Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 628459, 23 pages
doi:10.1155/2011/628459
Research Article
Asymptotic Behavior of Stochastic Partly
Dissipative Lattice Systems in Weighted Spaces
Xiaoying Han
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
Correspondence should be addressed to Xiaoying Han, xzh0003@auburn.edu
Received 22 June 2011; Accepted 3 September 2011
Academic Editor: I. Chueshov
Copyright q 2011 Xiaoying Han. 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.
We study stochastic partly dissipative lattice systems with random coupled coefficients and
multiplicative/additive white noise in a weighted space of infinite sequences. We first show
that these stochastic partly dissipative lattice differential equations generate a random dynamical
system. We then establish the existence of a tempered random bounded absorbing set and a global
compact random attractor for the associated random dynamical system.
1. Introduction
Stochastic lattice differential equations SLDE’s arise naturally in a wide variety of
applications where the spatial structure has a discrete character and random spatiotemporal
forcing, called noise, is taken into account. These random perturbations are not only
introduced to compensate for the defects in some deterministic models, but are also rather
intrinsic phenomena. SLDE’s may also arise as spatial discretization of stochastic partial
differential equations SPDE’s; however, this need not to be the case, and many of the most
interesting models are those which are far away from any SPDE’s.
The long term behavior of SLDE’s is usually studied via global random attractors. For
SLDE’s on regular spaces of infinite sequences, Bates et al. initiated the study on existence of
a global random attractor for a certain type of first-order SLDE’s with additive white noise
on 1D lattice Z 1. Continuing studies have been made on various types of SLDS’s with
multiplicative or additive noise, see 2–7.
Note that regular spaces of infinite sequences may exclude many important and
interesting solutions whose components are just bounded, considering that a weighted
space of infinite sequences can make the study of stochastic LDE’s more intensive. More
importantly, all existing works on SLDE’s consider either a noncoupled additive noise or
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a multiplicative white noise term at each individual node whereas in a realistic system
randomness appears at each node as well as the coupling mode between two nodes. Han et al.
initiated the asymptotic study of such SLDE’s in a weighted space of infinite sequences, with
not only additive/multiplicative noise but also coefficients which are randomly coupled 8.
In this work, following the idea of 8, we will investigate the existence of a global
random attractor for the following stochastic partly dissipative lattice systems with random
coupled coefficients and multiplicative/additive white noise in weighted spaces:
q
u̇i −λui j−q
ηi,j θt ωuij − fi ui − αvi hi ui ◦
v̇i −σvi μui gi ui ◦
u̇i −λui q
j−q
dwt
,
dt
dwi t
,
dt
1.1
i ∈ Z, t > 0,
ηi,j θt ωuij − fi ui − αvi hi ai
v̇i −σvi μui gi bi
dwt
,
dt
dwi t
,
dt
1.2
i ∈ Z, t > 0,
where ui , hi , gi , ai , bi ∈ R fi ∈ C1 R, R; i ∈ Z, λ, α, σ, μ > 0 are positive constants; A is the
coupling operator, ηi,−q ω, . . . , ηi,0 ω, . . . , ηi,q ω, i ∈ Z, q ∈ N, are random variables, and
wt, {wi t : i ∈ Z} are two-sided Brownian motions on proper probability spaces.
For deterministic partly dissipative lattice systems without noise, the existence of the
global attractor has been studied in 9–13. For stochastic lattice system 1.2 with additive
noises, when q 1, ηi,±1 ω ≡ 1, ηi,0 ω ≡ −2 for all i ∈ Z, Huang 4 and Wang et al. 14
proved the existence of a global random attractor for the associated RDS in the regular phase
space l2 × l2 . In this work we will consider the existence of a compact global random attractor
in the weighted space lρ2 ×lρ2 , which attracts random tempered bounded sets in pullback sense,
for stochastic lattice systems 1.1 and 1.2. Here we choose a positive weight function ρ :
Z → 0, M0 such that l2 ⊂ lρ2 . If i∈Z ρi < ∞, then lρ2 contains any infinite sequences
whose components are just bounded and l2 ⊂ l∞ ⊂ lρ2 . Note that when ρi ≡ 1, our results
recover the results obtained in 4, 14 while lρ2 is reduced to the standard l2 . Moreover, the
required conditions in this work for the existence of a random attractor for system 1.2-1.1
in weighted space lρ2 × lρ2 are weaker than those in l2 × l2 .
The rest of this paper is organized as follows. In Section 2, we present some
preliminary results for global random attractors of continuous random dynamical systems
in weighted spaces of infinite sequences. We then discuss the existence of random attractors
for stochastic lattice systems 1.1 and 1.2 in Sections 3 and 4, respectively.
2. Preliminaries
In this section, we present some concepts related to random dynamical systems RDSs and
random attractors 1, 8, 15 on weighted space of infinite sequences.
Let ρ be a positive function from Z to 0, M0 ⊂ R , where M0 is a finite positive
constant. Define for any i ∈ Z, ρi ρi and
lρ2
u ui i∈Z
2
:
ρi |ui | < ∞, ui ∈ R ,
i∈Z
2.1
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then lρ2 is a separable Hilbert space with the inner product u, vρ i∈Z ρi ui vi and norm
u
2ρ u, uρ i∈Z ρi |ui |2 for u ui i∈Z , v vi i∈Z ∈ lρ2 . Moreover, define
H ˙ lρ2 × lρ2
2.2
with inner product
u1 , v1 , u2 , v2
H
u1 , u2 v1 , v2 ,
ρ
ρ
for u1 , v1 , u2 , v2 ∈ H,
2.3
and norm
1/2
u, v
H u
2ρ v
2ρ
for u, v ∈ H,
2.4
then H is also a separable Hilbert space.
Let Ω, F, P be a probability space and {θt : Ω → Ω, t ∈ R} be a family of measurepreserving transformations such that t, Ω → θt Ω is BR × F, F-measurable, θ0 IdΩ and
θts θt θs for all s, t ∈ R. The space Ω, F, P, θt t∈R is called a metric dynamical system. In
the following, “property P holds for a.e. ω ∈ Ω with respect to θt t∈R ” means that there is
⊂ Ω with PΩ
1 and θt Ω
Ω
such that P holds for all ω ∈ Ω.
Ω
Recall the following definitions from existing literature.
i A stochastic process {St, ω}t≥0,ω∈Ω is said to be a continuous RDS over
Ω, F, P, θt t∈R with state space H, if S : R × Ω × H → H is BR × F ×
BH, BH-measurable, and for each ω ∈ Ω, the mapping St, ω : H → H, u →
St, ωu is continuous for t ≥ 0, S0, ωu u and St s, ω St, θs ωSs, ω for
all u ∈ H and s, t ≥ 0.
ii A set-valued mapping ω → Dω ⊂ H may be written as Dω for short is said
to be a random set if the mapping ω → distH u, Dω is measurable for any u ∈ H.
iii A random set Dω is called a closed (compact) random set if Dω is closed compact
for each ω ∈ Ω.
iv A random set Dω is said to be bounded if there exist u0 ∈ H and a random variable
rω > 0 such that Dω ⊂ {u ∈ H : u − u0 H ≤ rω} for all ω ∈ Ω.
v A random bounded set Dω is said to be tempered if for a.e. ω ∈ Ω,
lim e−βt sup{
u
H : u ∈ Dθ−t ω} 0,
t→∞
∀β > 0.
