Random attractors of a stochastic Hopfield neural network model with
  delays
The global asymptotic behavior of a stochastic Hopfield neural network model
(HNNM) with delays is explored by studying the existence and structure of
random attractors. It is first proved that the trajectory field of the
stochastic delayed HNNM admits an almost sure continuous version, which is
compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction
based on the random semiflow generated by the diffusion term. Then, this
version is shown to generate a random dynamical system (RDS) by piece-wise
linear approximation, after which the existence of a random absorbing set is
obtained by a careful uniform apriori estimate of the solutions. Subsequently,
the pullback asymptotic compactness of the RDS generated by the stochastic
delayed HNNM is proved and hence the existence of random attractors is
obtained. Moreover, sufficient conditions under which the attractors turn out
to be an exponential attracting stationary solution are given. Numerical
simulations are also conducted at last to illustrate the effectiveness of the
established results.
Wenjie Hu, Quanxin Zhu, Peter E. Kloeden
# Random attractors of a stochastic Hopfield neural network model with delays

###### Abstract

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for \(t>\tau\) (where \(\tau\) is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results.

**Key Words:** Random attractor, random dynamical systems, stochastic neural network, delay, piece-wise linear approximation

## 1 Introduction and examples

In [15], Hopfield proposed the following prominent HNNM

\[\frac{du_{i}(t)}{dt}=-c_{i}u_{i}(t)+\sum_{j=1}^{n}h_{ij}f_{j}\left(u_{j}\left( t\right)\right). \tag{1.1}\]

Here, \(u_{i}(t),i=1,2,\cdots,n\) represent the state of the \(i\)th neuron. \(c_{i}\) stands for the reset velocity of the \(i\)th neuron. \(\mathbf{H}=(h_{ij})\) corresponds to the connection matrix between neurons at time \(t\), where \(h_{ij},i,j=1,2,\cdots,n\) are the weight coefficients measuring the connection strength between different neurons. If the output of the \(j\)th neuron excites the neuron \(i\), then we have \(h_{ij}\geq 0\), otherwise, \(h_{ij}=0\). \(f_{i}:R\to R,i=1,2,\cdots,n\) are the neuron activation functions. Due to the significant roles that the HNNM 1.1 has played in the areas of signal processing, optimization, parallel computations, pattern recognition, associate memories and so forth, it has been extensively and intensively investigated including both its dynamics analysis and real world applications from the time that it was proposed. See, for instance [14], [15], [34] and the references therein.
In model 1.1, it is assumed that the updating and propagation of neuron signals occur instantaneously without time delay. Nevertheless, time delays are omnipresent both in neural processing and in signal transmission and must be taken into account in the communication channels. Hence, Marcus and Westervelt [25] incorporated time delays arising from distance between the neurons or axonal conduction time between neurons into system 1.1 leading to the following delayed neural network model

\[\frac{du_{i}(t)}{dt}=-c_{i}u_{i}(t)+\sum_{j=1}^{s}h_{ij}f_{j}\left(u_{j}\left( t\right)\right)+\,\sum_{j=1}^{n}b_{ij}g_{j}\left(u_{j}\left(t-\tau_{j}\right) \right), \tag{1.2}\]

where \(\tau_{j},j=1,\cdots,n\) denote the delay caused by the transmission along the axon of the \(j\)-th unit. \(b_{ij}\) represents the strength of the \(j\)-th unit on the \(i\)-th unit at time \(t-\tau_{j}\). The \(g_{j}\) term gives the corresponding output of the \(j\)-th unit at time \(t-\tau_{j}\). The dynamical behavior of solutions to 1.2 has drawn much research interest pertaining to both dynamics and applications. For example, [30] and [44] compared the memory capacity of 1.2 under different activation functions, while [9] and [35] obtained the global asymptotic stability of the steady state by the Lyapunov method.

In real nervous systems, signal transmission along axons is a noisy process caused by random fluctuations from the release of neurotransmitters and other random effects, as pointed by [13]. Therefore, a more realistic and accurate mathematical model should be the following stochastic delayed HNNM

\[\left\{\begin{array}{l}du_{i}(t)=\left[-c_{i}u_{i}(t)+\sum_{j=1}^{n}h_{ij}f_ {j}\left(u_{j}(t)\right)+\right.\\ \sum_{j=1}^{n}b_{ij}g_{j}\left(u_{j}\left(t-\tau_{j}\right)\right)]dt+\sum_{j= 1}^{n}\sigma_{ij}u_{j}(t)dw_{j}(t)\\ u_{i}(\xi)=\phi_{i}(\xi),\quad-\max_{1\leq k\leq n}\tau_{k}\leq\xi\leq 0,\end{array}\right. \tag{1.3}\]

which is obtained by adding an multiplicative noise \(\sum_{j=1}^{n}\sigma_{ij}u_{j}(t)dw_{j}(t)\) to 1.3. Here, \(\sigma_{ij}\) are parameters representing the density of stochastic effects, \(w_{j}(t),j=1,2,\cdots,n\) are mutually independent two-sided real-valued Wiener process on \((\Omega,\mathcal{F},P)\) with a filtration \(\{\mathcal{F}_{t},t\geq 0\}\). \(\phi_{i}(\xi),i=1,\ldots,n\) are the initial conditions. Let \(\tau=\max_{1\leq k\leq n}\tau_{k}\) and take

\[u(t)=\left(u_{1}(t),u_{2}(t),\ldots,u_{n}(t)\right)^{\mathrm{T}},\]

\[u_{t}(\tau)=\left(u_{1}\left(t-\tau_{1}\right),u_{2}\left(t-\tau_{2}\right), \ldots,u_{n}\left(t-\tau_{n}\right)\right)^{\mathrm{T}},\]

\[C=\mathrm{diag}\left(c_{1},c_{2},\ldots,c_{n}\right),\quad H=\left(h_{ij} \right)_{n\times n},\quad B=\left(b_{ij}\right)_{n\times n},\]

\[f(u)=\left(f_{1}\left(u_{1}\right),f_{2}\left(u_{2}\right),\ldots,f_{n}\left( u_{n}\right)\right)^{\mathrm{T}},\]

\[g(u)=\left(g_{1}\left(u_{1}\right),g_{2}\left(u_{2}\right),\ldots,g_{n}\left( u_{n}\right)\right)^{\mathrm{T}},\]

\[\Sigma=(\sigma_{ij})_{n\times n},\]

\[w(t)=\left(w_{1}(t),w_{1}(t),\ldots,w_{n}(t)\right)^{\mathrm{T}},\]

\[\phi(t)=\left(\phi_{1}(t),\phi_{2}(t),\ldots,\phi_{n}(t)\right)^{\mathrm{T}}.\]

Then, model 1.3 can be rewritten in the following matrix form

\[\left\{\begin{array}{l}du(t)=\left[-Cu(t)+Hf(u(t))+Bg\left(u(t-\tau)\right) \right]dt\\ +\Sigma u(t)\diamond w(t),\\ u(t)=\phi(t),\quad-\tau\leqslant t\leqslant 0,\end{array}\right. \tag{1.4}\]

in which \(\phi=\{\phi(s):-\tau\leqslant s\leqslant 0\}\) is \(\mathcal{C}=C([-\tau,0],\mathbb{R}^{n})\)-valued function, where \(\mathcal{C}\) is the space of all continuous \(\mathbb{R}^{n}\)-valued functions defined on \([-\tau,0]\) with a norm \(\|\xi\|=\sup\{|\xi(t)|:-\tau\leqslant t\leqslant 0\}\) and
\(|\cdot|\) is the Euclidean norm of a vector \(x\in\mathbb{R}^{n}\). Here \(u(t)\circ dw(t)\) is interpreted as the vector with \(j\)th component \(u_{j}(t)dw_{j}(t)\), \(j=1\), \(\cdots\), \(n\).

The local stability around equilibria of the stochastic HNNM and the stochastic delayed HNNM has been widely and thoroughly studied in the past decades by employing Ito's formula, stochastic analysis techniques and inequality techniques. See, for instance, [22, 32, 37] and [45] investigate stability of 1.4 with fixed or variant delays. On the other hand, there are also some other works concerning global stability for equilibrium of stochastic delayed HNNM by adopting the Lyapunov-Krasovskii functional methods together with stochastic analysis techniques, see for example [23, 40, 41] and the references therein.

Nevertheless, Lyapunov-Krasovskii functional methods depend critically on the construction of specified Lyapunov-Krasovskii functional which is different from equation to equation. Moreover, the criterion is in linear matrix inequality form that depends highly on numerical tools, and the condition on the drift term and the random coefficient is strict. From the dynamical systems point of view, if we can find a compact set that is invariant under the system and attracts all orbits of the system, i.e., if we can find a global attractor of the system, then we can obtain the asymptotic behavior of the system by investigating the structure of the attractor. This approach is effective especially for infinite dimensional dynamical systems generated by partial differential equations or delay differential equations. Generally, the attractors of dynamical systems may be fixed points, periodic orbits, invariant manifolds or chaotic attractors under certain conditions. Therefore, by investigating the attractors of dynamical systems, one can obtain a full view of complex dynamics of the systems. However, to our knowledge, the existence and structure of attractors for the stochastic delayed HNNM 1.4 have not been studied, although the existence and structure of attractors of the deterministic delayed HNNM, the delayed non-autonomous neural network models as well as random neural field lattice models have been studied in [12, 16, 24, 38, 31, 43]. It should be pointed out that, in the very recent work [31], the authors investigated the attractors of a random recurrent neural networks with delays. Unlike the delayed random neural networks that naturally generates an RDS, the existence of an RDS generated by stochastic delayed HNNM 1.4 needs much effort to established. Indeed, the main difficulty for obtaining the attractors of stochastic delayed HNNM 1.4 lies in showing the model generates a random dynamical system, which naturally holds for deterministic case.

