asymptotic behavior of stochastic discrete complex ginzburg–landau equations

Physica D 221 (2006) 157–169www.elsevier.com/locate/physd

Asymptotic behavior of stochastic discrete complex Ginzburg–Landauequations

Yan Lv∗, Jianhua Sun

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received 8 November 2005; received in revised form 17 June 2006; accepted 28 July 2006Available online 22 August 2006

Communicated by J. Lega

Abstract

The complex Ginzburg–Landau equation is one of the most-studied equations in applied mathematics. We consider the discretization ofcomplex Ginzburg–Landau equations on one dimensional lattice driven by a general Gaussian random field including the translation invariantone. The long time behavior of the sample paths and the distributions of solutions are studied respectively. Under the gauge nonlinear interaction,the dynamical behavior for the sample paths of the system is described by a global random attractor which is a random compact invariant set in aweighted Hilbert space. Furthermore the distributions of the system exponentially converge to the unique invariant measure of the system, that isthe system is ergodic. The asymptotic compactness and dissipative method are important in our approach.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Discrete Ginzburg–Landau equations; White noise; Random dynamical systems; Asymptotically compact; Random attractor; Ergodic; Invariant measure

1. Introduction

The complex Ginzburg–Landau equation is an important model in nonlinear science. It is encountered in several diverse branchesof physics, for example in the description of spatial pattern formation and of instabilities in non-equilibrium fluid dynamical systemsand chemical systems; superconductivity and superfluidity; nonlinear optics and Bose–Einstein condensates, see [1]. A most typicalequation with dissipative term is described as

∂u

∂t− (λ + iα)1u + (κ + iβ)|u|

pu = g(x) (1.1)

on a bounded open set of Rn . A large amount of work has been devoted to the study of the well-posedness of solutions, the existenceof global attractors and related dynamical issues [13,28].

Quite recently, spatially discrete systems have drawn considerable attention, especially in the study of biological systems[21], atomic chains [22,23], solid state physics [25], electrical lattices [15,16] and Bose–Einstein condensates [29]. And differentdynamical behavior of spatially discrete systems has been studied in many works, such as [7–9] for travelling waves solutions andchaos behavior and [3,30] for global attractors.

In this paper we focus on a discrete Ginzburg–Landau type model called a DGL lattice. It is quite often used to describe a numberof physical systems, such as Taylor and frustrated vortices in hydrodynamics [27] and laser arrays and semiconductor opticalamplifiers in optics [17,26]. However, scientific and engineering systems are often subject to uncertainty or random influence.Randomness can have a delicate impact on the overall evolution of these systems [2]. And in statistical mechanics the classical spin

∗ Corresponding author. Tel.: +86 1395 1685432.E-mail addresses: [email protected] (Y. Lv), [email protected] (J. Sun).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.07.023

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http://dx.doi.org/10.1016/j.physd.2006.07.023

158 Y. Lv, J. Sun / Physica D 221 (2006) 157–169

system is related to a differential equation on lattices driven by white noise, see Chapter 12 of [11]. Therefore, it is significant andof prime importance to introduce random effects in these models. For example [24] has studied the Bose–Einstein condensation farfrom thermal equilibrium by solving the complex Ginzburg–Landau equation with a stochastic term.

In this paper we concern ourselves with the following discrete Ginzburg–Landau equations with white noise

du j (t)

dt= (λ + iα)(u j−1 − 2u j + u j+1) − (γ + iδ)u j − F(u j ) + g j + a j

dw j (t)

dt, j ∈ Z, t > 0, (1.2)

u j (0) = u j,0, j ∈ Z (1.3)

where i =√

−1, λ, α, γ , δ ∈ R and u j , g j , a j ∈ C, j ∈ Z, with C is the set of complex numbers. w j : j ∈ Z are independent two-sided real valued standard Wiener processes. The nonlinear interaction is the gauge interaction of the form F(s) = (κ + iβ)|s|p−1sor the non-gauge interaction of the form F(s) = (κ + iβ)|s|p, for some p > 1, κ , β ∈ R. For more see [12] in which the asymptoticbehavior of system (1.2) with the gauge interaction on a one dimensional lattice Z for a j = 0 is studied and the global attractor isobtained in a weighted Banach space.

For the translation invariant assumption in statistical mechanics, a j should not decay away from the origin, which means thatW (t) = a jw j (t) j∈Z is not a Q-Wiener process in `2, the space of all summable complex sequences, see the definition in Section 2.In fact Q = diaga2

j can be any nonnegative definite operator on `2 in this paper. So noise in model (1.2) is a general Gaussianrandom field including the translation invariant one, see assumption (2.4) in Section 2.

Global attractor theory, which has been well developed, is an important tool to study the asymptotic behavior of a deterministicinfinite dimensional system, see [28] and others. The study of global random attractors was initiated by Ruelle [19]. The fundamentaltheory was developed in [5,6,20] and others. In [4] the random attractor for the system on a one dimensional lattice driven by whitenoise was obtained and [14] further generalized this result to higher dimensional lattices. On the other hand a random dynamicalsystem possessing a random attractor has at least one invariant measure [5]. Uniqueness of the invariant measure implies theergodicity of the system which is an important property in physics and mechanics. For a general method to prove the ergodicity ofinfinite dimensional stochastic systems, we refer to [11].

