Optimal Convergence of Slow–Fast Stochastic Reaction–Diffusion–Advection Equation with Hölder-Continuous Coefficients

Abstract

This paper investigates a slow–fast stochastic reaction–diffusion–advection equation with Hölder-continuous coefficients, where the irregularity of the coefficients presents significant analytical challenges. Our approach fundamentally relies on techniques from Poisson equations in Hilbert spaces, through which we establish optimal strong convergence rates for the approximation of the averaged solution by the slow component. The key advantage that this paper presents is that the coefficients are merely Hölder continuous yet the optimal rate can still be obtained, which is crucial for subsequent central limit theorems and numerical approximations.

1. Introduction

Many natural systems in physics, chemistry, and biology can be mathematically represented using advection–reaction–diffusion equations. The standard formulation of such equations is given by

$$\begin{array}{c}\hfill X_t-X_{\xi \xi }+X^3-X{X}_{\xi }=0,\end{array}$$

which can be used to describe the dynamics of spatially distributed biological, chemical, and physical processes, ecological models, genetic models, population distribution, etc.; see [1,2] and the references therein for more studies.

Let $T>0$ and $D=(0,1)$. Consider the following slow–fast stochastic reaction–diffusion–advection equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{𝜕X_t^{\epsilon }\left(\xi \right)}{𝜕t}}=\Delta X_t^{\epsilon }\left(\xi \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{𝜕}{𝜕\xi }}{\left(X_t^{\epsilon }\left(\xi \right)\right)}^2-{X_t^{\epsilon }}^3\left(\xi \right)+f(X_t^{\epsilon }\left(\xi \right),Y_t^{\epsilon }\left(\xi \right))+{\displaystyle \frac{𝜕W_t^1\left(\xi \right)}{𝜕t}},\hfill \\ {\displaystyle \frac{𝜕Y_t^{\epsilon }\left(\xi \right)}{𝜕t}}={\displaystyle \frac{1}{\epsilon }}\Delta Y_t^{\epsilon }\left(\xi \right)+{\displaystyle \frac{1}{\epsilon }}g(X_t^{\epsilon }\left(\xi \right),Y_t^{\epsilon }\left(\xi \right))+{\displaystyle \frac{1}{\sqrt{\epsilon }}}{\displaystyle \frac{𝜕W_t^2\left(\xi \right)}{𝜕t}},\hfill \\ X_t^{\epsilon }\left(0\right)=X_t^{\epsilon }\left(1\right)=Y_t^{\epsilon }\left(0\right)=Y_t^{\epsilon }\left(1\right)=0,t\in (0,T],\hfill \\ X_0^{\epsilon }\left(\xi \right)=x,Y_0^{\epsilon }\left(\xi \right)=y,\xi \in D,\hfill \end{array}\right.$$

(1)

where f and g are reaction coupling satisfying some appropriate conditions, and $W_t^1$ and $W_t^2$ are mutually independent $Q_1$- and $Q_2$-Wiener processes, both defined on some probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with a normal filtration ${\left\{{\mathcal{F}}_t\right\}}_{t⩾0}$. The parameter $0<\epsilon \ll 1$ characterizes the time-scale separation between the slow component $X_t^{\epsilon }$ and the fast component $Y_t^{\epsilon }$.

Such a multiscale model arises from describing multiscale phenomena and frequently appears in many real-world dynamical systems. Typical examples include but are not limited to phytoplankton dynamics (see, e.g., [3,4]), species competition (see, e.g., [5]), cell modeling (see, e.g., [6]), Hamiltonian systems (see, e.g., [7]), and electronic circuits (see, e.g., [8]), etc.

However, directly analyzing or simulating the multiscale system (1) remains challenging due to the strong time-scale separation between slow and fast modes and their nonlinear coupling effects. To address this, simplified models capturing the long-time system evolution are crucial for practical applications. This reduction methodology originates from the averaging principle, initially developed for deterministic systems by Bogoliubov [9] and later generalized to stochastic differential equations (SDEs) by Khasminskii [10]. Subsequent research has extensively explored averaging techniques for finite-dimensional stochastic systems, as documented in [11,12,13,14,15,16,17] and the references therein. The extension of averaging principles to infinite-dimensional systems presents significant mathematical challenges and has only been systematically developed in recent decades. Cerrai and Freidlin [18] established the averaging principle for stochastic reaction–diffusion equations, focusing on systems where the slow dynamics are deterministic. Later, Cerrai [19,20] extended this result to more general cases; see also [21,22,23,24,25,26,27,28] and the references therein for further developments regarding multiscale stochastic partial differential equations (SPDEs for short).

Note that all the works mentioned above focus on linear or semi-linear SPDEs. However, it is widely recognized that numerous models and equations in physics involve highly nonlinear terms. Consequently, multiscale SPDEs with highly nonlinear terms have garnered increasing attention in this field. Here are some representative references for different classes of nonlinear SPDEs: for parabolic-type SPDEs, we refer to [29,30,31], while hyperbolic-type systems are addressed in [32]; for applications in fluid mechanics with physical motivations, we point to [33,34], etc. We note that Gao [30] studied the strong convergence of the system (1) with Lipschitz continuous coefficients by using the techniques of time discretization and stopping time; however, the optimal convergence rate was not achieved. Very recently, Gao and Sun [31] studied the multiscale stochastic Burgers equation and obtained the optimal strong convergence rate, where the coefficients are assumed to be regular.

In the present paper, we focus our attention on system (1) with Hölder-continuous coefficients. Moreover, we shall establish a stronger convergence result in the averaging principle and provide the optimal convergence rate. To be precise, it is shown that, for every $T>0$, $q⩾1$ and $\gamma \in [0,1/2)$, there exists a constant $C_0>0$ such that

$$\underset{t\in [0,T]}{sup}\mathbb{E}{\parallel {(-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t)\parallel }^q⩽C_0{\epsilon }^{{\displaystyle \frac{q}{2}}},$$

and also see Theorem 1 below. The main contributions are summarized as follows:

• Our results significantly relax the regularity assumptions in [30,31], requiring only Hölder continuity in the fast variable while achieving stronger convergence regarding the ${\parallel ·\parallel }_{{(-A)}^{\gamma }}$ norm with any $\gamma \in [0,1/2)$. We note that the established convergence of ${(-A)}^{\gamma }{X}_t^{\epsilon }$ to ${(-A)}^{\gamma }{\overline{X}}_t$ proves particularly crucial for the analysis of the central limit theorem, which will be processed in the following work.
• Our method for establishing strong convergence fundamentally differs from existing approaches (see, e.g., [18,19,20,23,24,28]), where the time discretization procedure is used. Our method is based on the Poisson equation; see Theorem 2.
• Highly nonlinear terms, incluing the cubic term ${X_t^{\epsilon }}^3\left(\xi \right)$ and the advection term ${\displaystyle \frac{1}{2}}{\displaystyle \frac{𝜕}{𝜕\xi }}{\left(X_t^{\epsilon }\left(\xi \right)\right)}^2$, will also cause additional difficulties, which are addressed through some exponential moment estimates.
• We establish the characteristic ${\displaystyle \frac{1}{2}}$-order convergence rate, which is known to be sharp for finite-dimensional systems (when $\gamma =0$), as demonstrated in [35]. Under the regularity assumptions regarding noise (A2), we show that the averaging convergence is independent of the fast variable’s coefficient regularity.

In the following Section 2, we introduce the main theorem of this paper; in Section 3 and Section 4, we use the Poisson equation and exponential estimation technique to prove the result of strong convergence.