2.5
Denote by DH the set of all tempered random sets of H.
vi A random set Bω is said to be a random absorbing set in DH if for any
Dω ∈ DH and a.e. ω ∈ Ω, there exists TD ω such that St, θ−t ωDθ−t ω ⊂
Bω for all t ≥ TD ω.
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vii A random set Aω is said to be a random attracting set if for any Dω ∈ DH, we
have
lim distH St, θ−t ωDθ−t ω, Aω 0,
t→∞
a.e. ω ∈ Ω,
2.6
in which distH is the Hausdorff semidistance defined via distH E, F
supu∈E infv∈F u − v
ρ for any E, F ⊂ lρ2 .
viii A random compact set Aω is said to be a random global D attractor if it is a
compact random attracting set and St, ωAω Aθt ω for a.e. ω ∈ Ω and t ≥ 0.
Definition 2.1 see 8. {St, ω}t≥0,ω∈Ω is said to be random asymptotically null in DH, if for
any Dω ∈ DH, a.e. ω ∈ Ω, and any ε > 0, there exist T ε, ω, Dω > 0 and Iε, ω, Dω ∈
N such that
⎛
⎝
⎞1/2
ρi |St, θ−t ωuθ−t ωi |2 ⎠
≤ ε,
∀t ≥ T ε, ω, Dω, ∀uω ∈ Dω. 2.7
|i|>Iε,ω,Dω
Theorem 2.2 see 8. Let {St, ω}t≥0,ω∈Ω be a continuous RDS over Ω, F, P, θt t∈R with state
space H and suppose that
a there exists a random bounded closed absorbing set Bω ∈ DH such that for a.e. ω ∈ Ω
and any Dω ∈ DH, there exists TD ω > 0 yielding St, θ−t ωDθ−t ω ⊂ Bω for
all t ≥ TD ω;
b {St, ω}t≥0,ω∈Ω is random asymptotically null on Bω; that is, for a.e. ω ∈ Ω and for any
ε > 0, there exist T ε, ω, Bω > 0 and Iε, ω, Bω ∈ N such that
sup
ρi |St, θ−t ωuθ−t ωi |2 ≤ ε2 ,
∀t ≥ T ε, ω, Bω.
u∈Bω |i|>Iε,ω,Bω
2.8
Then the RDS {St, ω}t≥0,ω∈Ω possesses a unique global random D attractor Aω given by
Aω St, θ−t ωBθ−t ω.
τ≥TB ω t≥τ
2.9
3. Stochastic Partly Dissipative Lattice Systems with
Multiplicative Noise in Weighted Spaces
This section is devoted to the study of asymptotic behavior for system 1.1 in weighted space
H lρ2 × lρ2 . We first transform the stochastic lattice system 1.1 to random lattice system in
Section 3.1. We then show in Section 3.2 that 1.1 generates random dynamical system in H.
Finally we prove in Section 3.3 the existence of a global random attractor for system 1.1.
Throughout the rest of this paper, a positive weight function ρ : Z → R is chosen to
satisfy
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P0 0 < ρi ≤ M0 and ρi ≤ c · ρi ± 1, for all i ∈ Z for some positive constants M0
and c.
e.g., ρx 1/1 2 x2 q , q > 1/2 16, 17 and ρx e−|x| , x ∈ Z where > 0.
3.1. Mathematical Setting
Define Ω1 {ω ∈ CR, R : ω0 0} C0 R, R, and denote by F1 the Borel σ-algebra
on Ω1 generated by the compact open topology see 2, 15 and P1 the corresponding
Wiener measure on F1 . Defining θt t∈R on Ω1 via θt ω· ω· t − ωt for t ∈ R, then
Ω1 , F1 , P1 , θt t∈R is a metric dynamical system.
Consider the stochastic lattice system 1.1 with random coupled coefficients and
multiplicative white noise:
du −λu Aθt ωu − fu − αv h dt u ◦ dwt,
dv −σv μu g u ◦ dwt,
i ∈ Z, t > 0,
3.1
where u ui i∈Z , v vi i∈Z ; fu fi ui i∈Z with fi ∈ C1 R, R i ∈ Z, g gi i∈Z ,
h hi i∈Z ; λ, α, σ, μ are positive constants; ηi,−q ω, . . . , ηi,0 ω, . . . , ηi,q ω q ∈ N are
random variables on the probability space Ω1 , F1 , P1 ; A· is a linear operator on lρ2 defined
by
A·ui q
ηi,j ·uij ;
3.2
j−q
wt is a Brownian motion Wiener process on the probability space Ω1 , F1 , P1 ; ◦ denotes
the Stratonovich sense of the stochastic term.
For convenience, we first transform 3.1 into a random differential equation without
white noise. Let
δθt ω −
0
−∞
es θt ωsds,
t ∈ R, ω ∈ Ω1 ,
3.3
then δθt ω is an Ornstein-Uhlenbeck process on Ω1 , F1 , P1 , θt t∈R that solves the following
Ornstein-Uhlenbeck equation see 2, 15 for details
dδ δdt dwt,
t ≥ 0,
3.4
where wtω wt, ω ωt for ω ∈ Ω1 , t ∈ R, and possesses the following properties.
1 ∈ F1 of Ω1 of full P1 measure such that
Lemma 3.1 see 2, 15. There exists a θt -invariant set Ω
1 , one has
for ω ∈ Ω
i the random variable |δω| is tempered;
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ii the mapping δθt ω
t, ω −→ δθt ω −
0
−∞
3.5
es ωt sds wt
is a stationary solution of Ornstein-Uhlenbeck equation 3.4 with continuous trajectories;
iii
1
|δθt ω|
lim
t → ±∞
t → ±∞ t
t
t
lim
3.6
δθs ωds 0.
0
1 possesses same properties as the original one if we choose
The mapping of θ on Ω
1 to be denoted also by F1 . Therefore we can change our
the trace σ-algebra with respect to Ω
1 , still denoted by the symbols Ω1 , F1 , P1 , θt t∈R .
metric dynamical system with respect to Ω
Let
xt, ω e−δθt ω ut, ω,
yt, ω e−δθt ω vt, ω,
ω ∈ Ω1 ,
3.7
where ut, ω, vt, ω is a solution of 3.1, then ut, ω, vt, ω → xt, ω, yt, ω is a
homomorphism in H. System 3.1 can then be transformed to the following random system
with random coefficients but without white noise:
dx
−λx Aθt ωx − e−δθt ω f eδθt ω x δθt ωx − αy e−δθt ω h,
dt
dy
−σy δθt ωy μx e−δθt ω g
dt
t > 0,
3.8
Letting z x, y, 3.8 are equivalent to
dz
Fz, θt ω,
dt
3.9
t > 0,
where
Fz, θt ω −λx Aθt ωx − e−δθt ω f eδθt ω x δθt ωx − αy e−δθt ω h
.
