# Transportation Inequalities for Coupled Fractional Stochastic Evolution Equations Driven by Fractional Brownian Motion

Authors: Wentao Zhan; Yuanyuan Jing; Liping Xu; Zhi Li
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we consider the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameterH∈1/4,1/2. Our approach is based on Perov’s fixed-point theorem. Furthermore, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.


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## Body

## 1. Introduction

In the research of various fields of science and engineering, fractional stochastic differential equations (SDEs) play a significant role in the modeling of many complex phenomena in diverse areas. The intensive development in both theory and applications of fractional SDEs was investigated in [1–4]. In addition, many scholars have also developed interest in systems with memory or after effect (i.e., systems with finite delays in the state equation). Therefore, it is necessary to study stochastic evolution equations with finite delays. The equation is widely used in network flow analysis, mathematical finance, astrophysics, hydrology, image processing, and other directions [5, 6]. For the nature of the existence and uniqueness of the mild solutions, Sakthivel et al. [7] considered the nonlinear-type fractional SDE, Li [8] established stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space, and Mophou [9] was concerned about impulsive fractional semilinear differential equations.Moreover, under the research of many scholars, transportation inequalities are developed greatly in various SDEs and stochastic systems with respect to the different measure conditions. Among others, the Girsanov transformation argument introduced in [10] has been efficiently applied, e.g., Wu and Zhang [11] considered infinite-dimensional dynamical systems with respect to L2 metric; U¨stu¨nel [12] studied the multivalued SDE and singular SDE under uniform distance; besides, Bao et al. [13] investigated the neutral functional SDE with respect to both the uniform distance and the L2 distance; Saussereau [14] researched the SDE driven by a fractional Brownian motion; futhermore, Li and Luo [15] took it into account that stochastic delay evolution equations driven by fractional Brownian motion with the hurst parameter H > 1/2 under the L2 metric and the uniform metric; and Boufoussi and Hajji [16] established the transportation inequalities, with respect to the uniform distance, for the law of the mild solution for a neutral stochastic differential equation with finite delay, driven by a fractional Brownian motion with the Hurst parameter lesser than 1/2 in a Hilbert space.In connection with the aforementioned works, in this paper, we investigate the existence, uniqueness, and propertyT2C under the uniform distance for the law of mild solution of the coupled fractional stochastic delay evolution equations with finite delay driven by a fractional Brownian motion with the Hurst parameter 0 < H < 1/2:(1)Dα1cxt=A1xt+f1t,xt,ytdt+σ1tdB1Ht,t∈0,T,Dα2cyt=A2yt+f2t,xt,ytdt+σ2tdB2Ht,t∈0,T,xt=ϕ1t,t∈−r,0,yt=ϕ2t,t∈−r,0,where Dαic is the Caputo fractional derivative of order αi∈1/2,1, for each i=1,2, as for the state x⋅, y⋅ has values in a real and separable Hilbert X with an inner product ⋅,⋅X and norm ⋅X, where Ai,i=1,2 are the infinitesimal generators of analytic semigroups of bounded linear operators Tit,t≥0,BiH is the fractional Brownian motion on a real and separable Hilbert space Y, with the Hurst parameter H∈0,1/2, and let r>0 denote the constant. As for yt, we mean the segment solution which is defined in the usual way, that is, if y⋅,⋅:−r,T×Ω⟶X, then for any t>0(2)ytθ,ω=yt+θ,ω,θ∈−r,0,ω∈Ω.Before describing the properties fulfilled by operatorsfi,σi, we need to introduce some nontations and describe some spaces. Let D0 denote the space of all continuous functions φ:−r,0×Ω⟶X such that φθ,⋅ is ℱ0-measurable for each θ∈−r,0 and ∫−r0EφX2dt<∞. In the space D0, we endow with the following norm:(3)φD02=∫−r0EφtX2dt.Next, we denote byCa,b;L2Ω;X=Ca,b;L2Ω,ℱ,ℙ;X the Banach space of all continuous functions from a,b into L2Ω;X. Now, fixing T>0, we define(4)DT=y:y∈C−r,T;L2Ω;X,supt∈0,TEytX2<∞,∫−r0EytX2dt<∞,endowing with the following norm:(5)yDT=supt∈0,TEytX2+ytD0.We give initial dataφ1,φ2∈D0, and Y is another real and separable Hilbert space, and BQiH=BiH is a Y-valued fractional Brownian motion with increment covariance given by a nonnegative trace class operator Qi, and LY,X represents the space of all bounded, continuous, and linear operators from Y into X.We denotefi:J×D0×D0⟶X and σi:J⟶LQi0Y,X. Here, let us denote LQi0Y,X by the space of all Qi-Hilbert–Schmidt operators from Y into X for each i=1,2 which will also be introduced in the next section.Now, let us present the relevant knowledge of transportation inequalities. To connect the measure distances with the probability measures, we consider the transportation distance, also called as Wasserstein distance. LetE,d be a metric space provided with the σ-field ℬ, such that d⋅,⋅ is ℬ×ℬ-measurable. Fixing p≥1 and for any probability measures μ and ν on E, we define the Wasserstein distance of order p between μ and ν as(6)Wpdμ,v=infπ∈∏μ,v∫Ε×Εdx,ypdπx,y1/p,where Πμ,ν denotes the totality of probability measures on E×E with the marginal μ and ν. The relative entropy of ν with respect to μ is defined as(7)Hv,μ=∫logdνdμdv,ν≪μ,+∞,otherwise.The probability measureμ satisfies Lp-transportation inequality on E,d if there exists a constant C≥0 such that for any probability measure ν,(8)Wpdμ,ν≤2CHνμ.As usual, we writeμ∈TpC for this relation. The property T2C is of particular interest. We will investigate the property T2C for the law of the mild solution of stochastic delay evolution equations driven by fractional Brownian motion with the Hurst parameter 1/4 < H < 1/2 under the uniform distance.This paper is organized as follows. In Section2, we introduce some preliminaries used in this paper such as stochastic calculus, some properties of generalized Banach spaces, and fractional calculus. In Section 3, we state and prove the existence and uniqueness of the mild solution by using Perov’s fixed-point type in generalized Banach spaces. In Section 4, we investigate the property T2C for the law of the solution of fractional stochastic delay evolution equations driven by fractional Brownian motion with the Hurst parameter 1/4 < H < 1/2 under the uniform metric. In Section 5, we present an example to illustrate the efficiency of the obtained result.