There are only very few papers tackling the attractors for the stochastic delayed differential equations, see for example, [39], [42] and [10]. In [39], the authors proved that the stochastic delayed differential equation generates a mean square RDS processing a mean-square random attractor in the mean-square sense [19], which is not the random attractor in the classical pathwise sense. For a detail concept of the mean-square random attractors and mean-square RDSs, the readers are refereed to the review paper [6]. In [42] and [10] the authors obtained the existence and structure of random attractors for stochastic delayed differential equations with additive noise. It should be noted that the exponential Ornstein-Uhlenbeck process transformation as [7] did for stochastic (partial) partial differential equations may also be effective for transforming the stochastic delayed equation into random delayed equation, but a time delay then occurs in the noise term, which makes it difficult to establish the existence of a random dynamical system as well as that of tempered absorbing sets. This motivates us to propose a new method to investigate the existence of an RDS generated by 1.4 with general multiplicative noise, the existence and structure of random attractors of 1.4.

We prove that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous and measurable compact version for \(t>\tau\) which is constructed by random semiflow
generated by the linear multiplicative noise. Then, this version is shown to generate a RDS by the trick of piece-wise linear approximation of the version. The idea originates from the early works [29][26], [27] where stochastic delayed differential equations are recast into a Hilbert space to study the Lyapunov spectrum and invariant manifolds. Our work can be regarded as an attempt to study the random attractors of the HNNM 1.4 in Banach spaces. Very recently, [36] investigated the singular stochastic delay differential equations in Banach spaces by rough path theory, which is quite different from approach of the present work. Our main contributions are in the following three folds. Firstly, a novel method for proving the stochastic delayed HNNM generates an RDS in the natural phase space, i.e. \(\mathcal{C}=C([-\tau,0],\mathbb{R}^{n})\) is established, which is quite different from the situation in the early works [29] where a Hilbert space is taken as the phase space. Secondly, existence of random attractors of the model under appropriate conditions have been proved. Thirdly, optimal conditions under which that the attractor is a random stationary solution is obtained. vMoreover, the method can also be generalized to other stochastic evolution equations with delays, such as the stochastic delayed partial differential equations, for tackling the problem of topological dimension estimation, Lyapunov exponents, as well as the invariant manifolds that depends on showing the existence of RDSs.

This paper is organized as follows. In Section 2, we first introduce the theory of RDSs and random attractors as well as some notations and preliminary lemmas needed for the proof of our main results. In Section 3, based on the random semiflow generatated by the diffusion term of 1.4, we transform 1.4 into a random delayed differential equation admitting a measurable version that generates an RDS. For the purpose of showing the existence of random attractor for 1.4, we first give a uniform a priori estimate of the solution and then show the asymptotic compactness of the RDS generated by 1.4 in Section 4, which implies the existence of random attractor following the results in [4, 5]. In Section 5, we derive sufficient conditions for the random attractor being an exponential stationary solution. Finally, we illustrate the obtained results by numerical simulations.

## 2 Preliminaries

We first introduce the concepts of a random attractor and an RDS as well as some important lemmas following [1, 2, 4, 5] and [8].

**Definition 2.1**.: _Let \(\{\theta_{t}:\Omega\to\Omega,t\in\mathbb{R}\}\) be a family of measure preserving transformations such that \((t,\omega)\mapsto\theta_{t}\omega\) is measurable and \(\theta_{0}=\mathrm{id}\), \(\theta_{t+s}=\theta_{t}\theta_{s},\) for all \(s,t\in\mathbb{R}\). The flow \(\theta_{t}\) together with the probability space \(\big{(}\Omega,\mathcal{F},P,\left(\theta_{t}\right)_{t\in\mathbb{R}}\big{)}\) is called a metric dynamical system._

For a given separable Hilbert space \((X,\|\cdot\|_{X})\), denote by \(\mathcal{B}(X)\) the Borel-algebra of open subsets in \(X\).

**Definition 2.2**.: _A mapping \(\Phi:\mathbb{R}^{+}\times\Omega\times X\to X\) is said to be a random dynamcial system (RDS) on a complete separable metric space \((X,d)\) with Borel \(\sigma\)-algebra \(\mathcal{B}(X)\) over the metric dynamical system \(\big{(}\Omega,\mathcal{F},P,\left(\theta_{t}\right)_{t\in\mathbb{R}}\big{)}\) if (i) \(\Phi(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times X\to X\) is \((\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(X),\mathcal{B }(X))\)-measurable; (ii) \(\Phi(0,\omega,\cdot)\) is the identity on \(X\) for \(P\)-a.e. \(\omega\in\Omega\); (iii) \(\Phi(t+s,\omega,\cdot)=\Phi(t,\theta_{s}\omega,\cdot)\circ\Phi(s,\omega,\cdot),\) for all \(t,s\in\mathbb{R}^{+}\) for \(P\)-a.e. \(\omega\in\Omega\). A RDS \(\Phi\) is continuous or differentiable if \(\Phi(t,\omega,\cdot):X\to X\) is continuous or differentiable for all \(t\in\mathbb{R}^{+}\) and \(P\)-a.e. \(\omega\in\Omega\)._
**Definition 2.3**.: _A set-valued map \(\Omega\ni\omega\mapsto D(\omega)\in 2^{X}\) is said to be a random set in \(X\) if the mapping \(\omega\mapsto d(x,D(\omega))\) is \((\mathcal{F},\mathcal{B}(\mathbb{R}))\)-measurable for any \(x\in X,\) where \(d(x,D(\omega))\triangleq\inf_{y\in D(\omega)}\mathrm{d}(x,y)\) is the distance in \(X\) between the element \(x\) and the set \(D(\omega)\subset X\)._

**Definition 2.4**.: _A random set \(\{D(\omega)\}_{\omega\in\Omega}\) of \(X\) is called tempered with respect to \(\left(\theta_{t}\right)_{t\in\mathbb{R}}\) if for \(P\)-a.e. \(\omega\in\Omega\),_

\[\lim_{t\rightarrow\infty}e^{-\beta t}d\left(D\left(\theta_{-t}\omega\right) \right)=0,\quad\text{ for all }\beta>0,\]

_where \(d(D)=\sup_{x\in D}\|x\|_{X}\)._

**Definition 2.5**.: _Let \(\mathcal{D}=\{\{D(\omega)\}_{\omega\in\Omega}:D(\omega)\subset X,\omega\in\Omega\}\) be a family of random sets. A random set \(\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}\) is said to be a \(\mathcal{D}\)-pullback absorbing set for \(\Phi\) if for \(P\)-a.e. \(\omega\in\Omega\) and for every \(B\in\mathcal{D},\) there exists \(T=T(B,\omega)>0\) such that_

\[\Phi\left(t,\theta_{-t}\omega,B\left(\theta_{-t}\omega\right)\right)\subseteq K (\omega)\qquad\text{ for all }t\geq T.\]

_If, in addition, for all \(\omega\in\Omega,K(\omega)\) is a closed nonempty subset of \(X\) and \(K(\omega)\) is measurable in \(\Omega\) with respect to \(\mathcal{F},\) then we say \(K\) is a closed measurable \(\mathcal{D}\)-pullback absorbing set for \(\Phi\)._

**Definition 2.6**.: _An RDS \(\Phi\) is said to be \(\mathcal{D}\)-pullback asymptotically compact in \(X\) if for \(P\)-a.e. \(\omega\in\Omega\), \(\left\{\Phi\left(t_{n},\theta_{-t_{n}}\omega,x_{n}\right)\right\}_{n\geq 1}\) has a convergent subsequence in \(X\) whenever \(t_{n}\rightarrow\infty\) and \(x_{n}\in D\left(\theta_{-t_{n}}\omega\right)\) for any given \(D\in\mathcal{D}\)._

**Definition 2.7**.: _A compact random set \(\mathcal{A}(\omega)\) is said to be a \(\mathcal{D}\)-pullback random attractor associated to the RDS \(\Phi\) if it satisfies the invariance property_

\[\Phi(t,\omega)\mathcal{A}(\omega)=\mathcal{A}\left(\theta_{t}\omega\right), \quad\text{ for all }t\geq 0\]

_and the pullback attracting property_

\[\lim_{t\rightarrow\infty}\mathrm{dist}\left(\Phi\left(t,\theta_{-t}\omega \right)D\left(\theta_{-t}\omega\right),\mathcal{A}(\omega)\right)=0,\]

_for all \(t\geq 0,D\in\mathcal{D}\) P-a.e. \(\omega\in\Omega\).where \(\mathrm{dist}(\cdot,\cdot)\) denotes the Hausdorff semidistance_

\[\mathrm{dist}(A,B)=\sup_{x\in A}\inf_{y\in B}\mathrm{d}(x,y),\quad A,B\subset X.\]

**Lemma 2.1**.: _Let RDS \((\theta,\Phi)\) be a continuous RDS. Suppose that \(\Phi\) is pullback asymptotically compact and has a closed pullback \(\mathcal{D}\)-absorbing set \(K=\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}\). Then it possesses a random attractor \(\{\mathcal{A}(\omega)\}_{\omega\in\Omega},\) where_

\[\mathcal{A}(\omega)=\cap_{\tau\geq 0}\overline{\cup_{t\geq\tau}\Phi\left(t, \theta_{-t}\omega,K\left(\theta_{-t}\omega\right)\right)}.\]

F The following Gronwall inequality that will be frequently used in our subsequent proofs.

