2 GUNTER OCHS in the future, a pullback attractor takes only the dynamics of the past into account. However, in the framework of random dynamical syst

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1 WEAK RANDOM ATTRACTORS GUNTER OCHS Abstract. We dene point attractors and set attractors for random dynamical systems via convergence in probability forward in time. This denition of a set attractor is weaker than the usual one via almost sure pullback convergence. We derive basic properties of our weak random attractors such as uniqueness, support of invariant measures and invariance under (random) coordinate transformations. The notion of weak random attractors allows us to consider local and relative attractors. We discuss attractor{repeller decompositions and give a characterization of the basin of attraction.. Introduction Attractors play an important role in the theory of dynamical systems. A global attractor for an autonomous dynamical system given by a (semi{)ow or the iterates of a map is a compact invariant set which attracts all trajectories of points or compact resp. bounded sets as time tends to innity. That is, if a system possesses a global attractor A, then important information on its long term behavior is captured by A. In particular, A supports all invariant measures. A random dynamical system (for more information see the monograph of Ludwig Arnold []) models dynamics inuenced or perturbed by probabilistic noise. For a \typical" random dynamical system there is no deterministic subset of the phase space, which captures the long term dynamics. Therefore a reasonably dened attractor has to be a random set, i.e. a set valued random variable. Such random attractors were dened and investigated by several authors (e.g. [2, 6, 7, 8, 9, 0, 2, 4, 6]) using the idea of pullback convergence. That is, instead of letting time go forward to innity the evolution of the system from time?t to 0 is considered. The limit for t! then gives a xed object (a compact set) for time 0 which depends on the past of the system. However, this approach has two drawbacks. Firstly, the basin of attraction is typically a space of (set valued) random variables (a universe), which in particular makes it complicated to characterize local attractors. The second drawback is of more philosophical nature. Whereas an attractor should describe in some way the asymptotic dynamics of the system Date: October 5, 999..

2 2 GUNTER OCHS in the future, a pullback attractor takes only the dynamics of the past into account. However, in the framework of random dynamical systems there is a \link between future and past" due to the stationarity of the noise. Namely, pullback convergence to an attractor implies also attraction forward in time, although in a weaker sense via convergence in probability instead of almost sure convergence. This is the starting point of the present paper, where we use this forward attraction property as a denition of a (weak) random attractor. A random attractor is always a random compact set which is \invariant" under the random dynamical system (that is, it is allowed to \move" under time evolution, but in a stationary manner). We consider set attractors (which attract all random compact sets contained in their basin of attraction) and point attractors (which attract just random points). In Section 3 we derive basic properties of weak random attractors. We show that the set attractor is the maximal invariant random compact set in its basin and therefore unique. Every set attractor is also a point attractor but not vice versa. As in the deterministic case there exist point attractors which are proper subsets of set attractors (where certain parts with \transient dynamics" are removed). Therefore a point attractor is typically smaller than the set attractor and not unique. However, every point attractor already supports all invariant measures which assign full mass to its basin of attraction. For weak set attractors it often suces to verify attraction for a subset of the space of all random sets. In particular, if the basin of attraction is deterministic, then an invariant random compact set which attracts all deterministic compact sets is already a weak set attractor. We discuss the connection between weak random attractors and pullback attractors. It is shown that every pullback set attractor is also a weak set attractor, provided it attracts \enough" random sets. An important tool in the investigation of random dynamical systems are random coordinate transformations. In the framework of pullback attractors one has to look how the universe under consideration behaves under a coordinate change. Therefore one needs some assumptions (like growth conditions) to ensure the invariance of a pullback attractor under a random coordinate transformation. For weak (set and point) attractors this invariance holds without any additional assumptions. The basin of attraction of a weak random attractor is a random subset of the state space of the system (which may be the whole space in the case of a global attractor). This makes this concept exible in order to decompose the phase space into basins of dierent attractors. In particular, it is possible to decompose an attractor itself. We do not require attractors to be minimal (a deterministic attractor is minimal, if it contains a dense orbit). Therefore it is possible, that an attractor contains smaller invariant random compact sets which are also local or relative attractors.

3 WEAK RANDOM ATTRACTORS 3 In Section 4 we show that to every local attractor A sitting inside some invariant random compact set E there exists a corresponding repeller R. The set E splits into A, R, and a \transient" set, where all orbits converge to R as time t tends to? and to A for t! +. A repeated application of this procedure yields a Morse decomposition. Here an invariant random compact set splits into a nite number of smaller invariant random compact sets (the Morse sets) and a remaining transient set, which consists of \connecting orbits" between dierent Morse sets. In Section 5 we introduce the notion of stability (in the sense of Lyapunov) of a random invariant set. Under the assumption of stability a characterization of the basin of attraction of a weak random attractor is possible (see Theorem 8). Deterministic set attractors are always stable. In the case of random set attractors we are only able to prove stability for pullback attractors. It is not clear whether there exist weak random set attractors which are not stable (or even whether or not every weak random set attractor is a pullback attractor for some suitable universe). Finally, in Section 6, we give some examples. The primary goal of this paper is to introduce the concept of weak random attractors and not to investigate attractors for particular systems. Therefore our examples are mainly intended to serve as an illustration of our notions. 2. Preliminaries 2.. Random dynamical systems. A random dynamical system models dynamics under the inuence of noise. It consists of two ingredients: A measure preserving ow # = (# t ) t2t on a probability space (; F; P), which serves as a model for the noise. We always assume that # is invertible, i.e. T = R or Z. A measurable mapping ' : T (+) X! X; (t;!; x) 7! '(t;!)x; where the state space X is a metric space (with metric d), such that { x 7! '(t;!)x is continuous for xed t;!, { ' satises the cocycle property '(0;!) = id X and '(t + s;!) = '(s; # t!) '(t;!) for t; s 2 T (+) and! 2. We say ' is a random dynamical system on X over (; #). For technical reasons we assume X to be separable. Throughout this paper all assertions about! are assumed to hold on a # invariant set of full measure (unless otherwise stated). The cocycle property (which reduces to the ow property if ' is independent of the noise!) implies, that the skew product = ' : T (+) X! X; (t;!; x) 7! (t)(!; x) := (# t!; '(t;!)x) denes a measurable (semi-)ow on the product space X.

