Dynamic Behavior of Pseudo Almost Periodic Functions to Stochastic Differential Equations

Abstract

The main aim of this study is to create appropriate criteria for the existence of a unique $\mu$-pseudo almost periodic solution to a particular type of stochastic differential equation, utilizing Bochner’s double sequences criterion, improved Gronwall’s lemma, Hölder’s inequality, and measure theory techniques. By applying the inequalities analysis condition and the fixed point theorem for contraction mapping, we can establish the existence of a single $\mu$-pseudo almost periodic solution in distribution to the given stochastic equation. Finally, we use an example to demonstrate our stochastic processes.

1. Introduction

In the real world, there are events that are inevitable or impossible. Random events lie between inevitable and impossible events. Stochastic analysis and stochastic differential equations have experienced significant advancements in the past few years. The application of stochastic differential equations is prevalent across various disciplines, including economics, biology, engineering, and more. In the realm of economics, stochastic differential equations are employed to address the issue of determining the cost of options. In random events such as product sales and market prices, a random variable can be determined based on a large number of experimental data, and a mathematical model using stochastic differential equations can be established with initial conditions, so as to deduce the overall development law (see [1]). In the field of biology, it is utilized to uncover the patterns of disease occurrence, the transmission and prevalence processes of diseases, as well as the evolutionary mechanisms of tumors. Additionally, it can be employed to describe the growth, development, and reproduction processes within biological systems, and to investigate the dynamic behavior of these systems, thereby revealing certain biological laws (see [2]). In engineering, stochastic differential equations are used to analyze and design control systems, especially in the presence of random disturbances. These equations can help engineers design more robust and efficient control systems (see [3]). Stochastic differential equations have widespread applications in various fields, enabling researchers and practitioners to better understand and model complex systems with uncertainties, leading to advancements in many areas of science, technology, and industry.

The study of asymptotic periodic solutions for stochastic differential equations (SDEs) has been a significant area of research since the 1980s. Early developments included the work of Bezandry and Diagana, who investigated square-mean almost periodic solutions for a specific class of SDEs [4,5]. A foundational contribution from Chérif [6] later introduced the concept of quadratic-mean pseudo almost periodicity. Building on this, Diop et al. [7] presented the notion of a $\mu$-pseudo almost periodic process, which was subsequently applied to establish the existence of solutions for stochastic evolution equations [8,9,10].

More recently, scholarly attention has turned to almost periodic solutions in the distributional sense. For instance, Li et al. [11] utilized Lyapunov functions to analyze the stability and almost periodicity of SDE solutions by examining their inheritance properties. Yan et al. [12,13] employed interpolation theory alongside stochastic analysis methods to confirm the existence of distributional and weighted pseudo almost periodic mild solutions for impulsive SDEs. In a different approach, Wang and Fu [14] advanced operator theory and the Banach fixed point theorem to demonstrate the existence of p-th mean Besicovitch almost periodic solutions in distribution for a class of semilinear non-autonomous stochastic evolution equations. Further, Li et al. [15] established sufficient conditions for the existence and uniqueness of distributional Weyl almost periodic solutions for mean-field SDEs driven by both Brownian and fractional Brownian motion. Meanwhile, Zhao et al. [16] applied the concept of $S-$asymptotically $\omega$-periodic processes in distribution and the method of successive approximations to derive conditions for the existence and uniqueness of solutions to certain stochastic fractional-order functional differential equations. The existence of solutions related to pseudo almost periodicity and almost periodicity in distribution has been explored in a substantial body of additional literature, including [17,18,19,20,21,22], among others.

This study extends previous research by analyzing the conditions for the existence of a unique $\mu$-pseudo almost periodic solution in distribution for the following stochastic differential equation:

$$dv\left(t\right)=\left[Av\left(t\right)+L\left(v_t\right)+\gamma \left(t\right)\right]dt+\vartheta \left(t\right)dW\left(t\right),$$

(1)

here, the operator A generates a strongly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on a subset Q. The equation is driven by a standard two-sided Wiener process, $W\left(t\right)$. The model includes two stochastic processes, $\gamma$ and $\vartheta$, which map the real numbers into the Hilbert space $L^p(K,\mathcal{Q})$; their precise definitions will be specified as required. Furthermore, L is a bounded linear operator, with its domain being the space $C=C\left([-r,0];L^p(K,\mathcal{Q})\right)$, which is the set of continuous functions from $[-r,0]$ to $L^p(K,\mathcal{Q})$ equipped with the supremum norm. The operator L maps this space C into $L^p(K,\mathcal{Q})$.

This research is motivated by the fact that, to the best of our knowledge, the problem of proving the existence and uniqueness of $\mu$-pseudo almost periodic solutions in distribution for forced stochastic differential equations of the form (1) has not been previously addressed.

Our study is structured into the following sections. We begin in Section 2 by reviewing fundamental concepts and essential properties that form the basis for our work. The core contribution is presented in Section 3, where we prove the existence and uniqueness of such solutions for Equation (1). To conclude, the final section provides a case study that demonstrates the practical applicability of our main theorems.

2. Preliminaries

This section defines key concepts and assumptions, setting the foundation for the main results.

We begin by establishing our mathematical setting. Let L represent the Lebesgue $\sigma$-algebra on the real line R. We define $\mathcal{M}$ as the set of all positive measures $\mu$ on L that satisfy two specific conditions: the measure of the entire real line is infinite ($\mu \left(R\right)=+\infty$), while the measure of any closed, bounded interval $[c,d]$ (where $c\le d$) is finite.

Our analysis takes place in a real, separable Hilbert space $(K,\parallel ·{\parallel }_K)$ over a probability space $(\mathrm{\Theta },\mathcal{G},\mathbb{P})$. We will work with the Bochner space $L^p(K,\mathcal{Q})$, which consists of $\mathcal{Q}$-valued random variables Z. The norm in this space is defined as:

$$\parallel \widehat{\beta }{\parallel }_{L^p}={\left({\int }_{\mathrm{\Theta }}{\parallel \widehat{\beta }\parallel }_K^{p}d\mathbb{P}\right)}^{{\displaystyle \frac{1}{p}}}.$$

Definition 1

([7]). We introduce the following definitions for a stochastic process $\widehat{\beta }:R\to L^p(K,\mathcal{Q})$:

(1) 

Stochastic Boundedness:  A process $\widehat{\beta }$ is considered  stochastically bounded   if there is a positive constant $C_M$ for which the p-th moment of its norm is uniformly bounded across all time, i.e.,

$$\mathbb{E}\parallel \widehat{\beta }{\left(\tau \right)\parallel }_K^p\le C_M\mathrm{for}\mathrm{all}\tau \in R.$$

(2) 

Stochastic Continuity:   A process $\widehat{\beta }$ is  stochastically continuous   if it belongs to the space $C\left(R,L^p(K,\mathcal{Q})\right)$. This means that for any point $\sigma \in R$, the expected value of the p-th power of the norm difference approaches zero as τ converges to σ:

$$\underset{\tau \to \sigma }{lim}\mathbb{E}{\parallel \widehat{\beta }\left(\tau \right)-\widehat{\beta }\left(\sigma \right)\parallel }_K^p=0.$$

The space of all processes that are both stochastically bounded and continuous is denoted by $BC\left(R,L^p(K,\mathcal{Q})\right)$.

The initial value problem (IVP) of interest is formulated as:

$$\left\{\begin{array}{c}d{v}_{\tau }=\left[A{v}_{\tau }+\mathbf{L}\left(v_{\tau }\right)+\varphi \left(\tau \right)\right]d\tau +\gamma \left(\tau \right)dW\left(\tau \right)\mathrm{for}\tau \ge 0\hfill \\ v_0=\eta \in C=C\left([-h,0],L^p(K,\mathcal{Q})\right),\hfill \end{array}\right.$$

(2)

here, $\varphi :R^{+}\to L^p(K,\mathcal{Q})$ and $\gamma :R^{+}\to L^p(K,\mathcal{Q})$ are continuous stochastic processes.

Definition 2

([10]). A function v is classified as an integral solution to this IVP if it is continuous on $[-h,+\infty )$ and meets the following three conditions:

• The integral ${\int }_0^{\tau }v\left(\sigma \right)d\sigma$ is in the domain of A for all $\tau \ge 0$,
• The function satisfies the integral equation for all $\tau \ge 0$:

  $$v\left(\tau \right)=\eta \left(0\right)+A{\int }_0^{\tau }v\left(\sigma \right)d\sigma +{\int }_0^{\tau }\left(\mathbf{L}\left(v_{\sigma }\right)+\varphi \left(\sigma \right)\right)d\sigma +{\int }_0^{\tau }\gamma \left(\sigma \right)dW\left(\sigma \right),$$
• It satisfies the initial history condition $v_0=\eta$.

It is a standard result that when the domain of A is dense in $L^p(K,\mathcal{Q})$ (i.e., $\overline{Dom\left(A\right)}=L^p(K,\mathcal{Q})$), the concepts of mild and integral solutions coincide. Furthermore, any integral solution $v\left(\tau \right)$ must lie within the closure of the domain of A for all $\tau \ge 0$, which implies the initial state $\eta \left(0\right)$ must also be in $\overline{Dom\left(A\right)}$.

Based on these criteria, we define the operator $A_{\mathrm{sub}}$ on $\overline{Dom\left(A\right)}$. Its domain, $Dom\left(A_{\mathrm{sub}}\right)$, includes all $x\in Dom\left(A\right)$ such that $Ax$ is in $\overline{Dom\left(A\right)}$, and for these elements, we set $A_{\mathrm{sub}}x:=Ax$.

We impose the following proposition:

H0. 

The operator A satisfies the Hille–Yosida condition [See

Table 1. Summary of Stochastic Framework Elements.