**Lemma 2.2**.: _Let \(T>0\) and \(u,\alpha,f\) and \(g\) be non-negative continuous functions defined on \([0,T]\) such that_

\[u(t)\leq\alpha(t)+f(t)\int_{0}^{t}g(r)u(r)dr,\quad\text{ for }t\in[0,T].\]

_Then_

\[u(t)\leq\alpha(t)+f(t)\int_{0}^{t}g(r)\alpha(r)e^{\int_{r}^{t}f(\tau)g(\tau) dr}dr,\quad\text{ for }t\in[0,T].\]
Existence of an RDS

Throughout the remaining part of this paper, we always make the following assumptions on the nonlinear term \(f\) and \(g\).

**Hypothesis A1** The functions \(f(x)\) and \(g(x)\) are globally Lipschitz continuous with Lipschitz constants \(L_{f}\) and \(L_{g}\).

**Hypothesis A2**\(f(x)\) and \(g(x)\) are bounded with \(\mathbf{0}\) being a fixed point, that is \(f(\mathbf{0})=g(\mathbf{0})=0\) and there exists a \(M>0\) such that \(f(x)\leq M\) for any \(x\in\mathbb{R}^{n}\) and \(g(\phi)\leq M\) for any \(\phi\in\mathcal{C}\).

By [28] Theorem 2.1 and [10] Lemma 1, we have the following results concerning the existence of a continuous \(\mathcal{F}_{t}\)-measurable solution to (1.4).

**Lemma 3.1**.: _Assume that_ **Hypothesis A1**-**A2** _hold, then for P-a.e. \(\omega\in\Omega\), the stochastic delayed HNNM (1.4) admits a unique solution \(u(\cdot,\omega,\phi)\in C\left([-\tau,\infty),\mathbb{R}^{n}\right)\), which is adapted to \(\{\mathcal{F}_{t}\}_{t\in[0,\infty)}\) for any initial condition \(\phi\in\mathcal{C}\)._

In the following, we are devoted to showing the stochastic delayed HNNM (1.4) generates a RDS. For a measurable process \(u:[-\tau,\infty)\times\Omega\to\mathbb{R}^{n}\), the segment \(u_{t}:[-\tau,0]\times\Omega\to\mathbb{R}^{n},t\in\mathbb{R}^{+}\), is defined by

\[u_{t}(s):=u(t+s),\qquad s\in[-\tau,0].\]

Consider the following stochastic differential equation obtained by the diffusion term of (1.4)

\[\left\{\begin{array}{l}dv(t)=\Sigma v(t)\diamond dw(t),\\ v(0)=\mathrm{id}_{\mathbb{R}^{n}},\end{array}\right. \tag{3.1}\]

where \(\mathrm{id}_{\mathbb{R}^{n}}\) be the identity operator on \(\mathbb{R}^{n}\). We assume here and elsewhere that the components \(w_{i}(t)\) of the vector \(w(t)\) are mutually independent standard Wiener processes.

Let \(v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}\) be the solution of (3.1) and it is well known from [1] that (3.1) admits a linear continuous solution which is bounded for P-a.e. \(\omega\in\Omega\). Denote its bound by \(L_{v}\), i.e. \(\|v(t,\omega,\cdot)\|\leq L_{v}\) for all \(t\in\mathbb{R}^{n}\) and P-a.e. Moreover, it process a bounded continuous inverse \(v^{-1}(t,\omega,\cdot)\). Hence, the solution \(v(t,\omega,\cdot):\mathbb{R}\times\Omega\to L(\mathbb{R}^{n})\) is \(\{\mathcal{F}_{t}\}_{t\geq 0}\)-adapted.

Define the following operator

\[\begin{split} A(t,\omega,\phi)\triangleq& v^{-1}(t, \omega,\cdot)\circ[-Cv(t,\omega,\phi(0))+Hf(v(t,\omega,\phi(0)))\\ &+Bg\left(v_{t}(\cdot,\omega,\phi)\right)]\end{split} \tag{3.2}\]

with \(v_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}\) being defined by

\[v_{t}(s,\omega,\phi)=\left\{\begin{array}{ll}v(t+s,\omega,\phi(0)),&t+s \geq 0,\\ \phi(t+s),&-\tau\leq t+s<0.\end{array}\right. \tag{3.3}\]

Assume that \(g(\varphi)=L\varphi+\tilde{g}(\varphi)\) for any \(\varphi\in\mathcal{C}\), where \(L\) is the linear part of \(g\). (Hence, the Lipschitz constant of \(\tilde{g}\) denoted by \(L_{\tilde{g}}\) is \(L_{g}-\|L\|\)). Consider the linear random functional differential equation

\[\left\{\begin{array}{ll}\frac{d\bar{u}(t)}{dt}=-C\bar{u}(t)+v^{-1}(t,\omega, \cdot)\circ BLv_{t}(\cdot,\omega,\bar{u}_{t})\triangleq\tilde{L}(t,\omega) \bar{u}_{t},\\ \bar{u}_{0}=\phi\in\mathcal{C}\end{array}\right. \tag{3.4}\]

for any \(t>0\).
For the pathwise deterministic functional differential equation (3.4), it follows from [11] that the fundamental solution (3.4), denoted by \(S^{\phi}(\cdot,\omega):[0,\infty)\to\mathbb{R}^{n^{2}}\) is defined by

\[\begin{split} S^{\phi}(t,\omega)&=(S*\phi)(t,\omega )\triangleq S(t,\omega)\phi(0)\\ &+\int_{-\tau}^{0}\int_{\xi}^{0}S(t+\xi-s,\omega)\mathrm{d}\eta( \xi,t,\omega)\phi(s)\mathrm{d}s.\end{split} \tag{3.5}\]

for any \(t\geq 0\) and P-a.e. \(\omega\in\Omega\). Here, \(\eta:[-\tau,0]\times[0,\infty)\times\Omega\to\mathbb{R}^{n^{2}}\) is an \(n\times n\) matrix-valued function whose elements are of bounded variation such that

\[\tilde{L}(t,\xi)\phi=\int_{-\tau}^{0}\,\mathrm{d}\eta(\xi,t,\omega)\phi(\xi)\]

for P-a.e. \(\omega\in\Omega\) due to the Riez representation theorem.

For any fixed \(\omega\in\Omega\) and \(t>0\), the characteristic equation of (3.4) is given by

\[\det\left[\lambda\operatorname{Id}-\tilde{L}(t,\omega)\left(\mathrm{e}^{- \lambda(\cdot)}\right)\right]=0, \tag{3.6}\]

of which the spectrum is defined as

\[\varrho=\max\left\{\operatorname{Re}\lambda:\det\left[\lambda\operatorname{Id }-\tilde{L}(t,\omega)\left(\mathrm{e}^{-\lambda(\cdot)}\right)\right]=0 \right\}.\]

In the reminder of this paper, we always assume that \(\varrho<0\) and hence we have the following results by [11].

**Lemma 3.2**.: _If \(\varrho<0\), then there exist positive constants \(\gamma\) and \(K_{0}\) such that_

\[|(S*\phi)(t,\omega)|<K_{0}\mathrm{e}^{-\gamma t}\|\phi\| \tag{3.7}\]

_for all \(t\geq 0\), P-a.e. \(\omega\in\Omega\) and any \(\phi\in\mathcal{C}\). In particular, there exists a positive constant \(K_{1}\) such that_

\[|S(t,\omega)|\leq K_{1}\mathrm{e}^{\varrho t/2}\text{ for all }t\geq 0,\text{ P-a.e. }\omega\in\Omega. \tag{3.8}\]

We next show the global existence and properties of solution to a random functional differential equation generated by the operator (3.2).

**Lemma 3.3**.: _Assume that_ **Hypothesis A1 - A2** _hold and \(\varrho<0\), then the following random retarded functional differential equation_

\[\left\{\begin{array}{l}\frac{d\tilde{u}(t)}{dt}=A(t,\omega,\tilde{u}_{t}), \quad t>0,\\ \tilde{u}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.9}\]

_admits a global continuous mild solution \(\tilde{u}(\cdot,\omega,\phi):[-\tau,\infty)\to\mathbb{R}^{n}\) which is \(C^{1}\) in \(t\) for any \(\phi\in\mathcal{C}\) and P-a.e. \(\omega\in\Omega\). Moreover, \(\tilde{u}(t,\omega,\phi)\) is jointly continuous in \((t,\omega)\), \(\tilde{u}(t,\cdot,\phi)\) is \(\mathcal{F}_{t}\)-measurable for any \(\phi\in\mathcal{C}\) and the map \(\tilde{u}_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}\) is compact for any \(t>\tau\)._