−σy δθt ωy μx e−δθt ω g
3.10
We now make the following standing assumptions on fi , gi , hi , and ηi,j , j −q, . . . , q i ∈ Z and study in the following subsections asymptotic behavior of system 3.9.
H1 g gi i∈Z , h hi i∈Z ∈ lρ2 .
H2 Let
ηω sup ηi,−q ω, . . . , ηi,0 ω, . . . , ηi,q ω : i ∈ Z ≥ 0,
q ∈ N.
3.11
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ηθt ω < ∞ belongs to L1loc R with respect to t ∈ R for each ω ∈ Ω1 .
1
E η lim
t → ±∞ t
t
ηθt ωds < ∞,
3.12
0
and ηω is tempered, that is, there exists a θt -invariant set Ω10 ∈ F1 of full P1 measure
such that for ω ∈ Ω10 ,
lim e−βt supηθ−t ω 0,
t → ∞
∀β > 0.
3.13
t∈R
1 and still write Ω10 ∩ Ω
1 as Ω1 .
In the following, we will consider ω ∈ Ω10 ∩ Ω
t
q
q
H3 min{λ, σ} > qE|ηω|
limt → ±∞ 1/t 0 q k0 ck ηθt ωds, where q
q k0 ck .
H4 There exists a function R ∈ CR , R such that
sup max fi s ≤ Rr,
i∈Z s∈−r,r
∀r ∈ R .
3.14
H5 fi ∈ C1 R, R, fi 0 0, sfi s ≥ −bi2 , b bi i∈Z ∈ lρ2 , and there exists a constant a ≥ 0
such that fi s ≥ −a, for all s ∈ R, i ∈ Z.
3.2. Random Dynamical System Generated by Random Lattice System
In this subsection, we show that the random lattice system 3.9 generates a random
dynamical system on H.
Definition 3.2. We call z : 0, T → H a solution of the following random differential equation
dz
Fz, θt ω,
dt
z zi i∈Z ,
F Fi i∈Z ,
3.15
where ω ∈ Ω0 , if z ∈ C0, T , H satisfies
zi t zi 0 t
Fi zs, θs ωds,
for i ∈ Z, t ∈ 0, T .
3.16
0
Theorem 3.3. Let T > 0 and (P0), (H1), (H2), (H4), and (H5) hold. Then for any ω ∈ Ω1
and any initial data z0 x0, y0 ∈ H, 3.9 admits a unique solution z·; ω, z0 x·; ω, z0 , y·; ω, z0 ∈ C0, T , H with z0; ω, z0 z0 .
Proof. 1 Denote E l2 × l2 , we first show that if z0 ∈ E and h, g ∈ E, then 3.9 admits a
unique solution zt; ω, z0 , h, g ∈ E on 0, T with z0; ω, z0 , g, h z0 . Given z ∈ E, ω ∈ Ω1 ,
and h, g ∈ E, note that Fz, ω is continuous in z and measurable in ω from E × Ω1 to E.
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By 3.2 and H2,
⎡ ⎛
⎞2 ⎤1/2
q
⎥
⎢ ηi,j ωxij ⎠ ⎦ ≤ 2q 1 ηω · x
.
Aωx
⎣ ⎝
i∈Z
3.17
j−q
By H4,
"
! f eδω x ≤ max R eδω x
, a · eδω x
,
3.18
and therefore
"
! Fz, ω
E ≤ λ 2q 1 ηω max R eδω x
, a |δω| μ · x
α σ |δω| · y e−δω · h
g .
3.19
For any z1 x1 , y1 , z2 x2 , y2 ∈ E, and for some ϑ ∈ 0, 1
"
! ,a
≤ max R eδω 1 − ϑ x1 ϑ x2
f eδω x1 − f eδω x2
· eδω x1 − x2 .
3.20
Also
Aωx1 − Aωx2 ⎧
q
⎨ ⎩j−q
i∈Z
1
2
ηi,j ω xij − xij
⎫1/2
⎬
⎭
≤ 2q 1 ηω · x1 − x2 .
3.21
It then follows that
F z1 , ω − F z2 , ω
E
≤ α σ |δω| · y1 − y2
)
λ 2q 1 ηω μ |δω|
"* 3.22
! ,a
eδω max R eδω 1 − ϑ v1 ϑ v2
· x1 − x2 .
For any compact set D ⊂ E with supz∈D z
≤ r, defining random variable ζD ω ≥ 0 via
"
! ζD ω λ 2q 1 ηω μ |δω| max R eδω r , a · eδω r
α σ |δω|r e−δω · h
g ,
3.23
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we have
τ1
ζD θt ωdt < ∞,
3.24
∀τ ∈ R,
τ
and for any z, z1 , z2 ∈ D,
F z1 , ω − F z2 , ω
Fz, ω
E ≤ ζD ω,
E
≤ ζD ω z1 − z2
E
.
3.25
According to 15, 19, 20, problem 3.9 possesses a unique local solution z·; ω, z0 , g, h ∈
C0, Tmax , E 0 < Tmax ≤ T satisfying the integral equation
zt z0 t
Fzs, ωds,
for t ∈ 0, Tmax .
3.26
0
We will next show that Tmax T . Since the set C0 R of continuous random process in t
∞
m
is dense in the set L1 R see 18, 21, for each ω ∈ Ω1 , there exists a sequence {ηi,j t, ω}
m1
of continuous random process in t ∈ R such that
T
m
lim
ηi,j s, ω − ηi,j s, ω ds 0, ∀T > 0,
m→∞ 0
m
ηi,j t, ω ≤ ηi,j θt ω ≤ ηθt ω, ∀t ∈ R.
3.27
Consider the random differential equation with initial data z0 ∈ E :
⎛
⎞
−λxm Am t, ωxm − e−δθt ω f eδθt ω xm δθt ωxm − αym e−δθt ω h
dzm ⎝
⎠,
dt
m
m
m
−δθt ω
− σy δθt ωy μx e
g
3.28
where
Am t, ωxm
i
q
m
ηi,j t, ωxij .
