\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 117, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/117\hfil Existence of almost periodic solutions]
{Square-mean almost periodic solutions nonautonomous stochastic
differential equations}

\author[P. H. Bezandry, T. Diagana \hfil EJDE-2007/117\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 1, 2007. Published September 2, 2007.}
\subjclass[2000]{34K14, 60H10, 35B15, 34F05}
\keywords{Stochastic differential equation; stochastic processes;\hfill\break\indent
square-mean almost periodic; Wiener process;
Acquistapace-Terreni conditions}

\begin{abstract}
 This paper  concerns the square-mean almost periodic solutions
 to a class of nonautonomous stochastic differential equations
 on a separable real Hilbert space. Using the so-called
 `Acquistapace-Terreni' conditions, we establish the existence
 and uniqueness of a square-mean almost periodic mild solution
 to those nonautonomous stochastic differential equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

Let $(\mathbb{H}, \|\cdot\|)$ be a real (separable)
Hilbert space. The present paper is mainly concerned with the
existence of mean-almost periodic solutions to the class of
nonautonomous semilinear stochastic differential equations
\begin{equation}\label{I}
dX(t)= A(t) X(t) \,dt + F(t, X(t))\,dt + G(t, X(t))\,dW(t),
\quad t\in\mathbb{R},
\end{equation}
 where $A(t)$ for $t \in \mathbb{R}$ is a family of densely
defined closed linear operators satisfying the so-called
`Acquistapace-Terreni' conditions \cite{AT}, that is, there exist
constants $\lambda_0\geq 0, \theta\in (\frac{\pi}{2},\pi),  L,  K\geq 0$, and $\alpha$, $\beta\in (0, 1]$ with
$\alpha +\beta > 1$ such that
\begin{equation}\label{K}
\Sigma_\theta\cup\{0\}\subset \rho (A(t) - \lambda_0), \quad
\|R(\lambda,  A(t) - \lambda_0)\|\leq\frac{K}{1+|\lambda|}
\end{equation}
and
$$
\|(A(t) -\lambda_0) R(\lambda,  A(t) - \lambda_0)[R(\lambda_0,  A(t))
- R(\lambda_0,  A(s))]\|\leq L|t-s|^{\alpha}|\lambda|^{\beta}
$$
for $t,  s\in \mathbb{R}, \lambda\in \Sigma_\theta :=\{\lambda\in
{\bf C} - \{0\}:  |\mathop{\rm arg}\lambda|\leq\theta\}$,
 $F: \mathbb{R} \times L^2 ({\bf P}, \mathbb{H}) \to L^2 ({\bf P}, \mathbb{H})$ and
$G: \mathbb{R} \times L^2 ({\bf P}, \mathbb{H}) \to L^2 ({\bf P}, L_2^0)$
are jointly continuous satisfying some additional conditions,
 and $W(t)$ is a Wiener process.

The existence of almost periodic (respectively, periodic)
solutions to autonomous stochastic differential equations has been
studied by many authors, see, e.g. \cite{AT, BD,D,t3}.
In Da Prato-Tudor \cite{da}, the existence of an
almost periodic solution to \eqref{I} 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. In this paper, it goes back to study the
existence and uniqueness of a square-mean almost periodic solution
to \eqref{I} when  the operators $A(t)$ satisfy
`Acquistapace-Terreni' conditions (Theorem \ref{thm1}). Next, we make
extensive use of our abstract result to establish the existence of
mean-almost periodic solutions to a $n$-dimensional system of some
stochastic (parabolic) partial differential equations.

The organization of this work is as follows: in Section 2, we
recall some preliminary results that we will use in the sequel. In
Section 3, we give some sufficient conditions for the existence
and uniqueness of a square-mean almost periodic solution to
\eqref{I}. Finally, an example is given to illustrate our main
results.

\section{Preliminaries}

Throughout the rest of this paper, we assume that $(\mathbb{K},
\|\cdot\|_{K})$ and $(\mathbb{H}, \|\cdot\|)$ are real separable Hilbert
spaces, and $(\Omega, {\mathcal F}, {\bf P})$ is a probability space.
The space $L_2(\mathbb{K}, \mathbb{H})$ stands for the space of all
Hilbert-Schmidt operators acting from $\mathbb{K}$ into $\mathbb{H}$, 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 $\{W(t),  t\in\mathbb{R}\}$ is a $Q$-Wiener
process defined on $(\Omega, {\mathcal F}, {\bf P})$ with values in
$\mathbb{K}$. Recall that $W$ can obtained as follows: let 
$\{W_i(t),t\in\mathbb{R}\}$, $i=1, 2$, be independent $K$-valued $Q$-Wiener
processes, then
\[
W(t)=  \begin{cases}
   W_1(t)  & \mbox{if }t\geq 0, \\
   W_2(-t) & \mbox{if }t\leq 0,
 \end{cases}
\]
 is $Q$-Wiener process with $\mathbb{R}$ as time parameter.
We let
 $\mathcal{F}_t=\sigma\{W(s),  s\leq t\}$.