4 4 GUNTER OCHS 2.2. Random measures and random sets. The statistical behavior of dynamical systems is usually described by invariant measures. In the case of a random dynamical system an invariant (probability) measure is a probability measure on X with marginal P on, which is invariant under the skew product, i.e. (t) = for every t. An invariant measure can be identied with a random measure (! )!2 on X via the disintegration (d!; dx) =! (dx)p(d!): Then invariance is equivalent to '(t;!)! = #t!. The support S(!) of! denes a random invariant set. In general a random set is a family (B(!))!2 of subsets of X, such that its graph B := f(!; x) : x 2 B(!)g X is product measurable. We identify random sets with their graphs. A random set is forward invariant, if (t)b B (i.e. '(t;!)b(!) B(# t!)) for t > 0. It is invariant, if equality holds ((t)b = B). A random compact set (resp. random closed set) is a random set A such that all A(!) are compact (resp. closed) and! 7! d(x; A(!)) := inf d(x; y) y2a(!) is measurable for every x 2 X (the last condition is automatically fullled if the probability space (; F; P) is complete). We allow A(!) to be empty, i.e. the above inmum can be innity. Random compact sets are random variables! K(X), where K(X) denotes the space of all compact subsets of X (including ;) endowed with Hausdor metric d H and corresponding Borel {algebra. The following facts on random closed sets are well known (see e.g. Arnold [, Chapter.6] or Castaing and Valadier [4]). The union of two and the intersection of countably many random compact (closed) sets is again a random compact (closed) set. The same is true for the image (t)a of a random compact set with t 2 T (+). In the case of A a random closed set one has to take the!-wise closure of (t)a. For every random closed (compact) set A there exists a selector, i.e. a random variable x :! X with x(!) 2 A(!) for every! with A(!) 6= ;. Furthermore every invariant random compact set A with P(A(!) 6= ;) = supports an invariant measure. For later use we need the following regularity property of random measures: Lemma. Let (; F) be a measurable space and (X; d) be a Polish space with Borel {algebra B. If is a probability measure on ( X; F B), then for every " > 0 and every set B 2 F B there exist random compact sets K; L :! K(X) with K B, L \ B = ;, and (K [ L) >? ". Proof. Let C be the collection of all B 2 F B, for which for every " > 0 there exist random compact sets K and L with K B, B \ L = ;, and (K [ L) >? ".

5 WEAK RANDOM ATTRACTORS 5 We will show that C is a {algebra which contains all sets of the form E X and A with E 2 F and A a closed subset of X, which implies F B C. Let be the marginal of on X. Since X is a Polish space, there exists for every " > 0 a compact set C X with (C) >? ". If E is measurable dene K(!) := C and L(!) := ;, if! 2 E, and K(!) := ; and L(!) := C, if! 2 n E. This shows E X 2 C. If A X is closed, then, since is regular, there exists for " > 0 compact sets K 0 A and L 0 na with (K 0 ) > (A)? 2 " and (L 0) > (na)? 2 ". Choosing K K 0 and L L 0 we see A 2 C. By denition C contains the complement S of every B 2 C. Now assume B ; B 2 ; ::: 2 C, set B := j= B j, and x " > 0. Choose random compact sets K ; K 2 ; ::: and L ; L 2 ; ::: with K j B j, L j \ B j = ;, and (K j [ L j ) >? 2?j " for j 2 N. With Z j := n (K j [ L j ) we have for every n X = 0 n j= T Set L := j= L j, choose n with 00 n (Z j [ K j ) j= 0 \ A n j= L j A n L A < "; L j A : S n and set K := j= K j. Then K and L are random compact sets with K B, L \ B = ;, and 0 00 [ ( X) n (K [ n \ Z j A n L j A n L A j= j= ) (( X) n (K [ L)) " + 2?j " < 2"; j= i.e. B 2 C, which completes the proof Pullback attractors. A universe D is a space of random closed sets. A random compact set A 2 D is called a D (pullback) attractor, if A is invariant, i.e. '(t;!)a(!) = A(# t!) for t 0, lim t! dist('(t; #?t!)d(#?t!); A(!)) = lim t! dist((t)d; A) = 0 P almost surely for every D 2 D. Here dist(a 0 ; B 0 ) := sup a2a0 inf b2b0 d(a; b) denotes the semi Hausdor distance for deterministic subsets A 0 ; B 0 X (with dist(;; B 0 ) = 0) and dist(a; B) has to be interpreted as a function of! for random sets A and B, i.e. dist(a; B)(!) = dist(a(!); B(!)). Note that dist(a; B) is measurable if A and B are random closed sets. nx

