Comparison of Various Concepts of a Random Attractor: a Case Study

Arch. Math. 78 (2002) 233–240 Sonderdruck aus 0003-889X/02/030233-08 $ 3.10/0  Birkhäuser Verlag, Basel, 2002

Comparison of various concepts of a random attractor: A case study

MICHAEL SCHEUTZOW

Abstract. Several non-equivalent deﬁnitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with state space [0, ∞) which ﬁx 0. We examine conditions under which the set {0} is an attractor for three different notions of an attractor. It turns out that even in this simple case the various concepts are quite different. The purpose of this note is to highlight these differences and thus provide a basis for discussion about the “correct” concept of a random attractor.

1. Introduction. While the concept of an attractor for deterministic dynamical systems is well-established, the same can not be said for random dynamical systems, which have gained considerable attention during the past decade (for a comprehensive survey see [1]). All definitions of a random attractor A (ω) known to the author agree in that they require that A (ω) be a random compact set which is invariant under the random dynamical system (below we will give precise definitions). The definitions disagree however with respect to the class of sets which are attracted as well as the precise meaning of “attracted”. In order to keep things reasonably simple we will always insist that an attractor attracts all (deterministic) bounded subsets of the (metric) state space. Furthermore we will only discuss random dynamical systems in continuous time. Out of the three definitions we give below the notion of a forward attractor is closest to that of an attractor for a deterministic dynamical system. It is however believed to be the least appropriate one for random dynamical systems. The concept of a pullback attractor (also called strong attractor or just attractor) was proposed independently by H. Crauel and F. Flandoli [3] and B. Schmalfuß [11]. Weak attractors were recently introduced by G. Ochs [10]. In his paper he highlights differences between weak and pullback attractors e.g. concerning invariance properties under random transformations. It is not our aim to point out such different properties but rather to illustrate the different concepts within a particular class of one-dimensional random dynamical systems. For the related question of bifurcations of one-dimensional stochastic differential equations, see [4] and [12]. Before we provide precise definitions we recall the notion of a random dynamical system taking values in a space E. In the second section we will then consider the particular case E =[0, ∞).

Mathematics Subject Classiﬁcation (2000): Primary 37H05, 60H10; Secondary 60G44, 60J55. 234 M. SCHEUTZOW ARCH. MATH.

Throughout the paper we assume that (E,d) is a separable complete metric space. Whenever necessary we will equip E with the Borel-σ-algebra E (i.e. the smallest σ-algebra on E which contains all open subsets). D e f i n i t i o n 1.1. A random dynamical system (RDS) with state space E is a pair (ϑ, φ) consisting of the following two objects: 1. A metric dynamical system (MDS) ϑ ≡ (Ω, F, P, {ϑ(t), t ∈ R}), i.e. a probability space (Ω, F, P) with a family of measure preserving transformations {ϑ(t) : Ω → Ω, t ∈ R} such that (a) ϑ(0) = idΩ,ϑ(t) ◦ ϑ(s) = ϑ(t + s) for all t, s ∈ R; (b) the map (t,ω) → ϑ(t)ω is measurable and ϑ(t)P = P for all t ∈ R. 2. A (perfect) cocycle φ over ϑ of continuous mappings of X i.e. a measurable map φ : R+ × Ω × E → E,(t,ω,x) → φ(t,ω,x) such that (a) the mapping φ(·,ω,·) : (t, x) → φ(t,ω,x) is continuous for all ω ∈ Ω and (b) it satisﬁes the cocycle property:

φ(0,ω,·) = idE,φ(t + s,ω,·) = φ(t,ϑ(s)ω, φ(s,ω,·)) for all t, s м 0 and ω ∈ Ω.

D e f i n i t i o n 1.2. Let (ϑ, φ) be an RDS and ω → A (ω) satisfy (i) A (ω) is a compact subset of E for every ω ∈ Ω (ii) ω → d(x, A (ω)) is measurable for each x ∈ E (iii) φ(t,ω,A (ω)) = A (ϑ(t)ω) for all ω ∈ Ω, t м 0. Then A is called an invariant random compact set.