 The collection of all strongly measurable, square-integrable
 $\mathbb{H}$-valued random variables, denoted by $L^2({\bf P}, \mathbb{H})$, is a
 Banach space when it is equipped with norm $\|X\|_{L^2({\bf P}, \mathbb{H})}=({\bf
 E}\|X\|^2)^{1/2}$, where the expectation ${\bf E}$ is defined by
 $$
{\bf E}[g]=\int_{\Omega}g(\omega)d{\bf P}(\omega).
$$
 Let $\mathbb{K}_0=Q^{1/2}K$ and let $L^0_2=L_2(\mathbb{K}_0, \mathbb{H})$ with
 respect to the norm
 $$
\|\Phi\|^2_{\mathbb{L}^0_2}=\|\Phi\,Q^{1/2}\|_2^2=
\mathop{\rm Trace}  (\Phi\,Q \Phi^{*})\,.
$$
Throughout, we assume that
$A(t): D(A(t))\subset L^2({\bf P};\mathbb{H})\to L^2({\bf P}; \mathbb{H})$
is a family of densely defined closed
linear operators on a common domain $D = D(A(t))$, which is
independent of $t$ and dense in $L^2({\bf P}; \mathbb{H})$, and
 $F: \mathbb{R} \times L^2({\bf P}; \mathbb{H}) \mapsto L^2({\bf P}; \mathbb{H})$ and
$G: \mathbb{R} \times L^2({\bf P}; \mathbb{H}) \mapsto L^2({\bf P}; L^0_2)$ are jointly
continuous functions.

We suppose that the system
\begin{equation}\label{J}
 \begin{gathered}
  u'(t)=  A(t)u(t) \quad t\geq s, \\
  u(s) =x\in L^2({\bf P}; \mathbb{H}),
 \end{gathered}
\end{equation}
has an associated evolution family of operators
 $\{U(t,s): t\geq s \mbox{ with }  t,s\in \mathbb{R}\}$,
which is uniformly asymptotically stable.

If $\mathbb{B}_1, \mathbb{B}_2$ are Banach spaces, then the notation
${\mathcal L}(\mathbb{B}_1, \mathbb{B}_2)$ stands for the Banach space of bounded linear
operators from $\mathbb{B}_1$ into $B_2$. When $\mathbb{B}_1 = B_2$, this is simply
denoted ${\mathcal L}(\mathbb{B}_1)$.

\begin{definition} \label{def2.1} \rm
A family of bounded linear operators $\{U(t,s): t\geq s
\mbox{ with }  t,s\in \mathbb{R}\}$ on $L^2({\bf P}; \mathbb{H})$ is called an
evolution family of operators for \eqref{J} whenever the following
conditions hold:
\begin{itemize}
\item[(a)] $U(t,s)U(s,r)=U(t,r)$  for every  $ r \leq s \leq  t $;
\item[(b)] for each $x \in \mathbb{X}$ the function $(t,s)\to U(t,s)x$
is continuous and  $U(t,s)\in {\mathcal L}(L^2({\bf P}; \mathbb{H}), D)$
for every $t> s$; and
 \item[(c)] the function  $(s,t]\to {\mathcal L}(L^2({\bf P}; \mathbb{H})) $,
$t\to U(t,s)$  is differentiable with
 $$
\frac{\partial }{\partial t}U(t,s)= A(t)U(t,s).
$$
\end{itemize}
\end{definition}


For additional details on evolution families, we refer the reader to
the book by Lunardi \cite{Al2}.

For the reader's convenience, we review some basic definitions and
results for the notion of square-mean almost periodicity.

Let $(\mathbb{B}, \|\cdot\|)$ be a Banach space.

\begin{definition} \label{def2.2} \rm
A stochastic process $X: \mathbb{R} \to L^2({\bf P}; \mathbb{B})$ is said to be
continuous whenever
$$
\lim_{t\to s}{\bf E}\|X(t)-X(s)\|^2=0.
$$
\end{definition}

\begin{definition}\cite{BD} \rm
A continuous stochastic process $X: \mathbb{R} \to L^2({\bf P}; \mathbb{B})$
is said to be square-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
$$
\sup_{t\in {\bf R}}{\bf E}\|X(t+\tau) - X(t)\|^2 <\varepsilon.
$$
\end{definition}

The collection of all stochastic processes $X: \mathbb{R} \to L^2({\bf P}; \mathbb{B})$
which are square-mean almost periodic is then denoted by
$AP({\mathbb{R}};L^2({\bf P}; \mathbb{B}))$.

