Almost Periodic Solutions to Impulsive Stochastic Delay Differential Equations Driven by Fractional Brownian Motion With 1 2 < H < 1

In this article, we study the existence and uniqueness of square-mean piecewise almost periodic solutions to a class of impulsive stochastic functional differential equations driven by fractional Brownian motion. Moreover, the stability of the mild solution is obtained. To illustrate the results obtained in the paper, an impulsive stochastic functional differential equation driven by fractional Brownian motion is considered.


INTRODUCTION
Impulsive systems arise naturally in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time. Impulsive stochastic modeling has come to play an important role in many branches of science where more and more people have encountered impulsive stochastic differential equations. For example, a stochastic model for drug distribution in a biological system was described by Tsokos and Padgett [1] as a closed system with a simplified heart, one organ, or capillary bed, and recirculation of blood with a constant rate of flow, where the heart is considered as a mixing chamber of constant volume. Recently, there has been a significant development in impulsive stochastic differential equations (ISDEs). The existence and stability of ISDEs were investigated in [2][3][4][5][6][7][8][9][10][11] and the references therein.
On the other hand, in recent years, there has been considerable interest in studying fractional Brownian motions (fBms) due to their compact properties and applications in various scientific areas, including telecommunications [12,13], turbulence [14], image processing [15], and finance [16]. Stochastic differential equations (SDEs) driven by fBms attract the interest of researchers [2,3,[17][18][19][20][21]. Taking the time delay into account, the theory of stochastic differential equations has been generalized to stochastic functional differential equations; it makes the dynamics more complex and the system may lose stability and show almost periodicity. Arthi et al. [2] considered the existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by fractional Brownian motion (fBm), and Caraballo [3] studied the existence of mild solutions to stochastic delay evolution equations with fBm and impulses.
In this paper, we are concerned with the existence and stability of almost periodic mild solutions to the following impulsive stochastic functional differential system driven by fBm with Hurst index H ∈(1/2, 1): where Z is the set of integer, for any i, k ∈ Z, and the sequence {t i } is such that the derived sequence {t k i t i+k − t i } is equipotentially almost periodic. Moreover, A: D(A) ⊂ H → H is a linear bounded operator, ρ(A) is the resolvent set of A, and for λ ∈ ρ(A), R(λ, A) is the resolvent of A. In addition, b, σ, and I i are appropriate functions, x t (·): [−θ, 0] → H is given by x t (s) x(t + s), for any s ∈ [ − θ, 0], and ξ ∈ C θ is an F t0 −measurable random variable such that E ξ 2 < ∞. Let θ > 0 be a given constant and let C θ ϕ: [−θ, 0] × Ω → H, ϕ be continuous everywhere except for a finite number of points s at which ϕ(s − ) and ϕ(s + ) exist and satisfy ϕ(s − ) ϕ(s)}, endowed with the norm There are several difficulties with our problems. First, there is the delay for the impulsive stochastic differential equations. Second, about the stochastic differential equations driven by fractional Brownian motion, the classical stochastic integral failed for lack of the martingale property. Third, there is no strong solution for stochastic partial delay differential equations driven by fractional Brownian motion. The lifting space method, mild solutions, fixed point theorem, and semigroup theory will be used to overcome these difficulties.
The paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. Section 3 is devoted to stating the existence and uniqueness of the mild square-mean piecewise almost periodic solution to (1). In Section 4, we show the stability of the mild square-mean piecewise almost periodic solution. An example is provided to illustrate the effectiveness of the results.

PRELIMINARIES
Let (H, · H , (·, ·) H ) and (K, · K , (·, ·) K ) denote two real separable Hilbert spaces. We denote by L(H, K) the set of all linear bounded operators from H into K, equipped with the usual operator norm · and use |·| to denote the Euclidean norm of a vector. In this article, we use the symbol · to denote the norms of operators regardless of the spaces involved when no confusion possibly arises. Let (Ω, F , {F t } t ≥ 0 , P) be a filtered complete probability space satisfying the usual condition.

Fractional Brownian Motion
In this subsection, we briefly introduce some useful results about fBm and the corresponding stochastic integral taking values in a Hilbert space. For more details, refer to Hu [22], Mishura [23], Nualart [24], and references therein.
A real standard fractional Brownian motion {β H (t), t ∈ R} with Hurst parameter H ∈ (0, 1) is a Gaussian process with continuous sample paths such that E[β H (t)] 0 and for all s, t ≥ 0. It is known that fBm {β H (t), t ≥ 0} with H > 1 2 admits the following Wiener integral representation: where W is a standard Brownian motion and the kernel K H (t, s) is given by where c H > 0 is a constant satisfying E(β H 1 ) 2 1. For any function σ ∈ L 2 (0, T), the Wiener integral of σ with respect to β H is defined by where β H n , n 1, 2, · · · are independent fBms with the same Hurst parameter H ∈ ( 1 2 , 1), {e n , n ∈ N}, which is a complete orthonormal basis in K, {λ n , n ∈ N} that is a bounded sequence of non-negative real numbers satisfying Qe n λ n e n , and Q is non-negative self-adjoint trace class operator with TrQ  Remark. If {λ n } n∈N is a bounded sequence of non-negative real numbers such that the nuclear operator Q satisfies Qe n λe n , assuming that there exists a positive constant K σ such that

Piecewise Almost Periodic Stochastic Processes
In this subsection, we recall some notations about the squaremean piecewise almost periodic stochastic process and introduce Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 783125 some lemmas. For further details, we refer to Takens and Teissier [25] and Liu [26].
Recall that a stochastic process X: for all s ∈ R, and it is said to be bounded if there exists N > 0, such that E X(t) 2 ≤ N for all t ∈ R. For convenience, we list the following concepts and notations: ,and x(t + i ) represent the left and right limits of and t i < t ≤ t i+1 (see [27]).

