Almost Automorphic Solutions in Distribution for McKean–Vlasov SDEs Driven by Fractional Brownian Motion

Abstract

The main contribution of this paper is to prove the existence and uniqueness of almost automorphic solutions in distribution for a class of McKean–Vlasov SDEs driven by fractional Brownian motion, under appropriate conditions on the coefficients. The practical relevance of this result is illustrated by analyzing a stochastic heat equation on a bounded domain.

1. Introduction

Almost automorphy, which generalizes the classical notion of almost periodicity, is originally presented by Bochner [1] in the realm of differential geometry. We say that a continuous function f is almost automorphic if, given any sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$, we can extract a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ and find a function $\tilde{f}$ for which

$$\underset{n\to \infty }{lim}f(t+s_n)=\tilde{f}\left(t\right)\mathrm{and}\underset{n\to \infty }{lim}\tilde{f}(t-s_n)=f\left(t\right)$$

hold for all $t\in \mathbb{R}$. The theory of almost automorphy has attracted considerable attention over the years; classical treatments can be found in Veech [2,3], while more recent developments are due to the works [4,5,6], among others.

In the last few decades, considerable attention has been devoted to the development of almost periodic and almost automorphic theories for stochastic systems. Relevant contributions include the study [7], which explores the link between, on the one hand, the existence of almost periodic solutions and, on the other hand, exponential stability or exponential dichotomy within finite-dimensional linear inhomogeneous problems; the work [8], which addresses almost periodic solutions for affine stochastic evolution equations; authors [9] investigate periodic or almost periodic solutions in distribution for semilinear stochastic equations; researchers [10] focus on almost periodic solutions in distribution of affine SDEs; and references [11,12] examine square-mean almost periodic solutions for stochastic evolution equations; see also the more recent works [13,14]. The study of almost automorphic solutions to SDEs driven by Gaussian noise, as well as their broader extensions, commenced around 2010, initiated by [15]. Since then, this concept has attracted considerable interest from numerous researchers, including [16,17,18,19] and subsequent studies [20,21]. More recently, the authors of [22] examined almost automorphic solutions in the sense of distribution for mean-field SDEs driven by fBm with Hurst parameter $H\in (1/2,1)$. In a different line of research, the work of [23] employed variational methods to explore the recurrent behavior of solutions to SPDEs, offering an alternative to the conventional semigroup framework. Focusing on the mean-field setting, the authors of [24] investigated time-dependent stochastic evolution equations under fBm with Hurst parameter $H\in (1/2,1)$. Specifically, they proved the existence of mild solutions that are either almost automorphic or weighted pseudo almost automorphic.

Due to their extensive applicability across domains including stochastic control, queueing theory, mathematical finance, multi-factor stochastic volatility models, and statistical physics, McKean–Vlasov SDEs have attracted increasing attention in recent years. In such equations, both the drift and diffusion coefficients are allowed to depend not only on the current state of the process but also on its probability law; see [25,26,27,28].

Unlike standard Brownian motion, fractional Brownian motion is characterized by the property of long-range dependence and is neither a semimartingale nor a Markov process. One might therefore expect that the almost automorphic in distribution property could be lost when the driving noise is replaced by fBm. However, as we show in Theorem 1, this is not the case under the given Lipschitz and contraction conditions. The key observation is that the Wiener integral with respect to fBm, when applied to deterministic coefficient functions, preserves the recurrence of the coefficients through the Hardy–Littlewood–Sobolev inequality and the contraction estimates in the $L^2$ sense. In particular, the fractional noise does not introduce any additional drifting effect that would destroy the almost automorphy in distribution, provided the semigroup decays exponentially and the coefficients satisfy the standard Lipschitz conditions in the state and the measure argument.

2. Preliminaries

In the whole paper, $(\mathbb{H},\parallel ·\parallel )$ is assumed to be a real separable Hilbert space. Let $(\Omega ,\mathcal{F},\mathbf{P})$ be a complete filtered probability space, and denote by $L^2(\mathbf{P},\mathbb{H})$ the set of all $\mathbb{H}$-valued random variables X such that

$$\mathbf{E}{\parallel X\parallel }^2={\int }_{\Omega }{\parallel X\parallel }^{2}d\mathbf{P}<\infty .$$

It can be readily checked that $L^2(\mathbf{P},\mathbb{H})$ forms a Hilbert space under the norm $(\mathbf{E}{\parallel ·\parallel }^2{)}^{1/2}$. Let $\mathcal{P}\left(\mathbb{H}\right)$ stand for the space of Borel probability measures on $\mathbb{H}$ endowed with the weak topology, and for any $p>1$, let ${\mathcal{P}}_p\left(\mathbb{H}\right)$ denote the subspace of $\mathcal{P}\left(\mathbb{H}\right)$ consisting of those probability measures having finite moment of order p, i.e.,

$${\mathcal{P}}_p\left(\mathbb{H}\right)=\left\{\mu \in \mathcal{P}\left(\mathbb{H}\right):{\int }_{\mathbb{H}}{\parallel x\parallel }^p\mu \left(dx\right)<\infty \right\}.$$

It is a standard fact that ${\mathcal{P}}_p\left(\mathbb{H}\right)$ equipped with the p-Wasserstein distance produces

$$W_p(\mu ,\nu ):=\underset{\pi \in \prod (\mu ,\nu )}{inf}{\left({\int }_{\mathbb{H}\times \mathbb{H}}{|x-y|}^p\pi (dx,dy)\right)}^{1/p},\mu ,\nu \in {\mathcal{P}}_p\left(\mathbb{H}\right),$$

where $\prod (\mu ,\nu )$ denotes the collection of couplings between $\mu$ and $\nu$; in other words, $\pi \in \prod (\mu ,\nu )$ is a probability measure on $\mathbb{H}\times \mathbb{H}$ whose respective first and second marginals coincide with $\mu$ and $\nu$. In the particular case where $\mu ={\mu }_X$ and $\nu ={\mu }_Y$ correspond to the probability laws of two order-p random variables X and Y, then

$$W_p({\mu }_X,{\mu }_Y)\le (\mathbf{E}{\parallel X-Y\parallel }^p{)}^{1/p}.$$

(1)

The mathematical literature primarily utilizes the second-order Wasserstein metric $W_2$, with the first-order version $W_1$ being conventionally designated as the Kantorovich–Rubinstein distance, acknowledging its pivotal function in optimal transportation theory; see [29] for more details.

Definition 1

([15]). A stochastic process $X:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ is called stochastically continuous whenever

$$\underset{t\to s}{lim}\mathbf{E}{\parallel X_t-X_s\parallel }^2=0.$$

A stochastically continuous stochastic process $X:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ is said to be square-mean almost automorphic provided that for every sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$ there exists a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ and some stochastic process $Y:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ such that

$$\underset{n\to \infty }{lim}\mathbf{E}\parallel X_{t+s_n}-Y_t{\parallel }^2=0\mathit{and}\underset{n\to \infty }{lim}\mathbf{E}{\parallel Y_{t-s_n}-X_t\parallel }^2=0$$

hold for each $t\in \mathbb{R}$. The set consisting of all square-mean almost automorphic stochastic processes $X:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ is denoted by $AA(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$, which forms a Banach space when endowed with the norm ${\parallel X\parallel }_{\infty }:={sup}_{t\in \mathbb{R}}(\mathbf{E}\parallel X_t{{\parallel }^2)}^{\frac{1}{2}}$.

