# Synchronization of Coupled Stochastic Systems Driven byα-Stable Lévy Noises

Authors: Anhui Gu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The synchronization of the solutions to coupled stochastic systems ofN-Marcus stochastic ordinary differential equations which are driven by α-stable Lévy noises is investigated (N∈ℕ,1<α<2). We obtain the synchronization between two solutions and among different components of solutions under certain dissipative conditions. The synchronous phenomena persist no matter how large the intensity of the environment noises. These results generalize the  work of two Marcus canonical equations in X. M. Liu et al.' s (2010).


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## Body

## 1. Introduction

The synchronization of coupled systems is a ubiquitous phenomenon in the biological and physical science and is also known to occur in abundant of social science contexts; see for example [1–6] and references therein. In the recent book of Strogatz [4], a number of its diversity of occurrence and an extensive list of references can be found. Let u(t),v(t)∈ℝd be two functions defined in [t0,∞)(t0∈ℝ) and u(t),v(t) are said to be synchronized if limt→∞∥u(t)-v(t)∥=0. Synchronization of deterministic coupled systems has been investigated both for autonomous systems and nonautonomous systems (see, e.g., [7–10]).  For coupled systems of Ito¨ stochastic differential equations with various Gaussian noises (in the terms of Brownian motion), the synchronization of solutions has been considered in  the papers Caraballo and Kloeden [11], Caraballo et al. [12], Caraballo et al. [13] and Chueshov and Schmalfuß [14]. In [15], Shen et al. showed the synchronization of solutions for more general systems with multiplicative noise. Recently, Liu et al. [16, 17] studied the synchronization phenomenon for coupled systems driven by non-Gaussian noises (in terms of Lévy motion) and the analogous results also hold for the general systems with additive Lévy noises [18].A Lévy motionLt is a non-Gaussian process with independent and stationary increments; that is, increments ΔLt=Lt+Δt-Lt are stationary and independent for any non overlapping time lags Δt. Moreover, its sample paths are only continuous in probability, namely, ℙ(|Lt-Lt0|≥ϵ)→0 as t→t0 for any positive ϵ. With a suitable modification, these paths may be taken as càdlàg; that is, paths are continuous on the right and have limits on the left (see, e.g., [19, 20]). As a special case of Lévy processes, the symmetric α-stable Lévy motion plays an important role among stable processes just like Brownian motion among Gaussian processes. A stochastic process {Lt,t≥0} is called the α-stable Lévy motion if (1) L0=0 a.e., (2) L has independent increments, and (3) Lt-Ls~Sα((t-s)1/α,β,0) for 0≤s<t<∞ and for some 0<α≤2,-1≤β≤1, where Sα(σ,β,ν) denotes the α-stable distribution with index of stability α, scale parameter σ, skewness parameter β, and shift parameter ν; in particular, S2(σ,0,μ)=N(μ,2σ2) denotes the Gaussian distribution. For more details on α-stable distributions, we can refer to [21, 22].Let(Ω,ℱ,ℙ) be a probability space, where Ω=D(ℝ,ℝd) of càdlàg functions with the Skorohod metric (see [23]) as the canonical sample space and denote by ℱ:=𝔅(D(ℝ,ℝd)) the Borel σ-algebra on Ω. Let μL be the (Lévy) probability measure on ℱ which is given by the distribution of a two-sided Lévy process with paths in Ω, that is, ω(t)=Lt(ω). Define θ=(θt,t∈ℝ) on Ω the shift by (θtω)(s):=ω(t+s)-ω(t). Then, the mapping (t,ω)→θtω is continuous and measurable [24], and the (Lévy) probability measure is θ-invariant, that is, μL(θt-1(A))=μL(A), for all A∈ℱ; see [19] for more details.Consider the following Marcus stochastic ordinary differential equations (MSODEs) system driven byα-stable Lévy noises in ℝd:
(1)dXt(j)=f(j)(Xt(j))dt+∑i=1m‍ci(j)Xt(j)⋄dLt(i),j=1,…,N,
where ci(j)∈ℝ, Lt(i) are independent α-stable Lévy noises on (Ω,ℱ,ℙ), 1<α<2, ⋄ denotes the Marcus integral (see, e.g., [25]), and f(j),j=1,…,N are regular enough to ensure the existence and uniqueness of solutions and satisfy the one-sided dissipative Lipschitz condition
(2)〈x1-x2,f(j)(x1)-f(j)(x2)〉≤-L∥x1-x2∥2,j=1,…,N,
on ℝd for some L>0.Set(3)x(j)(t,ω)=e-Ot(j)Xt(j)(ω),t∈ℝ,ω∈Ω,j=1,…,N,
where
(4)Ot(j):=Ot(j)(ω)=∑i=1m‍ci(j)e-t∫-∞t‍esdLs(i),j=1,…,N,
are the stationary solutions of the Ornstein-Uhlenbeck stochastic differential equations
(5)dOt(j)=-Ot(j)dt+∑i=1m‍ci(j)⋄dLt(i),j=1,…,N.
