Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations d 𝑥(𝑡) = [𝐴(𝑡)𝑥(𝑡) + 𝐹 (𝑡,𝑥 (𝑡), 𝑥 𝑡 )] d 𝑡 + 𝐻(𝑡,𝑥(𝑡), 𝑥 𝑡 ) ∘ d 𝑊(𝑡) . A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup {𝑇 (𝑡)} 𝑡≥0 is essentially removed, which is generated by the linear densely defined operator 𝐴 : 𝐷(𝐴) ⊂ 𝐿 2 ( P , H ) → 𝐿 2 ( P , H ) , only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.


Introduction
The theory of almost periodic functions was first developed by the Danish mathematician H. Bohr in [1925][1926].Then Bohr's work was developed substantially by S. Bochner, J. Favard, V. V. Stepanov, and others.Generalization of the classical theory of almost periodic functions has been taken in several directions.These works were recapitulated in literatures [1] and [2].The concept of almost periodicity is important in probability for investigating stochastic processes [3][4][5][6][7].Such a notion is also of interest for applications arising in mathematical physics and statistics.Literature [8] developed the notion of -mean almost periodicity based on the concept of quadratic mean uniformly almost periodic utilized by [7] and pointed out that each -mean almost periodic process is uniformly continuous and stochastically bounded [9].Literature [8] also pointed that the collection of all -mean almost periodic processes is a Banach space when it is equipped with some norm obtained through the norm of   (P, B), where (B, ‖ ⋅ ‖) is a Banach space.
The asymptotically almost periodic functions were first introduced by Fréchet.In the modern theory of differential equations, many authors [1,2] applied successfully the asymptotic property to determine the existence of almost periodic solutions.Along with the development of such equations as the evolution partial differential equations, functional differential equations, and so forth, where the phase spaces are infinite, the theory of Banach valued asymptotically almost periodic functions had been developed [10][11][12].Some techniques in functional analysis and harmonic analysis were applied to such equations; for example, in [1,13], the authors applied spectrum theory to get almost periodic solutions for some linear abstract evolution differential equations.More recently, [14] developed the notion of -mean asymptotical almost periodicity for stochastic processes.Among others, it showed that each -mean asymptotically almost periodic stochastic process is stochastically bounded.
One should point that the following condition (C) is very much important in the above-mentioned literatures.
It is clear that the condition (C) is too strict [22] so that it cannot be satisfied even if for simple  = diag{1, −1} or  = diag{1, 0, −1}.Literature [22] presented some new criteria ensuring the existence and uniqueness of quadratic-mean almost periodic solution for stochastic differential equation (1), and only assumes that the linear system d () =  () d (3) admits exponential dichotomy.It is clear that when () = diag{1, −1}, the system (3) admits exponential dichotomy.More generally, in the case () ≡ , a constant matrix, the system (3) admits exponential dichotomy if and only if the eigenvalues of  have a nonzero real part.Literature [14,17] has obtained the existence and uniqueness of quadratic-mean almost automorphic solutions or asymptotically almost periodic solutions for stochastic functional differential equations under a hyperbolic and analytic semigroup {()} ≥0 .At the same time, one notices that the case that the eigenvalues of  have a zero real part is very common; for example, () = diag{1, 0, −1}.Therefore, it is interesting to ask, what is that, when the semigroup {()} ≥0 is not exponentially stable, which is generated by the family {() :  ∈ R}, or when the semigroup {()} ≥0 is nonhyperbolic?This question will be considered in the paper.
In the present paper, motivated by [8,14,15], we discuss the existence and uniqueness of quadratic-mean asymptotically almost periodic solution to the following stochastic functional differential equation on  2 (P, H), where P is the probability measure of the probability space (Ω, F, P) and (H, ‖ ⋅ ‖) is a real separable Hilbert space: We present the new criterion ensuring the existence of a unique quadratic-mean asymptotically almost periodic solution for the stochastic functional differential equation ( 1), by employing the properties of almost periodic function and the technique of inequality.We essentially remove the above conditions (C) and only assume that the linear system (2) admits exponential trichotomy (see Definition 11).We also point out that exponential trichotomy is the most complex asymptotic property of dynamical systems arising from the central manifold theory.Starting from the idea that the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold.The exponential trichotomy reflects a deeper analysis of the behavior of solutions of dynamical systems.In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow's domain.This paper is organized as follows.In Section 2, the relating notations, definitions, and the basic results are introduced, which would be used later.In Section 3, a new criterion ensuring the existence and uniqueness of a quadratic-mean asymptotically almost periodic mild solution for stochastic functional differential equations is presented.In Section 4, an example is given to illustrate our results.Finally, conclusions are drawn in Section 5.