The next lemma provides with some properties of the square-mean
almost periodic processes.


\begin{lemma}[\cite{BD}] \label{lemPH}
If $X$ belongs to $AP({\mathbb{R}};L^2({\bf P}; \mathbb{B}))$, then
\begin{itemize}
\item[(i)]  the mapping $t\to {\bf E}\|X(t)\|^2$ is
uniformly continuous;
\item[(ii)]  there exists a constant $M > 0$
such that ${\bf E}\|X(t)\|^2\le
    M$, for all $t\in \mathbb{R}$.
\end{itemize}
\end{lemma}

Let $\mbox{\bf CUB}(\mathbb{R}; L^2({\bf P}; \mathbb{B}))$ denote the collection of all
stochastic processes $X: \mathbb{R} \mapsto L^2({\bf P}; \mathbb{B})$, which are
continuous and uniformly bounded. It is then easy to check that
$\mbox{\bf CUB}(\mathbb{R};L^2({\bf P}; \mathbb{B}))$  is a Banach space when it is
equipped with the norm:
$$
\|X\|_{\infty}=\sup_{t \in \mathbb{R}}\left({\bf E}\|X(t)\|^2\right)^{1/2}.
$$

\begin{lemma}[\cite{BD}] \label{lem2.5}
$AP(\mathbb{R};L^2({\bf P}; \mathbb{B}))\subset \mbox{\bf CUB} (\mathbb{R};L^2({\bf P};\mathbb{B}))$
 is a closed subspace.
\end{lemma}

In view of the above, the space $AP(\mathbb{R};L^2({\bf P}; \mathbb{B}))$ of square-mean
almost periodic processes equipped with the norm $\|\cdot\|_\infty$
is a Banach space.

Let $(\mathbb{B}_1, \|\cdot\|_1)$ and $(\mathbb{B}_2, \|\cdot\|_2)$ be Banach spaces
and let $L^2({\bf P}; \mathbb{B}_1)$ and $L^2({\bf P}; \mathbb{B}_2)$ be their corresponding
$L^2$-spaces, respectively.

\begin{definition}\cite{BD}
A function $F: \mathbb{R} \times L^2({\bf P};\mathbb{B}_1) \to L^2({\bf P}; \mathbb{B}_2))$,
$(t, Y) \mapsto F(t, Y)$, which is
jointly continuous, is said to be square-mean almost periodic in
$t \in \mathbb{R}$ uniformly in $Y\in\mathbb{K}$ where $\mathbb{K} \subset L^2({\bf P}; \mathbb{B}_1)$
is a compact if for any $\varepsilon >0$, there exists
$l(\varepsilon, \mathbb{K}) >0$ such that any interval of length
$l(\varepsilon, \mathbb{K})$ contains at least a number $\tau$ for which
$$
\sup_{t\in {\bf R}}{\bf E}\|F(t+\tau, Y) - F(t, Y)\|^2_2
<\varepsilon
$$
for each stochastic process $Y: \mathbb{R} \to \mathbb{K}$.
\end{definition}


\begin{theorem}[\cite{BD}]\label{AB}
Let $F: \mathbb{R} \times L^2({\bf P}; \mathbb{B}_1) \to L^2({\bf P}; \mathbb{B}_2)$,
$(t, Y) \mapsto F(t, Y)$ be a square-mean almost periodic process
in $t \in \mathbb{R}$ uniformly in $Y\in\mathbb{K}$, where
$\mathbb{K} \subset L^2({\bf P}; \mathbb{B}_1)$ is
compact. Suppose that $F$ is Lipschitz in the following sense:
$$
{\bf E} \|F(t, Y) - F(t, Z)\|_2^2 \leq M {\bf E}\|Y -Z\|_1^2
$$
for all $Y, Z\in L^2({\bf P}; \mathbb{B}_1)$ and for each $t \in \mathbb{R}$,
where $M >0$. Then for any square-mean almost periodic process
$\Phi: \mathbb{R} \to L^2({\bf P}; \mathbb{B}_1)$, the stochastic process
$t \mapsto F(t, \Phi(t))$ is square-mean almost periodic.
\end{theorem}


\section{Main Result}

Throughout this section, we require the following assumptions:

\begin{itemize}
\item[(H0)] The operators $A(t)$, $U(r,s)$ commute and that the
evolution family $U(t,s)$ is asymptotically stable. Namely, there
exist some constants $M, \delta > 0$ such that
$$
\|U(t,s)\| \leq Me^{-\delta (t-s)} \quad \mbox{for every }  t\geq s.
$$
In addition,
$R(\lambda_0, A(\cdot))\in AP(\mathbb{R}; {\mathcal L}(L^2({\bf P}, \mathbb{H})))$ for
$\lambda_0$ in (\ref{K});


 \item[(H1)] The function $F: \mathbb{R}\times L^2({\bf P}; \mathbb{H})
\to L^2({\bf P}; \mathbb{H})$, $(t, X) \mapsto F(t,X)$ be a square-mean
almost periodic in $t \in \mathbb{R}$ uniformly in $X \in {\mathcal O}$
(${\mathcal O} \subset L^2({\bf P}; \mathbb{H})$ being a compact subspace).
Moreover, $F$ is Lipschitz in the following sense: there exists
$K > 0$ for which
    $$
{\bf E}\|F(t, X)-F(t, Y)\|^2\leq\,K {\bf E}\|X-Y\|^2
$$
    for all stochastic processes $X, Y \in L^2({\bf P}; \mathbb{H})$ and $t\in\mathbb{R}$;