Then system (1) can be translated into the following random ordinary differential equations (RODEs):
(6)dx(j)dt+=F(j)(x(j),Ot(j)):=e-Ot(j)f(j)(eOt(j)x(j))+Ot(j)x(j),j=1,…,N,
where dx(j)/dt+ is right-hand derivative of x(j) at t.Now we consider the linear coupled RODEs of (6), as follows:
(7)dx(j)dt+=F(j)(x(j),Ot(j))+ν(x(j-1)-2x(j)+x(j+1)),j=1,…,N,
with the coupled coefficient ν>0, where  x(0)=x(N) and  x(N+1)=x(1). Hence, (7) can be written as the following equivalent MSODEs:
(8)dXt(j)=f(j)(Xt(j))dt+ν(e-ΔOˇt(j)Xt(j-1)-2Xt(j)+e-ΔO^t(j)Xt(j+1))dt+∑i=1m‍ci(j)Xt(j)⋄dLt(i),j=1,…,N,
where  ΔOˇt(j)=Ot(j)-Ot(j-1),   ΔO^t(j)=Ot(j)-Ot(j+1),    Ot(0)=Ot(N), and Ot(N+1)=Ot(1).For synchronization of solutions (in the sense of Carathéodory [26]) to RODEs system (7), there are two cases: one for any two solutions and the other for components of solutions. When N=2, Liu et al. [17] consider both types of synchronization. Under the one-sided dissipative Lipschitz condition (2), they firstly proved that synchronization of any two solutions occurs and the random dynamical system generated by the solution of (7)  N=2 has a singleton sets random attractor, then they proved that the synchronization between any two components of solutions occurs as the coupled coefficient ν tends to infinity. The synchronization result implies that coupled dynamical systems share a dynamical feature in an  asymptotic sense. Based on the work of [15, 17], we consider the synchronization of solutions of (7) in the case of N≥3 and obtain the similar results. We show that the random dynamical system (RDS) generated by the solution of the coupled RODEs system (7) has a singleton sets random attractor which implies the synchronization of any two solutions of (7). Moreover, the singleton set random attractor determines a stationary stochastic solution of the equivalently coupled RODEs system (8). We also show that any two solutions of RODEs system (7) converge to a solution Z(t,ω) of the averaged RODEs  as follows:
(9)dZdt+=1N∑j=1N‍e-Ot(j)f(j)(eOt(j)Z)+1N∑j=1N‍Ot(j)Z,
as the coupling coefficient  ν→∞.Whenα=2, we have the standard Brownian motion, which the Marcus integral reduces to the Stratonovich stochastic integral, and both types of the  synchronization of system (8) have been considered in [15]. It is worth mentioning that the generalization is not trivial because new techniques similar to [15] are needed. We restrict here that α∈(1,2), only in this case, the solutions of the Ornstein-Uhlenbech equations based on α-stable Lévy noises are stationary, which is crucial to our purpose. When α∈(0,1), dealing with such values of the parameter seems to be a new challenging for us.The paper is organized as follows. In Section2, we recall some basic facts on random dynamical systems, and  then we give two lemmas which will be frequently used. In Section 3, we show the synchronization of two solutions to the coupled RODEs (7) and obtain the staionary stochastic solution to the equivalent MSODEs (8). In Section 4, we give the synchronization of components of solutions to the coupled RODEs (7), which implies that the equivalent MSODEs (8) share the similarly synchronous phenomenon when driven by the same α-stable Lévy noises.


## 2. Random Dynamical Systems and Auxiliary Lemmas

We will frequently use the following results.Lemma 1.
There exists a{θt}t∈ℝ-invariant subset Ω-∈ℱ of full measure for a.e. ω∈Ω-, and the sample paths ω(t) of Lt satisfy
(10)limt→±∞ω(t)t=0,t∈ℝ.
In addition, for j=1,…,N, there exist random variables O-(j)=Ot(j) and Tω>0 such that
(11)O-(j)(θtω)=Ot(j)(ω),limt→±∞1t∫0t‍O-(θsω)ds=0,ω∈Ω-,(12)e2∫τt‍Os(j)ds≤e(L/2)(t-τ)for-τ,t>Tω.Proof.
The equalities (10) and (11) can be found in [17, Lemma 2]. By (11), we have limt→∞(1/t)∫0t‍Os(j)ds=0, then there exists Tω(1)>0 such that ∫0t‍Os(j)ds≤(L/4)t for t>Tω(1). Similarly, limτ→-∞(1/τ)∫τ0‍Os(j)ds=0, which implies that there exists Tω(2)>0 such that ∫τ0‍Os(j)ds≤-(L/4)τ for τ<-Tω(2). Denoting  Tω=max{Tω(1),Tω(2)}, we have 2∫τt‍Os(j)ds≤(L/2)(t-τ) for -τ,t>Tω, which completes the proof.Lemma 2 (Gronwall type inequality).