Element	Description
Probability Space	$(\mathrm{\Theta },\mathcal{G},\mathbb{P})$ with Wiener process $W\left(t\right)$
& Filtration	${\mathcal{F}}_t$-adapted processes (Lemma 2)
H0	Operator A satisfies Hille-Yosida condition
H1	Semigroup $S\left(\tau \right)$ compact $\forall \tau >0$
H2	$\exists \beta >0$, bounded $I\subset \mathbb{R}:\mu \left(\right\{b+\alpha :b\in A\left\}\right)\le \beta \mu \left(A\right)$
	when $A\cap I=\varnothing$ (Section 2)
${\mathrm{\Xi }}^s$	Projection onto stable subspace $S_s$ (Proposition 4)
${\mathrm{\Xi }}^v$	Projection onto unstable subspace $S_v$ (Proposition 4)
$V\left(\tau \right)$	Solution semigroup on $C_{\mathit{sub}}$ (Proposition 2)
$\mu$	Positive Borel measure: $\mu \left(\mathbb{R}\right)=+\infty$, $\mu \left(\right[c,d\left]\right)<\infty$
	Satisfies condition (H2) (Section 2)
$C_p$	Burkholder-Davis-Gundy constant (Lemma 2):
	$\mathbb{E}{\left|{\int }_0^S\mathrm{\Phi }\left(r\right)dB\left(r\right)\right|}^p\le C_p\mathbb{E}{\left({\int }_0^S{\parallel \mathrm{\Phi }\left(r\right)\parallel }^{2}dr\right)}^{p/2}$
$\rho$	Exponential decay rate: $\left|V\left(\tau \right)\eta \right|\le \overline{K}{e}^{-\rho \tau }\left|\eta \right|$ for $\eta \in S_s$
	$\left|V\left(\tau \right)\eta \right|\le \overline{K}{e}^{\rho \tau }\left|\eta \right|$ for $\eta \in S_v$ (Proposition 4)
Lipschitz moduli	$L_{\gamma },L_{\vartheta }$: Constants for nonlinear terms $\gamma ,\vartheta$ satisfying
	${\mathbb{E}\parallel \gamma (t,\xi )-\gamma (t,\eta )\parallel }^p\le L_{\gamma }\mathbb{E}{\parallel \xi -\eta \parallel }^p$
	${\mathbb{E}\parallel \vartheta (t,\xi )-\vartheta (t,\eta )\parallel }^p\le L_{\vartheta }\mathbb{E}{\parallel \xi -\eta \parallel }^p$ (Theorem 5)
$\mathcal{M}$	Set of measures: $\{\mu :\mathcal{L}\to [0,+\infty ]\mid \mu (\mathbb{R})=+\infty ,$
	$\mu \left(\right[c,d\left]\right)<\infty \mathit{for}\mathit{bounded}[c,d]\}$ (Section 2)
Condition on $\mu$	Must satisfy (H2): Translation-boundedness
	$\mu \left(\right\{b+\alpha :b\in A\left\}\right)\le \beta \mu \left(A\right)$ when $A\cap I=\varnothing$

].

Proposition 1

([23]). Under assumption $H0$, the operator $A_{\mathit{sub}}$ generates a strongly continuous semigroup ${\left(S_{\mathit{sub}}\left(\tau \right)\right)}_{\tau \ge 0}\left(C_0\right)$ on the space $\overline{Dom\left(A\right)}$. The phase space for the IVP is defined as:

$$C_{\mathit{sub}}=\left\{\eta \in C:\eta \left(0\right)\in \overline{Dom\left(A\right)}\right\}.$$

For the corresponding homogeneous system,

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{d\tau }}{\widehat{v}}_{\tau }=A{\widehat{v}}_{\tau }+\mathbf{L}\left({\widehat{v}}_{\tau }\right),\tau \ge 0\hfill \\ {\widehat{v}}_0=\eta \in C,\hfill \end{array}\right.$$

we define a family of linear operators ${\left(\mathcal{V}\left(\tau \right)\right)}_{\tau \ge 0}$ on $C_{\mathit{sub}}$ where $\mathcal{V}\left(\tau \right)\eta ={\widehat{v}}_{\tau }(·,\eta )$ represents the state of the solution at time τ for the initial history η.

Proposition 2

([24]). The family ${\left(\mathcal{V}\left(\tau \right)\right)}_{\tau \ge 0}$ forms a strongly continuous semigroup on $C_{\mathit{sub}}$. It possesses a shift property for any $\tau \ge 0$ and $k\in [-h,0]$:

$$\left(\mathcal{V}\left(\tau \right)\eta \right)\left(k\right)=\left\{\begin{array}{cc}\left(\mathcal{V}\right(\tau +k\left)\eta \right)\left(0\right),\hfill & \mathit{if}\tau +k\ge 0\hfill \\ \eta (\tau +k),\hfill & \mathit{if}\tau +k\le 0.\hfill \end{array}\right.$$

Theorem 1

([24]). The infinitesimal generator of this semigroup, denoted ${\mathcal{G}}_{\mathcal{V}}$, is defined on $C_{\mathit{sub}}$ as follows:

$$\left\{\begin{array}{c}Dom\left({\mathcal{G}}_{\mathcal{V}}\right)=\left\{\eta \in C^1([-h,0];L^p(K,\mathcal{Q})):\eta \left(0\right)\in \overline{Dom\left(A\right)},{\eta }^{\prime }\left(0\right)=A\eta \left(0\right)+\mathbf{L}\left(\eta \right)\right\}\hfill \\ {\mathcal{G}}_{\mathcal{V}}\eta ={\eta }^{\prime },\mathrm{for}\eta \in Dom\left({\mathcal{G}}_{\mathcal{V}}\right)\hfill \end{array}\right.$$

We henceforth adopt an additional assumption:

H1. 

The semigroup $S\left(\tau \right)$ is compact as a map from $\overline{D\left(A\right)}$ (equipped with the graph norm) to X for every $\tau >0$ [See Table 1].

Proposition 3

([24]). If both assumptions $H0$ and $H1$ are true, then the solution semigroup ${\left(\mathcal{V}\left(\tau \right)\right)}_{\tau \ge 0}$ is compact for all $\tau >h$.

Definition 3

([24]). The semigroup ${\left(\mathcal{V}\left(\tau \right)\right)}_{\tau \ge 0}$ is called hyperbolic if its generator’s spectrum does not intersect the imaginary axis:

$$Sp\left({\mathcal{G}}_{\mathcal{V}}\right)\cap iR=\varnothing .$$

Proposition 4

([24]). If assumption $H1$ holds and the semigroup is hyperbolic, then the phase space $C_{\mathit{sub}}$ can be decomposed into a direct sum of two closed, $\mathcal{V}\left(\tau \right)$-invariant subspaces, $S_s$ and $S_v$:

$$C_{\mathit{sub}}=S_s\oplus S_v,$$

here, $S_s$ is the  stable subspace   and $S_v$ is the  unstable subspace. The restriction of the semigroup to $S_v$ forms a group. Furthermore, there exist positive constants $\overline{K}$ and ρ such that the following exponential estimates hold:

$$\begin{array}{cc}\hfill \left|\mathcal{V}\right(\tau \left)\eta \right|& \le \overline{K}{e}^{-\rho \tau }\left|\eta \right|,\mathit{for}\tau \ge 0\mathit{and}\eta \in S_s\hfill \\ \hfill \left|\mathcal{V}\right(\tau \left)\eta \right|& \le \overline{K}{e}^{\rho \tau }\left|\eta \right|,\mathit{for}\tau \le 0\mathit{and}\eta \in S_v,\hfill \end{array}$$

the corresponding projection operators onto these subspaces are denoted by ${\mathrm{\Xi }}^s$ and ${\mathrm{\Xi }}^v$[See Table 1].

Definition 4

([7]). Let μ be a measure belonging to a class of measures $\mathcal{M}$. A stochastic process, denoted by y, is defined as being$\mu$-ergodic if it fulfills the following two requirements:

• Regularity Condition:   The process y must be a bounded and continuous function from the set of real numbers, $\mathbb{R}$, into the Bochner space $L^p(K,\mathcal{Q})$. This is expressed as $y\in BC\left(\mathbb{R},L^p(K,\mathcal{Q})\right)$.
• Asymptotic Decay Condition:   The long-term time average of the process’s expected size must vanish. Specifically, the average of ${\mathbb{E}\parallel y\left(t\right)\parallel }^p$ over a symmetric interval $I=[-r,r]$, weighted by the measure μ, must converge to zero as the interval grows infinitely large. This is captured by the limit:

  $$\underset{r\to \infty }{lim}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{\parallel y\left(t\right)\parallel }^{p}d\mu \left(t\right)=0.$$

The space comprising all such μ-ergodic processes is designated by the notation $\mathcal{E}\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$.

H2. 

For any real number α, the existence of a positive number β and a bounded interval I satisfying the following conditions is guaranteed:

$$\mu \left(\{b+\alpha :b\in A\}\right)\le \beta \mu \left(A\right)ifA\in \mathcal{B}\mathit{such} \mathit{that}A\cap I=\varnothing$$

[See Table 1].

Example: Let $\mu$ be the measure defined by its Radon–Nikodym derivative: $d\mu \left(t\right)=e^{-\left|t\right|}dt$, where $dt$ is the Lebesgue measure.

Translation-Invariance (H2):

For any $\tau \in \mathbb{R}$, the translated measure ${\mu }_{\tau }\left(A\right)=\mu (A+\tau )$ satisfies: ${\mu }_{\tau }\left(A\right)={\int }_A{e}^{-|t-\tau |}dt$.

For $A\cap I=\varnothing$ (where I is a bounded interval), if $\left|t\right|$ is large, $|t-\tau |\approx \left|t\right|+\left|\tau \right|$, so:

$$e^{-|t-\tau |}\le e^{\left|\tau \right|}{e}^{-\left|t\right|},$$

$$e^{-|t-\tau |}\ge e^{-\left|\tau \right|}{e}^{-\left|t\right|}.$$

Thus,

$e^{-\left|\tau \right|}\mu \left(A\right)\le {\mu }_{\tau }\left(A\right)\le e^{\left|\tau \right|}\mu \left(A\right)$, which implies ${\mu }_{\tau }\approx \mu$ with $\alpha =e^{-\left|\tau \right|}$ and $\beta =e^{\left|\tau \right|}$, satisfying (H2).

Regularity (H1):

For $0\le a<b\le c$ and $\left|\tau \right|\to \infty$, the ratio:

$${\displaystyle \frac{\mu \left(\right(a+\tau ,b+\tau \left)\right)}{\mu \left(\right[\tau ,c+\tau \left]\right)}}={\displaystyle \frac{{\int }_{a+\tau }^{b+\tau }{e}^{-\left|t\right|}dt}{{\int }_{\tau }^{c+\tau }{e}^{-\left|t\right|}dt}},$$

simplifies (for $\tau \to +\infty$) to:

$${\displaystyle \frac{e^{-a}-e^{-b}}{1-e^{-c}}}>0,$$

which satisfies (H1).

Definition 5

([25]). If a continuous stochastic process $\widehat{\beta }:\mathbb{R}\to L^p(K,\mathcal{Q})$ satisfies the condition that for each $ϵ>0$ there exists $l>0$ such that for all $\gamma \in \mathbb{R}$, there exists $\alpha \in [\gamma ,\gamma +l]$ satisfying

$$\underset{t\in \mathbb{R}}{sup}\mathbb{E}{\parallel \widehat{\beta }(t+\alpha )-\widehat{\beta }\left(t\right)\parallel }^p<ϵ,$$

then, $\widehat{\beta }$ is called a p-mean almost periodic process(p-mean AP). The set of all p-mean almost periodic stochastic processes on the set of real numbers with values in $L^p(K,\mathcal{Q})$ is denoted by $AP\left(\mathbb{R},L^p(K,\mathcal{Q})\right)$.