3.29
j−q
Follow the same procedure as above, 3.28 has a unique solution zm ·; ω, z0 , g, h ∈
m
C0, Tmax , E, that is,
m
zi t
z0 t
0
m
Fi
zm s, ω ds,
)
m
for t ∈ 0, Tmax ,
3.30
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and by the continuity of Am s, ω in s, there holds
m
dzi
dt
⎛
⎜
⎝
m
−λxi
⎞
m
m
m
δθt ωxi − αyi e−δθt ω hi
Am θt ωxm i − e−δθt ω fi eδθt ω xi
⎟
⎠.
m
− σyi
m
δθt ωyi
m
μxi
e−δθt ω g
3.31
Note that
m m
m
m m
m m Am θt ωxm · xi ≤ ηθt ω · xi xi−q · · · xi xi · · · xi xiq ,
i
m 2
m
m
2δθt ω
· fi eδθt ω xi
· xi
≤ eδθt ω xi
−ae
m 2
≤ R eδθt ω xm · e2δθt ω xi
, t ∈ 0, T ,
multiplying 3.31 by
d
μ xm
dt
2
α y
m
μxi
0
0
αyi
m
2
and sum over i ∈ Z results in
m
3.32
.
≤ − min{λ, σ} 2a 2δθt ω 2 2q 1 ηθt ω
2
2
2μ
2α
h
2 g
2 e−2δθt ω .
· μ xm α ym
λ
σ
3.33
Applying Gronwall’s inequality to 3.33 we obtain that
μ xm
2
α ym
2
t
2
≤ e2at 0 2δθs ω22q1ηθs ωds μ
x0
2 α y0
t
2μ
2α
2
2
g
e2at 0 2δθs ω22q1ηθs ωds
h
λ
σ
/ t
0
·
e
0
.
κt,
s
min{λ,σ}−2as−2δθs ω 0 2δθr ω22q1ηθr ωdr
3.34
ds
)
m
t ∈ 0, Tmax ,
where κt ∈ C0, T , R is independent of m, which implies that the interval of existence of
zm t is 0, T , and zm ·; ω, z0 , g, h ∈ C1 0, T , E.
By 3.34,
m 2 m 2
xi yi ≤
κt
,
min μ, α
∀m ∈ N, t ∈ 0, T .
3.35
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11
2
m
Since |Fi zm t, θt ω| ≤ K 2 T, ω for some KT, ω > 0 and t ∈ 0, T , then for any t, τ ∈
0, T , m ∈ N,
t
m m
m
m
z s, θs ω ds ≤ KT, ω · |t − τ|.
zi t − zi τ Fi
3.36
τ
mk By the Arzela-Acoli Theorem, there exists a convergent subsequence {zi
m
{zi t, t ∈ 0, T } such that
mk zi
t, t ∈ 0, T } of
3.37
t −→ zi t as k −→ ∞, t ∈ 0, T , i ∈ Z,
and zi t is continuous on t ∈ 0, T . Moreover, |zi |2 ≤ κt/ min{μ, α} for t ∈ 0, T . By 3.27,
3.35, assumption H2, and the Lebesgue Dominated Convergence Theorem we have
lim
k→∞
t
0
t
m m ηi,j k s, ω − ηi,j θs ω ds lim ηi,j k s, ω − ηi,j θs ω ds 0,
0 k→∞
mk lim ηi,j s, ω − ηi,j θs ω 0, for a.e. s ∈ 0, T ,
k→∞
t
m m lim
ηi,j k s, ωxi k s − ηi,j θs ωxi s ds 0.
k→∞
3.38
0
Thus replacing m by mk in 3.31 and letting k → ∞ give
zi t z0 i t
3.39
Fi zi , θs ωds for t ∈ 0, T .
0
By the uniquness of the solutions of 3.9, we have zi t zi t for t ∈ 0, Tmax . By 3.34,
zt
2E ≤ κt/ min{μ, α} for t ∈ 0, Tmax , which implies that the solution zt of 3.9 exists
globally on t ∈ 0, T .
2 Next we prove that for any z0 ∈ H and h, g ∈ H, 3.9 has a solution
zt; ω, z0 , h, g on 0, T with z0; ω, z0 , h, g z0 . Let z1,0 , z2,0 ∈ E and h1 h1,i i∈Z ,
m
m
h2 h2,i i∈Z , g1 g1,i i∈Z ,g2 g2,i i∈Z ∈ l2 . Let z1 t, ω, z2 t, ω be two solutions
of 3.28 with initial data z1,0 , z2,0 and h, g replaced by h1 , h2 , g1 , g2 , respectively. Set
m
m
m
m
dm t z1 t − z2 t d1 t, d2 t ∈ E ⊂ H. Take inner product ·, ·H of d/dtdm
with dm and evaluate each term as follows. By P0, H1, H2, and H4,
m 2
Am θt ωdm , dm ≤ qηθ
t ω d1
,
1
1
ρ
ρ
m
m
m
m
f eδθt ω x1
− f eδθt ω x2 , d1
≥ −aeδθt ω d1
ρ
2
ρ
,
m
m
m
m
m
m
eδθt ω d1
f eδθt ω x1
− f eδθt ω x2 , d1
x2
≤ R eδθt ω x1
m
h1 − h2 , d1
ρ
m 2
ρ
≤ h1 − h2 2ρ · d1
ρ
;
m
g1 − g2 , d2
ρ
≤ g 1 − g2
2
ρ
m 2
· d2
ρ
2
ρ
,
.
3.40
12
International Journal of Differential Equations
It then follows that
d
m
μ d1
dt
2
ρ
α
m 2
d2
ρ
.
≤ − min{λ, σ} 2a 2δθt ω qηθ
t ω
m 2
m 2
· μ d1
α d2
ρ
ρ
2μ
2α
2
g1 − g2 ρ e−2δθt ω .
h1 − h2 2ρ λ
σ
3.41
For T > 0, applying Gronwall’s inequality to 3.41 on 0, T implies that
m
μ d1 t
2
ρ
m
α d2 t
2
ρ
m
≤ CT μ d1 0
2
ρ
m
α d2 0
2
ρ
h1 − h2 t2ρ g1 − g2
2
ρ
3.42
for some constant CT depending on T , and thus
m
m
z1 t − z2 t
2
H
T
≤C
m
m
z1 0 − z2 0
2
H
h1 − h2 2ρ g1 − g2
2
ρ
,
3.43
T is a constant depending on T . Denote by E
l2 × l2 , where l2 l2 with the
where C
C0, T , H such that
norm · ρ . By 3.43, there exists a mapping Φm ∈ CE
× E,
m
m
m
Φ z0 , g, h z t; ω, z0 , g, h, where z t; ω, z0 , g, h is the solution of 3.28 on 0, T with zm 0; ω, z0 , g, h z0 . Since l2 is dense in lρ2 , the mapping Φm can be extended uniquely
m : H × H → C0, T , H.
to a continuous mapping Φ
m z0 , g, h zm ·; ω, z0 , g, h ∈ C0, T , H for
For given z0 ∈ H and g, h ∈ H, Φ
{hn , gn } ⊂ E
such that
T > 0. There exist sequences {z0n } ⊂ E,
z0n − z0 E −→ 0,
hn − h
ρ −→ 0,
gn − g
ρ
−→ 0
as n −→ ∞.
