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

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 parameter H ∈ (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.


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. e intensive development in both theory and applications of fractional SDEs was investigated in [1][2][3][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). erefore, it is necessary to study stochastic evolution equations with finite delays. e 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 L 2 metric; € Ust € unel [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 L 2 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 L 2 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 property T 2 (C) 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: where c D α i is the Caputo fractional derivative of order α i ∈ (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 A i , i � 1, 2 are the infinitesimal generators of analytic semigroups of bounded linear operators T i (t), t ≥ 0 , B H i 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 y t , 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) Before describing the properties fulfilled by operators f i , σ i , we need to introduce some nontations and describe some spaces. Let D 0 denote the space of all continuous functions φ: In the space D 0 , we endow with the following norm: Next, we denote by C(a, b; L 2 (Ω; X)) � C(a, b; L 2 (Ω, F, P; X)) the Banach space of all continuous functions from [a, b] into L 2 (Ω; X). Now, fixing T > 0, we define endowing with the following norm: We give initial data φ 1 , φ 2 ∈ D 0 , and Y is another real and separable Hilbert space, and B H Q i � B H i is a Y-valued fractional Brownian motion with increment covariance given by a nonnegative trace class operator Q i , and L(Y, X) represents the space of all bounded, continuous, and linear operators from Y into X.
We denote Here, let us denote L 0 Q i (Y, X) by the space of all Q i -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. Let (E, 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 ] on E, we define the Wasserstein distance of order p between μ and ] as where Π(μ, ]) denotes the totality of probability measures on E × E with the marginal μ and ]. e relative entropy of ] with respect to μ is defined as e probability measure μ satisfies L p -transportation inequality on (E, d) if there exists a constant C ≥ 0 such that for any probability measure ], As usual, we write μ ∈ T p (C) for this relation. e property T 2 (C) is of particular interest. We will investigate the property T 2 (C) 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.
is paper is organized as follows. In Section 2, 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 T 2 (C) 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.

Preliminaries
In this section, we introduce some notations and recall definitions and preliminary results which are used throughout this paper.
Let (Ω, F, P, F t t≥0 ) be a complete probability space furnished with a normal filtration F t t≥0 . We postulate that the operator A i is self-adjoint and there exists the eigenvectors e k corresponding to eigenvalues c k such that is a square integrable kernel, for 0 < H < 1/2, and t > s; the formula is as follows (see [17]): where (14), we can infer that en, we obtain Taking the derivative of (14) with respect to t, we can have Apparently, we can obtain the following inequality: Let H be the Hilbert space defined as the closure of the vector space spanned by the set of step functions I [0,t] , t ∈ [0, T] with respect to the scalar product: Now, we consider the operator K * H,T from H to Furthermore, K * H,T is an isometry between H and L 2 ([0, T])(see [18]). Taking account for it turns out that B is a Wiener process. Moreover, for any φ ∈ H, with (13), we have For any 0 ≤ t ≤ T, we can also deduce Discrete Dynamics in Nature and Society where K * H,t is defined in the same way as in (20) with t instead of T. Next, we will use the notation K * H,t without specifying the parameter t ∈ [0, T].
Let (X, ‖ · ‖ X , 〈·, ·〉 X ) and (Y, ‖ · ‖ Y , 〈·, ·〉 Y ) be two real, separable Hilbert spaces and let L(Y, X) denote the space of all bounded linear operators from Y to X. Let Q ∈ L(Y, X) be a nonnegative self-adjoint operator i.e., Qe n � λ n e n with trace tr Q � ∞ n�1 λ n < ∞, where λ n ∈ R + and e n n≥1 is a complete orthonormal basis in Y. We define the infinitedimensional fBm on Y with covariance Q by the following formula: where B H n (t) n∈N be a sequence of one-dimensional mutually independent standard fractional Brownian motions on (Ω, F, P). B H n (t) is a Y-valued Gaussian process, starting from 0, and has zero mean and covariance: ). en, the Wiener integral of φ with respect to the fBm B H Q is defined as follows: where B n is the standard Brownian motion used to represent B H n as in (13), and the sum above is finite when ∞ n�1 λ n ‖K * H (φe n )‖ < ∞. e 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. Let Z be a nonempty set. We denote by a vector-valued metric on Z defined as a mapping d: Z × Z ⟶ R n with the following properties: Now, we consider a generalized metric space (Z, d). For r � (r 1 , . . . , r n ) ∈ R n + , we will define the open ball centered in x 0 with radius r: and the closed ball centered in x 0 with radius 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. If x, y ∈ R n , x � (x 1 , . . . , x n ), y � (y 1 , . . . , y n ), by x ≤ y then we mean x i ≤ y i , i � 1, . . . , n. Also, |x| � (|x 1 |, . . . , |x n |) and max(x, y) � max(max(x 1 , y 1 ), . . . , max(x n , y n )). If c ∈ R, then x ≤ c means x i ≤ c for each i � 1, . . . , n.
Definition 5. We denote that a real square matrix M is convergent to zero if and only if its spectral radius ρ(M) 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 det(M − λI) � 0, where I denotes the unit matrix of M n×n (R) ).

