Stability of the Stochastic Ginzburg–Landau–Newell Equations in Two Dimensions

Abstract

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple $H^2\subset H^1\subset H^0$ to obtain a bound on the difference between the original solution and the perturbed solution.

1. Introduction

The study of stability in physical systems is a cornerstone of modern theoretical physics, and at the forefront of this research lie the Ginzburg–Landau–Newell equations. This mathematical framework, an extension of the original Ginzburg–Landau theory, plays a crucial role in understanding phase transitions and pattern formation in a diverse range of systems, from superconductors to fluid dynamics. Ginzburg–Landau–Newell equations are an important model discussed in [1,2,3,4] as describing the wave pattern and turbulence phenomenon. In this paper, we are concerned with the 2D stochastic Ginzburg–Landau–Newell Equations (SGLNEs)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}u={[\mu u-\eta \nabla u+(1+i\alpha )\Delta u-(1+i\beta )|u|}^{2}u-{(r+i\delta )\left|v\right|}^{2}u]\mathrm{d}t+Q_1\mathrm{d}{W}_1\left(t\right),\hfill \\ \\ \mathrm{d}v={[\mu v+\eta \nabla v+(1+i\alpha )\Delta v-(1+i\beta )|v|}^{2}v-{(r+i\delta )\left|u\right|}^{2}v]\mathrm{d}t+Q_2\mathrm{d}{W}_2\left(t\right),\hfill \end{array}\right.\end{array}$$

(1)

with the zero Dirichlet boundary condition, where $\left(u\left(t\right),v\left(t\right)\right)$ is the complex-valued ($\mathbb{C}$-valued) unknown function defined on $D\subset {\mathbb{R}}^2$, and $\alpha ,\beta ,r,\delta ,\mu >0$ and $\eta >0$ are real numbers. The stochastic forcing is modeled by the white in time Gaussian noise $(Q_1\dot{W_1},Q_2\dot{W_2})$, which is to be specified in Section 2.

This model has been studied by Huang et al. [5,6], where the existence and uniqueness of $H^1$-strong solutions, the unique ergodicity, as well as the exponentially mixing property are proven. Here, we are mainly concerned with the issue of stability from the perspective of statistical mechanics. Roughly speaking, SGLNE (1) is perturbed so that all of its coefficients and even noise terms could be changed; then, the relationship between the original and perturbed systems is investigated by measuring the gap between their corresponding stationary distributions. In this way, the stability of the stationary distribution for SGLNE (1) is proven in the sense of Hairer et al. [7,8], which is in striking difference with its deterministic analogue and indicates the robustness of equilibrium for statistical hydrodynamics. Our research delves into the intricacies of the stochastic Ginzburg–Landau–Newell equations, unraveling their stability of the stationary distribution in the framework of statistical hydrodynamics, which reveals that the complex interplay between order and disorder in these dynamical systems is likely to be robust under disturbances. By pushing the boundaries of our current understanding, we aim to contribute novel insights that may pave the way for advancements in materials science, quantum computing, and beyond.

To put it more specifically, we continue to use the hypotheses on the coefficients of SGLNE (1) as in [5,6].

(H1) 

Assume $2\mu +{\eta }^2\le {\varrho }_1$, where ${\varrho }_1$ is the first eigenvalue of the Dirichlet problem

$$-\Delta u=\varrho u\mathrm{in}D,u=0\mathrm{on}𝜕D.$$

(H2) 

$r\ge 1,4{\beta }^2+3r^2+4{\delta }^2\le 5.$

For simplicity, denote $\Xi :=(\mu ,\eta ,\alpha ,\beta ,r,\delta ,Q_1,Q_2)$, and let ${\mu }^{\Xi }$ be the unique stationary distribution for SGLNE parameterized by $\Xi$ and ${\mu }^{\widehat{\Xi }}$ be any invariant probability measure for SGLNE associated with disturbed parameter $\widehat{\Xi }$. Range of $\Xi$ ($\widehat{\Xi }$, respectively) will be denoted by ${\Lambda }_0$ ($\Lambda$, respectively). Now, we are ready to state our main result.

Theorem 1.

Suppose the hypotheses (H1) and (H2) hold. Then, ${\mu }^{\widehat{\Xi }}\to {\mu }^{\Xi }$ as $\widehat{\Xi }\to \Xi .$

This is the simplified version of Theorem 3, and the exact meaning of convergences is explained in Section 2.

It is worth mentioning that Hairer, Mattingly, and Scheutzow [8] attached great importance to the stability of the stationary distribution and proposed a general framework, yet this is not suitable for our need. Hairer and Mattingly [7] obtained a deeper result by establishing the regular dependence of stationary distributions on the parameters for the 2D stochastic Navier–Stokes equations. Regretfully, with the nonlinear term being cubic in the case of SGLNE, we have to work on the Gelfand triple $H^2\subset H^1\subset H^0$ to make up for the deficiency of its lacking unique $L^2$-solutions to generate transition semigroups. Furthermore compared with [5,6], new nonlinear terms emerge and exponential moments therein need to be estimated more subtly. These difficulties lead us to obtain a weaker sense of stability, as opposed to [7,8].

The rest of the paper is organized as follows: in Section 2, we introduce some necessary notations, along with the degenerate noise construction. Then, we recall from [5,6] some main results related to SGLNE (1) including the well posedness, the existence and uniqueness of stationary distributions, and the spectral gap. In Section 3, we first give a sharp estimate of expectation of exponential moments and then establish the bound on the difference between solutions corresponding to different set of parameters. Finally, in Theorem 3, we obtain the stability of stationary distributions. In the last section, we discuss the application of this model and point out the direction of our future research.

2. Preliminaries

In this section, we introduce some necessary notations, then recall the main result established in [5,6]. Let $L^2$ and $H^1$ be the Hilbert spaces on D endowed with the complex inner products

$$⟨u,v⟩={\int }_{D}u\overline{v}\mathrm{d}x,{⟨u,v⟩}_1=⟨u,v⟩+{\int }_D\nabla u\nabla \overline{v}\mathrm{d}x.$$

For $s\in \mathbb{N}$, let $H^s$ be the usual Sobolev space on D. We denote by ${\parallel ·\parallel }_{L^p}$ the $L^p$ norm and by ${\parallel ·\parallel }_s$ the $H^s$ Sobolev-norm. $L^2$ is sometimes represented by $H^0$, and ${\parallel ·\parallel }_{L^2}$ is denoted as $\parallel ·\parallel$ for simplicity. The letter $C(·)$ with parameters or not inside the bracket denotes a positive constant, whose value may change from line to line.

