Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 409282, 27 pages
doi:10.1155/2012/409282
Research Article
Random Attractors for Stochastic Retarded Lattice
Dynamical Systems
Xiaoquan Ding1 and Jifa Jiang2
1
School of Mathematics and Statistics, Henan University of Science and Technology, Henan,
Luoyang 471023, China
2
School of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China
Correspondence should be addressed to Xiaoquan Ding, xqding@mail.haust.edu.cn
Received 27 August 2012; Accepted 16 September 2012
Academic Editor: Jinhu Lü
Copyright q 2012 X. Ding and J. Jiang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise.
We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove
the existence of a compact global random attractor within the set of tempered random bounded
sets.
1. Introduction
Lattice dynamical systems LDSs arise naturally in a wide variety of applications in science
and engineering where the spatial structure has a discrete character. Among such examples
are brain science 1, chemical reaction 2, material science 3, electrical engineering 4,
laser systems 5, pattern recognition 6, complex network 7, and many others. On the
other hand, LDSs also appear as spatial discretizations of partial differential equations on
unbounded domains.
There are many works concerning deterministic LDSs. For example, the traveling
wave solutions were studied in 8, 9, the chaotic properties of solutions were examined by
6, 10, the long-time behavior of LDSs was investigated by 11–17. In particular, Bates et
al. 11 established the first result on the existence of a global attractor for LDSs. Wang 13,
Zhou and Shi 14 used the idea of tail estimates on solutions and obtained, respectively, some
sufficient and necessary conditions for the existence of a global attractor for autonomous
LDSs. Later, the method of tail estimates is extended to nonautonomous LDSs 15–17.
It is noted that an evolutionary system in reality is usually affected by external
perturbations which in many cases are of great uncertainty or random influence. These
2
Abstract and Applied Analysis
random effects are not only introduced to compensate for the defects in some deterministic
models, but also are often rather intrinsic phenomena. Therefore, it is of prime importance
to take into account these random effects in some models, and this has led to stochastic
differential equations. Random attractors for stochastic partial differential equations were
first introduced by Crauel and Flandoli 18, Flandoli andSchmalfuss 19, with notable
developments given in 20–25 and others. Bates et al. 26 initiated the study of random
attractors for stochastic LDSs. Since then, many works have been done for the existence of
random attractors for stochastic LDSs, see, for example, 27–34 and the references therein.
Similarly to deterministic LDSs, the method of tail estimates also plays a key role in the study
of the existence of random attractors for stochastic LDSs.
On the other hand, in the natural world, the current rate of change of the state in
an evolutionary system always depends on the historical status of the system. Then, it is
more reasonable to describe the evolutionary systems by functional differential equations.
Many papers are devoted to the study of the asymptotic behavior of deterministic functional
differential equations, see, for example, 35–41 and the references therein. Especially, Zhao
and Zhou 40, 41 considered the asymptotic behavior of some deterministic retarded LDSs
and extended the method of tail estimates to deterministic retarded LDSs. More recently, Yan
et al. 42, 43 discussed the asymptotic behavior of some stochastic retarded LDSs with global
Lipschitz nonlinearities.
Consider the Hilbert space
2
u ui i∈Z : ui ∈ R,
2
|ui | < ∞ ,
1.1
u2i ,
1.2
i∈Z
whose inner product and norm are given by
u, v ui vi ,
i∈Z
u2 i∈Z
for all u ui i∈Z , v vi i∈Z ∈ 2 . For ν > 0, let C : C−ν, 0; 2 denote the Banach
space of all continuous functions ξ : −ν, 0 → 2 endowed with the supremum norm ξC sups∈−ν,0 ξs. For any real numbers a ≤ b, t ∈ a, b and any continuous function u :
a − ν, b → 2 , ut denotes the element of C given by ut s ut s for s ∈ −ν, 0.
In this paper, we investigate the long time behavior of the following stochastic retarded
LDS:
dui t ui−1 − 2ui ui1 − λi ui fi uti gi dt ai dwi t,
t > 0, i ∈ Z,
1.3
with initial data
ui t u0i t,
t ∈ −ν, 0, i ∈ Z,
1.4
where u ui i∈Z ∈ 2 , λi i∈Z is a bounded positive constant sequence, f fi i∈Z : C → 2
is a nonlinear mapping satisfying local Lipschitz condition, g gi i∈Z ∈ 2 , a ai i∈Z ∈ 2 ,
and {wi : i ∈ Z} are independent two-sided real-valued Wiener processes on a probability
space which will be specified later.
Abstract and Applied Analysis
3
It is worth mentioning that in the absence of the white noise, the existence of a global
attractor for 1.3-1.4 was established in 40. The main contribution of this paper is to
extend the method of tail estimates to stochastic retarded LDSs and prove the existence
of a random attractor for the infinite dimensional random dynamical system generated by
stochastic retarded LDS 1.3-1.4. It is clear that our method can be used for a variety of
other stochastic retarded LDSs, as it was for the nonretarded case.
The paper is organized as follows. In the next section, we recall some fundamental
results on the existence of a pullback random attractor for random dynamical systems. In
Section 3, we establish a necessary and sufficient condition for the relative compactness of
sequences in C−ν, 0; 2 . In Section 4, we define a continuous random dynamical system
for stochastic retarded LDS 1.3-1.4. The existence of the random attractor for 1.3-1.4 is
given in Section 5.
2. Preliminaries
In this section, we recall some basic concepts related to random attractors for random
dynamical systems. The reader is referred to 18–21, 26, 44, 45 for more details.
Let X, · X be a separable Banach space with Borel σ-algebra BX and Ω, F, P be
a probability space.
Definition 2.1. Ω, F, P, ϑt t∈R is called a metric dynamical system if ϑ : R × Ω → Ω is
BR ⊗ F, F-measurable, ϑ0 is the identity on Ω, ϑst ϑt ◦ ϑs for all s, t ∈ R, and ϑt P P
for all t ∈ R.
Definition 2.2. A set A ⊂ Ω is called invariant with respect to ϑt t∈R , if for all t ∈ R, it holds
ϑt−1 A A.