3.44
m
m z0n , hn , gn Φm z0n , hn , gn zm t; ω, z0n , hn , gn ∈ E
be the solution of
Let zn t Φ
3.28, then it satisfies the integral equation
m
zn
i
t z0n i t
m
Fi zn , θs ω ds.
3.45
0
m , we have for t ∈ 0, T ,
By the continuity of Φ
→ ∞ m m z0n , hn , gn n−→
zm t; ω, z0n , hn , gn Φ
Φ
z0 , h, g zm t; ω, z0 , h, g ∈ H.
3.46
Thus for each i ∈ Z,
m
m
m zn
t −→ zi t : zi t; ω, z0 , h, g
i
as n −→ ∞ uniformly on t ∈ 0, T .
3.47
International Journal of Differential Equations
13
m
Moreover, {zn i t} is bounded in n. Let n → ∞, then we have
m
zi t z0 i t
Fi zm , θs ω ds,
3.48
0
m
and zi t satisfies the differential equation 3.31.
m
μρi xi
Multiply equation 3.31 by
d
μ xm
dt
2
ρ
μ xm
α ym
2
ρ
2
ρ
α ym
2
ρ
0
0
m
αρi yi
and sum over i ∈ Z, we obtain
.
t ω
≤ − min{λ, σ} 2a 2δθt ω qηθ
2
2
m
m
· μ x
α y
ρ
ρ
2μ
2α
2
g ρ e−2δθt ω ,
h
2ρ λ
σ
t
2
s ωds
μ
x0
2ρ α y0 ρ
≤ e2at 0 2δθs ωqηθ
t
2μ
2α
2
2
s ωds
g ρ e2at 0 2δθs ωqηθ
h
ρ λ
σ
/ t
0
3.49
s
r ωdr
eλσ−2as−2δθs ω 0 2δθr ωqηθ
ds
·
0
.
κρ t,
t ∈ 0, Tmax Similar to the process 3.35–3.39 in part 1, we obtain the existence of a unique solution
zt; ω, z0 , g, h ∈ H of 3.9 with initial data z0 ∈ H, which is the limit function of a
subsequence of {zm t; ω, z0 , g, h} in H for t ∈ 0, T . In the latter part of this paper, we
may write zt; ω, z0 , h, g as zt; ω, z0 for simplicity.
Theorem 3.4. Assume that (P0), (H1), (H2), (H4), and (H5) hold. Then 3.9 generates a continuous
RDS {ψt, ω}t≥0,ω∈Ω1 over Ω1 , F1 , P1 , θt t∈R with state space H:
ψt, ωz0 : zt; ω, z0 for z0 ∈ H, t ≥ 0, ω ∈ Ω1 .
3.50
Moreover,
ϕt, ωu0 , v0 : eδθt ω ψt, ωe−δω u0 , v0 for u0 , v0 ∈ H, t ≥ 0, ω ∈ Ω1 ,
3.51
defines a continuous RDS {ϕt, ω}t≥0,ω∈Ω1 over Ω1 , F1 , P1 , θt t∈R associated with 3.1.
Proof. By Theorem 3.3, the solution zt; ω, z0 of 3.9 with z0; ω, z0 z0 exists globally on
0, ∞. It is then left to show that zt; ω, z0 zt; ω, z0 , h, g is measurable in t, ω, z0 .
In fact, for z0 ∈ E and h, g ∈ E, the solution of 3.9 zt; ω, z0 , h, g ∈ E for t ∈
0, ∞. In this case, function Fz, t, ω, h, g Fz, t, ω is continuous in z,h, g and measurable
in t, ω, which implies that z : 0, ∞ × Ω1 × E × E → E,t; ω, z0 , h, g → zt; ω, z0 , h, g is
B0, ∞ × F1 × BE × BE, BE-measurable.
14
International Journal of Differential Equations
For z0 ∈ H and h, g ∈ H, the solution zt; ω, z0 , h, g ∈ H for t ∈ 0, ∞. For any
given N > 0, define TN : H → E, u, v ui , vi i∈Z → TN u, v TN u, vi i∈Z by
TN u, vj uj , vj
0
if j ≤ N,
if j > N,
3.52
and write
zN t; ω, z0 , h, g z t; ω, TN z0 , TN h, g .
3.53
Then TN is continuous and for any z0 ∈ H, h, g ∈ H, and
z t; ω, z0 , h, g lim z t; ω, TN z0 , TN h, g .
N →∞
3.54
Thus z : 0, ∞ × Ω1 × E × E → H is B0, ∞ × F0 × BE × BE, BH-measurable. Observe
also that Id, Id, TN , TN : 0, ∞ × Ω1 × H × H → 0, ∞ × Ω0 × E × E is B0, ∞ × F0 ×
BH × BH, B0, ∞ × F0 × BE × BE-measurable. Hence zN z ◦ Id, Id, TN , TN :
0, ∞ × Ω1 × H × H → H is B0, ∞ × F0 × BH × BH, BH-measurable. It then
follows from 3.54 that z : 0, ∞ × Ω1 × H × H → H is B0, ∞ × F1 × BH × BH, BHmeasurable. Therefore, fix h, g ∈ H we have that zt; ω, z0 zt; ω, z0 , h, g is measurable
in t, ω, z0 . The other statements then follow directly.
Remark 3.5. If h, g ∈ E, system 3.1 defines a continuous RDS {ϕt}t≥0 over Ω1 , F1 , P1 ,
θt t∈R in both state spaces E and H.
3.3. Existence of Tempered Random Bounded Absorbing Sets and Global
Random Attractors in Weighted Space
In this subsection, we study the existence of a tempered random bounded absorbing set and
a global random attractor for the random dynamical system {ϕt, ω}t≥0,ω∈Ω1 generated by
3.1 in weighted space H.
Theorem 3.6. Assume that (P0), (H1)–(H5) hold, then there exists a closed tempered random
bounded absorbing set B1ρ ω ∈ DH of {ϕt, ω}t≥0,ω∈Ω1 such that for any Dω ∈ DH and
each ω ∈ Ω1 , there exists TD ω > 0 yielding ϕt, θ−t ωDθ−t ω ⊂ B1ρ ω for all t ≥ TD ω. In
particular, there exists T1ρ ω > 0 such that ϕt, θ−t ωB1ρ θ−t ω ⊂ B1ρ ω for all t ≥ T1ρ ω.
Proof. 1 For initial condition z0 ∈ E and h, g ∈ E, let zm t, ω zm t; ω, z0 ω, h, g be
a solution of 3.28 with z0 ω e−δω z0 ∈ E, where ω ∈ Ω1 , then zm t, ω ∈ E for all t ≥ 0.
Let 1 > 0 be such that
λ1 2 min{λ, σ} − 1 > 2qE
ηω.