Notations. To end this section, we introduce some notations: $\forall p\in [1,\infty ]$, and let $L^p:=L^p\left(D\right)$ denote the Banach space of functions on D equipped with the $L^p$-norm ${\parallel ·\parallel }_{L^p}$. For $p=2$, we denote the Hilbert space $L^2\left(D\right)$ endowed with scalar product $⟨·,·⟩$ and norm $\parallel ·\parallel .$ $\forall p,q\in [2,\infty )$, and we denote by $\mathcal{L}(L^p,L^q)$ the space of all the bounded linear operators from $L^p$ to $L^q.$ When $p=q=2$, we write $\mathcal{L}\left(H\right)=\mathcal{L}(H,H)$ for simplicity. An operator $Q\in \mathcal{L}\left(H\right)$ is called Hilbert–Schmidt if

$${\parallel Q\parallel }_{{\mathcal{L}}_2\left(H\right)}^2:=Tr\left(Q{Q}^{\ast }\right)<+\infty .$$

The space of all the Hilbert–Schmidt operators on H is denoted by ${\mathcal{L}}_2\left(H\right)$.

Next, we define the Gâteaux and Fréchet derivatives of $\varphi$ with respect to the x variable. A mapping $\varphi :H\times H\to \widehat{H}$, where $\widehat{H}$ is another Hilbert space, is Gâteaux differentiable at x if there exists a bounded linear operator $D_x\varphi (x,y)\in \mathcal{L}(H,\widehat{H})$ satisfying

$$\underset{\tau \to 0}{lim}{\displaystyle \frac{\varphi (x+\tau h,y)-\varphi (x,y)}{\tau }}=D_x\varphi (x,y).h,\forall h\in H.$$

The mapping $\varphi$ is Fréchet differentiable at x if it is Gâteaux differentiable and additionally satisfies the stronger condition:

$$\underset{\parallel h\parallel \to 0}{lim}{\displaystyle \frac{\parallel \varphi (x+h,y)-\varphi (x,y)-D_x{\varphi (x,y).h\parallel }_{\widehat{H}}}{\parallel h\parallel }}=0.$$

For $k⩾2,$ we inductively define the k times Gâteaux derivative $D_x^k\varphi (x,y)$ as an element of ${\mathcal{L}}^k(H,\widehat{H}):=\mathcal{L}(H,{\mathcal{L}}^{(k-1)}(H,\widehat{H}))$, endowed with the operator norm

$$\begin{array}{c}\hfill \parallel D_x^k{\varphi (x,y)\parallel }_{{\mathcal{L}}^k(H,\widehat{H})}:=\underset{\parallel h_1\parallel ⩽1,\parallel h_2\parallel ⩽1,\dots ,\parallel h_k\parallel ⩽1,\parallel h\parallel ⩽1}{sup}⟨D_x^k\varphi (x,y).(h_1,h_2,\dots ,h_k),h{⟩}_{\widehat{H}}.\end{array}$$

The above definitions of derivatives with respect to the variable x can be analogously defined for the y variable: $D_y\varphi (x,y)\in \mathcal{L}(H,\widehat{H})$, and for $k⩾2$, $D_y^k\varphi (x,y)\in {\mathcal{L}}^k(H,\widehat{H}):=\mathcal{L}(H,{\mathcal{L}}^{(k-1)}(H,\widehat{H}))$.

Finally, for $\varphi \in L^{\infty }(H\times H,\widehat{H})$ with norm ${\parallel \varphi \parallel }_{L^{\infty }\left(\widehat{H}\right)}:={sup}_{(x,y)\in H\times H}{\parallel \varphi (x,y)\parallel }_{\widehat{H}}<\infty ,$ we introduce several differentiable function spaces of importance for our analysis:

(1)

$\varphi \in C_b^{k,0}(H\times H,\widehat{H}),k\in \mathbb{N}:$$\varphi$ is $k-$ times Gâteaux differentiable at any $x\in H$ with bounded derivatives.

(2)

$\varphi \in C_b^{0,k}(H\times H,\widehat{H}),k\in \mathbb{N}:$$\varphi$ is k times Gâteaux differentiable at any $y\in H$ with bounded derivatives.

(3)

$\varphi \in {\mathbb{C}}_b^{0,k}(H\times H,\widehat{H}),k\in \mathbb{N}:$$\varphi$ is k times Fréchet differentiable at any $y\in H$ with bounded derivatives.

(4)

$\varphi \in C_b^{k,\theta }(H\times H,\widehat{H}),\theta \in (0,1):$$\varphi \in C_b^{k,0}(H\times H,\widehat{H})$, satisfying $\parallel \varphi (x,y_1)-\varphi (x,y_2){\parallel }_{\widehat{H}}⩽C_0{\parallel y_1-y_2\parallel }^{\theta }.$

When $\widehat{H}=\mathbb{R},$ we omit the symbol $\widehat{H}$ in the preceding notations for simplicity.

Throughout this paper, C, with or without subscripts, denotes a positive constant. Its value may vary from line to line, and its dependence on parameters will be clear from the context.

2. Assumptions and Main Results

Let $H:=L^2(0,1)$ be the usual space of square-integrable functions with scalar product and norm denoted by $⟨·,·⟩$ and $\parallel ·\parallel$, respectively. For $k\in \mathbb{N},W^{k,p}(0,1)$ is the Sobolev space of all functions in $L^p(0,1)$ whose differentials belong to $L^p(0,1)$ up to the order k. The usual Sobolev space $W^{k,p}(0,1)$ can be extended to the $W^{s,p}(0,1)$ for $s\in \mathbb{R}$. Set $H^k:=W^{k,2}(0,1)$ and denote by $H_0^1$ the subspace of $H^1$ of all functions whose trace at 0 and 1 vanishes.

Let

$$\begin{array}{c}\hfill Ax:={\displaystyle \frac{{𝜕}^2}{𝜕{\xi }^2}}x,x\in \mathcal{D}\left(A\right)=H^2\cap H_0^1.\end{array}$$

It is well-established that the Hilbert space H admits a complete orthonormal basis ${\left\{e_n\right\}}_{n\in \mathbb{N}}$ such that

$$\begin{array}{c}\hfill A{e}_n=-{\lambda }_n{e}_n,\end{array}$$

(2)

with $0<{\lambda }_1⩽{\lambda }_2⩽\dots {\lambda }_n⩽\dots .$ For $\forall \alpha \in \mathbb{R}$, denote by $H^{\alpha }:=\mathcal{D}({(-A)}^{{\displaystyle \frac{\alpha }{2}}})$ the Hilbert space equipped with the scalar product

$$⟨x,y{⟩}_{\alpha }:=⟨{(-A)}^{{\displaystyle \frac{\alpha }{2}}}x,{(-A)}^{{\displaystyle \frac{\alpha }{2}}}y⟩=\sum _{n=1}^{\infty }{\lambda }_n^{\alpha }⟨x,e_n⟩⟨y,e_n⟩,\forall x,y\in H^{\alpha },$$

and norm

$${\parallel x\parallel }_{\alpha }:={\left(\sum _{n=1}^{\infty }{\lambda }_n^{\alpha }⟨x,e_n{⟩}^2\right)}^{{\displaystyle \frac{1}{2}}},\forall x\in H^{\alpha }.$$

It is easy to deduce ${\parallel ·\parallel }_0=\parallel ·\parallel$, and the following inequalities hold:

$$\begin{array}{cc}\hfill & \parallel e^{tA}{x\parallel }_{{\alpha }_2}⩽C_{{\alpha }_1,{\alpha }_2}{t}^{-{\displaystyle \frac{{\alpha }_2-{\alpha }_1}{2}}}{e}^{-{\displaystyle \frac{{\lambda }_{1}t}{2}}}{\parallel x\parallel }_{{\alpha }_1},x\in H^{{\alpha }_2},{\alpha }_1⩽{\alpha }_2,t>0;\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \parallel e^{tA}x-x\parallel ⩽C{t}^{{\displaystyle \frac{\alpha }{2}}}{\parallel x\parallel }_{\alpha },x\in H^{\alpha },\alpha ⩾0,t⩾0.\hfill \end{array}$$

(4)

Define the bilinear operator $B(x,y):H(0,1)\times H_0^1(0,1)\to H_0^{-1}(0,1)$ by

$$\begin{array}{c}\hfill B(x,y)=x.{𝜕}_{\xi }y,\end{array}$$

and the trilinear operator $b(x,y,z):H(0,1)\times H_0^1(0,1)\times H(0,1)\to \mathbb{R}$ by

$$\begin{array}{c}\hfill b(x,y,z)={\int }_0^{1}x\left(\xi \right).{𝜕}_{\xi }y\left(\xi \right).z\left(\xi \right)d\xi .\end{array}$$

For convenience, let $B\left(x\right)=B(x,x)$, where $x\in H_0^1$.