Here, we focus on the degenerate stochastic forcing. To be precise, for $N\in \mathbb{N}$, let $\Omega =C_0({\mathbb{R}}^{+},{\mathbb{R}}^{2N})$ be the space of all continuous functions with initial values 0, $\mathbf{P}$, the standard Wiener measure on $\mathcal{F}:=\mathcal{B}\left(C_0({\mathbb{R}}^{+},{\mathbb{R}}^{2N})\right)$. Then, the coordinate processes

$$W_1(t,\omega ):=({\omega }_1\left(t\right),\dots ,{\omega }_N\left(t\right)),W_2(t,\omega ):=({\omega }_{N+1}\left(t\right),\dots ,{\omega }_{2N}\left(t\right)),\omega \in \Omega ,$$

(2)

are two independent standard Wiener processes on $(\Omega ,\mathcal{F},\mathbf{P})$. We define the linear map $Q_i:{\mathbb{R}}^N\to H^1\left(D\right),i=1,2$ in a natural way by

$$Q_i{e}_j=q_{i,j}{\mathbf{e}}_j,q_{i,j}>0,i=1,2,j=1,\dots ,N,$$

(3)

where ${\left\{e_i\right\}}_{i=1}^N$ is the canonical basis of ${\mathbb{R}}^N$, and ${\mathbf{e}}_i$’s are the normalized eigenfunctions corresponding to the eigenvalues $0<{\varrho }_1\le {\varrho }_2\le \cdots$ of $-\Delta$. Literally, $(Q_1{W}_1\left(t\right),Q_2{W}_2\left(t\right))$ is the considered degenerate noise. The noise is degenerate in the sense that it drives the system only in the first N Fourier modes.

This type of 3D SGLNE has been studied in [5] and the existence and uniqueness of strong solutions in $C([0,T];H^1\times H^1)\bigcap$$L^2([0,T];H^2\times H^2)$ with $H^1\times H^1$-initial data are obtained. For fixed initial data ${\Phi }_0=(u_0,v_0)\in H^1\times H^1$, we denote this unique solution by $\{u(t,{\Phi }_0),v(t,{\Phi }_0)\}$. Then, $\{(u(t,{\Phi }_0),v(t,{\Phi }_0));t\ge 0\}$ generates a strong Markov process on the state space $H^1\times H^1$. We denote by ${\mathcal{P}}_t$ the Markov semigroup associated with $\{u(t,{\Phi }_0),v(t,{\Phi }_0)\}$, and by ${\mathcal{P}}_t^{\ast }$ the dual semigroup of ${\mathcal{P}}_t$, which acts on the probability measure space $\mathcal{M}(H^1\times H^1)$. A distribution $\mu \in \mathcal{M}(H^1\times H^1)$ is called stationary provided ${\mathcal{P}}_t^{\ast }\mu =\mu$ for each $t\ge 0$. The sufficient yet not excessive conditions for the existence and the uniqueness of stationary distributions are also given in [5].

For $\nu >0$, we define a family of distances $d_{\nu }$ on $H^1\times H^1$ by

$$d_{\nu }(\mathbf{x},\mathbf{y})=\underset{\gamma }{inf}{\int }_0^1{exp(\nu \parallel \gamma \left(t\right)\parallel }_1^2)\parallel \dot{\gamma }{\left(t\right)\parallel }_{1}dt,$$

(4)

where the infimum runs over all paths $\gamma$ such that $\gamma \left(0\right)=\mathbf{x}$ and $\gamma \left(1\right)=\mathbf{y}$. Immediately, one obtains

$$d_{\nu }(\mathbf{x},\mathbf{y})\le {\parallel \mathbf{x}-\mathbf{y}\parallel }_1\left({exp(\nu \parallel \mathbf{x}\parallel }_1^2{)+exp(\nu \parallel \mathbf{y}\parallel }_1^2)\right).$$

(5)

For ${\mu }_1,{\mu }_2\in \mathcal{M}(H^1\times H^1)$, we use the notation $\mathcal{C}({\mu }_1,{\mu }_2)$ for the set of all measures $\Gamma$ on $(H^1\times H^1)\times (H^1\times H^1)$ such that $\Gamma (\mathbf{A}\times (H^1\times H^1))={\mu }_1\left(\mathbf{A}\right)$ and $\Gamma ((H^1\times H^1)\times \mathbf{A})={\mu }_2\left(\mathbf{A}\right)$ for any Borel set $\mathbf{A}\subset H^1\times H^1$. Measure $\Gamma$ is called a $coupling$ of ${\mu }_1$ and ${\mu }_2$. Define the 1-Wasserstein distance by

$$d_{\nu }({\mu }_1,{\mu }_2)=\underset{\pi \in \mathcal{C}({\mu }_1,{\mu }_2)}{inf}{\int }_{(H^1\times H^1)\times (H^1\times H^1)}{d}_{\nu }(\mathbf{x},\mathbf{y})\pi (d\mathbf{x},d\mathbf{y}).$$

(6)

The next theorem establishes the exponential ergodity.

Theorem 2

(Theorem 1.1 of [6]). Suppose that $\left(H1\right),\left(H2\right)$ hold. Let $\left\{{\mathcal{P}}_t\right\}$ be the Markov semigroup associated with (1), then for any N large enough, there exists positive constants ${\eta }_0=\frac{1}{{4\parallel Q\parallel }_1^2}$, γ and C such that

$$\begin{array}{c}\hfill d_{\eta }({\mathcal{P}}_t^{\ast }{\mu }_1,{\mathcal{P}}_t^{\ast }{\mu }_2)\le C{e}^{-\gamma t}{d}_{\eta }({\mu }_1,{\mu }_2),\end{array}$$

for any $t>0$, $\eta \le {\eta }_0$ and any two probability distributions ${\mu }_1$ and ${\mu }_2$ on $H^1\times H^1$.

3. Stability of Stochastic Systems

Refined from (Lemma 3.2 of [6]), the following lemma is somewhat technical. We will use it to estimate moments of ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ and in particular guarantee the validity of metric (6).

$\left(u\left(t\right),v\left(t\right)\right)$ is the unique solution of Equation (1) corresponding to initial data $\left(u_0,v_0\right)$.

Lemma 1.

$\left(u\left(t\right),v\left(t\right)\right)$ is the unique solution of Equation (1) corresponding to initial data $\left(u_0,v_0\right)$. Suppose that (H2) holds and $0<\nu <\frac{1}{{4\parallel Q\parallel }_1^2}$, then there holds

$$\begin{array}{cc}\hfill & \mathbf{E}\exp \left({\nu (\parallel u\left(t\right)\parallel }_1^2+{\parallel v\left(t\right)\parallel }_1^2)+\frac{\nu e^{-\frac{t}{4}}}{4}{\int }_0^t{(\parallel u\left(s\right)\parallel }_2^2+{\parallel v\left(s\right)\parallel }_2^2)ds\right)\hfill \\ \hfill \le & C\exp \left(\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)e^{-\frac{1}{4}t}\right).\hfill \end{array}$$

(7)

Proof. 