Definition 2.

A function $g:\mathbb{R}\times L^2(\mathbf{P},\mathbb{H})\times {\mathcal{P}}_2\left(\mathbb{H}\right)\to L^2(\mathbf{P},\mathbb{H}),(t,X,{\mu }_X)\mapsto g(t,X,{\mu }_X)$, which is jointly continuous, is said to be square-mean almost automorphic in $t\in \mathbb{R}$ for each fixed $X\in L^2(\mathbf{P},\mathbb{H})$ and its corresponding law ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$ if for every sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$, one can extract a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ such that for some function $\tilde{g}$,

$$\underset{n\to \infty }{lim}\mathbf{E}{∥g(t+s_n,X,{\mu }_X)-\tilde{g}(t,X,{\mu }_X)∥}^2=0\mathit{and}\underset{n\to \infty }{lim}\mathbf{E}{∥\tilde{g}(t-s_n,X,{\mu }_X)-g(t,X,{\mu }_X)∥}^2=0$$

hold for every $t\in \mathbb{R}$, $X\in L^2(\mathbf{P},\mathbb{H})$, and ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$.

Definition 3.

A function $h:\mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{H}\right)\to \mathbb{H},(t,{\mu }_X)\mapsto h(t,{\mu }_X)$, which is jointly continuous, is called almost automorphic in $t\in \mathbb{R}$ for each law ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$ if for every sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$, there exists a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ for which some function $\tilde{h}$ satisfies

$$\underset{n\to \infty }{lim}\parallel h(t+s_n,{\mu }_X)-\tilde{h}(t,{\mu }_X)\parallel =0\mathit{and}\underset{n\to \infty }{lim}\parallel \tilde{h}(t-s_n,{\mu }_X)-h(t,{\mu }_X)\parallel =0$$

for all $t\in \mathbb{R}$ and ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$.

Next, we turn to the notion of almost automorphy in distribution. For any $\mu ,\nu \in \mathcal{P}\left(\mathbb{H}\right)$, we define a metric as follows:

$$\mathbf{d}(\mu ,\nu ):=\underset{\parallel g\parallel \le 1}{sup}\left\{\int gd\mu -\int gd\nu \right\},$$

where g ranges over Lipschitz continuous real-valued functions on $\mathbb{H}$ with the norms

$$\parallel g\parallel ={\parallel g\parallel }_L+{\parallel g\parallel }_{\infty },{\parallel g\parallel }_L=\underset{x\ne y}{sup}{\displaystyle \frac{g\left(x\right)-g\left(y\right)}{\parallel x-y\parallel }},{\parallel g\parallel }_{\infty }=\underset{x\in \mathbb{H}}{sup}\left|g\left(x\right)\right|.$$

Definition 4

([20]). An $\mathbb{H}$-valued stochastic process $X_t$ is called almost automorphic in distribution whenever its law ${\mu }_{X_t}$ forms a $\mathcal{P}\left(\mathbb{H}\right)$-valued almost automorphic mapping. This means that for every sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$, one can find a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ and a $\mathcal{P}\left(\mathbb{H}\right)$-valued mapping ${\tilde{\mu }}_t$ such that

$$\underset{n\to \infty }{lim}\mathbf{d}({\mu }_{X_{t+s_n}},{\tilde{\mu }}_t)=0\mathit{and}\underset{n\to \infty }{lim}\mathbf{d}({\tilde{\mu }}_{t-s_n},{\mu }_{X_t})=0$$

hold for all $t\in \mathbb{R}$.

Remark 1.

A sequence ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }\subset \mathcal{P}\left(\mathbb{H}\right)$ is said to converge weakly to μ if $\int gd{\mu }_n\to \int gd\mu$ holds for every bounded continuous real-valued function g. It is a standard fact that the metric d is complete on $\mathcal{P}\left(\mathbb{H}\right)$, and that a sequence $\left\{{\mu }_n\right\}$ converges weakly to μ if and only if $\mathbf{d}({\mu }_n,\mu )\to 0$ as $n\to \infty$.

The above preliminaries provide the necessary functional setting and stability estimates, which play a crucial role in the subsequent contraction analysis.

3. Almost Automorphic Solutions in Distribution

Consider the following McKean–Vlasov SDEs driven by fBm

$$d{X}_t=A{X}_{t}dt+f(t,X_t,{\mu }_{X_t})dt+g(t,X_t,{\mu }_{X_t})d{B}_t+h(t,{\mu }_{X_t})d{B}_t^H,t\in \mathbb{R},$$

(2)

where A is an infinitesimal generator that gives rise to a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ such that

$$\parallel T_t\parallel \le K{e}^{-\omega t},\mathrm{for}\mathrm{all}t\ge 0$$

(3)

with $K>0$ and $\omega >0$; $f:\mathbb{R}\times L^2(\mathbf{P},\mathbb{H})\times {\mathcal{P}}_2\left(\mathbb{H}\right)\to L^2(\mathbf{P},\mathbb{H})$, $g:\mathbb{R}\times L^2(\mathbf{P},\mathbb{H})\times {\mathcal{P}}_2\left(\mathbb{H}\right)\to L^2(\mathbf{P},\mathbb{H})$, $h:\mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{H}\right)\to \mathbb{H}$, $B_t$ is a one-dimensional Brownian motion on a complete probability space $(\Omega ,\mathcal{F},\mathbf{P},{\mathcal{F}}_t)$, and $B_t^H$ is a one-dimensional fBm independent of $B_t$ with Hurst parameter $H\in (1/2,1)$ and covariance

$$\mathbf{E}\left[B_t^H{B}_s^H\right]={\displaystyle \frac{1}{2}}\left({\left|t\right|}^{2H}+{\left|s\right|}^{2H}-{|t-s|}^{2H}\right).$$

$Note$: Here f and g denote two stochastic processes, while h is a deterministic function.

Definition 5.

A process ${\left\{X_t\right\}}_{t\in \mathbb{R}}$ that is ${\mathcal{F}}_t$-progressively measurable is termed a mild solution to Equation (2) if the following stochastic integral equation is satisfied

$$X_t=T_{t-a}{X}_a+{\int }_a^t{T}_{t-s}f(s,X_s,{\mu }_{X_s})ds+{\int }_a^t{T}_{t-s}g(s,X_s,{\mu }_{X_s})d{B}_s+{\int }_a^t{T}_{t-s}h(s,{\mu }_{X_s})d{B}_s^H$$

(4)

holds for all $t\ge a$ and for each $a\in \mathbb{R}$, where the stochastic integral driven by fractional Brownian motion is understood in the sense of Wiener integral in the Hilbert space setting.