Preliminaries
Let (Ω, F, P) be a probability space, for a Banach space (B, ‖ ⋅ ‖) and  ≥ 1, denoted by   (P, B), the Banana space of all B-value random variable , such that It is then routine to check that   (P, B) is a Banach space when it is equipped with its natural norm ‖ ⋅ ‖  defined by (6) for each  ∈   (P, B).
This setting requires the following preliminary definitions.
Let CUB(R;   (P, B)) denote the collection of all stochastic processes  : R →   (P, B), which are continuous and uniformly bounded.It is easily to check that CUB(R;   (P, B)) is a Banach space when it is equipped with the norm The collection of all stochastic processes  : R →   (P, B) which are -mean almost periodic is denoted by (R;   (P, B)).(R;   (P, B)) is a closed subspace of CUB(R;   (P, B)).Therefore, (R;   (P, B)) is a Banach space when it is equipped with the norm ‖ ⋅ ‖ ∞ (see, e.g., [3]).
Definition 4 (see [8]).A function  : R ×   (P, B 1 ) →   (P, B 2 ), (, ) → (, ), which is jointly continuous, is said to be -mean almost periodic in  ∈ R uniformly in  ∈ K where K ⊂   (P, B 1 ) is compact if, for any  > 0, there exists (, K) > 0 such that any interval of length (, K) contains at least a number  for which sup for each stochastic process  : R → K.
The space of the restrictions of all -mean almost periodic stochastic processes on R is denoted by (R;   (P, B)), and -mean almost periodic stochastic processes in , uniformly for  in compact subset K of   (P, B 1 ) by A(R ×   (P, B 1 );   (P, B 2 )).
It is clear that when () = − tanh , the system (14) admits exponential trichotomy.More generally, in the case () ≡  = diag{1, 0, −1}, a constant matrix, the system (14) admits exponential trichotomy.Theorem 12. Let () be a continuous linear operator on R and let (14) have an exponential trichotomy (16) where Since the projection corresponding to an exponential trichotomy on R is uniquely determined it follows that   →  * ,   →  * without restriction to a subsequence.
Define the function (, ) as the form where () is the fundamental matrix solution of the linear differential system (29) with (0) = .
Then the system (26) has a unique quadratic-mean asymptotically almost periodic mild solution, which can be explicitly expressed as follows: We show that (26) exists as a mild solution.Note that (33) and (34) are well defined for each  ∈ R and satisfy (31) for all  ≥ , for each  ∈ R. Hence () given by ( 33) and ( 34) is a mild solution of (26).
Define a mapping L on (R,  2 (P, H)) by where the Φ  , Ψ  are defined in (34).
In order to prove Theorem 14, we first prove Lemmas 15 and 16.
Of course, this is more complicated than the previous case because of the involvement of the Brownian motion .
Next, let us show that () ∈  0 (R,  2 (P, H)).In fact where we make extensive use of the Itô isometry identity and the properties of W defined by W ≡ ( + ) − () for each .Note that W is also a Brownian motion and has the same distribution as .
In view of the above, it is clear that L maps (R, We first evaluate  1 as follows: Similar to the discussion given for  1 , for  2 ,  3 , and  4 , one has −(−)     (, (),   ) − (, (),   )     d) Then, As for the first term  1 , using Itô's isometry identity, one obtains Since Λ < 1, by (60), we know that L is a contraction mapping.The proof of Lemma 16 is complete.
Hence, combining Lemmas 15 and 16 and the contraction mapping principle, L has a unique fixed point (), which is obviously the unique quadratic-mean asymptotically almost periodic mild solution of (26).
This completes the proof of Theorem 14 due to Lemmas 15 and 16.
Remark 17.If the conditions of the main result of [14] and (H1) and (H2) hold, (26) admits exponential trichotomy with projections ,  and  = 0; hence system (26) has a unique quadratic-mean asymptotically almost periodic mild solution.So our main result improves the main result of [14].
In particular, for  =  = 0 one obtains the van der Pol equation and for  =  = 0 one obtains the Duffing equation.