\item[(H2)] The function $G: \mathbb{R}\times L^2({\bf P}; \mathbb{H}) \to
L^2({\bf P}; \mathbb{L}^0_2)$, $(t, X) \mapsto F(t,X)$ be a square-mean
almost periodic in $t \in \mathbb{R}$ uniformly in $X \in {\mathcal O}'$
(${\mathcal O}' \subset L^2({\bf P}; \mathbb{H})$ being a compact subspace).
Moreover, $G$ is Lipschitz in the following sense: there exists
$K'> 0$ for which
    $$
{\bf E}\|G(t, X)-G(t, Y)\|_{\mathbb{L}_2^0}^2\leq\,K' {\bf E}\|X-Y\|^2
$$
    for all stochastic processes $X, Y \in L^2({\bf P}; \mathbb{H})$ and
    $t\in\mathbb{R}$.
\end{itemize}

In order to study \eqref{I} we need the following lemma which can be
seen as an immediate consequence of \cite[Proposition 4.4]{MR}.

\begin{lemma} \label{lemC}
Suppose $A(t)$ satisfies the `Acquistapace-Terreni'
conditions, $U(t, s)$ is exponentially stable and $R(\lambda_0,
A(\cdot))\in AP(\mathbb{R}; {\mathcal L}(L^2({\bf P}, \mathbb{H})))$. Let $h>0$. Then,
for any $\varepsilon >0$, there exists $l(\varepsilon)>0$ such
that every interval of length $l$ contains at least a number
$\tau$ with the property that
$$
\|U(t+\tau, s+\tau)-U(t, s)\|\leq\varepsilon e^{-\frac{\delta}{2}(t-s)}
$$
for all $t-s\geq h$.
\end{lemma}

\begin{definition} \label{def3.2} \rm
A ${\mathcal F}_t$-progressively process $\{X(t)\}_{t\in \mathbb{R}}$ is
called a mild solution of $\eqref{I}$ on $\mathbb{R}$ if
\begin{equation}\label{L}
\begin{aligned}
X(t)&= U(t, s) X(s) + \int_{s}^t U(t, \sigma)
F(\sigma,X(\sigma))\,d\sigma \\
&\quad + \int_{s}^tU(t,\sigma) G(\sigma, X(\sigma))\,dW(\sigma)
\end{aligned}
\end{equation}
for all $t \geq s$ for each $s \in \mathbb{R}$.
\end{definition}

Now, we are ready to present our main result.

\begin{theorem}\label{thm1}
Under assumptions {\rm (H0)---(H2)}, then \eqref{I} has a unique
square-mean almost period mild solution, which can be explicitly
expressed as follows:
$$
X(t)= \int_{-\infty}^t U(t,\sigma) F(\sigma,X(\sigma))\,d\sigma +
\int_{-\infty}^t U(t,\sigma)G(\sigma, X(\sigma))\,dW(\sigma)
\quad \mbox{for each }  t \in \mathbb{R}
$$
whenever
$$
\Theta:= M^2\Big(2\frac{K}{\delta^2}\,+
\frac{K^\prime\cdot\mathop{\rm Tr}(Q)}{\delta}\Big) < 1.
$$
\end{theorem}

\begin{proof}
First of all, note that
\begin{equation}\label{ab}
 X(t)= \int_{-\infty}^t U(t, \sigma)
F(\sigma,X(\sigma))\,d\sigma + \int_{-\infty}^t U(t, \sigma)
G(\sigma, X(\sigma))\,dW(\sigma)
\end{equation}
is well-defined and satisfies
$$
X(t)= U(t, s) X(s) + \int_{s}^t U(t,\sigma) F(\sigma,X(\sigma))\,
d\sigma + \int_{s}^t U(t,\sigma)G(\sigma, X(\sigma))\, dW(\sigma)
$$
for all $t \geq s$ for each $s \in \mathbb{R}$, and hence $X$
given by (\ref{L}) is a mild solution to \eqref{I}.