## 2. Preliminaries

In this section, we introduce some notations and recall definitions and preliminary results which are used throughout this paper.LetΩ,ℱ,ℙ,ℱtt≥0 be a complete probability space furnished with a normal filtration ℱtt≥0. We postulate that the operator Ai is self-adjoint and there exists the eigenvectors ek corresponding to eigenvalues γk such that(9)Aiek=γkek,ek=2sinkπ,γk=π2k2,k∈ℕ+.For eachσ>0,Aiσ/2ek=γkσ/2,k=1,2,3,…, and let H˙σ be the domain of Aiσ/2 defined by(10)H˙σ=DAiσ/2=v∈L2X,s.t.vH˙σ2=∑k=0∞γkσ/2vk2<∞,where the vector vk=v,ek and the norm vH˙σ=Aiσ/2v. Let L2X be an X-valued Hilbert space with the inner product E⋅,⋅ and norm E⋅, and it is given by(11)L2X=χ:EχX2=∫ΩχωX2dℙω<∞,ω∈Ω.Definition 1.
ForH∈0,1, a continuous centered Gaussian process BHtt∈0,∞ with covariance function(12)RHt,s=EBHtBHs=12t2H+s2H−t−s2H,t,s∈0,∞,is called a one-dimensional fractional Brownian motion (fBm), and H is the Hurst parameter. In particular, when H = 1/2, BHt represents a standard Brownian motion.
Now, let us aim at the Wiener integral with respect to the fBm. To begin with,BHt has following integral expression (see [17]):(13)BHt=∫0tKHt,sdBs,where B=Bt:t∈0,T is a Wiener process, KHt,s is a square integrable kernel, for 0 < H < 1/2, and t>s; the formula is as follows (see [17]):(14)KHt,s=cHtsH−1/2t−sH−1/2−H−12s1/2−H∫stu−sH−1/2uH−3/2du,where cH=H/1−Hβ1−2H,H+1/2 and β⋅,⋅ denotes the Beta function. KHt,s=0,t≤s. Since 0 < H < 1/2, from (14), we can infer that(15)KHt,s≤cHt−sH−1/2+12−Hs1/2−H∫stu−sH−1/2u−H−1/2u2H−1du≤cHt−sH−1/2+12−Hs1/2−H∫stu−sH−1/2u−H−1/2du.
Then, we obtain(16)KHt,s≤2cHt−sH−1/2+sH−1/2.
Taking the derivative of (14) with respect to t, we can have(17)∂KH∂tt,s=cHH−1/2tsH−1/2t−sH−3/2.
Apparently, we can obtain the following inequality:(18)∂KH∂tt,s=cH12−Ht−sH−3/2.
Letℋ be the Hilbert space defined as the closure of the vector space spanned by the set of step functions I0,t,t∈0,T with respect to the scalar product:(19)I0,t,I0,s=RHt,s,∀t,s∈0,T.
Now, we consider the operatorKH,T∗ from ℋ to L20,T defined by(20)KH,T∗φs=KHT,sφs+∫sTφr−φs∂KH∂rr,sdr.
Furthermore,KH,T∗ is an isometry between ℋ and L20,T(see [18]). Taking account for B=Bt,t∈0,T defined by(21)Bt=BHKH∗−1Ι0,t,it turns out that B is a Wiener process. Moreover, for any φ∈ℋ, with (13), we have(22)∫0TφSdBHs:=BHφ=∫0TKH,Tφ∗tdBt.
For any0≤t≤T, we can also deduce(23)∫0tφsdBHs:=∫0TKH,T∗φΙ0,tsdBs=∫0tKH,t∗φsdBs,where KH,t∗ is defined in the same way as in (20) with t instead of T. Next, we will use the notation KH,t∗ without specifying the parameter t∈0,T.
LetX,⋅X,⋅,⋅X and Y,⋅Y,⋅,⋅Y be two real, separable Hilbert spaces and let ℒY,X denote the space of all bounded linear operators from Y to X. Let Q∈ℒY,X be a nonnegative self-adjoint operator i.e., Qen=λnen with trace trQ=∑n=1∞λn<∞, where λn∈ℝ+ and enn≥1 is a complete orthonormal basis in Y. We define the infinite-dimensional fBm on Y with covariance Q by the following formula:(24)BQHt=∑n=1∞λnenBnHt,t≥0,where BnHtn∈ℕ be a sequence of one-dimensional mutually independent standard fractional Brownian motions on Ω,ℱ,ℙ. BnHt is a Y-valued Gaussian process, starting from 0, and has zero mean and covariance:(25)EBQHt,xBQHs,y=Rs,tQx,y,for allx,y∈Yandt,s∈0,T.
