# Synchronization of Dissipative Dynamical Systems Driven by Non-Gaussian Lévy Noises

Authors: Xianming Liu; Jinqiao Duan; Jicheng Liu; Peter E. Kloeden
Journal: International Journal of Stochastic Analysis
 (2010)

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## Abstract

Dynamical systems driven by Gaussian noises have been considered
extensively in modeling, simulation, and theory. However, complex systems
in engineering and science are often subject to non-Gaussian fluctuations
or uncertainties. A coupled dynamical system under a class of Lévy noises is considered. After discussing cocycle property, stationary orbits, and random attractors, a synchronization phenomenon
is shown to occur, when the drift terms of the coupled system satisfy
certain dissipativity and integrability conditions. The synchronization
result implies that coupled dynamical systems share a dynamical
feature in some asymptotic sense.


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## Body

## 1. Introduction

Synchronization of coupled dynamical systems is an ubiquitous phenomenon that has been observed in biology, physics, and other areas. It concerns coupled dynamical systems that share a dynamical feature in an asymptotic sense. A descriptive account of its diversity of occurrence can be found in the recent book [1]. Recently Caraballo and Kloeden [2, 3] proved that synchronization in coupled deterministic dissipative dynamical systems persists in the presence of various Gaussian noises (in terms of Brownian motion), provided that appropriate concepts of random attractors and stochastic stationary solutions are used instead of their deterministic counterparts.In this paper we investigate a synchronization phenomenon for coupled dynamical systems driven by nonGaussian noises. We show that couple dissipative systems exhibit synchronization for a class of Lévy motions.This paper is organized as follows. We first recall some basic facts about random dynamical systems (RDSs) as well as formulate the problem of synchronization of stochastic dynamical systems driven by Lévy noises in Section2. The main result (Theorem 3.3) and an example are presented in Section 3.Throughout this paper, the norm of a vectorx in Euclidean space ℝd is always denote by |x|.


## 2. Dynamical Systems Driven by Lévy Noises

Dynamical systems driven by nonGaussian Lévy motions have attracted much attention recently [4, 5]. Under certain conditions, the SDEs driven by Lévy motion generate stochastic flows [4, 6], and also generate random dynamical systems (or cocycles) in the sense of Arnold [7]. Recently, exit time estimates have been investigated by Imkeller and Pavlyukevich [8], and Imkeller et al. [9], and Yang and Duan [10] for SDEs driven by Lévy motion. This shows some qualitatively different dynamical behavior between SDEs driven by Gaussian and nonGaussian noises.


### 2.1. Lévy Processes

A Lévy process or motion onℝd is characterized by a drift parameter γ∈ℝd, a covariance d×d matrix A, and a nonnegative Borel measure ν, defined on (ℝd,ℬ(ℝd)) and concentrated on ℝd∖{0}, which satisfies(2.1)∫ℝd∖{0}(|y|2⋀1)ν(dy)<∞,
or equivalently(2.2)∫ℝd∖{0}|y|21+|y|2ν(dy)<∞.
This measure ν is the so-called the Lévy jump measure of the Lévy process Lt. Moreover Lévy process Lt has the following Lévy-Itô decomposition:(2.3)Lt=γt+Bt+∫|x|<1xÑ(t,dx)+∫|x|≥1xN(t,dx),
where N(dt,dx) is Poisson random measure,(2.4)Ñ(dt,dx)=N(dt,dx)-ν(dx)dt,
is the compensated Poisson random measure of Lt, and Bt is an independent Brownian motion on ℝd with covariance matrix A (see [4, 10–12]). We call (A,ν,γ) the generating triplet.General semimartingales, especially Lévy motions, are thought to be appropriate models for nonGaussian processes with jumps [11]. Let us recall that a Lévy motion Lt is a nonGaussian process with independent and stationary increments. Moreover, its sample paths are only continuous in probability, namely, ℙ(|Lt-Lt0|≥δ)→0 as t→t0 for any positive δ. With a suitable modification [4], these paths may be taken as càdlàg, that is, paths are continuous on the right and have limits on the left. This continuity is weaker than the usual continuity in time. In fact, a càdlàg function has finite or at most countable discontinuities on any time interval (see, e.g., [4, page 118]). This generalizes the Brownian motion Bt, since Bt satisfies all these three conditions, but additionally, (i) almost every sample path of the Brownian motion is continuous in time in the usual sense, and (ii) the increments of Brownian motion are Gaussian distributed.The next useful lemma provides some important pathwise properties ofLt with two-sided time t∈ℝ.Lemma 2.1 (pathwise boundedness and convergence).
LetLt be a two-sided Lévy motion on ℝd for which 𝔼|L1|<∞ and 𝔼L1=0. Then we have the following.(i)
limt→±∞(1/t)Lt=0,a.s.(ii)
The integrals∫-∞te-λ(t-s)dLs(ω) are pathwisely uniformly bounded in λ>1 on finite time intervals [T1,T2] in ℝ.(iii)
The integrals∫T1te-λ(t-s)dLs(ω)→0 as λ→∞, pathwise on finite time intervals [T1,T2] in ℝ.Proof.
(i) This convergence result comes from the law of large numbers, in [11, Theorem 36.5].
(ii) Since the functionh(t)=e-λt is continuous in t, integrating by parts we obtain
(2.5)∫-∞te-λ(t-s)dLs(ω)=Lt(ω)-λ∫-∞te-λ(t-s)Ls(ω)ds.
Then by (i) and the fact that every càdlàg function is bounded on finite closed intervals, we conclude (ii).
(iii) Integrating again by parts, it follows that
(2.6)∫T1te-λ(t-s)dLs(ω)=(Lt-LT1)e-λ(t-T1)+λ∫T1te-λ(t-s)(Lt(ω)-Ls(ω))ds,
from which the result follows.Remark.
The assumptions onLt in the above lemma are satisfied by a wide class of Lévy processes, for instance, the symmetric α-stable Lévy motion on ℝd with 1<α<2. Indeed, in this case, we have ∫|x|>1|x|ν(dx)<∞, and then E|L1|<∞, see [11, Theorem 25.3].For the canonical sample space of Lévy processes, that is,Ω=D(ℝ,ℝd) of càdlàg functions which are defined on ℝ and taking values in ℝd is not separable, if we use the usual compact-open metric. However, it is complete and separable when endowed with the Skorohod metric (see, e.g., [13], [14, page 405]), in which case we call D(ℝ,ℝd) a Skorohod space.


