Existence and Uniqueness of Weak Solutions for the Stochastic Fractional Ginzburg–Landau Equation

Abstract

In this study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we prove that the initial-boundary value problem admits a unique global weak solution for any ${\mathcal{F}}_0$-measurable $L^2\left(I\right)$-valued random initial value with a finite second moment. We also utilize the properties of the fractional Laplacian and fractional Sobolev spaces to provide a proof of the existence of the uniqueness theorem. These results extend the analysis of the Ginzburg–Landau equations to models incorporating stochastic terms and the fractional Laplacian.

1. Introduction

The Ginzburg–Landau equation has become a research focus due to its wide applications in fields such as superconductivity, fluid flow, and optical soliton transmission (see [1,2]). The introduction of stochastic perturbations is more in line with the analysis needs of nonlinear systems that reflect uncertainties in real-world systems, such as reaction–diffusion equations, infectious disease models, and climate system models. It also provides key support for solving practical problems.

Recently, Qin, Jiang, and Chen [3] proposed a numerical scheme combining a temporal direction leap-frog method with a spatial finite difference scheme to solve this space-fractional Ginzburg–Landau–Schrödinger equation:

$$(a-ib)z_t+{(-\Delta )}^{{\displaystyle \frac{\gamma }{2}}}{z-(1-|z|}^2)z=0,x\in I,t>0.$$

They proved that the scheme converges while keeping both the discrete mass and energy conserved. Pino, Juneman, and Musso [4] investigated the linearized Ginzburg–Landau equation around a standard degree-one vortex solution in the plane. By utilizing explicit expressions for Fourier modes, they obtained precise estimates for the inverse of the linearization operator applicable to cases without orthogonality conditions. Zhang, Lu, and Liu [5] researched the stability and optimal controls for time–space fractional Ginzburg–Landau systems. In this study, we aim to extend the aforementioned results to fractional Ginzburg–Landau equations with stochastic perturbations.

Let $(S,\mathcal{F},\mathbb{P})$ be a complete probability space and ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ be a normal filtration in $\mathcal{F}$. U is a separable Hilbert space and $Q\in \mathcal{L}\left(U\right)$ is a symmetric nonnegative operator with $TrQ<+\infty$. Denote by $W={\left\{W\left(t\right)\right\}}_{t\ge 0}$ a U-valued Q-Wiener process on $(\mathsf{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$. Let $I\subset {\mathbb{R}}^n$ be a smooth bounded domain, let $\gamma \in (0,1)$, and let $\theta >0$. We consider the following stochastic fractional Ginzburg–Landau equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial z(t,x,w)}{\partial t}}+{(-\Delta )}^{\gamma }z(t,x,w)=\alpha \left(x\right)\left[z(t,x,w)-{\left|z(t,x,w)\right|}^{\theta }z(t,x,w)\right]\hfill \\ & +\beta (t,x,w)+h(t,x,w){\displaystyle \frac{\partial W(t,x,w)}{\partial t}},(t,x)\in (0,T)\times I,w\in S,\hfill \\ \hfill & z(t,x,w)=0,(t,x)\in (0,T)\times ({\mathbb{R}}^n\setminus I),w\in S,\hfill \\ \hfill & z(0,x,w)=\varphi (x,w),x\in I,w\in S.\hfill \end{array}\right.\end{array}$$

where $\alpha \in L^{\infty }\left(I\right)$ is a given deterministic function, $\varphi$ is an $L^2\left(I\right)$-valued ${\mathcal{F}}_0$-measurable random variable, and $\beta$ is an $L^2\left(I\right)$-valued adapted process. For each $t\in [0,T]$, h is an $\mathcal{L}\left(U,L^2\left(I\right)\right)$-valued process. Since the stochastic Itô integral of Hilbert-space-valued Wiener processes (see [6])

$${\int }_0^{t}h\left(\tau \right)\mathrm{d}W\left(\tau \right),t\in [0,T],$$

is well-defined, the above initial value problem is properly formulated as the following Itô stochastic differential equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \mathrm{d}z+{(-\Delta )}^{\gamma }z\mathrm{d}t={\alpha (z-|z|}^{\theta }z)\mathrm{d}t+\beta \left(t\right)\mathrm{d}t+h\left(t\right)\mathrm{d}W\left(t\right),t\in (0,T),\hfill \\ \hfill & z\left(0\right)=\varphi .\hfill \end{array}\right.\end{array}$$

(1)

Relevant nonlinear models are also a research focus in the field of Ginzburg–Landau equations. Some scholars have proven the existence and uniqueness of solutions to initial-boundary value problems in appropriate function spaces [7,8]. Juárez-Campos, Kaikina, and Ruiz-Paredes [9] proved the existence and uniqueness of solutions to the stochastic nonlinear Ginzburg–Landau equation with Neumann white-noise boundary conditions and analyzed its regularity near the origin, particularly when the boundary data are irregular. For semilinear equations with boundary noise and point noise, scholars used the fixed-point method to prove the existence and uniqueness of solutions (see [10,11]). Meanwhile, for real-valued stochastic Ginzburg–Landau equations driven by multiplicative noise in the Itô sense, scholars have obtained exact triangular and hyperbolic stochastic solutions (see [12,13]). For one-dimensional complex Ginzburg–Landau equations with multiplicative noise terms, they not only proved the existence and uniqueness of solutions but also proposed a new numerical approximation method and conducted error estimation (see [14,15]). For the existence of solutions to partial differential equations with fractional differential operators, see [16,17]. Inspired by the above research, this study proves the existence and uniqueness of weak solutions for problem (1) via the Galerkin method.