Proof.: Since \(v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}\) and \(v^{-1}(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}\) are both continuous, then (3.9) can be written as

\[\left\{\begin{array}{l}\frac{d\tilde{u}(t)}{dt}=-C\tilde{u}(t)+v^{-1}(t, \omega,\cdot)\circ Hf(v(t,\omega,\tilde{u}(t)))\\ +v^{-1}(t,\omega,\cdot)\circ Bg\left(v_{t}(\cdot,\omega,\tilde{u}_{t})\right),\quad t>0,\\ \tilde{u}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.10}\]
For each \(t\in\mathbb{R}^{+}\) and P-a.e. \(\omega\in\Omega\). Moreover, we have

\[\begin{split}\|A(t,\omega,\tilde{u}_{t})\|\leq& c| \tilde{u}(0)|+hL_{f}|\tilde{u}(0)|+bL_{g}\|\tilde{u}_{t}\|\\ \leq&(c+hL_{f}+bL_{g})\|\tilde{u}_{t}\|,\end{split} \tag{3.11}\]

where \(c,h,b\) represent the norm of matrix \(C,H\) and \(B\) respectively. Hence, the random retarded functional differential equation (3.9) admits a unique local solution with explosion time \(\rho=\rho(\phi,\omega)\). We next show that \(\rho=\infty\). By the variation-of-constants formula, we see (3.9) is equivalent to the following integral equation

\[\begin{cases}\tilde{u}(t,\omega,\phi)=S*\phi(t,\omega)+\int_{0}^{t}S(t-s, \omega)v^{-1}(s,\omega,\cdot)\circ\\ \qquad\qquad[Hf(v(s,\omega,\tilde{u}(s)))+B\tilde{g}\left(v_{s}(\cdot,\omega, \tilde{u}_{t})\right)]\mathrm{d}s,t\geq 0,\\ \tilde{u}(t)=\phi(t),\quad t\in[-\tau,0],\end{cases} \tag{3.12}\]

where \(S*\phi(t,\omega)\) and \(S(t,\omega)\) satisfy (3.7) and (3.8) respectively for \(t\geq 0\) and P-a.e. \(\omega\in\Omega\). Hence, we have

\[\begin{split}|\tilde{u}(t,\omega,\phi)|\leq&\,|(S* \phi)\left(t,\omega\right)|+\int_{0}^{t}|S(t-s)v^{-1}(s,\omega,\cdot)\circ\\ &[Hf(v(s,\omega,\tilde{u}(s)))+B\tilde{g}\left(v_{s}(\cdot,\omega, \tilde{u}_{s})\right)]\mathrm{d}s|\mathrm{d}s\\ \leq&\,|(S*\phi)\left(t,\omega\right)|+\int_{0}^{t}S(t-s )[hL_{f}\|\tilde{u}(s,\omega,\phi)\|\\ &+bL_{\tilde{g}}\|\tilde{u}_{s}(\cdot,\omega,\phi)\|]\mathrm{d}s\\ \leq& K_{0}\mathrm{e}^{-\gamma t}\left\|\phi\right\|\\ &+\left(hL_{f}+bL_{\tilde{g}}\right)\int_{0}^{t}\mathrm{e}^{\varrho(t-s )/2}\left\|\tilde{u}_{s}\left(\cdot,\omega,\phi\right)\right\|\mathrm{d}s, \end{split} \tag{3.13}\]

from which we can see for \(t\geq\tau,\xi\in[-\tau,0]\),

\[\begin{split}\|\tilde{u}(t+\xi,\omega,\phi)\|\leq& K_{0}\mathrm{e}^{\gamma(\tau-t)}\left\|\phi\right\|+(hL_{f}+bL_{\tilde{g}}) \mathrm{e}^{-\varrho\tau/2}\times\\ &\int_{0}^{t}\mathrm{e}^{\varrho/2(t-s)}\left\|\tilde{u}_{s} \left(\cdot,\omega,\phi\right)\right\|\mathrm{d}s.\end{split} \tag{3.14}\]

Namely,

\[\begin{split}\mathrm{e}^{-\varrho t/2}\left\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\|\leq& c_{0}\mathrm{e}^{-\gamma t} \mathrm{e}^{-\varrho t/2}\left\|\phi\right\|\\ &+c_{1}\int_{0}^{t}\mathrm{e}^{-\varrho s/2}\left\|\tilde{u}_{s} \left(\cdot,\omega,\phi\right)\right\|\mathrm{d}s\end{split} \tag{3.15}\]

where \(c_{0}=K_{0}\mathrm{e}^{\gamma\tau}\) and \(c_{1}=(hL_{f}+bL_{\tilde{g}})\mathrm{e}^{-\varrho\tau/2}\). Applying Gronwall's inequality yields

\[\begin{split}\mathrm{e}^{-\varrho t/2}\left\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\|\leq& c_{0}\|\phi\|[\mathrm{e}^{-( \gamma+\varrho/2)t}\\ &+c_{1}\mathrm{e}^{c_{1}t}\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s],\end{split} \tag{3.16}\]

and hence

\[\begin{split}\|\tilde{u}_{t}\left(\cdot,\omega,\phi\right)\|\leq& c_{0}\|\phi\|[\mathrm{e}^{-\gamma t}+c_{1}\mathrm{e}^{(c_{1}+ \varrho/2)t}\int_{0}^{t}\mathrm{e}^{-(\gamma+\varrho/2+c_{1})s}\mathrm{d}s] \end{split} \tag{3.17}\]

for P-a.e. \(\omega\in\Omega\) and \(0<t<\rho\). Thus, \(\{\|\tilde{u}(t,\omega,\phi)\|:0<t<\rho\}\) is bounded and so \(\tilde{u}(\cdot,\omega,\phi):[0,\rho)\to H\) is uniformly continuous for P-a.e. \(\omega\in\Omega\) and any \(\phi\in\mathcal{C}\). Taking the
limit \(\lim_{t\rightarrow\rho-}\tilde{u}_{t}(\cdot,\omega,\phi)\in\mathcal{C}\) as initial data for (3.9), then by local existence we may extend \(\tilde{u}\) to a larger interval than \([0,\rho)\). This contradicts the maximality of \(\rho.\) Hence \(\rho=\infty\).

It follows from the continuity of \(A(t,\omega,\tilde{u}_{t})\) with respect to \(t\) that \(\tilde{u}(\cdot,\omega,\phi)\) is \(C^{1}\) in \(t\) for any \(\phi\in\mathcal{C}\) and P-a.e. \(\omega\in\Omega\). For each \(t\in\mathbb{R}^{+}\) and \(\phi\in\mathcal{C}\), \(A(t,\omega,\tilde{u}_{t})\) is \(\mathcal{F}_{t}\)-measurable. Hence, a standard successive approximation argument shows that \(\tilde{u}_{t}(\cdot,\omega,\phi)\) is jointly measurable in \((t,\omega)\) and for each \(t\in\mathbb{R}^{+}\), \(\tilde{u}_{t}(\cdot,\omega,\phi)\) is \(\mathcal{F}_{t}\)-measurable for any \(\phi\in\mathcal{C}\).

Furthermore if we fixed \(t_{0}\geq\rho\) then the above argument also shows that for P-a.e. \(\omega\in\Omega\) the set \(\{d\tilde{u}(t,\omega,\phi)\mid:0\leq t\leq t_{0},\|\phi\|\leq 1\}\) is bounded. Hence by Ascoli's theorem the family

\[\{\tilde{u}_{t}(\cdot,\omega,\phi):\|\phi\|\leq 1\} \tag{3.18}\]

is relatively compact in \(\mathcal{C}\). 

Next, we show the existence of continuous measurable version of solutions to the stochastic delayed HNNM (1.4) constructed by the solution of (3.9).

**Theorem 3.1**.: _The trajectory field \(u(t,\omega,\phi)\), \(t\in[-\tau,\infty)\) of the stochastic delayed HNNM (1.4) has a Borel-measurable version \(U(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times\mathcal{C}\mapsto \mathcal{C}\) with the following properties: (I) For each \(\phi\in\mathcal{C},U(t,\cdot,\phi)=u(t,\cdot,\phi)\) a.s. on \(\Omega\). (II) For each \(t\in\mathbb{R}^{+}\) and \(\phi\in\mathcal{C},U(t,\cdot,\phi):\Omega\mapsto\mathcal{C}\) is \(\mathcal{F}_{t}\)-measurable. (III) There is a Borel set \(\Omega_{0}\subset\Omega\) of full P-measure such that for all \(\omega\in\Omega_{0}\) the map \(U(\cdot,\omega,\cdot):\mathbb{R}^{+}\times\mathcal{C}\mapsto\mathcal{C}\) is continuous. (IV) For each \(t\in\mathbb{R}^{+}\) and every \(\omega\in\Omega_{0}\), the map \(U(t,\omega,\cdot):\mathcal{C}\mapsto\mathcal{C}\) is continuous. (V) For each \(t>\tau\) and all \(\omega\in\Omega_{0}\), the map \(U(t,\omega,\cdot):\mathcal{C}\rightarrow\mathcal{C}\) is compact._

Proof.: Define \(u:[-\tau,\infty)\times\Omega\times\mathcal{C}\rightarrow\mathbb{R}^{n}\) by

\[u(t,\omega,\phi)=\left\{\begin{array}{ll}v\left(t,\omega,\tilde{u}(t,\omega,\phi)\right)&\quad t\geq 0,\\ \phi(t)&\quad t\in[-\tau,0),\end{array}\right. \tag{3.19}\]

We prove in the following that the above defined \(u\) is in fact the solution of the stochastic delayed HNNM (1.4). It follows from Lemma 3.3 that for each \(t\in\mathbb{R}^{+}\) and \(\phi\in\mathcal{C}\), \(\tilde{u}(t,\omega,\phi)\) is jointly measurable in \((t,\omega)\) and \(\tilde{u}(t,\cdot,\phi)\) is \(\mathcal{F}_{t}\)-adapted, which combined with the \(\mathcal{F}_{t}\)-adaptivity of \(v(t,\omega,\cdot)\) indicates that the process \(u(t,\cdot,\phi)\) is \(\mathcal{F}_{t}\)-adapted, for each \(t\geq 0\). Since \(\{\tilde{u}(t,\cdot,\phi):t\geq 0\}\) is \(\mathcal{F}_{t}\)-adapted and has C\({}^{1}\) sample paths, by taking Ito differentials of (3.19) and invoking Bismut's generalized Ito formula for stochastic flows, we have for any \(t>0\)

\[\begin{split} du\left(t,\omega,\phi\right)=& v\left(t, \omega,\frac{d\tilde{u}(t,\omega,\phi)}{dt}\right)dt\\ &+\sigma\left(v\left(t,\cdot,\tilde{u}(t,\omega,\phi)\right) \right)dw(t)\\ =&-Cv(t,\omega,\tilde{u}(t,\omega,\phi))+Hf(v(t,\omega, \tilde{u}(t,\omega,\phi)))\\ &+Bg(v_{t}(\cdot,\omega,\tilde{u}))+\sigma\left(v(t,\omega,\tilde{u }(t,\omega,\phi))\right)dw(t).\end{split} \tag{3.20}\]