For the drift coefficients f and g, we define the Nemytskii operators $F,G:H\times H\to H$ by

$$\begin{array}{c}\hfill F(x,y)\left(\xi \right):=f\left(x\right(\xi ),y(\xi \left)\right),G(x,y)\left(\xi \right):=g\left(x\right(\xi ),y(\xi \left)\right),\xi \in (0,1).\end{array}$$

Assume that

(A1): for some $\theta >0$,

$$\begin{array}{c}\hfill f,g\in C_b^{2,\theta }(\mathbb{R}\times \mathbb{R},\mathbb{R}).\end{array}$$

Then, we have that (see, e.g., [36])

$$\begin{array}{c}\hfill F,G\in C_b^{1,\theta }(H\times H,H).\end{array}$$

(5)

Moreover, if $\forall x,y\in H$ and $p,r_1,r_2\in [1,\infty ]$ satisfy ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{r_2}},$ there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \parallel D_x^{2}F(x,y).(h_1,h_2){\parallel }_{L^p}⩽C\parallel h_1{\parallel }_{L^{r_1}}{\parallel h_2\parallel }_{L^{r_2}}\end{array}$$

(6)

and

$$\begin{array}{c}\hfill \parallel D_x^{2}G(x,y).(h_1,h_2){\parallel }_{L^p}⩽C\parallel h_1{\parallel }_{L^{r_1}}{\parallel h_2\parallel }_{L^{r_2}}.\end{array}$$

(7)

To provide precise results, let $E\left(x\right)=-x^3$ and write the system (1) in the following abstract formulation in H:

$$\left\{\begin{array}{c}\mathrm{d}{X}_t^{\epsilon }=A{X}_t^{\epsilon }\mathrm{d}t+B\left(X_t^{\epsilon }\right)\mathrm{d}t+E\left(X_t^{\epsilon }\right)\mathrm{d}t+F(X_t^{\epsilon },Y_t^{\epsilon })\mathrm{d}t+d{W}_t^1,X_0^{\epsilon }=x,\hfill \\ \mathrm{d}{Y}_t^{\epsilon }={\displaystyle \frac{1}{\epsilon }}A{Y}_t^{\epsilon }\mathrm{d}t+{\displaystyle \frac{1}{\epsilon }}G(X_t^{\epsilon },Y_t^{\epsilon })\mathrm{d}t+{\displaystyle \frac{1}{\sqrt{\epsilon }}}d{W}_t^2,Y_0^{\epsilon }=y.\hfill \end{array}\right.$$

(8)

In present paper, we assume that system (8) is well-posed (see Remark 1 for an explanation). We further assume that

(A2): $Q_i$ are nonnegative symmetric operators, which have the same eigenfunctions ${\left\{e_n\right\}}_{n\in \mathbb{N}}$ as the operator A, i.e.,

$$\begin{array}{c}\hfill Q_i{e}_n={\beta }_{i,n}{e}_n,{\beta }_{i,n}>0,n\in \mathbb{N},\end{array}$$

(9)

and

$$\begin{array}{c}\hfill Tr\left((-A)Q_1\right)<+\infty \mathrm{and}Tr\left(Q_2\right):=\sum _{n\in \mathbb{N}}{\beta }_{2,n}<+\infty ,\end{array}$$

(10)

and, for any $T>0$ and $i=1,2$,

$$\begin{array}{c}\hfill {\int }_0^T{\mathsf{\Gamma }}_{i,t}^{{\displaystyle \frac{1+\vartheta }{2}}}\mathrm{d}t<+\infty ,\end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{1,t}:=\underset{n⩾1}{sup}{\displaystyle \frac{2{\lambda }_n}{{\beta }_{1,n}(e^{2{\lambda }_{n}t}-1)}}\mathrm{and}{\mathsf{\Gamma }}_{2,t}:=\underset{n⩾1}{sup}{\displaystyle \frac{2{\lambda }_n}{{\beta }_{2,n}(e^{2{\lambda }_{n}t}-1)}},\end{array}$$

${\lambda }_n$ is given by (2), and $\vartheta ⩾max(\theta ,1-\theta )$ with $\theta$ is the parameter in the assumption (5). Condition (11) is crucial to prove Theorem 2 below; see Lemma 9 in [37].

Given $x\in H$, consider the following frozen equation:

$$\begin{array}{c}\hfill \mathrm{d}{Y}_t^x=A{Y}_t^x\mathrm{d}t+G(x,Y_t^x)\mathrm{d}t+\mathrm{d}{W}_t^2,Y_0^x=y\in H^1.\end{array}$$

(12)

Under conditions (5), (9), (10), and (11), Equation (12) admits a unique strong solution $Y_t^x$ (see, e.g., [37]), which processes a unique invariant measure ${\mu }^x\left(\mathrm{d}y\right)$ (see, e.g., Theorem 4 and Proposition 4 in [38]). Thus, we shall have the averaged equation for system (8):

$$\begin{array}{c}\hfill \mathrm{d}{\overline{X}}_t=A{\overline{X}}_t\mathrm{d}t+B\left({\overline{X}}_t\right)\mathrm{d}t+E\left({\overline{X}}_t\right)\mathrm{d}t+\overline{F}\left({\overline{X}}_t\right)\mathrm{d}t+\mathrm{d}{W}_t^1,{\overline{X}}_0=x\in H,\end{array}$$

(13)

where

$$\begin{array}{c}\hfill \overline{F}\left(x\right):={\int }_{H}F(x,y){\mu }^x\left(\mathrm{d}y\right),\end{array}$$

(14)

By Theorem 2 below, we have $\overline{F}\in C_b^1(H,H)$. Thus, Equation (13) admits a unique solution ${\overline{X}}_t$. The following is the main result.

Theorem 1 (Strong convergence).

Let $x\in H^l\cap H_0^1,l\in (1/2,1]$ and $y\in H$. Assume that (A1)–(A2) hold. Then, for any $q⩾1$, $T>0$ and $\gamma \in [0,1/2)$, we have

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}\mathbb{E}{\parallel {(-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t)\parallel }^q⩽C{\epsilon }^{{\displaystyle \frac{q}{2}}},\end{array}$$

(15)

where ${\overline{X}}_t$ is the unique solution of Equation (13), and $C=C(T,x,y)>0$ is a constant independent of ε.

Remark 1. 

Note that the strong well-posedness of parabolic SPDEs with Hölder-continuous coefficients has been established in [37] through the application of Zvonkin’s transformation. To be precise, based on the associated infinite-dimensional Kolmogorov equation, we systematically eliminate the irregular drift components through an exact decomposition. This transformation yields a modified system with good Lipschitz properties. Similar techniques could potentially be adapted to prove the well-posedness of SPDE (8) under our assumptions. However, such an investigation falls outside the scope of the present work, and we therefore refrain from pursuing this direction here.

Remark 2. 

In contrast to previous works [30,31] that required smoother coefficients, our results hold under the weaker assumption of Hölder-continuous coefficients in the fast variable. Moreover, we establish the stronger convergence in the ${\parallel ·\parallel }_{{(-A)}^{\gamma }}$ norm for any $\gamma \in [0,1/2)$.