The structure of this paper is arranged as follows: Section 2 introduces some notations and Section 3 presents the global existence of weak solutions to problem (1) under the initial value $\varphi \in L_0^2(S,L^2\left(I\right))$. The subsequent sections elaborate on the research content of this paper in detail.

2. Some Notations

The main difficulty in studying problem (1) arises from the coexistence of a fractional Laplacian and a stochastic term. This section is devoted to establishing appropriate function spaces for problem (1).

The fractional Laplacian ${(-\Delta )}^{\gamma }$ is a class of nonlocal differential operators defined by singular integrals. Let z be a function in Schwartz space. For $\gamma \in (0,1)$,

$${(-\Delta )}^{\gamma }z\left(x\right)=C(n,\gamma )\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{z\left(x\right)-z\left(y\right)}{{|x-y|}^{n+2\gamma }}}\mathrm{d}y,x\in {\mathbb{R}}^n,$$

where P.V. denotes the Cauchy principal value, and C is a constant depending on n and $\gamma$. Let

$$L^p\left(I\right)=\{z\in L^p\left({\mathbb{R}}^n\right):z\left(x\right)=0\mathrm{a}.\mathrm{e}.\mathrm{on}{\mathbb{R}}^n\setminus I\}.$$

For a function $\alpha$ defined only on I, we say that $\alpha \in L^p\left(I\right)$ by identifying $\alpha$ with its zero extension to ${\mathbb{R}}^n$, namely, $\alpha \left(x\right)=0,\mathrm{a}.\mathrm{e}.\mathrm{on}{\mathbb{R}}^n\setminus I$. This convention is used throughout the paper.

Denote

$$H^{\gamma }\left({\mathbb{R}}^n\right)=\left\{z\in L^2\left({\mathbb{R}}^n\right):\underset{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\int \int }{\displaystyle \frac{{\left|z\left(x\right)-z\left(y\right)\right|}^2}{{|x-y|}^{n+2\gamma }}}\mathrm{d}x\mathrm{d}y<+\infty \right\}$$

and

$$B_0\left(I\right)=\left\{z\in H^{\gamma }\left({\mathbb{R}}^n\right):z=0,\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^n\setminus I\right\}.$$

Lemma 7 in [18] implies that $B_0\left(I\right)$ is a Hilbert space equipped with the following norm and inner product:

$${\parallel z\parallel }_{B_0\left(I\right)}={\left(\underset{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\int \int }{\displaystyle \frac{{\left|z\left(x\right)-z\left(y\right)\right|}^2}{{|x-y|}^{n+2\gamma }}}\mathrm{d}x\mathrm{d}y\right)}^{{\displaystyle \frac{1}{2}}},$$

$${(\xi ,\eta )}_{B_0\left(I\right)}=\underset{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\int \int }{\displaystyle \frac{\left(\xi \left(x\right)-\xi \left(y\right)\right)\left(\eta \left(x\right)-\eta \left(y\right)\right)}{{|x-y|}^{n+2\gamma }}}\mathrm{d}x\mathrm{d}y.$$

Moreover, by [19] (Proposition 9), if $\gamma \in (0,1)$ and $n>2\gamma$, then there exists a sequence of eigenfunctions of ${(-\Delta )}^{\gamma }$, denoted by

$$e_1,e_2,\cdots ,e_k,\cdots ,$$

(2)

which is an orthonormal basis for $L^2\left(I\right)$ and an orthogonal basis for $B_0\left(I\right)$.

Lemma 1 

([20], Proposition 1.18). Let $\gamma \in (0,1)$. Then, for every $z\in B_0\left(I\right)$,

$${⟨{(-\Delta )}^{\gamma }z,z⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}={\displaystyle \frac{1}{2}}C(n,\gamma ){\parallel z\parallel }_{B_0\left(I\right)}^2.$$

(3)

For $p>0$ with $n>p\gamma$, define $p_{\gamma }^{*}:={\displaystyle \frac{pn}{n-p\gamma }}.$ In particular, $2_{\gamma }^{*}={\displaystyle \frac{2n}{n-2\gamma }}$.

