Existence of almost periodic solutions to some nonautonomous higher-order stochastic di erence equations

The studyof almost periodicitywhich generalizes thenotionof periodicity is an area of interest in its own right and has sundry applications in elds like Physics. For a study of almost periodic and almost automorphic sequences we refer the reader to (Bezandry and Diagana [2], Bezandry et al. [3], Corduneanu [4], Diagana, Diagana et al. [6], Han and Hong [8], Hong and Nunez [10]) and references therein. Almost periodicity is also of importance in the study of stochastic processes. In Bezandry et al. [3], the notion of almost periodicity in mean was introduced and used to study the existence and uniqueness of almost periodic solutions to the stochastic Beverton-Holt equation. The main motivation of this paper comes from a paper by Diagana [5] in which discrete dichotomy techniques were utilized to nd su cient conditions for the existence of almost automorphic solutions to some higher-order nonautonomous systems of di erence equations. In this paper, we extend Diagana’s results to stochastic case. More precisely, we study the existence of almost periodic solutions to the class of higher-order nonautonomous stochastic di erence equations of the form:


Introduction
The study of almost periodicity which generalizes the notion of periodicity is an area of interest in its own right and has sundry applications in elds like Physics. For a study of almost periodic and almost automorphic sequences we refer the reader to (Bezandry and Diagana [2], Bezandry et al. [3], Corduneanu [4], Diagana, Diagana et al. [6], Han and Hong [8], Hong and Nunez [10]) and references therein. Almost periodicity is also of importance in the study of stochastic processes.
In Bezandry et al. [3], the notion of almost periodicity in mean was introduced and used to study the existence and uniqueness of almost periodic solutions to the stochastic Beverton-Holt equation.
The main motivation of this paper comes from a paper by Diagana [5] in which discrete dichotomy techniques were utilized to nd su cient conditions for the existence of almost automorphic solutions to some higher-order nonautonomous systems of di erence equations.
In this paper, we extend Diagana's results to stochastic case. More precisely, we study the existence of almost periodic solutions to the class of higher-order nonautonomous stochastic di erence equations of the form: for all U, V R k -valued random variables with nite expectation and t ∈ Z. We assume that for each xed r, the Ar(t) ′ s are independent and independent of X( ). This assumption together with Eq.(1.1) imply that the sequence {(A (t), . . . , A n− (t))} t∈Z is independent of the sequence {X(t)} t∈Z . For that, the main idea consists in rewriting Eq.(1.1) as a nonautonomous rst-order system of stochastic di erence equations on ( Indeed, setting Z(t) := ( X(t), X(t + ), . . . , X(t + n − ) T , where the symbol T stands for the transpose operation and if I denotes the identity matrix of R k , then Eq.(1.1) can be rewritten in (R k ) n in the following form and its corresponding homogeneous equation where A(t) is the family of time-dependent sequence matrices de ned by The paper is organized as follows. In Section 2, we recall a basic theory of almost periodic random sequences on Z. In Section 3, we apply the techniques developed in Section 2 to nd some su cient conditions for the existence of the almost periodic solution to some semi-linear system of stochastic di erence equations. In Section 4, we study some second-order stochastic di erence equations to illustrate our main result.

Preliminaries
In this section we review a basic theory for almost periodic random sequences. To facilitate our task, we rst introduce the notations needed in the sequel.
Let (B, · ) be a Banach space and let (Ω, F, P) be a complete probability space. Throughout the rest of the paper, Z denotes the set of all integers. De ne L (Ω; B) to be the space of all B-valued random variables V such that It is then routine to check that L (Ω; B) is a Banach space when it is equipped with its natural norm · de ned by, V := E V for each V ∈ L (Ω, B). Let X = {X(t)} t∈Z be a sequence of B-valued random variables satisfying E X(t) < ∞ for each t ∈ Z. Thus, interchangeably we can, and do, speak of such a sequence as a function, which goes from Z into L (Ω; B). This setting requires the following preliminary de nitions.
De nition 2.1. An L (Ω; B)-valued random sequence X = {X(t)} t∈Z is said to be Bohr almost periodic in mean if for each ε > there exists N (ε) > such that among any N consecutive integers there exists at least an integer p > for which An integer p > with the above-mentioned property is called an ε-almost period for X. The collection of all B-valued random sequences X = {X(t)} t∈Z which are Bohr almost periodic in mean is then denoted by AP(Z; L (Ω; B)).