In this paper, we first apply the random dynamical system theory to study the dynamical behavior of system (1.2) with the gaugeinteraction term, that is we prove the existence of a global random attractor in a weighted space `2

ρ . And since the nonlinear termis not locally Lipschitz in the space `2

ρ , a cut-off system and a random stopping time are introduced to obtain the existence ofglobal solutions. Since the flow defined by the system lacks smoothness on the infinite lattice, asymptotic compactness of the flowis needed in our approach. Second, we use the dissipative method to prove that all the distributions of the system converge to aninvariant measure of the system exponentially, which means the ergodicity of system (1.2).

The rest of the paper is organized as follows. Some notations and descriptions of the system are given in Section 2. In Section 3,the wellposedness and global existence of system (1.2) is obtained and then it is proved in Section 4 that the random dynamicalsystem generated by system (1.2) possesses a global random attractor. The ergodicity of the system is studied in the last section.

2. Stochastic discrete Ginzburg–Landau equations

In this section we give some notations and descriptions of the system with which we are concerned. For 2 ≤ q < ∞, let `q bethe space of sequences u j ∈ C : j ∈ Z with norm defined by

‖u‖q`q =

∑j∈Z

|u j |q < ∞.

For q = 2 we get the Hilbert space `2 with inner product

〈u, v〉 = Re∑j∈Z

u jv j , u, v ∈ `2,

and the norm is denoted by ‖ · ‖.For our purpose we consider a larger space in which our system is well defined.Introduce a family of weighted functions

ρ j =1

1 + ν| j |2, j ∈ Z, (2.1)

where ν > 0 will be determined later. Then we introduce the weighted complex functional space

`qρ =

u = (u j ) j∈Z : u j ∈ C, j ∈ Z and ‖u‖`qρ

=

∑j∈Z

ρ j |u j |q

1q

< ∞

.

Y. Lv, J. Sun / Physica D 221 (2006) 157–169 159

For q = 2, `qρ is a Hilbert space `2

ρ with inner product

〈u, v〉ρ = 〈u, ρv〉 = Re∑j∈Z

ρ j u jv j , u, v ∈ `2ρ,

and corresponding norm is denoted by

‖u‖ρ = 〈u, ρu〉12 .

Consider the following discrete Ginzburg–Landau equations with the gauge interaction on the infinite lattice Z driven by whitenoise (SDGL)

du j (t)

dt= (λ + iα)(u j−1 − 2u j + u j+1) − (γ + iδ)u j − (κ + iβ)|u j |

pu j + g j + a jdw j (t)

dt, j ∈ Z, t > 0, (2.2)

u j (0) = u j,0, j ∈ Z (2.3)

where p > 0, λ > 0, γ > 0, κ > 0, α, δ and β ∈ R, u = (u j ) j∈Z ∈ `2ρ , g = (g j ) j∈Z ∈ `2

ρ , a = (a j ) j∈Z ∈ `2ρ and w j : j ∈ Z

are mutually independent two-sided real Brownian motions.For u ∈ `2

ρ we define the operator A : `2ρ → `2

ρ as

(Au) j = −u j−1 + 2u j − u j+1, j ∈ Z.

It is easy to check that A is a bounded operator on `2ρ , see [14]. Then, by the perturbation theory of operator [18], −(λ + iα)A −

(γ + iδ) is a generator of a C0 semigroup S(t).Let

W (t) =

∑j∈Z

a jw j (t)e j , (a j ) j∈Z ∈ `2ρ .

Here e j denotes the element having 1 at position j and all the other components 0. Then W (·) is an `2ρ-valued Q-Wiener process

with Q = diag(. . . , a2j , . . .). In the physically natural viewpoint a j should not decay as j → ∞ and W (t) should be translation

invariant. More generally we assume

Q : `2→ `2 is a nonnegative bounded operator on `2. (2.4)

In our approach W (t) can be a more general Gaussian random field with Q not diagonal. For more general representations of Qsee [11].

By the above notations and assumption, (2.2) can be written as

du

dt= −(λ + iα)Au − (γ + iδ)u − (κ + iβ) f (u) + g + W (2.5)

u(0) = (u j,0) j∈Z = u0 (2.6)

where the nonlinear term f is defined as

f (u(t)) = |u(t)|pu(t) =|u j (t)|

pu j (t)

j∈Z .

We end this section by a result on W (t).

Lemma 2.1. W (t) is an `qρ-valued Wiener process for any q ≥ 1.

Proof. Since w j (t), j ∈ Z, are mutually independent real valued Wiener processes, by the Gaussian property for any q ≥ 1 thereis cq > 0 such that

E|a jw j (t)|q

= cq

(E|a jw j (t)|

2) q

2= cq t

q2 |a j |

q , t ≥ 0,

we have

E‖W (t)‖q`

qρ

= cq tq2∑

j

ρ j |a j |q , t ≥ 0.

By assumption (2.4) we have

maxj∈Z

|a j | < ∞,


then

E‖W (t)‖q`

qρ

< ∞

and W (t), t ≥ 0, is an `qρ-valued Wiener process.