Lemma 2 

([21], Corollary 7.2). Assume that $n>2\gamma$. If $p\in \left[2,2_{\gamma }^{*}\right]$, then $B_0\left(I\right)\hookrightarrow L^p\left(I\right)\hookrightarrow L^2\left(I\right).$ Hence, there exists a constant $c_0>0$ such that

$${\parallel z\parallel }_{L^2\left(I\right)}\le c_0{\parallel z\parallel }_{B_0\left(I\right)}.$$

(4)

Moreover, the embedding $B_0\left(I\right)\hookrightarrow \hookrightarrow L^p\left(I\right)$ is compact whenever $p\in \left[2,2_{\gamma }^{*}\right).$

Recall that $(S,\mathcal{F},{\mathcal{F}}_{t\ge 0},\mathbb{P})$ is a complete filtered probability space and U is a separable Hilbert space. Since $Q\in L\left(U\right)$ is a symmetric nonnegative operator with $TrQ<+\infty$, there exists a complete orthonormal basis $\left\{{\iota }_k\right\}$ of U, and a bounded sequence of nonnegative real numbers ${\lambda }_k$ such that

$$Q{\iota }_k={\lambda }_k{\iota }_k,k=1,2,\dots$$

(5)

A U-valued stochastic process $W\left(t\right),t\ge 0$, is called a Q-Wiener process on $[0,T]$ if

(i) $W\left(0\right)=0$;

(ii) W has continuous trajectories;

(iii) W has independent increments;

(iv) $\mathcal{L}\left(W\right(t)-W(s\left)\right)=\mathcal{N}(t-s)Q,T\ge t\ge s\ge 0$.

Lemma 3 

([6], Proposition 4.1). Assume that W is a Q-Wiener process, with $TrQ<+\infty$. Then, the following statements hold.

(i) W is a Gaussian process on U and

$$\mathbb{E}\left(W\right(t\left)\right)=0,Cov\left(W\right(t\left)\right)=tQ,t\ge 0.$$

(6)

(ii) For arbitrary $t,W$ has the expansion

$$W\left(t\right)=\sum _{j=1}^{\infty }\sqrt{{\lambda }_j}{\varrho }_j\left(t\right){\iota }_j$$

(7)

where

$${\varrho }_j\left(t\right)={\displaystyle \frac{1}{\sqrt{{\lambda }_j}}}<W\left(t\right),{\iota }_j>,j=1,2,\dots ,$$

are real-valued Brownian motions mutually independent on $(S,\mathcal{F},\mathbb{P})$ and the series in (7) is convergent in $L^2\left(S\right)$.

Lemma 4 

([6], Theorem 4.3). Let W be a Q-Wiener process such that (6) holds. Then, the series (7) is uniformly convergent on $[0,T]$ $\mathbb{P}$-a.s., for arbitrary $T>0$.

Let $(\mathbb{X},\parallel ·{\parallel }_{\mathbb{X}})$ be a Banach space. For any $p\in [1,+\infty )$, let

$$L_a^p(0,T;\mathbb{X})=\left\{z\mathrm{is}\mathrm{an}\mathbb{X}-\mathrm{valued}{\mathcal{F}}_{t\in [0,T]}\mathrm{adapted}\mathrm{random}\mathrm{process}:\mathbb{E}{\int }_0^T{\parallel z\left(t\right)\parallel }_{\mathbb{X}}^p\mathrm{d}t<\infty \right\}$$

endowed with the norm

$${\parallel z\parallel }_{L_a^p(0,T;\mathbb{X})}={\left(\mathbb{E}{\int }_0^T{\parallel z\left(t\right)\parallel }_{\mathbb{X}}^p\mathrm{d}t\right)}^{{\displaystyle \frac{1}{p}}}.$$

For $p=+\infty$, denote

$$L_a^{\infty }(0,T;\mathbb{X})=\left\{z\mathrm{is}\mathrm{a}\mathbb{X}-\mathrm{valued}{\mathcal{F}}_{t\in [0,T]}\mathrm{adapted}\mathrm{random}\mathrm{process}:z\mathrm{has}\mathrm{essentially}\mathrm{bounded}\sec \mathrm{ond}\mathrm{moments}\right\}$$

and

$${\parallel z\parallel }_{L_a^{\infty }(0,T;\mathbb{X})}=\underset{t\in [0,T]}{esssup}{\left({\mathbb{E}\parallel z\left(t\right)\parallel }_{\mathbb{X}}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

It is well-known that $(L_a^p(0,T;\mathbb{X}),\parallel ·{\parallel }_{L_a^p(0,T;\mathbb{X})})$ and $(L_a^{\infty }(0,T;\mathbb{X}),\parallel ·{\parallel }_{L_a^{\infty }(0,T;\mathbb{X})})$ are Banach spaces. Throughout this paper, $\mathbb{X}$ denotes $B_0\left(I\right)$, $L^2\left(I\right)$, or $L^{\theta +2}\left(I\right)$.