Notice that the quadratic variation of $Q_i{W}_i\left(t\right)$ in $H^1$ is given by ${\mathcal{E}}_i\triangleq {\sum }_{k=1}^N{q}_{i,k}^2$, then it follows from the Itô’s formula that

$$\begin{array}{c}\hfill {\parallel u\left(t\right)\parallel }_1^2=\parallel u_0{\parallel }_1^2+\sum _{i=1}^3{I}_i{\left(t\right),\parallel v\left(t\right)\parallel }_1^2={\parallel v_0\parallel }_1^2+\sum _{i=1}^3{J}_i\left(t\right),\end{array}$$

(8)

where

$$\begin{array}{ccc}\hfill I_1\left(t\right)& =& 2{\int }_0^t{⟨u,(1+i\alpha )\Delta u⟩}_1+2\mu {\int }_0^t{⟨u,u⟩}_1\mathrm{d}s-2\eta {\int }_0^t{⟨u,\nabla u⟩}_1\mathrm{d}s\hfill \\ & & -2{\int }_0^t{⟨u,(1+i\beta )\left|u\right|}^2{u⟩}_1\mathrm{d}s-2{\int }_0^t{⟨u,(r+i\delta )\left|v\right|}^2{u⟩}_1\mathrm{d}s=:I_1^1+I_1^2,\hfill \\ \hfill I_2\left(t\right)& =& 2\sum _{k=1}^N{q}_{1,k}{\int }_0^t{⟨u,{\mathbf{e}}_k⟩}_1\mathrm{d}{W}_{1,k}\left(s\right),I_3\left(t\right)={\mathcal{E}}_{1}t,\hfill \\ \hfill J_1\left(t\right)& =& 2{\int }_0^t{⟨v,(1+i\alpha )\Delta vs.⟩}_1+2\mu {\int }_0^t{⟨v,vs.⟩}_1\mathrm{d}s+2\eta {\int }_0^t{⟨v,\nabla vs.⟩}_1\mathrm{d}s\hfill \\ & & -2{\int }_0^t{⟨v,(1+i\beta )\left|v\right|}^2{vs.⟩}_1\mathrm{d}s-2{\int }_0^t{⟨v,(r+i\delta )\left|u\right|}^2{vs.⟩}_1\mathrm{d}s=:J_1^1+J_1^2,\hfill \\ \hfill J_2\left(t\right)& =& 2\sum _{k=1}^N{q}_{2,k}{\int }_0^t{⟨v,{\mathbf{e}}_k⟩}_1\mathrm{d}{W}_{2,k}\left(s\right),J_3\left(t\right)={\mathcal{E}}_{2}t,\hfill \end{array}$$

where $I_1^1$ and $I_1^1$ ( $J_1^1$ and $J_1^1$) are the first and second lines of $I_1$ ($J_1$), respectively. Recalling from $\left(H2\right)$ that $4{\beta }^2+3r^2+4{\delta }^2\le 5$ and noticing that

$$\begin{array}{ccc}& & {Re⟨u,(1+i\beta )\left|u\right|}^2{u⟩}_1={Re⟨u,(1+i\beta )\left|u\right|}^2{u⟩+Re⟨\nabla u,(1+i\beta )\nabla \left(\right|u|}^{2}u)⟩\hfill \\ & \ge & {3\parallel \left|u\right||\nabla u|\parallel }^2-2\left|Re\left[(1-i\beta )\int \nabla u·\overline{u}\left(Im(\nabla u·\overline{u})i\right)\mathrm{d}x\right]\right|+{\parallel u\parallel }_{L^4}^4\hfill \\ & \ge & \left(3-2\sqrt{1+{\beta }^2}\right){\parallel \left|u\right||\nabla u|\parallel }^2+{\parallel u\parallel }_{L^4}^4\ge {\parallel u\parallel }_{L^4}^4,\hfill \\ & & {Re⟨v,(1+i\beta )\left|v\right|}^2{vs.⟩}_1={Re⟨v,(1+i\beta )\left|v\right|}^2{vs.⟩+Re⟨\nabla v,(1+i\beta )\nabla \left(\right|v|}^{2}v)⟩\hfill \\ & \ge & {3\parallel \left|v\right||\nabla v|\parallel }^2-2\left|Re\left[(1-i\beta )\int \nabla v·\overline{v}\left(Im(\nabla vs.·\overline{v})i\right)\mathrm{d}x\right]\right|+{\parallel v\parallel }_{L^4}^4\ge {\parallel v\parallel }_{L^4}^4,\hfill \\ & & {Re⟨u,(r+i\delta )\left|v\right|}^2{u⟩}_1={Re⟨u,(r+i\delta )\left|v\right|}^2{u⟩+Re⟨\nabla u,(r+i\delta )\nabla \left(\right|v|}^{2}u)⟩\hfill \\ & \ge & {r\parallel \left|v\right||\nabla u|\parallel }^2-2\left|Re\left[(r-i\delta ){\int }_D\nabla u·\overline{u}\left(Re(\nabla vs.·\overline{v})\right)\mathrm{d}x\right]\right|+{r\parallel \left|u\right|\left|v\right|\parallel }^2\ge {r\parallel \left|u\right|\left|v\right|\parallel }^2,\hfill \\ & & {Re⟨v,(r+i\delta )\left|u\right|}^2{vs.⟩}_1={Re⟨v,(r+i\delta )\left|u\right|}^2{vs.⟩+Re⟨\nabla v,(r+i\delta )\nabla \left(\right|u|}^{2}v)⟩\hfill \\ & \ge & {r\parallel \left|u\right||\nabla v|\parallel }^2-2\left|Re\left[(r-i\delta ){\int }_D\nabla v·\overline{v}\left(Re(\nabla u·\overline{u})\right)\mathrm{d}x\right]\right|+{r\parallel \left|u\right|\left|v\right|\parallel }^2\ge {r\parallel \left|u\right|\left|v\right|\parallel }^2,\hfill \end{array}$$

we have

$$I_1^2+J_1^2\le -{\parallel u\parallel }_{L^4}^4-{\parallel v\parallel }_{L^4}^4-{2r\parallel \left|u\right|\left|v\right|\parallel }^2.$$

Interpolation inequality and Hölder’s inequality yield that

$${2\mu \parallel u\parallel }_1^2\le \frac{1}{2}{\parallel \nabla u\parallel }_1^2+{C\left(\mu \right)\parallel u\parallel }^2\le \frac{1}{2}{\parallel \nabla u\parallel }_1^2+\frac{1}{2}{\parallel u\parallel }_{L^4}^4+\frac{{C\left(\mu \right)}^2\left|D\right|}{2},$$

and

$$2\eta \left|{⟨u,\nabla u⟩}_1\right|\le \frac{1}{2}{\parallel \nabla u\parallel }_1^2+{C\left(\eta \right)\parallel u\parallel }^2\le \frac{1}{2}{\parallel \nabla u\parallel }_1^2+\frac{1}{2}{\parallel u\parallel }_{L^4}^4+\frac{{C\left(\eta \right)}^2\left|D\right|}{2},$$

and similarly,

$${2\mu \parallel v\parallel }_1^2+2\eta \left|{⟨v,\nabla vs.⟩}_1\right|\le {\parallel \nabla v\parallel }_1^2+{\parallel v\parallel }_{L^4}^4+{C\left(\eta \right)}^2\left|D\right|.$$

Consequently, taking real part and using Young’s inequality give

$$\begin{array}{ccc}\hfill {\parallel u\parallel }_1^2+{\parallel v\parallel }_1^2& \le & \parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2-{\int }_0^t{(\parallel \nabla u\parallel }_1^2+{\parallel \nabla v\parallel }_1^2)\mathrm{d}s+C(\mu ,\eta ,{\mathcal{E}}_1,{\mathcal{E}}_2)t\hfill \\ & & +2\sum _{k=1}^N{q}_{1,k}{\int }_0^t{⟨u,{\mathbf{e}}_k⟩}_1\mathrm{d}{W}_{1,k}\left(s\right)+2\sum _{k=1}^N{q}_{2,k}{\int }_0^t{⟨v,{\mathbf{e}}_k⟩}_1\mathrm{d}{W}_{2,k}\left(s\right).\hfill \end{array}$$