3. Wellposedness of the SDGL

We first give the global existence and uniqueness of a solution for system (2.2).For our purpose we first introduce the following probability space (Ω ,F, P) as

Ω = ω ∈ C(R; `2ρ) : ω(0) = 0

endowed with compact-open topology, P is the corresponding Wiener measure and F is the P-completeness of Borel σ -algebra onΩ . Let

θtω(·) = ω(· + t) − ω(t), t ∈ R,

then (Ω ,F, P, (θt )t∈R) is a metric dynamical system with the filtration

Ft :=

∨s≤tF t

s , t ∈ R,

where

F ts = σ W (t2) − W (t1) : s ≤ t1 ≤ t2 ≤ t

is the smallest σ -algebra generated by random variable W (t2) − W (t1) for all s ≤ t1 ≤ t2 ≤ t . Since the above probability space iscanonical we can define the Wiener process and its shift operator

W (t) = ω(t), W (t, θsω) = ω(t + s) − ω(s) = W (t + s) − W (s). (3.1)

For the details we refer to [2].Write system (2.5) in the mild sense

u(t) = S(t)u0 −

∫ t

0S(t − s)(κ + iβ)|u(s)|pu(s)ds +

∫ t

0S(t − s)gds +

∫ t

0S(t − s)dW (s). (3.2)

Notice that |u|pu is not locally Lipschitz on `2

ρ . So we cannot have the local existence of solution by the classic result of theexistence of a solution. For this we introduce a cutoff function. Let PM : R+

→ R+ be a smooth function satisfying PM (x) = 1 ifx < M and PM (x) = 0 if x > M + 1. Then we have the following system with globally Lipschitz nonlinear term

uM (t) = S(t)u0 −

∫ t

0S(t − s)(κ + iβ)PM (‖uM (s)‖∞)|uM (s)|puM (s)ds +

∫ t

0S(t − s)gds +

∫ t

0S(t − s)dW (s), (3.3)

with ‖uM (t)‖∞ = max j∈Z |uMj (t)|. Define the random stopping time τ(R) by

τ(R) = inft > 0 : ‖uM (t)‖∞ ≥ R.

Fix arbitrarily a positive number R < M and denote by χI the characteristic function of the set I . Consider the following integralequation for t < τ(R)

uM (t) = S(t)u0 −

∫ t

0S(t − s)(κ + iβ)PM (‖uM (s)‖∞)|uM (s)|puM (s)χs≤τ(R)ds

+

∫ t

0S(t − s)gχs≤τ(R)ds +

∫ t

0S(t − s)χs≤τ(R)dW (s). (3.4)

We have the following simple result for the above cutoff equation.

Lemma 3.1. For any T > 0 system (3.4) is wellposed and has a unique solution uM (t) ∈ C(0, T ; `2ρ) almost surely, which is

independent of M > R and satisfies (3.2) for t < τ(R).

The proof of the above lemma is standard, see [10].Now we prove the existence and uniqueness of a solution for system (2.5). The following two lemmas are necessary, for the

proof we refer to [11].


Lemma 3.2. For the weighted functions given by (2.1) the following result holds

limν→0

∣∣∣∣√ρ j+1

ρ j− 1

∣∣∣∣ = 0.

Lemma 3.3. For any ε > 0 there is ν > 0 such that

〈−(λ + iα)Au, u〉ρ ≤ ε‖u‖2ρ, u ∈ `2

ρ .

Now we take ν0 > 0 such that for ν ∈ (0, ν0)

〈−(λ + iα)Au, u〉ρ ≤γ

2‖u‖

2ρ, u ∈ `2

ρ .

The following property of the nonlinear term is also needed in our approach. Let u =(|u j |(cos η j + i sin η j )

)j∈Z, v =(

|v j |(cos ξ j + i sin ξ j ))

j∈Z with η j , ξ j are the arguments of complex numbers u j , v j respectively, then

〈 f (u) − f (v), u − v〉ρ = 〈|u|pu − |v|

pv, u − v〉ρ

=

∑j∈Z

ρ j |u j |p+2

+

∑j∈Z

ρ j |v j |p+2

− Re∑j∈Z

ρ j |u j |pu jv j − Re

∑j∈Z

ρ j |v j |pv j u j

=

∑j∈Z

ρ j |u j |p+2

+

∑j∈Z

ρ j |v j |p+2

− Re∑j∈Z

ρ j |u j |p|u j |(cos η j + i sin η j )|v j |(cos ξ j − i sin ξ j )

− Re∑j∈Z

ρ j |v j |p|v j |(cos ξ j + i sin ξ j )|u j |(cos η j − i sin η j )

=

∑j∈Z

ρ j |u j |p+2

+

∑j∈Z

ρ j |v j |p+2

−

∑j∈Z

ρ j |u j |p+1

|v j | cos(η j −ξ j )−∑j∈Z

ρ j |v j |p+1

|u j | cos(η j −ξ j )

≥

∑j∈Z

ρ j |u j |p+2

+

∑j∈Z

ρ j |v j |p+2

−

∑j∈Z

ρ j |u j |p+1

|v j | −

∑j∈Z

ρ j |v j |p+1

|u j |

=

∑j∈Z

ρ j

((|u j |

p+1− |v j |

p+1)(|u j | − |v j |))

≥ 0.

By the definition of cutoff function and the stopping time it is easy to see that uM is a F-measurable process on Ω . Then for thediscrete stochastic Ginzburg–Landau equations (2.2) we have the following well-posedness result.

Theorem 3.4. For any T > 0 and u0 ∈ `2ρ , there is a unique solution u ∈ L2(Ω; C(0, T ; `2

ρ)) of system (2.5). Moreover the mapu0 7→ u(·, ω, u0) is continuous a.s. under the norm of C(0, T ; `2

ρ).