3. Preliminaries and A-Priori Estimates

3.1. Poisson Equation in the Hilbert Space

Consider the following Poisson equation in the Hilbert space $H\times H$:

$$\begin{array}{c}\hfill {\mathcal{L}}_2(x,y)\psi (x,y)=-\varphi (x,y),y\in H,\end{array}$$

(16)

where ${\mathcal{L}}_2(x,y)$ is defined by

$$\begin{array}{cc}\hfill {\mathcal{L}}_2\phi (x,y)& :={\mathcal{L}}_2(x,y)\phi (x,y):=⟨Ay+G(x,y),D_y\phi (x,y)⟩\hfill \\ & +{\displaystyle \frac{1}{2}}Tr\left[D_y^2\phi (x,y)Q_2\right],\forall \phi \in C_b^{0,2}(H\times H),\hfill \end{array}$$

and $x\in H$ is a parameter, and $\varphi :H\times H\to H$ is measurable. Since Equation (16) is on the whole space, we define the following centering condition:

$$\begin{array}{c}\hfill {\int }_0^t\varphi (x,y){\mu }^x\left(\mathrm{d}y\right)=0,\forall x\in H,\end{array}$$

(17)

which arises naturally in our analysis and serves a role analogous to the centering condition in classical central limit theorems [39,40]. Furthermore, we assume that

(${\mathit{A}}_{\mathit{\varphi }}$): $\varphi (x,y)\in C_b^{1,\theta }(H\times H,H)$, and, for any $k\in H^{\kappa },x,h,y\in H,$

$$\parallel D_x^2{\varphi (x,y).(h,k)\parallel }_{L^p}⩽{C\parallel h\parallel }_{L^{r_1}}{\parallel k\parallel }_{L^{r_2}}.$$

The following results can be proved similarly to in Theorem 3.2 and Lemma 3.7 from [27]; we omit the details here.

Theorem 2. 

Assume that (9)–(11), (5), and (7) hold. Then, $\forall \varphi :H\times H\to H$ satisfies the centering condition (17) and (${\mathit{A}}_{\mathit{\varphi }}$), and Equation (16) admits a unique solution $\psi \in {\mathbb{C}}_b^{0,2}(H\times H,H)\cap C_b^{1,0}(H\times H,H)$, which is given by

$$\begin{array}{c}\hfill \psi (x,y)={\int }_0^{\infty }\mathbb{E}\left[\varphi (x,Y_t^x\left(y\right))\right]\mathrm{d}t,\end{array}$$

where $Y_t^x\left(y\right)$ satisfies the frozen Equation (12), and $\forall k\in H^{\kappa }$ and $x,y,h\in H,$

$$\begin{array}{c}\hfill \parallel D_x^2{\psi (x,y).(h,k)\parallel }_{⁠}⩽{C\parallel h\parallel }_{L^{r_1}}{\parallel k\parallel }_{L^{r_2}}.\end{array}$$

(18)

Furthermore, assume that F satisfies (5) and (6), and let $\overline{F}$ be defined by (14). Then, we have $\overline{F}\in C_b^1(H,H)$. Moreover, for any $k\in H^{\kappa },x,h\in H,$

$$\begin{array}{c}\hfill \parallel D_x^2\overline{F}{\left(x\right).(h,k)\parallel ⩽C\parallel h\parallel \parallel k\parallel }_{\kappa },\end{array}$$

where $C>0$ is a constant.

3.2. The A-Priori Estimates

To handle the analytical difficulties arising from the unbounded operators in our system, we employ a Galerkin approximation scheme to reduce the infinite-dimensional problem to finite-dimensional subspaces. This standard approach, whose detailed implementation can be found in Section 4 in [23], provides a rigorous foundation for our subsequent analysis. For notational simplicity and to maintain focus on the essential aspects of our arguments, we suppress explicit reference to the approximation procedure in this subsection and Section 4. We emphasize that all the estimates and bounds presented in what follows are to be understood as holding uniformly with respect to the approximation parameter. This uniformity ensures the validity of our results when passing to the infinite-dimensional limit.

Firstly, we provide the following estimates for the mild solution of the system (8).

Lemma 1. 

Let $x,y\in H,T>0$ and $q⩾1$. Then, the solution $(X_t^{\epsilon },Y_t^{\epsilon })$ of system (8) satisfies

$$\begin{array}{c}\hfill \underset{\epsilon \in (0,1)}{sup}\underset{t\in [0,T]}{sup}\mathbb{E}\parallel X_t^{\epsilon }{\parallel }^q⩽C_{q,T}{(1+\parallel x\parallel }^q),\end{array}$$

$$\begin{array}{c}\hfill \underset{\epsilon \in (0,1)}{sup}\underset{t\in [0,T]}{sup}\mathbb{E}\parallel Y_t^{\epsilon }{\parallel }^q⩽C_{q,T}{(1+\parallel y\parallel }^q),\end{array}$$

where $C_{q,T}>0$ is a constant.

Proof. 

Based on the boundedness of F and G and Lemma 5.2 from [30], Lemma 1 can be proved. □

Lemma 2. 

Let $T>0,\eta \in (1,3/2)$, $x\in H^l$ with $l\in [0,\eta ]$, and $y\in H$. Then, there exists $m>1$ such that, for every $q⩾1$ and $t\in (0,T]$, we have

$$\begin{array}{c}\hfill {\left(\mathbb{E}\parallel X_t^{\epsilon }{\parallel }_{\eta }^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C_{q,\eta ,l,T}{t}^{{\displaystyle -{\displaystyle \frac{\eta -l}{2}}}}({\displaystyle e^{-{\displaystyle \frac{{\lambda }_{1}t}{2}}}}{\parallel x\parallel }_l+{\parallel x\parallel }^m+1),\end{array}$$

where $C_{q,\eta ,l,T}>0$ is a constant.

Proof. 

The proof is in Appendix A. □

Remark 3. 

By Lemma 2 and the interpolation inequality, we have that, for every $T>0,⁠p⩾1,{\eta }_1\in (0,1],\eta \in (1,3/2)$, and $x\in H^l$ with $l\in [0,1]$, there exists $m>1$ such that, for every $t\in (0,T]$,

$$\begin{array}{c}\hfill {\left(\mathbb{E}\parallel X_s^{\epsilon }{\parallel }_{{\eta }_1}^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C{t}^{-{\displaystyle \frac{(\eta -l){\eta }_1}{2\eta }}}(e^{-{\displaystyle \frac{{\lambda }_1{\eta }_{1}t}{2\eta }}}{\parallel x\parallel }_l^{{\displaystyle \frac{{\eta }_1}{\eta }}}+{\parallel x\parallel }^{{\displaystyle \frac{m{\eta }_1}{\eta }}}+1).\end{array}$$

(19)

To derive the estimate for $\mathbb{E}\parallel X_t^{\epsilon }{\parallel }_2$, we need the following regularity results for $X_t^{\epsilon }$ and $Y_t^{\epsilon }$ with respect to the time variable.

Lemma 3. 

(i) Let $T>0$, $\alpha \in (1/2,2]$, $x\in H^l$ with $l\in (1/2,\alpha ]$ and $y\in H$. Then, for any $q⩾1$ and $0<s⩽t⩽T,$ we have

$$\begin{array}{cc}\hfill {\left(\mathbb{E}\parallel X_t^{\epsilon }-X_s^{\epsilon }{\parallel }^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C_{q,\alpha ,l,T}& ({(t-s)}^{{\displaystyle \frac{\alpha }{2}}}{s}^{-{\displaystyle \frac{\alpha -l}{2}}}{e}^{-{\displaystyle \frac{{\lambda }_{1}s}{2}}}{\parallel x\parallel }_l+{(t-s)}^{{\displaystyle \frac{1}{2}}});\hfill \end{array}$$

(20)

where $C_{q,\alpha ,l,T}>0$ is a constant.

(ii) Let $T>0$, $\beta \in [0,1]$, $x\in H$, $y\in H^l$ with $l\in [0,\beta ]$. Then, for any $q⩾1$ and $0<s⩽t⩽T,$ we have

$$\begin{array}{c}\hfill {\left(\mathbb{E}\parallel Y_t^{\epsilon }-Y_s^{\epsilon }{\parallel }^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C_{q,\beta ,l,T}({\displaystyle \frac{{(t-s)}^{\beta /2}}{s^{\beta /2-l/2}{\epsilon }^{l/2}}}{e}^{-{\displaystyle \frac{{\lambda }_{1}s}{2\epsilon }}}{\parallel y\parallel }_l+{\displaystyle \frac{{(t-s)}^{\beta /2}}{{\epsilon }^{\beta /2}}});\end{array}$$

(21)

where $C_{q,\beta ,l,T}>0$ is a constant.

