpx, n q ` P d ϕ t, ϑ tω B ϑ ˘tω R x, , R x, M ε . ` p ´ q p ´ qX p n q p n qˆ B ě xPD0 ÿ ´ ´ ¯ ´ ¯¯ ¯

13 1 Rpx, n q By definition of MB the sum converges to 0 as t . Further, by the definition of K and δ, Ñ8 ˚ δ lim sup P ϕ t, ϑ´tω B ϑ´tω K ε. tÑ8 p q p qĆ ă ` ˘ Since ε 0 was arbitrary it follows that En attracts B. ą

Step 4: Define Sn : En K. Then Steps 1, 2 and 3 and the definition of KB show “ X that KB attracts B.

Note that the statements in Step 1 and Step 4 imply that KB is a minimal closed random weak B-attracting set. Further we know that KB ω is compact for almost all ω Ω. p q P Changing KB ω on a set of measure 0 we can ensure that KB is a compact random set. p q In order to complete the proof that KB is a minimal weak B -attractor it remains to t u show strict invariance of KB.

Step 5: KB is strictly invariant.

The proof of this step is similar to that of Lemma 21. Fix t 0. Since KB attracts B in probability, we have ą

lim d ϕ t s, ϑ´sω B ϑ´sω ,KB ϑtω 0 in probability (9) sÑ8 p ` q p q p q “ ` ˘ and lim d ϕ s, ϑ´sω B ϑ´sω ,KB ω 0 in probability. (10) sÑ8 p q p q p q “

Using the cocycle property` and continuity of ϕ together˘ with compactness of KB, (10) implies

lim d ϕ t s, ϑ´sω B ϑ´sω ,ϕ t, ω KB ω 0 in probability. (11) sÑ8 p ` q p q p q p q “ ` ˘ Using the fact that ϕ t, ω KB is a random set (as in the proof of Lemma 21), and using p q minimality of KB, we see from (9) and (11) that KB ϑtω ϕ t, ω KB ω almost surely. Since (9) implies p qĂ p q p q

´1 lim d ϕ s, ϑ´sω B ϑ´sω , ϕ t, ω KB ϑtω 0 in probability, sÑ8 p q p q p q p q “ ` ` ˘ ˘ equation (10) and minimality of KB imply the converse inclusion.

The rest of the proof of the theorem is identical to that in the pullback case: just replace ˆ the set ΩB by KB.

14 5 Non-existence of a minimal forward attractor

In this section we provide an example of an RDS which has a forward attractor, but which fails to have a smallest forward attractor. Consider a stationary Ornstein-Uhlenbeck process Z, i.e. a real-valued centered Gaussian process defined on R with covariance 1 E Z t Z s 2 exp t s . We define Z on the canonical space C R, R of continuous functionsp p q p fromqq “ R to p´|R together´ |q with the usual shift and equippedp withq the law of Z. Then Z t, ω Z 0, ϑtω . Let g : 0, 1 1, 0 be continuous, non-decreasing such that g 1p 0q, “g xp 0 forq all x 0r, 1 sand Ñg r´x s 1 for all x 0, 1 2 , and let h t, y be thep uniqueq“ solutionp qă of the ordinaryPr q differentialp q“´ equation Pr { s p q

9 g h t if h t 0 h t p p qq p qą p q“ #0 otherwise with initial condition h 0 y 0, 1 . p q“ Pr s Next we define, for x R, y 0, 1 , and t 0, P Pr s ě x Z t, ω Z 0,ω , h t, y if t τ y ϕ t, ω x, y : p ` p q´ p q p qq ď p q p qp q “ # eτpyq´t x Z 0,ω Z t, ω , 0 if t τ y , p ´ p qq` p q ě p q where τ y : min s 0 : `h s,y 0 . Note that the motion˘ in the y-direction is deterministicp q “ and allt pointsě withp yq “ 0 umove in parallel in the x-direction. If y 1, then after the finite (deterministic)ą time τ y , the y-component arrives at 0 and staysă there, while the first coordinate is attractedp q by the process Z exponentially fast. It is straightforward to check that ϕ defines a continuous RDS on the Polish space E : R 0, 1 . If we consider E equipped with the Euclidean metric then the singleton“ Zˆ0 r,ω s, 0 is a forward attractor for the family B : C 0 : C R compact . tpThisp is noq qu longer true if we change the metric on E“in t theˆ tfollowingu Ă way (withoutu changing the topology of E):

d x, y , x,˜ y˜ : y˜ y Γ x˜ Γ x , p q p q “ | ´ | ` | p q´ p q| where Γ is strictly increasing,` odd,˘ continuous such that Γ x exp exp exp x for large x (the fact that this metric works can be checked byp q using “ thet factt thatp quuu the running maximum of a stationary Ornstein Uhlenbeck process up to time t is of the order ?log t). There are, however, many forward B-attractors with respect to the metric d, for example