For notational convenience, we denote by $L_0^2(S,L^2\left(I\right))$ the class of ${\mathcal{F}}_0$-measurable $L^2\left(I\right)$-valued random variables with finite second moments. Lastly, we set $C_T^1\left([0,T]\right)$ to be

$$C_T^1[0,T]=\{z\in C^1[0,T]\mathrm{with}z\mathrm{vanished}\mathrm{at}T\}.$$

3. Existence and Uniqueness of Weak Solutions

In this section, we employ the Galerkin method to prove the existence and uniqueness of weak solutions to initial-boundary value problem (1). We make the following assumptions:

(A1)

$\alpha \in L^{\infty }\left(I\right)$, and there exists a constant ${\alpha }_0>0$ such that

$$\alpha \left(x\right)\ge {\alpha }_0,\mathrm{a}.\mathrm{e}.x\in I.$$

(A2)

$\beta \in L_a^2(0,T;L^2\left(I\right))$.

(A3)

For each $t\in [0,T]$, $h\left(t\right)\in \mathcal{L}(U,L^2\left(I\right))$ is $\left\{{\mathcal{F}}_t\right\}$-adapted and uniformly bounded; namely, there exists a constant $h_0>0$ such that

$${\parallel h\left(t\right)\parallel }_{\mathcal{L}(U,L^2\left(I\right))}\le h_0,t\in [0,T].$$

Since $TrQ<\infty$, by (A3) and Lemma 3, we have

$${\parallel h\left(t\right)\parallel }_{L_0^2(U,L^2\left(I\right))}^2=\sum _{k=1}^{\infty }{\lambda }_k\parallel h\left(t\right){\iota }_k{\parallel }_{L^2\left(I\right)}^2\le {\parallel h\left(t\right)\parallel }_{\mathcal{L}(U,L^2\left(I\right))}^{2}TrQ\le h_0^{2}TrQ.$$

Hence, $h\in L_a^2\left(0,T;L_0^2(U,L^2\left(I\right))\right),$ and therefore the stochastic integral ${\int }_0^{t}h\left(\tau \right)\mathrm{d}W\left(\tau \right)$ is well-defined in $L^2\left(I\right)$.

Definition 1. 

A stochastic process $z\in L_a^2\left(0,T;B_0\left(I\right)\right)\cap L_a^{\theta +2}\left(0,T;L^{\theta +2}\left(I\right)\right)\cap L_a^{\infty }(0,T;L^2\left(I\right))$ is called a weak solution to initial-boundary value problem (1) if z satisfies

(1)

$z\left(0\right)=\varphi$, $\mathbb{P}-$a.s.;

(2)

For any $\hslash \in C_T^1[0,T]$ and any $ð\in B_0\left(I\right)$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -{\left(\varphi ,\hslash \left(0\right)ð\right)}_{L^2\left(I\right)}-{\int }_0^T{\left(z\left(t\right),{\hslash }^{\prime }\left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & +{\int }_0^T{⟨{(-\Delta )}^{\gamma }z\left(t\right),ð⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t+{\int }_0^T{\left({\alpha \left|z\left(t\right)\right|}^{\theta }z\left(t\right),\hslash \left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & ={\int }_0^T{\left(\alpha z\left(t\right),\hslash \left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t+{\int }_0^T{\left(\beta \left(t\right),\hslash \left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t+\hfill \\ \hfill & +{\int }_0^T{\left(\hslash \left(t\right)ð,h\left(t\right)\mathrm{d}W\left(t\right)\right)}_{L^2\left(I\right)},\mathbb{P}-\mathrm{a}.\mathrm{s}..\hfill \end{array}\end{array}$$

(8)

From Lemma 2, we deduce that if $n>2\gamma$ and $0<\theta <{\displaystyle \frac{4\gamma }{n-2\gamma }}$, then

$$B_0\left(I\right)\hookrightarrow L^{\theta +2}\left(I\right)\hookrightarrow L^2\left(I\right)\mathrm{and}{B}_0\left(I\right)\hookrightarrow \hookrightarrow L^{\theta +2}\left(I\right).$$

Next, we present the existence and uniqueness theorem of weak solutions for problem (1).

Theorem 1. 

Let $\gamma \in (0,1]$, $n>2\gamma$, and $0<\theta <{\displaystyle \frac{4\gamma }{n-2\gamma }}$. Provided that (A1), (A2), and (A3) are satisfied, then for any $\varphi \in L_0^2(S,L^2\left(I\right))$, there exists a unique weak solution to initial-boundary value problem (1) in $L_a^2\left(0,T;B_0\left(I\right)\right)\cap L_a^{\theta +2}\left(0,T;L^{\theta +2}\left(I\right)\right)\cap L_a^{\infty }(0,T;L^2\left(I\right))$.

Proof. 

We divide the proof into four steps.

Step 1. A priori estimate.