2.1
Definition 2.3. A continuous random dynamical system on X over a metric dynamical system
Ω, F, P, ϑt t∈R is a mapping
ϕ : R × Ω × X −→ X,
t, ω, x −→ ϕt, ω, x,
2.2
which is BR ⊗ F ⊗ BX, BX-measurable, and for all ω ∈ Ω,
i ϕt, ω, · : X → X is continuous for all t ∈ R ;
ii ϕ0, ω, · is the identity on X;
iii ϕt s, ω, · ϕt, ϑs ω, · ◦ ϕs, ω, · for all s, t ∈ R .
Definition 2.4. A random set D is a multivalued mapping D : Ω → 2X \ ∅ such that for every
x ∈ X, the mapping ω → dx, Dω is measurable, where dx, B is the distance between
the element x and the set B ⊂ X. It is said that the random set is bounded resp., closed or
compact if Dω is bounded resp., closed or compact for P-a.e. ω ∈ Ω.
4
Abstract and Applied Analysis
Definition 2.5. A random variable r : Ω → 0, ∞ is called tempered with respect to ϑt t∈R , if
for P-a.e. ω ∈ Ω
lim e−βt rϑ−t ω 0
t→∞
∀β > 0.
2.3
A random set D is called tempered if Dω is contained in a ball with center zero and
tempered radius rω for all ω ∈ Ω.
Remark 2.6. If r > 0 is tempered, then for any τ ∈ R, β > 0 and P-a.e. ω ∈ Ω
lim e−βt rϑ−tτ ω e−βτ · lim e−βt−τ rϑ−tτ ω 0.
t→∞
t→∞
2.4
Therefore, for any τ ∈ R, rϑτ · is also tempered. Moreover, if for P-a.e. ω ∈ Ω, rϑt ω is
continuous in t, then for any ν > 0, supσ∈−ν,0 rϑσ · is measurable and for all β > 0 and P-a.e.
ω∈Ω
lim e−βt sup rϑ−tσ ω ≤ lim eβ/2ν−t · sup
t→∞
t→∞
σ∈−ν,0
eβ/2s rϑs ω 0.
s∈−∞,0
2.5
Hence, for any ν > 0, supσ∈−ν,0 rϑσ · is also tempered.
Remark 2.7. If r > 0 is tempered, then for any α > 0 and P-a.e. ω ∈ Ω
Rω 0
−∞
eαs rϑs ωds < ∞.
2.6
Moreover, R is tempered, and if for P-a.e. ω ∈ Ω, rϑt ω is continuous in t, then Rϑt ω is
also continuous in t for such ω.
Hereafter, we always assume that ϕ is a continuous random dynamical system over
Ω, F, P, ϑt t∈R , and D is a collection of random subsets of X.
Definition 2.8. A random set K is called a random absorbing set in D if for every B ∈ D and
P-a.e. ω ∈ Ω, there exists tB ω > 0 such that
ϕt, ϑ−t ω, Bϑ−t ω ⊆ Kω
∀t ≥ tB ω.
2.7
Definition 2.9. A random set A is called a D-random attractor D-pullback attractor for ϕ if
the following hold:
i A is a random compact set;
ii A is strictly invariant, that is, for P-a.e. ω ∈ Ω and all t ≥ 0,
ϕt, ω, Aω Aϑt ω;
2.8
Abstract and Applied Analysis
5
iii A attracts all sets in D, that is, for all B ∈ D and P-a.e. ω ∈ Ω,
lim d ϕt, ϑ−t ω, Bϑ−t ω, Aω 0,
t→∞
2.9
where d is the Hausdorff semimetric given by dE, F supx∈E infy∈F x − yX for
any E ⊆ X and F ⊆ X.
Definition 2.10. ϕ is said to be D-pullback asymptotically compact in X if for all B ∈ D and
P-a.e. ω ∈ Ω, {ϕtn , ϑ−tn ω, xn }∞
n1 has a convergent subsequence in X whenever tn → ∞, and
xn ∈ Bϑ−tn ω.
The following existence result on a random attractor for a continuous random
dynamical system can be found in 19, 26. First, recall that a collection D of random subsets
of X is called inclusion closed if whenever E is an arbitrary random set, and F is in D with
Eω ⊂ Fω for all ω ∈ Ω, then E must belong to D.
Proposition 2.11. Let D be an inclusion-closed collection of random subsets of X and ϕ a continuous
random dynamical system on X over Ω, F, P, ϑt t∈R . Suppose that K ∈ D is a closed random
absorbing set for ϕ in D and ϕ is D-pullback asymptotically compact in X. Then ϕ has a unique
D-random attractor A which is given by
Aω ϕt, ϑ−t ω, Kϑ−t .
τ≥0 t≥τ
2.10
In this paper, we will take D as the collection of all tempered random subsets of C and
prove the stochastic retarded LDS has a D-random attractor.
3. Compactness Criterion in C−ν, 0; 2 In this section, we provide a necessary and sufficient condition for the relative compactness
of sequences in C−ν, 0; 2 , which will be used to establish the asymptotic compactness of
the retarded LDS.
Lemma 3.1. Let u ∈ C−ν, 0; 2 . Then for every ε > 0, there exists Nε > 0 such that for all
k ≥ Nε,
sup
|ui s|2 < ε.
s∈−ν,0|i|≥k
3.1
Proof. For every ε > 0, by virtue of the uniform continuity of u, there exist −ν s0 < s1 < s2 <
· · · < sp 0 such that
us − u sj <
√
ε
,
2
for s ∈ sj−1 , sj , j 1, 2, . . . , p.
3.2
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Abstract and Applied Analysis
Since for each sj , usj ∈ 2 , there exists Nj ε > 0 such that for all k ≥ Nj ε,
2 ε
ui sj < .
4
|i|≥k
3.3
Take Nε max1≤j≤p Nj ε. Then for each s ∈ −ν, 0, there exists j ∈ {1, 2, . . . , p} such that
s ∈ sj−1 , sj . Therefore, we get from 3.2 and 3.3 that for all k ≥ Nε,
|ui s|2 ≤ 2
|i|≥k
2
ui sj 2 ui s − ui sj 2
|i|≥k
≤2
|i|≥k
2
ui sj 2us − usj 2 < ε,
3.4
|i|≥k
which completes the proof.
Theorem 3.2. Let S ⊂ C−ν, 0; 2 . Then S is relative compact in C−ν, 0; 2 if and only if the
following conditions are satisfied:
i S is bounded in C−ν, 0; 2 ;
ii S is equicontinuous;
iii limk → ∞ supuui i∈Z ∈S sups∈−ν,0
|i|≥k
|ui s|2 0.
