1 Introduction

As we know, the recurrence properties (e.g., periodicity and almost periodicity ) are of great concern in qualitative theory of various kinds of differential equations and a lot of works have been done on these issues in the past decades. Particularly, in recent years, the existence of (asymptotic, almost) periodic solutions to semi-linear evolution equations has received much attention by many authors (see, e.g., [5, 11, 30] and the references therein) due to their significant applications in various fields such as physics, mathematical biology, and control theory. This interest also arises from a need to extend the well-known results on stochastic ordinary differential equations to stochastic partial differential equations. Likewise in these years, there are many papers investigating the existence and qualitative properties of (asymptotic, almost) periodic solutions for semi-linear stochastic evolution equations, see [9, 10, 12, 14, 23, 28] among others.

Bezandry and Diagana introduced in [2] the concept of quadratic-mean almost periodicity to the stochastic evolution equation

$$\begin{aligned} \textrm{d}x(t) = Ax(t)\textrm{d}t + f(t, x(t))\textrm{d}t + g(t, x(t))\textrm{d}W(t), t\in \mathbb R, \end{aligned}$$
(1)

where A is the infinitesimal generator of a uniformly stable semigroup. They investigated the existence and uniqueness of a quadratic-mean almost periodic solution by utilizing Banach contraction mapping principle. Then, they also studied in [4] the existence and uniqueness of quadratic-mean almost periodic solutions for non-autonomous stochastic evolution equation by replacing A in Eq. (1) with A(t), while Liu et al. [22] discussed the existence of asymptotically almost periodic mild solutions of Eq. (1) and Cao et al. [8] considered the existence and stability of quadratic-mean almost periodic mild solutions for the stochastic functional evolution equation

$$\begin{aligned} \textrm{d}x(t)=[Ax(t)+F(t,x(t),x_t)]\textrm{d}t+G(t,x(t),x_t)\textrm{d}W(t),\quad t\in [0,T] \end{aligned}$$
(2)

with \(x(t)=\phi (t),t\in [\sigma ,0]\) with \(\sigma <0\), where the linear operator \(A:D(A)\subset L^2(\mathbb P,\mathbb H)\rightarrow L^2(\mathbb P,\mathbb H)\) generates a uniformly exponentially stable \(C_0\)-semigroup on \(L^2(\mathbb P,\mathbb H)\). In addition, in Ref. [9] Cao et al. studied the existence and exponential stability of quadratic-mean asymptotically almost periodic mild solutions of Eq. (2). Huang considered in [7] the quadratic-mean asymptotically almost periodic mild solutions of Eq. (2) with A replaced by A(t).

Note that so far the existing results in the literature are mostly on the existence of quadratic-mean almost periodic solutions for semi-linear stochastic evolution equations. However, the counter examples presented in [25] show that nontrivial solutions of some stochastic differential equations with almost periodic coefficients may be never square-mean almost periodic. This means that the square-mean almost periodicity is frequently too strong for solutions of some stochastic differential equations and it is more suitable to introduce the concept of almost periodicity or almost automorphy in distribution sense for them. Consequently, the research of different kinds of almost periodic solutions in distribution for stochastic differential equations has become a research hotspot in recent years, see [15, 16, 18, 20] and the references therein. For instance, in [18] the authors studied the almost periodic solutions in distribution of the following equation applying Lyapunov function method:

$$\begin{aligned} \textrm{d}X(t)=F(t,X(t))\textrm{d}t+G(t,X(t))\textrm{d}W(t) \end{aligned}$$

where \(F:\mathbb R\times \mathbb R^d\rightarrow \mathbb R^d\) is a continuous function, \(G:\mathbb R\times \mathbb R^d\rightarrow \mathbb M^{d\times m}\) is a matrix-valued continuous function, and W(t) is a standard d-dimensional Brownian motion.

On the other hand, as a more abstract and general recurrent oscillation, Besicovitch almost periodic oscillation is a generalization of Bohr almost periodicity, Stepanov almost periodicity, Weyl almost periodicity, and so on. Because the space of Besicovitch functions is not complete, it is naturally more difficult to study the Besicovitch almost periodic solutions for evolution equations. In Li and Huang [17] and Li and Wang [21], the authors discussed the existence of Besicovitch almost periodic solutions of a stochastic model on Clifford-valued neural networks with time-varying delays. Up to now, however, the results on Besicovitch almost periodic solutions of stochastic evolution equations are very rare. For the work concerning existence of Besicovitch almost periodic solutions of deterministic evolution equations, one can see Li et al. [19]. It is clearly meaningful to extend the concept of p-th Besicovitch almost periodicity (see Definitions 2.4 and 2.6) for stochastic evolution equations, and meanwhile, the existence of p-th Besicovitch almost periodic solutions in distribution for non-autonomous stochastic evolution equations is still an untreated topic currently. Motivated by the above consideration, in this work we are going to discuss the p-th Besicovitch almost periodic solutions in distribution for a class of semi-linear non-autonomous stochastic evolution equations. Precisely, we shall consider the following semi-linear non-autonomous stochastic evolution equation:

$$\begin{aligned} \textrm{d}X(t)=A(t)X(t)\textrm{d}t+F(t,X(t))\textrm{d}t+G(t,X(t))\textrm{d}W(t),\,\,t\in \mathbb R \end{aligned}$$
(3)

where \(A(t):D(A(t))\subset \mathbb H\rightarrow \mathbb H\) (with \(\mathbb H\) a separable Hilbert space) is a family of closed linear operator satisfying the so-called Acquistapace–Terreni conditions, F and G are nonlinear functions to be specified later, and W(t) is a Q-Wiener process.

Our purpose here is to study the existence and stability problems of p-th Besicovitch almost periodic solutions in distribution for Eq. (3). We will first discuss the existence of p-th Besicovitch almost periodic solutions in distribution for Eq. (3) by applying Banach fixed principle. Since, as mentioned above, the set composed of all p-th Besicovitch almost periodic functions in distribution is not a Banach space, one cannot establish the existence result directly by Banach fixed point theorem in this space. To overcome this obstacle, we introduce instead the Banach space \(C_b(\mathbb R, L^p(\Omega ,\mathbb H))\) and prove firstly that Eq. (3) has a mild solution in this space. Particularly, to obtain the continuity of the solutions we employ the continuity of the evolution operator U(t, s) in uniform operator topology on finite interval. Then, we further verify that the obtained solution is p-th Besicovitch almost periodic in distribution employing a generalized Gronwall inequality. After that we also investigate the global exponential stability for the obtained p-th Besicovitch almost periodic solution. The obtained result shows that interestingly any solution of (3) whose initial value is close to that of the p-th Besicovitch almost periodic solution may tend to this p-th Besicovitch almost periodic solution exponentially as \(t \rightarrow +\infty \).

Let us mention here that this is seemingly the first paper to study existence and stability of p-th Besicovitch almost periodic solutions in distribution for semi-linear non-autonomous stochastic evolution equations. Moreover, since we discuss the problems on the larger space \(C_b(\mathbb R, L^p(\Omega ,\mathbb H))\), it enable us to impose fewer restrictions on the nonlinear functions (see (\(H_3\)) and (\(H_4\)) in Sect. 3) compared with the related works [3, 8, 10, 25] and therefore the obtained results of this note are easier to be applied in practice. In addition, it can also be seen that the method adopted in this paper is much different from that in [19] and our approach here can also be applied to study the existence of Besicovitch almost periodic solutions for other types of semi-linear stochastic evolution equations.

Subsequently, we will introduce briefly in Sect. 2 some definitions, preliminary lemmas on linear evolution operator, stochastic processes and p-th Besicovitch almost periodicity in distribution. We then study in Sect. 3 the existence and uniqueness of p-th Besicovitch almost periodic solutions in distribution for Eq. (3). Further, the globally exponential stability of this unique p-th Besicovitch almost periodic solution is investigated in Sect. 4. Finally, an example is presented in Sect. 5 to demonstrate the obtained theoretical results.

2 Preliminaries

In this section, let us first introduce some basic notations and preliminary results concerning linear evolution operator, stochastic processes and p-th Besicovitch almost periodicity in distribution.

Throughout the paper, \((\mathbb H,\Vert \cdot \Vert _{\mathbb H})\) and \((\mathbb K,\Vert \cdot \Vert _{\mathbb K})\) are both real separable Hilbert spaces, \(\mathscr {L}(\mathbb H,\mathbb K)\) denotes the space of all bounded linear operators from \(\mathbb H\) to \(\mathbb K\) equipped with the usual operator norm and it is abbreviated as \(\mathscr {L}(\mathbb H)\) if \(\mathbb H=\mathbb K\). We always assume that \(A(t):D(A(t))\subset \mathbb H\rightarrow \mathbb H\), \(t\in \mathbb {R}\) in Eq. (3) is a family of densely defined closed linear operators satisfying the so-called Acquistapace–Terreni conditions below.

Assumption 2.1

(see [1] Hypothesis I and II.) (Acquistapace–Terreni conditions) There exist constants \(\lambda _0\ge 0,\theta \in \left( \frac{\pi }{2},\pi \right) ,\) \(L,K\ge 0\) and \(\alpha ,\beta \in (0,1]\) with \(\alpha +\beta >1\) such that

$$\begin{aligned} \rho (A(t)-\lambda _0)\subset \sum \limits _\theta \cup \{0\},\,\, \Vert R(\lambda ,A(t)-\lambda _0)\Vert \le \frac{K}{1+|\lambda |} \end{aligned}$$

and

$$\begin{aligned} \Vert (A(t)-\lambda _0)R(\lambda ,A(t)-\lambda _0)[R(\lambda _0,A(t))-R(\lambda _0,A(s))]\Vert \le L|t-s|^\alpha |\lambda |^{-\beta } \end{aligned}$$

for \(t,s\in \mathbb R,\lambda \in \sum \limits _{\theta }:=\{\lambda \in \mathbb C-\{0\}:|\arg \lambda |\le \theta \}.\)

Definition 2.1

A family of bounded linear operators \(\{U(t,s):-\infty<s\le t<+\infty \}\) on \( \mathbb H \) associated with A(t) is said to be an evolution operator or evolution system if it verifies

  1. (i)

    \(U(t,s)U(s,r)=U(t,r)\) for all \(t,s,r\in \mathbb R\) such that \(r\le s\le t\);

  2. (ii)

    \(U(t,t)=I\) for all \(t\in \mathbb R\);

  3. (iii)

    \((t,s)\rightarrow U(t,s)\in \mathscr {L}(\mathbb H)\) is strongly continuous for \(s\le t\);

  4. (iv)

    for any \(x\in D(A(s)), U(t,s)x\in D(A(t))\), U(t, s)x is differentiable both in t and s, and

    $$\begin{aligned} \frac{\partial }{\partial t}U(t,s)x=A(t)U(t,s)x,~~\frac{\partial }{\partial s}U(t,s)x=-U(t,s)A(s)x. \end{aligned}$$

The following lemma shows that under Assumption 2.1\(\{A(t)\}_{t\in \mathbb {R}}\) generates an evolution operator \(\{U(t,s):-\infty<s\le t<+\infty \}\) on \( \mathbb H \).

