Stability analysis of delayed neural networks via compound-parameter -based integral inequality

1.   Introduction

The objective of this paper is to investigate the existence and uniqueness of invariant measures for the stochastic FitzHugh-Nagumo delay lattice system with long-range interactions on the integer set $\mathbb{Z}$:

where $u_{n}, v_{n}\in\mathbb{R}$, $t > 0$, $\alpha, \rho, \beta, \lambda > 0$, the coupling parameters $J(m)$ are real numbers satisfying $J(m) = J(-m)$ for all positive integer $m$, $a = (a_{n})_{n\in\mathbb{Z}}$, $c = (c_{n})_{n\in\mathbb{Z}}$, $b = (b_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$, and $l = (l_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$ are given deterministic sequences in $\ell^{2}_{\eta}$, $f_{n}, g_{j, n}, h_{j, n}$ are Lipschitz continuous functions for all $j\in\mathbb{N}, n\in\mathbb{Z}$, and $(W_{j}(t))_{j\in\mathbb{N}}$ is a sequence of independent two-sided real-valued Wiener processes defined on a complete filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}\}_{t\in\mathbb{R}}, P)$.

The subsequent changes should be observed while considering the transformation of $J(m)$,

where $k$ is any positive integer and $\delta_{m, n}$ is the Kronecker's delta. Then, lattice system (1.1) can be changed into

where $t > 0, n\in\mathbb{Z}$, $\triangle^{k} = \triangle\circ\cdots\circ\triangle, k$ times, and $\triangle$ is defined by $\triangle u_{n} = u_{n+1}+u_{n-1}-2u_{n}$.

The emergence of lattice equations from spatial discretization of partial differential equations is widely acknowledged. Lattice systems exhibiting long-range interactions have garnered significant attention in the literature. Of those, the dynamics of the DNA molecule were described by Schrödinger lattice systems in ^[1]. Subsequently, Pereira investigated the asymptotic behavior of Schrödinger lattice systems in ^[2] and delay lattice systems in ^[3], respectively. Recently, Chen et al. considered the long-term dynamics of stochastic complex Ginzburg-Landau systems in their study ^[4], and Wong-Zakai approximations of stochastic lattice systems in another study ^[5].

The FitzHugh-Nagumo systems were used to describe the transmission of signals across axons in neurobiology in ^[6]. The asymptotic behavior of FitzHugh-Nagumo systems were studied in both deterministic ^[7] and stochastic scenarios ^{[8,9,10,11,12]}. The FitzHugh-Nagumo lattice systems were employed to stimulate the propagation of action potentials in myelinated nerve axons in ^[13]. The attractors of FitzHugh-Nagumo lattice systems were investigated in the deterministic case by ^[7,14], and in the stochastic case by ^{[11,12,15,16,17,18]}. Among these studies, Wang et al. ^[11] derived the existence and upper semi-continuity of random attractors for FitzHugh-Nagumo lattice systems in $\ell^{2}\times\ell^{2}$, while Chen et al. ^[15] obtained the existence and uniqueness of weak pullback mean random attractors for FitzHugh-Nagumo lattice systems with nonlinear noises in weighted spaces $\ell^{2}_{\sigma}\times\ell^{2}_{\sigma}$.

Furthermore, time delays are a common occurrence in various systems, and can lead to instability, oscillation, and other changes in dynamical systems. Due to their practical and theoretical significance, there has been an increasing emphasis on the study of time-delay systems. Recent studies have delved into the exploration of random attractors for stochastic lattice systems featuring fixed delays in ^{[18,19,20,21,22]}. Additionally, investigations have also been carried out concerning systems with varying delays over time as documented in ^{[3,12,23,24,25]}.

Currently, there has been a significant amount of research conducted on the dynamical behavior of differential equations driven by linear noise. In order to effectively handle stochastic systems with nonlinear noise, Kloeden^[26] and Wang^[27,28] introduced the concept of weak pullback mean random attractors. The work described above has subsequently been widely applied in numerous studies on stochastic systems by a multitude of scholars in ^{[15,16,17,19,20,21,25,27,28,29,30,31,32,33,34,35,36,37,38]}. Among them, Wang et al ^[25] studied the stochastic delay modified Swift-Hohenberg lattice systems, as well as Chen et al.^[19] and Li et al.^[20] considered the stochastic delay lattice systems. However, to the best of our knowledge, the current state of literature on the invariant measures for stochastic FitzHugh-Nagumo delay lattice systems with long-range interactions driven by nonlinear noise in weighted space is regrettably scarce.