In the integral Equation (4), by letting $a\to -\infty$ and invoking the exponential dissipation condition (3) of T, we deduce that the stochastic process $X:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ is a mild solution to (2) if and only if it satisfies the stochastic integral equation

$$X_t={\int }_{-\infty }^t{T}_{t-s}f(s,X_s,{\mu }_{X_s})ds+{\int }_{-\infty }^t{T}_{t-s}g(s,X_s,{\mu }_{X_s})d{B}_s+{\int }_{-\infty }^t{T}_{t-s}h(s,{\mu }_{X_s})d{B}_s^H.$$

(5)

Let $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ denote the Banach space consisting of all stochastically continuous and uniformly bounded stochastic processes, equipped with the norm ${\parallel X\parallel }_{\infty }:={sup}_{t\in \mathbb{R}}(\mathbf{E}\parallel X_t{{\parallel }^2)}^{\frac{1}{2}}$. For any $X\in C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$, we define a nonlinear operator $\mathcal{T}$ on the Banach space $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ as follows:

$$\begin{array}{cc}\hfill {\left(\mathcal{T}X\right)}_t:& ={\int }_{-\infty }^t{T}_{t-s}f(s,X_s,{\mu }_{X_s})ds+{\int }_{-\infty }^t{T}_{t-s}g(s,X_s,{\mu }_{X_s})d{B}_s+{\int }_{-\infty }^t{T}_{t-s}h(s,{\mu }_{X_s})d{B}_s^H\hfill \\ \hfill :& ={\left({\mathcal{T}}_{1}X\right)}_t+{\left({\mathcal{T}}_{2}X\right)}_t+{\left({\mathcal{T}}_{3}X\right)}_t.\hfill \end{array}$$

(6)

If it can be shown that the operator $\mathcal{T}$ maps $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ into itself and is contractive, then by virtue of the Banach fixed-point theorem, we may deduce that there is a unique mild solution in the space $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ to Equation (2).

Lemma 1.

Assume that f and g are square-mean almost automorphic in $t\in \mathbb{R}$ for every $X\in L^2(\mathbf{P},\mathbb{H})$ and the corresponding law ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$ according to Definition 2, and h is almost automorphic in $t\in \mathbb{R}$ for every law ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$ as per Definition 3. In addition, f, g, and h fulfill the Lipschitz conditions: that is, for any $X,Y\in L^2(\mathbf{P},\mathbb{H})$, the associated laws ${\mu }_X,{\mu }_Y\in {\mathcal{P}}_2\left(\mathbb{H}\right)$, and all $t\in \mathbb{R}$,

$$\begin{array}{cc}\hfill \mathbf{E}\parallel f(t,Y,{\mu }_Y)-f(t,X,{\mu }_X){\parallel }^2& \le L_1\left({\mathbf{E}\parallel Y-X\parallel }^2+W_2{({\mu }_Y,{\mu }_X)}^2\right),\hfill \\ \hfill \mathbf{E}\parallel g(t,Y,{\mu }_Y)-g(t,X,{\mu }_X){\parallel }^2& \le L_2\left({\mathbf{E}\parallel Y-X\parallel }^2+W_2{({\mu }_Y,{\mu }_X)}^2\right),\hfill \\ \hfill \parallel h(t,{\mu }_Y)-h(t,{\mu }_X)\parallel & \le L_3{W}_2({\mu }_Y,{\mu }_X).\hfill \end{array}$$

Then the process $\mathcal{T}X:\mathbb{R}\to L^2(\mathbf{P},\mathbb{H})$ is stochastically continuous.

Proof. 

By the Lipschitz property of f, (1) and square-mean almost automorphic in t of f, we have

$$\begin{array}{cc}\hfill \underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel f(t,X_t,{\mu }_{X_t})\parallel }^2& \le 2\underset{t\in \mathbb{R}}{sup}\mathbf{E}\parallel f(t,X_t,{\mu }_{X_t})-f(t,0,{\mu }_0){\parallel }^2+2\underset{t\in \mathbb{R}}{sup}{\parallel f(t,0,{\mu }_0)\parallel }^2\hfill \\ \hfill & \le 2L_1\underset{t\in \mathbb{R}}{sup}\left(\mathbf{E}\parallel X_t{\parallel }^2+W_2{({\mu }_{X_t},{\mu }_0)}^2\right)+2\underset{t\in \mathbb{R}}{sup}{\parallel f(t,0,{\mu }_0)\parallel }^2\hfill \\ \hfill & \le 4L_1\underset{t\in \mathbb{R}}{sup}\mathbf{E}\parallel X_t{\parallel }^2+2\underset{t\in \mathbb{R}}{sup}{\parallel f(t,0,{\mu }_0)\parallel }^2<\infty ,\hfill \end{array}$$

i.e.,

$$\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel f(t,X_t,{\mu }_{X_t})\parallel }^2\le M_1<\infty .$$

(7)

Similarly, we can derive

$$\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel g(t,X_t,{\mu }_{X_t})\parallel }^2\le M_2<\infty ,$$

(8)

and

$$\underset{t\in \mathbb{R}}{sup}\parallel h(t,{\mu }_{X_t})\parallel \le M_3<\infty .$$

(9)

Now we show that $\mathcal{T}X$ is well-defined in $L^2(\mathbf{P},\mathbb{H})$. For the first term of (6), by (7),

$$\begin{array}{cc}\hfill \mathbf{E}{\parallel {\left({\mathcal{T}}_{1}X\right)}_t\parallel }^2& \le \mathbf{E}{\left({\int }_{-\infty }^t∥T_{t-s}f(s,X_s,{\mu }_{X_s})∥ds\right)}^2\hfill \\ \hfill & \le K^2\mathbf{E}{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}∥f(s,X_s,{\mu }_{X_s})∥ds\right)}^2\hfill \\ \hfill & \le K^2\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\mathbf{E}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}{∥f(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel f(t,X_t,{\mu }_{X_t})\parallel }^2{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)}^2\le {\displaystyle \frac{M_1{K}^2}{{\omega }^2}}.\hfill \end{array}$$

For the second term of (6), by Itô isometry and (8),

$$\begin{array}{cc}\hfill \mathbf{E}{\parallel {\left({\mathcal{T}}_{2}X\right)}_t\parallel }^2& =\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}g(s,X_s,{\mu }_{X_s})d{B}_s∥}^2\hfill \\ \hfill & =\mathbf{E}\left({\int }_{-\infty }^t{∥T_{t-s}g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2\mathbf{E}\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}{∥g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel g(t,X_t,{\mu }_{X_t})\parallel }^2\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}ds\right)\le {\displaystyle \frac{M_2{K}^2}{2\omega }}.\hfill \end{array}$$

For the third term of (6), by fractional Itô isometry and (9),

$$\begin{array}{cc}\hfill \mathbf{E}{\parallel {\left({\mathcal{T}}_{3}X\right)}_t\parallel }^2& =\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}h(s,{\mu }_{X_s})d{B}_s^H∥}^2\hfill \\ \hfill & =H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t{〈T_{t-s}h(s,{\mu }_{X_s}),T_{t-r}h(r,{\mu }_{X_r})〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le K^{2}H(2H-1){\left(\underset{t\in \mathbb{R}}{sup}\parallel h(t,{\mu }_{X_t})\parallel \right)}^2{\int }_{-\infty }^t{\int }_{-\infty }^t{e}^{-\omega (t-s)}{e}^{-\omega (t-r)}{|s-r|}^{2H-2}dsdr.\hfill \end{array}$$

Utilizing Hardy–Littlewood–Sobolev inequality,

$${\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\left|f\left(s\right)\right|\left|f\left(t\right)\right||s-t|}^{2H-2}dsdt\le C_H{\parallel f\parallel }_{L^{1/H}\left(\mathbb{R}\right)}^2,f\in L^{1/H}\left(\mathbb{R}\right),$$