Define
\begin{gather*}
\Phi X(t) := \int_{-\infty}^t U(t,\sigma)
F(\sigma,X(\sigma))\,d\sigma,\\
  \Psi X(t) :=
\int_{-\infty}^t U(t,\sigma) G(\sigma, X(\sigma))\,dW(\sigma).
\end{gather*}

Let us show that $\Phi X(\cdot)$ is square-mean almost periodic
whenever $X$ is. Indeed, assuming that $X$ is square-mean almost
periodic and using (H1), Theorem \ref{AB}, and Lemma \ref{lemC},
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
$$
{\bf E} \left\|F(\sigma + \tau, X(\sigma +\tau))
- F(\sigma, X(\sigma))\right\|^2 <\eta
$$
for each $\sigma \in \mathbb{R}$, where $\eta(\varepsilon)\to 0$ as
$\varepsilon\to 0$.
Moreover, it follows from Lemma \ref{lemPH} (ii) that there
exists a positive constant $K_1$ such that
$$
\sup_{\sigma\in\mathbb{R}}{\bf E}\|F(\sigma, X(\sigma))\|^2\leq K_1\,.
$$
Now
\begin{align*}
& \big\|(\Phi X)(t + \tau)-(\Phi X)(t)\big\|\\
& = \big\|\int_{-\infty}^{t+\tau} U(t+\tau,s)F(s,
X(s))\,ds-\int^t_{-\infty}U(t, s)F(s, X(s))\,ds\big\| \\
& = \|\int_{0}^{\infty}
U(t+\tau,t+\tau-s)\,F(t+\tau-s, X(t+\tau-s))\,ds\\
&\quad -\int_0^{\infty}U(t, t-s)\,F(t-s, X(t-s))\,ds\|\\
&\leq \big\|\int_{0}^{\infty}
U(t+\tau,t+\tau-s)[F(t+\tau-s, X(t+\tau-s))-F(t-s,
X(t-s))]\,ds\big\|\\
&\quad +
\big\|\Big(\int_{\varepsilon}^{\infty}+\int_0^{\varepsilon}\Big)
[U(t+\tau, t+\tau-s))-U(t, t-s)] F(t-s, X(t-s))\,ds \big\|.
\end{align*}
 Consequently,
\begin{align*}
& {\bf E} \|\Phi X(t+\tau) - \Phi X(t)\|^2 \\
&\leq 3 {\bf E}\Big[\int_0^{\infty}\|U(t+\tau, t
 +\tau-s)\| \|F(t+\tau-s,  X(t+\tau-s))\\
&\quad -F(t-s, X(t-s))\|\,ds\Big]^2\\
&\quad+ 3 {\bf E}\Big[\int_{\varepsilon}^{\infty}\|U(t+\tau, t+\tau-s) - U(t,
 t-s)\| \|F(t-s, X(t-s))\|\,ds\Big]^2\\
&\quad+ 3 {\bf E}\Big[\int_0^{\varepsilon}\|U(t+\tau, t+\tau-s) - U(t, t-s)\| \|F(t-s,
 X(t-s))\|\,ds\Big]^2\\
&\leq 3 M^2 {\bf E}
 \Big[\int_0^{\infty}e^{-\delta s}\|F(t+\tau-s, X(t+\tau-s))-F(t-s,
 X(t-s))\|\,ds\Big]^2\\
&\quad + 3 \varepsilon^2 {\bf E}
 \Big[\int_{\varepsilon}^{\infty}e^{-\frac{\delta}{2}s}\|F(t-s,
 X(t-s))\|\,ds\Big]^2 \\
&\quad + 3M^2{\bf E}
\Big[\int_0^{\varepsilon}2e^{-\delta s} \|F(t-s, X(t-s))\|\,
 ds\Big]^2.
\end{align*}
 Using Cauchy-Schwarz inequality it follows that
\begin{align*}
& {\bf E} \|\Phi X(t+\tau) - \Phi X(t)\|^2\\
&\leq 3 M^2 \Big(\int_0^{\infty}e^{-\delta s}\,ds\Big)\\
&\quad\times \Big(\int_0^{\infty}e^{-\delta s} {\bf E}\|F(t+\tau-s,
X(t+\tau-s))-F(t-s, X(t-s))\|^2\,ds\Big) \\
&\quad + 3 \varepsilon^2
 \Big(\int_{\varepsilon}^{\infty}e^{-\frac{\delta}{2}s}\,ds\Big)
 \Big(\int_{\varepsilon}^{\infty}e^{-\frac{\delta}{2}s}
 {\bf E}\|F(t-s, X(t-s))\|^2\,ds\Big) \\
&\quad + 12 M^2 \Big(\int_0^{\infty}e^{-\delta s}\,ds\Big)
 \Big(\int_0^{\varepsilon}e^{-\delta s} {\bf E}\|F(t-s, X(t-s))\|^2\,
 ds\Big)^2\\
&\quad \leq 3 M^2 \Big(\int_0^{\infty}e^{-\delta s}\,ds\Big)^2
 \sup_{\sigma\in\mathbb{R}}{\bf E} \|F(\sigma + \tau, X(\sigma +\tau)) - F(\sigma,
 X(\sigma))\|^2 \\
&\quad + 3 \varepsilon^2
 \Big(\int_{\varepsilon}^{\infty}e^{-\frac{\delta}{2}s}\,ds\Big)
 \sup_{\sigma\in\mathbb{R}}{\bf E}\|F(\sigma, X(\sigma))\|^2\\
&\quad + 12 M^2 \Big(\int_0^{\infty}e^{-\delta s}\,ds\Big)
  \sup_{\sigma\in\mathbb{R}}{\bf E}\|F(\sigma, X(\sigma))\|^2 \\
&\leq 3 \frac{M^2}{\delta^2}\eta
+3 \varepsilon^2\frac{4}{\delta^2}K_1+12M^2\varepsilon^2K_1\,,
\end{align*}
which implies that $\Phi X(\cdot)$ is square-mean almost
periodic.