The lattice system (1.1) is defined on $\mathbb{Z}$, which represents a spatially discrete analogue to stochastic partial differential equations (PDEs) defined on $\mathbb{R}$. Proving the existence of invariant measures for PDEs on unbounded domains poses a major challenge, primarily due to establishing the tightness of distribution laws of solutions caused by non-compactness in usual Sobolev embeddings on unbounded domains. Various approaches have been developed in literature to address the tightness of solution distributions for PDEs on unbounded domains, such as using weighted spaces in ^[39,40], weak Feller property of solutions in ^[41,42], and cut-off techniques in ^[43,44]. In this paper, the cut-off method will be employed to establish the existence of invariant measures for the stochastic lattice system (1.1) in $C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta})$. Specifically, we will demonstrate that when time is sufficiently large, the mean square of solution tails in $C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta})$ becomes uniformly small; based on this result, we can establish tightness in distribution laws for solutions in $C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta})$. The tail-estimates method has previously been used to prove existence of global attractors for deterministic PDEs ^[45,46] and stochastic PDEs with additive or linear multiplicative noise in ^[47,48]. In this paper, we will apply the tail-estimates approach to handle nonlinear noise involved in (1.1) in $C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta})$. For further information regarding existence of invariant measures for stochastic PDEs defined within bounded domains, please refer to ^[49] and its references.

The structure of this paper is organized as follows: Section 2 introduces the notations and discusses the well-posedness of lattice system (1.1). The subsequent section establishes necessary uniform estimates of solutions, which play a crucial role in demonstrating the main results in the following section. Sections 4 and 5 focus on establishing the existence and uniqueness of invariant measures for lattice system (1.1). Finally, we provide a summary and closing remarks in the last section.

2.   Preliminaries

In this section, we will investigate the well-posedness of the stochastic Fitzhugh-Nagumo delay lattice system (1.1) in weighted space $\ell_{\eta}^{2}\times\ell_{\eta}^{2}$, where $\ell_{\eta}^{2}$ is defined by

$\ell_{\eta}^{2}$ is a Hilbert space with the inner product and norm given by

We further assume that weights $\eta = (\eta_{n})_{n\in\mathbb{Z}}$ satisfy the conditions

and

To get the existence of invariant measures for lattice system (1.1) in $\ell^{2}_{\eta}$, the interaction $J(m)$ should decrease at a sufficiently rapid rate such that

For sequences $a = (a_{n})_{n\in\mathbb{Z}}$, $c = (c_{n})_{n\in\mathbb{Z}}$, $b = (b_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$, and $l = (l_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$ in lattice system (1.1), we assume

For the nonlinear term $f_{n}$ in lattice system (1.1), we assume that $f_{n}$ is a smooth function satisfying that there exists $\kappa\in\mathbb{R}$ such that for all $z\in\mathbb{R}$ and $n\in\mathbb{Z}$,

Moreover, for each $n\in\mathbb{Z}$ and $z\in\mathbb{R}$, we assume that there are positive constants $\iota_{n}$ and $\delta$ such that

where $\iota = (\iota_{n})_{n\in\mathbb{Z}}$ belongs to $\ell^{1}_{\eta}$ and its norm is denoted by $\|\iota\|_{1, \eta}$.

For every $j\in\mathbb{N}$ and $n\in\mathbb{Z}$, we assume that $g_{j, n}, h_{j, n}:\mathbb{R}\rightarrow \mathbb{R}$ is globally Lipschitz continuous; that is, there is a constant $L > 0$ such that for all $z_{1}, z_{2}, z_{1}^{\ast}, z_{2}^{\ast}\in\mathbb{R}$,

We further assume that for each $z, z^{\ast}\in\mathbb{R}$, $j\in\mathbb{N}$, and $n\in\mathbb{Z}$,

where $\gamma_{j, n} > 0$, $\|\gamma\|^{2} = \sum\limits_{j\in\mathbb{N}}\sum\limits_{n\in\mathbb{Z}}|\gamma_{j, n}|^{2} < \infty$, and $\|\gamma\|^{2}_{\eta} = \sum\limits_{j\in\mathbb{N}}\sum\limits_{n\in\mathbb{Z}}\eta_{n}|\gamma_{j, n}|^{2} < \infty$.