Suppose thatD(t) is an n×n matrix and Φ(t)andΨ(t) are n-dimensional vectors on [T0,T](T≥T0,T,T0∈ℝ) which are sufficiently regular. If the following inequality holds in the componentwise sense:
(13)ddt+Φ(t)≤D(t)Φ(t)+Ψ(t),t≥T0,
where (d/dt+)Φ(t):=limh↓0+((Φ(t+h)-Φ(t))/h) is right-hand derivative of Φ(t), then
(14)Φ(t)≤exp(∫T0t‍D(s)ds)Φ(T0)+∫T0t‍exp(∫τt‍D(s)ds)Ψ(τ)dτ,t≥T0.Proof.
See Lemma 2.8  in [27] and the proof of lemma  2.2  in [15].Proposition 3 (Random attractor for càdlàg RDS (see [16])).
Let(θ,φ) be an RDS on Ω×ℝd and let φ be continuous in space but càlàg in time. If there exists a family ℬ={ℬ(ω),ω∈Ω} of nonempty measurable compact subsets ℬ(ω) of ℝd and a TB,ω≥0 such that
(15)φ(t,θ-tω,B(θ-tω))⊂ℬ(ω),∀t≥TB,ω,
for all families B={B(ω),ω∈Ω} in a given attracting universe. then, the RDS (θ,φ) has a random attractor 𝒜={𝒜(ω),ω∈Ω} with the component subsets defined for each ω∈Ω by
(16)𝒜(ω)=⋂s>0‍⋃t≥s‍φ(t,θ-tω,B(θ-tω))¯.
Furthermore, if the random attractor consist of singleton sets, that is, 𝒜(ω)={X*(ω)} for some random variable X*, then Xt*(ω)=Xt*(θtω) is a stationary stochastic process.


## 3. Synchronization of Two Solutions

Consider the coupled system (7) with the following initial data:
(17)x(j)(0,ω)=x0(j)(ω)∈ℝd,ω∈Ω,j=1,…,N.
For asymptotic behavior of the difference between two solutions of RODEs system (7) with initial data (17) (omitting to RODEs system (7) for brevity), we get the following:Lemma 4.
For any two solutions(x1(1)(t),x1(2)(t),…,x1(N)(t))T  and  (x2(1)(t),x2(2)(t),…,x2(N)(t))T of RODEs system (7),
(18)limt→∞∥x1(j)(t)-x2(j)(t)∥=0,j=1,…,N,
that is, all solutions of the coupled RODEs system (7) converge pathwise to each other as time t tends to infinity.Proof.
By the dissipative Lipschitz condition (2), for j=1,…,N, we have
(19)ddt+∥x1(j)(t)-x2(j)(t)∥2=2〈x1(j)(t)-x2(j)(t),ddtx1(j)(t)-ddtx2(j)(t)〉=2e-Ot(j)〈f(j)(eOt(j)x1(j))-f(j)(eOt(j)x2(j)),x1(j)(t)=2e-Ot(j)-x2(j)(t)(eOt(j)x1(j))〉×(2eOt(j)-4ν)∥x1(j)(t)-x2(j)(t)∥2+2ν〈x1(j-1)(t)-x2(j-1)(t),x1(j)(t)-x2(j)(t)〉+2ν〈x1(j+1)(t)-x2(j+1)(t),x1(j)(t)-x2(j)(t)〉≤(2eOt(j)-2L-2ν)∥x1(j)(t)-x2(j)(t)∥2+ν∥x1(j-1)(t)-x2(j-1)(t)∥2+ν∥x1(j+1)(t)-x2(j+1)(t)∥2.
Define for t∈ℝ,
(20)x(t)=(∥x1(1)(t)-x2(1)(t)∥2,∥x1(2)(t)-x2(2)(t)∥2,…,∥x1(N)(t)-x2(N)(t)∥2)T,Dν(t)=(λν(1)(t)ν0⋯ννλν(2)(t)ν0⋯0νλν(3)(t)⋱⋱⋮⋱⋱⋱νν⋯0νλν(N)(t))N×N,
where λν(j)(t)=2eOt(j)-2L-2ν,j=1,…,N. Thus, the differential inequalities can be written as a simple form
(21)x˙(t)≤Dν(t)x(t),-componentwise.
By Lemma 2, it yields from (21) that
(22)x(t)≤exp(∫0t‍Dν(t)ds)x(0),-componentwise.
By [15, Lemma 3.2], we know that for t≥Tω defined in Lemma 1, and ν>0,
(23)∥exp(∫0t‍Dν(t)ds)x(0)∥≤e-Lt∥x(0)∥,
which leads to
(24)limt→∞∥x1(j)(t)-x2(j)(t)∥=0,j=1,…,N,
and completes the proof.Now, we use the theory of random dynamical systems which are  generated by stochastic differential equations driven byα-stable Lévy noise to find what the solutions of (7) will converge to. Obviously by condition (2) and [16, Lemma 4], we know that the solution
(25)φ(t,ω)=(x(1)(t,ω),x(2)(t,ω),…,x(N)(t,ω))T,ω∈Ω,
of system (7) generates a càdlàg RDS over (Ω,ℱ,ℙ,(θt)t∈ℝ) with state space Ω×ℝNd.Then, we have the result for this RDSφ.Theorem 5.