Similarly, assuming that $X$ is square-mean almost periodic and
using (H2), Theorem \ref{AB}, and Lemma \ref{lemC}, 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
$$
{\bf E} \left\|G(\sigma + \tau, X(\sigma +\tau))
  - G(\sigma, X(\sigma))\right\|_{\mathbb{L}_2^0}^2 <\eta
$$
for each $\sigma \in \mathbb{R}$, where $\eta(\varepsilon)\to 0$ as
$\varepsilon\to 0$. Moreover, it follows from Lemma \ref{lemPH} (ii)
that there exists a positive constant $K_2$ such that
$$
\sup_{\sigma\in\mathbb{R}}{\bf E}\|G(\sigma, X(\sigma))\|_{\mathbb{L}_2^0}^2\leq K_2.
$$
The next step consists of proving the square-mean almost
periodicity of $\Psi X(\cdot)$. Of course, this is more
complicated than the previous case because of the involvement of
the Wiener process $W$. To overcome such a difficulty, we make
extensive use of the properties of ${\tilde W}$ defined by
${\tilde W}(s):=W(s+\tau)-W(\tau)$ for each $s$. Note that
${\tilde W}$ is also a Wiener process and has the same
distribution as $W$.

Now, let us make an appropriate change of variables to get
\begin{align*}
&{\bf E}\|(\Psi X)(t + \tau)-(\Psi X)(t)\|^2 \\
&=\|\int_{0}^{\infty}
U(t+\tau,t+\tau-s)\,G(t+\tau-s, X(t+\tau-s))\,d{\tilde W}(s) \\
&\quad -\int_0^{\infty}U(t, t-s)\,G(t-s,
X(t-s))\,d{\tilde W}(s)\|^2 \\
&\leq 3 {\bf E}\|\int_0^{\infty}U(t+\tau,
t+\tau-s)\,[G(t+\tau-s, X(t+\tau-s))\\
&\quad -G(t-s, X(t-s))]\,d{\tilde W} (s)\|^2 \\
&\quad   + 3 {\bf E}\|\int_{\varepsilon}^{\infty}[U(t+\tau, t+\tau-s)
 - U(t, t-s)]\,G(t-s, X(t-s))\,d{\tilde W}(s)\|^2 \\
&\quad  + 3 {\bf E}\|\int_0^{\varepsilon}[U(t+\tau, t+\tau-s)
 - U(t, t-s)]\,G(t-s, X(t-s))\,d{\tilde W}(s)\|^2.
\end{align*}
 Then using an estimate on the Ito integral established in
\cite[Proposition 1.9]{I}, we obtain
\begin{align*}
&{\bf E}\|(\Psi X)(t + \tau)-(\Psi X)(t)\|^2 \\
& \leq3 \mathop{\rm Tr}Q \int_0^{\infty}\|U(t+\tau, t+\tau-s)\|^2 {\bf
E}\|G(t+\tau-s, X(t+\tau-s)) \\
&\quad -G(t-s, X(t-s))\|_{\mathbb{L}^0_2}^2\,ds \\
&\quad + 3 \mathop{\rm Tr}Q\int_{\varepsilon}^{\infty}\|U(t+\tau, t+\tau-s)
- U(t, t-s)\|^2 {\bf E}\|G(t-s, X(t-s))\|_{\mathbb{L}_2^0}^2\,ds \\
&\quad + 3 \mathop{\rm Tr}Q\int_0^{\varepsilon}\|U(t+\tau, t+\tau-s)
 - U(t, t-s)\|^2 {\bf E}\|G(t-s, X(t-s))\|_{\mathbb{L}_2^0}^2\,ds \\
&\leq 3 \mathop{\rm Tr} Q M^2 \Big(\int_0^{\infty}e^{-2\delta s}\,ds\Big)
 \sup_{\sigma\in\mathbb{R}}\|G(\sigma+\tau, X(\sigma+\tau))
 -G(\sigma, X(\sigma))\|_{\mathbb{L}_2^0}^2\\
&\quad + 3 \mathop{\rm Tr}Q \varepsilon^2
 \Big(\int_{\varepsilon}^{\infty}e^{-\delta\,
s}\,ds\Big) \sup_{\sigma\in\mathbb{R}}{\bf E}\|G(\sigma,
X(\sigma))\|_{\mathbb{L}_2^0}^2 \\
&\quad + 6 \mathop{\rm Tr}Q M^2 \Big(\int_0^{\varepsilon}e^{-2\delta
s}\,ds\Big) \sup_{\sigma\in\mathbb{R}}{\bf E}\|G(\sigma,
X(\sigma))\|_{\mathbb{L}_2^0}^2 \\
&\leq 3 \mathop{\rm Tr}Q\big[ \eta\frac{M^2}{2\delta}+\varepsilon\frac{K_2}{\delta}
+2 \varepsilon\,K_2\big],
\end{align*}
 which implies that $\Psi X(\cdot)$ is square-mean almost
periodic.
Define
$$
(\Lambda X)(t):=\int_{-\infty}^t U(t, s) F(s, X(s))\,ds
 + \int_{-\infty}^t U(t,s) G(s, X(s))\, dW(s)\,.
$$
In view of the above, it is clear that $\Lambda$ maps
$AP(\mathbb{R}; L^2({\bf P}; \mathbb{H}))$ into itself. To complete the proof, it
suffices to prove that $\Lambda$ has a unique fixed-point.
Clearly,
\begin{align*}
&\|(\Lambda X)(t)-(\Lambda Y)(t)\| \\
&=\|\int_{-\infty}^t U(t, s) [F(s,X(s))-F(s, Y(s))]\, ds \\
&\quad + \int_{-\infty}^t U(t, s) [G(s, X(s))-G(s, Y(s))]\,dW(s)\| \\
& \leq M \int_{-\infty}^t e^{-\delta
(t-s)} \|F(s,X(s))-F(s, Y(s))\|\,ds \\
&\quad +\|\int_{-\infty}^t U(t, s) [G(s, X(s))-G(s,
Y(s))]\,dW(s)\|\,.
\end{align*}
Since $(a+b)^2\leq 2a^2+2b^2$, we can write
\begin{align*}
&{\bf E}\|(\Lambda X)(t)-(\Lambda Y)(t)\|^2 \\
& \leq 2M^2 {\bf E}\Big(\int_{-\infty}^t e^{-\delta(t-s)}
 \|F(s, X(s))-F(s, Y(s))\|\,ds\Big)^2\\
&\quad  + 2{\bf E}\Big(\|\int_{-\infty}^t U(t, s)[G(s,
X(s))-G(s, Y(s))]\,dW(s)\|\Big)^2\,.
\end{align*}