For any $u = (u_{n})_{n\in\mathbb{Z}}\in\ell^{2}_{\eta}$ and $v = (v_{n})_{n\in\mathbb{Z}}\in\ell^{2}_{\eta}$, denote by $f(u) = (f_{n}(u_{n}))_{n\in\mathbb{Z}}$ and $f(v) = (f_{n}(v_{n}))_{n\in\mathbb{Z}}$. By (2.5), we get

where $\xi_{n} = \theta_{n}u_{n}+(1-\theta_{n})v_{n}$ for some $\theta_{n}\in(0, 1)$. Moreover, we can obtain that $f$ is locally Lipschitz continuous from $\ell^{2}_{\eta}$ to $\ell^{2}_{\eta}$; that is, there exists $L_{C} > 0$ such that for any $u, v\in\ell^{2}_{\eta}$ with $\|u\|^{2}_{\eta}\leq C$ and $\|v\|^{2}_{\eta}\leq C$,

For each $u^{1} = (u_{n}^{1})_{n\in\mathbb{Z}}, u^{2} = (u_{n}^{2})_{n\in\mathbb{Z}}, v^{1} = (v_{n}^{1})_{n\in\mathbb{Z}}, v^{2} = (v_{n}^{2})_{n\in\mathbb{Z}}\in \ell^{2}_{\eta}$, and $j\in\mathbb{N}$, denote by $g_{j}(u^{1}, v^{1}) = (g_{j, n}(u_{n}^{1}, v^{1}_{n}))_{n\in\mathbb{Z}}$ and $h_{j}(u^{1}, v^{1}) = (h_{j, n}(u_{n}^{1}, v^{1}_{n}))_{n\in\mathbb{Z}}$. It follows from (2.7) and (2.8) that

and

The system (1.1) can be reformulated as an abstract system in $\ell^{2}$, for $u = (u_{n})_{n\in\mathbb{Z}}\in \ell^{2}$, and we set

By Lemma 3.1 of ^[4], we have

By the above notation, system (1.1) can be rewritten as follows: For all $t > 0$,

Let $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta}))$ be $\mathcal{F}_{0}$ -measurable. Then, a continuous $\ell^{2}_{\eta}\times\ell^{2}_{\eta}$ -valued $\mathcal{F}_{t}$ -adapted stochastic process $(u(t), v(t))$ is called a solution of stochastic lattice system (2.15) if $(u_{0}, v_{0}) = (\phi, \varphi)$, $(u(t), v(t))\in L^{2}(\Omega, C([-\rho, T], \ell^{2}_{\eta}\times\ell^{2}_{\eta}))$ for all $T > -\rho$, $t\geq0$ and for almost all $\omega\in\Omega$,

By (2.1)–(2.8) and the theory of the functional differential equation, we can get that for any $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta}))$, stochastic lattice system (2.15) has a solution $(u(t), v(t))\in L^{2}(\Omega, C([-\rho, T], \ell^{2}_{\eta}\times\ell^{2}_{\eta}))$ for every $T\geq-\rho$. Moreover, this solution is unique if $(u^{\ast}(t), v^{\ast}(t))$ is any other solution of system (2.15), then

Actually, the stochastic lattice system (2.15) has a unique solution defined for $t\in[t_{0}-\rho, \infty)$, regardless of any initial time $t_{0}\geq0$ and any $\mathcal{F}_{t_{0}}$ -measurable $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell^{2}_{\eta}\times\ell^{2}_{\eta}))$.

Hereafter, for $t\in\mathbb{R}$, $(u_{t}, v_{t})$ is defined by

and let $C_{\rho, \eta} = C([-\rho, 0], \ell^{2}_{\eta})$ with the norm $\|\chi\|_{\rho, \eta} = \mathop{\sup}\limits_{-\rho\leq s\leq0}\|\chi(s)\|_{\eta}$, $\chi\in C_{\rho, \eta}$.

The establishment of Lipschitz continuity for solutions to stochastic lattice system (2.15) in relation to initial data will now be undertaken, which shall subsequently be employed.

Lemma 2.1. Suppose (2.1)–(2.8) hold and $(\phi_{1}, \varphi_{1}), (\phi_{2}, \varphi_{2})\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$. If $(u(t, \phi_{1}), v(t, \varphi_{1}))$ and $(u(t, \phi_{2}), v(t, \varphi_{2}))$ are the solutions of stochastic lattice system (2.15) with initial data $(\phi_{1}, \varphi_{1})$ and $(\phi_{2}, \varphi_{2})$, respectively, then for any $t\geq0$,

where $M_{1}$ is a positive constant independent of $(\phi_{1}, \varphi_{1})$, $(\phi_{2}, \varphi_{2})$, and $t$.