Lemma 2.1

(see [1] Theorem 3.3.) A family of densely defined closed operators \(\{A(t)\}_{t\in \mathbb {R}}\) satisfying the Acquistapace–Terreni conditions (Assumption 2.1) generates a unique evolution operator \(\{U(t,s):-\infty<s\le t<+\infty \}\) on \(\mathbb H\).

The next proposition, which is an immediate consequence of ([13], Lemma 14.1), shows that \(\{U(t,s):t>s\}\) is continuous in t in the uniform operator topology uniformly for s on a finite interval. This property will play an important role in the discussion in Sect. 3.

Proposition 2.1

The evolution operator \(\{U(t,s):t>s\}\) satisfies that, for any \(\theta \in (0,1)\), there is a \(K_1>0\) such that

$$\begin{aligned} \Vert U(t+h,s)-U(t,s)\Vert \le K_1h^\theta |t-s|^{-\theta }, ~\text { for }~t+h >s. \end{aligned}$$

Next we turn to state some notations on stochastic processes.

Let \((\Omega ,\mathscr {F},\{\mathscr {F}_t\}_{t\ge 0},P)\) be a complete probability space with a natural filtration \(\{\mathscr {F}_t\}_{t\ge 0}\) satisfying the usual conditions. Here, a filtration is a family \(\{\mathscr {F}_t\}_{t\ge 0}\) of increasing sub-\(\sigma \)-algebra of \(\mathscr {F}\) (i.e.,\(\mathscr {F}_t\subset \mathcal F_s\subset \mathscr {F}\) for \(0\le t<s<\infty \) ). A filtration \(\{\mathscr {F}_t\}_{t\ge 0}\) is said to be right continuous if \(\mathscr {F}_t=\cap _{s>t}\mathscr {F}_s\) for \(t\ge 0\). In addition, the usual conditions are right continuous and \(\mathscr {F}_0\) contains all the p-null sets. In the sequel, \({L}^p(\Omega ,\mathbb H)\) stands for the space of all \(\mathbb H\)-valued random variables with \(E\Vert X\Vert _{\mathbb H}^p:=\int _{\Omega }\Vert X\Vert _{\mathbb H}^pdP<\infty \).

Definition 2.2

A stochastic process \(X:\mathbb R\rightarrow {L}^p(\Omega ,\mathbb H)\) is said to be \( {L}^p\)-continuous, if, for any \(s\in \mathbb R\),

$$\begin{aligned} \lim \limits _{t\rightarrow s}E\Vert X(t)-X(s)\Vert ^p_{\mathbb H}=0. \end{aligned}$$

In addition, it is \( {L}^p\)-bounded if \(\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert ^p_{\mathbb H}<\infty \).

Assume \(\{\beta _n(t)\}_{t\ge 0}\) is a sequence of real-valued one-dimensional standard Brownian motions mutually independent over \((\Omega ,\mathcal {F},\{\mathcal {F}_t\}_{t\ge 0}, P)\). Set

$$\begin{aligned} W(t)=\sum \limits _{n=1}^{\infty }\sqrt{\lambda }_n\beta _n(t)e_n,\,\,t\in \mathbb R^+, \end{aligned}$$

where \(\lambda _n(n=1,2,\cdots )\) are nonnegative real numbers and \(\{e_n\}\) is a complete orthonormal basis in \(\mathbb K\). Let \(Q\in \mathscr {L}(\mathbb K)\) be a symmetric nonnegative operator defined by \(Qe_n=\lambda _n e_n\) with Tr\((Q)=\sum \limits _{i=1}^{\infty }\lambda _n<\infty \). Define \(\mathbb K_0=Q^{\frac{1}{2}}{\mathbb K}\), and let \(\mathscr {L}_2^0(\mathbb K_0,\mathbb H)\) represent the space of all Q-Hilbert–Schmidt operators acting from \(\mathbb K_0\) to \(\mathbb H\) endowed with the Hilbert–Schmidt norm \(\Vert \cdot \Vert _{\mathscr {L}_2^0}\). It then turns out that \(\mathscr {L}_2^0\) is a separable Hilbert space with respect to the norm

$$\begin{aligned} \Vert \Phi \Vert _{\mathscr {L}_2^0}^2=\text {Tr}(\Phi Q\Phi ^{*}) ~~ \text {for} ~~ \Phi \in \mathscr {L}_2^0. \end{aligned}$$

The above \(\mathbb K\)-valued stochastic process W(t) is called a Q-Wiener process. It is worth mentioning that a Q-Wiener process \(\{W(t):t\in \mathbb R\}\) can obtained as follows: Let \(\{W_i(t): t\in \mathbb R^+\},i=1,2\) be independent \(\mathbb K\)-valued Q-Wiener process, then

$$\begin{aligned} W(t)=\left\{ \begin{array}{ll} W_1(t), ~~ t\ge 0, &{} \\ W_2(-t), ~~ t\le 0, &{} \\ \end{array} \right. \end{aligned}$$
(4)

is a Q-Wiener process with the real number line as time parameter. We then let \(\mathscr {F}_t=\sigma \{W(s),s\le t\}.\) For more details on Q-Wiener processes, we refer to the book [27].

Now, let us present some concepts about almost periodic functions. First the following definition is well known.

Definition 2.3

Let \(C(\mathbb R,\mathbb H)\) be the set of continuous functions from \(\mathbb R\) to \(\mathbb H\). A function \(f\in C(\mathbb R,\mathbb H)\) is said to be almost periodic, if for any \(\varepsilon >0\), there exists \(\mathcal {l}>0\) such that every interval of length \(\mathcal {l}\) contains a number \(\tau \) such that

$$\Vert f(t+\tau )-f(t)\Vert _{\mathbb H}<\varepsilon , \,\, t\in \mathbb R.$$

The number \(\tau \) is called an \(\varepsilon \)-translation of f.

We denote by \(AP(\mathbb R,\mathbb H)\) the collection of all the almost periodic functions from \(\mathbb R\) to \(\mathbb H\). Clearly, any almost periodic function is bounded and continuous.

Then, we introduce, respectively, the definitions of p-th Besicovitch almost periodic in \(t\in \mathbb R\) uniformly, p-th mean Besicovitch almost periodic stochastic process and p-th Besicovitch almost periodic in distribution stochastic process (cf. [3, 6]).

Definition 2.4

A function \(f\in L_{loc}^{p}(\mathbb R,\mathbb H)\) is said to be p-th Besicovitch almost periodic, if for any \(\varepsilon >0\), there exists \(\mathcal {l}>0\) such that every interval of length \(\mathcal {l}\) contains a number \(\tau \) such that

$$\begin{aligned} \Vert f(t+\tau )-f(t)\Vert _{B^p}:=\bigg (\mathop {\lim \sup }\limits _{l\rightarrow \infty }\frac{1}{2l}\int _{-l}^{l}\Vert f(t+\tau )-f(t)\Vert _{\mathbb H}^p\textrm{d}t\bigg )^{\frac{1}{p}}<\varepsilon . \end{aligned}$$

Let \( L_\textrm{loc}^p(\mathbb R, L^{p}(\Omega ,\mathbb H))\) (\(2\le p<\infty \)) represents the space of all stochastic processes from \(\mathbb R\) to \( L^{p}(\Omega ,\mathbb H)\) which are locally p-integrable. For \(X\in L_\mathrm{{loc}}^p(\mathbb R, L^{p}(\Omega ,\mathbb H))\), we define

$$\begin{aligned} \Vert X(\cdot )\Vert _{S\!B^p}=\bigg (\mathop {\lim \sup }\limits _{l\rightarrow \infty }\frac{1}{2l}\int _{-l}^{l}E\Vert X(t))\Vert _{\mathbb H}^p\textrm{d}t\bigg )^{\frac{1}{p}}. \end{aligned}$$

Definition 2.5

A function \(f:\mathbb R\times \mathbb K\rightarrow \mathbb H\) is said to be p-th Besicovitch almost periodic in \(t\in \mathbb R\) uniformly on any bounded subset K of \(\mathbb K\), if, for any \(\varepsilon >0\), there exists \(\mathcal {l(\varepsilon ,\mathrm K)}>0\) such that every interval of length \(\mathcal {l(\varepsilon ,\mathrm K)}\) contains a number \(\tau \) such that

$$\Vert f(t+\tau ,x)-f(t,x)\Vert _{\!B^p}<\varepsilon , \,\, t\in \mathbb R.$$

Definition 2.6

A stochastic process \(X(\cdot )\in L_{loc}^p(\mathbb R, L^p(\Omega ,\mathbb H))\) is said to be p-th mean Besicovitch almost periodic, if, for any \(\varepsilon >0\), there exists \(\mathcal {l}>0\) such that every interval of length \(\mathcal {l}\) contains a number \(\tau \) such that

$$\begin{aligned} \Vert X(t+\tau )-X(t)\Vert _{S\!B^p}=\bigg (\mathop {\lim \sup }\limits _{l\rightarrow \infty }\frac{1}{2l}\int _{-l}^{l}E\Vert X(t+\tau )-X(t)\Vert _{\mathbb H}^p\textrm{d}t\bigg )^{\frac{1}{p}}<\varepsilon . \end{aligned}$$

To introduce the concept of p-th Besicovitch almost periodicity in distribution, for a metric space \((\mathbb E,d)\), we denote by \(\mathcal B(\mathbb E)\) the \(\sigma \)-algebra of Borel sets of \(\mathbb E\) and by \(\mathcal P(\mathbb E)\) the set of all probability measures on \(\mathbb E\). Let \( C_b(\mathbb E)\) be the space of all bounded and continuous functions \(f:\mathbb E\rightarrow \mathbb R\) with \(\Vert f\Vert _{\infty }:=\sup \limits _{x\in \mathbb E}\{|f(x)|\}\), we define that, for \(f\in C_b(\mathbb E)\), \(\mu ,\nu \in \mathcal P(\mathbb E)\),

$$\begin{aligned} \Vert f\Vert _L=\sup \limits _{a\ne b}\frac{| f(a)-f(b)|}{d(a,b)},\quad \Vert f\Vert _\mathrm{{BL}}=\max \{\Vert f\Vert _{\infty },\Vert f\Vert _L\}, \end{aligned}$$

and

$$\begin{aligned} d_\mathrm{{BL}}(\mu ,\nu ):=\sup \limits _{\Vert f\Vert _\mathrm{{BL}}\le 1}\bigg | \int _{\mathbb E}fd(\mu -\nu )\bigg |. \end{aligned}$$