Proof. The proof is divided into two steps. We first show the necessity of the conditions and
then prove the sufficiency.
1 Assume that S is relative compact in C−ν, 0; 2 . Then we want to show
conditions i, ii, and iii hold. Clearly, in this case, by the Ascoli-Arzelà theorem, S must
be bounded and equicontinuous. So we only need to prove condition iii.
Given ε > 0, since S is relative compact, there exists a finite subset E of S such that the
balls of radii ε/2 centered at E form a finite covering of S, that is, for each u ∈ S, there exists
v ∈ E such that
sup us − vs <
s∈−ν,0
ε
.
2
3.5
By Lemma 3.1, there exists K ∗ ε > 0 such that for all v ∈ E,
sup
|vi s|2 <
s∈−ν,0 |i|≥K ∗ ε
ε2
.
4
3.6
By 3.5 and 3.6, we find that for each u ∈ S, there exists v ∈ E such that
sup
|ui s|2 ≤ 2 sup
s∈−ν,0 |i|≥K ∗ ε
|ui s − vi s|2
s∈−ν,0 |i|≥K ∗ ε
2 sup
|vi s|2 < ε2 .
s∈−ν,0 |i|≥K ∗ ε
3.7
Abstract and Applied Analysis
7
Therefore, for all k ≥ K ∗ ε, we have
sup
sup
uui i∈Z ∈S s∈−ν,0 |i|≥k
|ui s|2 ≤ ε2 ,
3.8
which implies condition iii.
2 Assume that conditions i, ii, and iii are valid. We want to prove that S is
relative compact in C−ν, 0; 2 . That is, given ε > 0, we want to show that S has a finite
covering of balls of radii ε. By condition iii, we find that there exists Kε > 0 such that for
all u ui i∈Z ∈ S,
sup
|ui s|2 <
s∈−ν,0 |i|≥Kε
ε2
.
4
3.9
Consider the set S|K {u|K ui |i|≤Kε : u ui i∈Z ∈ S} in C−ν, 0; R2Kε1 .
By conditions i and ii, we know that S|K is bounded and equicontinuous in
C−ν, 0; R2Kε1 . Then, by the Ascoli-Arzelà theorem, we obtain that S|K is relative
compact in C−ν, 0; R2Kε1 and hence there exists a finite subset H of S|K such that the
balls of radii ε/2 centered at H form a finite covering of S|K , that is, for each u|K ∈ S|K , there
exists v|K ∈ H such that
sup
|ui s − vi s|2 <
s∈−ν,0 |i|≤Kε
ε2
.
4
3.10
Now for each v|K vi |i|≤Kε ∈ H, we choose v vi i∈Z such that vi vi for |i| ≤ Kε
and vi 0 for |i| > Kε. Then by 3.9 and 3.10, we find that for each u ∈ S, there exists
v ∈ H {
v : v|K ∈ H} such that
sup us − v s2 ≤ sup
s∈−ν,0
|ui s − vi s|2 sup
s∈−ν,0 |i|≤Kε
|ui s|2 < ε2 ,
s∈−ν,0 |i|>Kε
3.11
which implies that the set S has a finite covering of balls with radii ε. The proof is complete.
The next result is a variant of Theorem 3.2 which shows that condition iii in
Theorem 3.2 has an equivalent form which is easier to verify for asymptotic compactness
of dynamical systems associated with retarded LDSs.
∞
n
2
n ∞
Theorem 3.3. Let {un }∞
n1 {ui i∈Z }n1 ⊂ C−ν, 0; . Then {u }n1 is relative compact in
2
C−ν, 0; if and only if the following conditions are satisfied:
2
i {un }∞
n1 is bounded in C−ν, 0; ;
ii {un }∞
n1 is equicontinuous;
iii limk → ∞ lim supn → ∞ sups∈−ν,0
|i|≥k
|uni s|2 0.
8
Abstract and Applied Analysis
2
Proof. If {un }∞
n1 is relative compact in C−ν, 0; , then it follows from Theorem 3.2 that the
above conditions i, ii, and iii are satisfied. So, to complete the proof, we only need to
show that the above conditions i, ii, and iii imply the conditions in Theorem 3.2. Given
ε > 0, it follows from condition iii that there exists K1 ε > 0 such that
2
un s2 < ε ,
i
2
s∈−ν,0 |i|≥K ε
lim sup sup
n→∞
3.12
1
which implies that there exists Nε > 0 such that
un s2 < ε2 ,
i
sup
∀n > Nε.
s∈−ν,0 |i|≥K1 ε
3.13
By Lemma 3.1, we find that there exists K2 ε > 0 such that
sup
un s2 < ε2 ,
i
∀1 ≤ n ≤ Nε.
s∈−ν,0 |i|≥K2 ε
3.14
Take Kε max{K1 ε, K2 ε}. It follows from 3.13 and 3.14 that
sup
un s2 < ε2 ,
i
∀n ≥ 1,
3.15
∀k ≥ Kε.
3.16
s∈−ν,0 |i|≥Kε
which implies that
sup
sup
un s2 ≤ ε2 ,
i
{un }∞
n1 s∈−ν,0 |i|≥k
Therefore,
lim sup sup
k → ∞ n∈N s∈−ν,0
un s2 0,
i
|i|≥k
3.17
which together with conditions i and ii shows that the conditions in Theorem 3.2 are
satisfied with S {un }∞
n1 . The proof is complete.
4. Stochastic Retarded Lattice Differential Equations
In this section, we show that there is a continuous random dynamical system generated by
stochastic retarded LDS 1.3-1.4.
Abstract and Applied Analysis
9
For convenience, we now formulate 1.3-1.4 as a stochastic functional differential
equation in 2 . Define the linear operators A, B, B∗ , λ from 2 to 2 as follows. For u ui i∈Z ∈
2,
Aui −ui−1 2ui − ui1 ,
Bui ui1 − ui ,
λui λi ui ,
B∗ ui ui−1 − ui ,
4.1
for each i ∈ Z. Then A BB∗ B∗ B and B∗ u, v u, Bv for all u, v ∈ 2 . Therefore,
Au, u ≥ 0 for all u ∈ 2 . Let ei ∈ 2 denote the element having 1 at position i and all the
other components 0. Then
wt ai wi tei
with ai i∈Z ∈ 2 ,
i∈Z
4.2
is an 2 -valued two-sided Wiener process with a symmetric nonnegative finite trace
covariance operator Q such that Qei ai ei . For ξ ∈ C, let fξ fi ξi i∈Z . Then stochastic
retarded LDS 1.3-1.4 can be rewritten as a stochastic functional equation in 2
du −A λu f ut g dt dw,
t > 0,
4.3
with the initial data
ut u0 t,
t ∈ −ν, 0.