Proof. By (2.15), we get that for all $t\geq0$,

and

which along with (2.16) and Itô's formula shows that for all $t\geq0$,

where

and

By (2.13) and the fact of $J(m) = J(-m)$, we have

which along with (2.2) and (2.3) implies that

By (2.9), we obtain

By (2.12), we get

It follows from (2.17)–(2.21) that for all $t\geq0$,

which implies that for all $t\geq0$,

For the last two terms of (2.22), by (2.12), the Burkholder-Davis-Gundy (BDG) inequality, and the Minkowski inequality, we have

and

which along with (2.22) and (2.23) shows that

where $C_{2} = 2\big(1+2\rho L^{2}+ \rho C_{1}^{2} L^{2}\big)$, $C_{3} = 4\big(\tilde{\alpha}+2 L^{2}+|\kappa|+ C_{1}^{2} L^{2}\big)$. It follows from (2.24) and the Gronwall inequality that for all $t\geq0$,

This completes the proof. □

The existence of invariant measures of the stochastic lattice system (2.15) in the subsequent analysis necessitates the fulfillment of the following inequality.

where $\nu > 0, 2\delta-4\tilde{\alpha}-\frac{27\beta^{4}}{\lambda^{3}}-8 > 0, 2\lambda-\frac{27\alpha^{4}}{\delta^{3}}-6 > 0$.

3.   Uniform estimates

In this section, we obtain uniform estimates of the solutions to stochastic lattice system (2.15), which play a pivotal role in proving the existence of invariant measures. More specifically, we will showcase the compactness of a family of probability distributions pertaining to $(u_{t}, v_{t})$ in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Initially, our focus lies on discussing uniform estimates of solutions to stochastic lattice system (2.15) in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ for all $t\geq0$.

Lemma 3.1. Suppose (2.1)–(2.8) and (2.25) hold. Let $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$ be the initial data of stochastic lattice system (2.15), then the solution $(u, v)$ of the system (2.15) satisfies

where $M_{2}$ is a positive constant independent of $(\phi, \varphi)$.

Proof. By (2.15) and Itô's formula, we get that for all $t\geq0$,

Let $\nu$ be a positive constant which will be specified later. We get from (3.1) that for all $t\geq0$,

Taking the expectation, we obtain that for $t\geq0$,

Similar to (2.18) and (2.19), we get

By (2.6), we have

Note that

By (2.11), we obtain

and

For $t\geq 0$, it follows from (3.3)–(3.8) that

For $t\geq0$, by (2.25) and (3.9), we get that there exists $\nu_{1} > 0$ such that for all $\nu\in(0, \nu_{1})$,

Note that

which along with (3.10) implies the desired result. □

Lemma 3.2. Suppose (2.1)–(2.8) and (2.25) hold. Let $(\phi, \varphi)\in L^{4}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$ be the initial data of stochastic lattice system (2.15), then the solution $(u, v)$ of the system (2.15) satisfies

where $M_{3}$ is a positive constant independent of $(\phi, \varphi)$.

Proof. Given $n\in\mathbb{N}$, define $\tau_{n}$ by

and $\tau_{n} = \infty$ if the set $\{t\geq0:\|u(t)\|_{\eta}+\|v(t)\|_{\eta} > n\} = \emptyset$. By (3.1) and Itô's formula, we get for all $t\geq0$,

Let $\nu$ be a positive constant which will be specified later, and we get from (3.11) that for all $t\geq0$,

Similar to (2.18) and (2.19), we get

By (2.6) and Young's inequality, we have

Note that

and

By (2.8), we get

and

For $t\geq 0$, it follows from (3.12)–(3.18) that

which along with (2.25) implies that there exists $\nu_{2} > 0$ such that for all $\nu\in(0, \nu_{2})$,

Letting $n\rightarrow \infty$, we obtain from the above inequality that for all $t\geq0$,

Note that for all $t\in[-\rho, 0]$,

which along with (3.19) concludes the proof. □

Lemma 3.3. Suppose (2.1)–(2.8) and (2.25) hold. Let $(\phi, \varphi)\in L^{4}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$ be the initial data of stochastic lattice system (2.15), then the solution $(u, v)$ of the system (2.15) satisfies, for any $t > r\geq0$,

where $M_{4}$ is a positive constant depending on $(\phi, \varphi)$, but is independent of $t$ and $r$.

Proof. For $t > r\geq0$, by (2.15), we get

which together with (2.5), (2.10), and (2.14) implies that, for $t > r\geq0$,

where $C_{4} = \sqrt{2|J(0)|^{2}+8\Big(\sum\limits_{m = 1}^{\infty}|J(m)|\Big)^{2}}+L_{C}$. By (3.20), we get

By Schwarz's inequality and Lemma 3.2, we have

For the last two terms of (3.21), by (2.11), Lemma 3.2, and the BDG inequality, we get

and

which along with (3.21)–(3.23) implies the desired result. □

The subsequent step entails acquiring uniform estimates on the tails of solutions to stochastic lattice system (2.15), which play a pivotal role in proving the tightness of a family of solution distributions.