Then, in the case \(t>0\), we have

\[\begin{split} v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi) )&=\left\{\begin{array}{ll}v(t+\cdot,\omega,\tilde{u}_{t}( \cdot,\omega,\phi)),t+\cdot\geq 0\\ \phi(t+\cdot),-\tau\leq t+\cdot<0\end{array}\right.\\ &=u_{t}(\cdot,\omega,\phi),\end{split} \tag{3.21}\]
for any \(s\in[-\tau,0]\) and P-a.e. \(\omega\in\Omega\). Therefore (3.20) becomes

\[\begin{split} du\left(t,\omega,\phi\right)=&-Cu(t, \omega,\phi)+Hf(u(t,\omega,\phi))\\ &+Bg(u_{t}(\cdot,\omega,\phi))+\sigma\left(u(t,\omega,\phi) \right)dw(t),\end{split} \tag{3.22}\]

i.e., \(u(t,\omega,\phi)\) satisfies stochastic delayed HNNM (1.4) and

\[U(t,\omega,\phi):=v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi)) \tag{3.23}\]

is the version of its trajectory field we are seeking.

From the construction of \(U(t,\omega,\phi)\), one can easily see that statement (I) holds. It remains to show that \(U(t,\omega,\phi)\) satisfies all the requirements of the theorem. We begin by proving that \(U(t,\omega,\phi)\) is jointly (Borel)-measurable in \((t,\omega)\). By the definition of (3.3), one can clear see that

\[\begin{split}\mathbb{R}^{+}\times\Omega\to L\left(\mathcal{C} \right)\\ (t,\omega)\mapsto v_{t}(s,\omega,\cdot)\end{split} \tag{3.24}\]

is Borel-measurable for each \(s\in[-\tau,0]\). But the Borel field of \(C\left([-\tau,0]\times\mathbb{R}^{n},\mathbb{R}^{n}\right)\) is identified with that of \(C([-\tau,0],L(\mathbb{R}^{n}))\) and the latter is generated by evaluations, so it follows immediately that the map

\[\begin{split}\Lambda:\mathbb{R}^{+}\times\Omega\to L\left( \mathcal{C}\right)\\ (t,\omega)\mapsto v_{t}\left(\cdot,\omega,\cdot\right)\end{split} \tag{3.25}\]

is Borel-measurable and has continuous sample paths. It follows from Lemma 3.3 that \(\tilde{u}_{t}(\cdot,\omega,\phi)\) is Borel-measurable on \(\mathbb{R}^{+}\times\Omega\times\mathcal{C}\). Therefore the composition

\[U(t,\omega,\phi):=v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi)) \tag{3.26}\]

is measurable on \(\mathbb{R}^{+}\times\Omega\times\mathcal{C}\) and \(U\left(t,\omega,\cdot\right):\mathcal{C}\to\mathcal{C}\) is continuous for \(t\in\mathbb{R}^{+}\), P-a.e. \(\omega\in\Omega\). Also for P-a.e. \(\omega\in\Omega\), \(U(\cdot,\omega,\cdot)\) is continuous on \(\mathbb{R}^{+}\times\mathcal{C}\). This proves assertions (II), (III) and (IV) of the theorem.

At last, we prove statement (V) of the theorem. Fix \(t>\tau\) and pick any universal Borel set \(\Omega_{0}\subset\Omega\) of full \(P\)-measure such that \(v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}\) is linear for all \(\omega\in\Omega_{0}\). Take \(\omega\in\Omega_{0}\), then the composition (3.26) yields compactness of \(U(t,\omega,\cdot):\mathcal{C}\to\mathcal{C}\) as we have already established that the map \(\tilde{u}_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}\) is compact for any \(t>\tau\). This completes the proof of the theorem. 

In Theorem 3.1, we construct a continuous measurable version \(U(\cdot,\omega,\cdot)\) of (1.4) that generates a random semiflow. The idea originates from the early works [29], [26] and [27], where the authors recasted the stochastic delayed differential equations into a Hilbert space to study the Lyapunov spectrum and invariant manifolds. Our work can be regarded as a first attempt to study the random attractors of the HNNM (1.4) in Banach spaces.

### Cocycle property

The remaining part of this section is devoted to showing the cocycle property of the trajectory field version \(\mathrm{R}^{+}\times\Omega\times\mathcal{C}\to\mathcal{C}\) constructed in Theorem 3.1 using the Wong-Zakai approximation of the Wiener process, i.e., a piece-wise linear interpolation of its sample paths. See Twardowsky [18].
The Ito type (3.1) is equivalent to the following Stratonovich equation

\[\left\{\begin{array}{l}d\hat{v}(t,\cdot,\cdot)=\sum_{i=1}^{m}\Sigma\circ\hat{v}( t,\cdot,\cdot)\circ dw_{i}(t)\\ -\frac{1}{2}\sum_{i=1}^{m}\left(\Sigma\circ\left([\Sigma\circ\hat{v}(t,\cdot, \cdot)](\cdot)\left(e_{i}\right)\right\}\right)\left(\cdot\right)\left(e_{i} \right)dt,\\ \hat{v}(0,\cdot,\cdot)=\mathrm{id}_{\mathbb{R}^{n}},\end{array}\right. \tag{3.27}\]

where the notation \(\left[\Sigma\circ A\right]\left(\cdot\right)\left(e_{i}\right),A\in L\left( \mathbb{R}^{n}\right)\) stands for the map

\[\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}:v\rightarrow\Sigma\circ A(v)\left(e_ {i}\right).\]

For each \(k\geq 1\), let \(\hat{v}^{k}:\mathbb{R}^{+}\times\Omega\to L\left(\mathbb{R}^{n}\right)\) be the fundamental solution of the following random differential equation+

\[\left\{\begin{array}{l}d\hat{v}^{k}(t,\cdot,\cdot)=\sum_{i=1}^{m}\left[ \Sigma\circ\hat{v}^{k}(t,\cdot,\cdot)\right]\left(\cdot\right)\left(e_{i} \right)\left(W_{i}^{k}\right)^{\prime}(t)dt\\ -\frac{1}{2}\sum_{i=1}^{m}\left(\Sigma\circ\left([\Sigma\circ\hat{v}^{k}(t, \cdot,\cdot)](\cdot)\left(e_{i}\right)\right\}\right)\left(\cdot\right)\left(e _{i}\right)dt,\\ \varphi(0,\cdot,\cdot)=\mathrm{id}_{\mathbb{R}^{n}}.\end{array}\right. \tag{3.28}\]

Here \(W_{i}^{k}(t)\) is the piece-wise linear approximation of \(\omega_{i},i=1,2,\cdots,m\), which is defined by

\[\left(W_{i}^{k}\right)^{\prime}(t,\omega)=\left(\omega_{i}^{k}\right)^{ \prime}(t),\quad\omega\in\Omega,\quad t\geq 0,\]

where

\[\omega_{i}^{k}(t)=k\left[\omega\left(\frac{j+1}{k}\right)-\omega_{i}\left( \frac{j}{k}\right)\right]\left(t-\frac{j}{k}\right)+\omega_{i}\left(\frac{j}{ k}\right),\]

for \(\quad\frac{j}{k}\leq t<\frac{j+1}{k},\quad j=0,1,2,\ldots\) and \(\omega_{i}\) is standard mutually independent Brownian motions. It follows from [17] (see also Mohamed ref [29] and Twardowsky [18]) that for a.a. \(\omega\in\Omega\),

\[\lim_{k\rightarrow\infty}v^{k}(t,\omega,\cdot)=v(t,\omega,\cdot) \tag{3.29}\]

in \(L\left(\mathbb{R}^{n}\right)\) uniformly in \(t\in[0,T]\), for \(0<T<\infty\).