Definition 6

([7]). An p-mean μ-pseudo almost periodic process(p-mean μ-PAP) is a continuous stochastic process ψ that can be represented in the given format. The process maps from the set of real numbers to $L^p(K,\mathcal{Q})$.

$$\psi =\delta +\varrho ,$$

where $\delta \in AP\left(\mathbb{R},L^p(K,\mathcal{Q})\right)$ and $\varrho \in \mathcal{E}\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$.

The collection of stochastic processes that possess the properties described above is referred to as $PAP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$.

Theorem 2

([7]). $PAP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$ equipped with the norm. ${\parallel ·\parallel }_{\infty }$, is a Banach space.

Let $(M,d)$ be a metric space and let $h:M\to \mathbb{R}$ be a real-valued function. The Bounded Lipschitz norm of h is given by:

$${\parallel h\parallel }_{BL}={\parallel h\parallel }_{\infty }+{\mathrm{Lip}}_d\left(h\right),$$

where ${\parallel h\parallel }_{\infty }={sup}_{a\in M}\left|h\left(a\right)\right|$ and ${\mathrm{Lip}}_d\left(h\right)={sup}_{a\ne b}{\displaystyle \frac{\left|h\right(a)-h(b\left)\right|}{d(a,b)}}$.

For two probability measures ${\widehat{\mu }}_1$ and ${\widehat{\mu }}_2$ on M, the Bounded Lipschitz metric between them is defined as the largest difference in their expectations over the set of all test functions h with ${\parallel h\parallel }_{BL}\le 1$.

$$d_{BL}({\widehat{\mu }}_1,{\widehat{\mu }}_2)=\underset{{\parallel h\parallel }_{BL}\le 1}{sup}\left|{\int }_{M}hd{\widehat{\mu }}_1-{\int }_{M}hd{\widehat{\mu }}_2\right|.$$

Definition 7

(after [26]). A stochastic process $X:\mathbb{R}\to {\mathbb{R}}^n$ is called almost periodic in distribution(APD) if its statistical properties, as captured by its probability law, exhibit almost periodic behavior over time.

To formalize this, we consider the function that maps each time point t to the law of the process at that time, ${\widehat{\mu }}_t:=P\circ {\left[X\left(t\right)\right]}^{-1}$. The process X is almost periodic in distribution if this function $t\mapsto {\widehat{\mu }}_t$ is almost periodic in the Bohr sense. The distance between distributions is measured using the Bounded Lipschitz metric, $d_{\mathrm{BL}}$.

Formally, for any $\epsilon >0$, there must exist a number $\ell >0$ (a length of inclusion) such that every interval $[a,a+\ell ]$ contains at least one τ (an ε-almost period) satisfying:

$$\underset{t\in \mathbb{R}}{sup}{d}_{\mathrm{BL}}({\widehat{\mu }}_{t+\tau },{\widehat{\mu }}_t)<\epsilon .$$

Definition 8.

Let $\mu \in \mathcal{M}$. A Q-valued stochastic process f is considered to be $\mu$-pseudo almost periodic in distribution(μ-PAPD) if it is expressible as the sum $f=g+\phi$, where the component g is an almost periodic process in distribution, while the component φ is an element of the space $\mathcal{E}(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$.

Lemma 1.

The space of p-mean pseudo almost periodic functions, $PAP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$, is invariant under translation. That is, for any function h in this space and any real constant c, the shifted function $h(·-c)$ also belongs to the space.

Proof. 

Our proof relies on the fundamental decomposition of a pseudo almost periodic function. Any function $h\in PAP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$ can be uniquely written as the sum of an almost periodic part $h_1$ and an ergodic (or vanishing) part $h_2$. The shifted function is, thus, $h(·-c)=h_1(·-c)+h_2(·-c)$.

The almost periodic component $h_1$ poses no issue, as the space $AP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$ is closed under translation, ensuring $h_1(·-c)$ is also almost periodic. The core of the proof is to establish that the ergodic component $h_2(·-c)$ also satisfies the necessary conditions to belong to its space, $BC(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$.

We verify this by analyzing the defining integral condition for the $BC$ space. Using a change of variables ($t^{\prime }=t-c$), we find:

$$\begin{array}{cc}\hfill 0& \le {\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥h_2(t-c)∥}^{p}dt={\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_{I-c}\mathbb{E}{∥h_2\left(t^{\prime }\right)∥}^{p}d{t}^{\prime }.\hfill \end{array}$$

By bounding this expression, we can relate it back to the properties of the original function $h_2$. For an interval $I=[-r,r]$:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_{I-c}\mathbb{E}{∥h_2\left(t\right)∥}^{p}dt\le \left({\displaystyle \frac{\mu \left(I\right)+2\left|c\right|}{\mu \left(I\right)}}\right){\displaystyle \frac{1}{\mu \left(I\right)+2\left|c\right|}}{\int }_{I-c}\mathbb{E}{∥h_2\left(t\right)∥}^{p}dt.\end{array}$$

Since $h_2$ is in $BC(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$, its mean tends to zero over increasingly large intervals. The inequality confirms that this property is inherited by its translation, $h_2(·-c)$.

With both the translated almost periodic and ergodic parts remaining in their required spaces, their sum, $h(·-c)$, is therefore a member of $PAP(\mathbb{R},L^p(K,\mathcal{Q}),\mu )$. □

Lemma 2

([27]). Let $\mathrm{\Phi }:[0,S]\times \mathrm{\Sigma }\to \mathbb{L}\left(L^p(\mathrm{\Sigma },\mathbb{K})\right)$ be an ${\mathcal{F}}_t$-adapted measurable stochastic process satisfying ${\int }_0^S\mathbb{E}{\parallel \mathrm{\Phi }\left(r\right)\parallel }^{2}dr<\infty$ almost surely. Then, for any $p\in [1,\infty )$, there exists a constant $C_p>0$ such that

$$\mathbb{E}\underset{0\le r\le S}{sup}{∥{\int }_0^S\mathrm{\Phi }\left(r\right)d\mathbb{B}\left(r\right)∥}^p\le {\left(C_p{\int }_0^S\mathbb{E}{\parallel \mathrm{\Phi }\left(r\right)\parallel }^{2}dr\right)}^{p/2},$$

for all $S>0$.

Lemma 3

([28]). Let $h:\mathbb{R}\to \mathbb{R}$ be a bounded and continuous function satisfying, for all $\tau \in \mathbb{R}$,

$$h\left(\tau \right)\le \rho \left(\tau \right)+\sum _{i=1}^n{\kappa }_i{\int }_{-\infty }^{\tau }{e}^{-{\lambda }_i(\tau -s)}h\left(s\right)ds+\sum _{i=n+1}^{2n}{\kappa }_i{\int }_{\tau }^{\infty }{e}^{{\lambda }_i(\tau -s)}h\left(s\right)ds,$$

where

• $\rho :\mathbb{R}\to \mathbb{R}$ is a bounded function with $\overline{\rho }={lim\; sup}_{s\to +\infty }\rho \left(s\right)\le \rho \left(\tau \right)$ for all $\tau \in \mathbb{R}$,
• ${\kappa }_1,\dots ,{\kappa }_{2n}\ge 0$ are constants,
• ${\lambda }_1,\dots ,{\lambda }_{2n}>\eta +\zeta$ are positive constants, with $\eta ={\sum }_{i=1}^n{\kappa }_i$ and $\zeta ={\sum }_{i=n+1}^{2n}{\lambda }_i$.
  • Let $\lambda ={min}_{1\le i\le 2n}{\lambda }_i$. Then, for any $\theta \in (0,\lambda -\eta -\zeta )$,

    $$h\left(\tau \right)\le {\displaystyle \frac{\lambda }{\lambda -\eta -\zeta }}\widehat{\rho }\left(\tau \right)+{\left({\displaystyle \frac{\lambda }{\lambda -\eta -\zeta }}\right)}^2\eta {\int }_{-\infty }^{\tau }{e}^{-\theta (\tau -s)}\left(\widehat{\rho }\left(s\right)-\overline{\rho }\right)ds,\forall \tau \in \mathbb{R},$$

where $\widehat{\rho }\left(\tau \right)={sup}_{s\ge \tau }\rho \left(s\right)$ for each $\tau \in \mathbb{R}$.

Theorem 3

([10]). The existence and uniqueness of a bounded solution to Equation (1) on R is established under several key assumptions: $\left(H_0\right)$ and $\left(H_1\right)$ must hold, the semigroup ${\left(V\left(t\right)\right)}_{t\ge 0}$ must be hyperbolic, $\psi$ must be a continuous and bounded function, and $\vartheta$ must be a process with bounded moments. If these conditions are met, there is precisely one bounded solution v.

This solution is constructed for all $t\ge 0$ as the sum of deterministic and stochastic integrals over the stable and unstable subspaces, expressed as:

$$\begin{array}{cc}\hfill v_t& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{V}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0\gamma \left(s\right)\right)ds+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{V}^v(t-s){\mathrm{\Xi }}^v\left({\tilde{B}}_{\lambda }{X}_0\gamma \left(s\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{V}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0\vartheta \left(s\right)\right)dW\left(s\right)+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{V}^v(t-s){\mathrm{\Xi }}^v\left({\tilde{B}}_{\lambda }{X}_0\vartheta \left(s\right)\right)dW\left(s\right),\hfill \end{array}$$

here, ${\tilde{B}}_{\lambda }=\lambda {\left(\lambda I-\tilde{A}V\right)}^{-1}$, while ${\mathrm{\Xi }}^s$ and ${\mathrm{\Xi }}^v$ are the projectors from $C_0$ to its stable and unstable components.

Theorem 4

([7]). Let a measure $\mu$ be given that satisfies condition (H2). Consider a function

$$\widehat{\beta }:\mathbb{R}\times L^p(K,\mu )\to L^p(K,\mu ),$$

with the following properties:

• $\widehat{\beta }$ is pseudo almost periodic with respect to $\mu$, i.e., $\widehat{\beta }\in PAP\left(\mathbb{R}\times L^p(K,\mathcal{Q}),L^p(K,\mathcal{Q}),\mu \right)$.
• $\widehat{\beta }$ is uniformly Lipschitz in its second variable, meaning there is a constant $L>0$ such that for all $t\in \mathbb{R}$ and all $a,b\in L^p(K,\mathcal{Q})$, the following holds:

  $$\mathbb{E}\parallel \widehat{\beta }(t,a)-\widehat{\beta }{(t,b)\parallel }^p\le L·\mathbb{E}{\parallel a-b\parallel }^p.$$

Under these conditions, if x is any process in $PAP\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$, then the function defined by the composition $t\mapsto \widehat{\beta }(t,x\left(t\right))$ is also in $PAP\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$.