D e f i n i t i o n 1.3. Let A be an invariant random compact set of the RDS (ϑ, φ). A is called a forward attractor if for each bounded set B ֤ E .1 lim sup d(φ(t,ω,x), A (ϑ(t)ω)) = 0 a.s. →∞ t x∈B A is called a pullback attractor if for each bounded set B ֤ E .2 lim sup d(φ(t,ϑ(−t)ω, x), A (ω)) = 0 a.s. →∞ t x∈B A is called a weak attractor if for each bounded set B ֤ E .3 lim sup d(φ(t,ω,x), A (ϑ(t)ω)) = 0 in probability. →∞ t x∈B

Remark 1.4.Obviouslyallthreeconceptscoincideinthedeterministiccase(i.e. Ω is a singleton). In general each forward attractor and each pullback attractor is a weak attractor (the latter implication holds since ϑ(t) preserves P). No other implication holds as we will see in the next section. It follows from [10], Theorem 1 and Corollary 3.2, that an attractor (in whatever sense) – if it exists – is unique up to a set of measure 0. Some authors only require that an attractor attracts all compact subsets instead of all bounded ones or even just all singletons, e.g. [10]. Since we will only consider the special case E =[0, ∞) below, these distinctions will be irrelevant for us. Vol. 78, 2002 Random attractors 235

2. A case study. Let E =[0, ∞) be equipped with the euclidean metric. We will restrict our attention to RDSs on E which are generated by stochastic differential equations (SDEs) of the form (1) dX(t) = b(X(t))dt + σ(X(t)) ◦ dW(t) = ( ( )) + 1 σ( ( ))σ ( ( )) + σ( ( )) ( ), (2) b X t dt 2 X t X t X t dW t where W(t), t ∈ R is a two-sided standard (real-valued) Brownian motion. The symbol “◦” denotes the Stratonovich stochastic integral and the last equality sign holds due to the well- known transformation formula between the Itô and the Stratonovich integral ([7], p. 296). We will assume throughout that the following conditions on the coefficients b : E → R and σ : E → R are satisfied. • b(0) = σ(0) = 0 • b satisfies a Lipschitz condition on each compact subset of E and grows at most linearly • σ ∈ C2 with bounded first and second derivatives • σ(x)>0 for all x > 0. In order to arrive at an RDS, we choose specifically Ω = C(R, R) – the space of continuous functions from R to R, F the σ-field on Ω generated by the evaluations (or the topology of uniform convergence on compact subsets), P two-sided Wiener measure on (Ω, F) and define ϑ(t) : Ω → Ω by (ϑ(t)(ω)) (s) := ω(t + s) − ω(t), s, t ∈ R. Then ϑ ≡{Ω, F, P,ϑ(t), t ∈ R} is a metric dynamical system and W(t,ω)= ω(t), t ∈ R is a two-sided Brownian motion. It is shown in [2] and [1] that one can find a cocycle φ of homeomorphisms of E on the MDS ϑ such that for each x ∈ E, φ(t,ω,x), t м 0 solves (1) with initial condition X(0) = x. More generally, for every s ∈ R, φ(t − s,ϑ(s)(ω), x), t м s solves (1) with initial condition X(s) = x. Our assumptions on b and σ above imply (after changing φ on a set of measure 0 if necessary) that φ(t,ω,0) = 0 for all t,ω. We can extend φ to a map φ : R × Ω × E → E by defining (3) φ(−t,ω,x) := φ−1(t,ϑ(−t)(ω), .)(x), t м 0. Then φ satisfies the second part of Definition 1.1 even for all s, t ∈ R,i.e.φ is a cocycle of homeomorphisms with index set R. Note that this implies that φ is order preserving i.e. x > y м 0 implies φ(t,ω,x)>φ(t,ω,y) for all t ∈ R,ω∈ Ω. We will be interested in explicit necessary and/or sufficient conditions on the functions b and σ which guarantee that A (ω) ={0} is a (weak, pullback or forward) attractor of (ϑ, φ). Equation (1) does not only generate an RDS, it also generates a diffusion process on E. The probability law of a diffusion on E is uniquely characterized by its scale function p :[0, ∞] → R ∪{±∞}and its speed measure m(dx) on (0, ∞) defined as   x v 1 2b(u) (4) p(x) = exp − du dv σ(v) σ 2(u) 1  1  x 1 2b(u) (5) m(dx) = exp  du dx. σ(x) σ 2(u) 1 236 M. SCHEUTZOW ARCH. MATH.