By virtue of \(d_\mathrm{{BL}}(\mu ,\nu )\), we can give the definition of p-th Besicovitch almost periodic stochastic process in distribution as

Definition 2.7

A \(\mathbb H\)-valued stochastic process X(t) is called p-th Besicovitch almost periodic in distribution if the mapping \(t\rightarrow \gamma _t:=\gamma (X(t))\) is p-th Besicovitch almost periodic, where \(\gamma (X(t)):=P\circ [X(t)]^{-1}\) is the law for X(t) under P. That is to say, if, for any \(\varepsilon >0\), there exists \(\mathcal {l}>0\) such that every interval of length \(\mathcal {l}\) contains a number \(\tau \) satisfying

$$\begin{aligned} \bigg (\mathop {\lim \sup }\limits _{l\rightarrow \infty }\frac{1}{2l}\int _{-l}^{l}\bigg [d_\mathrm{{BL}}(P\circ [X(t+\tau )]^{-1},P\circ [X(t)]^{-1})\bigg ]^p\textrm{d}t\bigg )^{\frac{1}{p}}<\varepsilon . \end{aligned}$$

We now close this section by stating two important inequalities to be used in the later discussion. The first one is regarded as a generalized Gronwall inequality, that is,

Lemma 2.2

(see[15] Lemma 3.3.) Let \(u:\mathbb R\rightarrow \mathbb R\) be a continuous function such that, for every \(t\in \mathbb R\),

$$\begin{aligned} 0\le u(t)\le \alpha +\beta \int _{-\infty }^{t}e^{-\gamma (t-s)}u(s)\textrm{d}s. \end{aligned}$$

where \(\alpha ,\beta ,\gamma \ge 0\) are constants, and \(\gamma >\beta \). Then, there holds

$$u(t)\le \alpha \frac{\gamma }{\gamma -\beta },~~ for ~t\in \mathbb R.$$

Lemma 2.3

(see[27] Theorem 4.37.) For \(p\ge 2\), \(\varphi \in \big (\mathbb R,\mathscr {L}_2^0(\mathbb K,\mathbb H)\big )\), W(t) is a Q-Wiener process, and there exists a constant \(C_p>0\) such that for \(t\ge 0\),

$$\begin{aligned} E\left[ \sup \limits _{s\in [0,t]}\left\| \int _{0}^s\varphi (\tau )\textrm{d}W(\tau )\right\| ^p\right] \le C_p{E\left[ \int _{0}^t\Vert \varphi (\tau )\Vert _{\mathscr {L}_2^0}^2\textrm{d}\tau \right] ^{\frac{p}{2}}}, \end{aligned}$$

where \(C_p=\left[ \frac{p(p-1)}{2}\left( \frac{p}{p-1}\right) ^{p-2}\right] ^{\frac{p}{2}}.\)

3 p-th Besicovitch Almost Periodic Solutions in Distribution

In this section, we study the existence and uniqueness of p-th Besicovitch almost periodic solutions in distribution for Eq. (3) for \(p\ge 2\). To do this, we make the following assumptions. Let \(p\ge 2\).

\((H_{1})\):

The evolution operator \(\{U(t,s):-\infty<s\le t<+\infty \}\) generated by A(t) is a uniformly exponentially stable; that is, there exist constants \(M\ge 1\) and \(\delta >0\) such that

$$\begin{aligned} \Vert U(t,s) \Vert \le Me^{-\delta (t-s)},\,\, -\infty<s\le t<+\infty ; \end{aligned}$$
\((H_{2})\):

The family of resolvent operators \(R(\lambda _0,A(t))\in \mathscr {L}(\mathbb H)\) is almost periodic, where \(\lambda _0\) is required in Acquistapace–Terreni conditions;

\((H_{3})\):

The mappings \(F:\mathbb R\times L^p(\Omega ,\mathbb H)\rightarrow L^p(\Omega ,\mathbb H), G:\mathbb R\times L^p(\Omega ,\mathbb H)\rightarrow L^p(\Omega ,\mathscr {L}_2^0)\) are p-th Besicovitch almost periodic in \(t\in \mathbb R\) uniformly on any bounded subset of \(L^p(\Omega ,\mathbb H);\)

\((H_{4})\):

There exist positive constants \(L_1,L_2\) such that, for \(X,Y\in L^p(\Omega ,\mathbb H)\) and \(t\in \mathbb R\),

$$\begin{aligned}{} & {} \max \left\{ E\Vert F(t,X)\Vert ^p_\mathbb H,E\Vert G(t,X)\Vert _{\mathscr {L}_2^0}^p \right\} \le L_1(1+E\Vert X\Vert _\mathbb H)^p, \,\, \\{} & {} \max \left\{ E\Vert F(t,X)-F(t,Y)\Vert ^p_\mathbb H,E\Vert G(t,X)-G(t,Y)\Vert ^p_{\mathscr {L}_2^0}\right\} \le L_2E\Vert X-Y\Vert ^p_\mathbb H. \end{aligned}$$

The above conditions \((H_1)\) and \((H_2)\) imply that the evolution operator \(\{U(t,s):-\infty<s\le t<+\infty \}\) has almost periodic property. Namely,

Lemma 3.1

(see [5] Lemma 5.1) Suppose that \((H_1)\) and \((H_2)\) are satisfied. Then for any \(\varepsilon >0\) and \(\hat{h}>0\), there exists \(l=l(\varepsilon ,\hat{h})>0\) such that every interval of length l contains at least a number \(\tau >0\) with the property that

$$\begin{aligned} \Vert U(t+\tau ,s+\tau )-U(t,s)\Vert \le \varepsilon e^{-\frac{\delta }{2}(t-s)} \end{aligned}$$

for all \(t-s\ge \hat{h}.\)

We now give the definition of mild solutions of Eq. (3).

Definition 3.1

An \(\mathcal F_t\)-measurable stochastic process X(t) is called a mild solution of Eq. (3), if X(t) satisfies the stochastic integral equation

$$\begin{aligned} X(t)=U(t,a)X(a)+\int _{a}^tU(t,s)F(s,X(s))\textrm{d}s+\int _{a}^t U(t,s)G(s,X(s))\textrm{d}W(s), \end{aligned}$$

for all \(t\ge a\).

Now, we set about to study the existence of p-th Besicovitch almost periodic solutions in distribution. As stated in Sect. 1, the set formed by all the p-th Besicovitch almost periodic functions is not a Banach space, and we shall consider here this problem in \(\mathbb X\):= \(C_b(\mathbb R, L^{p}(\Omega ,\mathbb H))\) which is a Banach space endowed with the norm \(\Vert X\Vert _\mathbb {X}=\big (\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\big )^{\frac{1}{p}}\). The main result of this section is

Theorem 3.1

Let the assumptions \((H_1)\)-\((H_4)\) be fulfilled, and moreover, there holds

\((H_{5})\):

\(r_1\!:=\!6^{p-1} M^{p}L_2\frac{q}{p\delta }\left[ \left( \frac{p}{q\delta }\right) ^{\frac{p}{q}}\!+\! C_p \left( \frac{p\!-\!2}{2\delta }\right) ^{\frac{p-2}{2}} \right] \!<\!1\), for \( p\!>\!2\), \(\frac{1}{p}\!+\!\frac{1}{q}\!=\!1\); or \(r_2:= \frac{6\,M^2 L_2 (1+\delta )}{\delta ^2} <1\), for \( p=2\).

Then, Eq. (3) has a unique p-th Besicovitch almost periodic solution in distribution in the Banach space \(\mathbb X\).

Proof

From Definition 3.1, letting \(a\rightarrow -\infty \), we consider the stochastic integral equation

$$\begin{aligned} X(t)=\int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s+\int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s), \end{aligned}$$
(5)

and define the nonlinear operator \(\Phi :\mathbb X\rightarrow \mathbb X\) as, for \(X(\cdot )\in \mathbb {X},\)

$$\begin{aligned} \Phi (X)(t)=\int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s+\int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s). \end{aligned}$$

In the sequel, we shall first prove by Banach fixed point principle that \(\Phi \) has a fixed point \(X(\cdot )\) in \(\mathbb X\) which clearly is a solution of Eq. (3). Then, we further show that \(X(\cdot )\) is actually a p-th Besicovitch almost periodic solution in distribution. We begin with showing that \(\Phi \) is a self-mapping on \(\mathbb X\). In fact, let \(X\in \mathbb X\), then

$$\begin{aligned} E\Vert (\Phi X)(t)\Vert _{\mathbb H}^p=&E\bigg \Vert \int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s+\int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _{\mathbb H}^p\nonumber \\ \le&2^{p-1}E\bigg \Vert \int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s \bigg \Vert _{\mathbb H}^p \nonumber \\&\quad + 2^{p-1}E\bigg \Vert \int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s) \bigg \Vert _{\mathbb H}^p\nonumber \\ :=&2^{p-1}M_{1}(t)+2^{p-1}M_{2}(t) . \end{aligned}$$
(6)

By the H\(\ddot{o}\)lder inequality, we obtain that

$$\begin{aligned} M_{1}(t)=&E\bigg \Vert \int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s \bigg \Vert _{\mathbb H}^p\nonumber \\ \le&E\bigg \{\bigg [\int _{-\infty }^{t}\Vert U(t,s)\Vert ^{\frac{q}{p}}\textrm{d}s\bigg ]^{\frac{p}{q}}\bigg [\int _{-\infty }^{t}\Vert U(t,s)\Vert ^{\frac{p}{q}}\Vert F(s,X(s))\Vert _{\mathbb H}^{p}\textrm{d}s \bigg ]\bigg \}\nonumber \\ \le&M^{p} \bigg [\int _{-\infty }^{t} e^{-{\frac{q}{p}}\delta (t-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}\int _{-\infty }^{t}e^{-{\frac{p}{q}}\delta (t-s)} L_1E(1+\Vert X(s)\Vert _{\mathbb H})^{p}\textrm{d}s\nonumber \\ \le&2^{p-1}M^{p} L_1\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \int _{-\infty }^{t}e^{-{\frac{p}{q}}\delta (t-s)} {\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg )}\textrm{d}s\nonumber \\ =&2^{p-1}M^{p} L_1 \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}{\frac{q}{p\delta }\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg )} \nonumber \\ <&+\infty . \end{aligned}$$
(7)