3. Main Results

We now investigate the existence of a $\mu$-pseudo almost periodic solution in distribution for the following non-linear stochastic system:

$$dv\left(t\right)=\left[Av\left(t\right)+L\left(v_t\right)+\gamma \left(t,v\left(t\right)\right)\right]dt+\vartheta \left(t,v\left(t\right)\right)dW\left(t\right),t\in \mathbb{R}$$

(3)

In this context, the non-linear terms are defined by the stochastically continuous processes $\gamma :\mathbb{R}\times C\to L^p(K,\mathcal{Q})$ and $\vartheta :\mathbb{R}\times C\to L^p(K,\mathcal{Q})$. The following content assumes $p\ge 2$. To proceed, we introduce the following required hypotheses.

Theorem 5.

Assuming $\mu \in \mathcal{M}$ satisfies conditions $\mathit{H}\mathbf{0}$–$\mathit{H}\mathbf{2}$, we take γ and ϑ to be stochastic continuous processes belonging to $AP\left(\mathbb{R}\times L^p(K,\mathcal{Q}),L^p(K,\mathcal{Q}),\mu \right)$ and further suppose that there exist positive constants $L_{\gamma }$ and $L_{\vartheta }$ such that

$$\begin{array}{c}\hfill {\mathbb{E}\parallel \gamma (t,\xi )-\gamma (t,\eta )\parallel }^p\le L_{\gamma }·\mathbb{E}{\parallel \xi -\eta \parallel }^p,\\ \hfill {\mathbb{E}\parallel \vartheta (t,\xi )-\vartheta (t,\eta )\parallel }^p\le L_{\vartheta }·\mathbb{E}{\parallel \xi -\eta \parallel }^p,\end{array}$$

for every t belonging to the set of real numbers and for every ξ and η belonging to $L^p(K,\mathcal{Q})$, then the existence of a unique $L^p$-bounded mild solution for problem (3) is guaranteed under the condition that

$$\left(4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p\right)({\left|{\mathrm{\Xi }}^v\right|}^p+{\left|{\mathrm{\Xi }}^s\right|}^p)({\rho }^{-p}{L}_{\gamma }+{\left(2\rho \right)}^{(-p)/2}{C}_p{L}_{\vartheta })<1,$$

is satisfied. The property of almost periodicity in distribution is also guaranteed for this unique $L^p$-bounded mild solution, under the condition that

$$4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p(2{\left|{\mathrm{\Xi }}^s\right|}^p+{\left|{\mathrm{\Xi }}^v\right|}^p)({\rho }^{1-p}{L}_{\gamma }+C_p{L}_{\vartheta }{\left(2\rho \right)}^{(2-p)/2})<\rho .$$

Proof. 

Step 1. A bounded solution is $L^p$-continuous for $t\ge r$. To proceed, we introduce a non-linear operator $\mathrm{\Omega }$ that maps the space $C_b(\mathbb{R};L^p(K,\mathcal{Q}))$ into itself, given by the expression:

$$\begin{array}{cc}\hfill \left(\mathrm{\Omega }\zeta \right)\left(t\right)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right).\hfill \\ \hfill \mathbb{E}{∥\left(\mathrm{\Omega }\zeta \right)\left(t\right)-\left(\mathrm{\Omega }\zeta \right)\left(r\right)∥}^p& ⩽4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_r^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^r{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_r^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^r{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & ={\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3+{\mathrm{\Lambda }}_4.\hfill \end{array}$$

For the case where $t⩾r$, the result is a consequence of applying both the Cauchy–Schwarz inequality and pô’s isometry.

$$\begin{array}{cc}\hfill & {\mathrm{\Lambda }}_1⩽4^{p-1}\mathbb{E}{\left[\underset{\lambda \to +\infty }{lim}{\int }_r^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥∥\gamma \left(s,\zeta \left(s\right)\right)∥ds\right]}^p\hfill \\ \hfill & ⩽4^{p-1}{\left(\underset{\lambda \to +\infty }{lim}{\int }_r^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥ds\right)}^{p-1}\hfill \\ \hfill & \times \underset{\lambda \to +\infty }{lim}{\int }_r^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥\mathbb{E}{∥\gamma (s,\zeta (s\left)\right)∥}^{p}ds\hfill \\ \hfill & ⩽4^{p-1}{\left(\underset{\lambda \to +\infty }{lim}{\int }_r^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥ds\right)}^p\times {\int }_r^t\mathbb{E}{∥\gamma (s,\zeta (s\left)\right)∥}^{p}ds\hfill \\ \hfill & ⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\int }_r^t\mathbb{E}{∥\gamma (s,\zeta (s\left)\right)∥}^{p}ds.\hfill \\ \hfill & {\mathrm{\Lambda }}_2⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\int }_r^t\mathbb{E}{∥\gamma (s,\zeta (s\left)\right)∥}^{p}ds.\hfill \\ \hfill & {\mathrm{\Lambda }}_3⩽C_p{4}^{p-1}\mathbb{E}{\left[\underset{\lambda \to +\infty }{lim}{\int }_r^t{∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0\vartheta (s,\zeta \left(s\right))\right)∥}^{2}ds\right]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & ⩽C_p{4}^{p-1}{\left(\underset{\lambda \to +\infty }{lim}{\int }_r^t{∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥}^{2}ds\right)}^{{\displaystyle \frac{p-2}{2}}}\hfill \\ \hfill & \times \underset{\lambda \to +\infty }{lim}{\int }_r^t{∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥}^2\mathbb{E}{∥\vartheta (s,\zeta (s\left)\right)∥}^{p}ds\hfill \\ \hfill & ⩽C_p{4}^{p-1}{\left(\underset{\lambda \to +\infty }{lim}{\int }_r^t{∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥}^{2}ds\right)}^{{\displaystyle \frac{p}{2}}}\times {\int }_r^t\mathbb{E}{∥\vartheta (s,\zeta (s\left)\right)∥}^{p}ds\hfill \\ \hfill & ⩽C_p{4}^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\int }_r^t\mathbb{E}{∥\vartheta (s,\zeta (s\left)\right)∥}^{p}ds.\hfill \\ \hfill & {\mathrm{\Lambda }}_4⩽C_p{4}^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\int }_r^t\mathbb{E}{∥\vartheta (s,\zeta (s\left)\right)∥}^{p}ds.\hfill \end{array}$$

The $L^p$-boundedness of $\zeta (·)$, together with the Lipschitz continuity and p-mean almost periodicity of $\gamma$ in $\zeta$ and s, respectively, allows us to conclude that

$$\begin{array}{c}{\displaystyle \underset{s\in \mathbb{R}}{sup}}{\parallel \gamma (s,\zeta \left(s\right))\parallel }^p⩽{\displaystyle \underset{s\in \mathbb{R}}{sup}}{\parallel \gamma (s,\zeta \left(s\right))-\gamma (s,0)\parallel }^p+{\displaystyle \underset{s\in \mathbb{R}}{sup}}{\parallel \gamma (s,0)\parallel }^p\\ ⩽L{\displaystyle \underset{s\in \mathbb{R}}{sup}}{\parallel \zeta \left(s\right)\parallel }^p+{\displaystyle \underset{s\in \mathbb{R}}{sup}}{\parallel \gamma (s,0)\parallel }^p.\end{array}$$

(4)

Similarly to (4), we obtain ${sup}_{s\in \mathbb{R}}E{\parallel \vartheta (s,x\left(s\right))\parallel }^p$.

$${\mathbb{E}\parallel \left(\mathrm{\Omega }\zeta \right)\left(t\right)-\left(\mathrm{\Omega }\zeta \right)\left(r\right)\parallel }^p\to 0ast\to r^{+}.$$

Following an identical argument, it can be concluded that ${\mathbb{E}|\left(\mathrm{\Omega }\zeta \right)\left(t\right)-\left(\mathrm{\Omega }\zeta \right)\left(r\right)|}^p\to 0$ when t approaches r from the left. This is precisely the condition for the $L^p$-continuity of $\left(\mathrm{\Omega }\zeta \right)(·)$. It can be concluded from the above

$$\begin{array}{cc}\hfill & \mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds∥}^p\hfill \\ \hfill & ⩽{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\rho }^{-p}\underset{s\in \mathbb{R}}{sup}\mathbb{E}{∥\gamma \left(s,\zeta \left(s\right)\right)∥}^p.\hfill \\ \hfill & \mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & ⩽C_p{\displaystyle \frac{{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p}{{\left(2\rho \right)}^{p/2}}}\underset{s\in \mathbb{R}}{sup}\mathbb{E}{∥\vartheta \left(s,\zeta \left(s\right)\right)∥}^p.\hfill \end{array}$$

Similarly, the other two items can be obtained. A direct consequence of our previous steps is that $\left(\mathrm{\Omega }\zeta \right)\left(t\right)$ is $L^p$-bounded.