### 2.2. Random Dynamical Systems

Following Arnold [7], a random dynamical system (RDS) on a probability space (Ω,ℱ,ℙ) consists of two ingredients: a driving flow θt on the probability space Ω, that is, θt is a deterministic dynamical system, and a cocycle mapping φ:ℝ×Ω×ℝd→ℝd, namely, φ satisfies the conditions(2.7)φ(0,ω)=idℝd,φ(t+s,ω)=φ(t,θsω)∘φ(s,ω)
for all ω∈Ω and all s,t∈ℝ. This cocycle is required to be at least measurable from the σ-field ℬ(ℝ)×ℱ×ℬ(ℝd) to the σ-field ℬ(ℝd).For random dynamical systems driven by Lévy noise we takeΩ=D(ℝ,ℝd) with the Skorohod metric as the canonical sample space and denote by ℱ:=ℬ(D(ℝ,ℝd)) the associated Borel σ-field. Let μL be the (Lévy) probability measure on ℱ which is given by the distribution of a two-sided Lévy process with paths in D(ℝ,ℝd).The driving systemθ=(θt,t∈ℝ) on Ω is defined by the shift(2.8)(θtω)(s):=ω(t+s)-ω(t).
The map (t,ω)→θtω is continuous, thus measurable ([7, page 545]), and the (Lévy) probability measure is θ-invariant, that is,(2.9)μL(θt-1(A))=μL(A)
for all A∈ℱ, see [4, page 325].We say that a familyÂ={A(ω),ω∈Ω} of nonempty measurable compact subsets A(ω) of ℝd is invariant for a RDS (θ,φ), if φ(t,ω,A(ω))=A(θtω) for all t>0 and that it is a random attractor if in addition it is pathwise pullback attracting in the sense that(2.10)Hd*(φ(t,θ-tω,D(θ-tω)),A(ω))→0ast→∞
for all suitable families (called the attracting universe) of D̂={D(ω),ω∈Ω} of nonempty measurable bounded subsets D(ω) of ℝd, where Hd*(A,B)=supx∈Ainfy∈B|x-y| is the Hausdorff semi-distance on ℝd.The following result about the existence of a random attractor may be proved similarly as in [2, 15–18].Lemma 2.3 (random attractor for càdlàg RDS).
Let (θ,φ) be an RDS on Ω×ℝd and let φ be continuous in space, but càdlàg in time. If there exits a family B̂={B(ω),ω∈Ω} of nonempty measurable compact subsets B(ω) of ℝd and a TD̂,ω≥0 such that
(2.11)φ(t,θ-tω,D(θ-tω))⊂B(ω),∀t≥TD̂,ω,
for all families D̂={D(ω),ω∈Ω} in a given attracting universe, then the RDS (θ,φ) has a random attractor Â={A(ω),ω∈Ω} with the component subsets defined for each ω∈Ω by
(2.12)A(ω)=⋂s>0⋃t≥sφ(t,θ-tω,B(θ-tω))¯.
Forevermore if the random attractor consists of singleton sets, that is, A(ω)={X*(ω)} for some random variable X*, then Xt*(ω)=X*(θtω) is a stationary stochastic process.We also need the following Gronwall's lemma from [19].Lemma.
Letx(t) satisfy the differential inequality
(2.13)ddt+x≤g(t)x+h(t),
where (d/dt+)x:=limh↓0+((x(t+h)-x(t))/h) is right-hand derivative of x. Then
(2.14)x(t)≤x(0)exp[∫0tg(r)dr]+∫0texp[∫stg(r)dr]h(s)ds.