(9)

Next, we will adopt the strategy of (Lemma 5.1 of [7]). Denoting $G(t,\omega )=2\nu {\sum }_{k=1}^N{q}_{1,k}⟨u,$${\mathbf{e}}_k{⟩}_1{e}_k+2\nu {\sum }_{k=1}^N{q}_{2,k}{⟨v,{\mathbf{e}}_k⟩}_1{e}_k,$ for $0<\nu <1/(4\parallel Q{\parallel }_1^2)$, one then writes (9) in a concise way

$$\begin{array}{c}\hfill {\nu d(\parallel u(t,\omega )\parallel }_1^2+{\parallel v(t,\omega )\parallel }_1^2{)\le \nu (C-\parallel u(t,\omega )\parallel }_2^2-{\parallel v(t,\omega )\parallel }_2^2)dt+G(t,\omega )dW(t,\omega ).\end{array}$$

Fixing $t>0$ and $\forall s\in [0,t]$, set

$$Y\left(s\right)=\nu e^{\frac{1}{4}(s-t)}{(\parallel u\left(s\right)\parallel }_1^2+{\parallel v\left(s\right)\parallel }_1^2)+\frac{\nu }{4}{\int }_0^s{e}^{\frac{1}{4}(r-t)}{(\parallel u\left(r\right)\parallel }_2^2+{\parallel v\left(r\right)\parallel }_2^2)dr.$$

It is straightforward that $Y\left(0\right)=\nu e^{-\frac{1}{4}t}(\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)$ and

$$Y\left(t\right)\ge {\nu (\parallel u\left(t\right)\parallel }_1^2+{\parallel v\left(t\right)\parallel }_1^2)+\frac{\nu e^{-t/4}}{4}{\int }_0^t{(\parallel u\left(s\right)\parallel }_2^2+{\parallel v\left(s\right)\parallel }_2^2)ds.$$

Applying Itô’s formula to $Y\left(s\right)$ yields that

$$\begin{array}{ccc}\hfill dY\left(s\right)& =& \frac{\nu }{4}{e}^{\frac{1}{4}(s-t)}{(\parallel u\left(s\right)\parallel }_1^2+{\parallel v\left(s\right)\parallel }_1^2)+e^{\frac{1}{4}(s-t)}{\nu d(\parallel u\left(s\right)\parallel }_1^2+{\parallel v\left(s\right)\parallel }_1^2)\hfill \\ & & +\frac{\nu }{4}{e}^{\frac{1}{4}(s-t)}{(\parallel u\left(s\right)\parallel }_2^2+{\parallel v\left(s\right)\parallel }_2^2)\hfill \\ & \le & \nu exp\left(\frac{1}{4}(s-t)\right)\left(C-\frac{1}{2}{(\parallel u\left(s\right)\parallel }_2^2+{\parallel v\left(s\right)\parallel }_2^2)\right)ds+exp\left(\frac{1}{4}(s-t)\right)G\left(s\right)dW\left(s\right).\hfill \end{array}$$

The last inequality holds from the concise form of (9). Since $s\in [0,t]$, we have

$$Y\left(s\right)\le Y\left(0\right)+4\nu C-\frac{\nu }{2}{\int }_0^s{(\parallel u\left(r\right)\parallel }_2^2+{\parallel v\left(r\right)\parallel }_2^2)dr+{\int }_0^{s}exp\left(\frac{1}{4}(r-t)\right)G\left(r\right)dW\left(r\right).$$

Noting that $G{\left(t\right)}^2\le 4{\nu }^2{\parallel Q\parallel }_1^2{(\parallel u\left(t\right)\parallel }_2^2+{\parallel v\left(t\right)\parallel }_2^2)$, one obtains for any K that

$$\begin{array}{cc}& \mathbf{P}\left({\nu (\parallel u(t\parallel }_1^2+\parallel v\left(t{\parallel }_1^2\right)+\frac{\nu e^{-\frac{t}{4}}}{4}{\int }_0^t(\parallel u\left(s\right){\parallel }_2^2+\parallel u\left(s\right){\parallel }_2^2)ds-\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)e^{-\frac{1}{4}t}-4\nu C>K\right)\hfill \\ & \le \mathbf{P}\left(Y\left(t\right)-Y\left(0\right)-4\nu C>K\right)\hfill \\ & \le \mathbf{P}\left({\int }_0^{t}exp\left(\frac{1}{4}(r-t)\right)G\left(r\right)dW\left(r\right)-\frac{\nu }{2}{\int }_0^t{(\parallel u\left(r\right)\parallel }_2^2+{\parallel v\left(r\right)\parallel }_2^2)dr>K\right)\hfill \\ & \le \mathbf{P}\left({\int }_0^{t}exp\left(\frac{1}{4}(r-t)\right)G\left(r\right)dW\left(r\right)-\frac{1}{{8\nu \parallel Q\parallel }_1^2}⟨M⟩\left(t\right)>K\right)\le exp\left(-\frac{1}{{4\nu \parallel Q\parallel }_1^2}K\right),\hfill \end{array}$$

where $M\left(s\right)={\int }_0^{s}exp\left(\frac{1}{4}(r-t)\right)G\left(r\right)dW\left(r\right)$ and exponential martingale inequality is applied in the last inequality. Since $\mathbf{E}{e}^X={\int }_0^{\infty }\mathbf{P}(e^X>y)dy$ holds for any random variable X, one has

$$\begin{array}{ccc}& & \mathbf{E}\exp \left({\nu (\parallel u\left(t\right)\parallel }_1^2+{\parallel v\left(t\right)\parallel }_1^2)+\frac{\nu e^{-\frac{t}{4}}}{4}{\int }_0^t{(\parallel u\left(s\right)\parallel }_2^2+{\parallel v\left(s\right)\parallel }_2^2)ds\right)\hfill \\ & \le & \frac{{4\nu \parallel Q\parallel }_1^{2}exp\left(4\nu C\right)}{1-{4\nu \parallel Q\parallel }_1^2}\exp \left(\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)e^{-\frac{1}{4}t}\right),\hfill \end{array}$$

the claim then follows from $0<\nu <1/(4\parallel Q{\parallel }_1^2)$. □

Remark 1.

It is straightforward from Lemma 1 that

$$\mathbf{E}\exp \left(\nu \underset{0\le s\le t}{sup}{(\parallel u\left(s\right)\parallel }_1^2+{\parallel v\left(s\right)\parallel }_1^2)\right)\le C\exp \left(\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)\right).$$

(10)

We will use this bound to apply the Lebesgue dominated convergence theorem.

