\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 124, pp. 1--25.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/124\hfil Square-mean almost periodic solutions] {Existence of square-mean almost periodic mild solutions to some nonautonomous stochastic second-order differential equations} \author[P. H. Bezandry, T. Diagana\hfil EJDE-2010/124\hfilneg] {Paul H. Bezandry, Toka Diagana} % in alphabetical order \address{Paul H. Bezandry \newline Department of Mathematics, Howard University, Washington, DC 20059, USA} \email{pbezandry@howard.edu} \address{Toka Diagana \newline Department of Mathematics, Howard University, Washington, DC 20059, USA} \email{tdiagana@howard.edu} \thanks{Submitted May 4, 2010. Published August 30, 2010.} \subjclass[2000]{34K14, 60H10, 35B15, 34F05} \keywords{Stochastic differential equation; Wiener process} \begin{abstract} In this paper we use the well-known Schauder fixed point principle to obtain the existence of square-mean almost periodic solutions to some classes of nonautonomous second order stochastic differential equations on a Hilbert space. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\mathbb{B}$ be a Banach space. In Goldstein and N'Gu\'er\'ekata \cite{G}, the existence of almost automorphic solutions to the evolution $$ u'(t)=Au(t) + F(t, u(t)), \quad t \in \mathbb{R} $$ where $A: D(A)\subset \mathbb{B}\to\mathbb{B}$ is a closed linear operator on a Banach space $\mathbb{B}$ which generates an exponentially stable $C_0$-semigroup $\mathcal{T}=(T(t))_{t\geq 0}$ and the function $F:\mathbb{R}\times\mathbb{B}\to\mathbb{B}$ is given by $F(t, u)=P(t)Q(u)$ with $P$, $Q$ being some appropriate continuous functions satisfying some additional conditions, was established. The main tools used in \cite{G} are fractional powers of operators and the fixed-point theorem of Schauder. Recently Diagana \cite{TK} generalized the results of \cite{G} to the \emph{nonautonomous} case by obtaining the existence of almost automorphic mild solutions to \begin{equation}\label{11} u'(t) = A(t) u(t) + f(t, u(t)), \quad t \in \mathbb{R} \end{equation} where $A(t)$ for $t\in \mathbb{R}$ is a family of closed linear operators with domains $D(A(t))$ satisfying Acquistapace-Terreni conditions, and the function $f: \mathbb{R} \times \mathbb{B} \mapsto \mathbb{B}$ is almost automorphic in $t \in \mathbb{R}$ uniformly in the second variable. For that, Diagana utilized similar techniques as in \cite{G}, dichotomy tools, and the Schauder fixed point theorem. Let $\mathbb{H}$ be a Hilbert space. Motivated by the above mentioned papers, the present paper is aimed at utilizing Schauder fixed point theorem to study the existence of $p$-th mean almost periodic solutions to the nonautonomous stochastic differential equations \begin{equation}\label{B1} dX(t)= A(t) X(t) \,dt + F_1(t, X(t))\,dt + F_2(t, X(t))\,d\mathbb{W}(t), \quad t\in\mathbb{R}, \end{equation} where $(A(t))_{t \in \mathbb{R}}$ is a family of densely defined closed linear operators satisfying Acquistapace and Terreni conditions, the functions $F_1: \mathbb{R} \times L^p (\Omega, \mathbb{H}) \to L^p (\Omega, \mathbb{H})$ and $F_2: \mathbb{R} \times L^p (\Omega, \mathbb{H}) \to L^p (\Omega, \mathbb{L}_2^0)$ are jointly continuous satisfying some additional conditions, and $\mathbb{W}$ is a Wiener process. Then, we utilize our main results to study the existence of square-mean almost periodic solutions to the second order stochastic differential equations \begin{equation}\label{B2} \begin{aligned} &d X'(\omega, t) + a(t)\, d X(\omega, t)\\ &= \Big[- b(t)\, \mathcal{A} X(\omega, t) + f_1(t,X(\omega, t))\Big]\,dt \\ + f_2(t, X(\omega, t)) \,d\mathbb{W}(\omega, t), \end{aligned} \end{equation} for all $\omega\in\Omega$ and $t\in\mathbb{R}$, where $\mathcal{A}: D(\mathcal{A}) \subset \mathbb{H} \to \mathbb{H}$ is a self-adjoint linear operator whose spectrum consists of isolated eigenvalues $0 < \lambda_1 < \lambda_2 < \dots < \lambda_n \to \infty$ with each eigenvalue having a finite multiplicity $\gamma_j$ equals to the multiplicity of the corresponding eigenspace, the functions $a, b: \mathbb{R} \to (0, \infty)$ are almost periodic functions, and the function $f_i(i=1, 2): \mathbb{R} \times L^2(\Omega, \mathbb{H}) \to L^2(\Omega, \mathbb{H})$ are jointly continuous functions satisfying some additional conditions and $\mathbb{W}$ is a one dimensional Brownian motion. It should be mentioned the existence of almost periodic to \eqref{B1} in the case when $A(t)$ is periodic, that is, $A(t+T) = A(t)$ for each $t \in \mathbb{R}$ for some $T>0$ was established by Da Prato and Tudor in \cite{da}. In the paper by Bezandry and Diagana \cite{BD1}, upon assuming that the operators $A(t)$ satisfy Acquistapace-Terreni conditions and that $F_i$ ($i= 1, 2, 3,$) satisfy Lipschitz conditions, the Banach fixed point principle was utilized to obtain the existence of a square-mean almost periodic solutions to \eqref{B1}. In this paper is goes back to utilizing Schauder fixed theorem to establish the existence of $p$-th mean almost periodic solutions to \eqref{B1}. Next, we make extensive use of those abstract results to deal with the existence of square-mean almost periodic solutions to the second-order stochastic differential equations formulated in \eqref{B2}. \section{Preliminaries} In this section, $\mathcal{A}: D(\mathcal{A}) \subset \mathbb{H} \to \mathbb{H}$ stands for a self-adjoint linear operator whose spectrum consists of isolated eigenvalues $0 < \lambda_1 < \lambda_2 < \dots< \lambda_n \to \infty$ with each eigenvalue having a finite multiplicity $\gamma_j$ equals to the multiplicity of the corresponding eigenspace. Let $\{e_{j}^k\}$ be a (complete) orthonormal sequence of eigenvectors associated with the eigenvalues $\{\lambda_j\}_{j\geq 1}$. Clearly, for each \[ u \in D(\mathcal{A}) :=\Big\{x \in \mathbb{H}: \quad \sum_{j=1}^\infty \lambda_j^2 \| E_j x\|^2 < \infty\Big\}, \] $$ \mathcal{A}x = \sum_{j=1}^\infty \lambda_j \sum_{k=1}^{\gamma_j} \langle x, e_{j}^k \rangle e_{j}^k = \sum_{j=1}^\infty \lambda_j E_j x $$ where $ E_j x =\sum_{k=1}^{\gamma_j} \langle x, e_{j}^k \rangle e_{j}^k$. Note that $\{E_j\}_{j\geq1}$ is a sequence of orthogonal projections on $\mathbb{H}$. Moreover, each $x \in \mathbb{H}$ can written as follows: $$ x = \sum_{j=1}^\infty E_j x. $$ It should also be mentioned that the operator $-\mathcal{A}$ is the infinitesimal generator of an analytic semigroup $\{T(t)\}_{t \geq 0}$, which is explicitly expressed in terms of those orthogonal projections $E_j$ by, for all $x \in\mathbb{H}$, $$ T(t) x = \sum_{j=1}^\infty e^{-\lambda_j t} E_j x. $$ In addition, the fractional powers $\mathcal{A}^r$ ($r \geq 0$) of $\mathcal{A}$ exist and are given by $$ D(\mathcal{A}^r) = \Big\{x \in \mathbb{H}: \sum_{j=1}^\infty \lambda_j^{2r} \|E_j x\|^2 < \infty\Big\} $$ and $$ \mathcal{A}^r x = \sum_{j=1}^\infty \lambda_j^{2r} E_j x, \quad \forall x \in D(\mathcal{A}^r). $$ Let $\big(\mathbb{B}, \|\cdot\|\big)$ be a Banach space. If $L$ is a linear operator on the Banach space $\mathbb{B}$, then $D(L)$, $\rho (L)$, $\sigma (L)$, $N(L)$, $N(L)$, and $R(L)$ stand respectively for the domain, resolvent, spectrum, null space, and the range of $L$. also, we set $R(\lambda, L):= (\lambda I - L)^{-1}$ for all $\lambda\in\rho (L)$. If $P$ is a projection, we then set $Q=I-P$. If $\mathbb{B}_1$, $\mathbb{B}_2$ are Banach