Under the dissipative condition of (2), the RDS φ(t,ω),t∈ℝ,ω∈Ω, has a singleton sets random attractor given by
(26)𝒜ν(ω)=(x-ν(1)(ω),x-ν(2)(ω),…,x-ν(N)(ω))T,
which implies the synchronization of any two solutions of system (7). Furthermore,
(27)(x-ν(1)(θtω)eOt(1)(ω),x-ν(2)(θtω)eOt(2)(ω),…,x-ν(N)(θtω)eOt(N)(ω))T
is the stationary stochastic solution of the equivalent coupled MSODEs (8).Proof.
Forj=1,…,N, we have
(28)ddt∥x(j)(t)∥2=2〈x(j)(t),ddtx(j)(t)〉=2〈e-Ot(j)f(j)(eOt(j)x(j)(t)),x(j)(t)〉+2〈eOt(j)x(j)(t),x(j)(t)〉-4ν∥x(j)(t)∥2+2ν〈x(j)(t),x(j-1)(t)〉+2ν〈x(j)(t),x(j+1)(t)〉≤(2eOt(j)-2L-2ν)∥x(j)(t)∥2+ν∥x(j-1)(t)∥2+ν∥x(j+1)(t)∥2+2∥x(j)(t)∥∥f(j)(0)∥e-Ot(j)≤(2eOt(j)-L-2ν)∥x(j)(t)∥2+ν∥x(j-1)(t)∥2+ν∥x(j+1)(t)∥2+e-2Ot(j)L∥f(j)(0)∥2.
Analogous to (21), we get
(29)y˙(t)≤D~ν(t)y(t)+g(t),
where t∈ℝ,
(30)y(t)=(∥x(1)(t)∥2,∥x(2)(t)∥2,…,∥x(N)(t)∥2)T,g(t)=(e-2Ot(1)L∥f(1)(0)∥2,e-2Ot(2)L∥f(2)(0)∥2,…,g(t)e-2Ot(N)L∥f(N)(0)∥2e-2Ot(1)L)T,D~ν(t)=(λ~ν(1)(t)ν0⋯ννλ~ν(2)(t)ν0⋯0νλ~ν(3)(t)⋱⋱⋮⋱⋱⋱νν⋯0νλ~ν(N)(t))N×N,
where λ~ν(j)(t)=2eOt(j)-L-2ν,j=1,…,N. Then by Lemma 2,
(31)y(t)≤exp(∫t0t‍D~ν(t)ds)y(t0)+∫t0t‍exp(∫τt‍D~ν(t)ds)×g(τ)dτ,t≥t0.
By (23), we have
(32)∥exp(∫t0t‍D~ν(t)ds)y(t0)∥≤exp(-L2(t-t0))×∥y(t0)∥,t≥t0.
Define
(33)ρν(ω):=∫-∞0exp(∫τ0D~ν(s)ds)g(τ)dτ,(34)Rν2(ω)=1+∥ρν(ω)∥2,
and let 𝔹ν be a random ball in ℝNd centered at the origin with radius Rν(ω). Obviously, the infinite integrals on the right hand side of (33) and (34) are well defined by Lemma 1.
For a given attracting universe of tempered random bounded sets𝒟, that is, for any ω∈Ω, B∈𝒟, and all γ>0, we have limt→∞e-γtsupx∈B(θ-tω)∥x∥=0. Note that for all γ>0, if limt→∞e-γt∥y(t0)∥=0, then
(35)∑j=1N‍∥x(j)(0)∥2<Rν2(ω)ast0⟶-∞,
which implies that the closed random ball 𝔹ν(ω) is a pullback absorbing set at t=0 of the càdlàg RDS φ(t,ω); that is
(36)φ(t,θ-tω)B(θ-tω)⊂𝔹ν(ω),∀t≥t𝔹ν(ω),
in the attracting universe 𝒟. Hence by Proposition 3, the coupled system has a random attractor 𝒜ν={𝒜ν(ω),ω∈Ω} with 𝒜ν(ω)⊂𝔹ν satisfying that  𝒜ν(ω) is compact, φ-invariant, that is, φ(t,ω)𝒜ν(ω)=𝒜ν(θtω) for all t≥0, ω∈Ω, and attracting in 𝒟, that is, for all B∈𝒟,
(37)Hd*(φ(t,θ-tω,B(θ-tω)),𝒜ν(ω))⟶0ast⟶∞,
where Hd* is the Hausdorff semidistance on ℝNd. By Lemma 4, all solutions of (7) converge pathwise to each other; therefor, 𝒜ν(ω) consists of singleton sets, that is,
(38)𝒜ν(ω)=(x-ν(1)(ω),x-ν(2)(ω),…,x-ν(N)(ω))T.