LetL20Y,X be the space of all ξ∈LY,X such that ξQ1/2 is a Hilbert–Schmidt operator. The norm is given by(26)ξL20Y,X2:=∑n=1∞λnξsen2=trξQξ∗<∞.
Then,ξ is called a Q-Hilbert–Schmidt operator from Y to X.Definition 2.
Letφ:0,T⟶L20Y,X. Then, the Wiener integral of φ with respect to the fBm BQH is defined as follows:(27)∫0tφsdBQHs:=∑n=1∞∫0tλnφsendBnHs=∑n=1∞∫0tλnKH∗φensdBns,where Bn is the standard Brownian motion used to represent BnH as in (13), and the sum above is finite when ∑n=1∞λnKH∗φen<∞.
The classical Banach contraction principle was extended for contractive maps on spaces endowed with a vector-valued metric space by Perov [19] in 1964 and Precup [20, 21]. Now, we recall some useful definitions and results.Definition 3.
LetZ be a nonempty set. We denote by a vector-valued metric on Z defined as a mapping d:Z×Z⟶ℝn with the following properties:(1)
du,v≥0for allu,v∈Z;du,v=0,only ifu=v.(2)
du,v=du,vfor allu,v∈Z.(3)
du,v≤du,w+dw,vfor allu,v,w∈Z.
Now, we consider a generalized metric spaceZ,d. For r=r1,…,rn∈ℝ+n, we will define the open ball centered in x0 with radius r:(28)Bx0,r=x∈Z:dx0,x<r,and the closed ball centered in x0 with radius r:(29)Bx0,r¯=x∈Z:dx0,x≤r.
We state that for a generalized metric space, the notation of open and closed sets, convergence, Cauchy sequence, and completeness in a generalized metric space are similar to those in usual metric spaces. Ifx,y∈ℝn, x=x1,…,xn, y=y1,…,yn, by x≤y then we mean xi≤yi,i=1,…,n. Also, x=x1,…,xn and maxx,y=maxmaxx1,y1,…,maxxn,yn. If c∈ℝ, then x≤c means xi≤c for each i=1,…,n.Definition 4.
A generalized metric spaceZ,d where dx,y:=d1x,y⋮dnx,y is complete, if for every i=1,…,n, Z,di is a complete metric space.Definition 5.
We denote that a real square matrixM is convergent to zero if and only if its spectral radius ρM is strictly less than 1. In other words, it means that all the eigenvalues of M are in the open unit disc (i.e.,λ<1, for every λ∈ℂ with detM−λI=0, where I denotes the unit matrix of Mn×nℝ).Definition 6.
We denote that a nonsingular matrixA=aij1≤i,j≤n∈Mn×nℝ has the absolute value property if(30)A−1A≤I,where(31)A=aij1≤i,j≤n∈ℳn×nℝ.
Now, we need to use the following fixed-point theorem to prove the existence and uniqueness of mild solution for (1).Theorem 1 (see [19]).
LetZ,d be a complete generalized metric space with d:Z×Z⟶ℝn and let operator N:Z⟶Z be such that(32)dNx,Ny≤Mdx,y,for all x,y∈Z and some nonnegative square matrix M. If the matrix M is convergent to 0, that is, Mk⟶0 as k⟶0, then operator N has a unique fixed point x∗∈Z:(33)dNkx0,x∗≤MkI−M−1dNx0,x0,for every x0∈Z and k≥1.Definition 7.
The fractional integral of indexα with the lower limit 0 for a function f can be written as(34)Iαft=1Γα∫0tfst−s1−αds,t>0,α>0,provided the right-hand side is pointwise defined on 0,+∞, where Γ is the gamma function, which is defined by Γy:=∫0∞ty−1e−tdt.Definition 8.
The Caputo derivative of indexα for a function f∈Cn0,∞ is defined as(35)DCtαft=1Γn−α∫0tfnst−sα+1−nds=In−αfnt,t>0,n−1<α<n.