Proof. We just prove that τ(R) → ∞ as R → ∞.Let vM

= uM− W then for t < τ(R)

dvM (t)

dt= −(λ + iα)AvM

− (γ + iδ)vM− (κ + iβ)PM (‖uM

‖∞) f (vM+ W ) + g − (λ + iα)AW − (γ + iδ)W (3.5)

vM (0) = u0. (3.6)

We fix a ω ∈ Ω . Taking the inner product of (3.5) with vM in `2ρ , by the definition of 〈·, ·〉ρ and the choice of ν, we have

12

d‖vM‖

2ρ

dt= 〈−(λ + iα)AvM , ρvM

〉 − 〈(γ + iδ)vM , ρvM〉 − 〈(κ + iβ)PM (‖uM

‖∞) f (vM+ W ), ρvM

〉

+ 〈g, ρvM〉 − 〈(λ + iα)AW, ρvM

〉 − 〈(γ + iδ)W, ρvM〉

≤ −γ

2‖vM

‖2ρ − 〈(κ + iβ)PM (‖uM

‖∞) f (vM+ W ), ρvM

〉

+ 〈g, ρvM〉 − 〈(λ + iα)AW, ρvM

〉 − 〈(γ + iδ)W, ρvM〉. (3.7)

Further by the properties of A, Cauchy–Schwartz inequality and Young inequality, we have that there are some positive constantsc, c1, c2, c3, c4 depending on p, κ , β, γ , δ, λ, α such that the following estimates hold∣∣∣∣∣∣〈(κ + iβ)PM (‖uM

‖∞) f (vM+ W ), ρvM

〉 − κ PM (‖uM‖∞)

∑j∈Z

ρ j |vMj |

p+2

∣∣∣∣∣∣


= PM (‖uM‖∞)

∣∣∣∣∣∣∑j∈Z

ρ j Re[(κ + iβ)|vM

j + a jw j |p(vM

j + a jw j )vMj

]− κ

∑j∈Z

ρ j |vMj |

p+2

∣∣∣∣∣∣≤ PM (‖uM

‖∞)

∣∣∣∣∣∣κ∑j∈Z

ρ j |vMj |

2||vM

j + a jw j |p

− |vMj |

p| +

∑j∈Z

ρ j Re[(κ + iβ)|vM

j + a jw j |pa jw j v

Mj

]∣∣∣∣∣∣≤ PM (‖uM

‖∞)

∣∣∣∣∣∣c∑j∈Z

ρ j |vMj |

2∣∣∣|vM

j + a jw j | − |vMj |

∣∣∣ (|a jw j |p−1

+ |vMj |

p−1)

+ c∑j∈Z

ρ j

(|vM

j |p+1

|a jw j | + |a jw j |p+1

|vMj |

)∣∣∣∣∣∣≤ PM (‖uM

‖∞)

∣∣∣∣∣∣c∑j∈Z

ρ j |vMj |

2|a jw j |(|a jw j |

p−1+ |vM

j |p−1) + c

∑j∈Z

ρ j

(|vM

j |p+1

|a jw j | + |a jw j |p+1

|vMj |

)∣∣∣∣∣∣= PM (‖uM

‖∞)

∣∣∣∣∣∣c∑j∈Z

ρ j |vMj |

2|a jw j |

p+ c

∑j∈Z

ρ j |vMj |

p+1|a jw j | + c

∑j∈Z

ρ j |vMj |

p+1|a jw j | + c

∑j∈Z

ρ j |a jw j |p+1

|vMj |

∣∣∣∣∣∣≤ κ PM (‖uM

‖∞)∑j∈Z

ρ j |vMj |

p+2+ c1‖W‖

p+2

`p+2ρ

,

〈g, ρvM〉 ≤

γ

8‖vM

‖2ρ + c2‖g‖

2ρ,

−〈(λ + iα)AW, ρvM〉 ≤

γ

8‖vM

‖2ρ + c3‖W‖

2ρ,

−〈(γ + iδ)W, ρvM〉 ≤

γ

8‖vM

‖2ρ + c4‖W‖

2ρ .

Thus by (3.7) and Lemma 2.1 we have

d‖vM (t)‖2ρ

dt≤ −

γ

4‖vM (t)‖

2ρ + c0(‖g‖

2ρ + ‖W‖

2ρ + ‖W‖

p+2

`p+2ρ

) (3.8)

for some positive constant c0. Then Gronwall’s inequality yields

‖vM (t)‖2ρ ≤ e−

γ4 t

‖u0‖2ρ + c0

∫ t

0e−

γ4 (t−s)(‖g‖

2ρ + ‖W‖

2ρ + ‖W‖

p+2

`p+2ρ

)ds.

Since ‖W‖2ρ +‖W‖

p+2

`p+2ρ

grows at most polynomially, the righthand side of the above inequality is bounded by a continuous function.

Moreover for ω ∈ Ω

supt∈[0,T ]

‖vM (t)‖2ρ ≤ ‖u0‖

2ρ + c0

∫ T

0(‖g‖

2ρ + ‖W‖

2ρ + ‖W‖

p+2

`p+2ρ

)ds,

that is, the solution vM is bounded by a continuous function which is independent of M . Then τ(R) → ∞ as R → ∞. And for anyT > 0, we have a solution u(t) of (2.2) with u(t) = uM (t) for some M and R with M > R and τ(R) > T . The uniqueness andcontinuity on initial values follow those of uM .

The proof is complete.

4. Random dynamics of the SDGL

In this section we study the random dynamics of solutions for SDGL (2.2). It is proved that the long time behavior of (2.2) isdescribed by a global random attractor which is compact in `2

ρ .For convenience we give some basic knowledge of random dynamical systems; for more we refer to [2]. Given the metric

dynamical system (Ω ,F, P, (θt )t∈R) and Hilbert space (H, ‖ · ‖H ). Let E be the expectation operator with respect to P.