We transform the coupled RODEs (7) back to the coupled MSODEs (8), the corresponding pathwise singleton sets attractor is then equal to
(39)(eOt(N)(ω)x-ν(1)(θtω)eOt(1)(ω),x-ν(2)(θtω)eOt(2)(ω),…,x-ν(N)(θtω)eOt(N)(ω))T,
which is exactly a stationary stochastic solution of the coupled MSODEs (8) because the Ornstein-Uhlenbeck process is stationary (see [17]).


## 4. Synchronization of the Components of Solutions

It is known in Section3 that all solutions of the coupled RODEs system (7) converge pathwise to each other in the future for a fixed positive coupling coefficient ν. Here, we would like to discuss what will happen to solutions of the coupled RODEs system (7) as ν→∞. First, we will give some lemmas which play an important role in this section.Similar to [15, Section 4], we can set up the following estimations. Suppose that (xν(1)(t),xν(2)(t),…,xν(N)(t))T is a solution of the coupled RODEs system (7). For any two different components xν(j)(t),xν(k)(t) of the solution forallj,k∈{1,2,…,N},
(40)dνj,k(t)=2〈xν(j)(t)-xν(k)(t),F(j)(xν(j),Ot(j))-F(k)(xν(k),Ot(k))〉=2〈xν(j)(t)-xν(k)(t),e-Ot(j)f(j)(eOt(j)xν(j))-e-Ot(k)f(k)(eOt(k)xν(k))〉+2〈xν(j)(t)-xν(k)(t),Ot(j)xν(j)(t)-Ot(k)xν(k)(t)〉≤2∥xν(j)(t)-xν(k)(t)∥(e-Ot(j)∥f(j)(eOt(j)xν(j)(t))∥2∥xν(j)(t)-xν(k)(t)∥+|Ot(j)|∥xν(j)(t)∥(eOt(j)xν(j)(t)))+2∥xν(j)(t)-xν(k)(t)∥(e-Ot(k)∥f(k)(eOt(k)xν(k)(t))∥+2∥xν(j)(t)-xν(k)(t)∥+|Ot(k)|∥xν(k)(t)∥(eOt(k)xν(k)(t))),
thus, for fixed ϱ>0, we have
(41)-ϱν∥xν(j)(t)-xν(k)(t)∥2+dνk,j(t)≤1ν(4ϱe-2Ot(j)∥f(j)(eOt(j)xν(j)(t))∥2+4ν2ϱ|Ot(j)|2∥xν(j)(t)∥2)+1ν(4ϱe-2Ot(k)∥f(k)(eOt(k)xν(k)(t))∥2+4ν2ϱ|Ot(k)|2∥xν(k)(t)∥2∥f(k)(eOt(k)xν(k)(t))∥2).Let(42)CT1,T2j,k,ϱ(ν,ω)=4ϱsupt∈[T1,T2][(e-2Ot(j)∥f(j)(eOt(j)xν(j)(t))∥2=4ρsupt∈[T1,T2]+|Ot(j)|2∥xν(j)(t)∥2∥f(j)(eOt(j)xν(j)(t))∥2)=4ρsupt∈[T1,T2]+(e-2Ot(k)∥f(k)(eOt(k)xν(k)(t))∥2=4ρsupt∈[T1,T2]+|Ot(k)|2∥xν(k)(t)∥2(e-2Ot(j)∥f(j)(eOt(j)xν(j)(t))∥2)]
in any bounded interval [T1,T2]. Note that ρν(ω) in (33) satisfies
(43)ddν∥ρν(ω)∥2=2〈ρν(ω),ddνρν(ω)〉≤0,
and, consequently, ρν(ω)≤ρ1(ω) for ν≥1. Hence, CT1,T2j,k,ϱ(ν,ω) is uniformly bounded in ν and
(44)-ϱν∥xν(j)(t)-xν(k)(t)∥2+dνk,j(t)≤1νCT1,T2j,k,ϱ(ν,ω)
uniformly for t∈[T1,T2] with
(45)CT1,T2j,k,ϱ(ω)=supν≥1CT1,T2j,k,ϱ(ν,ω).Now let us estimate the difference between any two components of a solution of the coupled RODEs system (7) as ν→∞.Lemma 6.
Provided condition (2) is satisfied, then any two components of a solution (xν(1)(t),xν(2)(t),…,xν(N)(t))T of the coupled RODEs system (7) uniformly vanish in any bounded time interval when the coupling coefficient ν→∞; that is, for any bounded interval [T1,T2] and for allt∈[T1,T2], it yields
(46)limν→∞∥xν(j)(t)-xν(k)(t)∥=0,∀j,k∈{1,2,…,N}.Proof.
The proof is quite similar to the proof of Lemma 4.2 in [15]. To prove the result, we can equivalently estimate the difference between any two adjacent components only because the first and the last components of the solution are considered to be adjacent. We will notice that only one new term appears in each step which continues the process, except the last step that ends the process.