### 2.3. Dissipative Synchronization

Suppose that we have two autonomous ordinary differential equations inℝd,(2.15)dxdt=f(x),dydt=g(y),
where the vector fields f and g are sufficiently regular (e.g., differentiable) to ensure the existence and uniqueness of local solutions, and additionally satisfy one-sided dissipative Lipschitz conditions(2.16)max{〈x1-x2,f(x1)-f(x2)〉,〈x1-x2,g(x1)-g(x2)〉}≤-l|x1-x2|2
on ℝd for some l>0. These dissipative Lipschitz conditions ensure existence and uniqueness of global solutions. Each of the systems has a unique globally asymptotically stable equilibria, x¯ and y¯, respectively [18]. Then, the coupled deterministic dynamical system in ℝ2d(2.17)dxdt=f(x)+λ(y-x),dydt=g(x)+λ(x-y)
with parameter λ>0 also satisfies a one-sided dissipative Lipschitz condition and, hence, also has a unique equilibrium (x¯λ,y¯λ), which is globally asymptotically stable [18]. Moreover, (x¯λ,y¯λ)→(z¯,z¯) as λ→∞, where z¯ is the unique globally asymptotically stable equilibrium of the “averaged” system in ℝd(2.18)dzdt=12(f(z)+g(z)).
This phenomenon is known as synchronization for the coupled deterministic system (2.17). The parameter λ often appears naturally in the context of the problem under consideration. For example in control theory it is a control parameter which can be chosen by the engineer, whereas in chemical reactions in thin layers separated by a membrane it is the reciprocal of the thickness of the layers; see [20].Caraballo and Kloeden [2], and Caraballo et al. [3] showed that this synchronization phenomenon persists under Gaussian Brownian noise, provided that asymptotically stable stochastic stationary solutions are considered rather than asymptotically stable steady state solutions. Recall that a stationary solution X* of a SDE system may be characterized as a stationary orbit of the corresponding random dynamical system (θ,φ) (defined by the SDE system), namely, φ(t,ω,X*(ω))=X*(θtω).The aim of this paper is to investigate synchronization under nonGaussian Lévy noise. In particular, we consider a coupled SDE system inℝd, driven by Lévy motion(2.19)dXt=(f(Xt)+λ(Yt-Xt))dt+adLt1,dYt=(g(Yt)+λ(Xt-Yt))dt+bdLt2,
where a,b∈ℝd are constant vectors with no components equal to zero, Lt1, Lt2 are independent two-sided scalar Lévy motion as in Lemma 2.1, and f,g satisfy the one-sided dissipative Lipschitz conditions (2.16).In addition to the one-sided Lipschitz dissipative condition (2.16) on the functions f and g, as in [2] we further assume the following integrability condition. There exists m0>0 such that for any m∈(0,m0], and any càdlàg function u:ℝ→ℝd with subexponential growth it follows(2.20)∫-∞tems|f(u(s))|2ds<∞,∫-∞tems|g(u(s))|2ds<∞.
Without loss of generality, we assume that the one-sided dissipative Lipschitz constant l≤m0.In the next section we will show that the coupled system (2.19) has a unique stationary solution (X̃tλ,Ỹtλ) which is pathwise globally asymptotically stable with (X̃tλ,Ỹtλ)→(Zt∞,Zt∞) as λ→∞, pathwise on finite time intervals [T1,T2], where Zt∞ is the unique pathwise globally asymptotically stable stationary solution of the “averaged" SDE in ℝd(2.21)dZt=12[f(Zt)+g(Zt)]dt+12adLt1+12bdLt2.