(10)

where the optimal constant $C_H$ is given in Lieb’s paper [30]. Then we obtain that

$$\begin{array}{c}\hfill \mathbf{E}\parallel {\left({\mathcal{T}}_{3}X\right)}_t{\parallel }^2\le M_3^2{K}^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}ds\right)}^{2H}=M_3^2{K}^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H}.\end{array}$$

We now show that $t\mapsto {\left(\mathcal{T}X\right)}_t$ is stochastically continuous. For the first term ${\left({\mathcal{T}}_{1}X\right)}_t$,

$$\begin{array}{cc}\hfill & \mathbf{E}\parallel {\left({\mathcal{T}}_{1}X\right)}_{t_2}-{\left({\mathcal{T}}_{1}X\right)}_{t_1}{\parallel }^2=\mathbf{E}{∥{\int }_{-\infty }^{t_2}{T}_{t_2-s}f(s,X_s,{\mu }_{X_s})ds-{\int }_{-\infty }^{t_1}{T}_{t_1-s}f(s,X_s,{\mu }_{X_s})ds∥}^2\hfill \\ \hfill & \le 2\mathbf{E}{∥{\int }_{t_1}^{t_2}{T}_{t_2-s}f(s,X_s,{\mu }_{X_s})ds∥}^2+2\mathbf{E}{∥{\int }_{-\infty }^{t_1}{T}_{t_2-s}f(s,X_s,{\mu }_{X_s})-T_{t_1-s}f(s,X_s,{\mu }_{X_s})ds∥}^2\hfill \\ \hfill & \le 2\mathbf{E}{\left({\int }_{t_1}^{t_2}∥T_{t_2-s}f(s,X_s,{\mu }_{X_s})∥ds\right)}^2+2\mathbf{E}{\left({\int }_{-\infty }^{t_1}∥T_{t_2-s}f(s,X_s,{\mu }_{X_s})-T_{t_1-s}f(s,X_s,{\mu }_{X_s})∥ds\right)}^2\hfill \\ \hfill :& =2A_1+2A_2.\hfill \end{array}$$

As for the term $A_1$, by (7), we derive that as $t_2\to t_1$,

$$\begin{array}{cc}\hfill A_1& \le K^2\mathbf{E}{\left({\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)}∥f(s,X_s,{\mu }_{X_s})∥ds\right)}^2\hfill \\ \hfill & \le K^2\left({\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)}ds\right)\mathbf{E}\left({\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)}{∥f(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel f(t,X_t,{\mu }_{X_t})\parallel }^2{\left({\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)}ds\right)}^2\le {\displaystyle \frac{M_1{K}^2}{{\omega }^2}}{\left(1-e^{-\omega (t_2-t_1)}\right)}^2\to 0.\hfill \end{array}$$

As for the term $A_2$, by the semigroup property, we can obtain

$$\begin{array}{cc}\hfill A_2& \le K^2\mathbf{E}{\left({\int }_{-\infty }^{t_1}{e}^{-\omega (t_1-s)}∥T_{t_2-t_1}f(s,X_s,{\mu }_{X_s})-f(s,X_s,{\mu }_{X_s})∥ds\right)}^2\hfill \\ \hfill & \le K^2{\int }_{-\infty }^{t_1}{e}^{-\omega (t_1-s)}ds\mathbf{E}\left({\int }_{-\infty }^{t_1}{e}^{-\omega (t_1-s)}{∥T_{t_2-t_1}f(s,X_s,{\mu }_{X_s})-f(s,X_s,{\mu }_{X_s})∥}^{2}ds\right).\hfill \end{array}$$

Hence one can show that $A_2\to 0$ as $t_2\to t_1$ by the dominated convergence theorem with dominating function $F\left(s\right):=2e^{-\omega (t_1-s)}{M}_1$. For the second term ${\left({\mathcal{T}}_{2}X\right)}_t$, by Itô isometry, we have

$$\begin{array}{cc}\hfill & \mathbf{E}\parallel {\left({\mathcal{T}}_{2}X\right)}_{t_2}-{\left({\mathcal{T}}_{2}X\right)}_{t_1}{\parallel }^2=\mathbf{E}{∥{\int }_{-\infty }^{t_2}{T}_{t_2-s}g(s,X_s,{\mu }_{X_s})d{B}_s-{\int }_{-\infty }^{t_1}{T}_{t_1-s}g(s,X_s,{\mu }_{X_s})d{B}_s∥}^2\hfill \\ \hfill & \le 2\mathbf{E}{∥{\int }_{t_1}^{t_2}{T}_{t_2-s}g(s,X_s,{\mu }_{X_s})d{B}_s∥}^2+2\mathbf{E}{∥{\int }_{-\infty }^{t_1}{T}_{t_2-s}g(s,X_s,{\mu }_{X_s})-T_{t_1-s}g(s,X_s,{\mu }_{X_s})d{B}_s∥}^2\hfill \\ \hfill & =2\mathbf{E}\left({\int }_{t_1}^{t_2}{∥T_{t_2-s}g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)+2\mathbf{E}\left({\int }_{-\infty }^{t_1}{∥T_{t_2-s}g(s,X_s,{\mu }_{X_s})-T_{t_1-s}g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill :& =2B_1+2B_2.\hfill \end{array}$$

As for the term $B_1$, by (8), we derive that as $t_2\to t_1$,

$$\begin{array}{cc}\hfill B_1& \le K^2\mathbf{E}\left({\int }_{t_1}^{t_2}{e}^{-2\omega (t_2-s)}{∥g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel g(t,X_t,{\mu }_{X_t})\parallel }^2\left({\int }_{t_1}^{t_2}{e}^{-2\omega (t_2-s)}ds\right)\le {\displaystyle \frac{M_2{K}^2}{2\omega }}\left(1-e^{-2\omega (t_2-t_1)}\right)\to 0.\hfill \end{array}$$

As for the term $B_2$, by the semigroup property, we can obtain

$$\begin{array}{cc}\hfill B_2& \le K^2\mathbf{E}\left({\int }_{-\infty }^{t_1}{e}^{-2\omega (t_1-s)}{∥T_{t_2-t_1}g(s,X_s,{\mu }_{X_s})-g(s,X_s,{\mu }_{X_s})∥}^{2}ds\right).\hfill \end{array}$$

Hence one can show that $B_2\to 0$ as $t_2\to t_1$ by the dominated convergence theorem with dominating function $F\left(s\right):=2e^{-2\omega (t_1-s)}{M}_2$. For the third term ${\left({\mathcal{T}}_{3}X\right)}_t$, by fractional Itô isometry, we have

$$\begin{array}{cc}\hfill \mathbf{E}& \parallel {\left({\mathcal{T}}_{3}X\right)}_{t_2}-{\left({\mathcal{T}}_{3}X\right)}_{t_1}{\parallel }^2=\mathbf{E}{∥{\int }_{-\infty }^{t_2}{T}_{t_2-s}h(s,{\mu }_{X_s})d{B}_s^H-{\int }_{-\infty }^{t_1}{T}_{t_1-s}h(s,{\mu }_{X_s})d{B}_s^H∥}^2\hfill \\ \hfill \le & 2\mathbf{E}{∥{\int }_{t_1}^{t_2}{T}_{t_2-s}h(s,{\mu }_{X_s})d{B}_s^H∥}^2+2\mathbf{E}{∥{\int }_{-\infty }^{t_1}{T}_{t_2-s}h(s,{\mu }_{X_s})-T_{t_1-s}h(s,{\mu }_{X_s})d{B}_s^H∥}^2\hfill \\ \hfill =& 2H(2H-1){\int }_{t_1}^{t_2}{\int }_{t_1}^{t_2}{〈T_{t_2-s}h(s,{\mu }_{X_s}),T_{t_2-r}h(r,{\mu }_{X_r})〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & +2H(2H-1){\int }_{-\infty }^{t_1}{\int }_{-\infty }^{t_1}{〈T_{t_2-s}h(s,{\mu }_{X_s})-T_{t_1-s}h(s,{\mu }_{X_s}),T_{t_2-r}h(r,{\mu }_{X_r})-T_{t_1-r}h(r,{\mu }_{X_r})〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill :=& 2H(2H-1)C_1+2H(2H-1)C_2.\hfill \end{array}$$