**Theorem 3.2**.: _For each integer \(k\geq 1\), let \(\hat{v}^{k}(t,\omega,\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) be the solution of (3.28) and define the following operator_

\[\begin{array}{l}A^{k}(t,\omega,\phi)\triangleq[\hat{v}^{k}(t,\omega,\cdot)] ^{-1}[-C\hat{v}^{k}(t,\omega,\phi(0))+\\ Hf(\hat{v}^{k}(t,\omega,\phi(0)))+Bg\left(\hat{v}^{k}_{t}(\cdot,\omega,\phi) \right)],\end{array} \tag{3.30}\]

_with \(\hat{v}^{k}_{t}(\cdot,\omega,\cdot):[-\tau,0]\times\mathcal{C}\rightarrow \mathcal{C}\) being defined by_

\[\hat{v}^{k}_{t}(s,\omega,\phi)=\left\{\begin{array}{ll}\hat{v}^{k}(t+s, \omega,\phi(0)):&t+s\geq 0,\\ \phi(t+s)&-\tau\leq t+s<0.\end{array}\right. \tag{3.31}\]

_Let \(0<T<\infty\). Denote by \(B\) a bounded set in \(\mathcal{C}\) and \(\tilde{u}^{k}(t,\omega,\cdot):[-\tau,\infty)\rightarrow\mathbb{R}^{n}\) the solution of the following random retarded partial differential equation_

\[\left\{\begin{array}{l}\frac{d\tilde{u}^{k}(t)}{dt}=A^{k}(t,\omega,\tilde{u }^{k}_{t}),\quad t>0,\\ \tilde{u}^{k}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.32}\]

_Define_

\[U^{k}(t,\omega,\phi):=v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega, \phi)), \tag{3.33}\]

_then \(U^{k}(t,\omega,\phi)\) satisfies_

\[\lim_{k\rightarrow\infty}\sup_{0\leq t\leq T}\sup_{\phi\in B}\left\|U^{k}(t, \omega,\phi)-U(t,\omega,\phi)\right\|=0\]

_for P-a. e. \(\omega\in\Omega\)._
Proof.: Fix \(0<T<\infty\) and \(B\subset\mathcal{C}\) a bounded set throughout the proof. By Lemma 3.3, (3.32) admits a global solution for P-a. e. \(\omega\in\Omega\), which is defined as follows.

\[\begin{cases}\tilde{u}^{k}(t)=S(t)*\phi+\int_{0}^{t}S(t-s)[\hat{v}^{k}(s,\omega,\cdot)]^{-1}\circ\\ \qquad[Hf(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))+B\tilde{g}\left(\hat{v}^{k} _{s}(\cdot,\omega,\tilde{u}^{k}_{t})\right)]\mathrm{d}s,t\geq 0,\\ u(t)=\phi(t),\quad t\in[-\tau,0].\end{cases} \tag{3.34}\]

Therefore, by denoting \(I(t,\omega,\phi)=\|U^{k}(t,\omega,\phi)-U(t,\omega,\phi)\|\), we have for \(P\)-a.e. \(\omega\in\Omega\),

\[\begin{split} I(t,\omega,\phi)=&\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi))-v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}( \cdot,\omega,\phi))\|\\ \leq&\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega, \phi))-v_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\|\\ +&\|v_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega, \phi))-v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\|\\ \leq&\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega, \phi)-\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\|\\ +&\|[v_{t}(\cdot,\omega,\cdot)-v^{k}_{t}(\cdot,\omega, \cdot)]\tilde{u}^{k}_{t}(\cdot,\omega,\phi)]\|\\ \leq&\|v_{t}(\cdot,\omega,\cdot)\|\|\tilde{u}_{t}(\cdot, \omega,\phi)-\tilde{u}^{k}_{t}(\cdot,\omega,\phi)\|\\ +&\|v_{t}(\cdot,\omega,\cdot)-v^{k}_{t}(\cdot,\omega,\cdot)\|\|\tilde{u}^{ k}_{t}(\cdot,\omega,\phi)\|\\ \triangleq&\|v(t,\omega,\cdot)\|I_{1}+I_{2}\|\tilde{u}^{k}_{t}(\cdot, \omega,\phi)\|.\end{split} \tag{3.35}\]

Since \(\lim_{k\to\infty}v^{k}(t,\omega,\cdot)=v(t,\omega,\cdot)\), we have \(I_{2}\to 0\) as \(k\to\infty\). Because \(\|v(t,\omega,\cdot)\|\) is bounded, we only need to estimate \(I_{1}\).

\[\begin{split}|I_{1}|=&|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ Hf(v(s,\omega,u(s)))-\\ &\hat{v}^{k}(s,\omega,\cdot)^{-1}\circ Hf(\hat{v}^{k}(s,\omega, \tilde{u}^{k}(s)))]|\\ &+|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ B\tilde{g} \left(v_{s}(\cdot,\omega,u_{t})\right)-\\ &\hat{v}^{k}(s,\omega,\cdot)^{-1}\circ B\tilde{g}\left(\hat{v}^{ k}_{s}(\cdot,\omega,\tilde{u}^{k}_{t})\right)]\mathrm{d}s|\\ &\triangleq I_{11}+I_{12}.\end{split} \tag{3.36}\]

We estimate each term on the right hand side of (3.36) in the following by Lemmas 3.3. By the **Hypothesis A1**, we can see

\[\begin{split} I_{11}\leq&\|\int_{0}^{t}S(t-s)v^{-1 }(s,\omega,\cdot)\circ H[f(v(s,\omega,u(s)))\\ &-f(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))]\mathrm{d}s\|+\|\int_ {0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\\ &-\hat{v}^{k}(s,\omega,\cdot)^{-1}]\circ Hf(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))\mathrm{d}s\|\\ &\triangleq I_{111}+I_{112}.\end{split} \tag{3.37}\]

Since \(f\) is bounded, we have for any \(\rho\in\mathbb{R}^{n}\), \(Hf(\rho)\leq M\) which together with (3.29), implies that for any \(\varepsilon>0\), there exists sufficient large \(K_{0}\) such that for any \(k>K_{0}\)

\[I_{112}\leq M\varepsilon. \tag{3.38}\]
Moreover,

\[\begin{split} I_{111}\leq&\|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ H[f(v(s,\omega,u(s)))\\ &-f(\hat{v}^{k}(s,\omega,u(s)))]\mathrm{d}s\|+\|\int_{0}^{t}S(t-s )[v^{-1}(s,\omega,\cdot)\\ &\circ H[f(\hat{v}^{k}(s,\omega,u(s)))-f(\hat{v}^{k}(s,\omega, \tilde{u}^{k}(s)))]\mathrm{d}s\|\\ &\leq CKL_{f}\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\|v\left(s, \omega,u(s)\right)\\ &-\hat{v}^{k}(s,\omega,u(s))\|\mathrm{d}s+KL_{f}\int_{0}^{t} \mathrm{e}^{\varrho(t-s)/2}\\ &\left\|u(s)-\tilde{u}^{k}(s)\right\|\mathrm{d}s\triangleq I_{11 11}+I_{1112}.\end{split} \tag{3.39}\]

It follows from (3.29) that for any \(\varepsilon>0\), there exists sufficient large \(K_{0}\) such that for any \(k>K_{0}\)

\[I_{1111}\leq CKL_{f}\varepsilon\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\mathrm{ d}s\leq CKL_{f}\varepsilon. \tag{3.40}\]

Thus, we have for \(k>K_{0}\)

\[\begin{split} I_{1}\leq& M\varepsilon+CKL_{f} \varepsilon+KL_{f}\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\left\|u(s)-\tilde{ u}^{k}(s)\right\|\mathrm{d}s,\end{split} \tag{3.41}\]

which together with the Gronwall inequality indicates that \(I_{1}\to 0\) as \(k\to\infty\). 

We are now in the position to provide the cocycle property of \(U(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times\mathcal{C}\mapsto \mathcal{C}\). We directly give the following results for which the proof is the same as Theorem 3 in [29].

**Theorem 3.3**.: _(The Cocycle Property) Let \(\theta:\mathbb{R}^{+}\times\Omega\to\Omega\) be the shift corresponding Wiener measure on \((\Omega,\mathcal{F})\) which is defined by_

\[\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),t\in\mathbb{R}.\]

_Then, there is a universal set \(\Omega_{0}\subset\Omega\) of full Wiener measure such that_

\[U\left(t_{2},\theta_{t_{1}}\omega,U\left(t_{1},\omega,\phi\right)\right)=U \left(t_{1}+t_{2},\omega,\phi\right)\]

_for all \(\omega\in\Omega_{0},\phi\in\mathcal{C}\) and \(t_{1},t_{2}\geq 0\)._

## 4 Random attractors

In this section, we are concerned about the existence of random attractors of (1.4), which relies on showing the existence of an absorbing set and the asymptotic compactness of the RDS \((\theta,U)\). Throughout the reminder of this paper, we denote the family of tempered sets in the state space \(C\) by \(\mathcal{D}\). We first give the following uniform estimate results.

**Lemma 4.1**.: _Assume that conditions of Lemma 3.3, Theorem 3.2 hold and_

\[c_{1}+\varrho/2<0<c_{1}+\varrho/2+\gamma,\]
_then \(U\) possesses a random absorbing set \(\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}\), that is, for any \(\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}\) and \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), there exists \(t_{B}(\omega)>0\) such that_

\[U\left(t,\theta_{-t}\omega,\phi\left(\theta_{-t}\omega\right)\right)\in K( \omega)\quad\text{ for all }t\geq t_{B}(\omega)\]

_for all \(t\geq t_{B}(\omega)\) and \(\phi\left(\theta_{-t}\omega\right)\in B\left(\theta_{-t}\omega\right)\)._