spaces, then the space $B(\mathbb{B}_1, \mathbb{B}_2)$ denotes the collection of all bounded linear operators from $\mathbb{B}_1$ into $\mathbb{B}_2$ equipped with its natural topology. This is simply denoted by $B(\mathbb{B}_1)$ when $\mathbb{B}_1=\mathbb{B}_2$. \subsection{Evolution Families} Let $\mathbb{B}$ be a Banach space equipped with the norm $\|\cdot\|$. The family of closed linear operators $A(t)$ for $t\in \mathbb{R}$ on $\mathbb{B}$ with domain $D(A(t))$ (possibly not densely defined) is said to satisfy Acquistapace-Terreni conditions if: there exist constants $\omega \geq 0$, $\theta \in \Big(\frac{\pi}{2},\pi\Big)$, $K, L \geq 0$ and $\mu, \nu \in (0, 1]$ with $\mu + \nu > 1$ such that \begin{equation}\label{AT1} S_{\theta} \cup \{0\} \subset \rho\big(A(t)-\omega\big) \ni \lambda, \quad \|R\big(\lambda,A(t)-\omega\big)\|\le \frac{K}{1+|\lambda|} \end{equation} and \begin{equation}\label{AT2} \|\big(A(t)-\omega\big)R\big(\lambda,A(t)-\omega\big) \Big[R\Big(\omega,A(t)\Big)-R\Big(\omega,A(s)\Big)\Big]\| \le L |t-s|^\mu\,|\lambda|^{-\nu} \end{equation} for $t,s\in\mathbb{R}$, $ \lambda \in S_\theta:= \big\{\lambda\in\mathbb{C}\setminus\{0\}: |\arg \lambda|\le\theta\big\}$. It should mentioned that the conditions \eqref{AT1} and \eqref{AT2} were introduced in the literature by Acquistapace and Terreni in \cite{AFT, AT} for $\omega=0$. Among other things, it ensures that there exists a unique evolution family ${\mathcal{U}} = U(t,s)$ on $\mathbb{B}$ associated with $A(t)$ satisfying \begin{itemize} \item[(a)] $U(t,s)U(s,r)=U(t,r)$; \item[(b)] $U(t,t)=I$ for $t\geq s\geq r $ in $\mathbb{R}$; \item[(c)] $(t,s)\mapsto U(t,s)\in B(\mathbb{B})$ is continuous for $t>s$; \item[(d)] $U(\cdot,s)\in C^1((s,\infty),B(\mathbb{B}))$, $ \frac{\partial U}{\partial t}(t,s) =A(t)U(t,s)$ and \begin{equation}\label{au} \|A(t)^k U(t,s)\|\le K\,(t-s)^{-k} \end{equation} for $0< t-s\le 1$, $k=0,1$; and \item[(e)] $\partial_s^+ U(t,s)x=-U(t,s)A(s)x$ for $t>s$ and $x\in D(A(s))$ with $A(s)x \in \overline{D(A(s))}$. \end{itemize} It should also be mentioned that the above-mentioned properties were mainly established in \cite[Theorem 2.3]{Ac} and \cite[Theorem 2.1]{Ya2}, see also \cite{AT, Ya1}. In that case we say that $A(\cdot)$ generates the evolution family $U (\cdot, \cdot)$. One says that an evolution family $\mathcal{U}$ has an \emph{exponential dichotomy} (or is \emph{hyperbolic}) if there are projections $P(t)$ ($t\in\mathbb{R}$) that are uniformly bounded and strongly continuous in $t$ and constants $\delta>0$ and $N\ge1$ such that \begin{itemize} \item[(f)] $U(t,s)P(s) = P(t)U(t,s)$; \item[(g)] the restriction $U_Q(t,s):Q(s)\mathbb{B}\to Q(t)\mathbb{B}$ of $U(t,s)$ is invertible (we then set $\widetilde{U}_Q(s,t):=U_Q(t,s)^{-1}$); and \item[(h)] $\|U(t,s)P(s)\| \le Ne^{-\delta (t-s)}$ and $\|\widetilde{U}_Q(s,t)Q(t)\|\le Ne^{-\delta (t-s)}$ for $t\ge s$ and $t,s\in \mathbb{R}$. \end{itemize} This setting requires some estimates related to $U(t,s)$. For that, we introduce the interpolation spaces for $A(t)$. We refer the reader to the following excellent books \cite{EN}, and \cite{Lun} for proofs and further information on theses interpolation spaces. Let $A$ be a sectorial operator on $\mathbb{B}$ (for that, in \eqref{AT1}-\eqref{AT2}, replace $A(t)$ with $A$) and let $\alpha\in(0,1)$. Define the real interpolation space $$ \mathbb{B}^A_{\alpha}: = \Big\{x\in \mathbb{B}: \|x\|^A_{\alpha}:= \sup_{r>0} \|r^{\alpha}(A-\omega)R(r,A-\omega)x\|<\infty \Big\}, $$ which, by the way, is a Banach space when endowed with the norm $\|\cdot\|^A_{\alpha}$. For convenience we further write $$ \mathbb{B}_0^A:=\mathbb{B},\quad \|x\|_0^A:=\|x\|, \quad \mathbb{B}_1^A:=D(A) $$ and $$\|x\|^A_{1}:=\|(\omega-A)x\|. $$ Moreover, let $\hat{\mathbb{B}}^A:=\overline{D(A)}$ of $\mathbb{B}$. In particular, we have the following continuous embedding \begin{equation} \label{embeddings1} D(A)\hookrightarrow \mathbb{B}^A_{\beta}\hookrightarrow D((\omega-A)^{\alpha}) \hookrightarrow \mathbb{B}^A_{\alpha}\hookrightarrow \hat{\mathbb{B}}^A \hookrightarrow \mathbb{B}, \end{equation} for all $0<\alpha<\beta<1$, where the fractional powers are defined in the usual way. In general, $D(A)$ is not dense in the spaces $\mathbb{B}_\alpha^A$ and $\mathbb{B}$. However, we have the following continuous injection \begin{equation}\label{closure} \mathbb{B}_\beta^A \to \overline{D(A)}^{\|\cdot\|_\alpha^A} \end{equation} for $0<\alpha <\beta <1$. Given the family of linear operators $A(t)$ for $t\in \mathbb{R}$, satisfying \eqref{AT1}-\eqref{AT2}, we set $$ \mathbb{B}^t_\alpha:=\mathbb{B}_\alpha^{A(t)}, \quad \hat{\mathbb{B}}^t:=\hat{\mathbb{B}}^{A(t)} $$ for $0\le \alpha\le 1$ and $t\in\mathbb{R}$, with the corresponding norms. Then the embedding in \eqref{embeddings1} holds with constants independent of $t\in\mathbb{R}$. These interpolation spaces are of class $\mathcal{J}_{\alpha}$ \cite[Definition 1.1.1 ]{Lun} and hence there is a constant $c(\alpha)$ such that \begin{equation}\label{J} \|y\|_{\alpha}^t\leq c(\alpha)\|y\|^{1-\alpha} \|A(t)y\|^{\alpha}, \quad y\in D(A(t)). \end{equation} We have the following fundamental estimates for the evolution family $U(t,s)$. \begin{proposition}\cite{W}\label{pes} Suppose the evolution family $U = U(t,s)$ has exponential dichotomy. For $x \in \mathbb{B}$, $ 0\leq \alpha \leq 1$ and $t > s$, the following hold: \begin{itemize} \item[(i)] There is a constant $c(\alpha)$, such that %% \begin{equation}\label{eq1.1} \|U(t,s)P(s)x\|_{\alpha}^t\leq c(\alpha)e^{- \frac{\delta}{2}(t-s)}(t-s)^{-\alpha} \|x\|. \end{equation} \item[(ii)] There is a constant $m(\alpha)$, such that \begin{equation}\label{eq2.1} \|\widetilde{U}_{Q}(s,t)Q(t)x\|_{\alpha}^s\leq m(\alpha)e^{-\delta (t-s)}\|x\|. \end{equation} \end{itemize} \end{proposition} We need the following technical lemma. \begin{lemma}[{\cite[Diagana]{TK, BO}}] \label{pess} For each $x\in \mathbb{B}$, suppose that the family of operators $A(t)$ ($t \in \mathbb{R}$) satisfy Acquistapce-Terreni conditions, assumption {\rm (H.2)} holds, and that there exist real numbers $\mu, \alpha, \beta$ such that $ 0\leq \mu < \alpha < \beta < 1$ with $2\alpha> \mu + 1$. Then there is a constant $r(\mu, \alpha) > 0$ such that \begin{equation}\label{eq1.11} \|A(t) U(t,s)x\|_{\alpha}\leq r(\mu, \alpha)e^{- \frac{\delta}{4}(t-s)}(t-s)^{-\alpha} \|x\|. \end{equation} for all $t > s$. \end{lemma} \begin{proof} Let $x \in \mathbb{B}$. First of all, note that $\|A(t) U(t,s) \|_{B(\mathbb{B}, \mathbb{B}_\alpha)} \leq K (t-s)^{-(1-\alpha)}$ for all $t,s$ such that $0 < t-s \leq 1$ and $\alpha \in [0, 1]$. Letting $t-s \geq 1$ and using (H2) and the above-mentioned approximate, we obtain \begin{align*} \|A(t) U(t,s)x\|_\alpha &= \|A(t) U(t, t-1) U(t-1, s) x\|_\alpha \\ &\leq \|A(t) U(t, t-1) \|_{B(\mathbb{B}, \mathbb{B}_\alpha)} \| U(t-1, s) x\| \\ &\leq M K e^{\delta} e^{-\delta (t-s)}\|x\| \\ &= K_1 e^{-\delta (t-s)}\|x\| \\ &= K_1 e^{-\frac{3\delta}{4}(t-s)} (t-s)^{\alpha} (t-s)^{-\alpha} e^{-\frac{\delta}{4} (t-s)}\|x\|. \end{align*} Now since $e^{-\frac{3\delta}{4}(t-s)} (t-s)^{\alpha} \to 0$ as $t \to \infty$ it follows that there exists $c_4(\alpha) > 0$ such that $$ \|A(t) U(t,s) x\|_\alpha \leq c_4(\alpha) (t-s)^{-\alpha} e^{-\frac{\delta}{4} (t-s)}\|x\|. $$ Now, let $0 < t-s \leq 1$. Using \eqref{eq1.1} and the fact $2\alpha > \mu + 1$, we obtain \begin{align*} \|A(t) U(t,s)x\|_\alpha &= \|A(t) U(t, \frac{t+s}{2}) U(\frac{t+s}{2}, s) x\|_\alpha \\ &\leq \|A(t) U(t, \frac{t+s}{2}) \|_{B(\mathbb{B}, \mathbb{B}_\alpha)} \| U(\frac{t+s}{2}, s) x\| \\ &\leq k_1 \|A(t) U(t, \frac{t+s}{2}) \|_{B(\mathbb{B}, \mathbb{B}_\alpha)} \| U(\frac{t+s}{2}, s) x\|_\mu \\ &\leq k_1 K \Big(\frac{t-s}{2}\Big)^{\alpha - 1} c(\mu) \Big(\frac{t-s}{2}\Big)^{-\mu} e^{-\frac{\delta}{4} (t-s)} \|x\| \\ &\leq c_5 (\alpha, \mu) (t-s)^{\alpha -1 - \mu} e^{-\frac{\delta}{4} (t-s)} \|x\| \\ &\leq c_5 (\alpha, \mu) (t-s)^{-\alpha} e^{-\frac{\delta}{4} (t-s)} \|x\|. \end{align*} Therefore there exists $r(\alpha,\mu) > 0$ such that $$ \|A(t) U(t,s)x\|_{\alpha} \leq r(\alpha,\mu) (t-s)^{-\alpha} e^{-\frac{\delta}{4} (t-s)} \|x\| $$ for all $t, s \in \mathbb{R}$ with $t \geq s$. \end{proof} It should be mentioned that if $U(t,s)$ is exponentially stable, then $P(t) = I$ and $Q(t) = I- P(t) = 0$ for all $t\in \mathbb{R}$. In that case, \eqref{eq1.1} still holds and be rewritten as follows: for all $x \in \mathbb{B}$, \begin{equation}\label{eq111} \|U(t,s)x\|_{\alpha}^t\leq c(\alpha)e^{- \frac{\delta}{2}(t-s)}(t-s)^{-\alpha} \|x\|. \end{equation} \subsection{Wiener process and $P$-th mean almost periodic stochastic processes} For details of this subsection, we refer the reader to Bezandry and Diagana \cite{BD1}, Corduneanu \cite{COR89}, and the references therein. Throughout this paper, $\mathbb{H}$ and $\mathbb{K}$ will denote real separable Hilbert spaces with respective norms $\|\cdot\|$ and $\|\cdot\|_\mathbb{K}$. Let $(\Omega, {\mathcal F}, {\bf P})$ be a complete probability space. We denote by $L_2(\mathbb{K}, \mathbb{H})$ the space of all Hilbert-Schmidt operators acting between $\mathbb{K}$ and $\mathbb{H}$ equipped with the Hilbert-Schmidt norm $\|\cdot\|_2$. For a symmetric nonnegative operator $Q\in L_2(\mathbb{K}, \mathbb{H})$ with finite trace we assume that $\{\mathbb {W}(t), \; t\in\mathbb{R}\}$ is a $Q$-Wiener process defined on $(\Omega, {\mathcal F}, \textbf{P})$ and with values in $\mathbb{K}$. Recall that $\mathbb{W}$ can obtained as follows: let $\{W_i(t), \; t\in\mathbb{R}\}, \;\; i=1, 2$, be independent $\mathbb{K}$-valued $Q$-Wiener processes, then \[ \mathbb{W}(t)= \begin{cases} W_1(t) & \text{if } t\geq 0 \\ W_2(-t) & \text{if } t\leq 0 \end{cases} \] is $Q$-Wiener process with $\mathbb{R}$ as time parameter. We let $\mathcal{F}_t=\sigma\{\mathbb{W}(s), \; s\leq t\}$. Let $p\geq 2$. The collection of all strongly measurable, $p$-th integrable $\mathbb{H}$-valued random variables, denoted by $L^p(\Omega, \mathbb{H})$, is a Banach space equipped with norm $$ \|X\|_{L^p(\Omega, \mathbb{H})}=(\mathbf{E}\|X\|^p)^{1/p}\,, $$ where the expectation $\mathbf{E}$ is defined by $$ \mathbf{E}[g]=\int_{\Omega}g(\omega)d\textbf{P}(\omega)\,. $$ Let $\mathbb{K}_0=Q^{\frac{1}{2}}\mathbb{K}$ and $\mathbb{L}^0_2=L_2(\mathbb{K}_0, \mathbb{H})$ with respect to the norm $$ \|\Phi\|^2_{\mathbb{L}^0_2}=\|\Phi\,Q^{\frac{1}{2}}\|_2^2=\text{Tr} (\Phi Q \Phi^{*})\;. $$ \begin{definition} \rm A stochastic process $X: \mathbb{R} \to L^p(\Omega; \mathbb{B})$ is said to be continuous whenever $$ \lim_{t\to s}\mathbf{E}\|X(t)-X(s)\|^p=0. $$ \end{definition} \begin{definition} \rm A stochastic process $X: \mathbb{R} \to L^p(\Omega; \mathbb{B})$ is said to be stochastically bounded whenever $$ \lim_{N\to\infty}\sup_{t\in \textbf{R}}{\bf P}\Big\{\|X(t)\|> N\Big\}=0. $$ \end{definition} \begin{definition} \rm A continuous stochastic process $X: \mathbb{R} \to L^p(\Omega; \mathbb{B})$ is said to be $p$-th mean almost periodic if for each $\varepsilon >0$ there exists $l(\varepsilon)>0$ such that any interval of length $l(\varepsilon)$ contains at least a number $\tau$ for which \begin{equation}\label{3A40} \sup_{t\in \textbf{R}}\mathbf{E}\|X(t+\tau) - X(t)\|^p <\varepsilon\,. \end{equation} A continuous stochastic process $X$, which is $2$-nd mean almost periodic will be called \emph{square-mean almost periodic}. Like for classical almost periodic functions, the number $\tau$ will be called an $\varepsilon$-translation of $X$. \end{definition} The collection of all $p$-th mean almost periodic stochastic processes $X: \mathbb{R} \to L^p(\Omega;\mathbb{B})$ will be denoted by $AP({\mathbb{R}};L^p(\Omega;\mathbb{B}))$. The next lemma provides with some properties of $p$-th mean almost periodic processes. \begin{lemma}\label{AC} If $X$ belongs to $AP({\mathbb{R}};L^p(\Omega; \mathbb{B}))$, then \begin{itemize} \item[(i)] the mapping $t\to \mathbf{E}\|X(t)\|^p$ is uniformly continuous; \item[(ii)] there exists a constant $M > 0$ such that $\mathbf{E}\|X(t)\|^p\le M$, for each $t\in \mathbb{R}$; \item[(iii)] $X$ is stochastically bounded. \end{itemize} \end{lemma} \begin{lemma}\label{HHH} $AP(\mathbb{R};L^p(\Omega; \mathbb{B}))\subset BUC (\mathbb{R};L^p(\Omega; \mathbb{B}))$ is a closed subspace. \end{lemma} In view of Lemma \ref{HHH}, it follows that the space $AP(\mathbb{R};L^p(\Omega; \mathbb{B}))$ of $p$-th mean almost periodic processes equipped with the sup norm $\|\cdot\|_\infty$ is a Banach space. Let $\big(\mathbb{B}_1, \|\cdot\|_1\big)$ and $\big(\mathbb{B}_2, \|\cdot\|_2\big)$ be Banach spaces and let $L^p(\Omega; \mathbb{B}_1)$ and $L^p(\Omega; \mathbb{B}_2)$ be their corresponding $L^p$-spaces, respectively. \begin{definition} \rm A function $F: \mathbb{R} \times L^p(\Omega;\mathbb{B}_1) \to L^p(\Omega; \mathbb{B}_2))$, $(t, Y) \mapsto F(t, Y)$, which is jointly continuous, is said to be $p$-th mean almost periodic in $t \in \mathbb{R}$ uniformly in $Y\in K$ where $K \subset L^p(\Omega; \mathbb{B}_1)$ is a compact if for any $\varepsilon >0$, there exists $l_\varepsilon (K)>0$ such that any interval of length $l_\varepsilon(K)$ contains at least a number $\tau$ for which $$ \sup_{t\in \textbf{R}}\mathbf{E}\|F(t+\tau, Y) - F(t, Y)\|_2^p <\varepsilon $$ for each stochastic process $Y: \mathbb{R} \to K$. \end{definition} We have the following composition result. \begin{theorem}\label{U} Let $F: \mathbb{R} \times L^p(\Omega ; \mathbb{B}_1) \to L^p(\Omega ; \mathbb{B}_2)$, $(t,Y) \mapsto F(t, Y)$ be a $p$-th mean almost periodic process in $t \in \mathbb{R}$ uniformly in $Y\in K$, where $K \subset L^p(\Omega ; \mathbb{B}_1)$ is any compact subset. Suppose that $F (t, \cdot)$ is uniformly continuous on bounded subsets $K' \subset L^p(\Omega ; \mathbb{B}_1)$ in the following sense: for all $\varepsilon > 0$ there exists $\delta_\varepsilon > 0$ such that $X, Y \in K'$ and $\mathbf{E} \|X-Y\|_1^p < \delta_{\varepsilon}$, then $$ \mathbf{E} \|F(t, Y) - F(t, Z)\|_2^p < \varepsilon, \quad \forall t \in \mathbb{R}. $$ Then for any $p$-th mean almost periodic process $\Phi: \mathbb{R} \to L^p(\Omega ; \mathbb{B}_1)$, the stochastic process $t \mapsto F(t, \Phi(t))$ is $p$-th mean almost periodic. \end{theorem} \section{ Main Results} In this section, we study the existence of $p$-th mean almost periodic solutions to the class of nonautonomous stochastic differential equations of type \eqref{B1} where $(A(t))_{t \in \mathbb{R}}$ is a family of closed linear operators on $L^p(\Omega; \mathbb{H})$ satisfying \eqref{AT1}-\eqref{AT2}, and the functions $F_1: \mathbb{R} \times L^p(\Omega, \mathbb{H}) \to L^p(\Omega, \mathbb{H})$, $F_2: \mathbb{R} \times L^p(\Omega, \mathbb{H}) \to L^p(\Omega, \mathbb{L}_2^0)$ are $p$-th mean almost periodic in $t\in\mathbb{R}$ uniformly in the second variable, and $\mathbb{W}$ is $Q$-Wiener process taking its values in $\mathbb{K}$ with the real number line as time parameter. Our method for investigating the existence and uniqueness of a $p$-th mean almost periodic solution to \eqref{B1} consists of making extensive use of ideas and techniques utilized in \cite{G}, \cite{BO}, and the Schauder fixed-point theorem. To study the existence of $p$-th mean almost periodic solutions to \eqref{B1}, we suppose that the following assumptions hold: \begin{itemize} \item [(H1)] The injection $\mathbb{H}_\alpha\hookrightarrow \mathbb{H}$ is compact. \item [(H2)] The family of operators $A(t)$ satisfy Acquistapace-Terreni conditions and the evolution family $U(t, s)$ associated with $A(t)$ is exponentially stable; that is, there exist constant $M$, $\delta >0$ such that $$ \|U(t, s)\|\leq M e^{-\delta(t-s)} $$ for all $t\geq s$. \item [(H3)] Let $\mu, \alpha, \beta$ be real numbers such that $ 0\leq \mu < \alpha < \beta < 1$ with $2\alpha> \mu + 1$. Moreover, $\mathbb{H}_\alpha^t=\mathbb{H}_\alpha$ and $\mathbb{H}^t_\beta=\mathbb{H}_\beta$ for all $t\in\mathbb{R}$, with uniform equivalent norms. \item [(H4)] $R(\zeta, A(\cdot))\in AP(\mathbb{R}, L^p(\Omega; \mathbb{H}))$. \item [(H5)] The function $F_1: \mathbb{R}\times L^p(\Omega, \mathbb{H})\to L^p(\Omega, \mathbb{H})$ is $p$-th mean almost periodic in the first variable uniformly in the second variable. Furthermore, $X\to F_1(t, X)$ is uniformly continuous on any bounded subset $\mathcal{O}$ of $L^p(\Omega, \mathbb{H})$ for each $t\in\mathbb{R}$. Finally, $$ \sup_{t\in\mathbb{R}}\mathbf{E}\|F_1(t, X)\|^p \leq \mathcal{M}_1\big(\|X\|_\infty\big) $$ where $\mathcal{M}_1:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function satisfying $$ \lim_{r\to\infty}\frac{\mathcal{M}_1(r)}{r}=0\,. $$ \item [(H6)] The function $F_2: \mathbb{R}\times L^p(\Omega, \mathbb{H})\to L^p(\Omega, \mathbb{L}_2^0)$ is $p$-th mean almost periodic in the first variable uniformly in the second variable. Furthermore, $X\to F_2(t, X)$ is uniformly continuous on any bounded subset $\mathcal{O}'$ of $L^p(\Omega, \mathbb{H})$ for each $t\in\mathbb{R}$. Finally, $$ \sup_{t\in\mathbb{R}}\mathbf{E}\|F_2(t, X)\|^p\leq \mathcal{M}_2 \big(\|X\|_\infty\big) $$ where $\mathcal{M}_2:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function satisfying $\lim\limits_{r\to\infty} \mathcal{M}_2(r)/r=0$. \end{itemize} In this section, $\Gamma_1$ and $\Gamma_2$ stand respectively for the nonlinear integral operators defined by \begin{gather*} (\Gamma_1X)(t):=\int_{-\infty}^t U(t, s)F_1(s, X(s))\,ds,\\ (\Gamma_2X)(t):=\int_{-\infty}^t U(t, s) F_2(s, X(s))\,d\mathbb{W}(s)\,. \end{gather*} In addition to the above-mentioned assumptions, we assume that $\alpha\in \big(0, \frac{1}{2}-\frac{1}{p}\big)$ if $p>2$ and $\alpha\in \big(0, \frac{1}{2}\big)$ if $p=2$. \begin{lemma}\label{M1} Under assumptions {\rm (H2)--(H6)}, the mappings $\Gamma_i : BC(\mathbb{R}, L^p(\Omega, \mathbb{H}))\\ \to BC(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha))$ $(i=1, 2)$ are well defined and continuous. \end{lemma} \begin{proof} We first show that $\Gamma_i\big(BC\big(\mathbb{R}, L^p(\Omega, \mathbb{H})\big)\big)\subset BC\big(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha)\big)$ $(i=1,2)$. Let us start with $\Gamma_1X$. Using \eqref{eq111} it follows that for all $X\in BC (\mathbb{R}, L^p(\Omega, \mathbb{H}))$, \begin{align*} &\mathbf{E}\|\Gamma_1X(t)\|^p_\alpha\\ &\leq \mathbf{E}\Big[\int_{-\infty}^t c(\alpha)(t-s)^{-\alpha} e^{-\frac{\delta}{2} (t-s)}\|F_1(s, X(s))\|\,ds\Big]^p\\ &\leq c(\alpha)^p\Big(\int_{-\infty}^t (t-s)^{-\frac{p}{p-1}\alpha} e^{-\frac{\delta}{2} (t-s)}\,ds\Big)^{p-1} \Big(\int_{-\infty}^t e^{-\frac{\delta}{2}(t-s)}\mathbf{E}\|F_1(s, X(s))\|^p\,ds\Big)\\ &\leq c(\alpha)^p\Big(\Gamma\big(1-\frac{p}{p-1}\alpha\big) \Big(\frac{2}{\delta}\Big)^{1-\frac{p}{p-1}\alpha} \Big(\frac{2}{\delta}\Big)^{p-1}\mathcal{M}_1\big(\|X\|_\infty\big)\\ &\leq c(\alpha)^p\Big(\Gamma\big(1-\frac{p}{p-1}\alpha\big)\Big)^{p-1} \Big(\frac{2}{\delta}\Big)^{p(1-\alpha)}\mathcal{M}_1 \big(\|X\|_\infty\big)\,, \end{align*} and hence $$ \|\Gamma_1X\|^p_{\alpha, \infty}:=\sup_{t\in\mathbb{R}}\mathbf{E} \|\Gamma_1X(t)\|^p_\alpha\leq l(\alpha, \delta, p)\mathcal{M}_1 \big(\|X\|_\infty\big)\,, $$ where $l(\alpha, \delta, p)=c(\alpha)^p \Big(\Gamma\big(1-\frac{p}{p-1}\alpha\big)\Big)^{p-1} \Big(\frac{2}{\delta}\Big)^{p(1-\alpha)}$. As to $\Gamma_2X$, we proceed into two steps. For $p>2$, we need the following estimates. \begin{lemma}\label{2E80} Let $p>2$, $0<\alpha<1$, $\alpha +\frac{1}{p}<\xi <1/2$, and $\Psi: \Omega\times \mathbb{R}\to\mathbb{L}_2^0$ be an $(\mathcal{F}_t)$-adapted measurable stochastic process such that $$ \sup_{t\in\mathbb{R}}\mathbf{E}\|\Psi (t)\|^p_{\mathbb{L}_2^0}<\infty\,. $$ Then \begin{itemize} \item [(i)] $\mathbf{E}\|\int_{-\infty}^t(t-s)^{-\xi}U(t, s) \Psi (s) \,d\mathbb{W}(s)\|^p\leq s(\Gamma, \xi, \delta, p)\sup_{t\in\mathbb{R}}\mathbf{E} \|\Psi(t)\|^p_{\mathbb{L}_2^0}$; \item [(ii)] $ \mathbf{E}\|\int_{-\infty}^tU(t, s)\Psi (s) \,d\mathbb{W}(s)\|^p_\alpha\leq k(\Gamma, \alpha, \xi, \delta, p) \sup_{t\in\mathbb{R}}\mathbf{E}\|\Psi(t)\|^p_{\mathbb{L}_2^0}$ \end{itemize} where $s(\Gamma, \xi, \delta, p)$ and $k(\Gamma, \alpha, \xi, \delta, p)$ are positive constants with $\Gamma$ a classical Gamma function. \end{lemma} \begin{proof} (i) A direct application of a Proposition due to De Prato and Zabczyk \cite{DZ} and Holder's inequality allows us to write \begin{align*} &\mathbf{E}\|\int_{-\infty}^t (t-\sigma)^{-\xi}U(t, \sigma) \Psi (\sigma)\,d\mathbb{W}(\sigma)\|^p\\ &\leq C_p\mathbf{E}\Big[\int_{-\infty}^t(t-\sigma)^{-2\xi}\|U(t, \sigma) \Psi (\sigma)\|^2\,d\sigma\Big]^{p/2}\\ &\leq C_p N^p\mathbf{E}\Big[\int_{-\infty}^t(t-\sigma)^{-2\xi} e^{-2\delta (t-\sigma)}\|\Psi(\sigma)\|^2_{\mathbb{L}_2^0}\,d\sigma\Big]^{p/2}\\ &\leq C_p N^p\Big(\int_{-\infty}^t(t-\sigma)^{-2\xi} e^{-2\delta (t-\sigma)}\,d\sigma\Big)^{p-1} \Big(\int_{-\infty}^t e^{-2\delta (t-\sigma)}\mathbf{E}\|\Psi(\sigma) \|^p_{\mathbb{L}_2^0}\,d\sigma\Big)\\ &\leq C_p N^p\Big(\Gamma(1-\frac{2p\xi}{p-2}) (2\delta)^{\frac{2p\xi}{p-2} -1}\Big)^{\frac{p-2}{2}} \Big(\frac{1}{2\delta}\Big)\sup_{t\in\mathbb{R}}\mathbf{E}\|\Psi(t)\|^p_{\mathbb{L}_2^0}\\ &\leq s(\Gamma, \xi, \delta, p)\sup_{t\in\mathbb{R}}\mathbf{E} \|\Psi(t)\|^p_{\mathbb{L}_2^0}\,. \end{align*} To prove (ii), we use the factorization method of the stochastic convolution integral. \begin{equation}\label{2E70} \int_{-\infty}^tU(t, s)\Psi (s)\,d\mathbb{W}(s) =\frac{\sin\pi\xi}{\pi} (R_\xi \mathbb{S}_\Psi)(t)\quad \text{a.s.