Step 2. We now turn to proving the existence and uniqueness of an $L^p$-bounded solution. To do this, we work within the Banach space $C_b(\mathbb{R};L^p(K,\mathcal{Q}))$, which is the space of all $L^p$-continuous and bounded maps from $\mathbb{R}$ to $L^p(K,\mathcal{Q})$ under the supremum norm ${|·|}_{\infty }$. The core of our argument is to show that $\mathrm{\Omega }$ acts as a contraction on $C_b(\mathbb{R};L^p(K,\mathcal{Q}))$. Let ${\zeta }_1$ and ${\zeta }_2$ be any two functions in $C_b(\mathbb{R};L^p(K,\mathcal{Q}))$. For an arbitrary $t\in \mathbb{R}$, we have the following relation

$$\begin{array}{cc}\hfill & \mathbb{E}{∥\mathrm{\Omega }{\zeta }_1\left(t\right)-\mathrm{\Omega }{\zeta }_2\left(t\right)∥}^p\hfill \\ \hfill & \le 4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma \left(s,{\zeta }_2\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s,{\zeta }_2\left(s\right)\right)-\gamma \left(s,{\zeta }_1\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s,{\zeta }_1\left(s\right)\right)-\vartheta \left(s,{\zeta }_2\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s,{\zeta }_2\left(s\right)\right)-\vartheta \left(s,{\zeta }_1\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p.\hfill \end{array}$$

We calculated the second term

$$\begin{array}{cc}\hfill & 4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma \left(s,{\zeta }_2\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & \le 4^{p-1}\mathbb{E}{\left(\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }∥{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma \left(s,{\zeta }_2\left(s\right)\right)\right)\right)∥ds\right)}^p\hfill \\ \hfill & \le 4^{p-1}\mathbb{E}{\left(\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }∥{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v\left({\tilde{B}}_{\lambda }{X}_0\right)∥∥\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma \left(s,{\zeta }_2\left(s\right)\right)∥ds\right)}^p\hfill \\ \hfill & \le 4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p\mathbb{E}{\left({\int }_t^{+\infty }{e}^{\rho (t-s)}∥\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma \left(s,{\zeta }_2\left(s\right)\right)∥ds\right)}^p\hfill \\ \hfill & \le 4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left({\int }_t^{+\infty }{e}^{\rho (t-s)}ds\right)}^{p-1}\left({\int }_t^{+\infty }{e}^{\rho (t-s)}\mathbb{E}{∥\gamma \left(s,{\zeta }_1\left(s\right)\right)-\gamma (s,{\zeta }_2\left(s\right))∥}^{p}ds\right)\hfill \\ \hfill & \le 4^{p-1}{L}_{\gamma }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left({\int }_t^{+\infty }{e}^{\rho (t-s)}ds\right)}^{p-1}\left({\int }_t^{+\infty }{e}^{\rho (t-s)}\mathbb{E}{∥{\zeta }_{1s}-{\zeta }_{2s}∥}^{p}ds\right.\hfill \\ \hfill & \le 4^{p-1}{L}_{\gamma }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left({\int }_t^{+\infty }{e}^{\rho (t-s)}ds\right)}^p\mathbb{E}{∥{\zeta }_{1s}-{\zeta }_{2s}∥}^p\hfill \\ \hfill & \le 4^{p-1}{L}_{\gamma }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\rho }^{-p}\underset{s\in \mathbb{R}}{sup}\mathbb{E}{∥{\zeta }_{1s}-{\zeta }_{2s}∥}^p.\hfill \end{array}$$

The fourth expression evaluates to

$$\begin{array}{cc}\hfill & 4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s,{\zeta }_1\left(s\right)\right)-\vartheta \left(s,{\zeta }_2\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & \le 4^{p-1}{C}_p\mathbb{E}{\left(\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{∥{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s,{\zeta }_1\left(s\right)\right)-\vartheta \left(s,{\zeta }_2\left(s\right)\right)\right)\right)∥}^{2}ds\right)}^{p/2}\hfill \\ \hfill & \le 4^{p-1}{C}_p\mathbb{E}{\left(\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{∥{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v\left({\tilde{B}}_{\lambda }{X}_0\right)∥}^2{∥\vartheta \left(s,{\zeta }_1\left(s\right)\right)-\vartheta \left(s,{\zeta }_2\left(s\right)\right)∥}^{2}ds\right)}^{p/2}\hfill \\ \hfill & \le 4^{p-1}{C}_p{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left({\int }_t^{+\infty }{e}^{2\rho (t-s)}ds\right)}^{(p-2)/2}{\int }_t^{+\infty }{e}^{2\rho (t-s)}\mathbb{E}{∥\vartheta \left(s,{\zeta }_1\left(s\right)\right)-\vartheta \left(s,{\zeta }_2\left(s\right)\right)∥}^{p}ds\hfill \\ \hfill & \le 4^{p-1}{C}_p{L}_{\vartheta }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left(2\rho \right)}^{(2-p)/2}\left({\int }_t^{+\infty }{e}^{2\rho (t-s)}\mathbb{E}{∥{\zeta }_{1s}-{\zeta }_{2s}∥}^{p}ds\right)\hfill \\ \hfill & \le 4^{p-1}{C}_p{L}_{\vartheta }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left(2\rho \right)}^{(-p)/2}\underset{s\in \mathbb{R}}{sup}\mathbb{E}{∥{\zeta }_{1s}-{\zeta }_{2s}∥}^p.\hfill \end{array}$$

Similarly, the other two items can be obtained.

Therefore, by merging the previously mentioned inequality, we can deduce that, for every $t\in \mathbb{R}$,

$$\begin{array}{cc}\hfill & \mathbb{E}{∥\left(\mathrm{\Omega }{\zeta }_1\right)\left(t\right)-\left(\mathrm{\Omega }{\zeta }_2\right)\left(t\right)∥}^p\hfill \\ \hfill & \le \left(4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p\right)({\left|{\mathrm{\Xi }}^v\right|}^p+{\left|{\mathrm{\Xi }}^s\right|}^p)({\rho }^{-p}{L}_{\gamma }+{\left(2\rho \right)}^{(-p)/2}{C}_p{L}_{\vartheta })\underset{s\in \mathbb{R}}{sup}\mathbb{E}{∥{\zeta }_1\left(t\right)-{\zeta }_2\left(t\right)∥}^p.\hfill \\ \hfill & \mathrm{Hence},\hfill \\ \hfill & {∥\left(\mathrm{\Omega }{\zeta }_1\right)\left(t\right)-\left(\mathrm{\Omega }{\zeta }_2\right)\left(t\right)∥}_{L^p}^p\le N\underset{s\in \mathbb{R}}{sup}{∥{\zeta }_1\left(s\right)-{\zeta }_2\left(s\right)∥}_{L^p}^p\le N{\left(\underset{s\in \mathbb{R}}{sup}{∥{\zeta }_1\left(s\right)-{\zeta }_2\left(s\right)∥}_{L^p}\right)}^p,\hfill \\ \hfill & \mathrm{with}N:=\left(4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p\right)({\left|{\mathrm{\Xi }}^v\right|}^p+{\left|{\mathrm{\Xi }}^s\right|}^p)({\rho }^{-p}{L}_{\gamma }+{\left(2\rho \right)}^{(-p)/2}{C}_p{L}_{\vartheta }).\hfill \\ \hfill & \mathrm{Therefore},\hfill \\ \hfill & \parallel \left(\mathrm{\Omega }{\zeta }_1\right)\left(t\right)-\left(\mathrm{\Omega }{\zeta }_2\right){\left(t\right)\parallel }_{L^p}\le \sqrt[p]{N}{\parallel {\zeta }_1-{\zeta }_2\parallel }_{\infty }.\hfill \end{array}$$

This implies that

$${∥\mathrm{\Omega }{\zeta }_1-\mathrm{\Omega }{\zeta }_2∥}_{\infty }=\underset{t\in \mathbb{R}}{sup}{∥\left(\mathrm{\Omega }{\zeta }_1\right)\left(t\right)-\left(\mathrm{\Omega }{\zeta }_2\right)\left(t\right)∥}_{L^p}\le \sqrt[p]{N}{∥{\zeta }_1-{\zeta }_2∥}_{\infty }.$$

Because $N<1$, the operator $\mathrm{\Omega }$ is a strict contraction on $C_b(\mathbb{R};L^p(K,\mathcal{Q}))$. This guarantees the existence of a unique element $\zeta$ such that $\mathrm{\Omega }\zeta =\zeta$, which corresponds to the unique $L^p$-bounded mild solution of Equation (3). Constants ($\rho$, $L_{\gamma }$, $L_{\vartheta }$, $\kappa$, $\overline{\kappa }$, $|{\mathrm{\Xi }}^s|$, $|{\mathrm{\Xi }}^v|$, $C_p$,N) $>0$.

Step 3. Having established existence and uniqueness, we next prove that this solution $\zeta$ is APD. To do this, we employ Bochner’s criterion. We take arbitrary real sequences $\left(h_n^{\prime }\right)$ and $\left(q_n^{\prime }\right)$ and seek subsequences $\left(h_n\right)\subset \left(h_n^{\prime }\right)$ and $\left(q_n\right)\subset \left(q_n^{\prime }\right)$ with a common index set. These subsequences must satisfy the property that the limits ${lim}_{n\to \infty }{lim}_{m\to \infty }\widehat{\mu }(t+h_n+q_m)$ and ${lim}_{n\to \infty }\widehat{\mu }(t+h_n+q_n)$ both exist and are identical for every $t\in \mathbb{R}$, where $\widehat{\mu }\left(t\right)$ denotes the law of the solution $\zeta \left(t\right)$.

The almost periodicity of $\gamma$ and $\vartheta$ provides the necessary starting point, allowing us to find subsequences $\left(h_n\right)$ and $\left(q_n\right)$ such that

$$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}\gamma (t+h_n+q_m,x)=\underset{n\to \infty }{lim}\gamma (t+h_n+q_n,x)=:{\gamma }_0(t,x)$$

and

$$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}\vartheta (t+h_n+q_m,x)=\underset{n\to \infty }{lim}\vartheta (t+h_n+q_n,x)=:{\vartheta }_0(t,x).$$

The above limits exist under all three notions of convergence: pointwise, uniformly on compact intervals, and uniformly on the entire interval $\mathbb{R}$. Consequently, the criterion lets you deduce uniform convergence by verifying only pointwise convergence.