Lemma 3.4. Suppose (2.1)–(2.8) and (2.25) hold. For any compact subset $E\subset L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, the solution $(u, v)$ of stochastic lattice system (2.15) satisfies

Proof. Let $\vartheta$ be a smooth function which is defined on $\mathbb{R}^{+}$ such that $0\leq\vartheta(z)\leq1$ for all $z\in\mathbb{R}^{+}$, and

For $k\in N$, set $\vartheta_{k} = (\vartheta(\frac{|n|}{k}))_{n\in\mathbb{Z}}$, $\vartheta_{k}u = (\vartheta(\frac{|n|}{k})u_{n})_{n\in\mathbb{Z}}$, and $\vartheta_{k}v = (\vartheta(\frac{|n|}{k})v_{n})_{n\in\mathbb{Z}}$. By (2.15), we have

which along with Itô's formula implies that

Then, we get that for all $t\geq0$,

where $\nu$ is a positive constant which will be specified later. Furthermore, we find that

where

and

For any $n\in\mathbb{Z}$ and $m\in\mathbb{N}^{+}$, by the definition of $\vartheta(z)$ we can get that there exists a constant $C_{8} > 0$ such that

By (2.2), we have

which together with (3.27) implies that for any $p > 1$,

By (2.2), we obtain

which together with (3.27) and (2.3) implies that for any $p > 1$,

For any $p > 1$, it follows from (3.26), (3.28), and (3.29) that

By (2.6) and Young's inequality, we have

Note that

For the last two terms of (3.25), by (2.8), we get

Then, it follows from (3.25) and (3.30)–(3.33) that for $p > 1$,

Furthermore, it follows from (2.3) that there is a $K_{1} = K_{1}(\nu) > 0$ such that for all $k\geq K_{1}$,

By (2.3) again, we can choose $p = p(\nu)$ large enough such that

which along with (3.35) and (2.25) implies that there exists $\nu_{3} > 0$ such that for all $\nu\in(0, \nu_{3})$,

which together with (3.34) and (2.25) shows that for all $t\geq0$ and $k\geq K_{1}$,

Note that $(\phi, \varphi)\in E$ and $E$ is a compact subset in $L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$. Then, for each $\varepsilon > 0$, there exists $K_{2} = K_{2}(\varepsilon, \phi, \varphi)\geq1$ such that for all $k\geq K_{2}$,

It follows from (3.37) that for all $k\geq K_{2}$,

Since $a = (a_{n})_{n\in\mathbb{Z}}$, $c = (c_{n})_{n\in\mathbb{Z}}, b = (b_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}, l = (l_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}, \gamma = (\gamma_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}\in\ell_{\eta}^{2}$ and $\iota = (\iota_{n})_{n\in\mathbb{Z}}\in \ell^{1}_{\eta}$, we get that there exists $K_{3} = K_{3}(\varepsilon)\geq 1$ such that for all $t\geq0$ and $k\geq K_{3}$,

which along with (3.36) and (3.38) implies that for all $t\geq0$, $k\geq\max\{K_{1}, K_{2}, K_{3}\}$, and $(\phi, \varphi)\in E$,

Observe that $\{(\phi(s), \varphi(s))\in L^{2}(\Omega, \ell_{\eta}^{2}\times\ell_{\eta}^{2}):s\in[-\rho, 0]\}$ is a compact subset in $L^{2}(\Omega, \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Then, for each $\varepsilon > 0$, there are $s_{1}, s_{2}, \cdots s_{m}\in[-\rho, 0]$ such that

where $B((\phi(s_{j}), \varphi(s_{j})), \frac{1}{2}\sqrt{\varepsilon})$ is an open ball in $L^{2}(\Omega, \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ centered at $(\phi(s_{j}), \varphi(s_{j}))$ with radius $\frac{1}{2}\sqrt{\varepsilon}$. Since $(\phi(s_{j}), \varphi(s_{j}))\in L^{2}(\Omega, \ell_{\eta}^{2}\times\ell_{\eta}^{2})$, for $j = 1, \cdots, m$, there exists $K_{4} = K_{4}(\varepsilon, \phi, \varphi)\geq1$, such that for all $k\geq K_{4}$,