Definition 4.1. A stochastic process φ(t, ω) is called a random dynamical system (RDS) over (Ω ,F, P, (θt )t∈R) if φ is (B(R+) ×

F × B(H),B(H))-measurable

φ : R+× Ω × H → H, (t, ω, x) 7→ φ(t, ω, x)


and for all ω ∈ Ω

• φ(0, ω) = id (on H );• φ(t + s, ω, x) = φ(t, θsω, φ(s, ω, x)) for all t, s ≥ 0 and x ∈ H (cocycle property).

RDS φ is continuous or differentiable if φ(t, ω) : H → H is continuous or differentiable.

Definition 4.2. A random bounded set B(ω) ⊂ H is called tempered with respect to (θt )t∈R if for a.e. ω ∈ Ω and all ε > 0

limt→∞

e−εt d(B(θ−tω)) = 0

where d(B) = supx∈B ‖x‖H .

Consider a continuous random dynamical system φ(t, ω) over (Ω ,F, P, (θt )t∈R) and D the collection of all tempered randomsubsets of H .

Definition 4.3. A random set K (ω) is called an absorbing set in D if for all B ∈ D and a.e. ω ∈ Ω there exists tB(ω) > 0 such that∀t ≥ tB(ω)

φ(t, θ−tω, B(θ−tω)) ⊂ K (ω).

Definition 4.4. A random set A(ω) is a random D-attractor for RDS φ if

• A(ω) is a random compact set, i.e., ω 7→ d (x,A(ω)) is measurable for every x ∈ H and A(ω) is compact for a.e. ω ∈ Ω ;• A(ω) is strictly invariant, i.e., φ(t, ω,A(ω)) = A(θtω) ∀t ≥ 0 and for a.e. ω ∈ Ω ;• A(ω) attracts all sets in D, i.e., for all B ∈ D and a.e. ω ∈ Ω we have

limt→∞

d(φ (t, θ−tω, B(θ−tω)) ,A(ω)) = 0,

where d(X, Y ) = supx∈X infy∈Y ‖x − y‖H , X, Y ⊆ H .

The collection D is called the domain of attraction of A.

For our purpose we introduce the following asymptotically compact RDS.

Definition 4.5. Let φ be a RDS on Hilbert space H . φ is called asymptotically compact if for any bounded sequence xn ⊂ H andtn → ∞, the set φ(tn, θ−tn ω, xn) is precompact in H , for any ω ∈ Ω .

For an asymptotically compact RDS φ we have the following result, for the proof see [4].

Theorem 4.6. Let K ∈ D be an absorbing set for an asymptotically compact continuous RDS φ. Then φ has a unique globalrandom D attractor

A(ω) =

⋂τ≥tK (ω)

⋃t≥τ

φ(t, θ−tω, K (θ−tω)),

which is compact in H.

Let φ(t, ω, u0) = u(t, ω, u0), then φ is a continuous RDS on `2ρ by Theorem 3.4. For RDS φ we prove the existence of an

absorbing set in D. In fact we have

Theorem 4.7. There is a random ball K (0, R(ω)) = x ∈ `2ρ : ‖x‖ρ ≤ R(ω)(∈ D) which is an absorbing set in D for RDS φ.

Proof. We still use the estimates in Theorem 3.4. Consider (3.8). Let

p(θtω) = c0(‖g‖2ρ + ‖W (t)‖2

ρ + ‖W (t)‖p+2

`p+2ρ

).

Let v = u − W , then by the proof of Theorem 3.4 we have

‖v(t, ω, u0)‖2ρ ≤ e−

γ4 t

‖u0‖2ρ +

∫ t

0p(θsω)e−

γ4 (t−s)ds.

Replace ω by θ−tω in the above formula to construct the radius of the absorbing set and define

r2(ω) = 4 limt→∞

∫ t

0p(θs−tω)e−

γ4 (t−s)ds = 4 lim

t→∞

∫ 0

−tp(θsω)e

γ4 sds.


Since p grows not faster than polynomials, r(ω) < ∞ a.s. Let R2(ω) = r2(ω)+‖W (t)‖2ρ . Then K (ω) := K (0, R(ω)) is a tempered

ball by the property of W (t) and for any B ∈ D, ω ∈ Ω , there is tB(ω) > 0 such that

φ(t, θ−tω, B(θ−tω)) ⊂ K (ω), ∀t ≥ tB(ω).

This completes the proof.

To apply Theorem 4.6 we need the asymptotic compactness of φ. First we give the following lemma on the weak continuity ofthe random dynamical system φ.

Lemma 4.8. For T > 0, if un converges to u weakly, written as un u, in `2ρ , then ∀ω ∈ Ω , φ(T, ω, un) φ(T, ω, u) in `2

ρ andφ(·, ω, un) φ(·, ω, u) in L2(0, T ; `2

ρ).

The proof is similar to that of Lemma 4.5 in [4]. We omit it here.

Theorem 4.9. φ(t, ω) is an asymptotically compact random dynamical system.

Proof. Since K (ω) is an absorbing set for RDS φ, we just verify the result for sequences in K (ω).Let xn ∈ K (θ−tn ω) and tn ∞ as n → ∞. We show that (φ(tn, θ−tn ω, xn))n∈N has a convergent subsequence. Given T > 0,

we have

φ(tn, θ−tn ω, xn) ∈ K (ω), for all n with tn ≥ tK (ω)

and

φ(tn − T, θ−(tn−T )θ−T ω, xn) ∈ K (θ−T ω) for all n with tn − T ≥ tK (θ−T ω).