3.55
International Journal of Differential Equations
15
By H4 and H5, we have
−
b
2ρ ≤
d
μ xm
dt
2
ρ
α y
δθt ω m δθt ω m · fi e
< ∞,
ρi e
xi
xi
for fixed t ≥ 0,
3.56
i∈Z
m
2
ρ
2
.
t ω · μ xm α ym
≤ −λ1 2δθt ω 2qηθ
ρ
μ
α
2
g ρ .
2e−2δθt ω μ
b
2ρ h
2ρ 1
σ
2
ρ
3.57
Applying Gronwall’s inequality to 3.57, we obtain that for t > 0,
μ xm t, ω
2
ρ
α ym t, ω
2
ρ
t
2
s ωds
μ
x0
2ρ α y0 ρ
≤ e−λ1 t 0 2δθs ω2qηθ
t
μ
α
2
2
2
s ωds
g ρ e−λ1 t 0 2δθs ω2qηθ
2 μ
b
ρ h
ρ 1
σ
t
s
r ωdr
· eλ1 s−2δθs ω− 0 2δθr ω2qηθ
ds.
0
3.58
2 For any z0 ∈ H and h, g ∈ H, let {z0n } ⊂ E and {hn , gn } ⊂ E be sequences such
that
z0n − z0 E −→ 0,
hn − h
ρ −→ 0,
gn − g
ρ
−→ 0
as n −→ ∞.
3.59
Then zm t; ω, z0n , hn , gn → zm t; ω, z0 , h, g as n → ∞ in H, and 3.58 holds for z0 ∈ H.
Therefore,
μ xm t, θ−t ω, z0 θ−t ω
≤e
t
2
ρ
α ym t, θ−t ω, z0 θ−t ω
−λ1 t 0 2δθs−t ω2qηθ
s−t ωds
μ
x0 θ−t ω
2ρ
2
ρ
α y0 θ−t ω
2
ρ
1 2
r1ρ
ωe−2δω ,
2
3.60
where
μ
α
2
r1ρ
g
ω 4e2δω μ
b
2ρ h
2ρ 1
σ
2
ρ
0
−∞
0
r ωdr
eλ1 s−2δθs ω s 2δθr ω2qηθ
ds < ∞.
3.61
16
International Journal of Differential Equations
For any β > 0, since
−1
2 μ
α
2
g ρ
μ
b
2ρ h
2ρ 1
σ
−t
−t
t → ∞
r ω|dr
4e−2βtδθ−t ω
eλ1 st−2δθs ω s 2δθr ω2q|ηθ
ds −→ 0,
e−βt r0ρ θ−t ω
3.62
−∞
then r1ρ ω is tempered.
Let zt, ω ψt, ωz0 ω zt; ω, z0 ω, h, g be a solution of equation 3.9 with
z0 ω e−δω u0 , v0 ∈ H, where ω ∈ Ω1 and h, g ∈ H, then zt, ω ∈ H, and there exists
a subsequence zmk t, ω converging to zt, ω as mk → ∞ for all t ≥ 0. Inequality 3.60 still
holds after replacing zm t, ω by zt, ω since the right hand of 3.60 is independent of m.
Thus for u0 , v0 ∈ Dθ−t ω,
z0 θ−t ω x0 θ−t ω, y0 θ−t ω e−δθ−t ω u0 , v0 ,
μ
ut, θ−t ω, u0 , v0 2ρ α
vt, θ−t ω, u0 , v0 2ρ
t
s−t ωds
≤ e2δω e−λ1 t−2δθ−t ω 0 2δθs−t ω2qηθ
sup
u,v∈Dθ−t ω
μ
u
2ρ α
v
2ρ
3.63
1 2
r1ρ
ω.
2
> 0. By properties of ηθ±t ω and D ∈ DH, we have
Let 2 λ1 − 2qE|ηω|
0
s ωds
−t ω −t 2δθs ω2qηθ
e−2 /2 2qE|ηω|t−δθ
−→ 0
as t −→ ∞,
3.64
and hence
0
s ωds
lim e2δω e−λ1 t−δθ−t ω −t 2δθs ω2qηθ
t → ∞
sup
u,v∈Dθ−t ω
μ
u
2ρ α
v
2ρ 0.
3.65
1
Denote by r
1ρ r1ρ / min{μ, α}, it follows that
B1ρ ω u, v ∈ H : u, v
H ≤ r
1ρ ω BH 0, r
1ρ ω ⊂ H
3.66
is a tempered closed random absorbing set for {ϕt, ω}t≥0,ω∈Ω1 .
Theorem 3.7. Assume that (P0), (H1)–(H5) hold, then the RDS {ϕt, ω}t≥0,ω∈Ω1 generated by 3.1
possesses a unique global random D attractor given by
A1 ω τ≥TB1ρ ω t≥τ
ϕt, θ−t ωB1ρ θ−t ω ⊂ H.
3.67
Proof. According to Theorem 2.2, it remains to prove the asymptotically nullness of
{ϕt, ω}t≥0,ω∈Ω1 ; that is, for any ε > 0, there exists T ε, ω, B1ρ > T1ρ ω and Iε, ω ∈ N
International Journal of Differential Equations
17
such that when t ≥ T ε, ω, B1ρ , the solution ϕt, ωu0 , v0 ui , vi t; ω, u0 , v0 i∈Z ∈ H of
3.1 with u0 , v0 ∈ B1ρ θ−t ω satisfies
2
ρi ϕt, ωu0 , v0 i |i|≥Iε,ω
ρi |ui , vi t; ω, u0 , v0 |2 ≤ ε.
3.68
|i|≥Iε,ω
Choose a smooth increasing function ξ ∈ C1 R , 0, 1 such that
0 ≤ ξs ≤ 1,
ξs 0, 0 ≤ s ≤ 1,
ξ s ≤ C0 constant
1 ≤ s ≤ 2,
ξs 1,
for s ∈ R ,
3.69
s ≥ 2.
Let u, vt; ω, u0 , v0 , h, g ui , vi t; ω, u0 , v0 , h, gi∈Z be a solution of 3.1, then
z t; ω, z0 ω, h, g zi t; ω, z0 ω, h, g i∈Z e−δθt ω ϕt, ωu0 , v0 3.70
is a solution of 3.9 with z0 ω e−δω u0 , v0 ∈ H.
Let z0n Tn z0 , hn , gn Tn h, g, where Tn is as in 3.52. Then z0n ∈
E, hn , gn ∈ E and zt; ω, z0n , hn , gn → zt; ω, z0 , h, g in H. For any n ≥ 1, let zm t zm t; ω, z0n ω, hn , gn be the solution of 3.28, where zm 0 z0n ω. By Theorem 3.4,
1
zm · ∈ C0, ∞, E ∩ C
0, ∞, E. Let M be asuitable large integer will be specified
later; multiply 3.31 by
m
μρi ξ|i|/Mxi
0
0
m
αρi ξ|i|/Myi
and sum over i ∈ Z, we obtain
d |i|
m 2
m 2
ρi μ xi
ξ
α yi
dt i∈Z M
. |i|
m 2
m 2
≤ −λ1 2δθt ω 2qηθ
ρi μ xi
t ω · ξ
α yi
M
i∈Z
μ
cC0 m 2
α 2
x
ηθt ω ·
2e−2δθt ω
ρi μbi2 hn 2i gn i .