It follows from (3.40) and (3.41) that for all $k\geq K_{4}$ and $s\in[-\rho, 0]$,

which along with (3.39) implies the desired result. □

The tail estimates given by Lemma 3.4 have been enhanced to obtain uniform estimates on the tails of solutions, which are crucial for achieving tightness in the probability distributions of solution segments in the space $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Lemma 3.5. Suppose (2.1)–(2.8) and (2.25) hold. For any compact subset $E\subset L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, the solution $(u, v)$ of stochastic lattice system (2.15) satisfies

Proof. Let $\vartheta$ be the function defined in Lemma 3.4. For all $t\geq\rho$ and $t-\rho\leq r\leq t$, it follows from (3.24) that

which shows that for all $t\geq\rho$,

For any $\varepsilon > 0$, by Lemma 3.4, we get that there is a $K_{5} = K_{5}(\varepsilon, E)\geq1$ such that for all $k\geq K_{5}$ and $t\geq-\rho$,

which shows that for all $k\geq K_{5}$ and $t\geq-\rho$,

Then, for all $k\geq K_{5}$ and $t\geq\rho$,

Proceeding as in (3.30), we have

Then, by (3.43), we get that for all $k\geq K_{5}$ and $t\geq-\rho$,

Furthermore, it follows from (2.3) and Lemma 3.1 that there is a $K_{6} = K_{6}(\varepsilon, E)\geq K_{5}$, such that for all $k\geq K_{6}$ and $t\geq \rho$,

By (2.3) and Lemma 3.1 again, we can choose $p = p(\varepsilon)$ large enough such that for all $t\geq \rho$,

Since $\iota = (\iota_{n})_{n\in\mathbb{Z}}\in \ell_{\eta}^{1}$, we get that there exists $K_{7} = K_{7}(\varepsilon, E)\geq K_{6}$ such that for all $k\geq K_{7}$,

By (3.43), we get for all $k\geq K_{5}$ and $t\geq\rho$,

Since $a = (a_{n})_{n\in\mathbb{Z}}, c = (c_{n})_{n\in\mathbb{Z}}\in \ell_{\eta}^{2}$, it follows from (3.50) that there exists $K_{8} = K_{8}(\varepsilon, E)\geq K_{7}$ such that for all $k\geq K_{8}$ and $t\geq\rho$,

By (2.8), (3.43), and Lemma 3.4, we get for all $t\geq\rho$ and $k\geq K_{5}$,

and

Since $b = (b_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$, $l = (l_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$, and $\gamma = (\gamma_{j, n})_{j\in\mathbb{N}, n\in\mathbb{Z}}$ belong to $\ell_{\eta}^{2}$, we infer from (3.52) and (3.53) that there exists $K_{9} = K_{9}(\varepsilon, E)\geq K_{8}$ such that for all $k\geq K_{9}$ and $t\geq\rho$,

For the last two terms of (3.42), by the BDG inequality, (2.8), and (3.54), we have for all $k\geq K_{9}$ and $t\geq\rho$,

By (3.42)–(3.55), we get that for all $t\geq\rho$ and $k\geq K_{9}$,

where $C_{10} = 2\Big(1+2\tilde{\alpha}\rho+5\rho+\rho(\beta+\alpha)(2+16\|\gamma\|^{2})(1+2C^{2}_{9})\Big)\varepsilon$. This completes the proof. □

4.   Existence of invariant measures

The focus of this section is to establish the existence of invariant measures for lattice system (2.15) in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Initially, we introduce the transition operators of the lattice system and subsequently demonstrate the tightness of a family of probability distributions for solutions of the lattice system.

Given every $t_{0}\geq0$ and $\mathcal{F}_{t_{0}}$ -measurable $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, lattice system (2.15) possesses a distinct solution that is valid for all $t\geq t_{0}-\rho$. Given $t\geq t_{0}$ and $(\phi, \varphi)\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, we use $(u_{t}(t_{0}, \phi), v_{t}(t_{0}, \phi))$ to represent the segment of the solution $(u(t, t_{0}, \phi), v(t, t_{0}, \varphi))$ which is given by

Then, we have $(u_{t}(t_{0}, \phi), v_{t}(t_{0}, \varphi))\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$ for all $t\geq t_{0}$.

Suppose $\psi:C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})\rightarrow \mathbb{R}$ is a bounded Borel function. For $0\leq r\leq t$, we set

In addition, for $G\in\mathcal{B}(C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, $0\leq r\leq t$, and $(\phi, \varphi)\in C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$, we set

where $1_{G}$ is the characteristic function of $G$. Then, we can get that $p(r, \phi, \varphi; t, \cdot)$ is the probability distribution of $(u_{t}(r, \phi), v_{t}(r, \varphi))$ in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Furthermore, the transition operator $p_{0, t}$ is denoted as $p_{t}$ for the sake of convenience.