4.4
In the sequel, we consider the probability space Ω, F, P where
Ω ω ∈ C R, 2 : ω0 0 ,
4.5
F is the Borel σ-algebra induced by the compact-open topology of Ω, and P the corresponding
Wiener measure on Ω, F with respect to the covariance operator Q. Let
ϑt ω· ω· t − ωt,
t ∈ R.
4.6
Then Ω, F, P, ϑt t∈R is an ergodic metric dynamical system. Since the above probability
space is canonical, we have
wt, ω ωt,
wt, ϑs ω wt s, ω − ws, ω.
4.7
∈ F of full P-measure such
By Proposition A.1 in 26, there exists a {ϑt }t∈R -invariant set Ω
that
ωt
0
t → ±∞
t
lim
∀ω ∈ Ω.
4.8
10
Abstract and Applied Analysis
Let F be the P-completion of F and let
Ft Fst ,
t ∈ R,
4.9
s≤t
with
Fst σ{wτ2 − wτ1 : s ≤ τ1 ≤ τ2 ≤ t} ∨ N,
4.10
where σ{wτ2 − wτ1 : s ≤ τ1 ≤ τ2 ≤ t} is the smallest σ-algebra generated by the random
variable wτ2 − wτ1 for all τ1 , τ2 such that s ≤ τ1 ≤ τ2 ≤ t and N is the collection of P-null
sets of F. Note that
tτ
,
ϑτ−1 Fst Fsτ
4.11
so Ω, F, P, ϑt t∈R , Fst s≤t is a filtered metric dynamical system.
Note that problem 4.3-4.4 is interpreted as an integral equation as follows:
ut u 0 0
t
−A λu fus g ds wt,
0
ut u0 t,
t > 0,
4.12
t ∈ −ν, 0.
P-a.s. for any u0 ∈ C. By the theory in 46, we deal with 4.12 on the complete probability
space Ω, F, P. For λ and f, we make the following assumptions.
A1 There exist positive constants λl and λu such that
0 < λl ≤ λi ≤ λu < ∞,
i ∈ Z.
4.13
A2 f0 0.
A3 For any r > 0, there exists a constant lr > 0 such that
fξ − f η ≤ lrξ − η ,
C
4.14
for all ξ, η ∈ C−ν, 0; 2 with ξC , ηC ≤ r.
A4 There exist positive constants α0 and cf such that
t
e
0
2
fi usi ds
αs ≤
cf2
t
−ν
eαs |ui s|2 ds,
for all α ∈ 0, α0 , t > 0, u ∈ C−ν, t; 2 , i ∈ Z.
A5 λl > cf .
4.15
Abstract and Applied Analysis
11
We now associate a continuous random dynamical system with the stochastic retarded
lattice differential equations over Ω, F, P, ϑt t∈R . To this end, we introduce an auxiliary
Ornstein-Uhlenbeck process on Ω, F, P, ϑt t∈R and transform the stochastic retarded lattice
differential equations into a random one. Let
⎧ t
⎪
⎨
A λe−Aλt−s wt, ω − ws, ωds, ω ∈ Ω,
zt, ω −∞
⎪
⎩
0,
ω∈
/ Ω,
4.16
where e−Aλt is the uniformly continuous semigroup on 2 generated by bounded linear
operator −A − λ. Then by 4.8, 4.16 is well defined. The process zt, t ∈ R is a stationary,
Gaussian process. Moreover, the random variable z0, ω is tempered and for each ω ∈ Ω,
the mapping t → zt, ω is continuous. Furthermore, by Lemma 5.13 in 46, we find that for
all t ∈ R and P-a.s.,
zt t
−∞
e−Aλt−s dws.
4.17
Noticing that
t
−∞
e
−Aλt−s
dws e
−Aλt
z0 t
e−Aλt−s dws,
4.18
0
and using the Itô formula, we get from 4.17 that for all t > 0 and P-a.s.,
zt z0 −
t
4.19
A λzsds wt.
0
Setting vt ut−zt for t ≥ −ν in 4.12, then by 4.19, we obtain a deterministic equation,
P-a.s.
vt v 0 0
t
−A λv fvs zs g ds,
0
vt v t,
0
t > 0,
4.20
t ∈ −ν, 0,
which is equivalent to the functional differential equation
dv
−A λv f vt zt g,
dt
t > 0,
4.21
12
Abstract and Applied Analysis
with initial condition
vt v0 t,
4.22
t ∈ −ν, 0.
Here v0 t u0 t − z0 t, ω, t ∈ −ν, 0.
Problem 4.21-4.22 is a deterministic functional differential equation with random
coefficients, which can be solved pathwise. We now establish the following result for problem
4.21-4.22.
Theorem 4.1. Let T > 0 and ω ∈ Ω be fixed. Then the following properties hold.
1 For each v0 ∈ C, problem 4.21-4.22 has a unique solution v·, ω, v0 ∈ C−ν, T ; 2 .
2 Let v1 ·, ω, v10 and v2 ·, ω, v20 be the solutions of problem 4.21-4.22 for the initial data
v10 and v20 , respectively. Then there exists a constant cT > 0 such that for all t ∈ 0, T t
v1 ·, ω, v10 − v2t ·, ω, v20 ≤ v10 − v20 ecT t .
4.23
Ft, ξ, ω −Aξ0 − λξ0 f ξ zt ·, ω g,
4.24
C
C
Proof. 1 Denote
for all t ≥ 0, ξ ∈ C and ω ∈ Ω. Then by A1 –A3 , we have that
Ft, ξ, ω − Ft, ξ, ω ≤ 4 λu lf r ξ − ηC ,
4.25
for any ξ, η ∈ C with ξC ≤ r, ηC ≤ r. Therefore, F satisfies local Lipschitz condition and
maps the bounded sets of C into the bounded sets of 2 . Then by using a standard argument,
one can show that for each v0 ∈ C, there exists a Tmax ≤ ∞ such that problem 4.21-4.22 has
a unique solution v on 0, Tmax . Moreover, if Tmax < ∞ then
lim supvt C ∞.