## 3. Existence and Uniqueness

In this section, we investigate the existence and uniqueness of a mild solution for (1). First of all, we will give some hypotheses which will be used to prove our main result; for this question, we assume that the following conditions hold.H.1 There exists constants afi,bfi∈ℝ+ for each i=1,2,… such that(36)∫0tfis,xs,ys−fis,x¯s,y¯sX2≤afi∫−rtxs−x¯sX2ds+bfi∫−rtys−y¯sX2ds,for allx,y,x¯,y¯∈C−r,T;X.H.2 The function σ:0,T⟶LQ0Y,X satisfies the following Ho¨lder continuous conditions, that is, there exists a constant Cσ>0 such that for all t,s∈0,T,(37)σt−σsℒQ0≤Cσt−sγ,whereγ>1−2H.Now, we state the following definition of mild solution for our problem.Definition 9.
AH-valued process ut=xt,yt is called a mild solution of (1) with respect to the probability space Ω,F,P, if x,y∈C−r,T;L2Ω;X, xt,yt=ϕ1t,ϕ2tfort∈−r,0, and for each t∈0,T=J,ut it satisfies the following integral equation:(38)xt=Tα1ϕ10+∫0tt−sα1−1Eα1t−sf1s,xs,ysds+∫0tt−sα1−1Eα1t−sσ1sdB1Hs,P−a.s.,t∈J,yt=Tα2ϕ20+∫0tt−sα2−1Eα2t−sf2s,xs,ysds+∫0tt−sα2−1Eα2t−sσ2sdB2Hs,P−a.s.,t∈J,where(39)Tαit=∫0∞ηαiθTitαiθθ,i=1,2,Eαit=αi∫0∞θηαiθTitαiθθ,i=1,2,in which Tt=e−tAi,t≥0 is an analytic semigroup generated by the operator −Ai, and the Mainardi’s Wright-type function with αi∈0,1 is given by(40)ηαiz=∑0+∞−znn!Γ−αin+1−αi,αi∈0,1,z∈ℂ.
The operatorsTαitt≥0 and Eαitt≥0 in (38) have the following properties [22]:Lemma 1 (see [23]).
For anyt>0andχ∈X,Tαitt≥0 and Eαitt≥0 are linear and bounded operators. Moreover, for 0<αi<1 and 0≤v<2, there exists a constant C>0 such that(41)TαitχH˙v≤Ct−αiv/2χX,EαitχH˙v≤Ct−αiv/2χX.Lemma 2 (see [23]).
For anyT>0andχ∈X, the operator Eαit is strongly continuous. Moreover, for 0<αi<1 and 0≤v<2 and 0≤t1<t2≤T, there exists a constant C>0 such that(42)Eαit2−Eαit1χH˙v≤Ct2−t1αiv/2χX.Lemma 3 (see [23]).
LetSαit=tαi−1Eαit, for ∀χ∈X,0≤v<2, and 0<αi<1, there exists a constant C>0 such that(43)SαitχH˙v≤Ct2−vαi−2/2χX,Sαit2−Sαit1χH˙v≤Ct2−t12−2−vαi/2χX.The following lemma proves that the stochastic integral in (38) is well defined.Lemma 4.
Under the assumptions onA, Eαit, and σt, for 0≤v<2,0<αi<1, and 1/4 < H < 1/2, the stochastic integral in (38) is well defined and satisfies the following:(44)E∫0tt−sαi−1Eαit−sσ2sH˙v2≤Ctδ<∞,where the index should satisfy(45)δ=min2−vαi+4H−3,2−vαi+2H+γ−3,2H−2−vαi+1>0.Proof.
Using the Wiener integral with respect to fBm and noticing the expression ofKt∗ and the properties of the Ito^ integral, for 0<H<1/2, we get(46)E∫0tt−sαi−1Εαit−sσsdBHsH˙v2=∑k=1∞E∫0tλk1/2Kt∗Sαit−sekσsdβksH˙v2=∑k=1∞∫0tEλk1/2Kt∗Sαit−sekσsH˙v2ds=∑k=1∞∫0tEλk1/2Sαit−sekσsKHt,sek+∫stλk1/2Sαit−rσr−Sαit−sσs∂KH∂rr,sekdrH˙v2ds≤2∑k=1∞∫0tEλk1/2Sαit−sσsKHt,sekH˙v2ds+4∑k=1∞∫0tE∫stλk1/2Sαit−rσr−σs∂KH∂rr,sekdrH˙v2ds+4∑k=1∞∫0tE∫stλk1/2σsSαit−r−Sαit−s∂KH∂rr,sekdrH˙v2ds:=I1+I2+I3.