De nition 2.2.
A B-valued random sequence X = {X(t)} t∈Z is said to be almost periodic in probability if for each ε > , and η > there exists N (ε, η) > such that among any N consecutive integers there exists at least an integer p > for which Theorem 2.3. If X is almost periodic in mean, then it is almost periodic in probability and there also exists a constant M > such that E X(t) ≤ M for all t ∈ Z. Conversely, if X is almost periodic in probability and the sequence X(t) , t ∈ Z is uniformly integrable, then X is almost periodic in mean.

De nition 2.4.
A B-valued random sequence X = {X(t)} t∈Z satis es Bochner's almost sure uniform double sequence criterion if, for every pair of sequences (k ′ i ) and (l ′ i ), there exists a measurable subset Ω ⊂ Ω with P(Ω ) = and there exist subsequences k = (k i ) ⊂ (k ′ i ) and l = (l i ) ⊂ (l ′ i ) respectively, with the same indexes (independent of ω) such that, for every t ∈ Z, (In this case, Ω depends on the pair of sequences (k ′ i ) and (l ′ i ).

Theorem 2.5. The following properties of X are equivalent: (i) X satis es Bochner's almost sure uniform double sequence criterion. (ii) X is almost periodic in probability.
The proof of the theorem can be seen in Bedouhene et al. [1] for instance.
Theorem 2.6. If X satis es Bochner's almost sure uniform double sequence criterion and the sequence X(t) , t ∈ Z is uniformly integrable, then X is almost periodic in mean.

De nition 2.7. A function
is a compact if for any ε > , there exists a positive integer l(ε, K) such that among any l consecutive integers there exists at least a integer p with the following property Here again, the number p will be called an ε-translation of F and the set of all ε-translations of F is denoted by E(ε, F, K).
Let UB(Z; L (Ω; B)) denote the collection of all uniformly bounded L (Ω; B)-valued random sequences X = {X(t)} t∈Z . It is then easy to check that the space UB(Z; L (Ω; B)) is a Banach space when it is equipped with the norm: In view of the above, the space AP(Z; L (Ω; B)) of almost periodic random sequences equipped with the sup norm · ∞ is also a Banach space. We now state the following composition result.
then for any almost periodic random sequence X = {X(t)} t∈Z , then the L (Ω; B )-valued random sequence

Existence of almost periodic solutions
Let (L(B), · ) denote the Banach algebra of bounded linear operators on a Banach space B equipped with its operator-norm. Let {A(t)} t∈Z be a family of bounded linear invertible operators on B and consider the rst-order system of stochastic di erence equations given by where g : Ω × Z → B is almost periodic in mean, and its corresponding homogeneous equation Our settings requires the following assumptions: is almost periodic in mean.
The evolution family Φ(t, s) associated with Eq. (3.1) is given by and Φ(t, t) = I.
De nition 3.1. Eq.(3.2) is said to have a regular discrete dichotomy if there exist random projections P(t) ∈ L(B) for all t ∈ Z and positive constants M and β ∈ ( , ) such that the following four conditions are satis ed: By repeated application of [(i), De nition 3.1)], we obtain De ne the hull H(X) of a random sequence X as follows:

De nition 3.2. The set
Similarly, for a matrix function A(n), we de ne where T k A =Ã means that lim i→∞ A(t + l(i)) =Ã(t). H(A(t)). Then the system

Theorem 3.3. Suppose that Eq.(3.2) has a regular discrete dichotomy andÃ(t) ∈
satis es a regular discrete dichotomy with same projections and constants.
Let us now state the main results of this paper. For linear stochastic di erence equations, we obtain the following theorem.