} \end{equation} where $$ (R_\xi \mathbb{S}_\Psi)(t)=\int_{-\infty}^t(t-s)^{\xi -1}U(t, s) \mathbb{S}_\Psi(s)\,ds $$ with $$ \mathbb{S}_\Psi(s)=\int_{-\infty}^s(s-\sigma)^{-\xi}U(s, \sigma) \Psi (\sigma) \,d\mathbb{W}(\sigma)\,, $$ and $\xi$ satisfying $\alpha +\frac{1}{p} <\xi <1/2$. We can now evaluate \begin{align*}\label{2E83} & \mathbf{E}\|\int_{-\infty}^tU(t, s)\Psi (s)\,d\mathbb{W}(s)\|^p_\alpha \\ &\leq\big|\frac{\sin(\pi\xi)}{\pi}\big|^p\mathbf{E}\Big[\int_{-\infty}^t(t-s)^{-\xi}\|U(t, s)\mathbb{S}_\Psi(s)\|_\alpha\,ds\Big]^p \\ &\leq M(\alpha)^p\big|\frac{\sin(\pi\xi)}{\pi}\big|^p\mathbf{E}\Big[\int_{-\infty}^t(t-s)^{\xi-\alpha-1}e^{-\delta (t-s)}\|\mathbb{S}_\Psi(s)\|_\alpha\,ds\Big]^p \\ &\leq M(\alpha)^p\big|\frac{\sin(\pi\xi)}{\pi}\big|^p\Big(\int_{-\infty}^t(t-s)^{\frac{p}{p-1}(\xi-\alpha-1)}e^{-\delta (t-s)}\,ds\Big)^{p-1}\times \\ &\quad \times \Big(\int_{-\infty}^t e^{-\delta (t-s)}\mathbf{E}\|\mathbb{S}_\Psi(s)\|^p\,ds\Big) \\ &\leq r(\Gamma, \alpha, \xi, \delta, p)\sup_{s\in\mathbb{R}}\mathbf{E}\|\mathbb{S}_\Psi(s)\|^p\,. \end{align*} On the other hand, it follows from part (i) that \begin{equation}\label{2E81} \mathbf{E}\|\mathbb{S}_\Psi(t)\|^p \leq s(\Gamma, \xi, \delta, p)\sup_{t\in\mathbb{R}}\mathbf{E} \|\Psi(t)\|^p_{\mathbb{L}_2^0}\,. \end{equation} Thus, \begin{align*} &\mathbf{E}\|\int_{-\infty}^tU(t, s)\Psi (s)\,d\mathbb{W}(s)\|^p_\alpha \\ &\leq r(\Gamma, \alpha, \xi, \delta, p) s(\Gamma, \xi, \delta, p) \sup_{t\in\mathbb{R}}\mathbf{E}\|\Psi(t)\|^p_{\mathbb{L}_2^0}\\ &\leq k(\Gamma, \alpha, \xi, \delta, p) \sup_{t\in\mathbb{R}}\mathbf{E} \|\Psi(t)\|^p_{\mathbb{L}_2^0}\,. \end{align*} \end{proof} We now use the estimates obtained in Lemma \ref{2E80} (ii) to obtain \begin{align*} \mathbf{E}\|\Gamma_2X(t)\|^p_\alpha &\leq k(\alpha, \xi, \delta, p) \sup_{t\in\mathbb{R}}\mathbf{E} \|F_2(s, X(s))\|^p_{\mathbb{L}_2^0}\\ &\leq k(\alpha, \xi, \delta, p) \mathcal{M}_2\big(\|X\|_\infty\big)\,, \end{align*} and hence $$ \|\Gamma_2X\|^p_{\alpha, \infty}\leq k(\alpha, \xi, \delta, p) \mathcal{M}_2\big(\|X\|_\infty\big)\,, $$ where $k(\alpha, \xi, \delta, p)$ is a positive constant. For $p=2$, we have \begin{align*} \mathbf{E}\|\Gamma_2X(t)\|^2_\alpha &= \mathbf{E}\|\int_{-\infty}^t U(t, s) F_2(s, X(s))\,d \mathbb{W}(s)\|^2_\alpha\\ &\leq c(\alpha)^2\int_{-\infty}^t(t-s)^{-2\alpha} e^{-\delta (t-s)}\mathbf{E}\|F_2(s, X(s))\|^2_{\mathbb{L}^0_2}\\ &\leq c(\alpha)^2\Gamma\big(1-2\alpha\big) \delta^{1-2\alpha}\mathcal{M}_2\big(\|X\|_\infty\big)\,, \end{align*} and hence $$ \|\Gamma_2X\|^2_{\alpha, \infty}\leq s(\alpha, \delta) \mathcal{M}_2\big(\|X\|_\infty\big)\,, $$ where $s(\alpha, \delta)=c(\alpha)^2\Gamma\big(1-2\alpha\big) \delta^{1-2\alpha}$. For the continuity, let $X^n \in AP(\mathbb{R}; L^p(\Omega, \mathbb{H}))$ be a sequence which converges to some $X \in AP(\mathbb{R}; L^p(\Omega, \mathbb{H}))$; that is, $\|X^n -X\|_{\infty} \to 0$ as $n \to \infty$. It follows from the estimates in Proposition \ref{pes} that \begin{align*} & \mathbf{E}\|\int_{-\infty}^tU(t, s) [F_1(s, X^n(s))-F_1(s, X(s))]\,ds \|^p_{\alpha}\\ &\leq \mathbf{E}\Big[\int_{-\infty}^tc(\alpha) (t-s)^{-\alpha} e^{-\frac{\delta}{2} (t-s)} \|F_1(s, X^n(s)) - F_1(s, X(s))\|\,ds\Big]^p\,. \end{align*} Now, using the continuity of $F_1$ and the Lebesgue Dominated Convergence Theorem we obtain that \begin{align*} \mathbf{E}\|\int_{-\infty}^tU(t, s) [F_1(s, X^n(s))-F_1(s, X(s))]\,ds\|^p_{\alpha}\to 0\quad \text{as }n\to\infty\,. \end{align*} Therefore, $$ \|\Gamma_1 X^n -\Gamma_1 X\|_{\infty, \alpha} \to 0 \quad \text{as } n \to \infty. $$ For the term containing the Wiener process $\mathbb{W}$, we use the estimates in Lemma \ref{2E80} to obtain \begin{align*} &\mathbf{E}\|\int_{-\infty}^tU(t, s)[F_2(s, X^n(s)) -F_2(s, X(s))]\,d\mathbb{W}(s)\|^p_\alpha\\ &\leq k(\alpha, \xi, \delta, p) \sup_{t\in\mathbb{R}}\mathbf{E} \|F_2(t, X^n(t))-F_2(t, X(t))\|^p \end{align*} for $p>2$ and \begin{align*} &\mathbf{E}\|\int_{-\infty}^tU(t, s) [F_2(s, X^n(s))-F_2(s, X(s))] \,d\mathbb{W}(s)\|^2_{\alpha}\\ &\leq n(\alpha)^2\int_{-\infty}^t (t-s)^{-2\alpha} e^{-\delta (t-s)} \mathbf{E}\|F_2(s, X(s)^n) - F_2(s, X(s))\|^2\,ds \end{align*} for $p=2$. Now, using the continuity of $G$ and the Lebesgue Dominated Convergence Theorem we obtain that \begin{align*} \mathbf{E}\|\int_{-\infty}^t U(t, s) [F_2(s, X^n(s))-F_2(s, X(s))]\,d\mathbb{W}(s)\|^p_{\alpha}\to 0\quad \text{as } n\to\infty\,. \end{align*} Therefore, $$ \|\Gamma_2 X^n -\Gamma_2 X\|_{\infty, \alpha} \to 0 \quad \text{as } n \to \infty. $$ \end{proof} \begin{lemma}\label{5M2} Under assumptions {\rm (H2)--(H6)}, the integral operator $\Gamma_i$ $(i=1, 2)$ maps $AP\big(\mathbb{R}, L^p(\Omega, \mathbb{H})\big)$ into itself. \end{lemma} \begin{proof} Let us first show that $\Gamma_1X(\cdot)$ is $p$-th mean almost periodic et let $f_1(t)=F_1(t, X(t))$. Indeed, assuming that $X$ is $p$-th mean almost periodic and using assumption (H5), Theorem \ref{U}, and \cite[Proposition 4.4]{Man-Schn}, given $\varepsilon > 0$, one can find $l_\varepsilon >0$ such that any interval of length $l_\varepsilon$ contains at least $\tau$ with the property that $$ \|U(t+\tau, s+\tau) - U(t, s)\|\leq\varepsilon e^{-\frac{\delta}{2}(t-s)} $$ for all $t-s\geq\varepsilon$, and $$ \mathbf{E} \|f_1(\sigma +\tau) - f_1(\sigma)\|^p <\eta $$ for each $\sigma \in \mathbb{R}$, where $\eta(\varepsilon)\to 0$ as $\varepsilon\to 0$. Moreover, it follows from Lemma \ref{AC} (ii) that there exists a positive constant $K_1$ such that $$ \sup_{\sigma\in\mathbb{R}}\mathbf{E}\|f_1(\sigma)\|^p\leq K_1\,. $$ Now, using assumption (H2) and Holder's inequality, we obtain \begin{align*} &\mathbf{E} \|\Gamma_1X(t+\tau) -\Gamma_1X(t)\|^p\\ &\leq 3^{p-1} \mathbf{E}\Big [\int_0^{\infty}\|U(t+\tau, t+\tau-s)\|\|f_1(t+\tau-s)-f_1(t-s)\|\,ds\Big]^p \\ &\quad + 3^{p-1}\,\mathbf{E}\Big [\int_{\varepsilon}^{\infty}\|U(t+\tau, t+\tau-s) - U(t, t-s)\|\|f_1(t-s)\|\,ds\Big]^p \\ &\quad + 3^{p-1}\,\mathbf{E}\Big [\int_0^{\varepsilon}\|U(t+\tau, t+\tau-s) - U(t, t-s)\|\|f_1(t-s)\|\,ds\Big]^p \\ &\leq 3^{p-1} M^p \mathbf{E}\Big [\int_0^{\infty}e^{-\delta s}\|f_1(t+\tau-s)- f_1(t-s)\|\,ds\Big]^p \\ &\quad + 3^{p-1} \varepsilon^p\,\mathbf{E}\Big [\int_{\varepsilon}^{\infty}e^{-\frac{\delta}{2}s}\|f_1(t-s)\|\, ds\Big]^p + 3^{p-1} M^p\,\mathbf{E}\Big [\int_0^{\varepsilon}2e^{-\delta s}\|f_1(t-s)\|\, ds\Big]^p \\ &\leq 3^{p-1} M^p \Big(\int_0^\infty e^{-\delta s}\,ds\Big)^{p-1}\Big (\int_0^{\infty}e^{-\delta s} \mathbf{E}\|f_1(t+\tau-s)- f_1(t-s)\|^p\,ds\Big) \\ &\quad + 3^{p-1} \varepsilon^p\,\Big(\int_0^\infty e^{-\delta s}\,ds\Big)^{p-1}\Big (\int_{\varepsilon}^{\infty}e^{-\frac{\delta p s}{2}} \mathbf{E}\|f_1(t-s)\|^p\,ds\Big) \\ &\quad + 6^{p-1} M^p\,\Big(\int_0^\varepsilon e^{-\delta s}\,ds\Big)^{p-1}\Big (\int_0^{\varepsilon}e^{-\frac{\delta p s}{2}} \mathbf{E}\|f_1(t-s)\|^p\,ds\Big) \\ & \leq 3^{p-1} M^p \Big(\int_0^\infty e^{-\delta s}\,ds\Big)^{p} \sup_{s\in\mathbb{R}}\mathbf{E}\|f_1(t+\tau-s)- f_1(t-s)\|^p \\ &\quad +3^{p-1} \varepsilon^p\,\Big(\int_\varepsilon^\infty e^{-\delta s}\,ds\Big)^{p} \sup_{s\in\mathbb{R}}\mathbf{E}\|f_1(t-s)\|^p \\ &\quad + 6^{p-1} M^p\,\Big(\int_0^\varepsilon e^{-\delta s}\,ds\Big)^{p} \sup_{s\in\mathbb{R}}\mathbf{E}\|f_1(t-s)\|^p \\ & \leq 3^{p-1} M^p \Big(\frac{1}{\delta^p}\Big)\eta + 3^{p-1} M^p K_1 \Big(\frac{1}{\delta^p}\Big) \varepsilon^p + 6^{p-1}M^p\varepsilon^p K_1 \varepsilon^p. \end{align*} As for $\Gamma_2 X(\cdot)$, we split the proof in two cases: $p>2$ and $p=2$. To this end, we let $f_2(t)=F_2(t, X(t))$. Let us start with the case where $p>2$. Assuming that $X$ is $p$-th mean almost periodic and using assumption (H6), Theorem \ref{U}, and \cite [Proposition 4.4]{Man-Schn}, given $\varepsilon > 0$, one can find $l_\varepsilon>0$ such that any interval of length $l_\varepsilon$ contains at least $\tau$ with the property that $$ \|U(t+\tau, s+\tau) - U(t, s)\| \leq\varepsilon e^{-\frac{\delta}{2}(t-s)} $$ for all $t-s\geq\varepsilon$, and $$ \mathbf{E} \|f_2(\sigma +\tau) - f_2(\sigma)\|^p <\eta $$ for each $\sigma \in \mathbb{R}$, where $\eta(\varepsilon)\to 0$ as $\varepsilon\to 0$. Moreover, it follows from Lemma \ref{AC} (ii) that there exists a positive