In order to consider the perturbation of stochastic systems, we introduce more notations. Recall that $\Xi :=(\mu ,\eta ,\alpha ,\beta ,r,\delta ,Q_1,Q_2)$, and $\Lambda$ is the parameter space, which contains all $\Xi$’s that ensure the existence of stationary distributions for 2D SGLNE, which are not necessarily unique. We equip $\Lambda$ with the natural distance by

$$\begin{array}{cc}\hfill d{(\Xi ,\widehat{\Xi })}^2=& |\mu -\widehat{\mu }{|}^2+|\eta -\widehat{\eta }{|}^2+|\alpha -\widehat{\alpha }{|}^2+|\beta -\widehat{\beta }{|}^2+|r-\widehat{r}{|}^2+{|\delta -\widehat{\delta }|}^2\hfill \\ & +\parallel Q_1-{\widehat{Q}}_1{\parallel }_1^2+{\parallel Q_2-{\widehat{Q}}_2\parallel }_1^2.\hfill \end{array}$$

${\Lambda }_0$ isthe subset of $\Lambda$ that ensures the existence of a unique stationary distribution for 2D SGLNE. Then, $\Lambda$ and ${\Lambda }_0$ are not empty, see [5]. Then, for any $\Xi \in {\Lambda }_0$, we denote by ${\mu }^{\Xi }$ the unique stationary distribution for SGLNE with parameter $\Xi$ and by ${\mathcal{P}}_t^{\Xi }$ the corresponding semigroup. For $\widehat{\Xi }\in \Lambda$, ${\mu }^{\widehat{\Xi }}$ will simply denote any stationary probability distribution for ${\mathcal{P}}_t^{\widehat{\Xi }}$. For simplicity, we will simply write ${\mathcal{P}}_t^{\Xi \ast }$ for ${\left({\mathcal{P}}_t^{\Xi }\right)}^{\ast }$.

Let $(u_0,v_0)\in H^1\times H^1$, and for any two sets of parameters $\Xi$ and $\widehat{\Xi }$, denote by $(u_t,v_t)$ ($({\widehat{u}}_t,{\widehat{v}}_t)$, respectively) the unique solution to (1) starting from $(u_0,v_0)$ with respect to parameter $\Xi$ ($\widehat{\Xi }$, respectively). For $R>0$, define the stopping time

$${\tau }_R:=inf\{t⩾0:\parallel u(t,u_0){\parallel }_1\vee \parallel \widehat{u}(t,u_0){\parallel }_1\vee \parallel v(t,v_0){\parallel }_1\vee \parallel \widehat{v}(t,v_0){\parallel }_1⩾R\}.$$

The following result establishes the bound on the difference between solutions corresponding to different set of parameters, by which we will estimate the closeness of the time t transition distribution in the final theorem.

Proposition 1.

There exists a ${\nu }_0>0$ so that, for every positive $\nu \le {\nu }_0$, there exist $C>0$ and $C\left(R\right)>0$ so that

$$\mathbf{E}\parallel u_{t\wedge {\tau }_R}-{\widehat{u}}_{t\wedge {\tau }_R}{\parallel }_1^2+\mathbf{E}{\parallel v_{t\wedge {\tau }_R}-{\widehat{v}}_{t\wedge {\tau }_R}\parallel }_1^2\le C{e}^{C\left(R\right)t+\nu (\parallel u_0{\parallel }_1^2+\parallel u_0{\parallel }_1^2)}d{(\Xi ,\widehat{\Xi })}^2.$$

Proof. 

Set

$$\begin{array}{ccc}& {\tilde{u}}_t=u_t-{\widehat{u}}_t,{\tilde{v}}_t=v_t-{\widehat{v}}_t,\tilde{\mu }=\mu -\widehat{\mu },\tilde{\eta }=\eta -\widehat{\eta },\tilde{\alpha }=\alpha -\widehat{\alpha },& \\ & \tilde{\beta }=\beta -\widehat{\beta },\tilde{r}=r-\widehat{r},\tilde{\delta }=\delta -\widehat{\delta },{\tilde{Q}}_1=Q_1-{\widehat{Q}}_1,{\tilde{Q}}_2=Q_2-{\widehat{Q}}_2.\end{array}$$

Immediately, one has

$$\begin{array}{ccc}\hfill d\tilde{u}& =& [\mu \tilde{u}+\tilde{\mu }\widehat{u}-\eta \nabla \tilde{u}-\tilde{\eta }\nabla \widehat{u}+(1+i\alpha )\Delta \tilde{u}+i\tilde{\alpha }\Delta \widehat{u}-(1+i\beta )|u{|}^{2}u\hfill \\ & & +(1+i\widehat{\beta })|\widehat{u}{|}^2\widehat{u}-(r+i\delta ){\left|v\right|}^{2}u+(\widehat{r}+i\widehat{\delta })|\widehat{v}{|}^2\widehat{u}]dt+{\tilde{Q}}_{1}d{W}_1,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill d\tilde{v}& =& [\mu \tilde{v}+\tilde{\mu }\widehat{v}+\eta \nabla \tilde{v}+\tilde{\eta }\nabla \widehat{v}+(1+i\alpha )\Delta \tilde{v}+i\tilde{\alpha }\Delta \widehat{v}-(1+i\beta )|v{|}^{2}v\hfill \\ & & +(1+i\widehat{\beta })|\widehat{v}{|}^2\widehat{v}-(r+i\delta ){\left|u\right|}^{2}v+(\widehat{r}+i\widehat{\delta })|\widehat{u}{|}^2\widehat{v}]dt+{\tilde{Q}}_{2}d{W}_2.\hfill \end{array}$$

Introduce stochastic convolutions

$${\Psi }_1\left(t\right)={\int }_0^t{e}^{(1+i\alpha )\Delta (t-s)}{\tilde{Q}}_{1}d{W}_1\left(s\right),{\Psi }_2\left(t\right)={\int }_0^t{e}^{(1+i\alpha )\Delta (t-s)}{\tilde{Q}}_{2}d{W}_2\left(s\right),$$

and set ${\tilde{u}}_t^{{\Psi }_1}={\tilde{u}}_t-{\Psi }_1\left(t\right),{\tilde{v}}_t^{{\Psi }_2}={\tilde{v}}_t-{\Psi }_2\left(t\right).$ The Ornstein–Uhlenbeck transformation [9,10] yields that

$$\begin{array}{ccc}\hfill d{\tilde{u}}^{{\Psi }_1}& =& [\mu {\tilde{u}}^{{\Psi }_1}+\mu {\Psi }_1+\tilde{\mu }\widehat{u}-\eta \nabla {\tilde{u}}^{{\Psi }_1}-\eta \nabla {\Psi }_1-\tilde{\eta }\nabla \widehat{u}+(1+i\alpha )\Delta {\tilde{u}}^{{\Psi }_1}+i\tilde{\alpha }\Delta \widehat{u}\hfill \\ & & -{(1+i\beta )\left|u\right|}^{2}u+(1+i\widehat{\beta })|\widehat{u}{|}^2\widehat{u}-(r+i\delta ){\left|v\right|}^{2}u+(\widehat{r}+i\widehat{\delta })|\widehat{v}{|}^2\widehat{u}]dt,\hfill \end{array}$$