Proof. 

The proof is in Appendix A. □

By exactly the same arguments as in Lemma 3.6 in [31], we further have

Lemma 4. 

Let $T>0$, $\eta \in (1,3/2)$, $x\in H^l$ with $l\in (1/2,\eta )$ and $y\in H$. Then, there exists $m>1$ such that, for any $q⩾1$ and $0<s⩽t⩽T$, we have

$$\begin{array}{c}\hfill {\left(\mathbb{E}\parallel X_t^{\epsilon }-X_s^{\epsilon }{\parallel }_1^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C_{q,\eta ,l,T}{(t-s)}^{{\displaystyle \frac{\eta -1}{2}}}{s}^{-{\displaystyle \frac{\eta -l}{2}}}(1+e^{-{\displaystyle \frac{{\lambda }_{1}s}{2}}}{\parallel x\parallel }_l+{\parallel x\parallel }^m),\end{array}$$

(22)

Now, we provide the following uniform estimate for $\parallel X_t^{\epsilon }{\parallel }_2$.

Lemma 5. 

Let $T,\beta >0$, $\eta \in (1,3/2)$, $x\in H^l$ with $l\in (1/2,1]$ and $y\in H$. Then, there exists $m>1$ such that, for any $q⩾1$ and $t\in (0,T]$, we have

$$\begin{array}{c}\hfill {\left(\mathbb{E}\parallel X_t^{\epsilon }{\parallel }_2^q\right)}^{{\displaystyle \frac{1}{q}}}⩽C_{q,\beta ,l,T}(t^{-1+{\displaystyle \frac{3l-\eta }{2\eta }}}+{\epsilon }^{-{\displaystyle \frac{\beta }{2}}})(1+{\parallel x\parallel }_l+{\parallel x\parallel }^m+\parallel y\parallel ),\end{array}$$

where $C_{q,\beta ,l,T}>0$ is a constant.

Proof. 

The proof is in Appendix A. □

4. Strong Convergence in the Averaging Principle

Let

$$\begin{array}{cc}\hfill {\mathcal{L}}_1(x,y)\phi (x,y)& :=⟨Ax+B\left(x\right)+E\left(x\right)+F(x,y),D_x\phi (x,y)⟩\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}Tr\left(D_x^2\phi (x,y)Q_1\right),\forall \phi \in C_b^{2,0}(H\times H).\hfill \end{array}$$

(23)

We begin by deriving crucial strong fluctuation estimates for integral functionals of the coupled process $(X_r^{\epsilon },Y_r^{\epsilon })$ over arbitrary time intervals $[s,t].$ These estimates serve as fundamental tools for establishing our main convergence result in Theorem 1.

Lemma 6 (Strong fluctuation estimate).

Let $T>0$, $x\in H^l,l\in (1/2,1],y\in H$. Assume that (A1)–(A2) hold. Then, for any $q⩾1$, $\gamma \in [0,l/2)$, $0⩽s⩽t⩽T$, and $\varphi \in C_b^{2,\theta }(H\times H,H)$ satisfying (17), we have

$$\begin{array}{c}\hfill \mathbb{E}{∥{\int }_s^t{(-A)}^{\gamma }{e}^{(t-r)A}\varphi (X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r∥}^q⩽C_{q,\gamma ,T}{(t-s)}^{(l/2-\gamma )q}{\epsilon }^{q/2},\end{array}$$

where $C_{q,\gamma ,T}>0$ is a constant.

Proof. 

Consider the Poisson equation

$$\begin{array}{c}\hfill {\mathcal{L}}_2(x,y)\psi (x,y)=-\varphi (x,y),\end{array}$$

and let

$$\begin{array}{c}\hfill {\psi }_{t,\gamma }(r,x,y):={(-A)}^{\gamma }{e}^{(t-r)A}\psi (x,y).\end{array}$$

By the definition of ${\mathcal{L}}_2$, we have

$$\begin{array}{c}\hfill {\mathcal{L}}_2(x,y){\psi }_{t,\gamma }(r,x,y)=-{(-A)}^{\gamma }{e}^{(t-r)A}\varphi (x,y).\end{array}$$

(24)

Applying Itô’s formula, it follows that

$$\begin{array}{cc}\hfill {\psi }_{t,\gamma }(t,X_t^{\epsilon },Y_t^{\epsilon })& ={\psi }_{t,\gamma }(s,X_s^{\epsilon },Y_s^{\epsilon })+{\int }_s^t({𝜕}_r+{\mathcal{L}}_1){\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r\hfill \\ \hfill & +{\displaystyle \frac{1}{\epsilon }}{\int }_s^t{\mathcal{L}}_2{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r+M_{t,s}^1+{\displaystyle \frac{1}{\sqrt{\epsilon }}}{M}_{t,s}^2,\hfill \end{array}$$

(25)

where $M_{t,s}^1$ and $M_{t,s}^2$ are defined by

$$\begin{array}{c}\hfill M_{t,s}^1:={\int }_s^t{D}_x{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}{W}_r^{1}and{M}_{t,s}^2:={\int }_s^t{D}_y{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}{W}_r^2.\end{array}$$

Multiplying (25) by $\epsilon$ and applying (24), we get

$$\begin{array}{cc}\hfill & {\int }_s^t{(-A)}^{\gamma }{e}^{(t-r)A}\varphi (X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r\hfill \\ \hfill & =-{\int }_s^t{\mathcal{L}}_2{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r=\epsilon \left({\psi }_{t,\gamma }(s,X_s^{\epsilon },Y_s^{\epsilon })-{\psi }_{t,\gamma }(t,X_t^{\epsilon },Y_t^{\epsilon })\right)\hfill \\ \hfill & +\epsilon {\int }_s^t({𝜕}_r+{\mathcal{L}}_1){\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r+\epsilon M_{t,s}^1+{\epsilon }^{1/2}{M}_{t,s}^2.\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill {\int }_s^t{𝜕}_r{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r& ={\int }_s^t{𝜕}_r{\psi }_{t,\gamma }(r,X_t^{\epsilon },Y_t^{\epsilon })\mathrm{d}r\hfill \\ \hfill & +{\int }_s^t{𝜕}_r\left[{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })-{\psi }_{t,\gamma }(r,X_t^{\epsilon },Y_t^{\epsilon })\right]\mathrm{d}r\hfill \\ \hfill & ={\psi }_{t,\gamma }(t,X_t^{\epsilon },Y_t^{\epsilon })-{\tilde{\psi }}_{t,\gamma }(s,X_t^{\epsilon },Y_t^{\epsilon })\hfill \\ \hfill & +{\int }_s^t{𝜕}_r\left[{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })-{\psi }_{t,\gamma }(r,X_t^{\epsilon },Y_t^{\epsilon })\right]\mathrm{d}r,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {𝜕}_r{\psi }_{t,\gamma }(r,x,y)={(-A)}^{1+\gamma }{e}^{(t-r)A}\psi (x,y).\end{array}$$

Consequently, we further get

$$\begin{array}{cc}\hfill & {\int }_s^t{(-A)}^{\gamma }{e}^{(t-r)A}\varphi (X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r=\epsilon {(-A)}^{\gamma }{e}^{(t-s)A}\left[\psi (X_s^{\epsilon },Y_s^{\epsilon })-\psi (X_t^{\epsilon },Y_t^{\epsilon })\right]\hfill \\ \hfill & +\epsilon {\int }_s^t{(-A)}^{1+\gamma }{e}^{(t-r)A}\left(\psi (X_r^{\epsilon },Y_r^{\epsilon })-\psi (X_t^{\epsilon },Y_t^{\epsilon })\right)\mathrm{d}r\hfill \\ \hfill & +\epsilon {\int }_s^t{\mathcal{L}}_1{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r+\epsilon M_{t,s}^1+{\epsilon }^{1/2}{M}_{t,s}^2.\hfill \end{array}$$