constant $K_2$ such that $$ \sup_{\sigma\in\mathbb{R}}\mathbf{E}\|f_2(\sigma)\|^p\leq K_2\,. $$ Now \begin{align*} & \mathbf{E} \|f_2(t+\tau) - f_2(t)\|^p\\ & \leq 3^{p-1} \mathbf{E}\Big \|\int_0^{\infty}U(t+\tau, t+\tau-s) \Big [f_2(t+\tau-s)-f_2(t-s)\Big]\,d\mathbb{W}(s)\|^p\\ &\quad + 3^{p-1}\,\mathbf{E}\Big \|\int_{\varepsilon}^{\infty}\Big[U(t+\tau, t+\tau-s) - U(t, t-s)\Big] f_2(t-s)\,d\mathbb{W}(s)\|^p\\ &\quad + 3^{p-1}\,\mathbf{E}\Big \|\int_0^{\varepsilon}\Big[U(t+\tau, t+\tau-s) - U(t,t-s)\Big] f_2(t-s)\,d\mathbb{W}(s)\|^p. \end{align*} We then have \begin{align*} & \mathbf{E} \|\Gamma_2X(t+\tau) -\Gamma_2X(t)\|^p\\ & \leq 3^{p-1} C_p \mathbf{E}\Big [\int_0^{\infty}\|U(t+\tau, t+\tau-s)\|^2\|f_2(t+\tau-s)-f_2(t-s)\|^2_{\mathbb{L}_2^0}\,ds\Big]^{p/2}\\ &\quad + 3^{p-1} C_p\,\mathbf{E}\Big [\int_{\varepsilon}^{\infty}\|U(t+\tau, t+\tau-s) - U(t, t-s)\|^2\|f_2(t-s)\|^2_{\mathbb{L}_2^0}\,ds\Big]^{p/2} \\ &\quad + 3^{p-1} C_p\,\mathbf{E}\Big [\int_0^{\varepsilon}\|U(t+\tau, t+\tau-s) - U(t, t-s)\|^2\|f_2(t-s)\|^2_{\mathbb{L}_2^0}\,ds\Big]^{p/2}\\ &\leq 3^{p-1} C_p M^p \mathbf{E}\Big [\int_0^{\infty}e^{-2 \delta s}\|f_2(t+\tau-s)-f_2(t-s) \|^2_{\mathbb{L}_2^0}\,ds\Big]^{p/2}\\ &\quad + 3^{p-1} C_p \varepsilon^p \mathbf{E}\Big [\int_{\varepsilon}^{\infty}e^{-\delta s}\|f_2(t-s)\|^2_{\mathbb{L}_2^0}\,ds \Big]^{p/2}\\ &\quad + 3^{p-1} 2^{p/2} C_p\,\mathbf{E}\Big [\int_0^{\varepsilon}e^{-2\delta s}\|f_2(t-s)\|^2_{\mathbb{L}_2^0}\, ds\Big]^{p/2}\\ &\leq 3^{p-1} C_p M^p \Big(\int_0^\infty e^{-\frac{p \delta s}{p-2}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_0^{\infty}e^{-\frac{p \delta s}{2}} \|f_2(t+\tau-s)-f_2(t-s)\|^p_{\mathbb{L}_2^0}\,ds\Big)\\ &\quad + 3^{p-1} C_p \varepsilon^p \Big(\int_\varepsilon^\infty e^{-\frac{p \delta s}{2(p-2)}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_{\varepsilon}^{\infty}e^{-\frac{p \delta s}{4}} \mathbf{E}\|f_2(t-s)\|^p_{\mathbb{L}_2^0}\,ds\Big)\\ &\quad +3^{p-1} 2^{p/2} C_p\,M^p\Big(\int_0^\varepsilon e^{-\frac{p \delta s}{p-2}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_0^{\varepsilon}e^{-\frac{p \delta s}{2}}\mathbf{E} \|f_2(t-s)\|^p_{\mathbb{L}_2^0}\,ds\Big)\\ &\leq 3^{p-1} C_p M^p \eta\Big(\int_0^\infty e^{-\frac{p \delta s}{p-2}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_0^{\infty}e^{-\frac{p \delta s}{2}}\,ds\Big)\\ &\quad + 3^{p-1} C_p \varepsilon^p K_2 \Big(\int_\varepsilon^\infty e^{-\frac{p \delta s}{2(p-2)}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_{\varepsilon}^{\infty}e^{-\frac{p \delta s}{4}}\,ds\Big)\\ &\quad+ 3^{p-1} 2^{p/2} C_p\,M^p K_2\Big(\int_0^\varepsilon e^{-\frac{p \delta s}{p-2}}\,ds\Big)^{\frac{p-2}{2}}\Big (\int_0^{\varepsilon}e^{-\frac{p \delta s}{2}}\,ds\Big)\\ & \leq 3^{p-1} C_p M^p \eta\Big(\frac{p-2}{p \delta}\Big)^{p-2}\Big (\frac{2}{p \delta}\Big)\\ &\quad + 3^{p-1} C_p \varepsilon^p K_2 \Big(\frac{2(p-2)}{p \delta}\Big)^{\frac{p-2}{2}}\Big (\frac{4}{p \delta}\Big)+ 3^{p-1} 2^{p/2} C_p\,M^p K_2\varepsilon^p. \end{align*} As to the case $p=2$, we proceed in the same way an using isometry inequality to obtain \begin{align*} &\mathbf{E}\|\Gamma_2 X(t + \tau)-\Gamma_2 X(t)\|^2\\ &\leq 3\,M^2 \Big (\int_0^{\infty}e^{-2\delta s}\,ds\Big)\sup_{\sigma\in\mathbb{R}}\mathbf{E}\|f_2 (\sigma+\tau)- f-2(\sigma)\|_{\mathbb{L}_2^0}^2\\ &\quad + 3 \varepsilon^2 \Big (\int_{\varepsilon}^{\infty}e^{-\delta\, s}\,ds\Big) \sup_{\sigma\in\mathbb{R}}\mathbf{E}\|f_2(\sigma) \|_{\mathbb{L}_2^0}^2 + 6 M^2 \Big(\int_0^{\varepsilon}e^{-2\delta s}\,ds\Big) \sup_{\sigma\in\mathbb{R}}\mathbf{E}\|f_2(\sigma) \|_{\mathbb{L}_2^0}^2\\ &\leq 3 \Big[ \eta\frac{M^2}{2\delta}+\varepsilon\frac{K_2}{\delta} +2 \varepsilon K_2\Big]. \end{align*} Hence, $\Gamma_2 X(\cdot)$ is $p$-th mean almost periodic. \end{proof} Let $\gamma\in (0, 1]$ and let $$ BC^\gamma \big(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha)\big) =\Big\{X\in BC\big(\mathbb{R}, L^p(\Omega,\mathbb{H}_\alpha)\big): \|X\|_{\alpha, \gamma}<\infty\Big\}, $$ where $$ \|X\|_{\alpha, \gamma}=\sup_{t\in\mathbb{R}} \Big[\mathbf{E}\|X(t)\|^p_\alpha\Big]^{1/p} + \gamma\sup_{t, s\in\mathbb{R}, s\ne t}\frac{\Big[\mathbf{E} \|X(t)-X(s)\|^p_\alpha\Big]^{1/p}}{\big|t-s|^\gamma}\,. $$ Clearly, the space $BC^\gamma\big(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha)\big)$ equipped with the norm $\|\cdot\|_{\alpha, \gamma}$ is a Banach space, which is in fact the Banach space of all bounded continuous Holder functions from $\mathbb{R}$ to $L^p(\Omega, \mathbb{H}_\alpha)$ whose Holder exponent is $\gamma$. \begin{lemma}\label{5M3} Under assumptions {\rm (H1)--(H6)}, the mapping $\Gamma_1$ defined previously maps bounded sets of $BC\big(\mathbb{R}, L^p(\Omega, \mathbb{H})\big)$ into bounded sets of $BC^\gamma(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha))$ for some $0<\gamma <1$. \end{lemma} \begin{proof} Let $X\in BC(\mathbb{R}, L^p(\Omega, \mathbb{H}))$ and let $f_1(t)= F_1(t, X(t))$ for each $t\in\mathbb{R}$. Proceeding as before, we have \[ \mathbf{E}\|\Gamma_1X(t)\|^p_\alpha \leq c\mathbf{E}\|\Gamma_1X(t)\|^p_\beta \leq c \cdot l(\beta, \delta, p)\mathcal{M}_1\big(\|X\|_\infty\big)\,. \] Let $t_12$ and $p=2$. For $p>2$, we have \[ \mathbf{E}\|\Gamma_2X(t)\|^p_\alpha \leq c\mathbf{E}\|\Gamma_2X(t)\|^p_\beta\\ \leq c \cdot k(\beta, \xi, \delta, p) \mathcal{M}_2\big(\|X\|_\infty\big)\,. \] Let $t_1t_1$, it follows from Holder's inequality that \begin{align*} &N'_2\\ &\leq r(\mu, \alpha)\big|\frac{\sin(\pi\xi}{\pi}\big|^p\mathbf{E} \Big[\int_{t_1}^{t_2}(\tau-t_1)^{-\alpha}\Big(\int_{-\infty}^{t_1}(t_1-s)^{\xi-1} e^{-\frac{\delta}{4} (\tau -s)} \|\mathbb{S}_{f_2}(s)\|\,ds\Big)\,d\tau\Big]^p\\ &\leq r(\mu, \alpha)\big|\frac{\sin(\pi\xi}{\pi}\big|^p\mathbf{E} \Big[\Big(\int_{t_1}^{t_2}(\tau-t_1)^{-\alpha}\,d\tau\Big)^p\\ &\quad\times \Big(\int_{-\infty}^{t_1}(t_1-s)^{\xi-1} e^{-\frac{\delta}{4} (t_1 -s)} \|\mathbb{S}_{f_2}(s)\|\,ds\Big)^p\Big]\\ &\leq r(\mu, \alpha)\big|\frac{\sin(\pi\xi}{\pi}\big|^p (t_2-t_1)^{p(1-\alpha)}\Big(\int_{-\infty}^{t_1}(t_1-s)^{\frac{p}{p-1} (\xi-\alpha-1)} e^{\frac{\delta}{4} (t_1 -s)}\,ds\Big)^{p-1} \\ &\quad \times\Big(\int_{-\infty}^{t_1} e^{-\frac{\delta}{4} (t_1 -s)}\,ds\Big)\sup_{s\in\mathbb{R}}\mathbf{E}\|\mathbb{S}_{f_2}(s)\|^p\\ &\leq r(\xi, \beta, \delta, \Gamma, p) (1-\alpha)^{-p}\mathcal{M}_2\big(\|X\|_\infty\big)(t_2-t_1)^{p(1-\alpha)} \,. \end{align*} For $\gamma= 1-\alpha$, one has \begin{align*} &\Big[\mathbf{E}\|(\Gamma_2X)(t_2) -(\Gamma_2X)(t_1)\|^p_\alpha\Big]^{1/p} \\ &\leq r(\xi, \beta, \delta, \Gamma, p) (1-\alpha)^{-1} \Big[\mathcal{M}_2\big(\|X\|_\infty\big)\Big]^{1/p}(t_2-t_1)^{\gamma} \,. \end{align*} As for $p=2$, we have \[ \mathbf{E}\|\Gamma_2X(t)\|^2_{\alpha} \leq c \mathbf{E} \|\Gamma_2X(t)\|^2_{\beta} \leq c \cdot s(\beta, \delta)\mathcal{M}_2\big(\|X\|_\infty\big)\,. \] For $t_10$ such that for all $R\ge r$, the following hold \begin{align*} \big(\Gamma_1+\Gamma_2\big)\Big(B_{AP(\mathbb{R}, L^p(\Omega, \mathbb{H}))}(0, R)\Big) \subset B_{BC^{\gamma}(\mathbb{R}, L^p(\Omega, \mathbb{H}_\alpha))} \cap B_{AP(\mathbb{R}, L^p(\Omega, \mathbb{H}))}(0, R)\,. \end{align*} In view of the above, it follows that $\big(\Gamma_1+\Gamma_2\big): D\to D$ is continuous and compact, where $D$ is the ball in $AP(\mathbb{R}, L^p(\Omega, \mathbb{H}))$ of radius $R$ with $R\ge r$. Using the Schauder fixed point it follows that $\big(\Gamma_1+\Gamma_2\big)$ has a fixed point, which is obviously