The relevance of the function p is, that p(X(t)), t м 0 becomes a continuous local martingale. The ﬁniteness or not of p(0), p(∞), m(0, 1] and m[1, ∞) is closely linked with the behavior of the diffusion at or close to the boundary {0, ∞} of E, see e.g. [6] and [7]. We recall that p(0)>−∞ implies m(0, 1]=∞and p(∞)<∞ implies m[1, ∞) =∞due to the conditions on b and σ above, which exclude the possibility that φ(t,ω,x) hits the boundary of E in ﬁnite time.

Proposition 2.1. For each x ∈ E \{0} the cocycle φ deﬁned above has the following properties:

(i) lim φ(t,ω,x) = 0 almost surely iff p(0)>−∞ and p(∞) =∞. t→∞ (ii) If lim φ(t,ω,x) = 0 in probability, then p(∞) =∞and either t→∞ (α) p(0)>−∞ or (β) p(0) =−∞and m(0, 1]=∞. (iii) If p(∞) =∞and either (α) p(0)>−∞ or (β) p(0) =−∞and m(0, 1]=∞and m[1, ∞)<∞, then lim φ(t,ω,x) = 0 in probability. t→∞ (iv) lim φ(t,ω,x) =∞almost surely iff p(0) =−∞and p(∞)<∞. t→∞

In particular each property above holds for all x ∈ E \{0} if it holds for one such x.

P r o o f. (i) and (iv) follow immediately from [7], Proposition 5.5.22. To show (ii) assume that φ(t,ω,x) → 0 in probability. Again from [7], Proposition 5.5.22 it follows that p(∞) =∞. Assume that m(0, 1] < ∞. Denoting by λ Lebesgue measure on R, the ergodic theorem for additive functionals ([6], p. 228) implies for each x > 0 λ{ Ϲ : φ( ,ω, ) Ϲ } s T s x 1 → γ λ{s Ϲ T : φ(s,ω,x)>1} almost surely as T →∞for some (deterministic) γ ∈[0,∞) and therefore lim inf T −1λ{s Ϲ T : T→∞ φ(s,ω,x)>1} > 0 almost surely. Applying Fubini’s theorem we arrive at a contradiction to our assumption that φ(t,ω,x) → 0 in probability. Therefore m(0, 1]=∞. This proves part (ii). To show (iii) it remains to prove that the three properties p(0) =−∞, m(0, 1]=∞and m[1, ∞)<∞ together imply that φ(t,ω,x) → 0 in probability. Even though this looks pretty obvious, we did not find this result in the literature. One way to see it is as follows: fix x > 0 and >0 and choose δ>0 such that m(δ, ] > m[, ∞). Now choose a function b˜ м b on E which coincides with b on [δ, ∞) such that b˜ satisfies the general assumptions on b and such that the speed measure m˜ associated with b˜ and σ satisfies m˜ (0, 1] < ∞ (it is easy to see that such a b˜ exists). Let φ˜ be the cocycle associated with b˜ and σ (and the same Brownian motion W). Note that m˜ coincides with m on [δ, ∞). Now Theorem IV.4.7. in [9] on the asymptotics of the transition probabilities for diffusions with a finite speed measure shows that m˜ [, ∞) m˜ [, ∞) m[, ∞) lim sup P{φ(˜ t,ω,x)>} Ϲ Ϲ = Ϲ . t→∞ m˜ (0, ∞) m˜ [δ, ∞) m[δ, ∞) Vol. 78, 2002 Random attractors 237

Now the comparison theorem for one-dimensional diffusions ([5], Theorem VI.1.1) implies that φ(˜ t,ω,x) м φ(t,ω,x) almost surely. Therefore lim sup P{φ(t,ω,x)>} Ϲ lim sup P{φ(˜ t,ω,x)>} Ϲ . t→∞ t→∞ Since >0 was arbitrary, (iii) follows.

Remark 2.2.Thepreviousproposition does not make any claims as to whether φ(t,ω,x) converges to 0 in probability or not in case p(∞) =∞, p(0) =−∞, m(0, 1]=∞ and m[1, ∞) =∞. We will address this question in Example 2.8.