Thus, $\forall 0⩽s⩽t⩽T$ and $q⩾1$, so we get

$$\begin{array}{cc}\hfill & \mathbb{E}{∥{\int }_s^t{(-A)}^{\gamma }{e}^{(t-r)A}\varphi (X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r∥}^q\hfill \\ \hfill & ⩽C_q({\epsilon }^q\mathbb{E}\parallel {(-A)}^{\gamma }{e}^{(t-s)A}\left[\psi (X_s^{\epsilon },Y_s^{\epsilon })-\psi (X_t^{\epsilon },Y_t^{\epsilon })\right]{\parallel }^q\hfill \\ \hfill & +\mathbb{E}\epsilon {∥{\int }_s^t{(-A)}^{1+\gamma }{e}^{(t-r)A}\left(\psi (X_r^{\epsilon },Y_r^{\epsilon })-\psi (X_t^{\epsilon },Y_t^{\epsilon })\right)\mathrm{d}r∥}^q\hfill \\ \hfill & +\mathbb{E}\epsilon {∥{\int }_s^t{\mathcal{L}}_1{\psi }_{t,\gamma }(r,X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r∥}^q+{\epsilon }^q\mathbb{E}\parallel M_{t,s}^1{\parallel }^q+{\epsilon }^{q/2}\mathbb{E}{\parallel M_{t,s}^2\parallel }^q)\hfill \\ \hfill & =:\sum _{i=1}^5{\mathcal{J}}_i(t,s,\epsilon ).\hfill \end{array}$$

According to Theorem 2, Lemma 1, (20), and (21), we obtain

$$\begin{array}{cc}\hfill {\mathcal{J}}_1(t,s,\epsilon )& ⩽C_1{\epsilon }^q{(t-s)}^{-\gamma q}{\left(\mathbb{E}{\left(1+\parallel X_t^{\epsilon }{\parallel }^p+\parallel X_s^{\epsilon }{\parallel }^p+\parallel Y_t^{\epsilon }{\parallel }^p+{\parallel Y_s^{\epsilon }\parallel }^p\right)}^{2q}\right)}^{1/2}\hfill \\ \hfill & ·{\left(\mathbb{E}\parallel X_t^{\epsilon }-X_s^{\epsilon }{\parallel }^{2q}+\mathbb{E}{\parallel Y_t^{\epsilon }-Y_s^{\epsilon }\parallel }^{2q}\right)}^{1/2}\hfill \\ \hfill & ⩽C_1{(t-s)}^{(l/2-\gamma )q}{\epsilon }^{(1-l/2)q}⩽C_1{(t-s)}^{(l/2-\gamma )q}{\epsilon }^{q/2}.\hfill \end{array}$$

For the second term, by Minkowski’s inequality, we also get

$$\begin{array}{cc}\hfill {\mathcal{J}}_2(t,s,\epsilon )& ⩽C_2{\epsilon }^q\left({\int }_s^t{(t-r)}^{-1-\gamma }\right[{\left(\mathbb{E}\left[\parallel X_t^{\epsilon }-X_r^{\epsilon }{\parallel }^{2q}\right]\right)}^{1/2q}\hfill \\ \hfill & +{\left(\mathbb{E}\left[\parallel Y_t^{\epsilon }-Y_r^{\epsilon }{\parallel }^{2q}\right]\right)}^{1/2q}]\mathrm{d}r{)}^q\hfill \\ \hfill & ⩽C_2{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-1-\gamma }{(t-r)}^{l/2}{\epsilon }^{-l/2}\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_2{(t-s)}^{(l/2-\gamma )q}{\epsilon }^{(1-l/2)q}⩽C_2{(t-s)}^{(l/2-\gamma )q}{\epsilon }^{q/2}.\hfill \end{array}$$

For the third term, according to definition (23), we deduce that

$$\begin{array}{cc}\hfill {\mathcal{J}}_3(t,s,\epsilon )& ={\epsilon }^q\mathbb{E}{∥{\int }_s^t{(-A)}^{\gamma }{e}^{(t-r)A}{\mathcal{L}}_1\psi (X_r^{\epsilon },Y_r^{\epsilon })\mathrm{d}r∥}^q\hfill \\ \hfill & ⩽C_3{\epsilon }^q\mathbb{E}{\left({\int }_s^t\parallel {(-A)}^{\gamma }{e}^{(t-r)A}⟨A{X}_r^{\epsilon },D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & +C_3{\epsilon }^q\mathbb{E}{\left({\int }_s^t\parallel {(-A)}^{\gamma }{e}^{(t-r)A}⟨B\left(X_r^{\epsilon }\right),D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & +C_3{\epsilon }^q\mathbb{E}{\left({\int }_s^t\parallel {(-A)}^{\gamma }{e}^{(t-r)A}⟨E\left(X_r^{\epsilon }\right),D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & +C_3{\epsilon }^q\mathbb{E}{\left({\int }_s^t\parallel {(-A)}^{\gamma }{e}^{(t-r)A}⟨F(X_r^{\epsilon },Y_r^{\epsilon }),D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & +C_3{\epsilon }^q\mathbb{E}{\left({\int }_s^t\parallel {(-A)}^{\gamma }{e}^{(t-r)A}·{\displaystyle \frac{1}{2}}Tr\left[D_x^2\psi (X_r^{\epsilon },Y_r^{\epsilon })Q_1\right]\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & =:\sum _{i=1}^5{\mathcal{J}}_{3,i}(t,s,\epsilon ).\hfill \end{array}$$

According to Theorem 2, Lemmas 1 and 5, and Minkowski’s inequality, we obtain

$$\begin{array}{cc}\hfill {\mathcal{J}}_{3,1}(t,s,\epsilon )& ⩽C_{3,1}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-s)}^{-\gamma }\parallel ⟨A{X}_r^{\epsilon },D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,1}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{\left(\mathbb{E}\parallel A{X}_r^{\epsilon }{\parallel }^q\right)}^{{\displaystyle \frac{1}{q}}}\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,1}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }\left(r^{-1+{\displaystyle \frac{3l-\eta }{2\eta }}}+{\epsilon }^{-{\displaystyle \frac{l}{2}}}\right)\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,1}{(t-s)}^{({\displaystyle \frac{l}{2}}-\gamma )q}{\epsilon }^{(1-{\displaystyle \frac{l}{2}})q}⩽C_{3,1}{(t-s)}^{({\displaystyle \frac{l}{2}}-\gamma )q}{\epsilon }^{{\displaystyle \frac{q}{2}}}.\hfill \end{array}$$

By Theorem 2, (19), and Minkowski’s inequality, we have

$$\begin{array}{cc}\hfill {\mathcal{J}}_{3,2}(t,s,\epsilon )& ⩽C_{3,2}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }\parallel ⟨B\left(X_r^{\epsilon }\right),D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,2}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{\left(\mathbb{E}\parallel X_r^{\epsilon }{\parallel }_1^{2q}\right)}^{1/q}\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,2}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{r}^{-(\eta -l)/\eta }\mathrm{d}r\right)}^q⩽C_{3,2}{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q.\hfill \end{array}$$

Using Theorem 2, (19), Lemma 1, the Gagliardo–Nirenberg inequality, and Minkowski’s inequality, we obtain

$$\begin{array}{cc}\hfill {\mathcal{J}}_{3,3}(t,s,\epsilon )& ⩽C_{3,3}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }\parallel ⟨E\left(X_r^{\epsilon }\right),D_x\phi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,3}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }\parallel {X_r^{\epsilon }}^3\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,3}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }{\parallel X_r^{\epsilon }\parallel }_{L^6}^3\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,3}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{\left(\mathbb{E}\parallel X_r^{\epsilon }{\parallel }_1^{3q}\right)}^{1/q}\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,3}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{r}^{-3(\eta -l)/2\eta }\mathrm{d}r\right)}^q⩽C_{3,3}{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q.\hfill \end{array}$$