Aγ ω : Z 0,ω γ,Z 0,ω γ 0 p q “ r p q´ p q` s ˆ t u ` Z 0,ω γ 0, 1 ˘Z 0,ω γ 0, 1 t p q´ uˆr s Y t p q` uˆr s ď´` ˘ ` ˘¯ for an arbitrary γ 0 (note that this set is strictly invariant!). Now if there would be a smallest forwardą B-attractor A ω then A ω would have to be contained in the p q p q

15 intersection A1 ω A2 ω , which is a subset of R 0 . It is clear, however, that the set Z 0,ω , 0p isq the onlyp q strictly invariant compactˆ t subsetu of R 0 , and we already havetp notedp q thatqu thisŞ is not a forward attractor. This contradictsˆt theu assumption that there is a smallest forward B-attractor. Consequently, this RDS does not have a smallest B-attractor.

6 Another example

We construct an example of an RDS for which a minimal pullback point attractor ω ÞÑ A ω exists which, however, does not coincide with Ωx ω (writing Ωx ω instead of p q xPE p q p q Ωtxu ω for brevity). This shows that Theorem 4 in Crauel and Kloeden [9] is not entirely p q Ť correct. In the following example, A ω consists of a single point while Ωx ω p q xPE p q coincides with the whole space E almost surely. Even though we have Ωx ω A ω for almost all ω Ω ([6, Theorem 3.4] or Remark 11 (iii)), the (uncountable)pŤq uni Ă onp ofq P all Ωx ω will turn out to be considerably larger than A ω . p q p q Example 24. Let E : S1 be the unit circle which we identify with the interval 0, 2π equipped with the usual“ metric d x, y : x y 2π x y . Consider the SDEr q p q “ | ´ |^p ´ | ´ |q dX t cos X t dW1 t sin X t dW2 t p q“ p p qq p q` p p qq p q on E, where W1 and W2 are independent standard Brownian motions. Then there exists a ‘stable point’ ω S ω , measurable with respect to “the past” σ W t : t ÞÑ p q t p q ď 0 , W W1, W2 , which is the support of a random invariant measure, and whose Lyapunovu “ p exponentq is negative (see Baxendale [3]). The random one point set S ω is a (minimal) weak point attractor (even a forward point attractor) of the RDStϕpwhichqu is generated by the SDE. Recall that the system is reversible. Reverting time and using the same argument for the time inverted system gives existence of an ‘unstable point’ ω U ω , measurable with respect to W t : t 0 and therefore independent of S, whichÞÑ isp aq weak point repeller. The domaint p q of attractioně u of S ω is E U ω and that of U ω for the time-reverted flow is E S ω . t p qu zt p qu t p qu zt p qu When considering the system with continuous time, ω S ω is not a pullback point ÞÑ t p qu attractor of ϕ, though. In fact we even have Ωx ω E almost surely for each fixed x E since for each fixed y E, the process t pϕ q “t, ω y is a Brownian motion on E P P ÞÑ p´ q and therefore hits x for (some) arbitrarily large values of t showing that y Ωx ω for almost all ω Ω. In particular, the unique pullback point attractor of ϕ isP thep wholeq space E. P To obtain the required example we therefore evaluate the RDS ϕ at integer times only, i.e. we define ψn ω, x : ϕn ω, x , x E, n N0, p q “ p q P P and we work with T Z instead of T R. We denote the restriction of ϑ to T Z with the same symbol.“ We claim that now“ S ω is a minimal pullback point attractor,“ t p qu