Corollary 2.3. (i) A (ω) ≡{0} is a forward attractor of (ϑ, φ) iff p(0)>−∞ and p(∞) =∞. (ii) If A (ω) ≡{0} is a weak attractor of (ϑ, φ),thenp(∞) =∞and either (α) p(0)>−∞ (then {0} is even a forward attractor) or (β) p(0) =−∞and m(0, 1]=∞. (iii) If p(∞) =∞and either (α) p(0)>−∞ or (β) p(0) =−∞and m(0, 1]=∞and m[1, ∞)<∞, then A (ω) ≡{0} is a weak attractor of (ϑ, φ).

In order to ﬁnd conditions on b and σ which ensure that A (ω) ≡{0} is a pullback attractor, it is helpful to study the asymptotics of φ(t,ω,x) for t →−∞.

Proposition 2.4. A (ω) ≡{0} is a pullback attractor of (ϑ, φ) if and only if for some (and hence for each) x > 0, lim φ(t,ω,x) =∞almost surely. t→−∞

Proof.A (ω) ≡{0} is a pullback attractor of (ϑ, φ) if and only if for every y ∈ E we have lim φ(t,ϑ(−t)(ω), y) = 0 almost surely. Using the fact that φ is order preserving and equality t→−∞ (3), the assertion follows.

It remains to show how the property lim φ(t,ω,x) =∞can be veriﬁed in terms of b and t→−∞ σ. The following lemma is the key to the answer. Lemma 2.5. Let (ϑ, φ) be as above and deﬁne ϑ(¯ t)(ω) := ϑ(−t)(ω), W(t,ω):= W(−t,ω)= ω(−t) and φ(¯ t,ω,x) := φ(−t,ω,x), t ∈ R, x ∈ E.Then(ϑ,¯ φ)¯ is an RDS and φ(¯ t,ω,x) solves dX¯ (t) =−b(X¯ (t))dt + σ(X¯ (t)) ◦ dW(t) X¯ (0) = x.

P r o o f. It is straightforward to check that (ϑ,¯ φ)¯ is an RDS (of homeomorphisms of E). That φ¯ satisﬁes the SDE follows e.g. from [8], p. 117 or p. 175, see also [2]. Kunita requires that σ ∈ C2+δ for some δ>0, but it is easy to see that in our special case the “+δ”is unnecessary.

Corollary 2.6. A (ω) ≡{0} is a pullback attractor of (ϑ, φ) if and only if m(0, 1]=∞and m[1, ∞)<∞. 238 M. SCHEUTZOW ARCH. MATH.

P r o o f. Proposition 2.1(iv), Proposition 2.4 and Lemma 2.5 imply that {0} is a pullback attractor if and only if the scale function p¯ of the diffusion X¯ (t) satisﬁes p¯(0) =−∞and p¯(∞)<∞.Since   x v 1 2b(u) p¯(x) = exp  du dv, σ(v) σ 2(u) 1 1 the conditions p¯(0) =∞and p¯(∞)<∞ are equivalent to m(0, 1]=∞and m[1, ∞)<∞, so the assertion follows.

The table below combines Corollaries 2.3 and 2.6. For each of the nine possible combina- tions of ﬁnite or inﬁnite values of p(0), p(∞), m(0, 1] and m[1, ∞) the table tells us if {0} is a pullback attractor (p), a forward attractor (f), or a weak attractor (w). The ± indicates that both cases can occur (as far as the “+” is concerned, we will provide Example 2.8(ii) without giving a rigorous proof that it has {0} as a weak attractor).

p(0) p(∞) m(0, 1] m[1, ∞) pfwExample 1.>−∞ < ∞∞ ∞−−− 2.>−∞ ∞ ∞ < ∞ +++2.7(i) 3.>−∞ ∞ ∞ ∞ −++2.7(iii) 4. −∞ < ∞ < ∞∞−−−2.7(iv) 5. −∞ < ∞∞ ∞−−− 6. −∞ ∞ < ∞ < ∞ −−− 7. −∞ ∞ < ∞∞−−− 8. −∞ ∞ ∞ < ∞ +−+2.7(ii) 9. −∞ ∞ ∞ ∞ −−±2.8

E x a m p l e s 2.7. Using Corollaries 2.3 and 2.6 it is straightforward to check the following statements. Whenever we do not specify b on the interval [1, 2], it can be chosen in an arbitrary way subject to our general assumptions at the beginning of this section. In all examples we let σ(x) = x on E.