(11)

and

$$\begin{array}{ccc}\hfill d{\tilde{v}}^{{\Psi }_2}& =& [\mu {\tilde{v}}^{{\Psi }_2}+\mu {\Psi }_2+\tilde{\mu }\widehat{v}+\eta \nabla {\tilde{v}}^{{\Psi }_2}+\eta \nabla {\Psi }_2+\tilde{\eta }\nabla \widehat{v}+(1+i\alpha )\Delta {\tilde{v}}^{{\Psi }_2}+i\tilde{\alpha }\Delta \widehat{v}\hfill \\ & & -{(1+i\beta )\left|v\right|}^{2}v+(1+i\widehat{\beta })|\widehat{v}{|}^2\widehat{v}-(r+i\delta ){\left|u\right|}^{2}v+(\widehat{r}+i\widehat{\delta })|\widehat{u}{|}^2\widehat{v}]dt.\hfill \end{array}$$

(12)

Multiplying (11) with $(I-\Delta ){\tilde{u}}^{{\Psi }_1}$, integrating by parts and taking the real part give that

$$\begin{array}{ccc}\hfill \frac{1}{2}{𝜕}_t{\parallel {\tilde{u}}^{{\Psi }_1}\parallel }_1^2& =& \mu \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_1^2+\mu ⟨{\Psi }_1,(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩+\tilde{\mu }⟨\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\hfill \\ & & -\eta ⟨\nabla {\tilde{u}}^{{\Psi }_1},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩-\eta ⟨\nabla {\Psi }_1,(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩-\tilde{\eta }⟨\nabla \widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\hfill \\ & & +{⟨\Delta {\tilde{u}}^{{\Psi }_1},{\tilde{u}}^{{\Psi }_1}⟩}_1+\Re \left(i\tilde{\alpha }\right)⟨\Delta \widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\hfill \\ & & -{\Re (1+i\beta )⟨\left|u\right|}^{2}u-|\widehat{u}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩-\Re \left(i\tilde{\beta }\right)⟨|\widehat{u}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\hfill \\ & & -{\Re (r+i\delta )⟨\left|v\right|}^{2}u-|\widehat{v}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩-\Re (\tilde{r}+i\tilde{\delta })⟨|\widehat{v}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩,\hfill \end{array}$$

where we use ℜ to take the real part of the subsequent items. The linear terms are dealt with readily as follows

$$\begin{array}{ccc}& \mu ⟨{\Psi }_1,(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\le \frac{1}{8}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+2{\mu }^2{\parallel {\Psi }_1\parallel }^2,& \\ & \tilde{\mu }⟨\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\le \frac{1}{8}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+2{\tilde{\mu }}^2{\parallel \widehat{u}\parallel }^2,& \\ & \eta ⟨\nabla {\tilde{u}}^{{\Psi }_1},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\le \frac{1}{8}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+2{\eta }^2{\parallel {\tilde{u}}^{{\Psi }_1}\parallel }_1^2,& \\ & \eta ⟨\nabla {\Psi }_1,(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\le \frac{1}{8}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+2{\eta }^2{\parallel {\Psi }_1\parallel }_1^2,& \\ & \tilde{\eta }⟨\nabla \widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\le \frac{1}{8}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+2{\tilde{\eta }}^2{\parallel \widehat{u}\parallel }_1^2,& \\ & {⟨\Delta {\tilde{u}}^{{\Psi }_1},{\tilde{u}}^{{\Psi }_1}⟩}_1=-⟨(I-\Delta ){\tilde{u}}^{{\Psi }_1},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩+⟨{\tilde{u}}^{{\Psi }_1},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩=-\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+{\parallel {\tilde{u}}^{{\Psi }_1}\parallel }_1^2,\end{array}$$

and

$$\begin{array}{ccc}\hfill \Re \left(i\tilde{\alpha }\right)⟨\Delta \widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩& =& -\Re \left(i\tilde{\alpha }\right)⟨(I-\Delta )\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩+\Re \left(i\tilde{\alpha }\right)⟨\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩\hfill \\ & \le & 2\tilde{\alpha }\parallel \widehat{u}{\parallel }_2\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2\le \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+16{\tilde{\alpha }}^2{\parallel \widehat{u}\parallel }_2^2.\hfill \end{array}$$

Then, we tackle with the nonlinear terms. For the simple terms, one has

$$\begin{array}{ccc}\hfill |\Re \left(i\tilde{\beta }\right)⟨|\widehat{u}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩|& \le & \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+{\widetilde{4\beta }}^2\parallel \widehat{u}{\parallel }_{L^6}^6\le \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+{\widetilde{4\beta }}^{2}C{\parallel \widehat{u}\parallel }_1^6,\hfill \\ \hfill |\Re (\tilde{r}+i\tilde{\delta })⟨|\widehat{v}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩|& \le & \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+4({\tilde{r}}^2+{\tilde{\delta }}^2)\parallel |\widehat{v}{|}^2|\widehat{u}{|\parallel }^2\hfill \\ & \le & \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+4({\tilde{r}}^2+{\tilde{\delta }}^2)C\parallel \widehat{u}{\parallel }_1^2{\parallel \widehat{v}\parallel }_1^4.\hfill \end{array}$$

The tricky one can be dealt with through interpolation inequality,

$$\begin{array}{ccc}& & |\Re (1+i\beta )⟨|u{|}^{2}u-|\widehat{u}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩|\hfill \\ & \le & \sqrt{1+{\beta }^2}\parallel |{\tilde{u}}^{{\Psi }_1}+{\Psi }_1\left|\right(\left|u\right|+|\widehat{u}{\left|\right)}^2\parallel \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2\hfill \\ & \le & \frac{1}{32}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+8(1+{\beta }^2)\parallel {\tilde{u}}^{{\Psi }_1}+{\Psi }_1{\parallel }_{L^4}^2\parallel \left|u\right|+|\widehat{u}{|\parallel }_{L^8}^4\hfill \\ & \le & \frac{1}{32}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+(1+{\beta }^2)C_1\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2\parallel {\tilde{u}}^{{\Psi }_1}{\parallel (\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4)+(1+{\beta }^2)C_2\parallel {\Psi }_1{\parallel }_1^2{(\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4)\hfill \\ & \le & \frac{1}{16}{\parallel {\tilde{u}}^{{\Psi }_1}\parallel }_2^2+C\left(\beta \right)\left[\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }^2{(\parallel u\parallel }_1^8+\parallel \widehat{u}{\parallel }_1^8)+\parallel {\Psi }_1{\parallel }_1^2{(\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4)\right],\hfill \end{array}$$