a $p$-th mean almost periodic mild solution to \eqref{B1}. \end{proof} \section{Square-mean almost periodic solutions to some second order stochastic differential equations} In this section we study and obtain under some reasonable assumptions, the existence of square-mean almost periodic solutions to some classes of nonautonomous second-order stochastic differential equations of type \eqref{B2} on a Hilbert space $\mathbb{H}$ using Schauder's fixed-point theorem. For that, the main idea consists of rewriting \eqref{B2} as a nonautonomous first-order differential equation on $\mathbb{H} \times \mathbb{H}$ involving the family of 2$\times$2-operator matrices $\mathfrak{L}(t)$. Indeed, setting $ Z:=\begin{pmatrix}X \\ dX( t)\end{pmatrix}$, Equation \eqref{B2} can be rewritten in the Hilbert space $\mathbb{H} \times \mathbb{H}$ in the form \begin{equation}\label{6A41} dZ(\omega, t)= [\mathfrak{L}(t) Z(\omega, t) + F_1(t,Z(\omega, t))]\,dt + F_2(t, Z(\omega, t)) d \mathbb{W}(\omega, t), \end{equation} where $t\in \mathbb{R}$, $\mathfrak{L}(t)$ is the family of $2\times2$-operator matrices defined on $\mathcal{H} = \mathbb{H} \times \mathbb{H}$ by \begin{equation}\label{6A51} \mathfrak{L}(t) = \begin{pmatrix} 0 & I_{\mathbb{H}} \\ -b(t)\mathcal{A} & - a(t) I_{\mathbb{H}} \end{pmatrix} \end{equation} whose domain $D=D(\mathfrak{L}(t))$ is constant in $t \in \mathbb{R}$ and is given by $D(\mathfrak{L}(t)) = D(\mathcal{A})\times \mathbb{H}$. Moreover, the semilinear term $F_i (i=1, 2)$ appearing in \eqref{6A41} is defined on $\mathbb{R} \times \mathcal{H}_{\alpha}$ for some $\alpha \in (0, 1)$ by $$ F_i(t, Z)=\begin{pmatrix} 0\\ f_i(t, X)\end{pmatrix}, $$ where $\mathcal{H}_{\alpha} = \tilde {\mathcal{H}}_\alpha \times \mathbb{H}$ with $\tilde{\mathcal{H}}_\alpha$ is the real interpolation space between $\mathcal{B}$ and $D(\mathcal{A})$ given by $\tilde{\mathcal{H}}_{\alpha}:=\Big(\mathbb{H}, D(\mathcal{A})\Big)_{\alpha,\infty}$. First of all, note that for $0<\alpha <\beta <1$, then $$ L^2(\Omega, \mathcal{H}_\beta) \hookrightarrow L^2(\Omega, \mathcal{H}_\alpha)\hookrightarrow L^2(\Omega; \mathcal{H}) $$ are continuously embedded and hence therefore exist constants $k_1>0$, $k(\alpha)>0$ such that \begin{gather*} \mathbf{E}\|Z\|^2\leq k_1\mathbf{E}\|Z\|^2_\alpha\quad\text{for each } Z\in L^2(\Omega, \mathcal{H}_\alpha),\\ \mathbf{E}\|Z\|^2_\alpha\leq k(\alpha)\mathbf{E}\|Z\|^2_\beta\quad \text{for each } Z\in L^2(\Omega, \mathcal{H}_\beta). \end{gather*} To study the existence of square-mean solutions of \eqref{6A41}, in addition to (H1) we adopt the following assumptions. \begin{itemize} \item[(H7)] Let $f_i(i=1, 2):\mathbb{R}\times L^2(\Omega; \mathbb{H})\to L^2(\Omega; \mathbb{H})$ be square-mean almost periodic. Furthermore, $X\mapsto f_i(t, X)$ is uniformly continuous on any bounded subset $K$ of $L^2(\Omega; \mathbb{H})$ for each $t\in\mathbb{R}$. Finally, $$ \sup_{t\in\mathbb{R}}\mathbf{E}\|f_i(t, X)\|^2 \leq \mathcal{M}_i\big(\|X\|_\infty\big) $$ where $\mathcal{M}_i:\mathbb{R}^+\to \mathbb{R}^+$ is continuous function satisfying $$ \lim_{r\to\infty}\frac{\mathcal{M}_i(r)}{r}=0\,. $$ \end{itemize} Under the above assumptions, it will be shown that the linear operator matrices $\mathfrak{L}(t)$ satisfy the well-known Acquistapace-Terreni conditions, which does guarantee the existence of an evolution family $\mathfrak{U}(t,s)$ associated with it. Moreover, it will be shown that $\mathfrak{U}(t,s)$ is exponentially stable under those assumptions. \subsection{Square-Mean Almost Periodic Solutions} To analyze \eqref{6A41}, our strategy consists in studying the existence of square-mean almost periodic solutions to the corresponding class of stochastic differential equations of the form \begin{equation}\label{6A21} dZ(t)=[L(t) Z(t)+ F_1(t,Z(t))]dt+ F_2(t, Z(t)) d\mathbb{W}(t) \end{equation} for all $t \in \mathbb{R}$, where the operators $L(t): D(L(t))\subset L^2(\Omega, \mathcal{H})\to L^2(\Omega, \mathcal{H})$ satisfy Acquistapace-Terreni conditions, $F_i (i=1, 2)$ as before, and $\mathbb{W}$ is a one-dimensional Brownian motion. Note that each $Z \in L^2(\Omega, \mathcal{H})$ can be written in terms of the sequence of orthogonal projections $E_n$ as $$ X =\sum_{n=1}^{\infty}\sum_{k=1}^{\gamma_n}\langle X,e_{n}^{k }\rangle e_{n}^{k}=\sum_{n=1}^{\infty}E_n X. $$ Moreover, for each $X \in D(A)$, $$ AX = \sum_{j=1}^\infty \lambda_j \sum_{k=1}^{\gamma_j} \langle X, e_{j}^k \rangle e_{j}^k = \sum_{j=1}^\infty \lambda_j E_j X. $$ Therefore, for all $ Z:=\begin{pmatrix} X\\ Y \end{pmatrix} \in D (L) = D(A) \times L^2(\Omega, \mathcal{H})$, we obtain \begin{align*} L(t) Z&= \begin{pmatrix} 0 & I_{L^2(\Omega, \mathbb{H})}\\ -b(t) A & - a(t) I_{L^2(\Omega, \mathbb{H})} \end{pmatrix} \begin{pmatrix} X\\ Y \end{pmatrix}\\ &= \begin{pmatrix} Y \\ -b(t) A X - a(t) Y \end{pmatrix} = \begin{pmatrix} \sum_{n=1}^{\infty}E_n Y \\ -b(t) \sum_{n=1}^{\infty}\lambda_n E_n X - a(t) \sum_{n=1}^{\infty} E_n Y \end{pmatrix}\\ &= \sum_{n=1}^{\infty} \begin{pmatrix} 0 & 1\\ -b(t) \lambda_n & -a(t) \end{pmatrix} \begin{pmatrix} E_n & 0\\ 0 & E_n \end{pmatrix} \begin{pmatrix} X\\ Y \end{pmatrix} \\ &= \sum_{n=1}^{\infty} A_n (t) P_n Z, \end{align*} where \[ P_n := \begin{pmatrix} E_n & 0\\ 0 & E_n \end{pmatrix}, \quad n\geq 1, \] and \[ %4.4 A_n (t) := \begin{pmatrix} 0 & 1\\ -b(t) \lambda_n & -a(t) \end{pmatrix}, \quad n \geq 1. \] Now, the characteristic equation for $ A_n (t)$ is \begin{equation}\label{6A91} \lambda^2 + a(t) \lambda + \lambda_n b(t)=0 \end{equation} with discriminant $\Delta_n (t) = a^2(t) - 4 \lambda_n b(t)$ for all $t \in \mathbb{R}$. We assume that there exists $\delta_0, \gamma_0 > 0$ such that \begin{equation}\label{6A81} \inf_{t \in \mathbb{R}} a(t) > 2 \delta_0 > 0 ,\quad \inf_{t \in \mathbb{R}} b(t) > \gamma_0 > 0. \end{equation} From \eqref{6A81} it easily follows that all the roots of \eqref{6A91} are nonzero (with nonzero real parts) given by $$ \lambda_1^n (t) = \frac{-a(t) + \sqrt{\Delta_n (t)}}{2},\quad \lambda_2^n (t) = \frac{-a(t) - \sqrt{\Delta_n (t)}}{2}; $$ that is, $$ \sigma(A_n (t)) = \Big\{\lambda_1^n (t), \lambda_2^n (t) \Big\}. $$ In view of the above, it is easy to see that there exist $\gamma_0 \geq 0$ and $ \theta \in \Big( \frac{\pi}{2}, \pi\Big)$ such that $$ S_\theta \cup \{0\} \subset \rho\left(L(t) - \gamma_0I\right) $$ for each $t \in \mathbb{R}$ where $$ S_\theta = \Big\{z \in \mathbb{C} \setminus\{0\}: \big|\arg z\big| \leq \theta\Big\}. $$ On the other hand, one can show without difficulty that $A_n (t) = K_n^{-1} (t) J_n (t) K_n (t)$, where \[ J_n (t) = \begin{pmatrix} \lambda_1^n (t) & 0\\ 0 & \lambda_{2}^n (t) \end{pmatrix}, \quad K_n (t) = \begin{pmatrix} 1 & 1\\ \lambda_{1}^n (t) &\lambda_2^n (t) \end{pmatrix} \] and \[ K_n^{-1} (t)= \frac{1}{\lambda_1^n (t) - \lambda_2^n (t)} \begin{pmatrix} -\lambda_2^n (t) & 1\\ \lambda_{1}^n (t) &-1 \end{pmatrix}. \] For $\lambda \in S_{\theta}$ and $Z\in L^2(\Omega, \mathcal{H})$, one has \begin{align*} R(\lambda, L)Z &=\sum_{n=1}^{\infty}(\lambda-A_n (t))^{-1}P_nZ\\ &= \sum_{n=1}^{\infty}K_n (t)(\lambda- J_n (t)P_n)^{-1}K_n^{-1} (t)P_n Z. \end{align*} Hence, \begin{align*} \mathbf{E}\|R(\lambda, L) Z\|^2 &\leq \sum_{n=1}^{\infty}\|K_n (t) P_n(\lambda-J_n (t) P_n)^{-1}K_n^{-1} (t)P_n\|_{B(\mathcal{H})}^2 \mathbf{E} \|P_n Z\|^2\\ &\leq \sum_{n=1}^{\infty}\|K_n (t) P_n\|_{B(\mathcal{H})}^2 \|(\lambda-J_n (t) P_n)^{-1}\|_{B(\mathcal{H})}^2 \|K_n^{-1} (t) P_n \|_{B(\mathcal{H})}^2 \mathbf{E}\|P_n Z\|^2. \end{align*} Moreover, for $Z:=\begin{pmatrix}Z_1\\ Z_2\end{pmatrix}\in L^2(\Omega, \mathcal{H})$, we obtain \begin{align*} \mathbf{E}\|K_n (t) P_n Z\|^2 &= \mathbf{E}\|E_n Z_1+ E_n Z_2\|^2+ \mathbf{E}\|\lambda_1^n E_n Z_1 + \lambda_2^nE_n Z_2\|^2 \\ &\leq 3 \Big(1 + \big|\lambda_n^1(t)\big|^2\Big) \mathbf{E}\|Z\|^2. \end{align*} Thus, there exists $C_1>0$ such that \[ \mathbf{E}\|K_n (t) P_n Z\|^2 \leq C_1\big|\lambda_n^1 (t)\big| \mathbf{E}\|Z\|^2 \quad\text{for all }n \geq 1. \] Similarly, for $Z:=\begin{pmatrix}Z_1\\ Z_2\end{pmatrix}\in L^2(\Omega, \mathcal{H})$, one can show that there is $C_2>0$ such that \[ \mathbf{E}\|K_n^{-1}(t) P_n Z\|^2 \leq \frac{C_2}{\big|\lambda_n^1(t)\big|} \mathbf{E}\|Z\|^2 \quad\text{for all } n \geq 1. \] Now, for $ Z\in L^2(\Omega, \mathcal{H})$, we have \begin{align*} \mathbf{E} \|(\lambda - J_n (t) P_n)^{-1} Z\|^2 &= \mathbf{E} \Big\|\begin{pmatrix} \frac{1}{\lambda-\lambda_n^1(t)}&0\\ 0 & \frac{1}{\lambda-\lambda_n^2} \end{pmatrix} \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix}\Big\|^2\\ &\leq \frac{1}{|\lambda-\lambda_n^1(t)|^2}\mathbf{E}\|Z_1\|^2+ \frac{1}{|\lambda-\lambda_n^2(t)|^2}\mathbf{E}\|Z_2\|^2. \end{align*} Let $\lambda_0>0$. Define the function $$ \eta_t(\lambda):=\frac{1+|\lambda|}{|\lambda-\lambda_n^2(t)|}. $$ It is clear that $\eta_t$ is continuous and bounded on the closed set $$ \Sigma:=\{\lambda \in \mathbb{C}: |\lambda|\leq \lambda_0, \; |\arg \lambda|\leq \theta\}. $$ On the other hand, it is clear that $\eta$ is bounded for $|\lambda|>\lambda_0$. Thus $\eta$ is bounded on $ S_{\theta}$. If we take $$ N=\sup\big\{\frac{1+|\lambda|}{|\lambda-\lambda_n^j (t)|} : \lambda \in S_{\theta},\;n\geq 1,\; j=1,2,\big\}. $$ Therefore, \[ \mathbf{E} \|(\lambda - J_n (t)P_n)^{-1} Z\|^2 \leq \frac{N}{1+|\lambda|}\mathbf{E} \|Z\|^2,\quad \lambda \in S_{\theta}. \] Consequently, \[ \|R(\lambda,L(t))\| \leq \frac{K}{1+|\lambda|} \] for all $\lambda \in S_{\theta}$. First of all, note that the domain $D = D(L(t))$ is independent of $t$. Now note that the operator ${L}(t)$ is invertible with \[ L(t)^{-1}= \begin{pmatrix} -a(t)b^{-1}(t) A^{-1} & -b^{-1}(t) A^{-1}\\ I_\mathbb{H} & 0 \end{pmatrix}, \quad t\in \mathbb{R}. \] Hence, for $t,s,r\in\mathbb{R}$, computing $\big(L(t)-L(s)\big)L(r)^{-1}$ and assuming that there exist $L_a, L_b \geq 0$ and $\mu \in (0, 1]$ such that \begin{equation}\label{LP} \big|a(t) - a(s)\big|\leq L_a \big|t-s\big|^\mu,\quad \big|b(t) - b(s)\big|\leq L_b \big|t-s\big|^\mu, \end{equation} it easily follows that there exists $C > 0$ such that \[ \mathbf{E}\|(L(t)-L(s))L(r)^{-1}Z\|^2\leq C\big|t-s\big|^{2\mu}\mathbf{E}\|Z\|^2. \] In summary, the family of operators $\big\{L(t)\big\}_{t \in \mathbb{R}}$ satisfy Acquistpace-Terreni conditions. Consequently, there exists an evolution family $U(t,s)$ associated with it. Let us now check that $U(t,s)$ has exponential dichotomy. First of all note that For every $t\in \mathbb{R}$, the family of linear operators $L(t)$ generate an analytic semigroup $(e^{\tau L(t)})_{\tau\geq 0}$ on $L^2(\Omega, \mathcal{H})$ given by $$ e^{\tau L(t)}Z= \sum_{l=1}^{\infty} K_l(t)^{-1}P_l e^{\tau J_l}P_l K_l(t)P_l Z,\; Z \in L^2(\Omega, \mathcal{H}). $$ On the other hand, \[ \mathbf{E}\|e^{\tau L(t)}Z\|^2 = \sum_{l=1}^{\infty}\|K_l(t)^{-1}P_l\|^2_{B(\mathcal{H})} \|e^{\tau J_l}P_l\|^2_{B(\mathcal{H})}\|K_l(t)P_l\|^2_{B(\mathcal{H})} \mathbf{E} \|P_l Z\|^2, \] with for each $Z=\begin{pmatrix} Z_1\\ Z_2 \end{pmatrix}$, \begin{align*} \mathbf{E}\|e^{\tau J_l}P_l Z\|^2 &= \Big\| \begin{pmatrix} e^{\rho_1^l \tau}E_l&0\\ 0& e^{\rho_2^l \tau}E_l \end{pmatrix} \begin{pmatrix} Z_1\\ Z_2 \end{pmatrix} \Big\|^2\\ &\leq \mathbf{E}\| e^{\rho_1^l \tau}E_l Z_1\|^2+ \mathbf{E}\|e^{\rho_2^l\tau}E_l Z_2\|^2\\ &\leq e^{-2\delta_0 \tau}\mathbf{E}\|Z\|^2. \end{align*} Therefore, \begin{equation}\label{eq5} \|e^{\tau L(t)}\| \leq C e^{-\delta_0 \tau}, \quad \tau\geq0. \end{equation} Using the continuity of $a, b$ and the equality $$ R(\lambda, L(t))- R(\lambda, L(s))= R(\lambda, L(t))({L}(t)-L(s)) R(\lambda, L(s)), $$ it follows that the mapping $ J\ni t\mapsto R(\lambda, L(t))$ is strongly continuous for $\lambda\in S_{\omega}$ where $J\subset \mathbb{R}$ is an arbitrary compact interval. Therefore, $L(t)$ satisfies the assumptions of \cite[Corollary 2.3]{schn}, and thus the evolution family $(U(t, s))_{t\geq s}$ is exponentially stable. It remains to verify that $R(\gamma_0, L(\cdot)) \in AP(\mathbb{R}, B(L^2(\Omega; \mathcal{H})))$. For that we need to show that $L^{-1}(\cdot)\in AP(\mathbb{R}, B(L^2(\Omega, \mathcal{H})))$. Since $t \to a(t)$, $t\to b(t)$, and $t \to b(t)^{-1}$ are almost periodic it follows that $ t \to d(t) = - \frac{a(t)}{b(t)}$ is almost periodic, too. So for all $\varepsilon > 0$ there exists $l(\varepsilon) > 0$ such that every interval of length $l(\varepsilon)$ contains a $\tau$ such that $$ \big|\frac{1}{b(t+\tau)} - \frac{1}{b(t)}\big| < \frac{\varepsilon}{\|A^{-1}\| \sqrt{2}}, \quad \big|d(t+\tau) - d(t)\big| < \frac{\varepsilon}{\|A^{-1}\| \sqrt{2}} $$ for all $t\in \mathbb{R}$. Clearly, \begin{align*} \|L^{-1}(t+\tau) - L^{-1}(t) \| &\leq \Big(\big|\frac{1}{b(t+\tau)} - \frac{1}{b(t)}\big|^2 + \big|d(t+\tau) - d(t)\big|^2 \Big)^{1/2} \|A^{-1}\|_{B(\mathbb{H})} \\ &< \varepsilon \end{align*} and hence $t \to L^{-1}(t)$ is almost periodic with respect to $L^2(\Omega, \mathcal{H})$-operator topology. Therefore, $R(\gamma_0, L(\cdot)) \in AP(\mathbb{R}, B(L^2(\Omega; \mathcal{H})))$. To study the existence of square-mean almost periodic solutions of \eqref{6A21}, we use the general results obtained in Section 3. \begin{definition} \rm A continuous random function, $Z : \mathbb{R} \to L^2(\Omega; \mathcal{H})$ is said to be a bounded solution of \eqref{6A21} on $\mathbb{R}$ provided that \begin{align*}\label{TT} Z(t)&= \int_{s}^t U(t, s) F_1(s, Z(s))\,ds +\int_{s}^t U(t, s)P(s)\,F_2(s, Z(s))\,d\mathbb{W}(s) \end{align*} for each $t \geq s$ and for all $t,s \in \mathbb{R}$. \end{definition} \begin{remark}\label{6C90} \rm Note that it follows from (H7) that $F_i(i=1, 2):\mathbb{R}\times L^2(\Omega; \mathcal{H})\to L^2(\Omega; \mathcal{H})$ is square-mean almost periodic. Furthermore, $Z\mapsto F_i(t, Z)$ is uniformly continuous on any bounded subset $K$ of $L^2(\Omega; \mathcal{H})$ for each $t\in\mathbb{R}$. Finally, $$ \sup_{t\in\mathbb{R}}\mathbf{E}\|F_i(t, Z)\|^2 \leq \mathcal{M}_i\big(\|Z\|_\infty\big) $$ where $\mathcal{M}_i:\mathbb{R}^+\to \mathbb{R}^+$ is continuous function satisfying $$ \lim_{r\to\infty}\frac{\mathcal{M}_i(r)}{r}=0\,. $$ \end{remark} \begin{theorem}\label{6C91} Suppose assumptions {\rm (H1), (H3), (H7)} hold, then the nonauto\-nomous differential equation \eqref{6A21} has at least one square-mean almost periodic solution. \end{theorem} In view of Remark \ref{6C90}, the proof of the above theorem follows along the same lines as that of Theorem \ref{5M10} and hence it is omitted. \subsection*{Acknowledgments} The authors would like thanks the anonymous referee for the careful reading of the manuscript and insightful comments. \begin{thebibliography}{99} \bibitem{Ac} P. Acquistapace; Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations \textbf{1} (1988), 433--457. \bibitem{AFT} P. Acquistapace, F. 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