Definition 2.2. ([28]
). The family of the sequence {t k i t i+k − t i }, i ∈ Z, k ∈ Z will be called equipotentially almost periodic if for any ε > 0; there exists a relatively dense set Q ε of R and an integer q ∈ Z such that the inequality holds for each τ ∈ Q ε and i ∈ Z.
is said to be square-mean piecewise almost periodic if the following conditions are fulfilled: a) For any ε > 0, there exists a positive number δ δ(ε) such that if the points t′ and t″ belong to the same interval of continuity Let AP T (R, L 2 (Ω; H)) denote the space of all square-mean piecewise almost periodic functions. Obviously AP T (R, L 2 (Ω; H)) endowed with the supremum norm is a Banach space. Let UPC(R; L 2 (Ω; H)) be the space of all functions b ∈ PC(R, L 2 (Ω; H)) such that b satisfies the condition (a) in Definition 2.3. It is easy to check that UPC(R; L 2 (Ω; H)) is a Banach space with the norm for each X ∈ UPC(R; L 2 (Ω; H)). H) is called square-mean almost periodic if for any ε > 0, there exists a natural number N N(ε) such that, for each k ∈ Z, there is at least one integer p in the segment [k, k + N], for which inequality holds for all i ∈ Z.
there exits l(ε, Λ) > 0 such that any interval of length l(ε, Λ) contains at least a number τ for which The collection of all such processes is denoted by AP T (R × C θ , L 2 (Ω, H)).

EXISTENCE OF SQUARE-MEAN PIECEWISE ALMOST PERIODIC SOLUTION
In this section, we study the existence of the square-mean piecewise almost periodic solution to (1). We first present some assumptions as follows: (H1) Let the bounded linear operator A be an infinitesimal generator of an analytic semigroup {S(t), t ≥ 0} such that for any x,x ∈ C θ .
(H3) Let σ ∈ AP T (RL 0 2 (Ω, L 2 (Ω, H))) and let {I i x(t i ), i ∈ Z} be a square-mean piecewise almost periodic sequence satisfying for some positive constant M I . Recall the notion of a mild solution for Eq. 1.
Definition 3.1. An F t -progressive process {x(t)} t∈R is called a mild solution of the system (1) on R if it satisfies the corresponding stochastic integral equation for all t ≥ t 0 and for each t 0 ∈ R. Θd Consider the following equation: with t ≥ t 0 . It is easy to verify that the above equation is equivalent to (7). Define the operator L on AP T (R, L 2 (Ω, H)) by for all t ∈ R. To prove the theorem, it is sufficient to show that the next statements hold: is square-mean piecewise almost periodic. II) L admits a unique fixed point.
Proof of Statement (I)This will be done in two steps.
Step 2. We prove the almost periodicity of Lx(t).
Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 783125 For s ∈ [t j , t j + η], j ∈ Z, j ≤ i, by the mean value theorem of integral, we get that Similarly, we can show that where C 1 , C 2 are two positive constants. Thus, we have introduced the next estimate: where N 1 is a positive constant, which implies that Φ 1 (t) is square-mean piecewise almost periodic. We now show that Φ 2 (t) is square-mean piecewise almost periodic. Recall that t1σ(t) is piecewise almost periodic if for each ε > 0 there exists a real number l(ε) > 0 such that the estimate holds for every interval of length l(ε) containing a number τ. By using (H1) and the computation of fBm, we have Furthermore, by Hölder's inequality, we have where η min {ε, α 2 }. In the same way as that of handling Φ 1 (t), one can introduce the estimate where N 2 is a positive constant, and hence Φ 2 (t) is piecewise square-mean almost periodic. (2), one has t i+q+1 > t + τ > t i+q . From (H3), it follows that β i is a squaremean almost periodic sequence, for any ε > 0; there exists such a natural number N N(ε) that, for an arbitrary k ∈ Z, there is at least one integer p > 0 in the segment [k, k + N] such that the inequality holds for all i ∈ Z. We get which implies that Φ 3 (t) ∈ AP T (R, L 2 (Ω, H)). Thus, we have proved that Lx(t) ∈ AP T (R, L 2 (Ω, H)) and Lx(t) is squaremean piecewise almost periodic.
Proof of Statement (II). Given B {u ∈ AP T (R, L 2 (Ω, H))} and assuming that x(t), y(t) ∈ B are both almost periodic solutions of (1) and x(t) ≠ y(t), then we have From (H1), (H2), (H3) and the Cauchy-Schwarz inequality, we have that and It follows that for each t ∈ R, which implies that This means that L is a contraction when (8) holds and statement (II) follows.

ASYMPTOTIC STABILITY
In this section, we are interested in the asymptotical stability of the almost periodic mild solution to (1) with t 0 0. For convenience, we rewrite the equation as follows: for t ≥ t 0 , where C ≥ 0, u(σ) > 0, α i ≥ 0, i ∈ Z, and σ i , i ∈ Z are the first kind discontinuity points of the function v. Then, the following estimate holds: Theorem 4.1. Assume that (H1) − (H3) hold. The almost periodic solutions to (15) are asymptotically stable in the square-mean sense if Proof. Let x(t) and x*(t) be two square-mean piecewise almost periodic mild solutions of (15); we then have that for all t ≥ 0. By using Cauchy-Schwartz's inequality, Fubini's theorem, and assumptions (H1) − (H3), we deduce that for t ≥ 0. Multiplying both sides of the above inequality by e ct , we get t 0 e cs sup 0≤r≤s E x r ( ) − x * r ( ) 2 ds