## 2.1. Lévy Processes

A Lévy process or motion onℝd is characterized by a drift parameter γ∈ℝd, a covariance d×d matrix A, and a nonnegative Borel measure ν, defined on (ℝd,ℬ(ℝd)) and concentrated on ℝd∖{0}, which satisfies(2.1)∫ℝd∖{0}(|y|2⋀1)ν(dy)<∞,
or equivalently(2.2)∫ℝd∖{0}|y|21+|y|2ν(dy)<∞.
This measure ν is the so-called the Lévy jump measure of the Lévy process Lt. Moreover Lévy process Lt has the following Lévy-Itô decomposition:(2.3)Lt=γt+Bt+∫|x|<1xÑ(t,dx)+∫|x|≥1xN(t,dx),
where N(dt,dx) is Poisson random measure,(2.4)Ñ(dt,dx)=N(dt,dx)-ν(dx)dt,
is the compensated Poisson random measure of Lt, and Bt is an independent Brownian motion on ℝd with covariance matrix A (see [4, 10–12]). We call (A,ν,γ) the generating triplet.General semimartingales, especially Lévy motions, are thought to be appropriate models for nonGaussian processes with jumps [11]. Let us recall that a Lévy motion Lt is a nonGaussian process with independent and stationary increments. Moreover, its sample paths are only continuous in probability, namely, ℙ(|Lt-Lt0|≥δ)→0 as t→t0 for any positive δ. With a suitable modification [4], these paths may be taken as càdlàg, that is, paths are continuous on the right and have limits on the left. This continuity is weaker than the usual continuity in time. In fact, a càdlàg function has finite or at most countable discontinuities on any time interval (see, e.g., [4, page 118]). This generalizes the Brownian motion Bt, since Bt satisfies all these three conditions, but additionally, (i) almost every sample path of the Brownian motion is continuous in time in the usual sense, and (ii) the increments of Brownian motion are Gaussian distributed.The next useful lemma provides some important pathwise properties ofLt with two-sided time t∈ℝ.Lemma 2.1 (pathwise boundedness and convergence).
LetLt be a two-sided Lévy motion on ℝd for which 𝔼|L1|<∞ and 𝔼L1=0. Then we have the following.(i)
limt→±∞(1/t)Lt=0,a.s.(ii)
The integrals∫-∞te-λ(t-s)dLs(ω) are pathwisely uniformly bounded in λ>1 on finite time intervals [T1,T2] in ℝ.(iii)
The integrals∫T1te-λ(t-s)dLs(ω)→0 as λ→∞, pathwise on finite time intervals [T1,T2] in ℝ.Proof.
(i) This convergence result comes from the law of large numbers, in [11, Theorem 36.5].
(ii) Since the functionh(t)=e-λt is continuous in t, integrating by parts we obtain
(2.5)∫-∞te-λ(t-s)dLs(ω)=Lt(ω)-λ∫-∞te-λ(t-s)Ls(ω)ds.
Then by (i) and the fact that every càdlàg function is bounded on finite closed intervals, we conclude (ii).
(iii) Integrating again by parts, it follows that
(2.6)∫T1te-λ(t-s)dLs(ω)=(Lt-LT1)e-λ(t-T1)+λ∫T1te-λ(t-s)(Lt(ω)-Ls(ω))ds,
from which the result follows.Remark.
The assumptions onLt in the above lemma are satisfied by a wide class of Lévy processes, for instance, the symmetric α-stable Lévy motion on ℝd with 1<α<2. Indeed, in this case, we have ∫|x|>1|x|ν(dx)<∞, and then E|L1|<∞, see [11, Theorem 25.3].For the canonical sample space of Lévy processes, that is,Ω=D(ℝ,ℝd) of càdlàg functions which are defined on ℝ and taking values in ℝd is not separable, if we use the usual compact-open metric. However, it is complete and separable when endowed with the Skorohod metric (see, e.g., [13], [14, page 405]), in which case we call D(ℝ,ℝd) a Skorohod space.