It is important to note that the convergence of these limits is uniform with respect to $t\in \mathbb{R}$ and for x located in any bounded subset of $\mathcal{Q}$. Now, letting $\left(r_n\right)=(h_n+q_n)$ and performing the substitution $s=v-r_n$, the relevant process can be rewritten as

$$\begin{array}{cc}\hfill \zeta (t+r_n)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^{t+r_n}{\mathcal{V}}^s(t+r_n-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^{t+r_n}{\mathcal{V}}^v(t+r_n-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^{t+r_n}{\mathcal{V}}^s(t+r_n-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^{t+r_n}{\mathcal{V}}^v(t+r_n-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right),\hfill \end{array}$$

becomes

$$\begin{array}{cc}\hfill \zeta (t+r_n)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s+r_n,\zeta (s+r_n)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s+r_n,\zeta (s+r_n)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s+r_n,\zeta (s+r_n)\right)\right)d{\tilde{W}}_n\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s+r_n,\zeta (s+r_n)\right)\right)d{\tilde{W}}_n\left(s\right),\hfill \end{array}$$

in this context, we define the process ${\tilde{W}}_n\left(s\right)=W(s+r_n)-W\left(r_n\right)$. Due to the stationary increments of the Wiener process, $\tilde{W}n\left(s\right)$ is itself a Brownian motion that is distributionally identical to $W\left(s\right)$. Furthermore, leveraging the independence of the increments of W, we can conclude that the time-shifted process $\zeta (t+r_n)$ is equal in law to the process $\zeta n\left(t\right)$.

$$\begin{array}{cc}\hfill {\zeta }_n\left(t\right)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)\right)dW\left(s\right)\hfill \end{array}$$

the mild solution of

$$d{\zeta }_n\left(t\right)=\left[A{\zeta }_n\left(t\right)+L\left({\zeta }_n\left(t\right)\right)+\gamma \left(t+r_n,{\zeta }_n\left(t\right)\right)\right]dt+\vartheta \left(t+r_n,{\zeta }_n\left(t\right)\right)dW\left(t\right)\mathrm{and}t\in \mathbb{R}$$

furthermore,

$$\begin{array}{cc}\hfill {\zeta }_0\left(t\right)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)dW\left(s\right).\hfill \end{array}$$

Let us show that ${\zeta }_n\left(t\right)$ converges to ${\zeta }_0\left(t\right)$. We have

$$\begin{array}{cc}\hfill & \mathbb{E}{∥{\zeta }_n\left(t\right)-{\zeta }_0\left(t\right)∥}^p\hfill \\ \hfill & =4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & 4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)-\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_n\left(s\right)\right)-\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & +4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_t^{+\infty }{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\left(\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)\right)dW\left(s\right)∥}^p\hfill \\ \hfill & =K_1+K_2+K_3+K_4+K_5+K_6+K_7+K_8.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & K_1=4^{p-1}\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)\right)ds∥}^p\hfill \\ \hfill & ⩽4^{p-1}{\left(\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥ds\right)}^{p-1}\hfill \\ & \times \underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t∥{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥\mathbb{E}{∥\left(\gamma \left(s+r_n,{\zeta }_n\left(s\right)\right)-\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)\right)∥}^{p}ds\hfill \\ \hfill & ⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\rho }^{1-p}{L}_{\gamma }{\int }_{-\infty }^t{e}^{-\rho (t-s)}\mathbb{E}{∥{\zeta }_n\left(s\right)-{\zeta }_0\left(s\right)∥}^{p}ds.\hfill \\ \hfill & K_2⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\rho }^{-p}\mathbb{E}{∥\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)∥}^p.\hfill \\ \hfill & K_3⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\rho }^{1-p}{L}_{\gamma }{\int }_t^{+\infty }{e}^{\rho (t-s)}\mathbb{E}{∥{\zeta }_n\left(s\right)-{\zeta }_0\left(s\right)∥}^{p}ds.\hfill \\ \hfill & K_4⩽4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\rho }^{-p}\mathbb{E}{∥\gamma \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\gamma }_0\left(s,{\zeta }_0\left(s\right)\right)∥}^p.\hfill \end{array}$$

Applying Itô’s isometry, we get

$$\begin{array}{cc}\hfill & K_5\le 4^{p-1}{C}_p{L}_{\vartheta }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\left(2\rho \right)}^{(2-p)/2}{\int }_{-\infty }^t{e}^{-2\rho (t-s)}\mathbb{E}{∥{\zeta }_n\left(s\right)-{\zeta }_0\left(s\right)∥}^{p}ds;\hfill \\ \hfill & K_6\le 4^{p-1}{C}_p{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^s\right|}^p{\left(2\rho \right)}^{(-p)/2}\mathbb{E}{∥\left(\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)∥}^p;\hfill \\ \hfill & K_7\le 4^{p-1}{C}_p{L}_{\vartheta }{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left(2\rho \right)}^{(2-p)/2}{\int }_t^{+\infty }{e}^{2\rho (t-s)}\mathbb{E}{∥{\zeta }_n\left(s\right)-{\zeta }_0\left(s\right)∥}^{p}ds;\hfill \\ \hfill & K_8\le 4^{p-1}{C}_p{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p{\left(2\rho \right)}^{(-p)/2}\mathbb{E}{∥\left(\vartheta \left(s+r_n,{\zeta }_0\left(s\right)\right)-{\vartheta }_0\left(s,{\zeta }_0\left(s\right)\right)\right)∥}^p.\hfill \end{array}$$

Since the supremum ${sup}_{s\in \mathbb{R}}\mathbb{E}{\left|{\zeta }_0\left(s\right)\right|}^2$ is finite, the collection of variables ${{\zeta }_0\left(s\right)}_s$ exhibits tightness relative to bounded sets. This property ensures that the expression converges to zero as n approaches infinity. Similarly, the term on the right-hand side also tends to zero, based on the same reasoning established for the terms $K_2$, $K_4$, $K_6$, and $K_8$.

$$\begin{array}{cc}\hfill & \mathbb{E}{∥{\zeta }_n\left(t\right)-{\zeta }_0\left(t\right)∥}^p\hfill \\ \hfill & \le a_n+K_1+K_3+K_5+K_7\hfill \end{array}$$

for a sequence $\left({\alpha }_n\right)$ such that ${lim}_{n\to \infty }{\alpha }_n=0$. We conclude by Lemma 3 and

$$4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p(2{\left|{\mathrm{\Xi }}^s\right|}^p+{\left|{\mathrm{\Xi }}^v\right|}^p)({\rho }^{1-p}{L}_{\gamma }+C_p{L}_{\vartheta }{\left(2\rho \right)}^{(2-p)/2})<\rho ,$$

that

$$\underset{n\to \infty }{lim}\mathbb{E}{∥{\zeta }_n\left(t\right)-{\zeta }_0\left(t\right)∥}^p=0.$$

The convergence of ${\zeta }_n\left(t\right)$ to ${\zeta }_0\left(t\right)$ in distribution is established. As ${\zeta }_n\left(t\right)$ and $\zeta (t+r_n)$ share the same distribution, we can conclude that $\zeta (t+r_n)$ also converges in distribution to the same limit, ${\zeta }_0\left(t\right)$.

$$\underset{n\to \infty }{lim}\widehat{\mu }(t+h_n+q_n)=law\left({\zeta }_0\left(t\right)\right)={\widehat{\mu }}_t^0.$$

By analogy, we can easily deduce that

$$\underset{n\to \infty }{lim}\underset{m\to \infty }{lim}\widehat{\mu }(t+h_n+q_m)={\widehat{\mu }}_t^0.$$

Thus, $\zeta$ is APD. □

Establishing the existence of a $\mu$-pseudo almost periodic solution in distribution requires the following condition.

H3. 

The functions γ and ϑ satisfy the following conditions:

1. 

For each $x\in L^p(K,\mathcal{Q})$, the functions $t\mapsto \gamma (t,x)$ and $t\mapsto \vartheta (t,x)$ are μ-PAP. They can be expressed as sums $\gamma ={\gamma }_1+{\gamma }_2$ and $\vartheta ={\vartheta }_1+{\vartheta }_2$, where ${\gamma }_1,{\vartheta }_1\in SAP$ and ${\gamma }_2,{\vartheta }_2\in \mathcal{E}$.

2. 

Both the functions and their components are Lipschitz continuous with respect to their second argument, uniformly in $t\in \mathbb{R}$. This means that for all $\xi ,\eta \in L^p(K,\mathcal{Q})$, there exist positive constants such that:

–

${\mathbb{E}\parallel \gamma (t,\xi )-\gamma (t,\eta )\parallel }^p\le L_{\gamma }\mathbb{E}{\parallel \xi -\eta \parallel }^p$

–

${\mathbb{E}\parallel \vartheta (t,\xi )-\vartheta (t,\eta )\parallel }^p\le L_{\vartheta }\mathbb{E}{\parallel \xi -\eta \parallel }^p$

–

$\mathbb{E}\parallel {\gamma }_1(t,\xi )-{\gamma }_1{(t,\eta )\parallel }^p\le L_{{\gamma }_1}\mathbb{E}{\parallel \xi -\eta \parallel }^p$

–

$\mathbb{E}\parallel {\gamma }_2(t,\xi )-{\gamma }_2{(t,\eta )\parallel }^p\le L_{{\gamma }_2}\mathbb{E}{\parallel \xi -\eta \parallel }^p$

–

$\mathbb{E}\parallel {\vartheta }_1(t,\xi )-{\vartheta }_1{(t,\eta )\parallel }^p\le L_{{\vartheta }_1}\mathbb{E}{\parallel \xi -\eta \parallel }^p$

–

$\mathbb{E}\parallel {\vartheta }_2(t,\xi )-{\vartheta }_2{(t,\eta )\parallel }^p\le L_{{\vartheta }_2}\mathbb{E}{\parallel \xi -\eta \parallel }^p$

Theorem 6.

Assuming conditions (H0)–(H3) are satisfied, Equation (3) is guaranteed to have a unique $L^p$-bounded solution if the following two inequalities hold:

$$\left(4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p\right)\left({\left|{\mathrm{\Xi }}^v\right|}^p+{\left|{\mathrm{\Xi }}^s\right|}^p\right)\left[{\rho }^{-p}\left(L_{{\gamma }_1}+L_{{\gamma }_2}\right)+{\left(2\rho \right)}^{-p/2}{C}_p\left(L_{{\vartheta }_1}+L_{{\vartheta }_2}\right)\right]<1$$

(5)

and

$$4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p(2{\left|{\mathrm{\Xi }}^s\right|}^p+{\left|{\mathrm{\Xi }}^v\right|}^p)({\rho }^{1-p}{L}_{{\gamma }_1}+C_p{L}_{{\vartheta }_1}{\left(2\rho \right)}^{(2-p)/2})<\rho .$$

(6)

Moreover, this unique solution is also μ-PAPD((5) ensures existence/uniqueness and which (6) secures almost periodicity in distribution).

Proof. 