Then, we look at \(M_2(t)\), for \(p>2\), and from Lemma 2.3 it follows that

$$\begin{aligned} M_{2}(t)=&E\bigg \Vert \int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s) \bigg \Vert _{\mathbb H}^p\nonumber \\ \le&C_p \bigg [\int _{-\infty }^{t} \Vert U(t,s) \Vert ^{\frac{2p}{p-2}\cdot \frac{1}{p}} \textrm{d}s\bigg ]^{\frac{p-2}{p}\cdot \frac{p}{2}} E\bigg [\int _{-\infty }^{t} \Vert U(t,s) \Vert ^{\frac{2}{q}\cdot \frac{p}{2}}\bigg ( \Vert G(s,X(s))\Vert _{\mathscr {L}_0^2}^2\bigg )^{\frac{p}{2}} \textrm{d}s\bigg ]\nonumber \\ \le&C_pM^{p} \bigg [\int _{-\infty }^{t} e^{-{\frac{2}{p-2}\delta (t-s)}} \textrm{d}s\bigg ]^{\frac{p-2}{2}} \int _{-\infty }^{t} e^{-{\frac{p}{q}}\delta (t-s)} L_1E( 1+\Vert X(s)\Vert _\mathbb H)^p \textrm{d}s \nonumber \\ \le&2^{p-1}C_pM^{p} L_1 \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \int _{-\infty }^{t} e^{-\frac{p}{q}\delta (t-s)}{\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg )} \textrm{d}s\nonumber \\ <&+\infty . \end{aligned}$$
(8)

When \(p=2\), then \(C_2=1\) and hence

$$\begin{aligned} M_{2}(t)=&E\bigg \Vert \int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s) \bigg \Vert _{\mathbb H}^2\nonumber \\ \le&E\bigg [\int _{-\infty }^t \Vert U(t,s)\Vert ^2 \Vert G(s,X(s)) \Vert _{\mathscr {L}_0^2}^2\textrm{d}s \bigg ] \nonumber \\ \le&M^2E\bigg [\int _{-\infty }^t e^{-2\delta (t-s)} L_1(1+\Vert X(s) \Vert _\mathbb H)^2\textrm{d}s \bigg ]\nonumber \\ \le&2M^2L_1\int _{-\infty }^t e^{-2\delta (t-s)}{\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^2\bigg )} \textrm{d}s \nonumber \\ <&+\infty . \end{aligned}$$
(9)

Thus, substituting (7)–(9) into (6) gives that \(\Vert \Phi (X)\Vert _\mathbb {X}=\big (\sup \limits _{t\in \mathbb R}E\Vert (\Phi X)(t)\Vert _{\mathbb H}^p\big )^{\frac{1}{p}}<+\infty \).

Next we show the continuity of \((\Phi X)(t)\) in t for \(X\in \mathbb {X}\). Let \(h\in \mathbb R\) with |h| very small, then

$$\begin{aligned}&E\Vert (\Phi X)(t+h)-(\Phi X)(t)\Vert _{\mathbb H}^p=E\bigg \Vert \int _{-\infty }^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s \\&\qquad +\int _{-\infty }^{t+h} U(t+h,s)G(s,X(s))\textrm{d}W(s) \\&\qquad -\int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s-\int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _{\mathbb H}^p \\&\quad \le 2^{p-1}E\bigg \Vert \int _{-\infty }^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s -\int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&\qquad + 2^{p-1}E\bigg \Vert \int _{-\infty }^{t+h} U(t+h,s)G(s,X(s))\textrm{d}W(s) -\int _{-\infty }^t U(t,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _{\mathbb H}^p \\&\quad :=2^{p-1}M_{3}(t,h)+2^{p-1}M_{4}(t,h). \end{aligned}$$

We estimate \(M_3\) and \(M_4\) separately below. For \(M_3(t,h)\), one has that

$$\begin{aligned} M_{3}(t,h)&=E\bigg \Vert \int _{-\infty }^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s -\int _{-\infty }^tU(t,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&\le 2^{p-1}E\bigg \Vert \int _{-\infty }^{t}(U(t+h,s)-U(t,s))F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&\quad +2^{p-1}E\bigg \Vert \int _{t}^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&\le 2^{p-1}E\bigg [\bigg (\int _{-\infty }^{-\varepsilon }+\int _{-\varepsilon }^{0}+\int _{0}^{\varepsilon }+\int _{\varepsilon }^{t}\bigg ) \Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\quad +2^{p-1}E\bigg \Vert \int _{t}^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&\le 8^{p-1}\bigg \{E\bigg [\int _{-\infty }^{-\varepsilon }\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\quad + E\bigg [\int _{-\varepsilon }^{0}\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\quad + E\bigg [\int _{0}^{\varepsilon }\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\quad +E\bigg [\int _{\varepsilon }^{t}\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p\bigg \} \\&\quad +2^{p-1}E\bigg \Vert \int _{t}^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\&:=8^{p-1}\big [M_{311}(t,h,\varepsilon )+M_{312}(t,h,\varepsilon )+M_{313}(t,h,\varepsilon )+M_{314}(t,h,\varepsilon )\big ]\\&\quad +2^{p-1}M_{32}(t,h). \end{aligned}$$

Using the H\(\ddot{o}\)lder inequality and Proposition 2.1, we can derive that

$$\begin{aligned} \begin{aligned} M_{311}(t,h,\varepsilon )&= E\bigg [\int _{-\infty }^{-\varepsilon }\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&=E\bigg [\int _{-\infty }^{-\varepsilon }\big \Vert (U(t+h,t)-I)U\left( t,\frac{t+s}{2}\right) U\left( \frac{t+s}{2},s\right) F(s,X(s))\big \Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\le E\bigg [\int _{-\infty }^{-\varepsilon }\big \Vert U\left( t+h,\frac{t+s}{2}\right) -U\left( t,\frac{t+s}{2}\right) \big \Vert Me^{-\frac{t-s}{2}\delta }\Vert F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\le (K_1Mh^{\theta })^pL_1\bigg [\int _{-\infty }^{-\varepsilon } e^{-{\frac{q}{2p}}\delta (t-s)}\textrm{d}s\bigg ]^{\frac{p}{q}} \\&\quad \int _{-\infty }^{-\varepsilon }e^{-{\frac{p}{2q}}\delta (t-s)}\bigg |\frac{t-s}{2}\bigg |^{-p\theta } E(1+\Vert X(s)\Vert _{\mathbb H})^{p}\textrm{d}s \\&\le 2^{p+p\theta -1}(K_1Mh^{\theta })^pL_1{\bigg (\frac{2p}{q\delta }\bigg )}^{\frac{p}{q}} e^{-\frac{\delta }{2}(t+\varepsilon )} \\&\quad \int _{0}^{+\infty }e^{-{\frac{p}{2q}}\delta s}s^{-p\theta }\textrm{d}s\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg ) \\&\le 2^{p}(K_1Mh^{\theta })^pL_1{\bigg (\frac{2p}{q\delta }\bigg )}^{\frac{p}{q}} e^{-\frac{\delta }{2}(t+\varepsilon )}\left( \frac{q}{p\delta }\right) ^{1-p\theta } \\&\quad \Gamma (1-p\theta )\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg ), \\ \end{aligned} \end{aligned}$$

in which \(\theta \in (0,1)\) is from Proposition 2.1 so that \(1-p\theta >0\). Thus, \(M_{311}(t,h,\varepsilon )\rightarrow 0\) as \(\varepsilon ,h\rightarrow 0\). For \(M_{314}(t,h,\varepsilon )\), it can be similarly estimated that

$$\begin{aligned} M_{314}(t,h,\varepsilon )&= E\bigg [\int _{\varepsilon }^{t}\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\le 2^{p}(K_1Mh^{\theta })^pL_1{\bigg (\frac{2p}{q\delta }\bigg )}^{\frac{p}{q}} \bigg (1-e^{-\frac{q}{2p}\delta (t-\varepsilon )}\bigg )^{\frac{p}{q}}\left( \frac{q}{p\delta }\right) ^{1-p\theta }\\&\quad \Gamma (1-p\theta )\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg ), \end{aligned}$$

which implies \(M_{314}(t,h,\varepsilon )\rightarrow 0\) as \(\varepsilon ,h\rightarrow 0\).

Next we consider the estimate of \(M_{313}(t,h,\varepsilon )\) and \(M_{312}(t,h,\varepsilon )\).

$$\begin{aligned} M_{313}(t,h,\varepsilon )&=E\bigg [\int _{0}^{\varepsilon }\Vert (U(t+h,s)-U(t,s))F(s,X(s))\Vert _{\mathbb H}\textrm{d}s\bigg ]^p \\&\le \bigg [\int _{0}^{\varepsilon }1^q\textrm{d}s \bigg ]^{\frac{p}{q}}E\bigg [\int _{0}^{\varepsilon }\Vert U(t+h,s)-U(t,s)\Vert ^p\Vert F(s,X(s))\Vert ^p_{\mathbb H}\textrm{d}s\bigg ] \\&\le \varepsilon ^{p-1}\int _{0}^{\varepsilon }(2M)^pL_1e^{-p\delta (t-s)} E(1+\Vert X(s)\Vert _{\mathbb H})^{p}\textrm{d}s \\&\le \varepsilon ^{p-1} \frac{(4M)^pL_1}{p\delta }\bigg (e^{-p\delta (t-\varepsilon )}-e^{-p\delta t}\bigg )\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}^p\bigg ). \end{aligned}$$

Clearly, when \(\varepsilon \rightarrow 0^+\), \(M_{313}(t,h,\varepsilon )\rightarrow 0\). In the similar way, we get that \(M_{312}(t,h,\varepsilon )\rightarrow 0\) as \(h\rightarrow 0\).