With the help of the following inequality (see [24]):(47)KHt,s≤CHt−sH−1/2sH−1/2,furthermore, combining Lemma 3 and Ho¨lder inequality, we obtain(48)I1=2∑k=1∞∫0tEλk1/2Sαit−sσsKHt,sekH˙v2ds≤2CHσ^∫0tt−s2−vαi+2H−3S2H−1∑k=1∞Ελk1/2ek2ds≤2CHσ^TrQ∫0tt−s22−vαi+2H−3ds1/2∫0ts22H−1ds1/2≤CH,Qσ^t2−vαi+4H−3,where σ^:=sup0≤s≤TσsLQ0<+∞. On the contrary, utilizing H..2, expression (17), and Ho¨lder inequality, we get(49)I2=4∑k=1∞∫0tE∫stλk1/2Sαit−rσr−σs∂KH∂rr,sekdrH˙v2ds≤4cH2Cσ22−vTrQH−122∫0ts1−2H∫stt−r2−vαi−22r−sγr−sH−3/2γ1/2−HH˙v2drds≤CH,Q∫0ts1−2H∫stt−r22−vαi−4r−s22H−3+γdr1/2∫str4H−2dr1/2ds≤CH,Qβ22H−3+γ+1,22−vαi−31/2×β2−2H,2−vαi+4H+γ−4t2−vαi+2H+γ−3,where βp,q=∫01xp−11−xq−1dx is the standard Beta function, and we have used t2ω−t1ω≤Ct2−t1ω for 0≤ω≤1, in the above derivation.
Finally, forI3, applying Lemma 3 and expression (17), we have(50)I3=4∑k=1∞∫0tE∫stλk1/2σsSαit−r−Sαit−s∂KH∂rr,sekdrH˙v2ds≤4cH2H−122TrQσ^2∫0ts1−2H∫0tr−s2H−2−vαiγ2H−1r−s2H−3H˙vdrds≤CH,Qσ^2∫0ts1−2Ht−s4H−2−vαi−1ds≤CH,Qσ^2β2−2H,4H−2−vαit2H−2−vαi+1.
Then, whenδ=min2−vαi+4H−3,2−vαi+2H+γ−3,2H−2−vαi+1>0 and 1/4<H<1/2 and combining the above estimation inequalities of I1,I2, and I3, we can obtain(51)E∫0tt−sαi−1Eαit−sσsdBHsH˙v2≤Ctδ<∞,where C>0 is a constant depending only on H,Q,v,γ,αi, and function σs.Theorem 2.
Assume thatH.1−H.2 are satisfied and the matrix(52)Mtrice=B1B2B3B4,Bj≥0,j=1,2,3,4,where(53)B1=T2−vα1−1af12−vα1−1,B2=T2−vα1−1bf12−vα1−1,B3=T2−vα2−1af22−vα2−1,B4=T2−vα2−1bf22−vα2−1.
IfM converges to zero, then problem (1) has a unique solution.Proof.
We consider the operatorN:DT×DT⟶DT×DT defined by(54)Nx,y=N1x,y,N2x,y,x,y∈DT×DT,where(55)N1x,y=ϕ1t,t∈−r,0,Tα1tϕ10+∫0tt−sα1−1Eα1t−sf1s,xs,ysds+∫0tt−sα1−1Eα1t−sσ1sdB1Hs,ℙ−a.s,t∈J,N2x,y=ϕ2t,t∈−r,0,Tα2tϕ20+∫0tt−sα2−1Eα2t−sf2s,xs,ysds+∫0tt−sα2−1Eα2t−sσ2sdB2Hs,ℙ−a.s,t∈J.
Now, we prove thatNx,y has a fixed point by Theorem 1. Indeed, let x,y,x¯,y¯∈DT×DT, and by using Lemma 1 and Ho¨lder inequality, we obtain that(56)EN1xt,yt−N1x¯t,y¯tH˙v2≤∫0tt−s2α1−2Sα1t−sH˙v2dsE∫0tf1s,xs,ys−f1s,x¯s,y¯sX2ds≤t2−vα1−1af12−vα1−1∫0tExs−x¯sX2ds+t2−vα1−1bf12−vα1−1∫0tEys−y¯sX2ds≤t2−vα1−1af12−vα1−1∫0tsupτ∈JExτ−x¯τX2ds+t2−vα1−1bf12−vα1−1∫0tsupτ∈JEyτ−y¯τX2ds.
Therefore, sincex,y=x¯,y¯ over the interval −r,0, by taking supremum in the above inequality, we have(57)N1x,y−N1x¯,y¯DT2≤B12x−x¯DT2+B22y−y¯DT2,where(58)B1=T2−vα1−1af12−vα1−1,B2=T2−vα1−1bf12−vα1−1.
Repeating the above process, we can also obtain(59)EN2xt,yt−N2x¯t,y¯tH˙v2≤t2−vα2−1af22−vα2−1∫0tsupτ∈JExτ−x¯τX2ds+t2−vα2−1bf22−vα2−1∫0tsupτ∈JEyτ−y¯τX2ds.
Thus,(60)N2x,y−N2x¯,y¯DT2≤B32x−x¯DT2+B42y−y¯DT2,where(61)B3=T2−vα2−1af22−vα2−1,B2=T2−vα2−1bf22−vα2−1.
Hence,(62)Nx,y−Nx¯,y¯DT=N1x,y−N1x¯,y¯DTN2x,y−N2x¯,y¯DT≤B1B2B3B4x−x¯DTy−y¯DT.
Therefore,(63)Nx,y−Nx¯,y¯DT≤Mtricex−x¯DTy−y¯DT,for all,x,y,x¯,y¯∈DT×DT.
From Theorem1, the mapping N has a unique fixed x,y∈DT×DT which is a unique solution of equation (1).Remark 1.
Noticing thatB1,B2,B3,B4∈ℝ+, if B1B4−B2B3−B1+B4>0, then(64)B1B2B3B4,is convergent to zero.