Taking the inner product of the equation

$$\mathrm{d}z+{(-\Delta )}^{\gamma }z\mathrm{d}t+\alpha {\left|z\right|}^{\theta }z\mathrm{d}t=\alpha z\mathrm{d}t+\beta \left(t\right)\mathrm{d}t+h\left(t\right)\mathrm{d}W\left(t\right)$$

with z and then integrating by parts, by (3), we get

$$\begin{array}{cc}\hfill & {\mathbb{E}\parallel z\left(t\right)\parallel }_{L^2\left(I\right)}^2+C(n,\gamma )\mathbb{E}{\int }_0^t{\parallel z\left(\tau \right)\parallel }_{B_0\left(I\right)}^2\mathrm{d}\tau +2\mathbb{E}{\int }_0^t{\left(\alpha {\left|z\left(\tau \right)\right|}^{\theta }z\left(\tau \right),z\left(\tau \right)\right)}_{L^2\left(I\right)}\mathrm{d}\tau \hfill \\ \hfill =& {\mathbb{E}\parallel \varphi \parallel }_{L^2\left(I\right)}^2+2\mathbb{E}{\int }_0^t{\left(\alpha z\left(\tau \right),z\left(\tau \right)\right)}_{L^2\left(I\right)}\mathrm{d}\tau +2\mathbb{E}{\int }_0^t{\left(\beta \left(\tau \right),z\left(\tau \right)\right)}_{L^2\left(I\right)}\mathrm{d}\tau \hfill \\ & +2\mathbb{E}{\int }_0^t\left(z\left(\tau \right),h\left(\tau \right)\mathrm{d}W\left(\tau \right)\right)\hfill \end{array}$$

(9)

W is a U-valued Q-Wiener process with $TrQ<+\infty$ and (A3), deduced from the fact that

$$\begin{array}{c}\hfill \mathbb{E}{\left({\int }_0^t\left(h^{*}\left(\tau \right)z\left(\tau \right),\mathrm{d}W\left(\tau \right)\right)\right)}^2=\mathbb{E}{\int }_0^t{\left(Q{h}^{*}\left(\tau \right)z\left(\tau \right),h^{*}\left(\tau \right)z\left(\tau \right)\right)}_U\mathrm{d}\tau <\infty \end{array}$$

Hence, ${\int }_0^t(h^{*}\left(\tau \right)y\left(\tau \right),\mathrm{d}W\left(\tau \right))$ is a square-integrable martingale. In particular, its expectation is zero, that is,

$$\mathbb{E}{\int }_0^t(z\left(\tau \right),h\left(\tau \right)\mathrm{d}W\left(\tau \right))=0.$$

By (A2) and Young’s inequality,

$$2{(\beta \left(t\right),z\left(t\right))}_{L^2\left(I\right)}\le {\parallel \beta \left(t\right)\parallel }_{L^2\left(I\right)}^2+{\parallel z\left(t\right)\parallel }_{L^2\left(I\right)}^2.$$

Then, using (A1) and (9),

$$\begin{array}{cc}\hfill & {\mathbb{E}\parallel z\left(t\right)\parallel }_{L^2\left(I\right)}^2+C(n,\gamma )\mathbb{E}{\int }_0^t{\parallel z\left(\tau \right)\parallel }_{B_0\left(I\right)}^2\mathrm{d}\tau +2{\alpha }_0\mathbb{E}{\int }_0^t{\parallel z\left(\tau \right)\parallel }_{L^{\theta +2}\left(I\right)}^{\theta +2}\mathrm{d}\tau \hfill \\ \hfill \le & {\mathbb{E}\parallel \varphi \parallel }_{L^2\left(I\right)}^2+\left({2\parallel \alpha \parallel }_{L^{\infty }\left(I\right)}+1\right)\mathbb{E}{\int }_0^t{\parallel z\left(\tau \right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}\tau +\mathbb{E}{\int }_0^t{\parallel \beta \left(\tau \right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}\tau \hfill \end{array}$$

(10)

As $\varphi \in L_0^2(S,L^2\left(I\right))$, $\alpha \in L^{\infty }\left(I\right)$, and $\beta \in L_a^2(0,T;L^2\left(I\right))$, by Gronwall’s inequality, there exists a constant $C_1>0$, such that

$$\underset{t\in [0,T]}{sup}\mathbb{E}{\parallel z\left(t\right)\parallel }_{L^2\left(I\right)}^2\le C_1.$$

(11)

That is, $z\in L_a^{\infty }(0,T;L^2\left(I\right)).$ Substituting (11) into (10), we further obtain

$$\mathbb{E}{\int }_0^T{\parallel z\left(\tau \right)\parallel }_{B_0\left(I\right)}^2\mathrm{d}\tau +\mathbb{E}{\int }_0^T{\parallel z\left(\tau \right)\parallel }_{L^{\theta +2}\left(I\right)}^{\theta +2}\mathrm{d}\tau \le C_2,$$

where $C_2>0.$ Therefore, $z\in L_a^{\infty }(0,T;L^2\left(I\right))\cap L_a^2(0,T;B_0\left(I\right))\cap L_a^{\theta +2}(0,T;L^{\theta +2}\left(I\right)).$

Step 2. Galerkin approximation.