Let zn = φ(tn − T, θ−tn ω, xn). Since K (ω) is bounded, there exist x , xT ∈ `2ρ and subsequences (the indices we still denote by n),

such that

φ(tn, θ−tn ω, xn) x (4.1)

and

zn = φ(tn − T, θ−tn ω, xn) xT (4.2)

in `2ρ as n → ∞. We will show that ‖φ(tn, θ−tn ω, xn)‖ρ → ‖x‖ρ , then φ(tn, θ−tn ω, xn) strongly converges to x , that is φ is

asymptotically compact.First, by the lower semi-continuity of the norm with respect to the weak convergence,

‖x‖2ρ ≤ lim inf

n→∞‖φ(tn, θ−tn ω, xn)‖2

ρ . (4.3)

Next we prove that

‖x‖2ρ ≥ lim sup

n→∞

‖φ(tn, θ−tn ω, xn)‖2ρ . (4.4)

The cocycle property of φ implies

φ(tn, θ−tn ω, xn) = φ(T, θ−T ω, φ(tn − T, θ−tn ω, xn)), (4.5)

then by Lemma 4.8, (4.1) and (4.2) we have

x = φ(T, θ−T ω, xT ). (4.6)

Using Ito formula and the definition of inner product we have

eγ T‖φ(T )‖2

ρ = ‖φ(0)‖2ρ +

∫ T

02eγ s

[〈−(λ + iα)Aφ(s), φ(s)〉ρ +

⟨−

γ

2φ(s), φ(s)

⟩ρ

+ 〈−κ f (φ(s)), φ(s)〉ρ

+ 〈g, φ(s)〉ρ

]ds +

∫ T

0B0eγ sds + 2

∫ T

0eγ s

〈dW (s), φ(s)〉ρ (4.7)

where B0 =∑

j ρ j a2j . We replace ω by θ−T ω, take φ(0) = zn and use (4.5)

φ(T, θ−T ω, zn) = φ(tn, θ−tn ω, xn).


Then one gets

eγ T‖φ(tn, θ−tn ω, xn)‖2

ρ = ‖zn‖2ρ +

∫ T

02eγ s

[〈−(λ + iα)Aφ(s, θ−T ω, zn), φ(s, θ−T ω, zn)〉ρ

+

⟨−

γ

2φ(s, θ−T ω, zn), φ(s, θ−T ω, zn)

⟩ρ

+ 〈−κ f (φ(s, θ−T ω, zn)), φ(s, θ−T ω, zn)〉ρ

+ 〈g, φ(s, θ−T ω, zn)〉ρ

]ds +

∫ T

0B0eγ sds + 2

∫ T

0eγ s

〈dW (s), φ(s, θ−T ω, zn)〉ρ . (4.8)

From (4.2) and Lemma 4.8 we have

φ(·, θ−T ω, zn) φ(·, θ−T ω, xT ), in L2(0, T ; `2ρ)

and

φ(s, θ−T ω, zn) φ(s, θ−T ω, xT ), in `2ρ ∀s ∈ [0, T ]

as n → ∞.By the properties of A and the choice of ν0 we have

−〈(λ + iα)Ayn, yn〉ρ −γ

2〈yn, yn〉ρ

= −〈(λ + iα)A(yn − y), yn − y〉ρ − 〈(λ + iα)Ay, yn − y〉ρ − 〈(λ + iα)A(yn − y), y〉ρ

− 〈(λ + iα)Ay, y〉ρ −γ

2〈yn − y, yn − y〉ρ −

γ

2〈y, yn − y〉ρ −

γ

2〈yn − y, y〉ρ −

γ

2〈y, y〉ρ

≤ −〈(λ + iα)Ay, yn − y〉ρ − 〈(λ + iα)A(yn − y), y〉ρ − 〈(λ + iα)Ay, y〉ρ

−γ

2〈y, yn − y〉ρ −

γ

2〈yn − y, y〉ρ −

γ

2〈y, y〉ρ

and by the dissipativity of f we can write

−κ〈 f (yn), yn〉ρ = −κ〈 f (yn) − f (y), yn − y〉ρ − κ〈 f (yn) − f (y), y〉ρ − κ〈 f (y), yn − y〉ρ − κ〈 f (y), y〉ρ

≤ −κ〈 f (yn) − f (y), y〉ρ − κ〈 f (y), yn − y〉ρ − κ〈 f (y), y〉ρ .

Then we get

lim supn→∞

2∫ T

0eγ s

[〈−(λ + iα)Aφ(s, θ−T ω, zn), φ(s, θ−T ω, zn)〉ρ +

⟨−

γ

2φ(s, θ−T ω, zn), φ(s, θ−T ω, zn)

⟩ρ

]ds

≤ 2

∫ T

0eγ s

[〈−(λ + iα)Aφ(s, θ−T ω, xT ), φ(s, θ−T ω, xT )〉ρ +

⟨−

γ

2φ(s, θ−T ω, xT ), φ(s, θ−T ω, xT )

⟩ρ

]ds

and

lim supn→∞

2∫ T

0eγ s

〈−κ f (φ(s, θ−T ω, zn)), φ(s, θ−T ω, zn)〉ρds

≤ 2

∫ T

0eγ s

〈−κ f (φ(s, θ−T ω, xT )), φ(s, θ−T ω, xT )〉ρds.