As for the term $C_1$, by (9) and inequality (10), we derive that as $t_2\to t_1$,

$$\begin{array}{cc}\hfill C_1& \le K^2{\left(\underset{t\in \mathbb{R}}{sup}\parallel h(t,{\mu }_{X_t})\parallel \right)}^2{\int }_{t_1}^{t_2}{\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)}{e}^{-\omega (t_2-r)}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le M_3^2{K}^2{C}_H{\left({\int }_{t_1}^{t_2}{e}^{-\omega (t_2-s)/H}ds\right)}^{2H}=M_3^2{K}^2{C}_H{\left({\displaystyle \frac{H}{\omega }}\left(1-e^{-\omega (t_2-t_1)/H}\right)\right)}^{2H}\to 0.\hfill \end{array}$$

As for the term $C_2$, by inequality (10) and the semigroup property, we can obtain

$$\begin{array}{cc}\hfill C_2& \le {\int }_{-\infty }^{t_1}{\int }_{-\infty }^{t_1}∥T_{t_2-s}h(s,{\mu }_{X_s})-T_{t_1-s}h(s,{\mu }_{X_s})∥∥T_{t_2-r}h(r,{\mu }_{X_r})-T_{t_1-r}h(r,{\mu }_{X_r})∥{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le C_H{\left({\int }_{-\infty }^{t_1}{∥T_{t_2-s}h(s,{\mu }_{X_s})-T_{t_1-s}h(s,{\mu }_{X_s})∥}^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le K^2{C}_H{\left({\int }_{-\infty }^{t_1}{e}^{-\omega (t_1-s)/H}{∥T_{t_2-t_1}h(s,{\mu }_{X_s})-h(s,{\mu }_{X_s})∥}^{\frac{1}{H}}ds\right)}^{2H}.\hfill \end{array}$$

Hence one can show that $C_2\to 0$ as $t_2\to t_1$ by the dominated convergence theorem with dominating function $F\left(s\right):=e^{-\omega (t_1-s)/H}{\left(2M_3\right)}^{1/H}$. □

We now present the main results of this paper.

Theorem 1.

Under the same assumptions as in Lemma 1, Equation (2) has a unique mild solution in the space $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$, provided

$${\displaystyle \frac{6K^2{L}_1}{{\omega }^2}}+{\displaystyle \frac{3K^2{L}_2}{\omega }}+{\displaystyle \frac{3K^2{L}_3^{2}H(2H-1)C_H{H}^{2H}}{{\omega }^{2H}}}<1,$$

where the constant $C_H$ appears in the inequality (10). Moreover, this mild solution is almost automorphic in distribution, provided

$${\displaystyle \frac{12K^2{L}_1}{{\omega }^2}}+{\displaystyle \frac{12K^2{L}_2}{\omega }}+{\displaystyle \frac{6K^2{L}_3^{2}H(2H-1)C_H{H}^{2H-1}}{{\omega }^{2H}}}<1.$$

(11)

Proof. 

We now establish that the operator $\mathcal{T}$ acts as a contraction mapping on $C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$. Given $X,Y\in C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ and an arbitrary $t\in \mathbb{R}$, it follows that

$$\begin{array}{cc}\hfill {\mathbf{E}\parallel \left(\mathcal{T}X\right)}_t-{\left(\mathcal{T}Y\right)}_t{\parallel }^2=& \mathbf{E}∥{\int }_{-\infty }^t{T}_{t-s}\left(f(s,X_s,{\mu }_{X_s})-f(s,Y_s,{\mu }_{Y_s})\right)ds\hfill \\ \hfill & +{\int }_{-\infty }^t{T}_{t-s}\left(g(s,X_s,{\mu }_{X_s})-g(s,Y_s,{\mu }_{Y_s})\right)d{B}_s\hfill \\ \hfill & {+{\int }_{-\infty }^t{T}_{t-s}\left(h(s,{\mu }_{X_s})-h(s,{\mu }_{Y_s})\right)d{B}_s^H∥}^2\hfill \\ \hfill \le & 3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(f(s,X_s,{\mu }_{X_s})-f(s,Y_s,{\mu }_{Y_s})\right)ds∥}^2\hfill \\ \hfill & +3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(g(s,X_s,{\mu }_{X_s})-g(s,Y_s,{\mu }_{Y_s})\right)d{B}_s∥}^2\hfill \\ \hfill & +3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(h(s,{\mu }_{X_s})-h(s,{\mu }_{Y_s})\right)d{B}_s^H∥}^2\hfill \\ \hfill :=& 3D_1+3D_2+3D_3.\hfill \end{array}$$

For the term $D_1$, by the Lipschitz property of f and (1), we have

$$\begin{array}{cc}\hfill D_1& \le K^2\mathbf{E}{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}∥f(s,X_s,{\mu }_{X_s})-f(s,Y_s,{\mu }_{Y_s})∥ds\right)}^2\hfill \\ \hfill & \le K^2\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\mathbf{E}{∥f(s,X_s,{\mu }_{X_s})-f(s,Y_s,{\mu }_{Y_s})∥}^{2}ds\right)\hfill \\ \hfill & \le K^2{L}_1\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\left(\mathbf{E}\parallel X_s-Y_s{\parallel }^2+W_2{({\mu }_{X_s},{\mu }_{Y_s})}^2\right)ds\right)\hfill \\ \hfill & \le 2K^2{L}_1\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\mathbf{E}{\parallel X_s-Y_s\parallel }^{2}ds\right)\hfill \\ \hfill & \le 2K^2{L}_1{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)}^2\underset{t\in \mathbb{R}}{sup}\mathbf{E}\parallel X_t-Y_t{\parallel }^2={\displaystyle \frac{2K^2{L}_1}{{\omega }^2}}\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel X_t-Y_t\parallel }^2.\hfill \end{array}$$