Note that

$$\begin{aligned} M_{32}(t,h)=&E\bigg \Vert \int _{t}^{t+h}U(t+h,s)F(s,X(s))\textrm{d}s\bigg \Vert _{\mathbb H}^p \\ \le&M^pL_1\bigg [\int _{t}^{t+h} e^{-{\frac{q}{p}}\delta (t+h-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}\int _{t}^{t+h}e^{-{\frac{p}{q}}\delta (t+h-s)} E(1+\Vert X(s)\Vert _{\mathbb H})^{p}\textrm{d}s \\ \le&2^{p-1}(Mh)^pL_1\bigg (1+\sup \limits _{t\in \mathbb R}E\Vert X(t)\Vert _{\mathbb H}\bigg ), \end{aligned}$$

so we have readily \( M_3(t,h)\rightarrow 0\) as \(h\rightarrow 0\).

Applying Lemma 2.3 again, we find

$$\begin{aligned} M_{4}(t,h)=&E\bigg \Vert \int _{-\infty }^{t+h}U(t+h,s)G(s,X(s))\textrm{d}W(s)-\int _{-\infty }^tU(t,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _{\mathbb H}^p \\ =&E\bigg \Vert \int _{-\infty }^{t}\bigg (U\left( t+h,\frac{t+s}{2}\right) -U\left( t,\frac{t+s}{2}\right) \bigg )U\left( \frac{t+s}{2},s\right) G(s,X(s))\textrm{d}W(s) \\&\quad +\int _{t}^{t+h}U(t+h,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _{\mathbb H}^p \\ \le&2^{p-1}C_p\bigg \{E\bigg [\int _{-\infty }^{t}\bigg \Vert \bigg (U\left( t+h,\frac{t+s}{2}\right) \\&\quad -U\left( t,\frac{t+s}{2}\right) \bigg ) U\left( \frac{t+s}{2},s\right) G(s,X(s))\bigg \Vert _{\mathscr {L}_2^0}^2\textrm{d}s\bigg ]^{\frac{p}{2}} \\&\quad + E\bigg [\int _{t}^{t+h}\Vert U(t+h,s)G(s,X(s))\Vert _{\mathscr {L}_2^0}^2\textrm{d}s\bigg ]^{\frac{p}{2}}\bigg \}, \end{aligned}$$

which shows \(M_4(t,h)\rightarrow 0\) as \(h\rightarrow 0\). Hence, \(E\Vert (\Phi X)(t+h)-(\Phi X)(t)\Vert _{\mathbb H}^p \rightarrow 0\) as \(h\rightarrow 0\) indicating that \(\Phi (\mathbb {X})\subset \mathbb X\).

We now prove that \(\Phi \) is a contraction mapping. Let \(X_1,X_2\in \mathbb X\), then in the same way as estimates in (7)–(8) we have, for \(p>2\),

$$\begin{aligned}&E\Vert (\Phi X_1)(t)-(\Phi X_2)(t)\Vert _\mathbb H^p \\&\quad \le 2^{p-1}E\bigg \Vert \int _{-\infty }^tU(t,s)[F(s,X_1(s))-F(s,X_2(s))]\textrm{d}s\bigg \Vert _\mathbb H^p \\&\qquad +2^{p-1}E\bigg \Vert \int _{-\infty }^t U(t,s)[G(s,X_1(s))-G(s,X_2(s))]\textrm{d}W(s)\bigg \Vert _\mathbb H^p \\&\quad \le 2^{p-1}M^{p} \bigg [\int _{-\infty }^{t} e^{-{\frac{q}{p}}\delta (t-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}\int _{-\infty }^{t}e^{-{\frac{p}{q}}\delta (t-s)} L_2E\Vert X_1(s)-X_2(s)\Vert _{\mathbb H}^{p}\textrm{d}s \\&\qquad +2^{p-1}C_pM^{p} \bigg [\int _{-\infty }^{t} e^{{\frac{2}{p-2}\delta (t-s)}} \textrm{d}s\bigg ]^{\frac{p-2}{2}} \int _{-\infty }^{t} e^{-{\frac{p}{q}}\delta (t-s)} L_2E\Vert X_1(s)-X_2(s)\Vert _\mathbb H^p \textrm{d}s \\&\quad \le 2^{p-1}M^{p}L_2\frac{q}{p\delta } \bigg [ \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]\Vert X_1-X_2\Vert _\mathbb X^p. \end{aligned}$$

In the case of \(p=2\), similar to (9), there holds

$$\begin{aligned} \Vert \Phi X_1-\Phi X_2\Vert _\mathbb X^2\le&M^2L_2\frac{2+\delta }{\delta ^2}\Vert X_1-X_2\Vert _\mathbb X^2. \end{aligned}$$

Therefore, according to \((H_5)\), \(\Phi \) is a contraction mapping and, by virtue of Banach fixed point principle, \(\Phi \) has a unique fixed point \(X(\cdot )\) in \(\mathbb X\) which is obviously the unique solution of Eq. (3) in \(\mathbb X\).

Finally, it remains for us to show that \(X(\cdot )\) is a p-th Besicovitch almost periodic solution in distribution. From (5), it follows that

$$\begin{aligned} X(t+\tau )=&\int _{-\infty }^{t+\tau }U(t+\tau ,s)F(s,X(s))\textrm{d}s+\int _{-\infty }^{t+\tau } U(t+\tau ,s)G(s,X(s))\textrm{d}W(s)\\ =&\int _{-\infty }^{t}U(t+\tau ,s+\tau )F(s+\tau , X(s+\tau ))\textrm{d}s \\&+\int _{-\infty }^{t} U(t+\tau ,s)G(s+\tau , X(s+\tau ))\textrm{d}\big [W(s+\tau )-W(\tau )\big ]. \end{aligned}$$

Since \(W(s+\tau )-W(\tau )\) is a Brownian motion with the same distribution as W(s), we consider

$$\begin{aligned} \bar{X}(t+\tau )=&\int _{-\infty }^{t}U(t+\tau ,s+\tau )F(s+\tau , \bar{X}(s+\tau ))\textrm{d}s \nonumber \\&\quad +\int _{-\infty }^{t} U(t+\tau ,s)G(s+\tau , \bar{X}(s+\tau ))\textrm{d}W(s). \end{aligned}$$
(10)

Obviously, the above equation satisfies the hypotheses \((H_1)-(H_5)\) as well and so we can prove that \(\bar{X}(t+\tau )\) does exist and belongs to \( C_b(\mathbb R, L^{p}(\Omega ,\mathbb H))\); that is, there exists a constant \(\iota >0\) such that \(\sup \limits _{t\in \mathbb R}E\Vert \bar{X}(t)\Vert _{\mathbb H}^p<\iota ^p\). Combining now (5) and (10) implies that

$$\begin{aligned}&\frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _\mathbb H^p\textrm{d}t \nonumber \\&\quad =\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}U(t+\tau ,s+\tau )F(s+\tau ,\bar{X}(s+\tau ))\textrm{d}s \nonumber \\&\qquad +\int _{-\infty }^{t} U(t+\tau ,s)G(s+\tau ,\bar{X}(s+\tau ))\textrm{d}W(s)\nonumber \\&\qquad -\int _{-\infty }^{t}U(t,s)F(s,X(s))\textrm{d}s -\int _{-\infty }^{t} U(t,s)G(s,X(s))\textrm{d}W(s)\bigg \Vert _\mathbb H^p\textrm{d}t \nonumber \\&\quad =\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}[U(t+\tau ,s+\tau )-U(t,s)]F(s+\tau ,\bar{X}(s+\tau ))\textrm{d}s\nonumber \\&\qquad +\int _{-\infty }^{t}U(t,s)[F(s+\tau ,\bar{X}(s+\tau ))-F(s,\bar{X}(s+\tau ))]\textrm{d}s\nonumber \\&\qquad +\int _{-\infty }^{t}U(t,s)[F(s,\bar{X}(s+\tau ))-F(s,X(s))]\textrm{d}s\nonumber \\&\qquad +\int _{-\infty }^{t}[U(t+\tau ,s+\tau )-U(t,s)]G(s+\tau ,\bar{X}(s+\tau ))\textrm{d}W(s)\nonumber \\&\qquad +\int _{-\infty }^{t}U(t,s)[G(s+\tau ,\bar{X}(s+\tau ))-G(s,\bar{X}(s+\tau ))]\textrm{d}W(s)\nonumber \\&\qquad +\int _{-\infty }^{t}U(t,s)[G(s,\bar{X}(s+\tau ))-G(s,X(s))]\textrm{d}W(s)\bigg \Vert _\mathbb H^p\textrm{d}t \nonumber \\&\quad \le 6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t} [U(t+\tau ,s+\tau )-U(t,s)]F(s+\tau ,\bar{X}(s+\tau ))\textrm{d}s\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\qquad +6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}U(t,s)[F(s+\tau ,\bar{X}(s+\tau ))-F(s,\bar{X}(s+\tau ))]\textrm{d}s\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\qquad +6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}U(t,s)[F(s,\bar{X}(s+\tau ))-F(s,X(s))]\textrm{d}s\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\qquad +6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}[U(t+\tau ,s+\tau )-U(t,s)]G(s+\tau ,\bar{X}(s+\tau ))\textrm{d}W(s)\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\qquad +6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}U(t,s)[G(s+\tau ,\bar{X}(s+\tau ))-G(s,\bar{X}(s+\tau ))]\textrm{d}W(s)\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\qquad +6^{p-1}\frac{1}{2l}\int _{-l}^{l}E \bigg \Vert \int _{-\infty }^{t}U(t,s)[G(s,\bar{X}(s+\tau ))-G(s,X(s))]\textrm{d}W(s)\bigg \Vert _\mathbb H^p\textrm{d}t\nonumber \\&\quad :=\sum \limits _{i=1}^{6}6^{p-1}N_{i}. \end{aligned}$$
(11)

Next we estimate these terms, respectively. First applying Lemma 3.1 and the H\(\ddot{o}\)lder inequality, we get

$$\begin{aligned} N_1\le&\frac{1}{2l}\int _{-l}^{l}E\bigg \{\bigg [\int _{-\infty }^{t}\Vert U(t+\tau ,s+\tau )-U(t,s)\Vert ^{\frac{q}{p}}\textrm{d}s\bigg ]^{\frac{p}{q}}\nonumber \\&\times \bigg [\int _{-\infty }^{t}\Vert U(t+\tau ,s+\tau )-U(t,s)\Vert ^{\frac{p}{q}}\Vert F(s+\tau ,\bar{X}(s+\tau ))\Vert _{\mathbb H}^{p}\textrm{d}s \bigg ]\bigg \}\textrm{d}t\nonumber \\ \le&{\varepsilon }^{p}\frac{1}{2l}\int _{-l}^{l} \bigg [\int _{-\infty }^{t} e^{-{\frac{q}{2p}}\delta (t-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}E\bigg [\int _{-\infty }^{t}e^{-{\frac{p}{2q}}\delta (t-s)} L_1(1+\Vert \bar{X}(s+\tau )\Vert _{\mathbb H})^{p}\textrm{d}s\bigg ]\textrm{d}t \nonumber \\ \le&2^{p-1}{\varepsilon }^{p} L_1 \bigg (\frac{2p}{q\delta }\bigg )^{\frac{p}{q}} \frac{1}{2l}\int _{-l}^{l}\int _{-\infty }^{t}e^{-{\frac{p}{2q}}\delta (t-s)} {(1+\iota ^p)}\textrm{d}s\textrm{d}t\nonumber \\ =&2^{p-1}{\varepsilon }^{p} L_1 \bigg (\frac{2p}{q\delta }\bigg )^{\frac{p}{q}}\frac{2q}{p\delta } {(1+\iota ^p)} . \end{aligned}$$
(12)