(i) If b(x) =−x on E,then{0} is both a pullback and a forward attractor. (ii) If b(x) = 0 for x Ϲ 1 and b(x) =−x for x м 2,then{0} is a pullback but not a forward attractor. (iii) If b(x) =−x for x Ϲ 1 and b(x) = 0 for x м 2,then{0} is a forward but not a pullback attractor. (iv) If b(x) = x on E,then{0} is not a weak attractor (and hence neither a pullback nor a forward attractor).

The Examples 2.7 do not provide an answer to the question, if it is possible that {0} is a weak attractor but neither a pullback nor a forward attractor. It is clear from the table above, that if such an example exists, it must be true that p(0) =−∞and p(∞) = m(0, 1]=m[1, ∞) =∞. The following Examples 2.8(i) and (ii) deal with this case. Vol. 78, 2002 Random attractors 239

Examples2.8. (i) If b ≡ 0 and σ(x) = x on E,then p(0) =−∞and p(∞) = m(0, 1]=m[1, ∞) =∞. In this case we have the explicit formula φ(t,ω,x) = x exp{W(t)}.Since–forx > 0 – {φ( ,ω, ) Ϲ }= { ( ) Ϲ }= 1 { } P t x x P W t 0 2 , 0 is not a weak attractor. 2 3 (ii) If σ(x) = x and b(x) =−1 σ(x)σ (x) + 1 x ,thenp(x) = 2lnx and p(0) =−∞and x+1 2 2 (x+1)2 p(∞) = m(0, 1]=m[1, ∞) =∞. We would like to show that {0} is a weak attractor. This is equivalent to showing that for each x > 0, Z(t) := ln φ(t,ω,x) converges to −∞ in probability. From Itô’s formula we see that Z satisfies the SDE 1 dZ(t) = dW(t). 1 + e−Z(t) Being a continuous local martingale, Z is a time-changed Brownian motion [7], p. 173ff. From this fact and the SDE above, it is clear that Z can be obtained from 1 a Brownian motion by speeding it up by the factor 1+e−z when the current value is z (for a rigorous construction of Z via an additive functional of Brownian motion, see e.g. [6], p. 167ff). Since the speed-up factor converges to 0 as z →−∞, the process is very likely to spend an overwhelming proportion of time up to time T on the nega- tive half-axis (in fact even below −M for any fixed M)whenT is large. Therefore it certainly sounds plausible that for any given (positive) M, P{Z(t) Ϲ −M} converges to1ast converges to ∞, which is what we wanted to prove. We will not try to give a rigorous proof (which most likely would be too long to include it in this case study anyway). C o n c l u d i n g R e m a r k s 2.9. By definition, {0} is a forward attractor if all trajectories converge to 0 almost surely forward in time. If {0} is only a weak attractor, then this convergence takes place only in probability and therefore allows occasional excursions of the trajectories arbitrarily far away from 0, but the probability that such an excursion takes place at a given time t converges to 0 as t converges to ∞ (see Example 2.7(ii)). Corollary 2.6 shows – roughly speaking – that {0} is a pullback attractor if and only if the trajectories cannot converge to ∞ and the diffusion is slow close to zero (since m(0, 1]=∞) and quick (since m[1, ∞)<∞) away from zero. Cases in which {0} is a weak attractor, but not a pullback attractor, can only arise in two different ways cf. lines 3 and 9 in the table. In the first case trajectories spend a period with infinite expected duration outside any neighborhood of the attractor before they finally converge to the attractor. In the second case, all trajectories have infinitely many excursions with infinite expected duration both above and below each positive level. For example: every trajectory will hit level 1 for arbitrarily large times and the time after this until level 2 is reached is finite but has infinite expected value and vice versa. If – as in Example 2.8(ii) – the excursions above level 1 (say) are shorter than those below level 1 (in a suitable sense – after all both have an infinite expected value), then {0} is a weak attractor.

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Anschrift des Autors: Michael Scheutzow Fachbereich Mathematik, MA 7-5 Technische Universität Berlin Strasse des 17. Juni 136 10623 Berlin Germany