For the difference of the first part of the solution(xν(1)(t),xν(2)(t),…,xν(N)(t))T,
(47)ddt+∥xν(1)(t)-xν(2)(t)∥2=2〈xν(1)(t)-xν(2)(t),F(1)(xν(1),Ot(1))=2-F(2)(xν(2),Ot(2))〉-6ν∥xν(1)(t)-xν(2)(t)∥2+2ν〈xν(1)(t)-xν(2)(t),xν(N)(t)-xν(3)(t)〉≤-5∥xν(1)(t)-xν(2)(t)∥2+ν∥xν(N)(t)-xν(3)(t)∥2+dν1,2(t)≤-δν∥xν(1)(t)-xν(2)(t)∥2+ν∥xν(N)(t)-xν(3)(t)∥2+1νCT1,T21,2,5-δ(ω)
uniformly for t∈[T1,T2] by (44). Here, we can take
(48)δ={1-cosNπN+2,Niseven,1-cos(N-1)πN+1,Nisodd.
In fact, from [15, Lemma 4.1], we can take any δ∈(-2cos(Nπ/(N+2)),2) when N is even and any δ∈(-2cos((N-1)π/(N+1)),2) when N is odd.
We have seen that the estimations in (47) generate xν(3)(t)-xν(N)(t). Now, we have
(49)ddt+∥xν(3)(t)-xν(N)(t)∥2=2〈xν(3)(t)-xν(N)(t),F(3)(xν(3),Ot(3))-F(N)(xν(N),Ot(N))〉-4ν∥xν(3)(t)-xν(N)(t)∥2+2ν〈xν(3)(t)-xν(N)(t),xν(2)(t)-xν(1)(t)〉+2ν〈xν(3)(t)-xν(N)(t),xν(4)(t)-xν(N-1)(t)〉≤-δν∥xν(3)(t)-xν(N)(t)∥2+ν∥xν(1)(t)-xν(2)(t)∥2+ν∥xν(4)(t)-xν(N-1)(t)∥2+1νCT1,T23,N,2-δ(ω)
uniformly for t∈[T1,T2].
Note thatxν(1)(t)-xν(2)(t) has been fixed and xν(4)(t)-xν(N-1)(t) is generated. Similarly, it yields
(50)ddt+∥xν(4)(t)-xν(N-1)(t)∥2≤-δν∥xν(4)(t)-xν(N-1)(t)∥2+ν∥xν(3)(t)-xν(N)(t)∥2+ν∥xν(5)(t)-xν(N-2)(t)∥2+1νCT1,T24,N-1,2-δ(ω)
uniformly for t∈[T1,T2].
Continuing such estimations, forj=2,3,…, we get
(51)ddt+∥xν(j+3)(t)-xν(N-j)(t)∥2≤-δν∥xν(j+3)(t)-xν(N-j)(t)∥2+ν∥xν(j+2)(t)-xν(N-j+1)(t)∥2+ν∥xν(j+4)(t)-xν(N-j-1)(t)∥2+1νCT1,T2j+3,N-j,2-δ(ω)
uniformly for t∈[T1,T2].
We can divide the situation into two cases:N is even and N is odd, which is just as the  same as [15] did. When N is even, we can rewrite the inequalities in the matrix form
(52)u˙(t)≤Hνu(t)+1νC
uniformly for t∈[T1,T2], where for t∈ℝ,
(53)u(t)=(∥xν(1)(t)-xν(2)(t)∥2,∥xν(3)(t)-xν(N)(t)∥2,…,=∥xν(N/2+1)(t)-xν(N/2+2)(t)∥2)T,C=(CT1,T21,2,5-δ(ω),CT1,T23,N,2-δ(ω),…,==CT1,T2N/2,N/2+3,2-δ(ω),CT1,T2N/2+1,N/2+2,5-δ(ω))T
are (N/2)-dimensional vectors, and
(54)Hν=(-δνν0⋯0ν-δνν⋱⋮0ν⋱⋱0⋮⋱⋱-δνν0⋯0ν-δν)(N/2)×(N/2).
By Lemma 2, it follows from (52) that
(55)u(t)≤e(t-t0)Hνu(t0)+1ν∫t0t‍e(t-s)HνCds.
By [15, Lemma  4.1] again,  (1/ν)Hν is negative definite, then we have
(56)∥e(t-t0)Hνu(t0)∥≤e(t-t0)μmax∥u(t0)∥,
where μmax=-δ-2cos(Nπ/(N+2))<0 is the maximal eigenvalue of   (1/ν)Hν. Thus, (55) implies that
(57)u(t)⟶0asν⟶∞,∥xν(1)(t)-xν(2)(t)∥2⟶0,∥xν(N/2+1)(t)-xν(N/2+2)(t)∥2⟶0
uniformly for t∈[T1,T2] as ν→∞.