Definition 6.
We denote that a nonsingular matrix A � (a ij ) 1≤i,j≤n ∈ M n×n (R) has the absolute value property if where |A| � a ij 1≤i,j≤n ∈ M n×n (R).
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]). Let (Z, d) be a complete generalized metric space with d: Z × Z ⟶ R n and let operator N: Z ⟶ Z be such that for all x, y ∈ Z and some nonnegative square matrix M. If the matrix M is convergent to 0, that is, M k ⟶ 0 as k ⟶ 0, then operator N has a unique fixed point x * ∈ Z: for every x 0 ∈ Z and k ≥ 1. e fractional integral of index α with the lower limit 0 for a function f can be written as provided the right-hand side is pointwise defined on [0, +∞), where Γ is the gamma function, which is defined by Definition 8. e Caputo derivative of index α for a function f ∈ C n ([0, ∞)) is defined as

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.2) e function σ: [0, T] ⟶ L 0 Q (Y, X) satisfies the following H€ older continuous conditions, that is, there exists a constant C σ > 0 such that for all t, s ∈ [0, T], where c > 1 − 2H. Now, we state the following definition of mild solution for our problem.
e following lemma proves that the stochastic integral in (38) is well defined.

Lemma 4.
Under the assumptions on A, E α i (t), and σ(t), for 0 ≤ v < 2, 0 < α i < 1, and 1/4 < H < 1/2, the stochastic integral in (38) is well defined and satisfies the following: Discrete Dynamics in Nature and Society 5 where the index should satisfy Proof. Using the Wiener integral with respect to fBm and noticing the expression of K * t and the properties of the Ito integral, for 0 < H < (1/2), we get With the help of the following inequality (see [24]): furthermore, combining Lemma 3 and H€ older inequality, we obtain 6 Discrete Dynamics in Nature and Society where σ: � sup 0≤s≤T ‖σ(s)‖ L 0 Q < + ∞. On the contrary, utilizing (H..2), expression (17), and H€ older inequality, we get where β(p, q) � 1 0 x p− 1 (1 − x) q− 1 dx is the standard Beta function, and we have used t ω 2 − t ω 1 ≤ C(t 2 − t 1 ) ω for 0 ≤ ω ≤ 1, in the above derivation.
Finally, for I 3 , applying Lemma 3 and expression (17), we have and (1/4) <H <(1/2) and combining the above estimation inequalities of I 1 , I 2 , and I 3 , we can obtain where If M converges to zero, then problem (1) has a unique solution.
Proof. We consider the operator N: Discrete Dynamics in Nature and Society where Now, we prove that N(x, y) has a fixed point by eorem 1. Indeed, let (x, y),(x, y) ∈ D T × D T , and by using Lemma 1 and H€ older inequality, we obtain that erefore, since (x, y) � (x, y) over the interval [− r, 0], by taking supremum in the above inequality, we have where Repeating the above process, we can also obtain E N 2 (x(t), y(t)) − N 2 (x(t), y(t)) us, where Hence, (63) From eorem 1, the mapping N has a unique fixed (x, y) ∈ D T × D T which is a unique solution of equation (1).
is convergent to zero.

Transportation Inequalities
In this section, we consider the property T 2 (C), for the law of the mild solution of equation (1), on the space C � C([0, T], X) endowed with the uniform metric d ∞ .
Precisely, we have the following theorem.
Proof. Let P ϕ 1 , P ϕ 2 be the law of x(t, ϕ 1 ), y(t, ϕ 2 ), t ∈ [0, T] on C � C[(0, T), X] and Q i be any probability measure on C such that Q i ≪ P ϕ i . Define