and similarly,

$$\begin{array}{ccc}& & |\Re (r+i\delta )⟨|v{|}^{2}u-|\widehat{v}{|}^2\widehat{u},(I-\Delta ){\tilde{u}}^{{\Psi }_1}⟩|\hfill \\ & \le & \sqrt{r^2+{\delta }^2}(\parallel |\tilde{v}\left|\right(u\left|v\right|+\widehat{u}|\widehat{v}\left|\right)\parallel \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2+\parallel \tilde{u}\left|\right(\left|v\right||\widehat{v}\left|\right)\parallel \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2)\hfill \\ & \le & \sqrt{r^2+{\delta }^2}(\parallel |\tilde{v}{\left|\right(\left|u\right|}^2+{\left|v\right|}^2+|\widehat{u}{|}^2+|\widehat{v}{|}^2)\parallel \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2+\parallel \tilde{u}{\left|\right(\left|v\right|}^2+|\widehat{v}{|}^2)\parallel \parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2)\hfill \\ & \le & \frac{1}{32}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+C(r,\delta )[\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }^2{(\parallel u\parallel }_1^8+\parallel \widehat{u}{\parallel }_1^8+{\parallel v\parallel }_1^8+\parallel \widehat{v}{\parallel }_1^8)\hfill \\ & & +\parallel {\Psi }_2{\parallel }_1^2{(\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4+{\parallel v\parallel }_1^4+\parallel \widehat{v}{\parallel }_1^4\left)\right]+\frac{1}{16}\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }_2^2+\frac{1}{32}{\parallel {\tilde{u}}^{{\Psi }_1}\parallel }_2^2\hfill \\ & & +C(r,\delta )\left[\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }^2{(\parallel v\parallel }_1^8+\parallel \widehat{v}{\parallel }_1^8)+\parallel {\Psi }_1{\parallel }_1^2{(\parallel v\parallel }_1^4+\parallel \widehat{v}{\parallel }_1^4)\right]\hfill \\ & \le & \frac{1}{16}\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_2^2+\frac{1}{16}\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }_2^2+C(r,\delta )[\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }^2{(\parallel u\parallel }_1^8+\parallel \widehat{u}{\parallel }_1^8)+\parallel {\Psi }_1{\parallel }_1^2{(\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4)\hfill \\ & & +\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }^2{(\parallel u\parallel }_1^8+\parallel \widehat{u}{\parallel }_1^8+{\parallel v\parallel }_1^8+\parallel \widehat{v}{\parallel }_1^8)+\parallel {\Psi }_2{\parallel }_1^2{(\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4+{\parallel v\parallel }_1^4+\parallel \widehat{v}{\parallel }_1^4)].\hfill \end{array}$$

Applying a similar calculation to (12) and putting these bounds altogether, we yield that

$$\begin{array}{cc}& {𝜕}_t(\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_1^2+\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }_1^2)\hfill \\ \hfill \le & {C(\mu ,\eta ,\beta ,r,\delta )(1+\parallel u\parallel }_1^8+\parallel \widehat{u}{\parallel }_1^8+{\parallel v\parallel }_1^8+\parallel \widehat{v}{\parallel }_1^8\left)\right(\parallel {\tilde{u}}^{{\Psi }_1}{\parallel }_1^2+\parallel {\tilde{v}}^{{\Psi }_2}{\parallel }_1^2)\hfill \\ & {+C(\mu ,\eta ,\beta ,r,\delta )(1+\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4+{\parallel v\parallel }_1^4+\parallel \widehat{v}{\parallel }_1^4\left)\right(\parallel {\Psi }_1{\parallel }_1^2+\parallel {\Psi }_2{\parallel }_1^2)\hfill \\ & +C({\tilde{\mu }}^2+{\tilde{\eta }}^2+{\tilde{\alpha }}^2+{\tilde{\beta }}^2+{\tilde{r}}^2+{\tilde{\delta }}^2)(\parallel \widehat{u}{\parallel }_2^6+\parallel \widehat{v}{\parallel }_2^6).\hfill \end{array}$$

Noting that $(\mu ,\eta ,\beta ,r,\delta )$ inside $C(·)$ is fixed in the current circumstance, we shall omit them afterwards.

Therefore, applying Gronwall’s inequality yields

$$\begin{array}{ccc}\hfill \parallel {\tilde{u}}_{t\wedge {\tau }_R}^{{\Psi }_1}{\parallel }_1^2+{\parallel {\tilde{v}}_{t\wedge {\tau }_R}^{{\Psi }_2}\parallel }_1^2& \le & C{e}^{C\left(R\right)t}{\int }_0^{t\wedge {\tau }_R}{\left[\right(1+\parallel u\parallel }_1^4+\parallel \widehat{u}{\parallel }_1^4+{\parallel v\parallel }_1^4+\parallel \widehat{v}{\parallel }_1^4)\parallel {\Psi }_s{\parallel }_1^2\hfill \\ & & +({\tilde{\mu }}^2+{\tilde{\eta }}^2+{\tilde{\alpha }}^2+{\tilde{\beta }}^2+{\tilde{r}}^2+{\tilde{\delta }}^2)(\parallel {\widehat{u}}_s{\parallel }_2^6+\parallel {\widehat{v}}_s{\parallel }_2^6)]ds,\hfill \end{array}$$

where ${\Psi }_s:=({\Psi }_{1s},{\Psi }_{2s})$. Taking expectation, applying Cauchy–Schwartz inequality and using the fact that $\mathbf{E}\parallel {\Psi }_s{\parallel }_1^4\le C(\mathbf{E}\parallel {\Psi }_s{{\parallel }_1^2)}^2$, one obtains

$$\begin{array}{cc}& \mathbf{E}\parallel {\tilde{u}}_{t\wedge {\tau }_R}^{{\Psi }_1}{\parallel }_1^2+\mathbf{E}{\parallel {\tilde{v}}_{t\wedge {\tau }_R}^{{\Psi }_2}\parallel }_1^2\hfill \\ \hfill \le & C{e}^{C\left(R\right)t}{\int }_0^{t\wedge {\tau }_R}[\mathbf{E}\parallel {\Psi }_s{\parallel }_1^2(1+\mathbf{E}\parallel u_s{\parallel }_1^8+\mathbf{E}\parallel {\widehat{u}}_s{\parallel }_1^8+\mathbf{E}\parallel v_s{\parallel }_1^8+\mathbf{E}\parallel {\widehat{v}}_s{{\parallel }_1^8)}^{\frac{1}{2}}\hfill \\ & +({\tilde{\mu }}^2+{\tilde{\eta }}^2+{\tilde{\alpha }}^2+{\tilde{\beta }}^2+{\tilde{r}}^2+{\tilde{\delta }}^2)\mathbf{E}(\parallel {\widehat{u}}_s{\parallel }_2^6+\parallel {\widehat{v}}_s{\parallel }_2^6)]ds.\hfill \end{array}$$

Using the bound $x^n\le n!a^{-n}{e}^{ax}$, applying Lemma 1 and observing that $\mathbf{E}\parallel {\Psi }_t{\parallel }_1^2\le (\parallel {\tilde{Q}}_1{\parallel }^2+\parallel {\tilde{Q}}_2{\parallel }^2)/\left(2\sqrt{1+{\alpha }^2}\right)$, we obtain

$$\begin{array}{cc}& \mathbf{E}\parallel {\tilde{u}}_{t\wedge {\tau }_R}^{{\Psi }_1}{\parallel }_1^2+\mathbf{E}{\parallel {\tilde{v}}_{t\wedge {\tau }_R}^{{\Psi }_2}\parallel }_1^2\hfill \\ \hfill \le & C{e}^{C\left(R\right)t+\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)}\left(\parallel {\tilde{Q}}_1{\parallel }^2+{\parallel {\tilde{Q}}_2\parallel }^2+{\tilde{\mu }}^2+{\tilde{\eta }}^2+{\tilde{\alpha }}^2+{\tilde{\beta }}^2+{\tilde{r}}^2+{\tilde{\delta }}^2\right).\hfill \end{array}$$

The proof is thus finished. □

At this stage, we are ready to establish our main result, which is also the complete version of Theorem 1.

Theorem 3.