6 6 GUNTER OCHS An example for a universe is D := (D(!))!2 : lim sup t! t log d(x; D(# t!)g = 0 8 x 2 X ; the universe of all tempered random sets. For many systems it is possible to verify the existence of a D pullback attractor, which is then considered as the global attractor of the system (for examples see e.g. [0, 2, 6]). A dierent approach is due to Crauel [7], who denes random attractors by just attracting deterministic sets, i.e. he considers D := fd : D 2 Cg, where C is any given collection of (closed) subsets of X. In particular, his global set attractor corresponds to C = fc X compactg and his global point attractor to C = ffxg : x 2 Xg. 3. Weak point atrractors and weak set attractors Denition. Let ' be a random dynamical system over (; #) on a metric space X and B X be a forward invariant random set. An invariant random compact set A B is called a (weak) B point attractor, if A attracts random points in probability, i.e. n lim P! : d('(t;!)x(!); A(# t!)) > " = 0 t! for every " > 0 and every random variable x :! X with x(!) B(!). A is called a (weak) B set attractor, if it attracts random compact sets in probability, i.e. lim t! P! : dist('(t;!)c(!); A(# t!)) > " = lim t! P(dist((t)C; A) > ") = 0 for every random compact set C B and every " > 0. Remarks. () The random variables dist('(t; )C(); A(# t )) and dist((t)c; A) = dist('(t; #?t )C(#?t ); A()) have the same distribution due to the # invariance of P. (2) A global (point or set) attractor corresponds to the case B(!) X. We speak of a local attractor, if the basin B(!) contains a neighborhood of A(!). In general an attractor according to our denition need not be a global or local attractor. In this case we speak of a relative attractor. (3) It is also possible to dene a bounded set attractor, which attracts all random closed sets C for which C(!) is bounded. A bounded set attractor is clearly a global (compact) set attractor. The converse is false (if X is not locally compact), as already deterministic examples show. Our rst (elementary, although called a theorem) result shows, that a set attractor is the maximal invariant random compact set within its basin.

7 WEAK RANDOM ATTRACTORS 7 Theorem. Let B be a forward invariant random set. If A is a weak B set attractor, then any invariant random compact set E B is with probability one contained in A, i.e. Pf! : E(!) A(!)g =. In particular, there exists (modulo null sets) at most one weak B set attractor. Proof. Let E B be a random compact set with (t)e = E for t > 0. For " > 0 we obtain P(dist(E; A) > ") = P(dist((t)E; A) > ")! 0 as t!, since A is an attractor. Hence P(dist(E; A) ") = for every " > 0, which implies P(E A) =. Now we are going to show that a point attractor supports all invariant measure on its basin. Since every set attractor is also a point attractor, the result holds also for set attractors. Theorem 2. Let ' be a random dynamical system over (; #) on a Polish state space X. Assume A is a weak B point attractor for some B X and is a ' invariant measure with (B) =. Then (A) =. Proof. Assume (A) <. For " > 0 dene a random set A " via A " (!) := fx 2 X : d(x; A(!)) "g. Then A = T ">0 A" and hence lim "!0 (A " ) = (A) <. Thus there exists " > 0 with (A " ) <? ". By the ergodic theorem there exists a measurable function f : X! R + with R f d =? (A " ) > " > 0 and lim t! t fs < t : (s)(!; x) 62 A" g = f (!; x) almost surely, where denotes Lebesgue measure if T = R and counting measure if T = Z. With G := f(!; x) 2 B : f (!; x) "g it follows (G) > 0 (using (B) = ). By Lemma there exist a random compact set K G with (K) > 0. This implies the existence of a random variable x :! X with P(x(!) 2 K(!)) =: > 0. Since K G B we can choose x such that x(!) 2 B(!) for every!. By construction of x lim inf t! t P f(!; s) 2 [0; t[: (s)(!; x(!)) 62 A" g " > 0: On the other hand, since A is an attractor, lim P s! ((s)(!; x(!)) 62 A" ) = 0 ) lim t! t P f(!; s) 2 [0; t[: (s)(!; x(!)) 62 A" g = 0: Hence the assumption (A) < leads to a contradiction.