## 2.2. Random Dynamical Systems

Following Arnold [7], a random dynamical system (RDS) on a probability space (Ω,ℱ,ℙ) consists of two ingredients: a driving flow θt on the probability space Ω, that is, θt is a deterministic dynamical system, and a cocycle mapping φ:ℝ×Ω×ℝd→ℝd, namely, φ satisfies the conditions(2.7)φ(0,ω)=idℝd,φ(t+s,ω)=φ(t,θsω)∘φ(s,ω)
for all ω∈Ω and all s,t∈ℝ. This cocycle is required to be at least measurable from the σ-field ℬ(ℝ)×ℱ×ℬ(ℝd) to the σ-field ℬ(ℝd).For random dynamical systems driven by Lévy noise we takeΩ=D(ℝ,ℝd) with the Skorohod metric as the canonical sample space and denote by ℱ:=ℬ(D(ℝ,ℝd)) the associated Borel σ-field. Let μL be the (Lévy) probability measure on ℱ which is given by the distribution of a two-sided Lévy process with paths in D(ℝ,ℝd).The driving systemθ=(θt,t∈ℝ) on Ω is defined by the shift(2.8)(θtω)(s):=ω(t+s)-ω(t).
The map (t,ω)→θtω is continuous, thus measurable ([7, page 545]), and the (Lévy) probability measure is θ-invariant, that is,(2.9)μL(θt-1(A))=μL(A)
for all A∈ℱ, see [4, page 325].We say that a familyÂ={A(ω),ω∈Ω} of nonempty measurable compact subsets A(ω) of ℝd is invariant for a RDS (θ,φ), if φ(t,ω,A(ω))=A(θtω) for all t>0 and that it is a random attractor if in addition it is pathwise pullback attracting in the sense that(2.10)Hd*(φ(t,θ-tω,D(θ-tω)),A(ω))→0ast→∞
for all suitable families (called the attracting universe) of D̂={D(ω),ω∈Ω} of nonempty measurable bounded subsets D(ω) of ℝd, where Hd*(A,B)=supx∈Ainfy∈B|x-y| is the Hausdorff semi-distance on ℝd.The following result about the existence of a random attractor may be proved similarly as in [2, 15–18].Lemma 2.3 (random attractor for càdlàg RDS).
Let (θ,φ) be an RDS on Ω×ℝd and let φ be continuous in space, but càdlàg in time. If there exits a family B̂={B(ω),ω∈Ω} of nonempty measurable compact subsets B(ω) of ℝd and a TD̂,ω≥0 such that
(2.11)φ(t,θ-tω,D(θ-tω))⊂B(ω),∀t≥TD̂,ω,
for all families D̂={D(ω),ω∈Ω} in a given attracting universe, then the RDS (θ,φ) has a random attractor Â={A(ω),ω∈Ω} with the component subsets defined for each ω∈Ω by
(2.12)A(ω)=⋂s>0⋃t≥sφ(t,θ-tω,B(θ-tω))¯.
Forevermore if the random attractor consists of singleton sets, that is, A(ω)={X*(ω)} for some random variable X*, then Xt*(ω)=X*(θtω) is a stationary stochastic process.We also need the following Gronwall's lemma from [19].Lemma.
Letx(t) satisfy the differential inequality
(2.13)ddt+x≤g(t)x+h(t),
where (d/dt+)x:=limh↓0+((x(t+h)-x(t))/h) is right-hand derivative of x. Then
(2.14)x(t)≤x(0)exp[∫0tg(r)dr]+∫0texp[∫stg(r)dr]h(s)ds.


## 2.3. Dissipative Synchronization

Suppose that we have two autonomous ordinary differential equations inℝd,(2.15)dxdt=f(x),dydt=g(y),
where the vector fields f and g are sufficiently regular (e.g., differentiable) to ensure the existence and uniqueness of local solutions, and additionally satisfy one-sided dissipative Lipschitz conditions(2.16)max{〈x1-x2,f(x1)-f(x2)〉,〈x1-x2,g(x1)-g(x2)〉}≤-l|x1-x2|2
on ℝd for some l>0. These dissipative Lipschitz conditions ensure existence and uniqueness of global solutions. Each of the systems has a unique globally asymptotically stable equilibria, x¯ and y¯, respectively [18]. Then, the coupled deterministic dynamical system in ℝ2d(2.17)dxdt=f(x)+λ(y-x),dydt=g(x)+λ(x-y)
with parameter λ>0 also satisfies a one-sided dissipative Lipschitz condition and, hence, also has a unique equilibrium (x¯λ,y¯λ), which is globally asymptotically stable [18]. Moreover, (x¯λ,y¯λ)→(z¯,z¯) as λ→∞, where z¯ is the unique globally asymptotically stable equilibrium of the “averaged” system in ℝd(2.18)dzdt=12(f(z)+g(z)).
This phenomenon is known as synchronization for the coupled deterministic system (2.17). The parameter λ often appears naturally in the context of the problem under consideration. For example in control theory it is a control parameter which can be chosen by the engineer, whereas in chemical reactions in thin layers separated by a membrane it is the reciprocal of the thickness of the layers; see [20].Caraballo and Kloeden [2], and Caraballo et al. [3] showed that this synchronization phenomenon persists under Gaussian Brownian noise, provided that asymptotically stable stochastic stationary solutions are considered rather than asymptotically stable steady state solutions. Recall that a stationary solution X* of a SDE system may be characterized as a stationary orbit of the corresponding random dynamical system (θ,φ) (defined by the SDE system), namely, φ(t,ω,X*(ω))=X*(θtω).The aim of this paper is to investigate synchronization under nonGaussian Lévy noise. In particular, we consider a coupled SDE system inℝd, driven by Lévy motion(2.19)dXt=(f(Xt)+λ(Yt-Xt))dt+adLt1,dYt=(g(Yt)+λ(Xt-Yt))dt+bdLt2,
where a,b∈ℝd are constant vectors with no components equal to zero, Lt1, Lt2 are independent two-sided scalar Lévy motion as in Lemma 2.1, and f,g satisfy the one-sided dissipative Lipschitz conditions (2.16).In addition to the one-sided Lipschitz dissipative condition (2.16) on the functions f and g, as in [2] we further assume the following integrability condition. There exists m0>0 such that for any m∈(0,m0], and any càdlàg function u:ℝ→ℝd with subexponential growth it follows(2.20)∫-∞tems|f(u(s))|2ds<∞,∫-∞tems|g(u(s))|2ds<∞.
Without loss of generality, we assume that the one-sided dissipative Lipschitz constant l≤m0.In the next section we will show that the coupled system (2.19) has a unique stationary solution (X̃tλ,Ỹtλ) which is pathwise globally asymptotically stable with (X̃tλ,Ỹtλ)→(Zt∞,Zt∞) as λ→∞, pathwise on finite time intervals [T1,T2], where Zt∞ is the unique pathwise globally asymptotically stable stationary solution of the “averaged" SDE in ℝd(2.21)dZt=12[f(Zt)+g(Zt)]dt+12adLt1+12bdLt2.