For the term $D_2$, by Itô isometry, the Lipschitz condition of g and (1), we have

$$\begin{array}{cc}\hfill D_2& =\mathbf{E}\left({\int }_{-\infty }^t{∥T_{t-s}\left(g(s,X_s,{\mu }_{X_s})-g(s,Y_s,{\mu }_{Y_s})\right)∥}^{2}ds\right)\hfill \\ \hfill & \le K^2{\int }_{-\infty }^t{e}^{-2\omega (t-s)}\mathbf{E}{∥g(s,X_s,{\mu }_{X_s})-g(s,Y_s,{\mu }_{Y_s})∥}^{2}ds\hfill \\ \hfill & \le K^2{L}_2{\int }_{-\infty }^t{e}^{-2\omega (t-s)}\left(\mathbf{E}\parallel X_s-Y_s{\parallel }^2+W_2{({\mu }_{X_s},{\mu }_{Y_s})}^2\right)ds\hfill \\ \hfill & \le 2K^2{L}_2{\int }_{-\infty }^t{e}^{-2\omega (t-s)}\mathbf{E}{\parallel X_s-Y_s\parallel }^{2}ds\hfill \\ \hfill & \le {\displaystyle \frac{K^2{L}_2}{\omega }}\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel X_t-Y_t\parallel }^2.\hfill \end{array}$$

For the term $D_3$, by fractional Itô isometry, inequality (10), the Lipschitz condition of h and (1), we have

$$\begin{array}{cc}\hfill D_3& =H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t{〈T_{t-s}\left(h(s,{\mu }_{X_s})-h(s,{\mu }_{Y_s})\right),T_{t-r}\left(h(r,{\mu }_{X_r})-h(r,{\mu }_{Y_r})\right)〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t∥T_{t-s}\left(h(s,{\mu }_{X_s})-h(s,{\mu }_{Y_s})\right)∥∥T_{t-r}\left(h(r,{\mu }_{X_r})-h(r,{\mu }_{Y_r})\right)∥{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le H(2H-1)C_H{\left({\int }_{-\infty }^t{∥T_{t-s}\left(h(s,{\mu }_{X_s})-h(s,{\mu }_{Y_s})\right)∥}^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le K^2{L}_3^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}{W}_2{({\mu }_{X_s},{\mu }_{Y_s})}^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le K^2{L}_3^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}{\left(\mathbf{E}\parallel X_s-Y_s{\parallel }^2\right)}^{{\displaystyle \frac{1}{2H}}}ds\right)}^{2H}\hfill \\ \hfill & \le K^2{L}_3^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H}\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel X_t-Y_t\parallel }^2.\hfill \end{array}$$

It can be deduced from the above estimates that for each $t\in \mathbb{R}$,

$${\mathbf{E}\parallel \left(\mathcal{T}X\right)}_t-{\left(\mathcal{T}Y\right)}_t{\parallel }^2\le \left({\displaystyle \frac{6K^2{L}_1}{{\omega }^2}}+{\displaystyle \frac{3K^2{L}_2}{\omega }}+3K^2{L}_3^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H}\right)\underset{t\in \mathbb{R}}{sup}\mathbf{E}{\parallel X_t-Y_t\parallel }^2.$$

Then we can easily obtain that

$$\parallel \mathcal{T}{X}_t{-\mathcal{T}Y\parallel }_{\infty }\le \sqrt{\alpha }{\parallel X-Y\parallel }_{\infty },$$

with

$$\alpha :={\displaystyle \frac{6K^2{L}_1}{{\omega }^2}}+{\displaystyle \frac{3K^2{L}_2}{\omega }}+3K^2{L}_3^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H}>0.$$

Given that $\alpha <1$, we conclude that $\mathcal{T}$ is a contraction mapping. Hence, applying the Banach fixed-point theorem yields a unique $X\in C_b(\mathbb{R};L^2(\mathbf{P},\mathbb{H}))$ such that $\mathcal{T}X=X$. This X is precisely the unique mild solution of Equation (2) and satisfies the stochastic integral Equation (5).

Lastly, we need to check that this unique mild solution is almost automorphic in distribution. Owing to the almost automorphy of f, g, and h, for any sequence of real numbers ${\left\{s_n^{\prime }\right\}}_{n=1}^{\infty }$, there exists a subsequence ${\left\{s_n\right\}}_{n=1}^{\infty }$ for which some functions $\tilde{f}$, $\tilde{g}$, and $\tilde{h}$

$$\underset{n\to \infty }{lim}\mathbf{E}\parallel f(t+s_n,X,{\mu }_X)-\tilde{f}(t,X,{\mu }_X){\parallel }^2=0\mathrm{and}\underset{n\to \infty }{lim}\mathbf{E}{\parallel \tilde{f}(t-s_n,X,{\mu }_X)-f(t,X,{\mu }_X)\parallel }^2=0,$$

$$\underset{n\to \infty }{lim}\mathbf{E}{∥g(t+s_n,X,{\mu }_X)-\tilde{g}(t,X,{\mu }_X)∥}^2=0\mathrm{and}\underset{n\to \infty }{lim}\mathbf{E}{∥\tilde{g}(t-s_n,X,{\mu }_X)-g(t,X,{\mu }_X)∥}^2=0,$$

$$\underset{n\to \infty }{lim}\parallel h(t+s_n,{\mu }_X)-\tilde{h}(t,{\mu }_X)\parallel =0\mathrm{and}\underset{n\to \infty }{lim}\parallel \tilde{h}(t-s_n,{\mu }_X)-h(t,{\mu }_X)\parallel =0$$

hold for each $t\in \mathbb{R}$, $X\in L^2(\mathbf{P},\mathbb{H})$ and ${\mu }_X\in {\mathcal{P}}_2\left(\mathbb{H}\right)$. Let $\tilde{X}$ satisfy the integral equation

$${\tilde{X}}_t={\int }_{-\infty }^t{T}_{t-s}\tilde{f}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})ds+{\int }_{-\infty }^t{T}_{t-s}\tilde{g}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})d{B}_s+{\int }_{-\infty }^t{T}_{t-s}\tilde{h}(s,{\mu }_{{\tilde{X}}_s})d{B}_s^H.$$

Applying the change of variable $\sigma -s_n=s$ in Equation (5) yields

$$X_{t+s_n}={\int }_{-\infty }^t{T}_{t-s}f(s+s_n,X_{s+s_n},{\mu }_{X_{s+s_n}})ds+{\int }_{-\infty }^t{T}_{t-s}g(s+s_n,X_{s+s_n},{\mu }_{X_{s+s_n}})d{\tilde{B}}_s+{\int }_{-\infty }^t{T}_{t-s}h(s+s_n,{\mu }_{X_{s+s_n}})d{\tilde{B}}_s^H,$$

where ${\tilde{B}}_s=B_{s+s_n}-B_{s_n}$ is a standard Brownian motion with the same distribution as $B_s$ and ${\tilde{B}}_s^H=B_{s+s_n}^H-B_{s_n}^H$ is a fBm with the same distribution as $B_s^H$. On the other hand, let process $X_t^n$ satisfy the following equation

$$X_t^n={\int }_{-\infty }^t{T}_{t-s}f(s+s_n,X_s^n,{\mu }_{X_s^n})ds+{\int }_{-\infty }^t{T}_{t-s}g(s+s_n,X_s^n,{\mu }_{X_s^n})d{B}_s+{\int }_{-\infty }^t{T}_{t-s}h(s+s_n,{\mu }_{X_s^n})d{B}_s^H.$$