Then, combining the derivation methods of (8) and (9) there hold, for \(p>2\),

$$\begin{aligned} N_{4}&\le 2^{p-1}{\varepsilon }^{p}L_1C_p\bigg (\frac{p-2}{\delta }\bigg )^{\frac{p-2}{2}} \frac{2q}{{p}\delta } {(1+\iota ^p)}, \end{aligned}$$
(13)

and, for \(p=2\),

$$\begin{aligned} N_{4}=&\frac{1}{2l}\int _{-l}^{l} E\bigg \Vert \int _{-\infty }^t [U(t+\tau ,s+\tau )-U(t,s)]G(s+\tau ,\bar{X}(s+\tau ))\textrm{d}W(s) \bigg \Vert _{\mathbb H}^2\textrm{d}t\nonumber \\ \le&\frac{1}{2l}\int _{-l}^{l}E\bigg [ \int _{-\infty }^t \Vert U(t+\tau ,s+\tau )-U(t,s)\Vert ^2 \Vert G(s+\tau ,\bar{X}(s+\tau ))\Vert _{\mathscr {L}_0^2}^2\textrm{d}s \bigg ] \textrm{d}t \nonumber \\ \le&\varepsilon ^2\frac{1}{2l}\int _{-l}^{l}E\bigg [\int _{-\infty }^t e^{-\frac{\delta }{2}(t-s)} L_1(1+\Vert \bar{X}(s+\tau ) \Vert _\mathbb H)^2\textrm{d}s \bigg ]\textrm{d}t\nonumber \\ \le&2\varepsilon ^2L_1 \frac{1}{2l}\int _{-l}^{l}\int _{-\infty }^t e^{-\frac{\delta }{2}(t-s)} {(1+\iota ^2)} \textrm{d}s\textrm{d}t \nonumber \\ \le&{{\frac{4}{\delta }}} {\varepsilon ^2}L_1{(1+\iota ^2)}. \end{aligned}$$
(14)

Note that, due to \((H_3)\), for \(\varepsilon >0\), there exists a \(\mathcal {l}>0\) such that every interval of length \(\mathcal {l}\) contains a number \(\tau \) such that

$$\begin{aligned}&\frac{1}{2l}\int _{-l}^{l}E\big \Vert F(t,\bar{X}(t+\tau ))-F(t+\tau ,\bar{X}(t+\tau ))\big \Vert ^p_\mathbb H\textrm{d}t<\varepsilon ,\,\, \nonumber \\&\frac{1}{2l}\int _{-l}^{l}E\big \Vert G(t,\bar{X}(t+\tau ))-G(t+\tau ,\bar{X}(t+\tau ))\big \Vert ^p_{\mathscr {L}_2^0}\textrm{d}t<\varepsilon . \end{aligned}$$
(15)

From this, it gives that

$$\begin{aligned} N_2\le&\frac{1}{2l}\int _{-l}^{l}E\bigg \{\bigg [\int _{-\infty }^{t}\Vert U(t,s)\Vert ^{\frac{q}{p}}\textrm{d}s\bigg ]^{\frac{p}{q}} \nonumber \\&\quad \bigg [\int _{-\infty }^{t}\Vert U(t,s)\Vert ^{\frac{p}{q}}\Vert F(s+\tau ,\bar{X}(s+\tau ))-F(s,\bar{X}(s+\tau ))\Vert _{\mathbb H}^{p}\textrm{d}s \bigg ]\bigg \}\textrm{d}t \nonumber \\ \le&M^{p}\frac{1}{2l}\int _{-l}^{l} \bigg [\int _{-\infty }^{t} e^{-{\frac{q}{p}}\delta (t-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}E \nonumber \\&\quad \bigg [\int _{-\infty }^{t}e^{-{\frac{p}{q}}\delta (t-s)} \Vert F(s+\tau ,\bar{X}(s+\tau ))-F(s,\bar{X}(s+\tau ))\Vert _{\mathbb H}^{p} \textrm{d}s\bigg ]\textrm{d}t \nonumber \\ \le&M^{p} \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}\frac{1}{2l}\int _{-l}^{l}\int _{-\infty }^{t}e^{-{\frac{p}{q}}\delta (t-s)} E\Vert F(s+\tau ,\bar{X}(s+\tau ))-F(s,\bar{X}(s+\tau ))\Vert _{\mathbb H}^{p} \textrm{d}s \textrm{d}t\nonumber \\ =&M^{p} \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}\int _{-\infty }^l e^{-\frac{p}{q}\delta (l-s) }\bigg (\frac{1}{2l}\int _{s-2l}^{s}E\Vert F(t+\tau ,\bar{X}(t+\tau ))-F(t,\bar{X}(t+\tau ))\Vert _{\mathbb H}^{p} \textrm{d}t\bigg )\textrm{d}s\nonumber \\ \le&M^p\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}\frac{q}{p\delta }\varepsilon . \end{aligned}$$
(16)

Also using (15), we obtain, for \(N_5\),

$$\begin{aligned} N_5&\le M^p\frac{q}{p\delta } \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}}\varepsilon ,\,\,(p>2), \end{aligned}$$
(17)

and

$$\begin{aligned} N_5&\le \frac{3{M^2\varepsilon } }{\delta } ,\,\,(p=2). \end{aligned}$$
(18)

For \(N_3\), it follows from the Lipschitz condition \((H_{4})\) that

$$\begin{aligned} N_{3} \le&M^{p} L_2 \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \frac{1}{2l}\int _{-l}^{l}\int _{-\infty }^te^{-\frac{p}{q}\delta (t-s) }E\Vert \bar{X}(s+\tau )- X(s)\Vert _{\mathbb H}^p \textrm{d}s\textrm{d}t . \end{aligned}$$

which from Fubini’s theorem implies

$$\begin{aligned} N_{3}&\le M^{p} L_2 \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \int _{-\infty }^l e^{-\frac{p}{q}\delta (l-s) }\bigg (\frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}(t+\tau )- X(t)\Vert _{\mathbb H}^p\textrm{d}t\bigg )\textrm{d}s . \end{aligned}$$
(19)

At last, let us estimate \(N_6\) by \((H_{4})\) again. If \(p>2\), then

$$\begin{aligned} N_{6}&\le M^{p}L_2C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \int _{-\infty }^le^{-\frac{p}{q}\delta (l-s)}\bigg (\frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}(t+\tau )- X(t)\Vert _{\mathbb H}^p\textrm{d}t\bigg )\textrm{d}s. \end{aligned}$$
(20)

For the case \(p=2\), one has

$$\begin{aligned} N_{6}&\le M^{2}L_2 \int _{-\infty }^le^{-\delta (l-s)}\bigg (\frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}(t+\tau )- X(t)\Vert _{\mathbb H}^2\textrm{d}t\bigg )\textrm{d}s. \end{aligned}$$
(21)

Thus, substituting (12)–(21) into (11) yields, for \(p>2\),

$$\begin{aligned}{} & {} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb {H}}^p\textrm{d}t\le \Delta _1{\varepsilon } \\{} & {} \quad +\rho _1\int _{-\infty }^{l}e^{-\frac{p}{q}\delta (l-s)}\bigg (\frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}( {t}+\tau )-X({t})\Vert _{\mathbb H}^p\textrm{d} {t}\bigg )\textrm{d} {s}, \end{aligned}$$

where

$$\begin{aligned} \Delta _1&= 12^{p-1}{\varepsilon }^{p-1}L_1{(1+\iota ^p)} \frac{2q}{ {p}\delta }\bigg [\bigg (\frac{2p}{q\delta }\bigg )^{\frac{p}{q}}+C_p \bigg (\frac{p-2}{\delta }\bigg )^{\frac{p-2}{2}}\bigg ] \\&\quad +6^{p-1}M^{p} \frac{q}{p\delta }\bigg [\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} +\bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ], \\ \rho _1&=6^{p-1}M^{p} L_2\bigg [ \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]. \end{aligned}$$

Since \(\rho _1<\frac{p}{q}\delta \) due to \((H_5)\), we see, for every \(\gamma \in (0,\frac{p}{q}\delta -\rho _1)\),

$$\begin{aligned} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}^p\textrm{d}t{} & {} \le \Delta _1\varepsilon +\rho _1\int _{-\infty }^{l}e^{-\gamma (l-s)}\\{} & {} \quad \left( \frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}( {t}+\tau )-X( {t})\Vert _{\mathbb H}^p\textrm{d} {t}\right) \textrm{d} {s}. \end{aligned}$$

Thus, we deduce by Lemma 2.2 that

$$\begin{aligned} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}^p\textrm{d}t< {\frac{\Delta _1\frac{p}{q}\delta }{\frac{p}{q}\delta -\rho _1}\varepsilon }. \end{aligned}$$
(22)

Likewise, if \(p=q=2\), then

$$\begin{aligned} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )- X(t)\Vert _{\mathbb H}^2\textrm{d}t{} & {} \le \Delta _2{\varepsilon }+\rho _2\int _{-\infty }^{l}e^{-\delta (l-s)}\\{} & {} \quad \left( \frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}({t}+\tau )-X({t})\Vert _{\mathbb H}^2d{t}\right) \textrm{d}{s}, \end{aligned}$$

where

$$\begin{aligned} \Delta _2=&12\varepsilon L_1 {(1+\iota ^2)}\frac{4+\delta }{\delta ^2} +\frac{3M^2(2+\delta )}{\delta ^2}, \\ \rho _2=&6M^2 L_2\frac{1+\delta }{\delta }. \end{aligned}$$