8 8 GUNTER OCHS Remark. (4) It suces to assume that X is separable and a Borel subset of its completion X. In this case Lemma applies to G interpreted as a measurable subset of X. (5) The point attractor dened by Crauel [7] (which attracts deterministic points in the pullback sense) typically needs not to contain \random unstable sets", i.e. random invariant sets whose basin of attraction has zero Lebesgue measure. Therefore not every pullback point attractor is a weak point attractor (this is dierent for set attractors, see below). However, it is also possible to construct examples of invariant sets, which are global point attractors in our sense but not in the sense of Crauel (cf. Example ). Theorem 3. Let B be a forward invariant random set and D a collection of random closed sets with D B for every D 2 D. Assume that for every random compact set C B and every > 0 there exists D 2 D with P(C(!) D(!)) >?. If lim t! P(dist((t)D; A) > ") = 0 for every D 2 D and every " > 0, then A is a weak B set attractor. Proof. Given a random compact set C B and > 0 choose D 2 D with P(C(!) D(!)) >? ) P('(t;!)C(!) '(t;!)d(!)) >? for t > 0. Using the # t invariance of P we have for " > 0 lim sup t! P(dist((t)C; A) > ") lim sup P(dist((t)D; A) > ") + = : t! Since > 0 was arbitrary it follows lim t! P(dist((t)C; A) > ") = 0. A straightforward consequence is Corollary 3.. Let B and D satisfy the assumption of Theorem 3. Then any D (pullback) attractor is also a weak B attractor. Furthermore, in the case of a deterministic basin B it suces to consider attraction of deterministic sets. A related result in the framework of pullback attractors is due to Crauel [6]. Corollary 3.2. Let X be a Polish space and B 0 X a deterministic set, such that B := B 0 is a forward invariant random set. Assume that B := fc 2 K(X) : C B 0 g is a Borel subset of K(X). If lim t! dist('(t;!)c 0 ; A(# t!)) = 0 in probability for every (deterministic) compact set C 0 B 0, then A is a B set attractor. Proof. We show that for every random compact set C and every > 0 there exists a deterministic compast set C 0 B 0 with P(C(!) C 0 ) >? (cf. Crauel [5, Proposition 3.5]). Observe that (K(X); d H ) is a Polish space if X is. Hence there exists a compact set C B mit P(C(!) 2 C) >?.

9 WEAK RANDOM ATTRACTORS 9 S It remains to show that C 0 := C2C C is compact. To do this take a sequence (x i ) C 0. For every i there exists C i 2 C with x i 2 C i. By compactness of C there exists a converging subsequence, w.l.o.g. there exists D = lim i! C i 2 C, i.e. D C 0. This implies lim i! d(x i ; D) = 0, i.e. there exists y i 2 D with d(x i ; y i )! 0. Any accumulation point of (y i ) (which exists by compactness of D) is also an accumulation point of (x i ). Remarks. (6) A similar assertion holds for bounded set attractors. (7) A weak point attractor (even with deterministic basin) is typically not determined by just attracting deterministic points (cf. Crauel [7]). Our next result is concerned with the invariance of weak random attractors under coordinate transformations. Theorem 4. Let ' be a random dynamical system on a metric space X over (; #). Furthermore, let fh! g!2 a family of homeomorphisms from X to a metric space Y. Dene h : X! Y by h(!; x) := (!; h! (x)) and assume that h and h? are measurable with respect to F B(X) resp. F B(Y ). Then a continuous random dynamical system on Y over (; #) is dened by (t;!) = h #t! '(t;!) h?!. Suppose A is a weak random B (point or set) attractor for ' with some forward invariant B X. Then h(a) is a weak random h(b) (point or set) attractor for. Proof. Let C h(b) be a random compact (resp. one point) set and denote D := h? (C) B. We have lim t! dist( ' (t)d; A) = 0 in probability, which is equivalent to the assertion that for every sequence (t n ) of times with lim t n = there exists a subsequence (t n(k) ) k2n with lim k! dist( ' (t n(k) )D; A) = 0 P a.s., i.e. there exists 0 with P( 0 ) = and lim dist('(t n(k); #?tn(k)!)d(#?tn(k)!); A(!)) = 0 k! for every! 2 0. By the continuity of h! and the compactness of A(!) it follows (applying the map h! ) for! 2 0 lim k! dist( (t n(k); #?tn(k)!)c(#?tn(k)!); (ha)(!)) = 0; which implies lim t! dist( (t)c; h(a)) = 0 in probability. Remarks. (8) A D pullback attractor for ' corresponds to a D 0 pullback attractor for, where D 0 is the \image" of D under h. Therefore pullback attractors for tempered sets are only invariant under coordinate transformations which satisfy some growth condition (see e.g. Keller and Schmalfu [4] or Imkeller and Schmalfu [2]). There is no general result which ensures the invariance of pullback attractors for deterministic sets under coordinate changes. In particular,