Note that $X_{t+s_n}$ has the same distribution as $X_t^n$ for each $t\in \mathbb{R}$. Now we show that $X_t^n$ converges in square-mean to ${\tilde{X}}_t$ for given $t\in \mathbb{R}$. Note that

$$\begin{array}{cc}\hfill \mathbf{E}{\parallel X_t^n-{\tilde{X}}_t\parallel }^2& \le 3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(f(s+s_n,X_s^n,{\mu }_{X_s^n})-\tilde{f}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)ds∥}^2\hfill \\ \hfill & +3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(g(s+s_n,X_s^n,{\mu }_{X_s^n})-\tilde{g}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)d{B}_s∥}^2\hfill \\ \hfill & +3\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(h(s+s_n,{\mu }_{X_s^n})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s})\right)d{B}_s^H∥}^2\hfill \\ \hfill & \le 6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(f(s+s_n,X_s^n,{\mu }_{X_s^n})-f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)ds∥}^2\hfill \\ \hfill & +6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{f}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)ds∥}^2\hfill \\ \hfill & +6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(g(s+s_n,X_s^n,{\mu }_{X_s^n})-g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)d{B}_s∥}^2\hfill \\ \hfill & +6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{g}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right)d{B}_s∥}^2\hfill \\ \hfill & +6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(h(s+s_n,{\mu }_{X_s^n})-h(s+s_n,{\mu }_{{\tilde{X}}_s})\right)d{B}_s^H∥}^2\hfill \\ \hfill & +6\mathbf{E}{∥{\int }_{-\infty }^t{T}_{t-s}\left(h(s+s_n,{\mu }_{{\tilde{X}}_s})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s})\right)d{B}_s^H∥}^2\hfill \\ \hfill :& ={\displaystyle \sum _{i=1}^6}6F_i.\hfill \end{array}$$

For the terms $F_1$, by the Lipschitz property of f and (1), we have

$$\begin{array}{cc}\hfill F_1& \le K^2\mathbf{E}{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\parallel f(s+s_n,X_s^n,{\mu }_{X_s^n})-f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\parallel ds\right)}^2\hfill \\ \hfill & \le K^2\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\mathbf{E}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\parallel f(s+s_n,X_s^n,{\mu }_{X_s^n})-f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s}){\parallel }^{2}ds\right)\hfill \\ \hfill & \le {\displaystyle \frac{K^2{L}_1}{\omega }}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\left(\mathbf{E}\parallel X_s^n-{\tilde{X}}_s{\parallel }^2+W_2{({\mu }_{X_s^n},{\mu }_{{\tilde{X}}_s})}^2\right)ds\right)\hfill \\ \hfill & \le {\displaystyle \frac{2K^2{L}_1}{\omega }}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right).\hfill \end{array}$$

And for the terms $F_2$, we obtain

$$\begin{array}{cc}\hfill F_2& \le K^2\mathbf{E}{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\parallel f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{f}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\parallel ds\right)}^2\hfill \\ \hfill & \le K^2\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}ds\right)\mathbf{E}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\parallel f(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{f}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s}){\parallel }^{2}ds\right).\hfill \end{array}$$

One can show that $F_2\to 0$ as $n\to \infty$ by (7) and the dominated convergence theorem. For the terms $F_3$, by Itô isometry, the Lipschitz condition of g and (1), we have

$$\begin{array}{cc}\hfill F_3& =\mathbf{E}\left({\int }_{-\infty }^t\parallel T_{t-s}\left(g(s+s_n,X_s^n,{\mu }_{X_s^n})-g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right){\parallel }^{2}ds\right)\hfill \\ \hfill & \le K^2\mathbf{E}\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}\parallel g(s+s_n,X_s^n,{\mu }_{X_s^n})-g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s}){\parallel }^{2}ds\right)\hfill \\ \hfill & \le K^2{L}_2\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}\left(\mathbf{E}\parallel X_s^n-{\tilde{X}}_s{\parallel }^2+W_2{({\mu }_{X_s^n},{\mu }_{{\tilde{X}}_s})}^2\right)ds\right)\hfill \\ \hfill & \le 2K^2{L}_2\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right).\hfill \end{array}$$

And for the terms $F_4$, we obtain

$$\begin{array}{cc}\hfill F_4& =\mathbf{E}\left({\int }_{-\infty }^t\parallel T_{t-s}\left(g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{g}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})\right){\parallel }^{2}ds\right)\hfill \\ \hfill & \le K^2\mathbf{E}\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}\parallel g(s+s_n,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s})-\tilde{g}(s,{\tilde{X}}_s,{\mu }_{{\tilde{X}}_s}){\parallel }^{2}ds\right).\hfill \end{array}$$

One can show that $F_4\to 0$ as $n\to \infty$ by (8) and the dominated convergence theorem. For the term $F_5$, by fractional Itô isometry, inequality (10), the Lipschitz condition of h and (1), we have

$$\begin{array}{cc}\hfill F_5& =H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t{〈T_{t-s}\left(h(s+s_n,{\mu }_{X_s^n})-h(s+s_n,{\mu }_{{\tilde{X}}_s})\right),T_{t-r}\left(h(r+s_n,{\mu }_{X_r^n})-h(r+s_n,{\mu }_{{\tilde{X}}_r})\right)〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill \le & H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t\parallel T_{t-s}\left(h(s+s_n,{\mu }_{X_s^n})-h(s+s_n,{\mu }_{{\tilde{X}}_s})\right)\parallel \parallel T_{t-r}\left(h(r+s_n,{\mu }_{X_r^n})-h(r+s_n,{\mu }_{{\tilde{X}}_r})\right)\parallel {|s-r|}^{2H-2}dsdr\hfill \\ \hfill \le & H(2H-1)C_H{\left({\int }_{-\infty }^t\parallel T_{t-s}\left(h(s+s_n,{\mu }_{X_s^n})-h(s+s_n,{\mu }_{{\tilde{X}}_s})\right){\parallel }^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill \le & K^2{L}_3^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}{W}_2{({\mu }_{X_s^n},{\mu }_{{\tilde{X}}_s})}^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill \le & K^2{L}_3^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}{\left(\mathbf{E}\parallel X_s^n-{\tilde{X}}_s{\parallel }^2\right)}^{{\displaystyle \frac{1}{2H}}}ds\right)}^{2H}\hfill \\ \hfill \le & K^2{L}_3^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}ds\right)}^{2H-1}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right)\hfill \\ \hfill =& K^2{L}_3^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H-1}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right).\hfill \end{array}$$

And for the terms $F_6$, we obtain

$$\begin{array}{cc}\hfill F_6& =H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t{〈T_{t-s}\left(h(s+s_n,{\mu }_{{\tilde{X}}_s})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s})\right),T_{t-r}\left(h(r+s_n,{\mu }_{{\tilde{X}}_r})-\tilde{h}(r,{\mu }_{{\tilde{X}}_r})\right)〉}_{\mathbb{H}}{|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le H(2H-1){\int }_{-\infty }^t{\int }_{-\infty }^t\parallel T_{t-s}\left(h(s+s_n,{\mu }_{{\tilde{X}}_s})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s})\right)\parallel \parallel T_{t-r}\left(h(r+s_n,{\mu }_{{\tilde{X}}_r})-\tilde{h}(r,{\mu }_{{\tilde{X}}_r})\right)\parallel {|s-r|}^{2H-2}dsdr\hfill \\ \hfill & \le H(2H-1)C_H{\left({\int }_{-\infty }^t\parallel T_{t-s}\left(h(s+s_n,{\mu }_{{\tilde{X}}_s})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s})\right){\parallel }^{\frac{1}{H}}ds\right)}^{2H}\hfill \\ \hfill & \le K^{2}H(2H-1)C_H{\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}\parallel h(s+s_n,{\mu }_{{\tilde{X}}_s})-\tilde{h}(s,{\mu }_{{\tilde{X}}_s}){\parallel }^{\frac{1}{H}}ds\right)}^{2H}.\hfill \end{array}$$