Similarly, whenN is odd, we can rewrite the inequalities in the matrix form
(58)v˙(t)≤H~νv(t)+1νC~
uniformly for t∈[T1,T2], where for t∈ℝ,
(59)v(t)=(∥xν(1)(t)-xν(2)(t)∥2,∥xν(3)(t)-xν(N)(t)∥2,…,=∥xν((N+1)/2)(t)-xν((N+1)/2+2)(t)∥2)T,C~=(CT1,T21,2,5-δ(ω),CT1,T23,N,2-δ(ω),…,=CT1,T2(N-1)/2,(N+1)/2+3,2-δ(ω),CT1,T2(N+1)/2,(N+1)/2+2,5-δ(ω))T
are ((N-1)/2)-dimensional vectors, and
(60)H~ν=(-δνν0⋯0ν-δνν⋱⋮0ν⋱⋱0⋮⋱⋱-δνν0⋯0ν-δν)((N-1)/2)×((N-1)/2).
By Lemma 2, it follows from (58) that
(61)v(t)≤e(t-t0)H~νv(t0)+1ν∫t0t‍e(t-s)H~νC~ds.
Just like the even case, for uniform t∈[T1,T2], we have
(62)∥xν(1)(t)-xν(2)(t)∥2⟶0,asν⟶∞.
For other adjacent components, the process above can be repeated. Hence, we can draw a conclusion that the difference between any adjacent components of a solution of the coupled RODEs system (7) tends to zero uniformly for t∈[T1,T2] as the coupling coefficient goes to infinity which completes the proof.We know that all components of a solution of system (7) have the same limit uniformly for t∈[T1,T2] as ν→∞. Now, we are in the position to find what they converge to.Lemma 7.
If assumptions (2) and (6) hold, the càdlàg random dynamical system φ(t,ω) generated by the solution of the averaged RODEs system
(63)dZdt+=1N∑j=1N‍eOt(j)f(j)(eOt(j)Z)+1N∑j=1N‍Ot(j)Z
has a singleton sets random attractor denoted by {Z-(ω)}. Furthermore,
(64)Z-(θtω)exp(1N∑j=1N‍Ot(j)(ω))
is the stationary stochastic solution of the equivalently averaged SODE system
(65)dz=1N∑j=1N‍e-ζt(j)f(j)(eζt(j)z)dt+1N∑i=1m‍∑j=1N‍ci(j)z⋄dLt(i),
where ζt(j)=∑k=1N‍(Ot(j)-Ot(k)),j=1,…,N.Proof.
Assume thatZ1(t) and Z2(t) are two solutions of (63), we have
(66)ddt+∥Z1(t)-Z2(t)∥2≤(-2L+2N∑j=1N‍Ot(j))∥Z1(t)-Z2(t)∥2.
It follows from Gronwall's lemma (see [27, Lemma 2.8]) that
(67)∥Z1(t)-Z2(t)∥2≤e-2t(L-(1/N)∑j=1N(1/t)∫0t‍Os(j)ds)∥Z1(0)-Z2(0)∥2,
which implies that
(68)limt→∞∥Z1(t)-Z2(t)∥2=0.
Then, all solutions of (63) converge pathwise to each other.
Now, we have to give what they converge to based on the theory of càdlàg random dynamical systems. LetZ(t) be a solution of (63), we get
(69)ddt+∥Z(t)∥2≤(-2L+2N∑j=1N‍Ot(j))∥Z(t)∥2+1LN∑j=1N‍e-2Ot(j)∥f(j)(0)∥2.
From Gronwall's lemma in [27], again, it yields for -t0,t>Tω,
(70)∥Z(t)∥2≤e-L(t-t0)+(2/N)∑j=1N∫t0tOs(j)ds∥Z(t0)∥2+1LN×∑j=1N‍∥f(j)(0)∥2∫t0t‍e-2Oτ(j)-L(t-τ)+(2/N)∑k=1N∫τtOs(k)dsdτ.
By pathwise pullback convergence with t0→-∞, the random closed ball centered as the origin with random radius R~(ω) is a pullback absorbing set of φ(t,ω), where
(71)R~2(ω)=1+1LN∑j=1N‍∥f(j)(0)∥2∫-∞0‍eLτ-2Oτ(j)+(2/N)∑k=1N∫τ0Os(k)dsdτ.
Obviously, by Lemma 1, the integral defined in the right hand side is well defined.
By Proposition3, there exists a random attractor {Z-(ω)} for φ(t,ω). Since all solutions of (63) converge pathwise to each other, the random attractor {Z-(ω)} is composed of singleton sets.
Note that the averaged RODE (63) is transformed from the averaged SODE (65) by the following  transformation:
(72)Z(t,ω)=zexp(-1N∑j=1N‍Ot(j)(ω)),
so the pathwise singleton sets attractor Z-(θtω)exp((1/N)∑j=1N‍Ot(j)(ω)) is a stationary solution of the averaged SODE (65) since the Ornstein-Uhlenbeck process is stationary.Now, we will present another main solution of this work.Theorem 8.