As in (2), ${\left\{e_k\right\}}_{k\ge 1}$ is a basis of the Gelfand triple $\left\{B_0\left(I\right),L^2\left(I\right),{\left\{B_0\left(I\right)\right\}}^{*}\right\}$ orthogonal in $B_0\left(I\right)$ and ${\left\{B_0\left(I\right)\right\}}^{*}$ and orthonormal in $L^2\left(I\right)$. Set $X_m:=\mathrm{span}\{e_1,\dots ,e_m\},$ and denote by $P_m:L^2\left(I\right)\to X_m$ the orthogonal projection. As in (5), $\left\{{\iota }_k\right\}$ is a complete orthonormal basis of U. Denote

$$W_m\left(t\right)=\sum _{k=1}^m{\left(W\left(t\right),{\iota }_k\right)}_U{\iota }_k,\forall t\in [0,T].$$

We look for an approximate solution in the form $z_m\left(t\right)={\sum }_{k=1}^m{g}_{k,m}\left(t\right)e_k.$ The coefficients $g_{k,m}$ are defined by the finite-dimensional stochastic system

$$\left\{\begin{array}{cc}\hfill & {(\mathrm{d}{z}_m\left(t\right),e_k)}_{L^2\left(I\right)}+{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_k⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\mathrm{d}t\hfill \\ \hfill =& {(\alpha z_m\left(t\right),e_k)}_{L^2\left(I\right)}\mathrm{d}t-\left(\alpha \right|z_m\left(t\right){{|}^{\theta }{z}_m\left(t\right),e_k)}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & +{(\beta \left(t\right),e_k)}_{L^2\left(I\right)}\mathrm{d}t+{(h^{*}\left(t\right)e_k,\mathrm{d}{W}_m\left(t\right))}_U,k=1,\dots ,m,\hfill \\ \hfill & z_m\left(0\right)=P_m\varphi =\sum _{k=1}^m{(\varphi ,e_k)}_{L^2\left(I\right)}{e}_k.\hfill \end{array}\right.$$

(12)

As $X_m$ is finite-dimensional and the map $r\mapsto {\left|r\right|}^{\theta }r$ is continuous and monotone on $\mathbb{R}$, while the diffusion coefficient is square-integrable, the above system admits a unique local strong solution up to an explosion time. Next, we establish uniform a priori estimates for the approximate solutions $\left\{z_m\right\}$.

As in Step 1, by applying Itô’s formula to (12), we obtain

$$\begin{array}{cc}\hfill \mathrm{d}\parallel z_m{\left(t\right)\parallel }_{L^2\left(I\right)}^2& =-2{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),z_m\left(t\right)⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\mathrm{d}t+2{(\alpha z_m\left(t\right),z_m\left(t\right))}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & -2\left(\alpha \right|z_m\left(t\right){{|}^{\theta }{z}_m\left(t\right),z_m\left(t\right))}_{L^2\left(I\right)}\mathrm{d}t+2{(\beta \left(t\right),z_m\left(t\right))}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & +\parallel P_m{h\left(t\right)\parallel }_{L_2^0(U,L^2\left(I\right))}^2\mathrm{d}t+2{(z_m\left(t\right),h\left(t\right)\mathrm{d}{W}_m\left(t\right))}_{L^2\left(I\right)}.\hfill \end{array}$$

(13)

As $P_m$ is an orthogonal projection in $L^2\left(I\right)$, then $\parallel P_m{h\left(t\right)\parallel }_{L_0^2(U,L^2\left(I\right))}\le {\parallel h\left(t\right)\parallel }_{L_0^2(U,L^2\left(I\right))}.$ Therefore, from Lemma 1, (A1), Young’s inequality, and (13),

$$\begin{array}{cc}\hfill & \mathrm{d}\parallel z_m{\left(t\right)\parallel }_{L^2\left(I\right)}^2+C(n,\gamma )\parallel z_m{\left(t\right)\parallel }_{B_0\left(I\right)}^2\mathrm{d}t+2{\alpha }_0{\parallel z_m\left(t\right)\parallel }_{L^{\theta +2}\left(I\right)}^{\theta +2}\mathrm{d}t\hfill \\ \hfill \le & {(2\parallel \alpha \parallel }_{L^{\infty }\left(I\right)}+1)\parallel z_m{\left(t\right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}t+{\parallel \beta \left(t\right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}t\hfill \\ \hfill & +{\parallel h\left(t\right)\parallel }_{L_0^2(U,L^2\left(I\right))}^2\mathrm{d}t+2{(z_m\left(t\right),h\left(t\right)\mathrm{d}{W}_m\left(t\right))}_{L^2\left(I\right)}.\hfill \end{array}$$

Integrating over $[0,t]$ and taking the expectation, we get

$$\begin{array}{cc}\hfill \mathbb{E}\parallel z_m{\left(t\right)\parallel }_{L^2\left(I\right)}^2& +C(n,\gamma )\mathbb{E}{\int }_0^t\parallel z_m{\left(\tau \right)\parallel }_{B_0\left(I\right)}^2\mathrm{d}\tau +2{\alpha }_0\mathbb{E}{\int }_0^t{\parallel z_m\left(\tau \right)\parallel }_{L^{\theta +2}\left(I\right)}^{\theta +2}\mathrm{d}\tau \hfill \\ \hfill & \le \mathbb{E}\parallel P_m{\varphi \parallel }_{L^2\left(I\right)}^2+{(2\parallel \alpha \parallel }_{L^{\infty }\left(I\right)}+1){\int }_0^t\mathbb{E}{\parallel z_m\left(\tau \right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}\tau \hfill \\ \hfill & +\mathbb{E}{\int }_0^t{\parallel \beta \left(\tau \right)\parallel }_{L^2\left(I\right)}^2\mathrm{d}\tau +\mathbb{E}{\int }_0^t{\parallel h\left(\tau \right)\parallel }_{L_0^2(U,L^2\left(I\right))}^2\mathrm{d}\tau .\hfill \end{array}$$