To establish the existence and uniqueness of a solution, we introduce an operator $\mathrm{\Omega }:C_b(\mathbb{R},L^p(K,\mathcal{Q}))\to C_b(\mathbb{R},L^p(K,\mathcal{Q}))$ and seek its fixed point. The operator is defined as follows:

$$\begin{array}{cc}\hfill \left(\mathrm{\Omega }\zeta \right)\left(t\right)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,\zeta \left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,\zeta \left(s\right)\right)\right)dW\left(s\right).\hfill \end{array}$$

As demonstrated in Step 2 of the proof of Theorem 5, $\mathrm{\Omega }$ is a contraction. Therefore, it admits a unique fixed point ${\zeta }^{*}\left(t\right)$. By hypothesis, for any such fixed point ${\zeta }^{*}\in {\mathcal{C}}_b(\mathbb{R},L^p(K,\mathcal{Q}))$, the following holds

$$\begin{array}{cc}\hfill \left(\mathrm{\Omega }{\zeta }^{*}\right)\left(t\right)& =\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,{\zeta }^{*}\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\gamma \left(s,{\zeta }^{*}\left(s\right)\right)\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,{\zeta }^{*}\left(s\right)\right)\right)dW\left(s\right)\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0\vartheta \left(s,{\zeta }^{*}\left(s\right)\right)\right)dW\left(s\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left(\mathrm{\Omega }{\zeta }^{*}\right)\left(t\right)& =\left[\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_1(s,{\zeta }^{*}\left(s\right))\right)ds\right.\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\gamma }_1(s,{\zeta }^{*}\left(s\right))\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_1(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\hfill \\ \hfill & \left.+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_1(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\right]\hfill \\ \hfill & +\left[\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds\right.\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\hfill \\ \hfill & \left.+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\right].\hfill \end{array}$$

Since

$${\zeta }^{*}\left(t\right)={\zeta }_1^{*}\left(t\right)+{\zeta }_2^{*}\left(t\right),$$

therefore,

$$\begin{array}{cc}\hfill \left(\mathrm{\Omega }{\zeta }_1^{*}\right)\left(t\right)& =\left[\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_1(s,{\zeta }^{*}\left(s\right))\right)ds\right.\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\gamma }_1(s,{\zeta }^{*}\left(s\right))\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_1(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\hfill \\ \hfill & \left.+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_1(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\right],\hfill \\ \hfill \left(\mathrm{\Omega }{\zeta }_2^{*}\right)\left(t\right)& =\left[\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds\right.\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds\hfill \\ \hfill & +\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\hfill \\ \hfill & \left.+\underset{\lambda \to +\infty }{lim}{\int }_{+\infty }^t{\mathcal{V}}^v(t-s){\mathrm{\Xi }}^v{\tilde{B}}_{\lambda }\left(X_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)\right].\hfill \end{array}$$

Using (H0)–(H3), it follows from Theorem 5 that ${\zeta }_1^{*}\left(t\right)$ is almost periodicity in distribution. In order to show that ${\zeta }^{*}\left(t\right)$ is a $\mu$-pseudo almost periodic process in distribution, it is sufficient to prove that ${\zeta }_2^{*}\left(t\right)\in \mathcal{E}(\mathbb{R},L^p(K,\mathcal{Q}))$. The properties of $L^p$-continuity and $L^p$-boundedness for ${\zeta }_2^{*}\left(t\right)$ follow from reasoning identical to that presented in Step 1 of the proof of Theorem 5. The conclusion of the proof is reached by employing Hölder’s inequality and Fubini’s theorem. Specifically, considering the interval $I=[-r,r]$ for an arbitrary $r>0$, we obtain the following:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds∥}^{p}d\mu \left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0{\gamma }_2(t-s,{\zeta }^{*}(t-s))\right)ds∥}^{p}d\mu \left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\left(\mathbb{E}{\left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥·∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥ds\right]}^p\right)d\mu \left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{\mu \left(I\right)}}\left({\int }_{I}d\mu \left(t\right)\right)\left\{{\left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥ds\right]}^{p-1}\right.\hfill \\ \hfill & \left.\times \left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥·\mathbb{E}{∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}ds\right]\right\}\hfill \\ \hfill & \le {\displaystyle \frac{{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p}{\mu \left(I\right){\rho }^{p-1}}}{\int }_{I}d\mu \left(t\right){\int }_0^{+\infty }{e}^{-\rho s}\mathbb{E}{∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}ds\hfill \\ \hfill & ={\displaystyle \frac{{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p}{{\rho }^{p-1}}}{\int }_0^{+\infty }{e}^{-\rho s}\left({\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)\right)ds.\hfill \end{array}$$

Since

$$\left|e^{-\rho s}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)\right|\le e^{-\rho s}{∥\begin{array}{c}{\gamma }_2\end{array}∥}_{\infty }^p$$

and

$${\int }_0^{\infty }{e}^{-\rho s}{∥\begin{array}{c}{\gamma }_2\end{array}∥}_{\infty }^p\mathrm{d}s<\infty .$$

By employing the Lebesgue dominated convergence theorem and noting that the function space $\mathcal{E}\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$ remains invariant under translations, we obtain

$$\underset{r\to \infty }{lim}{\int }_0^{+\infty }{e}^{-\rho s}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\gamma }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)ds=0.$$

Which implies that

$${\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }\left(X_0{\gamma }_2(s,{\zeta }^{*}\left(s\right))\right)ds∥}^{p}d\mu \left(t\right)=0.$$

Let I be the interval $[-r,r]$ for a positive value r. By employing Hölder’s inequality in conjunction with Fubini’s theorem, we can deduce that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)∥}^{p}d\mu \left(t\right)\hfill \\ \hfill & \le C_p{\displaystyle \frac{1}{\mu \left(I\right)}}\left({\int }_I\mathbb{E}{\left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }{∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{2}ds\right]}^{p/2}d\mu \left(t\right)\right)\hfill \\ \hfill & \le C_p{\displaystyle \frac{1}{\mu \left(I\right)}}\left({\int }_{I}d\mu \left(t\right)\right)\left\{{\left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥ds\right]}^{(p-2)/2}\right.\hfill \\ \hfill & \left.\times \left[\underset{\lambda \to +\infty }{lim}{\int }_0^{+\infty }{∥{\mathcal{V}}^s\left(s\right){\mathrm{\Xi }}^s{\tilde{B}}_{\lambda }{X}_0∥}^2·\mathbb{E}{∥{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}ds\right]\right\}\hfill \\ \hfill & \le C_p{\displaystyle \frac{{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p}{\mu \left(I\right){\left(2\rho \right)}^{(p-2)/2}}}{\int }_{I}d\mu \left(t\right){\int }_0^{+\infty }{e}^{-2\rho s}\mathbb{E}{∥{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}ds\hfill \\ \hfill & =C_p{\displaystyle \frac{{\tilde{\kappa }}^p{\overline{\kappa }}^p{\left|{\mathrm{\Xi }}^v\right|}^p}{{\left(2\rho \right)}^{(p-2)/2}}}{\int }_0^{+\infty }{e}^{-2\rho s}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)ds.\hfill \end{array}$$

Since

$$\left|e^{-2\rho s}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)\right|\le e^{-2\rho s}{∥\begin{array}{c}{\vartheta }_2\end{array}∥}_{\infty }^p$$

and

$${\int }_0^{+\infty }{e}^{-2\rho s}{∥\begin{array}{c}{\vartheta }_2\end{array}∥}_{\infty }^p\mathrm{d}s<\infty .$$

Given the translation invariance of the space $\mathcal{E}\left(\mathbb{R},L^p\left(K,\mathcal{Q}\right),\mu \right)$, an application of the Lebesgue dominated convergence theorem yields the result,

$$\underset{r\to \infty }{lim}{\int }_0^{+\infty }{e}^{-2\rho s}{\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\vartheta }_2(t-s,{\zeta }^{*}(t-s))∥}^{p}d\mu \left(t\right)ds=0.$$

Which implies that

$${\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥\underset{\lambda \to +\infty }{lim}{\int }_{-\infty }^t{\mathcal{V}}^s(t-s){\mathrm{\Xi }}^s\left({\tilde{B}}_{\lambda }{X}_0{\vartheta }_2(s,{\zeta }^{*}\left(s\right))\right)dW\left(s\right)∥}^{p}d\mu \left(t\right)=0.$$

By analogy, we can easily deduce that

$${\displaystyle \frac{1}{\mu \left(I\right)}}{\int }_I\mathbb{E}{∥{\zeta }_2^{*}\left(t\right)∥}^{p}d\mu \left(t\right)=0.$$

The proof is concluded by establishing that the unique $L^p$-bounded solution of Equation (3) is, in fact, $\mu$-PAPD. Please refer to the following

Table 2. Comparison between $\mu$-pseudo almost periodic in distribution and $\mu$-pseudo almost automorphic in distribution.

Dimension	$\mathit{\mu }$-Pseudo Almost Periodic in Distribution	$\mathit{\mu }$-Pseudo Almost Automorphic in Distribution
Core intuition	The distribution repeats in time with an approximate periodic pattern	The distribution reappears via subsequences and backward subsequences (recurrence without strict periodicity)
Strictness	Stronger requirement: emphasizes uniform near-periodicity	More flexible: allows irregular, non-uniform recurrence
Mathematical foundation	Based on the definition of almost periodicity	Based on the definition of almost automorphy
Residual handling	Decomposable into “periodic-type distribution + $\mu$-small perturbation”	Decomposable into “automorphic-type distribution + $\mu$-small perturbation”
Inclusion relation	A subclass of the automorphic case (⊆)	Contains the periodic case (⊇)
Application scenario	When the solution’s distribution shows a stable near-periodic structure	When the solution’s distribution shows non-uniform, but still recurrent structure
Metaphor	Like a clock bell—rings at regular intervals, predictable and rigid	Like meeting an old friend—the timing is irregular, but encounters always recur
Typical features	Strong regularity, suited to symmetric or periodic disturbances	High flexibility, well-suited to irregular or asymmetric disturbances

for the differences between $\mu$-pseudo almost periodic in distribution and $\mu$-pseudo almost automorphic in distribution. □

4. Example

As an application of the result in Equation (1), we analyze a model based on the stochastic partial differential equation with delay presented below:

$$\left\{\begin{array}{c}dv(z,\eta )=\left[\mathrm{\Delta }v(z,\eta )+\tilde{\gamma }v(z,\eta )+sinz+sin\left(2\pi \sqrt{2}z\right)+e^{-\left|z\right|}\right]dz\hfill \\ +\left[{\displaystyle \frac{sin\left(2\pi z\right)+sin\left(\sqrt{2}z\right)}{2}}+e^{-\left|z\right|}\right]dW\left(z\right),z\in \mathbb{R},\eta \in \mathrm{\Theta },\hfill \\ v(z,\eta )=0,z\in \mathbb{R},\eta \in \partial \mathrm{\Theta },\hfill \\ v(\theta ,\xi )=\varrho (\theta ,\xi ),\theta \in [-r,0],\xi \in \mathrm{\Theta }.\hfill \end{array}\right.$$

(7)

The framework for this system is established on a filtered probability space $(\mathrm{\Theta },\mathcal{G},K,{\left\{{\mathcal{G}}_z\right\}}_{z\ge 0})$, where $\mathrm{\Theta }$ is the sample space. The function $v(z,\eta )$ evolves over a temporal variable $z\in [0,\infty )$ and a spatial variable $\eta \in D$, where D is a bounded, open subset of ${\mathbb{R}}^n$ with a smooth boundary $\partial D$.