4.26
t↑Tmax
We prove now that this local solution is a global one. Let T ∈ 0, Tmax . By A5 , we can choose
β > 0 small enough such that 2λl > 2cf β. Taking the inner product of 4.21 with v in 2 , we
get
1 d
v2 Av, v λv, v f vt zt , v g, v .
2 dt
4.27
Clearly,
Av, v Bv, Bv ≥ 0,
λv, v λi vi2 ≥ λl v2 .
i∈Z
4.28
Abstract and Applied Analysis
13
Using the Young inequality, we find that
cf
t
1 fvt zt 2 ,
f v zt , v ≤ f vt zt v ≤ v2 2
2cf
β
1
g 2 .
g, v ≤ g v ≤ v2 2
2β
4.29
Then it follows from 4.27, 4.28, 4.29 that
2 1 2
d
1
v2 ≤ − 2λl − cf − β v2 fvt zt ϑ· ω g .
dt
cf
β
4.30
Choose α ∈ 0, α0 small enough such that 2λl > 2cf α β. Then by 4.30, we obtain
αt d αt
eαt fvt zt 2 e g 2 .
e v2 ≤ − 2λl − cf − α − β eαt v2 dt
cf
β
4.31
Now, we can also choose γ > 0 small enough such that 2λl > 2 γcf α β. Integrating
4.31 over 0, t t ∈ 0, T leads to
t
eαt vt2 ≤ v02 − 2λl − cf − α − β
eαs vs2 ds
0
1
cf
t
e
αs 0
2 t
g 2
fv z ds eαs ds.
β
0
s
4.32
s
Using the Young inequality and A4 , we find that
1
cf
t
2
eαs fvs zs ds ≤ cf
0
≤ cf
t
−ν
t
0
cf
eαs vs zs2 ds
eαs 1 γ vs2 1 γ −1 zs2 ds
0
−ν
eαs vs zs2 ds.
4.33
14
Abstract and Applied Analysis
Then by 4.32 and 4.33, we obtain
t
eαs vs2 ds v02
eαt vt2 ≤ − 2λl − 2 γ cf − α − β
0
2
0
t
g eαt cf
eαs vs zs2 ds c1 eαs zϑs ω2 ds
αβ
−ν
0
2
t
2
g 0 2
0
2
αs
eαt ,
≤ 1 2νcf v 2νcf z c1 e zs ds C
C
αβ
0
4.34
where c1 cf 1 γ −1 . Consequently,
2
vt ≤
2
t
2 g 0 2
0
2
−αt
αs−t
.
1 2νcf v 2νcf z e c1 e
zs ds C
C
αβ
0
4.35
Hence, for fixed σ ∈ −ν, 0, we get that for t ∈ −σ, T ,
2
tσ
2 g 0 2
0
2
−αtσ
αs−t−σ
c1
e
vt σ ≤ 1 2νcf v 2νcf z e
zs ds C
C
αβ
0
2
t
2 g 0 2
0
2
αν−t
αν
αs−t
,
≤ 1 2νcf v 2νcf z e
c1 e
e
zs ds C
C
αβ
0
4.36
2
and for t ∈ 0, −σ,
2 2 2
vt σ2 ≤ v0 ≤ 1 2νcf v0 2νcf z0 eαν−t .
C
C
C
4.37
In view of 4.36 and 4.37, we find that for all t ∈ 0, T ,
2
t
2
2 g t 2
2
0
0
αν−t
αν
αs−t
v ≤ 1 2νcf v 2νcf z e
.
c1 e
e
zs ds C
C
C
αβ
0
4.38
Therefore, for all t ∈ 0, T ,
2
T
2 g t 2
0 2
0
2
αν
αν
v ≤ 1 2νcf ,
2νc
c
e
ds
e
zs
v
z
f
1
C
C
C
αβ
0
4.39
which, together with 4.26, implies that Tmax ∞. This proves the property 1.
2 Let v t, ω v1 t, ω, v10 − v1 t, ω, v20 . By 4.38, there exists a constant rT > 0
such that
t
v ≤ rT ,
1 C
t
v ≤ rT .
2 C
4.40
Abstract and Applied Analysis
15
Then from 4.20 and 4.25, we have that for t ∈ 0, T vt ≤ v0 4 λu lf rT t
0
vs C ds.
4.41
Hence, for fixed σ ∈ −ν, 0, we get that for t ∈ −σ, T ,
vt σ ≤ v0 4 λu lf rT tσ
0
vs C ds
t s
0
≤ v 4 λu lf rT v C ds,
C
4.42
0
and for t ∈ 0, −σ,
0
vt σ ≤ v .
4.43
C
In view of 4.42 and 4.43, we find that for all t ∈ 0, T ,
t
t s
0
u
v ≤ v
4
λ
l
v C ds.
rT
f
C
C
4.44
0
The Gronwall inequality implies that for all t ∈ 0, T ,
t
u
v ≤ v0 e4λ lf rT t .
C
C
4.45
This proves the property 2. The proof is complete.
Conversely, if for each ω ∈ Ω, vt, ω, v0 is a solution of problem 4.21-4.22 with
v · u0 · − z0 ·, ω, then the process
0
u t, ω, u0 v t, ω, v0 zt, ω
4.46
is a solution of problem 4.3-4.4. And if u0 is a C-valued F0 -measurable random variable,
then ut, ω, u0 is an Ft -adapted process.
Theorem 4.2. Problem 4.21-4.22 generates a continuous random dynamical system φ over
Ω, F, P, ϑt∈R , where
φ t, ω, v0 vt ·, ω, v0 ,
for t ≥ 0, ω ∈ Ω, v0 ∈ C.
4.47
16
Abstract and Applied Analysis
Moreover, if one defines ψ by
ψ t, ω, u0 ut ·, ω, u0 ,
for t ≥ 0, ω ∈ Ω, u0 ∈ C,
4.48
then ψ is another continuous random dynamical system associated to problem 4.3-4.4.
Proof. From property 2 of Theorem 4.1, it follows that φ·, ω, · : 0, ∞ × C → C is
continuous for all ω ∈ Ω. By 4.20, we have that for s, t ≥ 0 and σ ∈ −ν, 0,
tσ φ t, ϑs ω, φ s, ω, v0 σ φ s, ω, v0 0 F τ, φ τ, ϑs ω, φ s, ω, v0 , ϑs ω dτ.