## 4. Transportation Inequalities

In this section, we consider the propertyT2C, for the law of the mild solution of equation (1), on the space ℂ=C0,T,X endowed with the uniform metric d∞. Precisely, we have the following theorem.Theorem 3.
Assume thatH.1andH.2 holds, and let ℙϕ1,ℙϕ2 be the law of xϕ1,⋅,yϕ2,⋅, the solution process of equation (1). Using the metric(65)d∞x,y=supt∈Jx−yX,x,y∈C0,T,X,the probability measure ℙϕ1,ℙϕ2 satisfies “T2C” in the sense that(66)W2d∞ℚ1,ℙϕ12+W2d∞ℚ1,ℙϕ22≤2CH1ℚ2ℙϕ1+H1ℚ2ℙϕ2,on the metric space C0,T,X with the metric d∞.Proof.
Letℙϕ1,ℙϕ2 be the law of xt,ϕ1,yt,ϕ2,t∈0,T on C=C0,T,X and ℚi be any probability measure on ℂ such that ℚi≪ℙϕi. Define(67)ℚ˜i:=dℚidℙϕix⋅,ϕiℙ,which is a probability measure on Ω,F. Recalling the definition of entropy and adopting a measure-transformation argument,(68)Hℚ˜ℙ=H1ℚ˜2ℙH2ℚ˜2ℙ,Hℚℙϕ=H1ℚ1ℙϕ1H2ℚ2ℙϕ2,Hiℚ˜iℙ=∫Ωlogdℚ˜idℙdℚ˜i=∫Ωlogdℚidℙϕx⋅,ϕdℚidℙϕix⋅,ϕidℙ=∫clogdℚidℙϕdℚidℙϕidℙϕi=Hℚiℙϕi,i=1,2.
Following [25], then there exists a predictable process h1t,h2t∈X,t∈J with(69)∫0ThisX2ds<+∞,i=1,2,ℙ−a.s,such that(70)H1ℚ1ℙϕ1,H2ℚ2ℙϕ2=12Eℚ˜1∫0Th1sX2ds,12Eℚ˜1∫0Th2sX2ds.
By the Girsanov theorem, the processB˜1t and B˜2t which are defined by(71)B˜1t=B1t−∫0th1sds,B˜2t=B1t−∫0th2sds,are two Brownian motions with respect to Ftt≥0 on the probability space Ω,F,ℚ˜i. Let us consider the ℚ˜i-fractional Brownian motion B˜tHtt∈J defined by(72)B˜tHt=∫0tKHt,sdB˜is=∫0tKHt,sdBis−KHhit,i=1,2,where KHhi is defined by KHhit=∫0tKHt,shisds. By the Fubini theorem, we obtain(73)KHhit=∫0tcHtsH−1/2t−sH−1/2−H−12s1/2−H∫stuH−3/2u−sH−1/2duhisds=∫0tcHtuH−1/2t−uH−1/2hiu−H−12uH−3/2∫0uu1/2−Hu−sH−1/2hisdsdu:=∫0tgiudu,i=1,2.
Consequently, under the measureℚ˜i, the process ut,ϕ=xt,ϕ1,yt,ϕ2t∈J satisfies that(74)xt=ϕ1t,t∈−r,0,Tα1tϕ10+∫0tt−sα1−1Eα1t−sf1s,xs,ysds+∫0tt−sα1−1Eα1t−sσ1sdB˜1Hs+∫0tt−sα1−1Eα1t−sσ1sg1sds,ℙ−a.s.t∈J,yt=ϕ2t,t∈−r,0,Tα2tϕ20+∫0tt−sα2−1Eα2t−sf2s,xs,ysds+∫0tt−sα2−1Eα2t−sσ2sdB˜2Hs+∫0tt−sα2−1Eα2t−sσ2sg2sds,ℙ−a.s.t∈J.
We now consider the solutionx¯,y¯ (under ℚ˜1,ℚ˜2) of the following equation:(75)Dα1cxt=A1x¯t+f1t,x¯t,y¯tdt+σ1tdB˜1Ht,t∈0,T,Dα2cyt=A2y¯t+f2t,x¯t,y¯tdt+σ2tdB˜2Ht,t∈0,T,xt=ϕ1t,t∈−r,0,yt=ϕ2t,t∈−r,0.
By Theorem2, under ℚ˜1,ℚ˜2, the law of x¯t,ϕ1,y¯t,ϕ1,t∈J is ℙϕ1,ℙϕ2; therefore, xt,x¯t,yt,y¯t,t∈0,T under ℚ˜ is a coupling of ℚ,ℙϕ and it follows that(76)W2d∞ℚ,ℙϕ2≤Eℚ˜1d∞x,x¯2Eℚ˜2d∞y,y¯2=Eℚ˜1supt∈Jxt−x¯tX2Eℚ˜2supt∈Jyt−y¯tX2,where we also use the Cauchy inequality(77)a+b2≤2a2+2b2.