10 0 GUNTER OCHS the point attractor dened by Crauel [7] is typically not invariant. (9) Theorem 4 shows that an attractor only depends on the topology on X but not on the metric d. This follows by considering the homeomorphism id : (X; d)! (X; d 0 ), where d 0 is a dierent metric generating the same topology. (0) Sometimes it is useful to consider bundle random dynamical systems and/or random metrics (see e.g. Arnold [, Chapters.9 and 4.3]), that is the state space and/or the metric on it depend on!. Weak random attractors can also be dened in this setup. Again an attractor does not depend on the particular choice of the random metric. 4. Decomposition of attractors In this section we provide a tool to analyze the structure of an attractor. In many cases a random attractor (in particular a global attractor) is not minimal. Then the supports of ergodic invariant measures are only subsets of the attractor. A deterministic attractor, on which there is a \gradient like" structure of the dynamics on it, allows a Morse decomposition. That is, there are smaller compact invarint subsets (the Morse sets) and connecting orbits between them. The union of the Morse sets is a point attractor, which supports all invariant measures. Here we prove an analogous statement for weak random attractors. We start with a decomposition with two Morse sets, one of which is a local attractor and the other one the \corresponding repeller". Theorem 5 (attractor{repeller decomposition). Let E be an invariant random compact set for a random dynamical system ' over (; #). Assume that ' restricted to E is invertible, i.e. '(t;!) : E(!)! E(# t!) is a homeomorphism for t 0. Further let A E be a U set attractor for some forward invariant random set U E, such that U(!) is an open neighborhood of A(!) relative to E(!). Then there exists an invariant random compact set R E with the following properties: R \ A = ;, A is an E n R set attractor for ', R is an E n A set attractor for the inverse of ', i.e. lim t! P(dist((?t)C; R) > ") = 0 for every " > 0 and every random compact set C E n A. Proof. With '(?t;!) := '(t; #?t!)? : E(!)! E(#?t!)

11 WEAK RANDOM ATTRACTORS for t > 0 we have '(?t;!) (E(!) n U(!)) E(#?t!) n U(#?t!) by the forward invariance of U. Hence \ \ R(!) := '(?t; # t!) (E(# t!) n U(# t!)) = '(?t; # t!) (E(# t!) n U(# t!)) t0 t2n is a countable intersection over a decreasing sequence of compact sets. Thus R is a random compact set. By denition '(?t;!)r(!) = R(#?t!), i.e. R is invariant. Since A U we have R \ A = ;. Now assume C E n R is a random compact set. By denition of R for every x 2 C(!) there exists t = t (!; x) > 0 with '(t;!)x 2 U(# t!) for every t t. Since '(t ;!) is a homeomorphism and U() E() is open we have '(t ;!)y 2 U(# t!) for every y from an open neighborhood of x in E(!). By compactness of C(!) there exists t 2 = t 2 (!) with '(t;!)c(!) U(# t!) for every t t 2. Given " > 0 choose t 3 with P(t 2 (!) > t 3 ) < " 2. With C0 := (t 3 )C and C 00 (!) := C 0 (!) if C 0 (!) U(!) and C 00 (!) := ; else there exists t 4 > 0 with P(dist((t)C 00 ; A)) > ") < " 2 for t t 4. Since P(C 00 (!) 6= C 0 (!)) < " 2 it follows P(dist((t)C; A) > ") = P(dist((t? t 3 )C 0 ; A) > ") < P(dist((t? t 3 )C 00 ; A) > ") + P(C 00 (!) 6= C 0 (!)) < " for t t 3 + t 4. This shows that A is an E n R set attractor. In order to prove the last property dene for given " > 0 a random compact set D by D(!) := fy 2 E(!) : d(y; R(!)) "g. For any random compact set C E n A and t 0 we have P(dist((?t)C; R) ") = P(((t)D)(!) \ C(!) 6= ;) (using the invertibility of on E). Since C \ A = D \ R = ; we can nd > 0 with P(C(!) \ fx : d(x; A(!)) g 6= ;) < 2 " and t 0 > 0 with P(dist((t)D; A) ) < 2 " for t t 0. This implies P(((t)D)(!) \ C(!) 6= ;) < ", which completes the proof. To proceed further we need to know how weak random attractors behave under intersection and unication of their basins. Theorem 6. Let A be a B set (resp. point) attractor and A 2 be a B 2 set (resp. point) attractor. Then A \ A 2 is a B \ B 2 set (resp. point) attractor. Proof. Let " > 0 and a random compact set C B \ B 2 (in the case of point attractors a one point set) be given. We have \ >0 f(x; y) 2 A (!) A 2 (!) : d(x; y) 2g = (A (!) \ A 2 (!)) 2 ; which implies by compactness, that there exists (!) > 0 with d(z; A (!)) < (!); d(z; A 2 (!)) < (!) ) d(z; A (!) \ A 2 (!)) < ":

12 2 GUNTER OCHS Choose > 0 with P((!) < ) < " 3 and t 0 with P(dist((t)C; A i ) ) < " 3 for i = ; 2 and t t 0. Then P(dist((t)C; A \ A 2 ) ") < " for t t 0. Theorem 7. Let A be a B point attractor and A 2 be a B 2 point attractor. Then A [ A 2 is a B [ B 2 point attractor. Proof. Let x be a random variable with x(!) 2 B (!) [ B 2 (!). We have for t; " > 0 P(d('(t;!)x(!); (A [ A 2 )(# t!)) > ") Pf! : x(!) 2 B (!) and d('(t;!)x(!); A (# t!)) > "g + Pf! : x(!) 2 B 2 (!) and d('(t;!)x(!); A 2 (# t!)) > "g; which converges to 0 as t!. Theorems 5,6 and 7 yield the following Morse decomposition, if there is a \hierarchy" of local attractors. Corollary 7.. Let A ::: A p E be invariant random compact sets for the random dynamical system ' such that each A k is a U k set attractor, with U k (!) an open neighborhood of A k (!) relative to E(!). Assume that ' restricted to E is invertible. Then there exist invariant random compact sets R p ::: R, such that: Each A k is an E n R k set attractor for ', each R k is an E n A k set attractor for the inverse of ', if k l p, then R k \ A l is an R k n R l set attractor for ' and an A l n A k set attractor for the inverse of '. Sp? A [ R p [ k= (A k+ \ R k ) is an E point attractor. Remarks. () By Theorem 2 it follows that in the situation of Corollary 7. every invariant measure with (E) = is supported by the Morse sets A, R p and A k+ \ R k for k = ; :::; p?. (2) A similar Morse decomposition is possible, if there is a nite number of local attractors which are only partially ordered by inclusion. (3) If E is a B attractor for some B E, then it is clear that the Morse sets support every invarinat measure with (B) =, but it does not follow that the union of the Morse sets is a B point attractor. However, we conjecture that this holds true. (4) For set attractors a statement analogous to Theorem 7 does not hold. In the situation of Theorem 5 the E set attractor is equal to E by Theorem. (5) As already mentioned a B point attractor is in general not unique. However, Theorem 6 shows that the intersection of two B point attractors is again a B point attractor. Thus it makes sense to look for a point attractor which is minimal with respect to inclusion.