(14)

As $P_m$ is contractive in $L^2\left(I\right)$, we have $\mathbb{E}\parallel P_m{\varphi \parallel }_{L^2\left(I\right)}^2\le \mathbb{E}{\parallel \varphi \parallel }_{L^2\left(I\right)}^2.$ Hence, by Gronwall’s inequality, there exists a constant $C_3>0$, independent of m, such that

$$\underset{t\in [0,T]}{sup}\mathbb{E}{\parallel z_m\left(t\right)\parallel }_{L^2\left(I\right)}^2\le C_3.$$

(15)

Substituting (15) into (14), we further obtain a constant $C_4>0$, independent of m, such that

$$\mathbb{E}{\int }_0^T\parallel z_m{\left(\tau \right)\parallel }_{B_0\left(I\right)}^{2}d\tau +\mathbb{E}{\int }_0^T{\parallel z_m\left(\tau \right)\parallel }_{L^{\theta +2}\left(I\right)}^{\theta +2}d\tau \le C_4.$$

Therefore, $\left\{z_m\right\}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}_a^{\infty }(0,T;L^2\left(I\right))\cap L_a^2(0,T;B_0\left(I\right))\cap L_a^{\theta +2}(0,T;L^{\theta +2}\left(I\right)).$ Hence, up to a subsequence, still denoted by $\left\{z_m\right\}$, there exist $z\in L_a^{\infty }(0,T;L^2\left(I\right))\cap L_a^2(0,T;B_0\left(I\right))\cap L_a^{\theta +2}(0,T;L^{\theta +2}\left(I\right))$ such that

$$\begin{array}{cc}\hfill z_m& \rightharpoonup z\mathrm{weakly}\mathrm{in}{L}_a^2(0,T;B_0\left(I\right)),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill z_m& \stackrel{*}{\rightharpoonup }z\mathrm{weakly-}*\mathrm{in}{L}_a^{\infty }(0,T;L^2\left(I\right)),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill z_m& \rightharpoonup z\mathrm{weakly}\mathrm{in}{L}_a^{\theta +2}(0,T;L^{\theta +2}\left(I\right)).\hfill \end{array}$$

(18)

Furthermore, since $B_0\left(I\right)\hookrightarrow \hookrightarrow L^2\left(I\right)$ compactly and the continuous embedding $L^2\left(I\right)\hookrightarrow B_0^{*}\left(I\right),$ the Aubin–Lions compactness argument yields, up to a further subsequence,

$$z_m\to z\mathrm{strongly}\mathrm{in}{L}_a^2(0,T;L^2\left(I\right)).$$

(19)

In particular, taking a subsequence if necessary,

$$z_m(t,x,\omega )\to z(t,x,\omega )\mathrm{for}\mathrm{a}.\mathrm{e}.(t,x,\omega )\in (0,T)\times I\times \mathsf{\Omega }.$$

(20)

Next, we prove that z is a weak solution to initial-boundary value problem (1).

Step 3. Passage to the limit.

Now, we take the limit in the Galerkin formulation to show that z is a weak solution to problem (1). To this end, it suffices to show that z satisfies the requirements of Definition 1. For each fixed $\hslash \in C_T^1[0,T]$ and $i\in \left\{1,2,\dots ,m\right\}$, choosing as a test function $\eta \in L^{\infty }\left(S\right)$ any arbitrary $\mathcal{F}$ measurable random variable, multiplying (12) by $\eta$, integrating over $(0,T)$, using integration by parts in time, and taking the expectation, we obtain

$$\begin{array}{cc}\hfill 0=& \mathbb{E}\left\{\eta {(P_m\varphi ,\hslash \left(0\right)e_i)}_{L^2\left(I\right)}\right\}\hfill \\ \hfill & +\mathbb{E}\left\{\eta {\int }_0^T{(z_m\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill & -\mathbb{E}\left\{\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right\}\hfill \\ & -\mathbb{E}\left\{\eta {\int }_0^T\left(\alpha \right|z_m\left(t\right){{|}^{\theta }{z}_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill & +\mathbb{E}\left\{\eta {\int }_0^T{(\alpha z_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill & +\mathbb{E}\left\{\eta {\int }_0^T{(\beta \left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill & +\mathbb{E}\left\{\eta {\int }_0^T{(\hslash \left(t\right)e_i,h\left(t\right)\mathrm{d}{W}_m\left(t\right))}_{L^2\left(I\right)}\right\}\hfill \\ \hfill :=& E_1+E_2-E_3-E_4+E_5+E_6+E_7.\hfill \end{array}$$

(21)

Note that for any $t\in [0,T]$, $\hslash \left(t\right)e_i$ is a deterministic function. Next, we let $m\to \infty$ in Equation (21) and compute the corresponding limits.