## 3. Systems Driven by Lévy Noise

For the coupled system (2.19), we have the following two lemmas about its stationary solutions.Lemma 3.1 (existence of stationary solutions).
If the Assumption (2.20) holds, f and g are continuous and satisfy the one-sided Lipschitz dissipative conditions (2.16) with Lipschitz constant l, then the coupled stochastic system (2.19) has a unique stationary solution.Proof.
First, the stationary solutions of the Langevin equations [4, 21]
(3.1)dXt=-λXtdt+adLt1,dYt=-λYtdt+bdLt2
are given by
(3.2)X̅tλ=ae-λt∫-∞teλsdLt1,Y̅tλ=be-λt∫-∞teλsdLt2.
The differences of the solutions of (2.19) and these stationary solutions satisfy a system of random ordinary differential equations, with right-hand derivative in time
(3.3)ddt+(Xt-X̅tλ)=f(Xt)+λ(Yt-Xt)+λX̅tλ,ddt+(Yt-Y̅tλ)=g(Yt)+λ(Xt-Yt)+λY̅tλ.
The equations (3.3) are equivalent to
(3.4)ddt+Utλ=f(Xt)+λ(Vtλ-Utλ)+λY̅tλ,ddt+Vtλ=g(Yt)+λ(Utλ-Vtλ)+λX̅tλ,
where Utλ=Xt-X̅tλ and Vtλ=Yt-Y̅tλ. Thus,
(3.5)12ddt+(|Utλ|2+|Vtλ|2)=(Utλ,f(Utλ+X̅tλ)-f(X̅tλ))+(Vtλ,g(Vtλ+Y̅tλ)-g(Y̅tλ))+(Utλ,f(X̅tλ)+λY̅tλ)+(Vtλ,g(Y̅tλ)+λX̅tλ)-λ|Utλ-Vtλ|2≤-l2(|Utλ|2+|Vtλ|2)+2l|f(X̅tλ)+λY̅tλ|2+2l|g(Y̅tλ)+λX̅tλ|2.
Hence, by Lemma 2.4,
(3.6)|Utλ|2+|Vtλ|2≤(|Ut0λ|2+|Vt0λ|2)el(t-t0)+4e-ltl∫t0tels[|f(X̅tλ)+λY̅tλ|2+|g(Y̅tλ)+λX̅tλ|2]ds.
Define
(3.7)|Rλ(ω)|2=1+4l∫-∞0els[|f(X̅λ(θsω))+λY̅λ(θsω)|2+|g(Y̅λ(θsω))+λX̅λ(θsω)|2]ds
and let B2dλ(ω) be a random closed ball in ℝ2d centered on the origin and of radius Rλ(ω).
Now we can use pathwise pullback convergence (i.e., witht0→-∞) to show that |Utλ|2+|Vtλ|2 is pathwise absorbed by the family B̂2dλ={B2dλ(ω),ω∈Ω}, that is, for appropriate families D̂, there exists TD̂,ω≥0 such that
(3.8)φ(t,θ-tω,D(θ-tω))⊂B2dλ(ω),∀t≥TD̂,ω.
Hence, by Lemma2.3, the coupled system has a random attractor Âλ={Aλ(ω),ω∈Ω} with Aλ(ω)⊂B2dλ(ω).
Note that, by Lemma2.1, it can be shown that the random compact absorbing balls B2dλ(ω) are contained in the common compact ball for λ≥1.
However, the difference(ΔXt,ΔYt)=(Xt1-Xt2,Yt1-Yt2) of any pair of solutions satisfies the system of random ordinary differential equations
(3.9)ddt+ΔXt=f(Xt1)-f(Xt2)+λ(ΔYt-ΔXt),ddt+ΔYt=g(Yt1)-g(Yt2)-λ(ΔYt-ΔXt),
so
(3.10)ddt+(|ΔXt|2+|ΔYt|2)=2(ΔXt,f(Xt1)-f(Xt2))+2(ΔYt,g(Yt1)-g(Yt2))-2λ|ΔXt-ΔYt|2≤-2l(|ΔXt|2+|ΔYt|2)
from which we obtain
(3.11)|ΔXt|2+|ΔYt|2≤(|ΔX0|2+|ΔY0|2)e-2lt
which means all solutions converge pathwise to each other as t→∞. Thus the random attractor consists of a singleton set formed by an ordered pair of stationary processes (X̃tλ(ω),Ỹtλ(ω)) or equivalently (X̃λ(θtω),Ỹλ(θtω)).Lemma 3.2 (a property of stationary solutions).
The stationary solutions of the coupled stochastic system (2.19) have the following asymptotic behavior:
(3.12)X̃tλ(ω)-Ỹtλ(ω)→0asλ→∞
pathwise on any bounded time interval [T1,T2] of ℝ.Proof.
Since(3.13)d(X̃tλ-Ỹtλ)=(-2λ(X̃tλ-Ỹtλ)+f(X̃tλ)-g(Ỹtλ))dt+adLt1-bdLt2,
we have