16 but that the closure of the union over all Ω-limit sets of the points in E equals E almost surely. To see the first claim, consider the neighbourhood I S ω ε,S ω ε (mod 2π) of S ω for some ε 0, 1 . The claim follows once we“ show p p thatq´ for eachp q`x qE we have p q Pp q P Ωx ω I almost surely. Since the Lyapunov exponent of ψ is negative, the normalized p qĂ ´1 c Lebesgue measure of the set ψn ϑ´nω, I converges to 0 geometrically fast as n almost surely. The first Borel-Cantellip lemma¨qp q now implies that the Lebesgue measureÑ8 of ´1 c the set C ω of all x E which are contained in the set ψn ϑ´nω, I for infinitely many n pNqis zero. SinceP the distribution of C ω is invariantp under¨qp rotationsq of E this implies thatP for each fixed x E we have P x p Cq ω 0. This being true for every ε P p P p qq “ of the form 1 m we obtain Ωx ω S ω almost surely. { p q “ t p qu To see the second claim, take any non-empty (deterministic) compact interval J E. ´1 Ă Then the set ψn ϑ´nω, J is a non-trivial interval for each n. Note that for the centre p ¨qp q ´1 point x J, the process n ψn ϑ´nω, x performs a random walk on E (Brownian P ÞÑ p ¨qp q ´1 motion evaluated at discrete time steps) and therefore, the set n ψn ϑ´nω, x is almost surely dense in E. Moreover, for any y E and any δ 0, we almostp surely¨qp findq P ą Ť´1 a sequence of integer random times tn such that y δ, y δ n ψtn ϑ´tn ω, J . Therefore, for any z in that set, wep haveq r ´ ` sX p ¨qp q ‰ H Ş

Ωz ω J almost surely, p qX ‰ H showing the second claim. Note that we have actually proved more than we claimed insofar that for every non-empty open subset I of E we have

Ωz ω E for P-almost every ω. zPI p q“ ď 7 Perfection

Proposition 25. Let ϕ, ϑ be an RDS as in Definition 1. Then there exists an RDS ψ p q on the same measurable dynamical system and a set Ω1 F of measure 1 such that ψ P agrees with ϕ on Ω1 and ψ satisfies (i) and (ii) of Definition 1 without exceptional sets.

Proof. By assumption, there exists a set N F such that for all ω N c the following P P hold: x ϕ t, ω x is continuous for every t T2, ϕ 0,ω id, and ϕ t s,ω ÞÑ p q P p q “ p ` q “ ϕ t, ϑsω ϕ s,ω for all s, t T2. Define p q˝ p q P c Ω1 : ω Ω: ϑsω N for almost all s T1 “ t P P P u (here, “almost all” refers to Lebesgue measure in the continuous case and counting mea- sure in the discrete case). Clearly, Ω1 has full measure and is invariant under ϑt for every t T1. Then ψ t, ω x : ϕ t, ω x in case ω Ω1 and ψ t, ω id in case ω Ω1 satisfiesP the claims inp theq proposition.“ p q P p q “ R

17 References

[1] L. Arnold, Random Dynamical Systems, Springer, New York 1998 [2] L. Arnold and M. Scheutzow, Perfect cocycles through stochastic differential equa- tions, Probab. Theory Relat. Fields 101 (1995) 65–88 [3] P. H. Baxendale, Asymptotic behaviour of stochastic flows of diffeomorphisms, pp. 1–19 in Stochastic processes and their applications (Nagoya, 1985), Lecture Notes in Math., 1203, Springer, Berlin 1986 [4] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant mea- sures for a class of monotone random systems, Dyn. Syst., 19 (2004) 127–144 [5] H. Crauel, Global random attractors are uniquely determined by attracting deter- ministic compact sets, Ann. Mat. Pura Appl., IV. Ser., Vol. CLXXVI (1999) 57–72 [6] H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., II. Ser., 63 (2001) 413–427 [7] H. Crauel, Random Probability Measures on Polish Spaces, Series Stochastics Mono- graphs, Volume 11, Taylor & Francis, London and New York 2002 [8] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. The- ory Relat. Fields 100 (1994) 365–393 [9] H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver. 117 (2015) 173–206 [10] H. Crauel, G. Dimitroff, and M. Scheutzow, Criteria for strong and weak random attractors, J. Dynam. Differential Equations 21 (2009) 233–247 [11] C. Dellacherie and P.A. Meyer, Probabilities and Potential, North-Holland, Ams- terdam 1978 [12] J. Dugundji, Topology, Allyn and Bacon, Boston 1966 [13] F. Flandoli, B. Gess, and M. Scheutzow, Synchronization by noise, Probab. Theory Relat. Fields 168 (2017) 511-556 [14] F. Flandoli, B. Gess, and M. Scheutzow, Synchronization by noise for order- preserving random dynamical systems, Ann. Probab. 45 (2017) 1325-1350 [15] G. Kager and M. Scheutzow, Generation of one-sided random dynamical systems by stochastic differential equations, Electronic J. Prob. 2 (1997) 17 pages [16] G. Ochs, Weak random attractors, Report 449, Institut für Dynamische Systeme, Universität Bremen, 1999 [17] M. Scheutzow, On the perfection of crude cocycles, Random and Comp. Dynamics 4 (1996) 235–255

18 [18] M. Scheutzow, Comparison of various concepts of a random attractor: a case study. Arch. Math. (Basel) 78 (2002) 233–240

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