One can show that $F_6\to 0$ as $n\to \infty$ by (9) and the dominated convergence theorem. It can be deduced from the above estimates that for each $t\in \mathbb{R}$,

$$\begin{array}{cc}\hfill \mathbf{E}\parallel X_t^n-{\tilde{X}}_t{\parallel }^2\le {\alpha }_n& +{\displaystyle \frac{12K^2{L}_1}{\omega }}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right)+12K^2{L}_2\left({\int }_{-\infty }^t{e}^{-2\omega (t-s)}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right)\hfill \\ \hfill & +6K^2{L}_3^{2}H(2H-1)C_H{\left({\displaystyle \frac{H}{\omega }}\right)}^{2H-1}\left({\int }_{-\infty }^t{e}^{-\omega (t-s)/H}\mathbf{E}{\parallel X_s^n-{\tilde{X}}_s\parallel }^{2}ds\right),\hfill \end{array}$$

where the sequence ${\alpha }_n\to 0$ as $n\to \infty$. By generalized Gronwall lemma in [31] (Lemma 3.3) and (11), we have that

$$\underset{n\to \infty }{lim}\mathbf{E}{\parallel X_t^n-{\tilde{X}}_t\parallel }^2=0,\mathrm{for}\mathrm{each}t\in \mathbb{R}.$$

Since $X_{t+s_n}$ has the same distribution as $X_t^n$, it follows that ${lim}_{n\to \infty }\mathbf{d}({\mu }_{X_{t+s_n}},{\mu }_{{\tilde{X}}_t})=0$ by Remark 1. And we can show in a similar way that ${lim}_{n\to \infty }\mathbf{d}({\mu }_{{\tilde{X}}_{t-s_n}},{\mu }_{X_t})=0$. The proof of Theorem 1 is complete. □

Thus, the contraction mapping principle not only guarantees the existence and uniqueness of the mild solution but also preserves the almost automorphy in distribution under the given Lipschitz conditions.

4. An Example

We investigate the following stochastic heat equation under homogeneous Dirichlet boundary conditions

$$\left\{\begin{array}{cc}\hfill d{X}_t\left(x\right)& ={\partial }_x^2{X}_t\left(x\right)dt+f(t,X_t\left(x\right),{\mu }_{X_t\left(x\right)})dt+g(t,X_t\left(x\right),{\mu }_{X_t\left(x\right)})d{B}_t+h(t,{\mu }_{X_t\left(x\right)})d{B}_t^H,(t,x)\in \mathbb{R}\times (0,1),\hfill \\ \hfill X_t\left(0\right)& =X_t\left(1\right)=0,t\in \mathbb{R}.\hfill \end{array}\right.$$

(12)

Consider the operator A defined on $\mathcal{D}\left(A\right)$ by

$$\begin{array}{cc}\hfill \mathcal{D}\left(A\right)& =\left\{\phi \in C^1[0,1]\mid {\phi }^{\prime }\mathrm{is}\mathrm{absolutely}\mathrm{continuous}\mathrm{on}[0,1],{\phi }^{″}\in L^2[0,1],\phi \left(0\right)=\phi \left(1\right)=0\right\},\hfill \\ \hfill A\phi \left(x\right)& ={\phi }^{″}\left(x\right),x\in (0,1),\phi \in \mathcal{D}\left(A\right).\hfill \end{array}$$

Then the operator A gives rise to a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ on $\mathbb{H}=H_0^1(0,1)$ given by

$$T_t\phi ={\displaystyle \sum _{n=1}^{\infty }}{e}^{-n^2{\pi }^{2}t}\langle \phi ,e_n\rangle e_n,$$

where $e_n\left(x\right)=\sqrt{2}sin\left(n\pi x\right)$, $n=1,2,\dots$ and $\parallel T_t\parallel \le e^{-{\pi }^{2}t}$, $t\ge 0$. Analogously to Lemma 1 and Theorem 1, we have the following conclusions.

Lemma 2.

Suppose that f and g are square-mean almost automorphic in $t\in \mathbb{R}$ for every $X\in L^2(\mathbf{P},H_0^1(0,1))$ and the corresponding law ${\mu }_X\in {\mathcal{P}}_2\left(H_0^1(0,1)\right)$ according to Definition 2, and that h is almost automorphic in $t\in \mathbb{R}$ for every law ${\mu }_X\in {\mathcal{P}}_2\left(H_0^1(0,1)\right)$ as per Definition 3. In addition, f, g, and h fulfill the Lipschitz conditions: that is, for any $X,Y\in L^2(\mathbf{P},H_0^1(0,1))$, the associated laws ${\mu }_X,{\mu }_Y\in {\mathcal{P}}_2\left(H_0^1(0,1)\right)$, and all $t\in \mathbb{R}$,

$$\begin{array}{cc}\hfill \mathbf{E}\parallel f(t,Y,{\mu }_Y)-f(t,X,{\mu }_X){\parallel }^2& \le L_1\left({\mathbf{E}\parallel Y-X\parallel }^2+W_2{({\mu }_Y,{\mu }_X)}^2\right),\hfill \\ \hfill \mathbf{E}\parallel g(t,Y,{\mu }_Y)-g(t,X,{\mu }_X){\parallel }^2& \le L_2\left({\mathbf{E}\parallel Y-X\parallel }^2+W_2{({\mu }_Y,{\mu }_X)}^2\right),\hfill \\ \hfill \parallel h(t,{\mu }_Y)-h(t,{\mu }_X)\parallel & \le L_3{W}_2({\mu }_Y,{\mu }_X).\hfill \end{array}$$

Then the process $\mathcal{T}X:\mathbb{R}\to L^2(\mathbf{P},H_0^1(0,1))$ defined in (6) is stochastically continuous.

Theorem 2.

Under the same assumptions as in Lemma 2, Equation (12) has a unique mild solution in the space $C_b(\mathbb{R};L^2(\mathbf{P},H_0^1(0,1)))$, provided

$${\displaystyle \frac{6L_1}{{\pi }^4}}+{\displaystyle \frac{3L_2}{{\pi }^2}}+{\displaystyle \frac{3L_3^{2}H(2H-1)C_H{H}^{2H}}{{\pi }^{4H}}}<1,$$

Moreover, this mild solution is almost automorphic in distribution, provided

$${\displaystyle \frac{12L_1}{{\pi }^4}}+{\displaystyle \frac{12L_2}{{\pi }^2}}+{\displaystyle \frac{6L_3^{2}H(2H-1)C_H{H}^{2H-1}}{{\pi }^{4H}}}<1.$$

5. Conclusions

In this paper, we have studied a class of McKean–Vlasov SDEs driven by fBm with Hurst parameter $H\in (1/2,1)$. By applying the Banach fixed-point theorem on the space of stochastically continuous and uniformly bounded processes, we established the existence and uniqueness of a mild solution. Moreover, under suitable Lipschitz conditions and almost automorphy assumptions on the coefficients, this unique mild solution is shown to be almost automorphic in distribution. A stochastic heat equation is provided to illustrate the applicability of our main results.