(3.14)d(Dtλe2λt)=e2λt(f(X̃tλ)-g(Ỹtλ))+ae2λtdLt1-be2λtdLt2,
where Dtλ=X̃tλ-Ỹtλ, so pathwise
(3.15)|Dtλ|≤e-2λ(t-T1)|DT1λ|+∫T1te-2λ(t-s)(|f(X̃sλ)|+|g(Ỹsλ)|)ds+|a||∫T1te-2λ(t-s)dLt1|+|b||∫T1te-2λ(t-s)dLt2|.
By Lemma 2.1 we see that the radius Rλ(θtω) is pathwise uniformly bounded on each bounded time interval [T1,T2], so we see that the right hand of above inequality converge to 0 as λ→∞ pathwise on the bounded time interval [T1,T2].We now present the main result of this paper.Theorem 3.3 (synchronization under Lévy noise).
Suppose that the coupled stochastic system inℝ2d(3.16)dXt=(f(Xt)+λ(Yt-Xt))dt+adLt1,dYt=(g(Yt)+λ(Xt-Yt))dt+bdLt2
defines a random dynamical system (θ,φ). In addition, assume that the continuous functions f,g satisfy the integrability condition (2.20) as well as the one-sided Lipschitz dissipative condition (2.16), then the coupled stochastic system (3.16) is synchronized to a single averaged SDE in ℝd(3.17)dZt=12[f(Zt)+g(Zt)]dt+a2dLt1+b2dLt2,
in the sense that the stationary solutions of (3.16) pathwise converge to that of (3.17), that is, (X̃tλ,Ỹtλ)→(Zt∞,Zt∞) pathwise on any bounded time interval [T1,T2] as parameter λ→∞.Proof.
It is enough to demonstrate the result for any sequenceλn→∞. Define
(3.18)Ztλ:=12[X̃tλ+Ỹtλ],t∈ℝ.
Note that Ztλ(ω)=Zλ(θtω) satisfies the equation
(3.19)dZtλ=12[f(X̃tλ)+g(Ỹtλ)]dt+a2dLt1+b2dLt2.
Also we define
(3.20)Z̅t(ω)=Z̅(θtω):=12[X̅t(ω)+Y̅t(ω)],t∈ℝ,
where X̅t and Y̅t are the (stationary) solutions of the Langevin equations
(3.21)dXt=-Xtdt+adLt1,dYt=-Ytdt+bdLt2,
that is,
(3.22)X̅t=ae-t∫-∞tesdLt1,Y̅t=be-t∫-∞tesdLt2.
The difference Ztλ-Z̅t satisfies
(3.23)2(Ztλ-Z̅t)=2(Zλ-Z̅)+∫0t(f(X̃sλ)+g(Ỹsλ)+X̅s+Y̅s)ds.
By Lemma 2.1, and the fact that these solutions belong to the common compact ball and every càdlàg function is bounded on finite closed intervals, we obtain
(3.24)|f(X̃tλ(ω))+g(Ỹtλ(ω))|+|X̅t(ω)+Y̅t(ω)|≤MT1,T2(ω)<∞,
which implies uniform boundedness as well as equicontinuity. Thus by the Ascoli-Arzela theorem [13], we conclude that for any sequence λn→∞, there is a random subsequence λnj(ω)→∞, such that Ztλnj(ω)-Z̅t(ω)→Zt∞(ω)-Z̅t(ω) as j→∞. Thus Ztλnj(ω)→Zt∞(ω) as j→∞. Now, by Lemma 3.2(3.25)Ztλnj(ω)-Ỹtλnj(ω)=X̃tλnj(ω)-Ỹtλnj(ω)2→0,Ztλnj(ω)-X̃tλnj(ω)=Ỹtλnj(ω)-X̃tλnj(ω)2→0,
as λnj→∞, so
(3.26)X̃tλnj(ω)=2Ztλnj(ω)-Ỹtλnj(ω)→Zt∞(ω),Ỹtλnj(ω)=2Ztλnj(ω)-X̃tλnj(ω)→Zt∞(ω),
as λnj→∞.
Using the integral representation of the equation, it can be verified thatZt∞ is a solution of the averaged random differential equation (3.17) for all t∈ℝ. The drift of this SDE satisfies the dissipative one-sided condition (2.16). It has a random attractor consisting of a singleton set formed by a stationary orbit, which must be equal to Zt∞.
Finally, we note that all possible subsequences ofZtλn have the same pathwise limit. Thus the full sequence Ztλn converges to Zt∞, as λn→∞. This completes the proof.