Definition 4.1. A probability measure $\mu$ on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ is called an invariant measure of lattice system (2.15) if

Now, we show the properties of transition operators $\{p_{r, t}\}_{0\leq r\leq t}$ as follows.

Lemma 4.1. Suppose (2.1)–(2.8) and (2.25) hold. Then, we have

(i) The family $\{p_{r, t}\}_{0\leq r\leq t}$ is Feller; that is, if $\psi:C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})\rightarrow \mathbb{R}$ is bounded and continuous, then $p_{r, t}\psi:C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})\rightarrow \mathbb{R}$ is bounded and continuous.

(ii) The family $\{p_{r, t}\}_{0\leq r\leq t}$ is homogeneous; that is,

(iii) Given $r\geq0$ and $(\phi, \varphi)\in C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$, the process $\{(u_{t}(r, \phi), v_{t}(r, \varphi))\}_{t\geq r}$ is a $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ -value Markov process. Consequently, if $\psi:C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})\rightarrow \mathbb{R}$ is a bounded Borel function, then for any $0\leq s\leq r\leq t$, P-a.s.

for all $(\phi, \varphi)\in C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ and $G\in\mathcal{B}(C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, the Chapman-Kolmogorov equation is valid:

Proof. By Lemma 2.1 and the standard arguments as in ^[49], we can get the Feller property (ⅰ)–(ⅲ). □

Lemma 4.2. Suppose (2.1)–(2.8) and (2.25) hold. Then, the distribution laws of the process $\{(u_{t}(0, 0), v_{t}(0, 0))\}_{t\geq0}$ is tight on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Proof. For all $t\geq0$, by Lemma 3.1 and Chebyshev's inequality, we have

Then, for each $\varepsilon > 0$, there exists a constant $R_{1} = R_{1}(\varepsilon) > 0$ such that

By Lemma 3.3, we get that for all $r, s\in[-\rho, 0]$ and $t\geq \rho$,

for some $C_{12} > 0$. Given $\varepsilon > 0$, it follows from the usual technique of diadic division and (4.2) that there exists a constant $R_{2} = R_{2}(\varepsilon) > 0$ such that for all $t\geq0$,

By Lemma 3.5, we obtain that for every $\varepsilon > 0$ and $m\in\mathbb{N}$, there exists an integer $k_{m} = k_{m}(\varepsilon, m)\geq1$ such that for all $t\geq0$,

Then, for all $t\geq0$ and $m\in\mathbb{N}$,

which shows that for all $t\geq0$,

For $\varepsilon > 0$, set $\mathcal{Z}_{\varepsilon} = \mathcal{Z}_{1, \varepsilon}\bigcap\mathcal{Z}_{2, \varepsilon}\bigcap\mathcal{Z}_{3, \varepsilon}$, where

It follows from (4.1), (4.3), and (4.5)–(4.8) that for all $t\geq0$,

By Arzela-Ascoli theorem, we can establish the pre-compactness of $\mathcal{Z}_{\varepsilon}$ in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Specifically, by (4.7), we get that $\mathcal{Z}_{\varepsilon}$ is equi-continuous in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. On the other hand, by (4.6) and (4.7), we have for every $r\in[-\rho, 0]$,

which along with (4.8) shows that $\{(u(r), v(r)), (u, v)\in\mathcal{Z}_{\varepsilon}\}$ is pre-compact in $\ell_{\eta}^{2}\times\ell_{\eta}^{2}$. This completes the proof. □

Now, the main outcome of this paper has been showed: The existence of invariant measures for lattice system (2.15) on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Theorem 4.1. Suppose (2.1)–(2.8) and (2.25) hold. Then, lattice system (2.15) has an invariant measure on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Proof. By using Krylov-Bogolyubov's method, for each $n\in\mathbb{N}$, the probability measure $\mu_{n}$ is defined by

It follows from Lemma 4.2 that the sequence $(\mu_{n})_{n = 1}^{\infty}$ is tight on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$. Consequently, there exists a probability measure $\mu$ on $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$ and a subsequence (still denoted by $(\mu_{n})_{n = 1}^{\infty}$) such that

By (4.10)–(4.11) and Lemma 4.1, we can get for every $t\geq0$ and every bounded and continuous function $\psi:C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})\rightarrow \mathbb{R}$,

which implies that $\mu$ is an invariant measure of lattice system (2.15). This completes the proof. □

