Arch. Math. 78 (2002) 233–240
0003-889X/02/030233-08 $ 3.10/0
 Birkhäuser Verlag, Basel, 2002
Sonderdruck aus
Comparison of various concepts of a random attractor:
A case study
By
M ICHAEL S CHEUTZOW
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. S CHEUTZOW
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, ·) = id E ,
φ(t + s, ω, ·) = φ(t, ϑ(s)ω, φ(s, ω, ·))
for all t, s ⭌ 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 ⭌ 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).
1. A is called a forward attractor if for each bounded set B Ⲵ E
lim sup d(φ(t, ω, x), A (ϑ(t)ω)) = 0 a.s.
t→∞ x∈B
2. A is called a pullback attractor if for each bounded set B Ⲵ E
lim sup d(φ(t, ϑ(−t)ω, x), A (ω)) = 0 a.s.
t→∞ x∈B
3. A is called a weak attractor if for each bounded set B Ⲵ E
lim sup d(φ(t, ω, x), A (ϑ(t)ω)) = 0 in probability .
t→∞ x∈B
R e m a r k 1.4. Obviously all three concepts coincide in the deterministic case (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.
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Random attractors
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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)
(2)
dX(t) = b(X(t))dt + σ(X(t)) ◦ dW(t)
= b(X(t))dt + 12 σ(X(t))σ (X(t)) + σ(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 wellknown 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 ∈ C 2 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 ⭌ 0 solves (1)
with initial condition X(0) = x . More generally, for every s ∈ R, φ(t − s, ϑ(s)(ω), x), t ⭌ 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 ⭌ 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 ⭌ 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
 v

x
1
2b(u)
exp −
du  dv
(4)
p(x) =
σ(v)
σ 2 (u)
1
1
 x

1
2b(u)
exp 
du  dx.
(5)
m(dx) =
σ(x)
σ 2 (u)
1
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M. S CHEUTZOW
ARCH . MATH .
The relevance of the function p is, that p(X(t)), t ⭌ 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{s ⬉ T : φ(s, ω, x) ⬉ 1}
→γ
λ{s ⬉ T : φ(s, ω, x) > 1}
almost surely as T → ∞ for some (deterministic) γ ∈ [0, ∞) and therefore lim inf T −1 λ{s ⬉ 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 b̃ ⭌ b on
E which coincides with b on [δ, ∞) such that b̃ satisfies the general assumptions on b and
such that the speed measure m̃ associated with b̃ and σ satisfies m̃(0, 1] < ∞ (it is easy to see
that such a b̃ exists). Let φ̃ be the cocycle associated with b̃ and σ (and the same Brownian
motion W ). Note that m̃ 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[, ∞)
m̃[, ∞)
m̃[, ∞)
=
lim sup P{φ̃(t, ω, x) > } ⬉
⬉
⬉ .
m[δ, ∞)
m̃(0, ∞)
m̃[δ, ∞)
t→∞
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Random attractors
237
Now the comparison theorem for one-dimensional diffusions ([5], Theorem VI.1.1) implies
that φ̃(t, ω, x) ⭌ φ(t, ω, x) almost surely. Therefore
lim sup P{φ(t, ω, x) > } ⬉ lim sup P{φ̃(t, ω, x) > } ⬉ .
t→∞
t→∞
Since > 0 was arbitrary, (iii) follows.
R e m a r k 2.2. The previous proposition 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), then p(∞) = ∞ 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→−∞
P r o o f. 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 ϑ̄(t)(ω) := ϑ(−t)(ω), W(t, ω) := W(−t, ω) =
ω(−t) and φ̄(t, ω, x) := φ(−t, ω, x), t ∈ R, x ∈ E . Then (ϑ̄, φ̄) is an RDS and φ̄(t, ω, x) solves
d X̄(t) = −b( X̄(t))dt + σ( X̄(t)) ◦ dW(t)
X̄(0) = x.
P r o o f. It is straightforward to check that (ϑ̄, φ̄) is an RDS (of homeomorphisms of E ).
That φ̄ satisfies the SDE follows e.g. from [8], p. 117 or p. 175, see also [2]. Kunita requires
that σ ∈ C 2+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, ∞) < ∞.
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M. S CHEUTZOW
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 p̄ of the diffusion X̄(t) satisfies p̄(0) = −∞ and
p̄(∞) < ∞. Since
 v

x
1
2b(u)
exp 
du  dv,
p̄(x) =
σ(v)
σ 2 (u)
1
1
the conditions p̄(0) = ∞ and p̄(∞) < ∞ 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 combinations 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, ∞) p f w Example
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 ⬉ 1 and b(x) = −x for x ⭌ 2, then {0} is a pullback but not a forward
attractor.
(iii) If b(x) = −x for x ⬉ 1 and b(x) = 0 for x ⭌ 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
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E x a m p l e s 2.8.
(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 – for x > 0 –
P{φ(t, ω, x) ⬉ x} = P{W(t) ⬉ 0} = 12 , {0} is not a weak attractor.
x2
x3
and b(x) = − 12 σ(x)σ (x) + 12 (x+1)
(ii) If σ(x) = x+1
2 , then p(x) = 2 ln x and p(0) = −∞ and
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
a Brownian motion by speeding it up by the factor 1+e1−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 negative half-axis (in fact even below −M for any fixed M ) when T is large. Therefore it
certainly sounds plausible that for any given (positive) M , P{Z(t) ⬉ −M} converges
to 1 as t 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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Eingegangen am 8. 5. 2000
Anschrift des Autors:
Michael Scheutzow
Fachbereich Mathematik, MA 7-5
Technische Universität Berlin
Strasse des 17. Juni 136
10623 Berlin
Germany