0
4.49
Then again by 4.20 and noticing that
Ft, ξ, ϑs ω Ft s, ξ, ω,
∀s, t ≥ 0, ξ ∈ C,
4.50
we get that
φ t, ϑs ω, φ s, ω, v
0
s σ v 0 F τ, φ τ, ω, v0 , ω dτ
0
0
tsσ F τ, φ τ − s, ϑs ω, φ s, ω, v0 , ω dτ.
4.51
s
For each ω ∈ Ω consider
Φ τ, ω, v0 φ τ, ω, v0 ,
φ τ − s, ϑs ω, φ s, ω, v0 ,
if 0 ≤ τ ≤ s,
if s < τ ≤ t s.
4.52
for s, t ≥ 0.
4.53
Then for τ t s, we have
Φ t s, ω, v0 φ t, ϑs ω, φ s, ω, v0
It follows from 4.51 that
tsσ 0
0
Φ t s, ω, v σ v 0 F τ, Φ τ, ω, v0 , ω dτ,
4.54
0
for all σ ∈ −ν, 0. By the uniqueness of the solution of 4.20, we find that
Φ t s, ω, v0 φ t s, ω, v0 ,
4.55
Abstract and Applied Analysis
17
while 4.53 implies
φ t s, ω, v0 φ t, ϑs ω, φ s, ω, v0
for s, t ≥ 0.
4.56
Hence, φ is a continuous random dynamical system.
As for ψ, noticing that
ψ t, ω, u0 φ t, ω, u0 − z0 ω zt ω,
for t ≥ 0, ω ∈ Ω and u0 ∈ C,
4.57
we get from 4.56 that for s, t ≥ 0,
φ t, ϑs ω, φ s, ω, u0 − z0 ω zt ϑs ω
ψ t, ϑs ω, ψ s, ω, u0
φ t s, ω, u0 − z0 ω zts ω
4.58
ψ t s, ω, u0 .
Therefore, ψ is also a continuous random dynamical system. Furthermore, φ and ψ are
conjugated random dynamical systems, that is
ψt, ω, T ω, ξ T ϑt ω, φt, ω, ξ ,
for any ξ ∈ C,
4.59
where for every ω ∈ Ω, T ω, ξ ξ z0 ω is a homeomorphism of C. The proof is complete.
5. Existence of Random Attractors
In this section, we prove the existence of a D-random attractor for the random dynamical
system ψ associated with 4.3-4.4. We first establish the existence of a D-random attractor
for its conjugated random dynamical system φ, then the existence of a D-random attractor for
ψ follows from the conjugation relation between φ and ψ. To this end, we will derive uniform
estimates on the solutions of problem 4.21-4.22 when t → ∞ with the purpose of proving
the existence of a bounded random absorbing set and the asymptotic compactness for φ.
From now on, we always assume that D is the collection of all tempered subsets of C
with respect to Ω, F, P, ϑt t∈R . The next lemma shows that φ has a random absorbing set in
D.
Lemma 5.1. There exists K ∈ D such that K is a random absorbing set for φ in D, that is, for any
B ∈ D and P-a.e. ω ∈ Ω, there exists TB ω > 0 such that
φt, ϑ−t ω, Bϑ−t ω ⊆ Kω ∀t ≥ TB ω.
5.1
18
Abstract and Applied Analysis
Proof. Replacing ω by ϑ−t ω in 4.38, we get that for all t ≥ 0,
2 2
2 t
v ϑ−t ω, v0 ϑ−t ω ≤ 1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω eαν−t
C
C
t
c1 eαν
≤
C
eαs−t zs, ϑ−t ω2 ds 2
g αβ
2 2
1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω eαν−t
0
C
5.2
C
2
g 2
αs
.
e z0, ϑs ω ds αβ
−∞
0
c1 eαν
By assumption, B ∈ D is tempered. On the other hand, by Remark 2.6, z0 ω2C is also
tempered. Therefore, if v0 ϑ−t ω ∈ Bϑ−t ω, then there exists TB ω > 0 such that for all
t ≥ TB ω,
2
2 1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω eαν−t ≤ 1 rω,
C
C
5.3
where
rω 0
−∞
eαs z0, ϑs ω2 ds
5.4
is tempered by Remark 2.7. Then it follows from 5.2 and 5.3 that for all t ≥ TB ω,
2
2
g t
0
αν
1.
v ϑ−t ω, v ϑ−t ω ≤ c1 e 1rω C
αβ
5.5
Given ω ∈ Ω, denote by
Kω ξ ∈ C : ξ2C ≤ r1 ω ,
5.6
where
r1 ω c1 e
αν
2
g 1
1rω αβ
5.7
is tempered. Then K ∈ D. Further, 5.5 indicates that K is a random absorbing set for φ in D,
which completes the proof.
Abstract and Applied Analysis
19
Lemma 5.2. Let B ∈ D and v0 ω ∈ Bω. Then for every ε > 0 and P-a.e. ω ∈ Ω, there exist
T ∗ T ∗ B, ω, ε > 0 and N ∗ N ∗ ω, ε > 0 such that the solution vt, ω, v0 ω of problem
4.21-4.22 satisfies, for all t ≥ T ∗ ,
sup
2
vit s, ϑ−t ω, u0 ϑ−t ω ≤ ε.
5.8
s∈−ν,0 |i|≥N ∗
Proof. Let ρ be a smooth function defined on R such that 0 ≤ ρs ≤ 1 for all s ≥ 0, and
0,
ρs 1,
0 ≤ s ≤ 1,
s ≥ 2.
5.9
Then there exists a positive deterministic constant c2 such that |ρ s| ≤ c2 for all s ≥ 0. Taking
the inner product of 4.21 with x ρ|i|/kvi in 2 , we obtain that
1 d |i|
ρ
|vi |2 Av, x λv, x f vt zt , x g, x .