ρ
M
1
σ
|i|≥M
3.71
Applying Gronwall’s inequality to 3.71 from T1ρ TB1ρ ω to t gives
|i| m 2
2
m ξ
α yi
t; ω, z0n , hn , gn
ρi μ xi t; ω, z0n , hn , gn
M
i∈Z
t
2
2
−λ1 t−T1ρ T 2δθs ω2qηθ
s ωds
1ρ
≤e
· μ xm T1ρ
α ym T1ρ ρ
ρ
t
t
μ
α
2
s ωds−2δθτ ω
ρi μbi2 hn 2i e−λ1 t−τ τ 2δθs ω2qηθ
dτ
gn i
σ
1
T1ρ
|i|≥M
2
cC0 t −λ1 t−τ t 2δθs ω2q|ηθ
s ω|ds−2δθτ ω
τ
e
ηθτ ω μ xm τ α ym τ
ρ
M T1ρ
2
ρ
dτ.
3.72
18
International Journal of Differential Equations
Therefore for u0 , v0 ∈ B1ρ θ−t ω ∩ H,
|i| m 2
2
m ρi μ xi t; θ−t ω, z0n , hn , gn
ξ
α yi
t; θ−t ω, z0n , hn , gn
M
i∈Z
t
2
2
−λ1 t−T1ρ T 2δθs−t ω2q|ηθ
s−t ω|ds
1ρ
≤e
μ xm T1ρ α ym T1ρ ρ
ρ
2
34
5
ρi
|i|≥M
2
cC0
M
2
i
μ
α 2
μbi2 hn 2i gn i
1
σ
t
T1ρ
t
s−t ω|ds−2δθτ−t ω
e−λ1 t−τ τ 2δθs−t ω2q|ηθ
dτ
34
5
ii
t
e
t
−λ1 t−τ τ 2δθs−t ω2q|ηθ
s−t ω|ds−2δθτ−t ω
T1ρ
ηθτ−t ω μ x
m
τ
34
2
ρ
α y
m
τ
2
ρ
dτ .
iii
5
3.73
We next estimate terms i, ii, iii on the right-hand side of 3.73. By 3.61,
e
−λ1 t−T1ρ t
s−t ωds
T1ρ 2δθs−t ω2qηθ
μ xm T1ρ
2
ρ
α ym T1ρ
2
ρ
−→ 0 as t −→ ∞,
3.74
which implies that for all ε > 0, there exists T1 ε, ω, B1ρ ≥ T1ρ such that for t ≥ T1 ε, ω, B1ρ ,
i ≤
ε
min μ, α e−2δω .
3
3.75
By h, g, b ∈ lρ2 and
t
e
t
−λ1 t−τ τ 2δθs−t ω2qηθ
s−t ωds−2δθτ−t ω
T1ρ
dτ ≤
0
−∞
0
s ωds−2δθτ ω
eλ1 τ− τ δθs ω2qηθ
dτ < ∞,
3.76
there exists I1 ε, ω ∈ N such that for M > I1 ε, ω,
ii ≤
ε
min μ, α e−2δω .
3
3.77
Note that 2 λ1 − 2qE|ηω|
> 0, then by H2, ηω is tempered and it follows that
there exists a T1 > 0 such that
ηθτ−t ω ≤ e2 /3t−τ ,
∀t − τ ≥ T1 .
3.78
International Journal of Differential Equations
19
Therefore
t
t
s−t ωds−2δθτ−t ω
e−λ1 t−τ τ 2δθs−t ω2qηθ
ηθτ−t ω μ xm τ
T1ρ
≤ μ
x0 2ρ α y0
2
ρ
1 2
r1ρ
ωe−2δω
2
0
s ωds
e−λ1 −2 /3t−4δθ−t ω −t 2δθs ω2qηθ
t−T1ρ
2
ρ
α ym τ
t
e−2 /3τ dτ
2
ρ
dτ
3.79
T1ρ
τ
s ωds
e−λ1 −ε2 /3τ 0 2δθs ω2qηθ
dτ.
0
For t T1ρ , by H2 and 3.6, there exists 0 < T2 T2 ω < t − T1ρ such that
1
τ
τ
.
22
,
s ω ds < λ1 −
2δθs ω 2qηθ
3
0
for τ ≥ T2 .
3.80
Let T2 max{T1 , T2 } < ∞, and t > T2 T1ρ , write
t−T1ρ
e
τ
−λ1 −2 /3τ 0 2δθs ω2qηθ
s ωds
dτ / T
0
2
t−T1ρ 0
0
τ
s ωds
e−λ1 −2 /3τ 0 2δθs ω2qηθ
dτ,
T2
3.81
of which
T2
t−T1ρ
τ
s ωds
e−λ1 −2 /3τ 0 2δθs ω2qηθ
dτ < ∞,
0
e
τ
−λ−2 /3τ 0 2δθs ω2qηθ
s ωds
T2
3
dτ ≤ e−2 /3T2 .
2
3.82
Equation 3.79 together with 3.82 implies that there exist T3 ε, ω, B1ρ ≥ T2 and I2 ε, ω ∈ N
such that for M > I2 ε, ω, t ≥ T3 ε, ω, B1ρ ,
iii ≤
ε
min μ, α e−2δω .
3
3.83
In summary, let
T ε, ω, B1ρ max T1 ε, ω, B1ρ , T2 ε, ω, B1ρ , T3 ε, ω, B1ρ ,
Iε, ω max{I1 ε, ω, I2 ε, ω}
3.84
Then for t > T ε, ω, B1ρ and M ≥ Iε, ω, we have
|i|≥2M
2
2
m m ρi μ xi t; θ−t ω, z0n , hn , gn
α yi
≤ ε min μ, α e−2δω .
t; θ−t ω, z0n , hn , gn
3.85
20
International Journal of Differential Equations
mk Since zi
|i|≥2M
t; θ−t ω, z0n θ−t ω, hn , gn → zi t; θ−t ω, z0n θ−t ω, hn , gn as mk → ∞, by 3.85,
2
2
ρi ϕt, ωu0 , v0 i ρi e2δω zi t; θ−t ω, z0n , gn , hn |i|≥2M
≤
2δω
e
min μ, α |i|≥2M
2
2
m m ρi μ xi t; θ−t ω, z0n , hn , gn
α yi
t; θ−t ω, z0n , hn , gn
≤ ε.
3.86
Letting n → ∞ in 3.86, we obtain
|i|≥2M
2
ρi ϕt, ωu0 , v0 i ≤ ε
3.87
That is, {ϕt, ω}t≥0,ω∈Ω1 is asymptotically null on B1ρ ω, which completes the proof.