Now, we can use the result above to estimate the distance betweenu and u¯ with respect to d∞:(78)xt−x¯tX2=∫0tt−sα1−1Eα1t−sf1s,xs,ys−f1s,x¯s,y¯sds+∫0tt−sα1−1Eα1t−sσ1sg1sdsX2≤∫0tt−sα1−1Eα1t−sf1s,xs,ys−f1s,x¯s,y¯sdsX2+∫0tt−sα1−1Eα1t−sσ1sg1sdsX2:=2J1+J2.
By using theHo¨lder inequality, condition H.1, and Lemma 3, we obtain(79)J1≤∫0tt−sα1−1Eα1t−sH˙v2ds∫0tf1s,xs,ys−f1s,xs,ysX2ds≤t2−vα1−1af12−vα1−1∫0txs−x¯sX2ds+t2−vα1−1bf12−vα1−1∫0tys−y¯sX2ds≤t2−vα1−1af12−vα1−1∫0tsupτ∈0,sxτ−x¯τX2ds+t2−vα1−1bf12−vα1−1∫0tsupτ∈0,syτ−y¯τX2ds.
For the second term, using the Fubini theorem andHo¨lder inequality, we obtain(80)J2≤2cH2σ^∫0tSα1t−stsH−1/2t−sH−1/2h1sds2+2H−12cH2σ^∫0tSα1t−ssH−3/2∫0su1/2−Hs−uH−1/2h1ududs2≤2cH2σ^t2−vα1+2H−22−vα1+2H−2∫0th1sX2ds+41−HH−122cH2σ^β2H,2−vαi−1×β2−2H,2−vαi+4H+4t2−vαi+4H−3∫0th1uX2du:=C1∗∫0th1sX2ds.
Combining (78)–(80), we have(81)xs−x¯sX2≤t2−vα1−1af12−vα1−1∫0tsupτ∈Jxτ−x¯τX2ds+t2−vα1−1bf12−vα1−1∫0tsupτ∈Jyτ−y¯τX2ds+C1∗∫0th1sX2ds.
Similarly, we have(82)ys−y¯sX2≤t2−vα1−1af22−vα2−1∫0tsupτ∈Jxτ−x¯τX2ds+t2−vα1−1bf22−vα2−1∫0tsupτ∈Jyτ−y¯τX2ds+C2∗∫0th2sX2ds.
Adding (81) and (82), we obtain(83)sups∈Jxs−x¯sX2+ys−y¯sX2≤C^∫0tsupτ∈0,sxτ−x¯τX2+yτ−y¯τX2ds+C∗∫0th1sX2+h2sX2ds,where C^=max2t2−vα1−1af1+af2/2−vα1−1,2t2−vα1−1bf1+bf2/2−vα1−1and C∗=maxC1∗,C2∗. By using the Gronwall inequality, we have(84)sups∈0,txs−x¯sX2+ys−y¯sX2≤C∗expC^T∫0Th1sX2+h2sX2ds.
Hence, it follows that(85)W2d∞ℚ1,Ρϕ12+W2d∞ℚ2,Ρϕ22≤C∗expC^TEQ˜1∫0Th1sX2+h2sX2ds≤2CH1ℚ1pϕ1+H2ℚ2pϕ2,where C=C∗expC^T. The proof is complete.Remark 2.
In [15], by using the Girsanov theorem for fractional Brownian motion, the authors established the transportation inequalities for the law of the mild solution to stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈1/2,1. Besides, Boufoussi and Hajji [16] established the transportation inequalities for the law of the mild solution to stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈0,1/2. However, the transportation inequalities for fractional stochastic evolution equations driven by the fractional Brownian motion are more complicated. On the contrary, for a coupled system, we have to consider the transportation inequalities for the law of the random vector Xt,Yt, which is more difficult. So, our results generalize and improve the results in [15, 16].