Proof.: For any \(\phi(\omega)\in B(\omega)\), it follows from (3.16) in the proof of Lemma 3.3 that

\[\begin{split}\mathrm{e}^{-\varrho t/2}\left\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\|\leq& c_{0}\|\phi\|[\mathrm{e}^{-( \gamma+\varrho/2)t}\\ &+c_{1}\mathrm{e}^{c_{1}t}\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s],\end{split} \tag{4.1}\]

where \(c_{0}=l\mathrm{e}^{\gamma\tau}\) and \(c_{1}=(hL_{f}+bL_{\tilde{g}})\mathrm{e}^{-\varrho\tau/2}\). This implies

\[\begin{split}\left\|\tilde{u}_{t}\left(\cdot,\omega,\phi \right)\right\|\leq& c_{0}\|\phi\|[\mathrm{e}^{-\gamma t}+c_{1} \mathrm{e}^{(c_{1}+\varrho/2)t}\int_{0}^{t}\mathrm{e}^{-(\gamma+\varrho/2+c_{1 })s}\mathrm{d}s]\end{split} \tag{4.2}\]

for P-a.e. \(\omega\in\Omega\) and \(0<t<\rho\) by Gronwall's inequality. Thus, we have

\[\begin{split}\left\|U\left(t,\omega,\phi\right)\right\|=& \|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi))\|\\ \leq& c_{0}\|\phi\|\|v_{t}(\cdot,\omega,\cdot)\| \mathrm{e}^{-\gamma t}+c_{0}c_{1}\mathrm{e}^{(c_{1}+\varrho/2)t}\|\phi\|\times \\ &\|v_{t}(\cdot,\omega,\cdot)\|\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s\end{split} \tag{4.3}\]

for P-a.e. \(\omega\in\Omega\). Therefore,

\[\begin{split}\left\|U\left(t,\theta_{-t}\omega,\phi\right)\right\| =&\|v_{t}(\cdot,\theta_{-t}\omega,\tilde{u}_{t}(\cdot,\theta_{-t} \omega,\phi))\|\\ \leq& c_{0}\mathrm{e}^{-\gamma t}\|\phi(\theta_{-t} \omega)\|\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\|+c_{0}c_{1}\times\\ &(\frac{\mathrm{e}^{(c_{1}+\varrho/2)t}}{c_{1}+\varrho/2+\gamma}- \frac{\mathrm{e}^{-\gamma t}}{c_{1}+\varrho/2+\gamma})\times\\ &\|\phi(\theta_{-t}\omega)\|\|v_{t}(\cdot,\theta_{-t}\omega, \cdot)\|.\end{split} \tag{4.4}\]

Since \(c_{1}+\varrho/2<0,\gamma>0\), there must exits a \(\lambda\) and \(T_{B}>0\) such that for all \(t>T_{B}\)

\[c_{0}\mathrm{e}^{-\gamma t}+c_{0}c_{1}(\frac{\mathrm{e}^{(c_{1}+\varrho/2)t}} {c_{1}+\varrho/2+\gamma}-\frac{\mathrm{e}^{-\gamma t}}{c_{1}+\varrho/2+\gamma })<\lambda.\]

Moreover, since \(\|v_{t}(\cdot,\omega,\cdot)\|\) is bounded for all \(t\in\mathbb{R}^{+}\) and P-a.e. \(\omega\in\Omega\), if \(\phi\left(\theta_{-t}\omega\right)\in B\left(\theta_{-t}\omega\right)\) then for all \(t\geq t_{B}(\omega)\),

\[\|U\left(t,\theta_{-t}\omega\right),\phi\left(\theta_{-t}\omega\right)\|\leq \lambda\|\phi(\theta_{-t}\omega)\|\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\|.\]

Therefore, \(K(\omega)=\{\vartheta\in\mathcal{C}:\|\vartheta\|\leq\lambda\|\phi(\theta_{-t }\omega)\|\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\|\}\) form a tempered bounded closed random absorbing set of RDS \(U\) in \(\mathcal{D}\), which completes the proof. 

In the next lemma, we show that we can take initial data in \(\{K(\omega)\}_{\omega\in\Omega}\) to obtain the pullback asymptotic compactness of \(U\).

**Lemma 4.2**.: _Assume that \(\{K(\omega)\}_{\omega\in\Omega}\) is a random absorbing in \(\mathcal{C}\) for the RDS \(U\). Then \((\theta,U)\) is pullback asymptotically compact._
Proof.: Take an arbitrary random set \(\left\{B(\omega)\right\}_{\omega\in\Omega}\in\mathcal{D}\), a sequence \(t_{n}\rightarrow+\infty\) and \(\phi_{n}\in B\left(\theta_{-t_{n}}\omega\right)\). We have to prove that \(\left\{U\left(t_{n},\theta_{-t_{n}}\omega,\phi_{n}\right)\right\}\) is precompact. Since \(\left\{K(\omega)\right\}\) is a random absorbing for \(\Psi\), then there exists \(T>0\) such that, for all \(\omega\in\Omega\),

\[U\left(t,\theta_{-t}\omega\right)B\left(\theta_{-t}\omega\right)\subset K(\omega) \tag{4.5}\]

for all \(t\geq T\). Because \(t_{n}\rightarrow+\infty\), we can choose \(n_{1}\geq 1\) such that \(t_{n_{1}}-1\geq T\). Applying (4.5) for \(t=t_{n_{1}}-1\) and \(\omega=\theta_{-1}\omega\), we find that

\[\eta_{1}\triangleq\Psi\left(t_{n_{1}}-1,\theta_{-t_{n_{1}}}\omega,\phi_{n_{1 }}\right)\in K\left(\theta_{-1}\omega\right) \tag{4.6}\]

Similarly, we can choose a subsequence \(\left\{n_{k}\right\}\) of \(\left\{n\right\}\) such that \(n_{1}<n_{2}<\cdots<n_{k}\rightarrow+\infty\) such that

\[\eta_{k}\triangleq\Psi\left(t_{n_{k}}-k,\theta_{-t_{n_{k}}}\omega,\phi_{n_{k }}\right)\in K\left(\theta_{-k}\omega\right) \tag{4.7}\]

Hence, by the assumption we conclude that the sequence

\[\left\{U\left(k,\theta_{-k}\omega,\eta_{k}\right)\right\} \tag{4.8}\]

is precompact. On the other hand by (4.7), we have

\[\begin{split} U(k,\theta_{-k}\omega,\eta_{k})&=U(k,\theta_{-k}\omega,U(t_{n_{k}}-k,\theta_{-t_{n_{k}}}\omega,U_{n_{k}})\\ &=U\left(t_{n_{k}},\theta_{-t_{n_{k}}}\omega,\phi_{n_{k}}\right),\end{split} \tag{4.9}\]

for all \(k\geq 1\). Combining (4.8) and (4.9), we obtain that the sequence \(\left\{U\left(t_{n_{k}},\theta_{-t_{n_{k}}}\omega,\phi_{n_{k}}\right)\right\}\) is precompact. Therefore, \(\left\{U\left(t_{n},\theta_{t_{n}}\omega,\phi_{n_{k}}\right)\right\}\) is precompact, which completes the proof. 

Lemma (4.1) says that the continuous RDS \(\Psi\) has a random absorbing set while Lemma (4.2) tells us that \(\left(\theta,U\right)\) is pullback asymptotically compact in \(\mathcal{C}\). Thus, it follows from Lemma 2.1 that the continuous RDS \(\left(\theta,U\right)\) possesses a random attractor. Namely, we obtain the following result.

**Theorem 4.1**.: _Assume that conditions of Lemma 4.1 hold, then the continuous RDS \(U\) generated by (1.4) admits a unique \(\mathcal{D}\)-pullback attractor in \(\mathcal{C}\) belonging to the class \(\mathcal{D}\)._

## 5 Exponentially attracting stationary solution

In this section, we derive sufficient conditions that guarantee the random attractor is an exponentially attracting random fixed point \(\xi^{*}\) by a general Banach fixed point theorem. n particular we use the following fixed point theorem in probability analysis, which is often called the uniform strictly contracting property, originates from the early work [33] and was then extended to various versions in [20, 3, 21] and [7]. Here, we adopt the version in [7] for infinite dimensional case.

**Lemma 5.1**.: _Let \(\left(\mathcal{G},d_{\mathcal{G}}\right)\) be a complete metric space with bounded metric. Suppose that_

\[U(t,\omega,\mathcal{G})\subset\mathcal{G}\]

_for \(\omega\in\Omega,t\geq 0\), and that \(x\to U(t,\omega,x)\) is continuous. In addition, we assume the contraction condition: There exists a constant \(k<0\) such that, for \(\omega\in\Omega\),_

\[\sup_{x\neq y\in\mathcal{G}}\log\frac{d_{\mathcal{G}}(U(1,\omega,x),U(1, \omega,y))}{d_{\mathcal{G}}(x,y)}\leq k.\]
_Then \(U\) has a unique generalized fixed point \(\gamma^{*}\) in \(\mathcal{G}\). Moreover, the following convergence property holds:_

\[\lim_{t\to\infty}U\left(t,\theta_{-t}\omega,x\right)=\gamma^{*}(\omega)\]

_for any \(\omega\in\Omega\) and \(x\in\mathcal{G}\)._

**Theorem 5.1**.: _Assume that conditions of Lemma 4.1 hold. Moreover, assume that_

\[\frac{\varrho}{2}e^{[\frac{\varrho}{2}+L_{v}(hL_{f}+bL_{\tilde{g}})]}+\tau-1<0.\]

_Then the RDS \(U\) generated by SNDRDE (1.4) possess a tempered random fixed point \(\chi^{*}\), which is unique under all tempered random variables in \(\mathcal{C}\) and attracts exponentially fast every random variable in \(\mathcal{C}\)._

Proof.: If \(c_{1}+\varrho/2<0<c_{1}+\varrho/2+\gamma\), then the conditions of Theorem 4.1 hold and hence (1.4) possess random attractors in \(\mathcal{C}\). We will prove that (1.4) admits a unique globally exponentially attracting random stationary solution in \(\mathcal{C}\), which immediately implies the random attractor in \(\mathcal{C}\) obtained in Theorem 4.1 is the random fixed point \(\chi^{*}\).

By (4.3), we can see that for any \(\phi\in\mathcal{C}\)

\[\begin{split}\|U\left(t,\omega,\phi\right)\|\leq& c_{0}\|\phi\|\|v_{t}(\cdot,\omega,\cdot)\| \mathrm{e}^{-\gamma t}+c_{0}c_{1}\mathrm{e}^{(c_{1}+\varrho/2)t}\|\phi\|\times \\ &\|v_{t}(\cdot,\omega,\cdot)\|\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s\end{split} \tag{5.1}\]

which implies that for any \(\phi\in\mathcal{C}_{co}\), \(U(t,\omega,\psi)\in\mathcal{C}\), i.e. \(\mathcal{C}\) is invariant under the random semiflow \(U\). Moreover, it follows from Lemma 3.3 that \(U\) is continuous in \(\mathcal{C}\).