In this case, one has \(\rho _2<\delta \) by \((H_5)\). Hence, for \(\gamma \in (0, \delta -\rho _2)\),

$$\begin{aligned} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}^2\textrm{d}t{} & {} \le \Delta _2\varepsilon +\rho _2\int _{-\infty }^{l}e^{-\gamma (l-s)}\\{} & {} \quad \left( \frac{1}{2l}\int _{s-2l}^{s}E\Vert \bar{X}({t}+\tau )-X({t})\Vert _{\mathbb H}^2\textrm{d}{t}\right) \textrm{d}{s}. \end{aligned}$$

Therefore, we infer for this case that

$$\begin{aligned} \frac{1}{2l}\int _{-l}^{l}E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}^2\textrm{d}t< {{\frac{\Delta _2\delta }{\delta -\rho _2}}\varepsilon }. \end{aligned}$$
(23)

Then,

$$\begin{aligned}&\frac{1}{2l}\int _{-l}^{l}\bigg [d_\mathrm{{BL}}\big (\mu (\bar{X}(t+\tau )),\mu ( X(t))\big )\bigg ]^p\textrm{d}t \\&\quad = \frac{1}{2l}\int _{-l}^{l}\bigg [d_\mathrm{{BL}}\big (P\circ \bar{X}^{-1}(t+\tau ),P\circ X^{-1}(t)\big )\bigg ]^p\textrm{d}t \\&\quad = \frac{1}{2l}\int _{-l}^{l}\bigg [\sup \limits _{\Vert f\Vert _\mathrm{{BL}}\le 1}\bigg |\int _{\mathbb H}fd(P\circ \bar{X}^{-1}(t+\tau )-P\circ X^{-1}(t)) \bigg |\bigg ]^p\textrm{d}t \\&\quad = \frac{1}{2l}\int _{-l}^{l}\bigg [\sup \limits _{\Vert f\Vert _\mathrm{{BL}}\le 1}\bigg |\int _{\Omega }\bigg (f \bar{X}(t+\tau )-fX(t)\bigg )dP \bigg | \bigg ]^p\textrm{d}t \\&\quad \le \frac{1}{2l}\int _{-l}^{l}\bigg [\int _{\Omega }\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}dP\bigg ]^p\textrm{d}t \\&\quad =\frac{1}{2l}\int _{-l}^{l} E\Vert \bar{X}(t+\tau )-X(t)\Vert _{\mathbb H}^p \textrm{d}t, \end{aligned}$$

from which, together with the fact that the distribution of \(\bar{X}(t+\tau )\) is the same as that of \(X(t+\tau )\), we conclude that, as desired, X(t) is a p-th Besicovitch almost periodic solution in distribution of Eq. (3). \(\square \)

4 Global Exponential Stability

In this section, we use some inequality techniques to further prove that the p-th Besicovitch almost periodic solution in distribution of Eq. (3) obtained above is globally exponentially stable in the sense that

Definition 4.1

Let X(t) be a p-th Besicovitch almost periodic solution in distribution of Eq. (3) with the initial value \(X(t_0)\). If there exist constants \(C>0\) and \(\lambda >0\) such that for any solution Y(t) of (3) with initial value \(Y(t_0)\) there holds

$$\begin{aligned} E\Vert Y(t)-X(t)\Vert _{\mathbb H}^p \le CE\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^pe^{\lambda (t-t_0)},\,\, t>t_0, \end{aligned}$$

then X(t) is said to be globally exponentially stable.

We establish the following result for the global exponential stability.

Theorem 4.1

Let \((H_1)-(H_5)\) hold. Then, the p-th Besicovitch almost periodic solution in distribution \(X(\cdot )\) of Eq. (3) is globally exponentially stable.

Proof

Let \(X(\cdot )\) be the p-th Besicovitch almost periodic solution in distribution with initial value \(X(t_0)\) of Eq. (3) obtained in Theorem 3.1. Suppose that Y is an arbitrary solution of Eq. (3) with the initial value \(Y(t_0)\). Set \(Z=Y-X\), then it verifies

$$\begin{aligned} d{Z}(t)= A(t)Z(t)\textrm{d}t+[F(t,X(t))-F(t,Y(t))]\textrm{d}t+[G(t,X(t))-G(t,Y(t))]\textrm{d}W(t). \end{aligned}$$
(24)

We first consider two real linear continuous functions \(\Gamma _1(\cdot )\) and \(\Gamma _2(\cdot ): [0,+\infty )\rightarrow \mathbb R\) given, respectively, by

$$\begin{aligned} \Gamma _1(x)=&\frac{p(\delta -x)}{q}-3^{p-1}M^{p}L_2 \bigg [\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]\,\,(p>2), \\ \Gamma _2(x)=&\delta -x-\frac{6M^2L_2 (1+\delta )}{\delta }\,\,(p=2). \end{aligned}$$

Sine \(\Gamma _1(0),\Gamma _2(0)>0\) by \((H_5)\), and

\(\Gamma _1(\varsigma _1)=\Gamma _2(\varsigma _2)=0\), where

$$\begin{aligned}&\varsigma _1=\delta - \frac{q}{p}3^{p-1}M^{p}L_2\bigg [ \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]>0,\\&\varsigma _2=\delta -\frac{6M^2L_2 (1+\delta )}{\delta } >0, \end{aligned}$$

and there is \(\varsigma =\min \{\varsigma _1,\varsigma _2,\delta \}\) such that \(\Gamma _1(x),~\Gamma _2(x)>0\) for \(x\in (0,\varsigma )\). So we can choose \(0<\lambda <\varsigma \) such that \(\Gamma _1(\lambda ),~\Gamma _2(\lambda )>0\) and hence

$$\begin{aligned}&\frac{q}{p(\delta -\lambda )} \gamma ^1:=\frac{q}{p(\delta -\lambda )} 3^{p-1}M^{p}L_2\bigg [ \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]<1 ~~(p>2),\\&\frac{1}{\delta -\lambda } \gamma ^2:=\frac{6 L_2 M^2(1+\delta )}{\delta (\delta -\lambda )}<1 ~~(p=2). \end{aligned}$$

Put \(C_1= \frac{p\delta }{q\gamma ^1}\) and \(C_2= \frac{\delta }{\gamma ^2}\), then \(C_1\), \(C_2>1\), and it is easily seen that

$$\begin{aligned}&\frac{1}{C_1}-\frac{q}{p(\delta -\lambda )}3^{p-1} L_2 M^{p}\bigg [ \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]<0~~(p>2),\\&\frac{1}{C_2}-\frac{6 L_2 M^2(1+\delta )}{\delta (\delta -\lambda )}<0~~ (p=2). \end{aligned}$$

Now, we return to estimate \(E\Vert Z(t)\Vert _{\mathbb H}^{p}\). Let \(C=\max \left\{ M^p C_1, M^2C_2\right\} \), then \(C>1\). We will prove that, for any \(\varepsilon >0\),

$$\begin{aligned} E\Vert Z(t)\Vert _{\mathbb H}^p < C\left( E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon \right) e^{-\frac{p}{q}\lambda (t-t_0)},\,\, \text{ for } \text{ all }~ t\ge t_0. \end{aligned}$$
(25)

Apparently,

$$E\Vert Z(t_0)\Vert _{\mathbb H}^p< E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon .$$

Suppose that there is some \(t_1>t_0\) such that

$$\begin{aligned} E\Vert Z(t_1)\Vert _{\mathbb H}^p = C\left( E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon \right) e^{-\frac{p}{q}\lambda (t_1-t_0)}, \end{aligned}$$
(26)

and

$$\begin{aligned} E\Vert Z(t)\Vert _{\mathbb H}^p< C\left( E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon \right) e^{-\frac{p}{q}\lambda (t-t_0)},\,\,\text {for}~~ t_0\le t<t_1. \end{aligned}$$
(27)

Because Z(t) satisfies

$$\begin{aligned} Z(t)=&Z(t_0)U(t,t_0)+\int _{t_0}^tU(t,s)[F(s,X(s))-F(s,Y(s))]\textrm{d}s\\&+\int _{t_0}^tU(t,s)[G(s,X(s))-G(s,Y(s))]\textrm{d}W(s), \end{aligned}$$

one has

$$\begin{aligned} { {E\Vert Y(t_1)-X(t_1)\Vert _\mathbb H^p}} \le&3^{p-1}E\big \Vert (Y(t_0)-X(t_0))U(t_1,t_0) \big \Vert _\mathbb H^p\nonumber \\&+3^{p-1} E\bigg \Vert \int _{t_0}^{t_1}U(t_1,s)[F(s,X(s))-F(s,Y(s))]\textrm{d}s \bigg \Vert _\mathbb H^p\nonumber \\&+3^{p-1}E\bigg \Vert \int _{t_0}^{t_1}U(t_1,s)[G(s,X(s))-G(s,Y(s))]\textrm{d}W(s) \bigg \Vert _\mathbb H^p\nonumber \\ :=&3^{p-1}[\Lambda _{1}+\Lambda _{2}+\Lambda _{3}]. \end{aligned}$$
(28)

From (26) and (27), it follows easily that

$$\begin{aligned} \Lambda _{2}\le&M^{p} E\bigg \{\bigg [\int _{t_0}^{t_1}e^{-\frac{q}{p}\delta (t_1-s)}\textrm{d}s\bigg ]^{\frac{p}{q}}\bigg [\int _{t_0}^{t_1} e^{-\frac{p}{q}\delta (t_1-s)}\Vert F(s,X(s))-F(s,Y(s))\Vert _\mathbb H^p\textrm{d}s\bigg ]\bigg \} \nonumber \\ \le&M^{p}L_2\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \int _{t_0}^{t_1}e^{-\frac{p}{q}\delta (t_1-s)}E\Vert Y(s)-X(s) \Vert _{\mathbb H}^p\textrm{d}s\nonumber \\ \le&M^{p}L_2\bigg (\frac{p}{q \delta }\bigg )^{\frac{p}{q}} \int _{t_0}^{t_1}e^{-\frac{p}{q}\delta (t_1-s)}e^{-\frac{p}{q}\lambda (s-t_0)}dsC( E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon )\nonumber \\ \le&M^{p}L_2\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \int _{t_0}^{t_1}e^{-\frac{p}{q}(t_1-s)(\delta -\lambda )}dsC(E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon )e^{-\frac{p}{q}\lambda (t_1-t_0)}. \end{aligned}$$
(29)