2 dt i∈Z
k
5.10
We now estimate terms in 5.10 as follows. First, we get from A1 that
λv, x i∈Z
λi ρ
|i| 2
|i| 2
vi ≥ λl ρ
|vi | .
k
k
i∈Z
5.11
Secondly, by the property of the cutoff function ρ, we estimate
Av, x Bv, Bx
≥
≥
|i|
|i 1|
vi1 − ρ
vi
vi1 − vi ρ
k
k
i∈Z
|i|
|i 1|
|i|
−ρ
vi1 ρ
vi1 − vi ρ
vi1 − vi k
k
k
i∈Z
|i|
|i 1|
|i|
ρ
−ρ
vi1 − vi vi1 ρ
vi1 − vi 2
k
k
k
i∈Z
|i 1| |i|
−ρ
ρ
vi1 − vi vi1
k
k
i∈Z
ρ ξi −
|vi1 − vi ||vi1 |
k
i∈Z
≥ −
2c2
c2 |vi1 |2 |vi ||vi1 | ≥ −
v2 .
k i∈Z
k
5.12
20
Abstract and Applied Analysis
Thirdly, using the Young inequality and A3 , we find that
t
|i| t
f vi zti ϑ· ω
f v zt , x ρ
k
i∈Z
2
cf |i|
1 |i|
≤
ρ
ρ
|f vit zti ϑ· ω .
|vi |2 2 i∈Z
k
2cf i∈Z
k
5.13
Finally, using the Young inequality again, we obtain that
|i|
β |i| 2 1 |i| 2
gi vi ≤
vi gi .
g, x ρ
ρ
ρ
k
2 i∈Z
k
2β i∈Z
k
i∈Z
5.14
Taking into account 5.10, 5.11, 5.12, 5.13, and 5.14, we obtain that
|i| d |i|
2
l
ρ
ρ
|vi | ≤ − 2λ − cf − β
|vi |2
dt i∈Z
k
k
i∈Z
2
1 |i| t
f vi zti ρ
cf i∈Z
k
5.15
1 |i| 2 4c2
gi ρ
v2 ,
β i∈Z
k
k
which implies
d
dt
eαt
|i| 2
ρ
|vi |
k
i∈Z
|i| ≤ − 2λl − cf − α − β eαt ρ
|vi |2
k
i∈Z
2
1 αt |i| t
f vi zti e
ρ
cf
k
i∈Z
1 αt |i| 2 4c2 αt
e
gi e v2 .
ρ
β i∈Z
k
k
5.16
Abstract and Applied Analysis
21
Using the Young inequality and A4 , we get that
1
cf
t
eαs
0
|i| f vs zs 2 ds
ρ
i
i
k
i∈Z
t
≤ cf
eαs
−ν
|i| ρ
|vi s zi s|2 ds
k
i∈Z
t
|i| αs
e
ρ
1 γ |vi s|2 1 γ −1 |zi s|2 ds
k
0
i∈Z
≤ cf
0
5.17
|i| e
ρ
|vi s zi s|2 ds.
k
−ν
i∈Z
cf
αs
Integrating 5.16 over 0, t t ≥ 0 leads to
eαt
|i| |i| ρ
|vi t|2 − ρ
|vi 0|2
k
k
i∈Z
i∈Z
t
|i| ≤ − 2λl − cf − α − β
eαs ρ
|vi s|2 ds
k
0
i∈Z
t
|i| 2
e
ρ
|f vis zsi ds
k
0
i∈Z
1
cf
1
β
t
5.18
αs
eαs
0
|i| 4c2 t αs
gi2 ds ρ
e vs2 ds.
k
k
0
i∈Z
It follows from 5.17 and 5.18 that
eαt
|i| |i| |vi t|2 − ρ
ρ
|vi 0|2
k
k
i∈Z
i∈Z
≤ − 2λ − 2 γ cf − α − β
l
cf
c1
≤ cf
0
|i| e
ρ
|vi s|2 ds
k
0
i∈Z
αs
|i| eαt |i| 2
2
gi
e
ρ
ρ
|vi s zi s| ds k
αβ i∈Z
k
−ν
i∈Z
t
αs
4c2
e
|zi s| ds k
0
|i|>k−1
0
−ν
c1
t
αs
eαs
t
2
t
2
e vs ds
αs
0
|i| eαt |i| 2
gi
ρ
ρ
|vi s zi s|2 ds k
αβ i∈Z
k
i∈Z
4c2
e
|zi s| ds k
0
|i|>k−1
αs
2
t
0
eαs vs2 ds,
5.19
22
Abstract and Applied Analysis
which implies
2 |i| 1 |i| 2
0 2
0
2
−αt
gi
ρ
ρ
|vi t| ≤ 1 2νcf v 2νcf z e C
C
k
αβ i∈Z
k
i∈Z
c1
t
e
αs−t
0
4c2
|zi s| ds k
|i|>k−1
2
t
e
αs−t
5.20
2
vs ds.
0
If we take t ≥ ν, we have that, for all σ ∈ −ν, 0,
|i| ρ
|vi t σ|2
k
i∈Z
2 1 |i| 2
2
gi
≤ 1 2νcf v0 2νcf z0 e−αtσ ρ
C
C
αβ i∈Z
k
c1
tσ
e
αs−t−σ
0
≤
4c2
|zi s| ds k
|i|>k−1
2
tσ
eαs−t−σ vs2 ds
0
5.21
2 1 |i| 2
0 2
0
αν−t
gi
ρ
1 2νcf v 2νcf z e
C
C
αβ i∈Z
k
c1 e
αν
t
e
αs−t
0
4c2 eαν
|zi s| ds k
|i|>k−1
2
t
eαs−t vs2 ds,
0
whence for all t ≥ ν,
2 |i| 0 2
0
αν−t
vt σ2 ≤ 1 2νcf ρ
2νc
v
f z e
i
C
C
k
σ∈−ν,0 i∈Z
sup
t
1 |i| 2
ρ
gi c1 eαν eαs−t
|zi s|2 ds
αβ i∈Z
k
0
|i|>k−1
4c2 eαν
k
t
0
2
eαs−t vs, ω, v0 ω ds.
5.22
Abstract and Applied Analysis
23
Replacing ω by ϑ−t ω, we find that
2
|i| ρ
vit σ, ϑ−t ω, v0 ϑ−t ω k
σ∈−ν,0 i∈Z
2
2 ≤ 1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω eαν−t
sup
C
C
5.23
t
1 |i| 2
ρ
gi c1 eαν eαs−t
|zi s, ϑ−t ω|2 ds
αβ i∈Z
k
0
|i|>k−1
4c2 eαν
k
t
2
eαs−t vs, ϑ−t ω, v0 ϑ−t ω ds.