4. Stochastic Partly Dissipative Lattice Systems with Additive
White Noise in Weighted Spaces
This section is devoted to the study of asymptotic behavior for system 1.2 in weighted space
H lρ2 × lρ2 . The structure and the idea of proofs are similar to that of Section 3, and we will
present our major results without elaborting the details of proofs in this section.
4.1. Mathematical Setting
Define Ω2 {ω ∈ CR, l2 : ω0 0}, and denote by F2 the Borel σ-algebra on Ω2 generated
by the compact open topology 1 and P2 is the corresponding Wiener measure on F2 , then
Ω2 , F2 , P2 , θt t∈R is a metric dynamical system. Let the infinite sequence ei i ∈ Z denote
the element having 1 at position i and 0 for all other components. Write
W 1 t, ω ai wi tei ,
W 2 t, ω i∈Z
bi wi tei ,
i∈Z
4.1
where {wi t : i ∈ Z} are independent two-sided Brownian motions on probability space
Ω2 , F2 , P2 ; then W 1 t, ω and W 2 t, ω are Wiener processes with values in l2 defined on the
probability space Ω2 , F2 , P2 .
Consider stochastic lattice system 1.2 with random coupled coefficients and additive
independent white noises:
u̇ −λu Aθt ωu − fu αv h v̇ −σv μu g dW 2 t
,
dt
dW 1 t
,
dt
i ∈ Z, t > 0,
4.2
International Journal of Differential Equations
21
where ui , vi , hi , gi , ai , bi ∈ R; u ui i∈Z , v vi i∈Z , fu fi ui i∈Z , g gi i∈Z , h hi i∈Z ,
a ai i∈Z ∈ l2 , b bi i∈Z ∈ l2 , fi ∈ C1 R, R i ∈ Z; λ, α, σ, μ are positive constants;
ηi,−q ω,. . ., ηi,0 ω,. . .,ηi,q ω, i ∈ Z, q ∈ N are random variables; A is defined as in 3.2.
To transform 4.2 into a random equation without white noise, let
δ1 θt ω −λ
0
−∞
eλs θt ωsds,
δ2 θt ω −σ
0
−∞
eσs θt ωsds,
t ∈ R, ω ∈ Ω2 .
4.3
Then δ1 θt ω, δ2 θt ω are both Ornstein-Uhlenbeck processes on Ω2 , F2 , P2 and solve the
following Ornstein-Uhlenbeck equations see 1, respectively,
dδ1 λδ1 dt dW 1 t,
dδ2 σδ2 dt dSW 2 t,
t ≥ 0.
4.4
2 ∈ F2 of Ω2 of full P measure such that for
Lemma 4.1 see 1. There exists a θt -invariant set Ω
ω ∈ Ω2 ,
i limt → ±∞ ωt
/t 0;
ii the random variables δj ω
are tempered and the mappings
t, ω −→ δj θt ω ∈ l2 ,
4.5
j 1, 2,
are stationary solutions of Ornstein-Uhlenbeck equations 4.4 in l2 with continuous
trajectories;
iii
lim
t → ±∞
δj θt ω
1
lim
t → ±∞ t
t
t
δj θs ωds 0,
j 1, 2.
4.6
0
2 , still
In the following, we consider the completion of the probability space ω ∈ Ω
denoted by Ω2 , F2 , P2 ,
Let
xt,
ω ut, ω − δ1 θt ω,
yt,
ω vt, ω − δ 2 θt ω,
ω ∈ Ω2 , t ∈ R.
4.7
then system 4.2 becomes the following random system with random coefficients but
without white noise:
dx
−λx
Aθt ωx
− f x
δ1 θt ω − αx
Aθt ωδ1 θt ω − αδ2 θt ω h,
dt
dy
−σ y
μx
μδ1 θt ω g, i ∈ Z, t > 0.
dt
In addition, we make the following assumptions on functions fi , i ∈ Z:
4.8
22
International Journal of Differential Equations
H6 fi ∈ C1 R, R satisfy
sfi s ≥ μs2p1 − di2 ,
fi s ≤ df |s| |s|2p 1 ,
∀s ∈ R, i ∈ Z,
4.9
where μ, di , df are positive constants, p ∈ N, and d di i∈Z ∈ l2 .
4.2. Random Dynamical System Generated by Random Lattice System
Denote by z
x,
y,
we have the following.
Theorem 4.2. Let T > 0 and assume that (P0), (H1), (H2), (H4), and (H6) hold. Then for every
ω ∈ Ω2 and any initial data z
0 x
0 , y
0 ∈ H, problem 4.8 admits a unique solution z
·; ω, z
0 ∈
C0, T , H with z
0; ω, z
0 z
0 .
Proof. Similar to the proof of Theorem 3.3.
Theorem 4.3. Assume that (P0), (H1), (H2), (H4), and (H6) hold. Then system 4.8 generates a
continuous RDS {ψt,
ω}t≥0,ω∈Ω2 over Ω2 , F2 , P2 , θt t∈R with state space H:
ψt,
ωz
0 : z
t; ω, z
0 ,
for z
0 ∈ H, t ≥ 0, ω ∈ Ω2 .
4.10
1
u0 − δ1 ω
δ θt ω
,
δ2 θt ω
v0 − δ2 ω
4.11
Moreover,
ϕt,
ωu0 , v0 : ψt,
ω
ω}t≥0,ω∈Ω2 over
where u0 , v0 ∈ H,t ≥ 0, ω ∈ Ω2 , defines a continuous RDS {ϕt,
Ω2 , F2 , P2 , θt t∈R associated with 4.2.
Proof. It follows immediately from similar arguments to the proof of Theorem 3.4.
4.3. Existence of Tempered Bounded Random Absorbing Set and Random
Attractor in Weighted Space
In this subsection, we study the existence of a tempered random bounded absorbing set and
a global random attractor for the random dynamical system {ϕt,
ω}t≥0,ω∈Ω2 generated by
4.2 in weighted space H.
Theorem 4.4. Assume that (P0), (H1)–(H4), and (H6) hold. Then
a there exists a closed tempered bounded random absorbing set B2ρ ω ∈ DH of RDS
{ϕt,
ω}t≥0,ω∈Ω2 such that for any D ∈ DH and each ω ∈ Ω2 , there exists T
D ω >
0 yielding ϕt,
θ−t ωDθ−t ω ⊂ B2ρ ω, for all t ≥ T
D ω. In particular, there exists
θ−t ωB2ρ θ−t ω ⊂ B2ρ ω, for all t ≥ T2ρ ω;
T2ρ ω > 0 such that ϕt,
International Journal of Differential Equations
23
b the RDS {ϕt,
ω2 }t≥0,ω∈Ω2 generated by equations 4.2 possesses a unique global random
D attractor given by
A2ρ ω ϕt,
θ−t ωB2ρ θ−t ω ∈ H.
τ≥T2ρ ω t≥τ
4.12
Proof. Similar to the proofs of Theorems 3.6 and 3.7.
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