According to Theorem 2 and Lemma 1, we obtain

$$\begin{array}{cc}\hfill {\mathcal{J}}_{3,4}(t,s,\epsilon )& ⩽C_{3,4}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }\parallel ⟨F(X_r^{\epsilon },Y_r^{\epsilon }),D_x\psi (X_r^{\epsilon },Y_r^{\epsilon })⟩\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,4}{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q.\hfill \end{array}$$

For the last term, by (18), Lemma 1, and assumption (A2), we get

$$\begin{array}{cc}\hfill {\mathcal{J}}_{3,5}(t,s,\epsilon )& ⩽C_{3,5}{\epsilon }^q\mathbb{E}{\left({\int }_s^t{(t-r)}^{-\gamma }·{\displaystyle \frac{1}{2}}Tr\left(Q_1\right)\parallel D_x^2\psi (X_r^{\epsilon },Y_r^{\epsilon })\parallel \mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,5}{\epsilon }^q{\left({\int }_s^t{(t-r)}^{-\gamma }{\left(\mathbb{E}(1+\parallel X_r^{\epsilon }{\parallel }^q+\parallel Y_r^{\epsilon }{\parallel }^q)\right)}^{1/q}\mathrm{d}r\right)}^q\hfill \\ \hfill & ⩽C_{3,5}{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q.\hfill \end{array}$$

According to Theorem 2, Burkholder–Davis–Gundy’s inequality, and assumption (A2), we have

$$\begin{array}{cc}\hfill {\mathcal{J}}_4(t,s,\epsilon )& ⩽C_4{\epsilon }^q{\left({\int }_s^t\mathbb{E}{\parallel {(-A)}^{\gamma }{e}^{(t-r)A}{D}_x\psi (X_r^{\epsilon },Y_r^{\epsilon })Q_1^{1/2}\parallel }_{{\mathcal{L}}_2\left(H\right)}^2\mathrm{d}r\right)}^{q/2}\hfill \\ \hfill & ⩽C_4{(t-s)}^{(1/2-\gamma )q}{\epsilon }^q⩽C_4{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q,\hfill \end{array}$$

and similarly

$$\begin{array}{c}\hfill {\mathcal{J}}_5(t,s,\epsilon )⩽C_5{(t-s)}^{(1/2-\gamma )q}{\epsilon }^q⩽C_5{(t-s)}^{(l/2-\gamma )q}{\epsilon }^q.\end{array}$$

Thus, the proof is finished. □

Now, we are in a position to provide the following:

Proof of Theorem 1. 

Let $T>0$. By (8) and (13), $\forall t\in [0,T]$, so we have

$$\begin{array}{cc}\hfill {(-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t)& ={\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(B\left(X_s^{\epsilon }\right)-B\left({\overline{X}}_s\right)\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(E\left(X_s^{\epsilon }\right)-E\left({\overline{X}}_s\right)\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(\overline{F}\left(X_s^{\epsilon }\right)-\overline{F}\left({\overline{X}}_s\right)\right)\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\delta F(X_s^{\epsilon },Y_s^{\epsilon })\mathrm{d}s,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \delta F(x,y):=F(x,y)-\overline{F}\left(x\right).\end{array}$$

Thus, $\forall q⩾1$, and we have

$$\begin{array}{cc}\hfill {\parallel (-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t){\parallel }^q& ⩽C_q{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(B\left(X_s^{\epsilon }\right)-B\left({\overline{X}}_s\right)\right)\mathrm{d}s∥}^q\hfill \\ \hfill & +C_q{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(E\left(X_s^{\epsilon }\right)-E\left({\overline{X}}_s\right)\right)\mathrm{d}s∥}^q\hfill \\ \hfill & +C_q{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\left(\overline{F}\left(X_s^{\epsilon }\right)-\overline{F}\left({\overline{X}}_s\right)\right)\mathrm{d}s∥}^q\hfill \\ \hfill & +C_q{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\delta F(X_s^{\epsilon },Y_s^{\epsilon })\mathrm{d}s∥}^q=:\sum _{i=1}^4{\mathcal{Q}}_i(t,\epsilon ).\hfill \end{array}$$

For the first term, by Corollary 2.2. in [41], we have

$$\begin{array}{cc}\hfill {\mathcal{Q}}_1(t,\epsilon )& ⩽C_1{\left({\int }_0^t{(t-s)}^{-(\gamma +{\displaystyle \frac{1}{2}})}\parallel B\left(X_s^{\epsilon }\right)-B\left({\overline{X}}_s\right){\parallel }_{-1}\mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_1{\left({\int }_0^t{(t-s)}^{-(\gamma +{\displaystyle \frac{1}{2}})}\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel \left(\parallel X_s^{\epsilon }{\parallel }_1+{\parallel {\overline{X}}_s\parallel }_1\right)\mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_1{\left({\int }_0^t{(t-s)}^{-(\gamma +{\displaystyle \frac{1}{2}})·{\displaystyle \frac{q}{q-1}}}\mathrm{d}s\right)}^{q-1}·\left({\int }_0^t{\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel }^q{\left(\parallel X_s^{\epsilon }{\parallel }_1+{\parallel {\overline{X}}_s\parallel }_1\right)}^q\mathrm{d}s\right)\hfill \\ \hfill & ⩽C_1{\int }_0^t\parallel X_s^{\epsilon }-{\overline{X}}_s{\parallel }^q(\parallel X_s^{\epsilon }{\parallel }_1^q+\parallel {\overline{X}}_s{\parallel }_1^q)\mathrm{d}s,\hfill \end{array}$$

where we hold that, if $q>2$, $-{\displaystyle \frac{q(2\gamma +1)}{2(q-1)}}>-1$, and then

$${\int }_0^t{(t-s)}^{-{\displaystyle \frac{q(2\gamma +1)}{2(q-1)}}}\mathrm{d}s⩽C_T.$$

According to Hölder’s inequality, we deduce

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2(t,\epsilon )& ⩽C_2{\left({\int }_0^t\parallel {(-A)}^{\gamma }{e}^{(t-s)A}\left(E\left(X_s^{\epsilon }\right)-E\left({\overline{X}}_s\right)\right)\parallel \mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_2{\left({\int }_0^t{(t-s)}^{-\gamma }\parallel {-X_s^{\epsilon }}^3+{\overline{X}}_s^3\parallel \mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_2{\left({\int }_0^t{(t-s)}^{-\gamma }\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel \parallel {X_s^{\epsilon }}^2+X_s^{\epsilon }{\overline{X}}_s+{{\overline{X}}_s}^2{\parallel }_{L^{\infty }}\mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_2{\left({\int }_0^t{(t-s)}^{-\gamma }\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel \left(\parallel X_s^{\epsilon }{\parallel }_1^2+{\parallel {\overline{X}}_s\parallel }_1^2\right)\mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_2{\int }_0^t{\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel }^q\left(\parallel X_s^{\epsilon }{\parallel }_1^{2q}+{\parallel {\overline{X}}_s\parallel }_1^{2q}\right)\mathrm{d}s.\hfill \end{array}$$

For the third term, by Theorem 2, it is evident that

$$\begin{array}{cc}\hfill {\mathcal{Q}}_3(t,\epsilon )& ⩽C_3{\left({\int }_0^t{(t-s)}^{-\gamma }\parallel \overline{F}\left(X_s^{\epsilon }\right)-\overline{F}\left({\overline{X}}_s\right)\parallel \mathrm{d}s\right)}^q\hfill \\ \hfill & ⩽C_3{\left({\int }_0^t{(t-s)}^{-\gamma }\parallel X_s^{\epsilon }-{\overline{X}}_s\parallel \mathrm{d}s\right)}^q⩽C_3{\int }_0^t\parallel X_s^{\epsilon }-{\overline{X}}_s{\parallel }^q\mathrm{d}s.\hfill \end{array}$$