The dynamics are governed by both deterministic and stochastic influences. The deterministic part includes spatial diffusion via the Laplacian $\mathrm{\Delta }$, a linear reaction term with a positive coefficient $\tilde{\gamma }>0$, and external forcing functions. Randomness is introduced by a standard one-dimensional Brownian motion $W\left(z\right)$, indexed by $z\in \mathbb{R}$, with respect to the filtration ${\mathcal{G}}_z=\sigma \{W\left(s\right)\mid s\le z\}$.

The system is constrained by a zero Dirichlet boundary condition on $\partial \mathrm{\Theta }$. A memory effect is incorporated through a fixed time delay $r>0$. Consequently, the initial state is specified by a continuous history function $\varrho \in C([-r,0]\times \overline{\mathrm{\Theta }},\mathbb{R})$. In line with this delay structure, the solution’s history segment at any time z is captured by the function $v_z$, defined by the relation $v_z(\theta ,\eta ):=v(z+\theta ,\eta )$ for all $\theta \in [-r,0]$ and $\eta \in \mathrm{\Theta }$.

The initial value problem can be used to reformulate the partial functional problem (7) as follows

$$\left\{\begin{array}{c}dv\left(z\right)=\left[Av\right(z)+L(v\left(z\right))+e(z\left)\right]dz+\delta \left(z\right)dW\left(z\right),z\in \mathbb{R},\hfill \\ v_0=\varrho \in {\mathcal{C}}_X,\hfill \end{array}\right.$$

where

$$\left\{\begin{array}{c}D\left(A\right)=\{v\in \mathcal{C}(\overline{\mathrm{\Theta }},\mathbb{R});\mathrm{\Delta }v\in \mathcal{C}(\overline{\mathrm{\Theta }},\mathbb{R})\mathrm{and}v=0\mathrm{on}\partial \mathrm{\Theta }\},\hfill \\ Av=\mathrm{\Delta }v.\hfill \end{array}\right.$$

Let $e:\mathbb{R}\to X$ be a continuous function. We also consider a bounded linear operator L from ${\mathcal{C}}_X$ to X, which is defined as a point-wise multiplication by $\tilde{\gamma }$, such that for any function $\varrho \in {\mathcal{C}}_X$, we have $\left(L\varrho \right)\left(\eta \right)=\tilde{\gamma }\varrho \left(\eta \right)$. $e\left(z\right)=sinz+sin2\pi \sqrt{2}z+e^{-\left|z\right|}$, for $z\in \mathbb{R},\varrho \in {\mathcal{C}}_X$ and $\eta \in \mathrm{\Theta }$. This function e is $\mu$-PAPD $\left(PAP\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)\right)$ and $\delta \left(z\right)$ is $PAP\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$.

Under the condition that the operator A gives rise to a semigroup ${\left(T\left(z\right)\right)}_{z\ge 0}$ on X that possesses exponential stability, it follows that

$$\parallel T\left(z\right)\parallel \le exp\left({\displaystyle \frac{{\gamma \left|\mathrm{\Theta }\right|}^{\frac{2}{i}}}{4\pi }}\right)exp(-\gamma z),\forall z\ge 0,$$

the parameter $\gamma >0$ is the smallest eigenvalue of $-\mathrm{\Delta }$ on the space $H_0^1\left(\mathrm{\Theta }\right)$. For every $z>0$, $T\left(z\right)$ is compact on the closure of the domain $D\left(A\right)$.

Consider $\mu \in \mathcal{M}$, where $d\mu \left(z\right)$ is given by ${\tilde{\rho }}_{\zeta }\left(z\right)dz$, where ${\tilde{\rho }}_{\zeta }\left(z\right)=e^{\zeta z}$ and $\zeta$ belongs to $\mathbb{R}$.

Suppose that the semigroup ${\left(V\left(z\right)\right)}_{z⩾0}$ is hyperbolic. According to Theorem 6, Equation (3) has a single mild solution in $PAP\left(\mathbb{R},L^p(K,\mathcal{Q}),\mu \right)$. As a result, the system (7) has a unique solution that is $\mu$-pseudo almost periodic in distribution.

Concluding example (explicit data and verification of N < 1).

• Linear part A and semigroup:

  –

  Spatial domain $D=(0,\pi )\subset \mathbb{R}$ with Dirichlet boundary; state space $X=L^2\left(D\right)$.

  –

  $Av=\mathrm{\Delta }v$ with $D\left(A\right)=H^2\left(D\right)\cap H_0^1\left(D\right)$.

  –

  The semigroup $T\left(t\right)=e^{tA}$ is analytic, compact for $t>0$, and exponentially stable:

  $$\parallel T\left(t\right)\parallel \le e^{-t},t\ge 0.$$

• Delay operator L:

  –

  Memory horizon $r=1$; history space $C:=C\left(\right[-1,0],X)$ with the sup norm ${\parallel ·\parallel }_C$.

  –

  $L:C\to X$, $L\varphi =\tilde{\gamma }\varphi \left(0\right)$, with $\tilde{\gamma }=0.2$.

• Measure $\mu$ and the space $L^p(K,\mathcal{Q})$:

  –

  Fix $p=2$.

  –

  $K=[0,1]$, $\mathcal{Q}=\mathcal{B}\left(\right[0,1\left]\right)$, with the Lebesgue probability measure.

  –

  $d\mu \left(t\right)=e^{\zeta t}dt$ with $\zeta =\frac{1}{4}$ (so $\zeta <1$, the decay rate of T).

• Nonlinearities $\gamma ,\vartheta$ ($\mu$-PAP in t, Lipschitz in the history/state):

  –

  Define the almost periodic plus decaying profiles

  $$e\left(t\right)=sint+sin\left(2\pi \sqrt{2}t\right)+e^{-\left|t\right|},\delta \left(t\right)=\frac{1}{2}\left(sin\left(2\pi t\right)+sin\left(\sqrt{2}t\right)\right)+e^{-\left|t\right|}.$$

  –

  For $\varphi \in C$ and $k\in K$, set

  $$\gamma (t,\varphi )\left(k\right):=e\left(t\right)+{\epsilon tanh(\parallel \varphi \parallel }_C{),\vartheta (t,\varphi )\left(k\right):=\delta \left(t\right)+\epsilon tanh(\parallel \varphi \parallel }_C).$$

  –

  Lipschitz bounds (since $|tanha-tanhb|\le |a-b|$):

  $${\parallel \gamma (t,\varphi )-\gamma (t,\psi )\parallel }_{L^2\left(K\right)}\le {\epsilon \parallel \varphi -\psi \parallel }_C{,\parallel \vartheta (t,\varphi )-\vartheta (t,\psi )\parallel }_{L^2\left(K\right)}\le \epsilon {\parallel \varphi -\psi \parallel }_C.$$

  Consequently, in the mean-square sense of Theorem 5, we can take

  $$L_{\gamma }={\epsilon }^2,L_{\vartheta }={\epsilon }^2.$$

• Hyperbolicity-related constants and auxiliary parameters:

  –

  Purely stable dichotomy: ${\mathrm{\Xi }}^v=0$.

  –

  Weighted stable bound:

  $${\mathrm{\Xi }}^s={\int }_0^{\infty }\parallel T\left(s\right)\parallel e^{\zeta s}ds\le {\int }_0^{\infty }{e}^{-(1-\zeta )s}ds={\displaystyle \frac{1}{1-\zeta }}={\displaystyle \frac{4}{3}}.$$

  –

  Take $\tilde{\kappa }=\overline{\kappa }=1$, Burkholder–Davis–Gundy constant $C_2=2$, and $\rho =1$.

Verification that N < 1.

Recall the criterion from Theorem 5:

$$N:=4^{p-1}{\tilde{\kappa }}^p{\overline{\kappa }}^p\left(|{\mathrm{\Xi }}^v{|}^p+{\left|{\mathrm{\Xi }}^s\right|}^p\right)\left({\rho }^{-p}{L}_{\gamma }+{\left(2\rho \right)}^{-p/2}{C}_p{L}_{\vartheta }\right)<1.$$

With $p=2$, $\tilde{\kappa }=\overline{\kappa }=1$, ${\mathrm{\Xi }}^v=0$, ${\mathrm{\Xi }}^s=\frac{4}{3}$, $\rho =1$, $C_2=2$, and $L_{\gamma }=L_{\vartheta }={\epsilon }^2$,

$$N=4·{\left(\frac{4}{3}\right)}^2·\left({\epsilon }^2+\frac{1}{2}·2·{\epsilon }^2\right)=4·{\displaystyle \frac{16}{9}}·\left(2{\epsilon }^2\right)={\displaystyle \frac{128}{9}}{\epsilon }^2.$$

Choosing $\epsilon =0.12$ gives

$$N={\displaystyle \frac{128}{9}}\times 0.{12}^2\approx 0.205<1.$$

Conclusion.

With the explicit parameters

$$A=\mathrm{\Delta }\mathrm{on}{H}^2\left(D\right)\cap H_0^1\left(D\right),L\varphi =0.2\varphi \left(0\right),K=[0,1],\mathcal{Q}=\mathcal{B}\left([0,1]\right),p=2,d\mu \left(t\right)=e^{t/4}dt,$$

and

$$\gamma (t,\varphi )\left(k\right)=e\left(t\right)+{0.12tanh(\parallel \varphi \parallel }_C{),\vartheta (t,\varphi )\left(k\right)=\delta \left(t\right)+0.12tanh(\parallel \varphi \parallel }_C),$$

the inequality $N<1$ holds, ensuring (by Theorem 5) a unique $L^2$-bounded mild solution to (3); by Theorem 6, this solution is $\mu$-pseudo almost periodic in distribution.

5. Conclusions

The existence and uniqueness of solutions of stochastic systems is a hot topic of research in the field of stochastic systems. Due to the importance and prevalence of periodic and almost periodic phenomena in nature, the periodic and almost periodic solutions of stochastic systems have always been emphasized by researchers. Since stochastic systems can have very complex periodic and almost periodic motions, it leads to the fact that the identification of the structure of periodic and almost periodic solutions for specific systems is still a relatively difficult problem. This paper focuses on the creation of appropriate criteria for the existence of unique $\mu$-pseudo almost periodic solutions in distribution to stochastic equations with forcing terms, using Bochner’s double sequences criterion, improved Gronwall’s lemma, Hölder’s inequality, and measure-theoretic techniques. By applying analytic conditions for inequalities and the fixed point theorem for contraction mappings, we can establish the existence of a unique $\mu$-pseudo almost periodic solution to a given stochastic equation in distribution. Powerful tools are given to the study of pseudo almost periodic solutions of given stochastic equations in distribution. These studies not only refine and develop the theory of stochastic systems, but also promote a deeper understanding of the dynamical properties of these physically important equations.