13 WEAK RANDOM ATTRACTORS 3 5. The basin of attraction In this section we assume our random dynamical system ' to be continuous in time, i.e. (t; x) 7! '(t;!)x is continuous for xed!. Note that this is automatically true if T = Z and is also satised for continuous time random dynamical systems generated by stochastic or random dierential equations (see Arnold [, Chapter 2]). Continuity in time implies that t 7! A(# t!) is continuous for every invariant random compact set A. To obtain a characterization of the basin of attraction of an attractor A we need the notion of stability. A compact invariant set A for a deterministic continuous ow (f t ) t2t on a metric space is called stable (in the sense of Lyapunov), if for each " > 0 there exists a > 0 such that dist(x; A) < implies dist(f t (x); A) < " for every t > 0. We have the following generalization for random dynamical systems: Denition. (i) Let ' be a random dynamical system over (; #) on a metric space X. An invariant random compact set A is called stable if there exist sequences (" n ) and ( n ) of random variables from to ]0; [ with lim n! " n = 0 P almost surely, such that the following holds: Whenever d(x; A(!)) < n (!), then d('(t;!)x; A(# t!)) < " n (# t!) for every t > 0. (ii) A point x 2 X is called attracted by A in the! ber, if for every " > 0 lim t! t fs 2 [0; t[: dist('(s;!)x; A(# s!)) > "g = 0; where denotes Lebesgue measure if T = R and counting measure if T = Z. Theorem 8. Assume A is stable with t 7! n (# t!) continuous. (i) A is a weak random B set attractor with B(!) = fx 2 X : x is attracted by A in the! berg: (ii) If A is a weak random B point attractor for some B X, then for every random variable x :! X with x(!) 2 B(!) the probability that x(!) is attracted by A in the! ber is equal to. Proof. Set ^B(!) := fx : 8 n 9 t n = t n (!; x) 0 with d('(t n ;!)x; A(# tn!)) < n (# tn!)g = \ n= [ q2q\t + fx : d('(q;!)x; A(# q!)) < n (# q!)g : We are going to show that there exists a set with P( ) = such that x 2 ^B(!) if and only if x is attracted by A in the! ber whenever! 2. For n 2 N and > 0 let f n; (resp. g n; ) be the conditional expectation of the characteristic function of f" n > g (resp. f n > g) with respect to the {algebra of # invariant sets. Then, if is xed, lim n! f n; (!) = 0

14 4 GUNTER OCHS for! from a set of full measure. Furthermore, by the ergodic theorem with probability one lim t! t fs 2 [0; t[: " n(# s!) > g = f n; (!) for every n 2 N. An analogous statement holds with " n replaced by n and f n; replaced by g n;. Let be the set of full measure where all this holds. Assume! 2 and x 2 ^B(!). For given > 0 choose n with fn; (!) <. By the denition of ^B(!) and the stability of A we have lim sup t! lim sup t! t fs 2 [0; t[: d('(s;!)x; A(# s!)) > g t (t n + fs 2 [t n ; t[: " n (# s!) > g) = f n; (!) < ; which shows that x is attracted by A in the! ber. Now assume! 2 and x is attracted in the! ber. Fix n 2 N. Since n > 0 there exists > 0 with g n; (!) > 0, i.e. lim t! t fs 2 [0; t[: n(# s!) > g = g n; (!) > 0 = lim t! t fs 2 [0; t[: d('(s;!)x; A(# s!)) > g: Hence there exists s =: t n > 0 with d('(s;!)x; A(# s!)) < n (# s!), i.e. x 2 ^B(!). To prove (i) observe rst that the forward invariance of B = ^B is clear from the denition. Let C ^B be a random compact set. Given > 0 choose n 2 N with P(" n ) < 2. For T > 0 set = [ q2q\t + \[0;T ] B T := f(!; x) 2 B : t(!; x) T g f(!; x) : d('(q;!)x; A(# q!)) < n (# q!)g and D T := C n B T. Then D T (!) is compact, i.e. D T is a random compact set with respect to the completion F of F, on which P is dened. By denition of ^B and the compactness of C(!) we have sup x2c(!) t n (!; x) <, which implies lim T! P(D T (!) = ;) =. Choose T with P(D T (!) 6= ;) < 2. If D T (!) = ;, then dist('(t;!)c(!); A(# t!)) < " n (# t!) for every t T by the denition of stability. Hence we have for t T P(dist((t)C; A) > ) P(dist((t)C; A) > " n ) + 2 which shows that A is a ^B set attractor. P(D T (!) 6= ;) + 2 < ;