For any $\Xi \in {\Lambda }_0$, and positive constant $\nu \le \frac{1}{{12\parallel Q\parallel }_1^2}$, there holds

$$\underset{\widehat{\Xi }\to \Xi }{lim}{d}_{\nu }({\mu }^{\Xi },{\mu }^{\widehat{\Xi }})=0,$$

where $\widehat{\Xi }\to \Xi$ is taken inside Λ and measured by the distance $d(\Xi ,\widehat{\Xi })$.

Proof. 

By (Theorem 1.1 of [6]), there exists a $t_0>0$ so that

$$d_{\nu }(P_t^{\Xi \ast }{\mu }^{\Xi },P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }})\le \frac{1}{2}{d}_{\nu }({\mu }^{\Xi },{\mu }^{\widehat{\Xi }}),$$

for any $t\ge t_0$. Next, we fix a $t\ge t_0$, then

$$\begin{array}{ccc}\hfill d_{\nu }({\mu }^{\Xi },{\mu }^{\widehat{\Xi }})& =& d_{\nu }(P_t^{\Xi \ast }{\mu }^{\Xi },P_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }})\le d_{\nu }(P_t^{\Xi \ast }{\mu }^{\Xi },P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }})+d_{\nu }(P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }},P_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }})\hfill \\ & \le & \frac{1}{2}{d}_{\nu }({\mu }^{\Xi },{\mu }^{\widehat{\Xi }})+d_{\nu }(P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }},P_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }}).\hfill \end{array}$$

(13)

Next, we carry out the estimate on $d_{\nu }(P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }},P_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }})$, which is exactly the integral of $d_{\nu }(P_t^{\Xi \ast }{\delta }_{(u_0,v_0)},P_t^{\widehat{\Xi }\ast }{\delta }_{(u_0,v_0)})$, with $\delta$ being the $\delta$-measure. The upper bound (5) yields that

$$\begin{array}{cc}& d_{\nu }(P_t^{\Xi \ast }{\delta }_{(u_0,v_0)},P_t^{\widehat{\Xi }\ast }{\delta }_{(u_0,v_0)})\le \mathbf{E}{d}_{\nu }\left((u_t,v_t),({\widehat{u}}_t,{\widehat{v}}_t)\right)\hfill \\ \hfill \le & {\left(\mathbf{E}(\parallel u_t-{\widehat{u}}_t{\parallel }_1^2+\parallel v_t-{\widehat{v}}_t{\parallel }_1^2)2\mathbf{E}\left(e^{2\nu (\parallel u_t{\parallel }_1^2+\parallel v_t{\parallel }_1^2)}+e^{2\nu (\parallel {\widehat{u}}_t{\parallel }_1^2+\parallel {\widehat{v}}_t{\parallel }_1^2)}\right)\right)}^{\frac{1}{2}}.\hfill \end{array}$$

One derives from Lemma 1 and Proposition 1 that

$$\begin{array}{ccc}\hfill d_{\nu }(P_t^{\Xi \ast }{\delta }_{(u_0,v_0)},P_t^{\widehat{\Xi }\ast }{\delta }_{(u_0,v_0)})& \le & C{e}^{3\nu (\parallel u_0{\parallel }_1^2+\parallel v_0{\parallel }_1^2)}(e^{C\left(R\right)t}d(\Xi ,\widehat{\Xi })+\mathbf{E}{\parallel u_t-u_{t\wedge {\tau }_R}\parallel }_1\hfill \\ & & +\mathbf{E}\parallel {\widehat{u}}_t-{\widehat{u}}_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}\parallel v_t-v_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}{\parallel {\widehat{v}}_t-{\widehat{v}}_{t\wedge {\tau }_R}\parallel }_1).\hfill \end{array}$$

Taking the integration, one has

$$\begin{array}{ccc}\hfill d_{\nu }(P_t^{\Xi \ast }{\mu }^{\widehat{\Xi }},P_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }})& \le & C{\int }_{H^1\times H^1}{e}^{3\nu (\parallel u{\parallel }_1^2+\parallel v{\parallel }_1^2)}{\mu }^{\widehat{\Xi }}\left(d(u,v)\right)(e^{C\left(R\right)t}d(\Xi ,\widehat{\Xi })+\mathbf{E}{\parallel u_t-u_{t\wedge {\tau }_R}\parallel }_1\hfill \\ & & +\mathbf{E}\parallel {\widehat{u}}_t-{\widehat{u}}_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}\parallel v_t-v_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}{\parallel {\widehat{v}}_t-{\widehat{v}}_{t\wedge {\tau }_R}\parallel }_1).\hfill \end{array}$$

Since $\widehat{\Xi }\to \Xi$, $d(\Xi ,\widehat{\Xi })$ is bounded. Then, one applies Lemma 1 to obtain uniformly that

$$\begin{array}{cc}& {\int }_{H^1\times H^1}{e}^{3\nu (\parallel u{\parallel }_1^2+\parallel v{\parallel }_1^2)}{\mu }^{\widehat{\Xi }}\left(d(u,v)\right)\hfill \\ \hfill =& {\int }_{H^1\times H^1}{e}^{3\nu (\parallel u{\parallel }_1^2+\parallel v{\parallel }_1^2)}{P}_t^{\widehat{\Xi }\ast }{\mu }^{\widehat{\Xi }}\left(d(u,v)\right)=\mathbf{E}{e}^{3\nu (\parallel {\widehat{u}}_t{\parallel }_1^2+\parallel {\widehat{v}}_t{\parallel }_1^2)}\le C.\hfill \end{array}$$

Combining (13), gives that

$$\begin{array}{ccc}\hfill d_{\nu }({\mu }^{\Xi },{\mu }^{\widehat{\Xi }})& \le & C{e}^{C\left(R\right)t}d(\Xi ,\widehat{\Xi })+C(\mathbf{E}{\parallel u_t-u_{t\wedge {\tau }_R}\parallel }_1\hfill \\ & & +\mathbf{E}\parallel {\widehat{u}}_t-{\widehat{u}}_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}\parallel v_t-v_{t\wedge {\tau }_R}{\parallel }_1+\mathbf{E}{\parallel {\widehat{v}}_t-{\widehat{v}}_{t\wedge {\tau }_R}\parallel }_1)\le \mathrm{I}+\mathrm{II}.\hfill \end{array}$$

In view of Remark 1, for R sufficiently large, term II can be arbitrarily small after using the dominated convergence theorem. Term I can also be sufficiently small by letting $d(\Xi ,\widehat{\Xi })\to 0$. Hence, the claim is proven. □

4. Conclusions

Since ${\mu }^{\Xi }$ and ${\mu }^{\widehat{\Xi }}$ correspond to equilibrium states of original and disturbed systems, respectively, we learn from Theorem 3 that 2D stochastic Ginzburg–Landau–Newell equations are stable under parameter and noise disturbances. This is of great practical significance, for instance, helping to simulate and observe the vortex motion of superconducting materials under strong external magnetic fields. The study of vortex motion will enhance the understanding of complex interactions within superconductors and guide innovation in superconducting technology, such as magnet design and improvement of power transmission systems. However, the stability results we obtained are not yet satisfactory, and in particular, we cannot provide an estimate of convergence speed. In future work, we will consider developing numerical methods and gaining inspiration before moving forward with theoretical research.