We evaluate the first term of the right-hand side as follows:
\begin{align*}
& {\bf E}\Big(\int_{-\infty}^te^{-\delta(t-s)}\|F(s, X(s))-F(s, Y(s))\|\,
ds\Big)^2 \\
&\leq  {\bf E}\Big[\Big(\int_{-\infty}^t\,e^{-\delta(t-s)} ds\Big)
\Big(\int_{-\infty}^te^{-\delta(t-s)}\|F(s, X(s))-F(s, Y(s))\|^2\,
ds\Big)\Big] \\
& \leq \Big(\int_{-\infty}^te^{-\delta(t-s)}\,ds\Big)
\Big(\int_{-\infty}^te^{-\delta(t-s)} {\bf E}\| F(s, X(s))-F(s,
Y(s))\|^2\,ds\Big) \\
& \leq  K  \cdot \Big(\int_{-\infty}^te^{-\delta(t-s)}\,
ds\Big)\Big(\int_{-\infty}^te^{-\delta(t-s)}  {\bf E}\|
X(s))-Y(s))\|^2\,ds\Big) \\
&\leq K \cdot \Big(\int_{-\infty}^te^{-\delta(t-s)}\,
ds\Big)^2 \sup_{t\in\mathbb{R}}{\bf E}\|X(t)-Y(t)\|^2 \\
&= K \cdot \Big(\int_{-\infty}^te^{-\delta(t-s)}\,
ds\Big)^2\|X-Y\|_{\infty} \\
& \leq \frac{K}{\delta^2} \cdot \|X-Y\|_{\infty}\,.
\end{align*}
As to the second term, we use again an estimate on the Ito integral
established in \cite{I} to obtain:
\begin{align*}
&{\bf E}\Big(\|\int_{-\infty}^t U(t, s)\,
[G(s, X(s))-G(s, Y(s))]\,dW(s)\|\Big)^2 \\
&\quad \leq \mbox{Tr Q} \cdot {\bf E}
\Big[\int_{-\infty}^t\|U(t, s)\,[G(s, X(s))-G(s, Y(s))]\|^2\, ds\Big]\\
& \leq \mathop{\rm Tr} Q \cdot {\bf E}
\Big[\int_{-\infty}^t\|U(t, s)\|^2\|G(s, X(s))-G(s,
Y(s))\|_{\mathbb{L}_2^0}^2\,ds\Big] \\
& \leq \mathop{\rm Tr} Q \cdot
M^2\int_{-\infty}^te^{-2\delta(t-s)}{\bf E}\|G(s,X(s))-G(s,
Y(s))\|_{\mathbb{L}_2^0}^2\,ds\\
& \leq  \mathop{\rm Tr} Q \cdot M^2
K' \cdot \Big(\int_{-\infty}^te^{-2\delta(t-s)}\,ds\Big)
 \sup_{t\in R}{\bf E}\| X(s))-Y(s))\|^2\\
& \leq  \mathop{\rm Tr} Q \cdot \frac{M^2
K'}{2\delta} \cdot \|X-Y\|_{\infty}\,.
\end{align*}
Thus, by combining, it follows that
$$
{\bf E}\|(\Lambda X)(t)-(\Lambda Y)(t)\|\leq M^2
 \Big(2\frac{K}{\delta^2}+
\frac{K^\prime\cdot\mathop{\rm Tr} Q}{\delta}\Big)\|X-Y\|_{\infty},
$$
and therefore,
\[
\|\Lambda X-\Lambda Y\|_{\infty}\leq M^2
 \Big(2\frac{K}{\delta^2}\,+ \frac{K^\prime\cdot\mathop{\rm Tr}
Q}{\delta}\Big)\|X-Y\|_{\infty}= \Theta  \cdot
\|X-Y\|_\infty.
\]
Consequently, if $\Theta < 1$, then \eqref{I} has a unique
fixed-point, which obviously is the unique square-mean almost
periodic solution to \eqref{I}.
\end{proof}