15 WEAK RANDOM ATTRACTORS 5 Now assume that (ii) does not hold, i.e. there exists a random variable x :! X with x(!) 2 B(!) and P(x(!) 62 ^B(!)) > 0. Then there exists n 2 N with := Pf! : '(t;!)x(!) n (# t!) for every t 0g = Pf! : '(q;!)x(!) n (# q!) for every q 2 Q \ T + g > 0: Choose > 0 with P( n ) < 2. Then P(d('(t;!)x(!); A(# t!)) > ) P(d('(t;!)x(!); A(# t!)) n (# t!))? 2 2 > 0 for every t > 0, which contradicts the assumption that A is a B point attractor. Corollary 8.. Every stable local point attractor, which satises the assumptions of Theorem 8 (i.e. t 7! n (# t!) is continuous), is a local set attractor. Proof. Let A be a B point attractor. Assume A is stable and B(!) contains an open neighborhood of A(!). Dene ^B as in the proof of Theorem 8. By denition ^B is the intersection of random sets ^Bn with ^Bn (!) open. Take a random closed set C B such that C(!) contains an open neighborhood of B(!). Assume that the probability that ^B(!) contains an open neighborhood of A(!) is strictly smaller than. Then D(!) := C(!) n ^Bn (!) 6= ; with positive probability for n suciently large. Since D(!) is closed D is a random closed set with respect to the completion F of F. Hence there exists an F-measurable function x :! X with P(x(!) 2 D(!)) > 0. We can choose x such that x(!) 2 B(!). Furthermore, x can be replaced by an F-maesurable x :! X with P(x = x) =, i.e. there exists a random variable x with P(x(!) 2 ^B(!)) < = P(x(!) 2 B(!)). This contradicts Theorem 8.(ii). For deterministic systems the converse of Corollary 8. is also true, i.e. local set attractors are always stable. Although we conjecture that this is also the case for local random set attractors, we are not able to prove it in full generality. At least we have the following result, which applies to pullback attractors on locally compact spaces. Theorem 9. Let A be an invariant random compact set for an invertible random dynamical system '. Assume that there exists a random set N such that N(!) contains an open neighborhood of A(!), with lim t! dist((t)n; A) = 0 P almost surely. Suppose that t 7! dist('(t; #?t!)n(#?t!); A(!)) is lower semi{continuous. Then A is stable.

16 6 GUNTER OCHS Proof. Set " n := sup dist((t)n; A) = supfdist((q)n; A) : q 2 Q \ T; q ng: tn By assumption " n! 0 almost surely. Since ' is invertible, '(n; #?n!)n(#?n!) contains an open neighborhood of A(!). By compactness of A(!) there exists n (!) > 0 with x 2 '(n; #?n!)n(#?n!) for every x with d(x; A(!)) < n (!). Hence A is stable. Remarks. (6) To obtain continuity of t 7! n (# t!) (which is needed for Theorem 8 and Corollary 8.) one has to assume that N(!) contains the (!) neighborhood of A(!) for some (!) > 0 with t 7! (# t!) continuous. (7) By Theorem 8 A is a B set attractor for some B. The argument in the proof of Theorem 3 implies N B, i.e. in particular A is a local set attractor. Vice versa, the assumptions of Theorem 9 are met if A is a D pullback attractor for a universe D which contains a random neighborhood of A. (8) Of course it is also possible to consider a relative version of stability. Then one has to work with the restriction of a random dynamical system to some forward invariant random set. In this case the assertions of Theorem 8 and Corollary 8. remain valid. 6. Examples The goal of this section is to illustrate our notions. The rst example shows that there exists a global weak random attractor, which does not attract deterministic sets in the sense of almost sure pullback convergence, i.e. it is not an attractor in the sense of Crauel [7]. Our second example shows, how the results of this paper apply to an intensively investigated random attractor, namely the attractor of the stochastic Dung{van der Pol equation. In Example 3 we show how the multiplicative ergodic theorem leads to a random Morse decomposition on the projective space for a random dynamical system generated by a linear cocycle. Example. Set X = R and '(t;!) = ' t, where (' t ) t2r is the (deterministic) ow generated by the ordinary dierential equation _x =? 2 arctan x. The invariant set f0g attacts every bounded deterministic set and is thus by Corollary 3.2 a global weak set attractor. For t > 0 and x 2 R we have j'(t;!)xj jxj? t. Hence for a random closed set D it is necessary (and sucient, as a simple calculation shows) for attraction in the pullback sense that lim t! t supfjxj : x 2 D(#?t!)g = 0: Now apply a random coordinate transformation of the form h! : x 7! x + a(!), where a :! R is any random variable. By Theorem 4 A(!) =