Combining the above computations, we have

$$\begin{array}{cc}\hfill {\parallel (-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t){\parallel }^q& ⩽C_4{\int }_0^t\parallel X_s^{\epsilon }-{\overline{X}}_s{\parallel }^q\left(1+\parallel X_s^{\epsilon }{\parallel }_1^{2q}+{\parallel {\overline{X}}_s\parallel }_1^{2q}\right)\mathrm{d}s\hfill \\ \hfill & +C_4{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\delta F(X_s^{\epsilon },Y_s^{\epsilon })\mathrm{d}s∥}^q.\hfill \end{array}$$

Then, by Gronwall’s inequality, we get

$$\begin{array}{cc}\hfill & {\mathbb{E}\parallel (-A)}^{\gamma }(X_t^{\epsilon }-{\overline{X}}_t){\parallel }^q\hfill \\ \hfill & ⩽C_5{\left(\mathbb{E}{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\delta F(X_s^{\epsilon },Y_s^{\epsilon })\mathrm{d}s∥}^{2q}\right)}^{1/2}·{\left(\mathbb{E}{e}^{C_6{\int }_0^t(1+\parallel X_s^{\epsilon }{\parallel }_1^{2q}+\parallel {\overline{X}}_s{\parallel }_1^{2q})\mathrm{d}s}\right)}^{1/2}.\hfill \end{array}$$

According to Lemma 1.1 in [42], one can check that

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}\mathbb{E}\left(e^{C{\int }_0^t(\parallel X_s^{\epsilon }{\parallel }_1^{2q}+\parallel {\overline{X}}_s{\parallel }_1^{2q})\mathrm{d}s}\right)⩽C_T.\end{array}$$

Since $\delta F(x,y)$ satisfies the centering condition (17), using Lemma 6, we can directly obtain

$$\begin{array}{c}\hfill {\left(\mathbb{E}{∥{\int }_0^t{(-A)}^{\gamma }{e}^{(t-s)A}\delta F(X_s^{\epsilon },Y_s^{\epsilon })\mathrm{d}s∥}^{2q}\right)}^{1/2}⩽C_7{\epsilon }^{q/2}.\end{array}$$

Thus, we obtain the desired result. □

5. Example

Consider a practical application: let $f(x,y)=x+y,g(x,y)=-y$ in (1), and then we have

$$\left\{\begin{array}{c}{\displaystyle \frac{𝜕X_t^{\epsilon }\left(\xi \right)}{𝜕t}}=\Delta X_t^{\epsilon }\left(\xi \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{𝜕}{𝜕\xi }}{\left(X_t^{\epsilon }\left(\xi \right)\right)}^2-{X_t^{\epsilon }}^3\left(\xi \right)+X_t^{\epsilon }\left(\xi \right)+Y_t^{\epsilon }\left(\xi \right)+{\displaystyle \frac{𝜕W_t^1\left(\xi \right)}{𝜕t}},\hfill \\ {\displaystyle \frac{𝜕Y_t^{\epsilon }\left(\xi \right)}{𝜕t}}={\displaystyle \frac{1}{\epsilon }}\Delta Y_t^{\epsilon }\left(\xi \right)-{\displaystyle \frac{1}{\epsilon }}{Y}_t^{\epsilon }\left(\xi \right)+{\displaystyle \frac{1}{\sqrt{\epsilon }}}{\displaystyle \frac{𝜕W_t^2\left(\xi \right)}{𝜕t}},\hfill \\ X_t^{\epsilon }\left(0\right)=X_t^{\epsilon }\left(1\right)=Y_t^{\epsilon }\left(0\right)=Y_t^{\epsilon }\left(1\right)=0,t\in (0,T],\hfill \\ X_0^{\epsilon }\left(\xi \right)=x,Y_0^{\epsilon }\left(\xi \right)=y,\xi \in (0,1),\hfill \end{array}\right.$$

(26)

where

• $X_t^{\epsilon }$ represents the concentration of a slowly varying chemical substance;
• $Y_t^{\epsilon }$ represents the concentration of a rapidly varying chemical substance;
• the nonlinear term $-{X_t^{\epsilon }}^3$ represents self-inhibition;
• The term ${\displaystyle \frac{1}{2}}{\displaystyle \frac{𝜕}{𝜕\xi }}{\left(X_t^{\epsilon }\left(\xi \right)\right)}^2$ represents a convective effect;
• The functions $f(x,y)=x+y$ and $g(x,y)=-y$ represent the reaction coupling.

It is easy to check that $f(x,y)=x+y$ and $g(x,y)=-y$ satisfy the Lipschitz condition and our assumption (A1). Note that, in this special case, the frozen equation is

$$\begin{array}{c}\hfill {\displaystyle \frac{𝜕Y_t\left(\xi \right)}{𝜕t}}=\Delta Y_t\left(\xi \right)-Y_t\left(\xi \right)+{\displaystyle \frac{𝜕W_t^2\left(\xi \right)}{𝜕t}},\end{array}$$

(27)

which has a unique invariant measure $\mu .$ Moreover, we can have exactly

$$\overline{f}\left(x\right)={\int }_{\mathbb{R}}(x+y)\mu \left(\mathrm{d}y\right)=x+0=x.$$

Thus, by our result regarding Theorem 1, we determine that $X_t^{\epsilon }$ converges strongly (with the optimal convergence rate $1/2$) to ${\overline{X}}_t$, thereby satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{𝜕{\overline{X}}_t\left(\xi \right)}{𝜕t}}=\Delta {\overline{X}}_t\left(\xi \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{𝜕}{𝜕\xi }}\left({\overline{X}}_{(}\xi \right){)}^2-{{\overline{X}}_t}^3\left(\xi \right)+{\overline{X}}_t\left(\xi \right)+{\displaystyle \frac{𝜕W_t^1\left(\xi \right)}{𝜕t}}.\end{array}$$

(28)

From the above example, we can see that, as $\epsilon$ approaches zero, the original fast–slow system (26) converges strongly (with the optimal convergence rate $1/2$) to an averaged system (28), which no longer depends on the fast variable $Y_t^{\epsilon }$. Intuitively, the averaged system is much simpler than the original one. When $\epsilon$ is small, directly simulating the original system numerically requires extremely small time steps. In contrast, simulating the effective averaged equation significantly improves computational efficiency; see [43] and the references therein for the Heterogeneous Multiscale Methods (simulating the effective averaged equation instead of the slow–fast system) and studies on numerical analysis regarding slow–fast SPDEs based on the optimal strong convergence in the averaging principle. The main application of the optimal rates of convergence obtained in the present manuscript is the construction of efficient numerical schemes for the slow–fast system based on HMMs. However, obtaining the explicit invariant measure ${\mu }^x$ for the averaged Equation (13) is highly challenging, primarily because of the complexity in the dynamics of the frozen Equation (12). Hence, the explicit form of the averaged Equation (13) is generally inaccessible, making it difficult for us to intuitively see how the approximate solution of the averaged equation approximates the solution of the original system. We chose Example (26) because it directly validates our hypothesis and yields the explicit form of the averaged Equation (27) thanks to the explicit form of the invariant measure $\mu$ of the frozen Equation (27). For the averaged Equation (27), we shall simulate it via the general solver for SPDEs. This clearly demonstrates that the averaged equation is significantly simpler, both in terms of numerical simulation and theoretical analysis, compared to the original fast–slow system (26).

6. Conclusions

This work establishes a framework for analyzing multiscale stochastic reaction–diffusion–advection systems with Hölder-continuous coefficients. Compared with the existing work [30], we obtain the optimal strong convergence rate in the averaging principle, which is primarily motivated by developing efficient numerical schemes using Heterogeneous Multiscale Methods (HMMs), as detailed in [43] and the references therein. Thus, the optimal convergence rate established in our work is crucial for numerical approximation, which shall be studied in future work.