0
We now estimate terms in 5.23 as follows. Since B ∈ D is tempered set, and z0 ω2C is
tempered function, if v0 ϑ−t ω ∈ Bϑ−t ω, then for every ε > 0, there exists T1 T1 B, ω, ε > 0
such that for all t ≥ T1 ,
2
2 ε
1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω eαν−t ≤ .
C
C
4
5.24
Secondly, since g ∈ 2 , there exists N1 N1 ω, ε > 0 such that, for all k ≥ N1 ,
1 |i| 2
1 |i| 2 ε
gi ≤
gi ≤ .
ρ
ρ
αβ i∈Z
k
αβ |i|>k
k
4
5.25
Thirdly, note that
0
−∞
5.26
eαs z0, ϑs ω2 ds < ∞.
Then by the Lebesgue theorem of dominated convergence, there exists N2 N2 ω, ε > 0
such that for all k ≥ N2 ,
0
−∞
eαs
|zi 0, ϑs ω|2 ds ≤
|i|>k−1
ε
.
4c1 eαν
5.27
Then it follows from 5.27 that for all t ≥ 0 and k ≥ N2 ,
c1 e
αν
t
e
0
αs−t
2
|zi s, ϑ−t ω| ds ≤ c1 e
|i|>k−1
αν
0
−∞
eαs
|zi ϑs ω|2 ds ≤
|i|>k−1
ε
.
4
5.28
24
Abstract and Applied Analysis
Next, we get from 4.35 that
4c2 eαν
k
t
2
eαs−t v s, ϑ−t ω, v0 ϑ−t ω ds
0
2
4c2 g eαν 4c1 c2 eαν t s ατ−t
e
≤
zτ, ϑ−t ω2 dτds
k
kα2 β
0 0
2
2 4c2 eαν 1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω te−αt .
C
C
k
5.29
For the integral on the right side of 5.29, we have that for all t ≥ 0,
t s
e
0
ατ−t
2
zτ, ϑ−t ω dτds 0
t
se−αs z0, ϑ−s ω2 ds
0
≤
5.30
∞
se
−αs
2
z0, ϑ−s ω ds < ∞.
0
Since B ∈ D is tempered set, and z0 ω2C is tempered function, there exists T3 T3 B, ω, ε >
0 such that, for all t ≥ T3 and k ∈ N,
2
2 4c2 eαν ε
1 2νcf v0 ϑ−t ω 2νcf z0 ϑ−t ω te−αt ≤ .
C
C
k
8
5.31
At the same time, there exists N3 N3 ω, ε > 0 such that, for all k ≥ N3 ,
2
4c2 g eαν 4c1 c2 eαν ∞ −αs
ε
se z0, ϑ−s ω2 ds ≤ .
k
8
kα2 β
0
5.32
It follows from 5.29, 5.30, 5.31, and 5.32 that, for all t ≥ T3 ω, ε and k ≥ N3 ω, ε,
4c2 eαν
k
t
2
ε
eαs−t v s, ϑ−t ω, v0 ϑ−t ω ds ≤ .
4
0
5.33
Let T4 T4 B, ω, ε max{T1 , T2 , T3 }, N4 N4 ω, ε max{N1 , N2 , N3 }. Then it follows
from 5.23, 5.24, 5.25, 5.28, and 5.33 that, for all t ≥ T4 and k ≥ N4 ,
sup
2
vit σ, ϑ−t ω, v0 ϑ−t ω σ∈−ν,0 |i|≥2k
2
|i| < sup
ρ
vit σ, ϑ−t ω, v0 ϑ−t ω ≤ ε.
k
σ∈−ν,0 i∈Z
The proof is complete.
5.34
Abstract and Applied Analysis
25
Lemma 5.3. The random dynamical system φ is D-pullback asymptotically compact in C, that is, for
P-a.e. ω ∈ Ω, the sequence {φtn , ϑ−tn ω, vn0 ϑ−tn ω}∞
n1 has a convergent subsequence in C provided
tn → ∞, B ∈ D and vn0 ϑ−tn ω ∈ Bϑ−tn ω.
Proof. By 4.21 and Lemma 5.1, we find that, for every t ≥ TB ω ν, and σ1 , σ2 ∈ −ν, 0,
φ t, ϑ−t ω, v0 ϑ−t ω σ1 − φ t, ϑ−t ω, v0 ϑ−t ω σ2 v t σ1 , ϑ−t ω, v0 ϑ−t ω − v t σ2 , ϑ−t ω, v0 ϑ−t ω 5.35
≤ v t ξ, ϑ−t ω, v0 ϑ−t ω |σ1 − σ2 | ≤ r2 ω|σ1 − σ2 |,
where
r2 ω 4 λ lf
u
sup
!"
#
r1 ϑσ ω z0, ϑσ ω
g ,
5.36
σ∈−ν,0
and ξ is between σ1 and σ2 .
By Lemma 5.1, 5.35, and Lemma 5.2, {φtn , ϑ−tn ω, vn0 ϑ−tn ω}∞
n1 satisfies conditions
∞
0
i–iii in Theorem 3.3. Therefore, {φtn , ϑ−tn ω, vn ϑ−tn ω}n1 is relative compact in C and
hence has a convergent subsequence in C.
We are now in a position to present our main result about the existence of a D-random
attractor for ψ in C.
Theorem 5.4. The random dynamical system ψ has a unique D-random attractor in C.
Proof. Notice that φ has a closed absorbing set K in D by Lemma 5.1 and is D-pullback
asymptotically compact in C by Lemma 5.3. Hence, the existence of a unique D-random
attractor {A1 ω}ω∈Ω for φ follows from Proposition 2.11 immediately.
Since ψ and φ are conjugated by the random homeomorphism T ω, ξ ξ z0 ω, and
0
z ω ∈ C is tempered, then by Proposition 1.8.3 in 45, ψ has a unique D-random attractor
{A2 ω}ω∈Ω in C which is given by
A2 ω ξω z0 ω : ξω ∈ A1 ω .
5.37
The proof is complete.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants
11071166 and 11271110, the Key Programs for Science and Technology of the Education
Department of Henan Province under Grant 12A110007, and the Scientific Research Start-up
Funds of Henan University of Science and Technology.
26
Abstract and Applied Analysis
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