Now we consider the stochastic integral term 2∫ T

0 eγ s〈dW (s), φ(s, θ−T ω, zn)〉ρ . By the definition of W (t), see (3.1), we define

the stochastic integral by every sample path. Given ω ∈ Ω∫ T

0eγ s

〈dω(s), yn(s)〉ρ = lim|sk+1−sk |→0

∑0≤sk<sk+1≤T

k

eγ sk 〈ω(sk+1) − ω(sk), yn(sk)〉ρ in L2(Ω).

It is well known the above convergence is uniform on [0, T ] and independent of the choice of sk . So if yn(s) y(s) in `2ρ , we have∫ T

0eγ s

〈dω(s), yn(s)〉ρ →

∫ T

0eγ s

〈dω(s), y(s)〉ρ, as n → ∞.

Then

2∫ T

0eγ s

〈dW (s), φ(s, θ−T ω, zn)〉ρ → 2∫ T

0eγ s

〈dW (s), φ(s, θ−T ω, xT )〉ρ

as n → ∞.


For sufficiently large n we have

zn ∈ K (θ−T ω),

then

lim supn→∞

‖zn‖2ρ ≤ R2(θ−T ω).

Hence by all the above analysis we have

lim supn→∞

eγ T‖φ(tn, θ−tn ω, xn)‖2

ρ ≤ R2(θ−T ω) +

∫ T

02eγ s

[〈−(λ + iα)Aφ(s, θ−T ω, xT ), φ(s, θ−T ω, xT )〉ρ

+

⟨−

γ

2φ(s, θ−T ω, xT ), φ(s, θ−T ω, xT )

⟩ρ

+ 〈−κ f (φ(s, θ−T ω, xT )), φ(s, θ−T ω, xT )〉ρ

+ 〈g, φ(s, θ−T ω, xT )〉ρ

]ds +

∫ T

0B0eγ sds + 2

∫ T

0eγ s

〈dW (s), φ(s, θ−T ω, xT )〉ρ

≤ R2(θ−T ω) + eγ T‖φ(T, θ−T ω, xT )‖2

ρ − ‖φ(0, θ−T ω, xT )‖2ρ

≤ R2(θ−T ω) + eγ T‖x‖

2ρ,

so

lim supn→∞

‖φ(tn, θ−tn ω, xn)‖2ρ ≤ e−γ T R2(θ−T ω) + ‖x‖

2ρ .

Since K (θ−T ω) is a tempered set, by taking T → ∞ we have

lim supn→∞

‖φ(tn, θ−tn ω, xn)‖2ρ ≤ ‖x‖

2ρ .

Then (4.3) and (4.4) yield the strong convergence

φ(tn, θ−tn ω, xn) → x in `2ρ .

Therefore φ is an asymptotically compact random dynamical system.

Now we can draw our main result as follows.

Theorem 4.10. System (2.2) possesses a unique global random D-attractor, which is a compact random invariant set in theweighted space `2

ρ with ν ∈ (0, ν0).

5. Ergodic behavior of the SDGL

We have proved that the SDGL has a global random attractor which is compact in `2ρ . By the theory of random attractor [5],

there is at least an invariant measure for the random dynamical system generated by the SDGL. In this section we will prove that asβ = 0 such an invariant measure is unique which means the SDGL is ergodic. And all the distributions of the solutions convergeexponentially to the invariant measure.

We follow the method of [11]. Write Pt , t ≥ 0, as the Markov semigroup associated with the solution process u of (2.5), that is,

Ptϕ(x) = E(ϕ(u(t, 0, x))), t ≥ 0, x ∈ `2ρ, ϕ ∈ Cb(`

2ρ),

where Cb(`2ρ) denotes the space of all uniformly continuous and bounded functions on `2

ρ . And denote the transition probability ofthe solution process u as Pt (x, ·), for x ∈ `2

ρ .Let W A(t), t ≥ 0 be the solution to the linear equation

Z(t) = AZ(t) + W (t),

Z(0) = 0,

given by

Z(t) = W A(t) =

∫ t

0S(t − r)dW (r), t ≥ 0,

where S(t), t ≥ 0, is the semigroup generated by A := −(λ + iα)A − (γ + iδ) on `2ρ .


Lemma 5.1. The process W A has bounded q-moments in `qρ spaces, for all q ≥ 1, that is,

supt>0

E‖W A(t)‖q`

qρ

< +∞. (5.1)

Proof. Denote

W A(t) = (W A, j (t)) j∈Z, S(t) =(S jl(t)

)j,l∈Z ,

then

W A, j (t) =

∑l∈Z

al

∫ t

0S jl(t − r)dwl(r)

and

E‖W A(t)‖q`

qρ

=

∑j∈Z

ρ j E(|W A, j (t)|q).

By Gaussianity and Q = diag(a2l )l∈Z, for some constant cq ,

E|W A, j (t)|q

= cq(E|W A, j (t)|2)

q2 ≤ cq

(∫ t

0

∑l∈Z

|al S jl(t − r)|2dr

) q2

.

Since∑l∈Z

|S jl(t − r)|2 ≤ ‖S(t − r)‖2L(`2

ρ ),

E‖W A(t)‖q`

qρ

≤ cq

∑j∈Z

ρ j

(∫ t

0maxl∈Z

|al |2· ‖S(t − r)‖2

L(`2ρ )

dr

) q2

.

Here ‖ · ‖L(`2ρ ) denotes the norm of the linear operator on `2

ρ .And by the choice of ν0

‖S(t − r)‖L(`2ρ ) ≤ e−

γ2 (t−r), t ≥ r,

estimate (5.1) follows.