When \(p>2\), we can get

$$\begin{aligned} \Lambda _{3}&\le M^{p}L_2C_p \bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \int _{t_0}^{t_1} e^{-\frac{p}{q}(t_1-s) (\delta -\lambda ) }dsC(E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon )\nonumber \\&\quad e^{-\frac{p}{q}\lambda (t_1-t_0)}, \end{aligned}$$
(30)

and for \(p=q=2\), it can be readily shown that

$$\begin{aligned} \Lambda _{3} \le&M^2 L_2 \int _{t_0}^{t_1} e^{-(t_1-s)(\delta -\lambda )}dsC(E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^2+\varepsilon )e^{-\lambda (t_1-t_0)}. \end{aligned}$$
(31)

Then substituting (29)–(31) into (28), one obtains, for \(p>2\),

$$\begin{aligned}&{ {E\Vert Y(t_1)-X(t_1)\Vert _\mathbb H^p}} \\&\quad \le 3^{p-1} \bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^p+\varepsilon \bigg )e^{-p\delta (t_1-t_0)}M^p \\&\qquad + C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^p+\varepsilon \bigg )e^{-\frac{p}{q}\lambda (t_1-t_0)}\int _{t_0}^{t_1}e^{-\frac{p}{q}(t_1-s)(\delta -\lambda )}\textrm{d}s \\&\qquad \times 3^{p-1}M^{p}L_2\bigg [\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+C_p\bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ]\\&\quad \le 3^{p-1}C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^p+\varepsilon \bigg )e^{- {\frac{p}{q}}\lambda (t_1-t_0)} \\&\qquad \bigg \{\frac{e^{ ( {\frac{p}{q}} \lambda -p\delta )(t_1-t_0)}M^p}{C}+\frac{q M^pL_2}{p(\delta -\lambda )}\bigg (1-e^{\frac{p}{q}(\lambda -\delta )(t_1-t_0)}\bigg ) \bigg [\bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}} \\&\qquad +C_p\bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg ] \bigg \} \\&\quad \le 3^{p-1}C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^p+\varepsilon \bigg )e^{- {\frac{p}{q}}\lambda (t_1-t_0)} \\&\qquad \bigg \{e^{ {\frac{p}{q}}(\lambda -\delta )(t_1-t_0)}\bigg [\frac{1}{C_1}-\frac{qM^{p}L_2}{p(\delta -\lambda )}\bigg ( \bigg (\frac{p}{q\delta }\bigg )^{\frac{p}{q}}+C_p\bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \bigg )\bigg ] \\&\qquad +\frac{qL_2M^{p}}{p(\delta -\lambda )}\bigg [ \bigg ( \frac{p}{q\delta }\bigg )^{\frac{p}{q}}+ C_p\bigg (\frac{p-2}{2\delta }\bigg )^{\frac{p-2}{2}} \ \bigg ]\bigg \} \\&\quad <C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon \bigg )e^{{-}{\frac{p}{q}}\lambda (t_1-t_0)}, \end{aligned}$$

and for \(p=2\),

$$\begin{aligned}&{ {E\Vert Y(t_1)-X(t_1)\Vert _\mathbb H^2}} \\&\quad \le 3 \bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^2+\varepsilon \bigg )e^{-2\delta (t_1-t_0)}M^2 \\&\qquad + C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^2+\varepsilon \bigg )e^{- \lambda (t_1-t_0)}\int _{t_0}^{t_1}e^{- (t_1-s)(\delta -\lambda )}\textrm{d}s \frac{1+\delta }{\delta }3M^2L_2 \\&\quad \le 6C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^2+\varepsilon \bigg )e^{- { \lambda (t_1-t_0)}} \\&\qquad \quad \bigg [\frac{e^{ ( \lambda -2\delta )(t_1-t_0)}M^2}{C}+\frac{M^2 L_2 }{(\delta -\lambda )}\bigg (1-e^{ (\lambda -\delta )(t_1-t_0)}\bigg )\frac{1+\delta }{\delta } \bigg ]\\&\quad \le 6C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _\mathbb H^2+\varepsilon \bigg )e^{-\lambda (t_1-t_0)} \\&\qquad \quad \bigg \{e^ {(\lambda -\delta )(t_1-t_0)}\bigg [\frac{1}{C_2}-\frac{M^2 L_2(1+\delta )}{(\delta -\lambda )\delta }\bigg ] +\frac{ M^2L_2(1+\delta )}{(\delta -\lambda )\delta } \bigg \} \\&\quad < C\bigg (E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^2+\varepsilon \bigg )e^{- \lambda (t_1-t_0)}. \end{aligned}$$

Therefore,

$$\begin{aligned} E\Vert Z(t_1)\Vert _{\mathbb H}^p<C\bigg ( E\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^p+\varepsilon \bigg )e^{-\frac{p}{q}\lambda (t_1-t_0)}, \end{aligned}$$

which contradicts (26). Hence, (25) holds. Letting \(\varepsilon \rightarrow 0^+\), we obtain immediately

$$\begin{aligned} E\Vert Z(t_1)\Vert _{\mathbb H}^p\le CE\Vert X(t_0)-Y(t_0)\Vert _{\mathbb H}^pe^{-\frac{p}{q}\lambda (t_1-t_0)},\,\, t>t_0. \end{aligned}$$

Consequently, the p-th Besicovitch almost periodic solution in distribution of Eq. (3) is globally exponentially stable. \(\square \)

5 An Example

In this part, we provide an example to illustrate the above obtained results. Consider the following stochastic boundary value problem:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial }{\partial t}X(t,x)=a(t)\Delta X(t,x)+\frac{\cos \sqrt{5}t+\sin \sqrt{3}t}{400}\sin (X(t,x))\textrm{d}t \\ {}&\qquad +\frac{\cos (X(t,x)+\sin 5t)}{400}\textrm{d}W(t),~~t\in \mathbb R, ~~x\in \mathscr {O}, \\&X(t,x)=0,~~t\in \mathbb R, ~~x\in \partial \mathscr {O}, \end{aligned}\right. \end{aligned}$$
(32)

where \(\mathscr {O}\subset \mathbb R^n\) is a bounded domain with boundary \(\partial \mathscr {O}\) of Class \(C^2\) being locally on one side of \(\mathscr {O}\). The function \(a(t):\mathbb R\rightarrow \mathbb R^+\) is H\(\ddot{o}\)lder continuous in t with parameter \(0<\mu <1\), \(a(t_0)<1\) (for some \(t_0\in \mathbb R\)) and \(a(t)\in AP(\mathbb R, \mathbb R)\). In addition, W(t) is a real-valued Brownian motion.

To apply the preceding obtained results to Eq. (32), we need to rewrite it into the abstract form of Eq. (3). For this, let \(\mathbb H=L^2(\mathscr {O})\) and take \(\mathbb K=\mathbb K_0=\Omega =\mathbb R,Q=I\) (the identity operator on \(\mathbb R\)). Define the family of operators A(t) by

$$\begin{aligned} A(t)f(x)=a(t)\Delta f(x) \end{aligned}$$

with the domain \(D(A(t))=H^{2}(\mathscr {O})\cap H_0^1(\mathscr {O})\). We also introduce the functions \(F(\cdot ,\cdot ):\mathbb R\times L^p(\Omega ,\mathbb H)\rightarrow L^p(\Omega ,\mathbb H)\) and \(G(\cdot ,\cdot ):\mathbb R\times L^p(\Omega ,\mathbb H)\rightarrow L^p(\Omega ,\mathscr {L}_2^0)\) as

$$\begin{aligned} F(t,X(t,x))=\frac{\cos \sqrt{5}t+\sin \sqrt{3}t}{400}\sin (X(t,x)), \end{aligned}$$

and

$$\begin{aligned} G(t,X(t,x))=\frac{\cos (X(t,x)+\sin 5t)}{400}. \end{aligned}$$

There by these notations (32) turns out to be the form of non-autonomous stochastic evolution Eq. (3) on the probability space \((\mathbb R,\mathscr {B}(\mathbb R),\mathscr {F}_t, P)\) as

$$\begin{aligned} \frac{\partial }{\partial t}X(t,x)=A(t)X(t,x)+F(t,X(t,x))\textrm{d}t+G(t,X(t,x))\textrm{d}W(t), ~~t\in \mathbb {R},~~u\in \mathscr {O}. \end{aligned}$$

In what follows, we examine for Eq. (32) that all the conditions of Theorems 3.1 and 4.1 are well fulfilled so that we are able to apply them to obtain the existence and global exponential stability of the p-th Besicovitch almost periodic solution in distribution for this system.

First of all, according to Prato et al. [26] and Yagi [29], under the assumptions about a(t), A(t) generates a unique evolution operator \(\{U(t,s):-\infty<s\le t<+\infty \}\) on \(\mathbb H\) which satisfies \(\Vert U(t,s) \Vert \le Me^{-\delta (t-s)}\) with \(M=\delta =1\) and \(R(\lambda _0,A(t)) \) is almost periodic. Thus, \((H_1)\) and \((H_2)\) hold true.

On the other hand, by a simple calculation we find that, for \(X_1,X_2\in L^p(\Omega ,\mathbb H)\),

$$\begin{aligned}{} & {} \Vert F(t,X_1)-F(t,X_2)\Vert _{\mathbb H}\le \bigg |\frac{\cos \sqrt{5}t+\sin \sqrt{3}t}{400}\bigg | \Vert \sin (X_1(t,x)) \\{} & {} \quad -\sin (X_2(t,x))\Vert _{\mathbb H}\le \frac{1}{200}\Vert X_1-X_2\Vert _{\mathbb H}, \end{aligned}$$

and

$$\begin{aligned} \Vert F(t,X)\Vert _{\mathbb H}\le \bigg |\frac{\cos \sqrt{5}t+\sin \sqrt{3}t}{400}\bigg |\Vert \sin (X(t,x))\Vert _{\mathbb H}\le \frac{1}{200}\left( 1+\Vert X\Vert _{\mathbb H}\right) , \end{aligned}$$

which show that \(L_1=L_2=\frac{1}{200}\). In addition, F(t, X) and G(t, X) are clearly p-th Besicovitch almost periodic in \(t\in \mathbb R\) uniformly on any compact subset of \(L^p(\Omega ,\mathbb H)\). Hence, the assumptions \((H_3)\) and \((H_4)\) are satisfied as well.

Finally, it is easy to compute that for \(p=3\), \(r_1=0.9675<1\) and for \(p=2\), \(r_2=0.06 <1.\)

Therefore, all of the conditions of Theorem 3.1 and Theorem 4.1 are satisfied. This allows us to infer that Eq. (32) has a unique p-th Besicovitch almost periodic solution in distribution \(X(\cdot ,\cdot )\) and it is globally exponentially stable.