\section{Example}

Let ${\mathcal O} \subset \mathbb{R}^n$ be a bounded subset whose boundary
$\partial {\mathcal O}$ is of class $C^2$ and being locally on one
side of ${\mathcal O}$.

Consider the parabolic stochastic partial differential equation
\begin{gather}\label{H}
 d_tX(t, \xi)= \{A(t, \xi)X (t, \xi) + F(t, X(t,
\xi))\}\,d_t + G (t, X(t, \xi))\,dW(t),\\
\sum_{i,j = 1}^n n_i(\xi) a_{ij}(t, \xi) d_i X(t, \xi) =0, \quad
t \in \mathbb{R}, \;  \xi \in \partial {\mathcal O}\label{H'},
\end{gather}
where $ d_t = \frac{d}{dt}$, $ d_i = \frac{d}{d\xi_i}$,
$n(\xi) = (n_1(\xi), n_2(\xi), \dots , n_n(\xi))$
is the outer unit normal vector, the family of operators
$A(t, \xi)$ are formally given by
$$
A(t, \xi)=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}
\Big(a_{ij}(t, \xi) \frac{\partial}{\partial x_j}\Big)
+  c(t, \xi), \quad t \in \mathbb{R}, \; \xi \in {\mathcal O},
$$
$W$ is a real valued Brownian motion, and
$a_{ij}, c$ ($i,j = 1, 2, \dots , n$) satisfy the following
conditions: \\
(H3)
\begin{itemize}
  \item [(i)] The coefficients $(a_{ij})_{i,j=1,\dots ,n}$ are symmetric,
that is,   $a_{ij}=a_{ji}$ for all $i,j =1, \dots , n$.
 Moreover, $a_{ij} \in C_b^\mu (\mathbb{R}, L^2 ({\bf P},
   C(\overline{\mathcal O}))) \cap
C_b(\mathbb{R}, L^2 ({\bf P}, C^1(\overline{\mathcal O}))) \cap AP(\mathbb{R}; L^2
({\bf P}, L^2({\mathcal O})))$ for all $i,j =1,\dots n$, and $c \in
C_b^\mu (\mathbb{R}, L^2 ({\bf P}, L^2({\mathcal O}))) \cap C_b(\mathbb{R}, L^2
({\bf P}, C(\overline{\mathcal O}))) \cap AP(\mathbb{R}; L^2 ({\bf P},
L^1({\mathcal O})))$ for some $\mu \in (1/2, 1]$.

  \item [(ii)] There exists $\varepsilon_0 >0$ such
  that
  $$
\sum_{i, j=1}^na_{ij} (t, \xi)\eta_i \eta_j\geq \varepsilon_0
  |\eta|^2,
 $$
  for all $(t, \xi)\in\mathbb{R}\times \overline{\mathcal O}$ and $\eta\in\mathbb{R}^n$.
\end{itemize}
Under above assumptions, the existence of an evolution family
$U(t,s)$ satisfying (H0) is obtained, see, eg., \cite{MR}.

Set $\mathbb{H}=L^2({\mathcal O})$. For each $t\in\mathbb{R}$ define an operator
$A(t)$ on $L^2({\bf P}; H)$ by
\[
{\mathcal D(A(t))}=\{X \in L^2({\bf P}, H^2({\mathcal O})):
\sum_{i,j = 1}^n n_i(\cdot) a_{ij}(t, \cdot)
d_i X(t, \cdot) =0\quad \mbox{on }  \partial{\mathcal O}\}
\]
and
$A(t)X=A(t, \xi) X(\xi)$ for all $ X\in {\mathcal D}(A(t))$.

 Thus under assumptions
(H1)--(H3), then the system \eqref{H}--\eqref{H'} has a unique
mild solution, which obviously is square-mean almost periodic,
whenever $M$ is small enough.


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\end{document}