## 5. An Example

In this section, we present an example to illustrate the usefulness and applicability of our results. We consider the following fractional stochastic partial differential equation with delay effects:(86)Dα1cut,x=∂2∂x2ut,x+1−a1ut,xt−τsint+sin2t−b1vt,xt−τcost+cos2t+e−tdBH/dt,t∈0,T,0≤x≤π,Dα1cut,x=∂2∂x2ut,x+1−a2ut,xt−τsint+sin2t−b2vt,xt−τcost+cos2t+e−tdBH/dt,t∈0,T,0≤x≤π,ut,0=ut,π=0,t∈0,T,vt,0=vt,π=0,t∈0,T,ut,x=ϕ1t,x,t∈−r,0,0≤x≤π,vt,x=ϕ2t,x,t∈−r,0,0≤x≤π,where ai,bi>0,αi∈2/3,1 and τ>0,BH denotes a fractional Brownian motion. To rewrite this system into the abstract form (1), we set(87)f1t,ϕ1t,ϕ2tη=1−a1ϕ1ηtsint+sin2t−b1ϕ2ηtcost+cos2t,f2t,ϕ1t,ϕ2tη=1−a2ϕ1ηtsint+sin2t−b2ϕ1ηtcost+cos2t,σ1t=e−t,σ2t=2e−t,and K=H=L20,π. We denote the operator A by Au=u″, with domain DA=u∈H,u″∈Handu0=uπ=0.Then, it is easy to obtain(88)Az=−∑n=1∞e−n2tz,enen,z∈H,and A is the infinitesimal generator of an analytic semigroup Stt≥0 on H, which has following the following formula:(89)Stu=∑n=1∞e−n2tu,enen,u∈H,where enu=2/π1/2sinnu,n=1,2,…, is the orthogonal set of eigenvectors of A. If the analytic semigroup St,t∈J, is compact, then there exists a constant K≥1 such that St2≤K.In order to define the operatorQ:K⟶K, we choose a sequence Qen=σnen and assume that(90)trQ=∑n=1∞σn<∞.Define the processBHs by(91)BH=∑n=1∞σnBnHten,where 1/4 < H < 1/2 and BnHn∈ℕ is a sequence of two-sided one-dimensional mutually independent fractional Brownian motions. Thus, one has(92)f1t,x,y−f2t,x¯,y¯2≤8a1x−x¯D0+8b1y−y¯D0,f2t,x,y−f2t,x¯,y¯2≤8a2x−x¯D0+8b2y−y¯D0.On account of the conditions, it is straightforward to check thatH.1 and H.2 hold. Let(93)M=22T2−vα1−1a12−vα1−1T2−vα1−1b12−vα1−1T2−vα2−1a22−vα2−1T2−vα2−1b22−vα1−1.IfM converges to zero, then assumptions in Theorem 2 are fulfilled, then we can conclude that the law of the unique mild solution of system (86) on 0,T satisfies the property T2C.


## 6. Conclusion

In this paper, by Perov’s fixed-point theorem, some stochastic analysis technique, and the properties of operator semigroup, we show the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameterH∈1/4,1/2. Furthermore, we establish the transportation inequalities for the law of the mild solution, with respect to the uniform distance. In our next paper, we will explore the existence, uniqueness, and the transportation inequalities of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈0,1/4.



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