Therefore, we only need to prove the contraction property. That is, there exists \(k<0\) such that

\[\sup_{\varphi\neq\psi\in\mathcal{C}}\|U(1,\omega,\varphi)-U(1,\omega,\psi)\| \leq e^{k}\|\varphi-\psi\|. \tag{5.2}\]

Hence, it suffices to prove that for any \(\varphi,\psi\in\mathcal{C}\)

\[\begin{split}\|U(1,\omega,\varphi)-U(1,\omega,\psi)\|=& \|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\varphi))-\\ & v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\psi))\|\leq e^{ k}\|\varphi-\psi\|.\end{split} \tag{5.3}\]

By Eq. (3.12) and Eq. (3.26), we have for any \(\varphi,\psi\in\mathcal{C}\)

\[\begin{split}\|G(t,\omega)\|&\leq\|v_{t}(\cdot, \omega,\tilde{u}_{t}(\cdot,\omega,\varphi)-\tilde{u}_{t}(\cdot,\omega,\psi))\| \\ &\leq\|v_{t}(\cdot,\omega,\cdot)\|\{\|S(t,\omega)*[\varphi-\psi] \|+\\ &\sup_{\zeta\in[-\tau,0]}\int_{0}^{t+\zeta}S(t+\zeta-r,\omega)v^{- 1}(s,\omega,\cdot)\circ\\ &[H(f(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega,\varphi)))-f( v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega,\psi))))\\ &+B(\tilde{g}\left(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega, \varphi))\right)-\\ &\tilde{g}\left(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega, \psi))\right)))]\mathrm{d}s\}\\ &\leq L_{v}e^{\gamma\tau}e^{-\gamma t}\|\phi-\psi\|+L_{v}e^{ \frac{\sigma\tau}{2}}(hL_{f}+bL_{\tilde{g}})\times\\ &\int_{0}^{t}e^{-\frac{\varrho}{2}(t-r)}\left\|U(r,\omega,\phi)-U (r,\omega,\psi)\right\|\mathrm{d}r.\end{split} \tag{5.4}\]
Multiply both sides of (5.4) by \(e^{\frac{gt}{2}}\) and denote by \(H(t,\omega)=e^{\frac{gt}{2}}G(t,\omega)\), \(c_{3}(t)=L_{v}e^{\gamma\tau}e^{-\gamma t}e^{\frac{gt}{2}}\), \(c_{4}(t)=L_{v}e^{\frac{\rho(\tau+t)}{2}}(hL_{f}+bL_{\tilde{g}})\), we obtain

\[\|H(t,\omega)\|\leq c_{3}(t)\|\phi-\psi\|+c_{4}(t)\int_{0}^{t}\|H(r,\omega)\| \mathrm{d}r. \tag{5.5}\]

Again, the Gronwall inequality gives rise to

\[\|H(t,\omega)\|\leq\|\phi-\psi\|(c_{3}(t)+\int_{0}^{t}c_{3}(r)e^{\int_{r}^{t}c _{4}(s)ds}\mathrm{d}r). \tag{5.6}\]

Dividing both sides of (5.6) by \(e^{\frac{gt}{2}}\) and take \(t=1\) lead to

\[\|G(1,\omega)\|\leq\|\phi-\psi\|(c_{3}(1)e^{-\frac{\rho}{2}}+e^{-\frac{\rho}{ 2}}\int_{0}^{1}c_{3}(r)e^{\int_{r}^{1}c_{4}(s)ds}\mathrm{d}r). \tag{5.7}\]

Therefore, if there exists \(\varsigma<0\) such that \(c_{3}(1)e^{-\frac{\rho}{2}}<e^{\varsigma}\) and \(e^{-\frac{\rho}{2}}\int_{0}^{1}c_{3}(r)e^{\int_{r}^{1}c_{4}(s)ds}\mathrm{d}r<e ^{\varsigma}\) then result of the theorem hold. By a simple computation, we see that \(\frac{\rho}{2}e^{[\frac{\rho}{2}+L_{v}(hL_{f}+bL_{\tilde{g}})]}+\tau-1<0\) implies the above claim holds and hence completes the proof. 

## 6 Numerical Simulations

This section is devoted to some numerical simulations in order to demonstrate the effectiveness and efficiency of the established theoretical results. In our numerical examples, we consider the following stochastic delayed HNNM.

\[\left\{\begin{aligned} & du_{1}(t)=[-c_{1}u_{1}(t)+h_{11}f_{1} \left(u_{1}(t)\right)+h_{12}f_{2}\left(u_{2}(t)\right)+\\ & b_{11}g_{1}\left(u_{1}\left(t-\tau_{1}\right)\right)+b_{12}g_{2} \left(u_{2}\left(t-\tau_{2}\right)\right)]dt\\ &+\sigma_{11}u_{1}(t)dw_{1}(t)+\sigma_{12}u_{2}(t)dw_{2}(t)\\ & du_{2}(t)=[-c_{2}u_{2}(t)+h_{21}f_{1}\left(u_{1}(t)\right)+h_{ 22}f_{2}\left(u_{2}(t)\right)+\\ & b_{21}g_{1}\left(u_{1}\left(t-\tau_{1}\right)\right)+b_{22}g_{2 }\left(u_{2}\left(t-\tau_{2}\right)\right)]dt\\ &+\sigma_{21}u_{1}(t)dw_{1}(t)+\sigma_{22}u_{2}(t)dw_{2}(t).\end{aligned}\right. \tag{6.1}\]

Take

\[u(t)=\left(u_{1}(t),u_{2}(t)\right)^{\mathrm{T}},\]

\[u_{t}(\tau)=\left(u_{1}\left(t-\tau_{1}\right),u_{2}\left(t-\tau_{2}\right) \right)^{\mathrm{T}},\]

\[C=\left(\begin{array}{cc}c_{1}&0\\ 0&c_{2}\end{array}\right)=\left(\begin{array}{cc}5&0\\ 0&5\end{array}\right),\]

\[H=\left(\begin{array}{cc}h_{11}&h_{12}\\ h_{21}&h_{22}\end{array}\right)=\left(\begin{array}{cc}0.2&0.1\\ 0.3&0.1\end{array}\right),\]

\[B=\left(\begin{array}{cc}b_{11}&b_{12}\\ b_{21}&b_{22}\end{array}\right)=\left(\begin{array}{cc}-0.3&0.2\\ 0.1&0.3\end{array}\right),\]

\[\sigma=\left(\begin{array}{cc}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{array}\right)=\left(\begin{array}{cc}0.01&0\\ 0&0.02\end{array}\right),\]
\(f_{j}(u_{j})=g_{j}(u_{j})=\tanh(u_{j})=\frac{e^{u_{j}^{\prime}-e^{-u_{j}}}}{e^{u _{j}^{\prime}+e^{-u_{j}}}_{j}}\) for \(j=1,2\).

Hence the characteristic equation of (6.1) is

\[\det\left[\left(\begin{array}{ll}\lambda-5-0.3e^{-\lambda\tau_{1}}&0.2e^{- \lambda\tau_{1}}\\ 0.1e^{-\lambda\tau_{2}}&\lambda-5+0.3e^{-\lambda\tau_{2}}\end{array}\right) \right]=0, \tag{6.2}\]

We first choose \(\tau_{1}=\tau_{2}=0.1\) and hence **Hypothesis A1 - A2** hold and \(\rho<0\). Thus it follows from 4.1 that the equation admits a random attractor. Moreover, \(\frac{\rho}{2}e^{[\frac{\rho}{2}+L_{v}(\hbar L_{f}+bL_{\bar{g}})]}+\tau-1\)\(<0\) also holds and it follows from Theorem 5.1 the attractor is a unique exponentially attractive tempered random fixed point, i.e. the trivial fixed point \((0,0)\), which is shown in Fig 1. In Fig 1a, the initial condition is chosen to be \(\phi(t)=(0.1,0.2)\) for \(t\in[-0.1,0]\) while in In Fig 1b, the initial condition is first chosen to be \(\phi(t)=(10,20)\) for \(t\in[-0.1,0]\). From which we can see no matter how large the initial condition is, the state will tend to \((0,0)\).

## 7 Summary and discussions

In this paper, we have obtained the existence and qualitative property of random attractors for (1.3), which first required showing that the system generates a random dynamicla system.. We showed under certain conditions that the random attractor is a globally exponentially attracting random stationary solution. From dynamical system theory, the conditions for the attractors to be fixed point are so strong that could be hardly met in the real world applications. Indeed, from the dissipative system theory, if estimation on the dimension of random attractors can provide researchers with useful information about the structure of the random attractor. Hence, the topological dimension for the random attractors of (1.3) will be investigated in a future paper. Furthermore, in order to obtain the global complex dynamics and nonlocal analysis of the qualitative properties of the system, the existence and structure of the associated invariant manifolds of the stationary solutions, the existence of connecting orbits (including the heteroclinic orbits or homoclinic orbits) are all of significance and deserve more attention.
Acknowledgement

This work was jointly supported by the National Natural Science Foundation of China (62173139), China Postdoctoral Science Foundation(2019TQ0089), Hunan Provincial Natural Science Foundation of China (2020JJ5344, 2019RS1033) the Science and Technology Innovation Program of Hunan Province (2021RC4030), the Scientific Research Fund of Hunan Provincial Education Department (20B353).

## Acknowledgments

The author thanks Dr Ding Kui for assistance of numerical simulations.