Theorem 5.2. Assume that β = 0, there exists a unique invariant measure µ for the semigroup Pt , t ≥ 0. Moreover, for all boundedand Lipschitz continuous functions ϕ on `2

ρ with Lipschitz constant ‖ϕ‖Lip one has∣∣∣∣∣Ptϕ(x) −

∫`2ρ

ϕ(y)µ(dy)

∣∣∣∣∣ ≤ (c5 + 2‖x‖ρ)e−γ2 t

‖ϕ‖Lip,

where

c5 = supt≥0

E(

‖W A(t)‖ρ +2κ

γ‖W A(t)‖p+1

`2p+2ρ

)+

2γ

‖g‖ρ .

Proof. Denote by u(t, τ, uτ ), t ≥ τ , τ ∈ R, uτ ∈ `2ρ , the mild solution of

du(t)

dt= −(λ + iα)Au − (γ + iδ)u − κ f (u) + g + W , t > τ, (5.2)

u(τ ) = uτ .

Then, by Theorem 3.4,

u(t, τ, uτ ) = S(t − τ)uτ −

∫ t

τ

S(t − r) (κ f (u(t, τ, uτ )) − g) dr + W τ

A(t),

where

W τ

A(t) =

∫ t

τ

S(t − r)dW (r), t > τ.


We show now that

E‖u(t, τ, uτ )‖ρ ≤ ‖uτ‖ρ + c5, t ≥ τ, uτ ∈ `2ρ, (5.3)

where c5 is the constant from the theorem. Remark first that

zτ (t) = u(t, τ, uτ ) − W τ

A(t), t ≥ τ

is the mild solution of the problem

z(t) = Az(t) − κ f (z(t) + W τ

A(t)) + g,

z(τ ) = uτ .

Denote

z∗τ (t) =

zτ (t)

‖zτ (t)‖ρ

zτ (t) 6= 0,

0 zτ (t) = 0.

Then, by the chain rule (see [10]),

d−

dt‖zτ (t)‖ρ ≤ 〈zτ (t), z∗

τ (t)〉ρ = 〈 Azτ (t) − κ f (zτ (t) + W τ

A(t)) + g, z∗

τ (t)〉ρ

= 〈−(λ + iα)Azτ (t), z∗τ (t)〉ρ − κ〈 f (zτ (t) + W τ

A(t)) − f (W τ

A(t)), z∗

τ (t)〉ρ

+ 〈g − κ f (W τ

A(t)), z∗

τ (t)〉ρ + 〈−(γ + iδ)zτ (t), z∗τ (t)〉ρ

≤ 〈g − κ f (W τ

A(t)), z∗

τ (t)〉ρ −γ

2‖zτ (t)‖ρ

≤ ‖g − κ f (W τ

A(t))‖ρ −

γ

2‖zτ (t)‖ρ, t ≥ τ.

Consequently

‖zτ (t)‖ρ ≤ e−γ2 (t−τ)

‖uτ‖ρ +

∫ t

τ

e−γ2 (t−r)

‖g − κ f (W τ

A(r))‖ρdr, t ≥ τ,

then

E‖zτ (t)‖ρ ≤ ‖uτ‖ρ +2γ

supt≥τ

E‖g − κ f (W τ

A(t))‖ρ, t ≥ τ,

and

E‖u(t, τ, uτ )‖ρ ≤ ‖uτ‖ρ + c5, t ≥ τ, uτ ∈ `2ρ .

In a similar way we can obtain for arbitrary u(1)τ , u(2)

τ ∈ `2ρ

‖u(t, τ, u(1)τ ) − u(t, τ, u(2)

τ )‖ρ ≤ e−γ2 (t−τ)

‖u(1)τ − u(2)

τ ‖ρ, t ≥ τ.

Consequently for ς ≤ τ ≤ t

E‖u(t, τ, uτ ) − u(t, ς, uτ )‖ρ = E‖u(t, τ, uτ ) − u(t, τ, u(τ, ς, uτ ))‖ρ

≤ e−γ2 (t−τ)E‖uτ − u(τ, ς, uτ )‖ρ,

and, by (5.3)

E‖u(t, τ, uτ ) − u(t, ς, uτ )‖ρ ≤ e−γ2 (t−τ)(2‖uτ‖ρ + c5).

Therefore there exists a random variable ζ , the same for all uτ ∈ `2ρ , such that

limτ→−∞

E‖u(0, τ, uτ ) − ζ‖ρ = 0. (5.4)

We claim that the law µ = L(ζ ) is the unique invariant measure for Pt , t ≥ 0. To see this it is enough to remark that, by (5.4), forarbitrary uτ ∈ `2

ρ ,

Pt (uτ , ·) = L(u(t, uτ )) = L(u(0, −t, uτ )) → µ, weakly as t → +∞.


Finally, let ϕ be a bounded Lipschitz function on `2ρ , then, by (5.4), for s ≥ t ≥ 0

|Ptϕ(uτ ) − Psϕ(uτ )| = |E(ϕ(u(t, 0, uτ )) − ϕ(u(s, 0, uτ )))|

= |E(ϕ(u(0, −t, uτ )) − ϕ(u(0, −s, uτ )))|

≤ ‖ϕ‖LipE‖u(0, −t, uτ ) − u(0, −s, uτ )‖ρ

≤ ‖ϕ‖Lipe−γ2 t (2‖uτ‖ρ + c5).

Letting s → ∞ we obtain the desired inequality.

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (No. 10171044) and the Natural ScienceFoundation of Jiangsu Province (No. BK2001024).

The authors would like to thank the referees for the suggestions and comments as they have improved the exposition of thismanuscript.

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