17 WEAK RANDOM ATTRACTORS 7 fa(!)g is the global set attractor for the random dynamical system dened by (t;!) = h #t! '(t;!) h?! : For a random set D it is necessary to be attracted in the pullback sense that lim t! t sup fjx? a(#?t!)j : x 2 D(#?t!)g = 0: In particular, if exp(a) is not tempered (i.e. lim sup t! t ja(# t!)j = ), then A does not attract deterministic points and deterministic compact sets by almost sure pullback convergence. Example 2 (The stochastic Dung{van der Pol attractor). It is known, that the random dynamical system ' generated by the stochastically perturbed Dung{van der Pol equation d x = y dt d y =??x + y? x 2 (x + y) dt + xdw possesses a global set attractor A (Keller and Schmalfu [4]). Furthermore, f0g is an invariant set, which is a repeller for suciently large > 0 (Imkeller and Lederer []). By Theorem 5 there exists a corresponding repeller A for the inverse of ' restricted to A, which is the R 2 n f0g attractor for ' (cf. Keller and Ochs [3, Theorem 2]). It is numerically observed (see Arnold et al. [3], [, Chapter 9.4]) that there is a random two point set A 0 (!) = fx(!)g A (!) which is a local attractor. The existence of A 0 would imply (with Theorem 5 and Corollary 7.) the existence of corresponding repellers R A and R = R \ A. This would give us a Morse decomposition of A with Morse sets f0g, A 0, and R. All invariant measures are supported by these Morse sets (see Remark ). The dynamics on R should be chaotic, as numerical approximations of A indicate [3]. However, there exist no numerical images of the random set R itself. Example 3. An invertible linear random dynamical system ' on R n (i.e. '(t;!) 2 Gl(n; R)) induces a random dynamical system on the projective space P n?. Assume ' satises the intergrability condition of the multiplicative ergodic theorem of Oseledets [5], see also [, Chapter 3]. Then there exist invariant random subspaces E (!); :::; E p (!). For l k p( n) denote by F l;k (!) the linear span of E l (!); :::; E k (!). The F l;k can be interpreted as invariant random compact sets in P n?. The assertions of the multiplicative ergodic theorem then imply that A = F ;k and R = F k+;p give for every k 2 f; :::; p? g an attractor{repeller decomposition in the sense of Theorem 5 of P n?, or, more general, A = F l;m and R = F m+;k are an attractor{repeller decomposition of F l;k whenever l m < k p. The sets A k := F ;k for k = ; :::; p? satisfy the

18 8 GUNTER OCHS assumptions of Corollary 7.. In particular, all invariant measures on P n? are supported by the global point attractor E [ ::: [ E p. Acknowledgments I would like to thank Peter Ashwin, Hans Crauel (who brought my attention to point attractors), and Bjorn Schmalfu for stimulating discussions on the subject of (random) attractors. References [] L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, 998. [2] L. Arnold and B. Schmalfu. Fixed points and attractors for random dynamical systems. In A. Naess and S. Krenk, editors, IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, pages 9{28. Kluwer, Dordrecht, 996. [3] L. Arnold, N. Sri Namachchivaya, and K. R. Schenk{Hoppe. Toward an understanding of stochastic Hopf bifurcation: a case study. International Journal of Bifurcation and Chaos, 6:947{975, 996. [4] C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. Volume 580 of Springer Lecture Notes in Mathematics. Springer, Berlin Heidelberg New York, 977. [5] H. Crauel. Random Probability Measures on Polish Spaces. Habilitationsschrift, Bremen, 995. [6] H. Crauel. Global random attractors are uniquely determined by attracting deterministic compact sets. Ann. Mat. Pura Appl. IV. Ser., Vol. CLXXVI, 999. [7] H. Crauel. Random point attractors versus random set attractors. Preprint, 999. [8] H. Crauel, A. Debussche, and F. Flandoli. Random attractors. Journal of Dynamics and Dierential Equations, 9(2):307{34, 997. [9] H. Crauel and F. Flandoli. Attractors for random dynamical systems. Probab. Theory Relat. Fields, 00:365{393, 994. [0] F. Flandoli and B. Schmalfu. Random attractors for the 3D stochastic Navier{Stokes equation with multiplicative white noise. Stochastics and Stochastics Reports, 59:2{ 45, 996. [] P. Imkeller and C. Lederer. An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator. Preprint 998. [2] P. Imkeller and B. Schmalfu. The conjugacy of stochastic and random dierential equations and the existence of global attractors. Preprint, 998. [3] H. Keller and G. Ochs. Numerical approximation of random attractors. In Stochastic Dynamics. Springer, Berlin Heidelberg New York, 999. [4] H. Keller and B. Schmalfu. Attractors for stochastic dierential equations with nontrivial noise. Buletinul A.S. a R.M. Mathematica, (26):43{54, 998. [5] V. I. Oseledec. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 9:97{23, 968. [6] K. R. Schenk-Hoppe. Random attractors { general properties, existence, and applications to stochastic bifurcation theory. Discrete and Continuous Dynamical Systems, 4:99{30, 998. Institut fur Dynamische Systeme, Universitat Bremen, Postfach , D Bremen, Germany address: gunter@math.uni-bremen.de