### 3.1. An Example

Consider two scalar SDEs:(3.27)dXt=-(Xt+1)dt+dLt1,dYt=-(Yt+3)dt+2dLt2,
which we rewrite as(3.28)dXt=-Xtdt+dLt3,dYt=-Ytdt+2dLt4,
where Lt3=-t+Lt1 and Lt4=-3t/2+Lt2.The corresponding coupled system (3.16) is(3.29)dXt=-Xtdt+λ(Yt-Xt)dt+dLt3,dYt=-Ytdt+λ(Xt-Yt)dt+2dLt4
with the stationary solutions(3.30)X̃tλ=∫-∞te-(λ+1)(t-s)coshλ(t-s)dLs3+2∫-∞te-(λ+1)(t-s)sinhλ(t-s)dLs4,Ỹtλ=∫-∞te-(λ+1)(t-s)sinhλ(t-s)dLs3+2∫-∞te-(λ+1)(t-s)coshλ(t-s)dLs4.
Let λ→∞, then(3.31)(X̃tλ,Ỹtλ)→(Zt∞,Zt∞),
where Zt∞, given by(3.32)Zt∞=∫-∞t12e-(t-s)dLs3+∫-∞te-(t-s)dLs4,
is the stationary solution of the following averaged SDE:(3.33)dZt=-Ztdt+12dLt3+dLt4,
which is equivalent to the following SDE, in terms of the original Lévy motions L1 and L2,(3.34)dZt=-(Zt+2)dt+12dLt1+dLt2.


## 3.1. An Example

Consider two scalar SDEs:(3.27)dXt=-(Xt+1)dt+dLt1,dYt=-(Yt+3)dt+2dLt2,
which we rewrite as(3.28)dXt=-Xtdt+dLt3,dYt=-Ytdt+2dLt4,
where Lt3=-t+Lt1 and Lt4=-3t/2+Lt2.The corresponding coupled system (3.16) is(3.29)dXt=-Xtdt+λ(Yt-Xt)dt+dLt3,dYt=-Ytdt+λ(Xt-Yt)dt+2dLt4
with the stationary solutions(3.30)X̃tλ=∫-∞te-(λ+1)(t-s)coshλ(t-s)dLs3+2∫-∞te-(λ+1)(t-s)sinhλ(t-s)dLs4,Ỹtλ=∫-∞te-(λ+1)(t-s)sinhλ(t-s)dLs3+2∫-∞te-(λ+1)(t-s)coshλ(t-s)dLs4.
Let λ→∞, then(3.31)(X̃tλ,Ỹtλ)→(Zt∞,Zt∞),
where Zt∞, given by(3.32)Zt∞=∫-∞t12e-(t-s)dLs3+∫-∞te-(t-s)dLs4,
is the stationary solution of the following averaged SDE:(3.33)dZt=-Ztdt+12dLt3+dLt4,
which is equivalent to the following SDE, in terms of the original Lévy motions L1 and L2,(3.34)dZt=-(Zt+2)dt+12dLt1+dLt2.



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*Source: 502803-2010-02-23.xml*