Let(73)(x-νn(1)(t,ω),x-νn(2)(t,ω),…,x-νn(N)(t,ω))T=(x-νn(1)(θtω),x-νn(2)(θtω),…,x-νn(N)(θtω))T
be the singleton sets random attractor of the càdlàg random dynamical system φ(t,ω) generated by the solution of RODEs system (7), then
(74)((x-νn(1)(t,ω),x-νn(2)(t,ω),…,x-νn(N)(t,ω))T)⟶(Z-(t,ω),Z-(t,ω),…,Z-(t,ω))T
pathwise uniformly for t belongs to any bounded time interval [T1,T2] for any sequence νn→∞, where Z-(t,ω)=Z-(θtω) is the solution of the averaged RODE (63) and Z-(ω) is the singleton sets random attractor of the càdlàg random dynamical system φ(t,ω) which is generated by the solution of averaged RODE (63).Proof.
Define(75)Z-ν(ω)=1N∑j=1N‍x-ν(j)(ω),
where {x-ν(1)(ω),x-ν(2)(ω),…,x-ν(N)(ω)} is the singleton sets random attractor of the càdlàg RDS generated by RODEs system (7). Thus, Z-ν(t,ω)=Z-ν(θtω) satisfies
(76)dZ-ν(t,ω)dt+=1N∑j=1N‍(e-Ot(j)f(j)(eOt(j)x-ν(j)(t,ω))+Ot(j)x-ν(j)(t,ω)).
Then, we get
(77)∥dZ-ν(t,ω)dt+∥2≤2N∑j=1N‍(e-2Ot(j)∥f(j)(eOt(j)x-ν(j)(t,ω))∥22N∑j=1N‍++|Ot(j)|2∥x-ν(j)(t,ω)∥2∥f(j)(eOt(j)x-ν(j)(t,ω))∥2),
by the continuous property of the solutions and the fact that these solutions belong to the compact ball 𝔹1(ω), it follows that
(78)supt∈[T1,T2]∥dZ-ν(t,ω)dt+∥≤(2N∑j=1Nϱ4CT1,T2j,•,ϱ(ω)‍)1/2<∞.
By the Ascoli-Arzelà theorem in a Skorohod space of bounded time intveral D([T1,T2],ℝd) in [23], there exists a subsequence νnk→∞ such that Z-νnk(t,ω) converges to Z-(t,ω) as nk→∞.
Since difference between any two components of a solution of the coupled RODEs system (7) tends to zero uniformly for t∈[T1,T2] as ν→∞, from (75), we have
(79)x-νnk(j)(t,ω)=Z-νnk(t,ω)+1N∑j′≠j‍∑j′′≠j′‍(x-νnk(j′′)(t,ω)-x-νnk(j′)(t,ω))⟶Z-(t,ω)
uniformly for t∈[T1,T2] as νnk→∞ for j=1,…,N. Furthermore, it follows from (76) that, for t≥T1,
(80)Z-ν(t,ω)=Z-ν(T1,ω)+1N∑j=1N‍∫T1t‍(e-Os(j)f(j)(eOs(j)x-ν(j)(s,ω))Z-ν(T1,ω)+1N∑j=1N‍∫T1t‍+Os(j)x-ν(j)(s,ω)eOs(j))ds.
Thus,
(81)Z-(t,ω)=Z-(T1,ω)+1N∑j=1N‍∫T1t‍(e-Os(j)f(j)(eOs(j)Z-(s,ω))Z-(T1,ω)+1N∑j=1N‍∫T1t‍++Os(j)Z-(s,ω)eOs(j))ds,
uniformly for t∈[T1,T2] as νnk→∞, which implies that Z-ν(s,ω) solves RODE (63). Then, we note that all possible sequences of Z-νnk(t,ω) converge to the same limit Z-(t,ω) uniformly for t∈[T1,T2] as νn→∞. Since the RDS φ generated by the solutions of RODE (63) has a singleton sets random attractor {Z-(ω)}, the stationary stochastic process Z-(θtω) must be equal to Z-(t,ω), that is, Z-(t,ω)=Z-(θtω), which completes the proof.As an obvious result of Theorem8, we get the following.Corollary 9.
(82)
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in Skorohod metric pathwise uniformly for t∈[T1,T2] as ν→∞.Remark 10.
The results in this paper hold just in almost everywhere becauseω∈Ω- in Lemma 1,  and we still use Ω instead of Ω-.Remark 11.
Although the same results hold when the systems perturbed by non-Gaussian noises (see, e.g., [17] and this paper for α-stable Lévy noises and  [16, 18] for additive Lévy noises), there exists some difference between dealing with such stochastic systems when driven by Bronian motions and Lévy motions. Firstly, to some extent, the cases of Lévy noises have more general sense than the Bronian motions. For example, when α=2, the Lévy noise is the standard Brownian motion and the Marcus integral is reduced to the Stratonovich stochastic integral, that is, the case of multiplicative white noise (see [11, 15]). Here we only are restricted to 1<α<2. Secondly, We need to consider the càdlàg functions in the Skorohod metric, which are different from the continuous cases in the metric under the compact-open topology. Last but not least, we have to consider the solutions in the sense of Carthéodory and the right hand derivatives.



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*Source: 685798-2013-03-19.xml*