5.   Uniqueness of invariant measures

In this section, we examine the uniqueness of invariant measures for system (2.15) under additional constraints on the diffusion and drift terms. Specifically, we impose the following assumption:

which implies that there exists a small number $\varsigma > 0$ such that

From now on, we fix such a $\varsigma > 0$ satisfying (5.2). We will demonstrate that, subject to condition (5.2), any two solutions of Eq (2.15) converge toward each other at an exponential rate, which immediately implies the uniqueness of invariant measures. To begin with, we establish uniform estimates in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Lemma 5.1. Suppose (2.1)–(2.8) and (5.1) hold, and $(\phi_{1}, \varphi_{1}), (\phi_{2}, \varphi_{2})\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$. If $(u(t, \phi_{1}), v(t, \varphi_{1}))$ and $(u(t, \phi_{2}), v(t, \varphi_{2}))$ are the solutions of system (2.15) with initial data $(\phi_{1}, \varphi_{1})$ and $(\phi_{2}, \varphi_{2})$, respectively, then for any $t\geq-\rho$,

where $M_{5}$ is a positive constant depending on $(\phi, \varphi)$.

Proof. By (2.17), we get that for $t\geq0$,

Similar to (2.18) and (2.19), we obtain

By (2.9), we have

By (2.12), we get

It follows from (5.2)–(5.6) that for all $t\geq0$,

which implies that for all $t\geq0$,

On the other hand, for $t\in[-\rho, 0]$, we have

which along with (5.7) concludes the proof. □

Lemma 5.2. Suppose (2.1)–(2.8) and (5.1) hold, and $(\phi_{1}, \varphi_{1}), (\phi_{2}, \varphi_{2})\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$. If $(u(t, \phi_{1}), v(t, \varphi_{1}))$ and $(u(t, \phi_{2}), v(t, \varphi_{2}))$ are the solutions of system (2.15) with initial data $(\phi_{1}, \varphi_{1})$ and $(\phi_{2}, \varphi_{2})$, respectively, then for any $t\geq\rho$,

where $M_{6}$ is a positive constant independent of $(\phi_{1}, \varphi_{1})$ and $(\phi_{2}, \varphi_{2})$.

Proof. By (2.17), we get that for $t\geq\rho$ and $r\geq t-\rho$,

where

and

By (5.8), we get that for all $t\geq\rho$,

By Lemma 5.1, we see that for all $t\geq\rho$,

Similar to (2.18) and (2.19), we obtain

which along with Lemma 5.1 implies that

By (2.9) and (5.1), we have

By (2.12) and Lemma 5.1 we get

For the last two terms of (5.9), by the BDG inequality and (5.13), we get

where $C_{14} = \frac{3C_{13}^{2}L^{2}(1+2\rho L^{2})}{\varsigma}$. It follows from (5.9)–(5.14) that for all $t\geq0$,

where $C_{15} = 2\big((1+2\rho L^{2})(1+\frac{2\tilde{\alpha}+6L^{2}}{\varsigma})+C_{14}\big)$. This completes the proof. □

Theorem 5.1. Suppose (2.1)–(2.8) and (5.1) hold. Then, stochastic lattice system (2.15) has a unique invariant measure in $C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2})$.

Proof. For any $(\phi_{1}, \varphi_{1}), (\phi_{2}, \varphi_{2})\in L^{2}(\Omega, C([-\rho, 0], \ell_{\eta}^{2}\times\ell_{\eta}^{2}))$, by Lemma 5.2, we see that the segments of the solutions $(u_{t}(\phi_{1}), v_{t}(\varphi_{1}))$ and $(u_{t}(\phi_{2}), v_{t}(\varphi_{2}))$ of (2.15) satisfy, for all $t\geq\rho$,

which along with the standard arguments (see, e.g., ^[50]) implies the uniqueness of invariant measures for the lattice system (2.15). This completes the proof. □

6.   Conclusions

The current focus lies in the theoretical proof of the well-posedness of solutions and the existence and uniqueness of invariant measures for these stochastic delay lattice systems. In the future, our research group will investigate the convergence and approximation of invariant measures for the systems under noise perturbation, as well as explore large deviation principles for the systems. Additionally, we will employ finite-dimensional numerical approximation methods to address both the existence of numerical solutions and numerical invariant measures.

Author contributions

Xintao Li and Lianbing She: Conceptualization, Writing original draft and writing-review and editing; Rongrui Lin: Writing original draft and writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Scientific Research and Cultivation Project of Liupanshui Normal University (LPSSY2023KJYBPY14).

Conflict of interest

The authors declare no conflict of interest.