Proof. It is not hard to show thatX(t) de ned by Eq.(3.3) is a solution of Eq. (3.2). Moreover,
This implies that X (t) , t ∈ Z is uniformly integrable. Now, to prove the almost periodicity ofX(·), it su ces by Theorem 2.6 to show thatX(·) satis es Bochner's almost sure uniform double sequence criterion. To this end, let k ′ = (k ′ i ) and l ′ = (l ′ i ) be arbitrary sequences of nonnegative integers and then choose a measurable set Ω ⊂ Ω with P(Ω ) = . Let (k i ) ⊂ (k ′ i ) and (l ′ i ) ⊂ (l ′ i ) be their common subsequences such that for each ω ∈ Ω , (T k+l A(ω) = (T l T k A)(ω) and (T k+l g(ω) = (T l T k g)(ω). For simplicity, we omit ω in what follows. Then we havē Thus, taking the limit of the above expression as i → ∞ and recalling the fact that lim i→∞X (t + k i ) = (T kX )(t), we can then write Moreover, as desired.
Consider the semilinear stochastic di erence equations given by and its corresponding homogeneous equation where F : Z × B n → B n . In order to state similar results for the nonlinear case (3.4), we need the following assumption: (H. ) F : (t, w) → F(t, w) is almost periodic in mean in t ∈ Z uniformly in w in O where O ⊂ B n is an arbitrary bounded subset. In addition, we assume that there exists a constant L > such that Under these conditions on A and F, we have the following theorem Thus, Γ is a contraction provided that ML β+ −β < . Using the Banach xed point theorem, we obtain that Γ has a unique xed pointZ, which is the unique almost periodic solution of Eq.(3.4).
Let B = R k be the k-dimensional space of real numbers equipped with Euclidean topology.

Almost periodic solutions to a second-order stochastic di erence equations
Let B = R the set of real numbers equipped with natural absolute value. To illustrate Corollary 3.6, we study the existence of almost periodic solutions to a second-order stochastic di erence equations of the form: where the function f : Z × R → R is almost periodic and satis es (H. ) The function (t, x) → f (t, x) is Lipschitz in x ∈ R uniformly in t ∈ Z, that is, there exists L > such that for all X, X ′ ∈ L (Ω, R) and t ∈ Z.
We also assume that the real random variables a(t) ′ s, b(t) ′ s appearing in Eq.(4.1) are independent and independent of X( ). This assumption together with Eq.(4.1) imply that the sequence {(a(t), b(t))} t∈Z is independent of the sequence {X(t)} t∈Z .
Setting Z(t) := (X(t), X(t + )), note that Eq.(4.1) can be rewritten in R as follows (4.2) and its corresponding homogeneous equation where A(t) is the family of time-dependent sequence matrices de ned by and the function F appearing in Eq.(4.2) is de ned by F(t, Z) = , f (t, X) T .
We adopt the following assumptions: (H. ) The sequences a, b : Z → R are periodic in the following sense: there exists T ∈ Z+ such that for all t ∈ Z, almost surely.
a(t) for all t ∈ Z , almost surely. Next, we show that Eq.(4.3) has a regular discrete dichotomy. For that, let's compute the eigenvalues of A(t).
for all t ∈ Z. Then, the characteristic equation is given by Clearly, (H.7) yields either D(t) > or D(t) < for all t ∈ Z. Under assumptions (H.6)-(H.7), we have: 1. If D(t) > for all t ∈ Z, then the eigenvalues of A(t) are given by and Moreover, it can be shown easily that λ (t), λ (t) < for all t ∈ Z. for all (t, s) ∈ T, where T = {(t, s) ∈ Z × Z : t ≥ s}, has an exponential dichotomy which yields (see Henry [9]) that Eq.(4.3) has a discrete dichotomy. Also, using (H. ), one can easily show that almost surely which, in turn, implies that for all (t, s) ∈ T almost surely. The techniques used in the proof of Corollary 3.6 allow us to obtain the following.  where L is a positive number such that E|ξ t | ≤ L for all t ∈ Z.
We can now conclude that all hypotheses of Theorem 4.1 are satis ed.