Starting with the term $E_1=\mathbb{E}\left\{\eta {(P_m\varphi ,\hslash \left(0\right)e_i)}_{L^2\left(I\right)}\right\}$, by (19),

$$P_m\varphi \to \varphi ,\mathrm{strongly}\mathrm{in}{L}_0^2(S,L^2\left(I\right)).$$

Therefore,

$$\mathbb{E}\left\{\eta {(P_m\varphi ,\hslash \left(0\right)e_i)}_{L^2\left(I\right)}\right\}⟼\mathbb{E}\left\{\eta {(\varphi ,\hslash \left(0\right)e_i)}_{L^2\left(I\right)}\right\},m\to \infty .$$

As for $E_2=\mathbb{E}\left\{\eta {\int }_0^T{(z_m\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}$, the Hölder inequality yields the bound

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{\eta {\int }_0^T{(z_m\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill \le & {\parallel \eta \parallel }_{L^{\infty }\left(S\right)}{\int }_0^T(\mathbb{E}\parallel z_m{\left(t\right)\parallel }_{L^2\left(I\right)}^2{)}^{{\displaystyle \frac{1}{2}}}(\mathbb{E}\parallel {\hslash }^{\prime }\left(t\right)e_i{{\parallel }_{L^2\left(I\right)}^2)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}t\hfill \\ \hfill \le & c_1{\parallel z_m\parallel }_{L_a^{\infty }(0,T;L^2\left(I\right))}\hfill \end{array}$$

with $c_1=c_1{(\parallel \eta \parallel }_{L^{\infty }\left(S\right)},\parallel {\hslash }^{\prime }{\parallel }_{C[0,T]},T)>0$. Consequently, by (17) and the Dominated Convergence Theorem, we arrive at

$$\mathbb{E}\left\{\eta {\int }_0^T{(z_m\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}⟶\mathbb{E}\left\{\eta {\int }_0^T{(z\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\},m\to \infty .$$

Next, we turn to $E_3=\mathbb{E}\left\{\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right\}$. By (4) and (16), as $m\to \infty$,

$$\begin{array}{cc}\hfill & {\int }_0^T{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\hfill \\ \hfill =& {\displaystyle \frac{C(n,\gamma )}{2}}{\int }_0^T\underset{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\int \int }{\displaystyle \frac{(z_m(t,x)-z_m(t,y))(\hslash \left(t\right)e_i\left(x\right)-\hslash \left(t\right)e_i\left(y\right))}{{|x-y|}^{n+2\gamma }}}\mathrm{d}x\mathrm{d}y\mathrm{d}t\hfill \\ \hfill \to & {\displaystyle \frac{C(n,\gamma )}{2}}{\int }_0^T\underset{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\int \int }{\displaystyle \frac{(z(t,x)-z(t,y))(\hslash \left(t\right)e_i\left(x\right)-\hslash \left(t\right)e_i\left(y\right))}{{|x-y|}^{n+2\gamma }}}\mathrm{d}x\mathrm{d}y\mathrm{d}t.\hfill \end{array}$$

Hence, the term is bounded by

$$\mathbb{E}\left\{\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right\}\le c_2{\parallel \eta \parallel }_{L^{\infty }\left(S\right)},$$

where $c_2=c_2(c_0,\parallel z_m{\parallel }_{B_0\left(I\right)},T)>0$. From this estimate, it follows immediately that

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }{z}_m\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right\}\hfill \\ \hfill ⟶& \mathbb{E}\left\{\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }z\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right\}.\hfill \end{array}$$

In view of $E_4=\mathbb{E}\left\{\eta {\int }_0^T\left(\alpha \right|z_m\left(t\right){{|}^{\theta }{z}_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}$, from (18), we deduce that

$${\left|z_m\right|}^{\theta }{z}_m\rightharpoonup {\left|z_m\right|}^{\theta }{z}_m,\mathrm{weakly}\mathrm{in}{L}_a^{{\displaystyle \frac{\theta +2}{\theta +1}}}(0,T;L^{{\displaystyle \frac{\theta +2}{\theta +1}}}\left(I\right)).$$

Assumption (A1) ensures the convergence

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{\eta {\int }_0^T\left(\alpha \right|z_m\left(t\right){{|}^{\theta }{z}_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}\hfill \\ \hfill ⟶& \mathbb{E}\left\{\eta {\int }_0^T{\left(\alpha \right|z\left(t\right)|}^{\theta }z\left(t\right),\hslash \left(t\right)e_i{)}_{L^2\left(I\right)}\mathrm{d}t\right\}.\hfill \end{array}$$

Referring to $E_5=\mathbb{E}\left\{\eta {\int }_0^T{(\alpha z_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}$, by (A1) and (19), we deduce that

$$\begin{array}{c}\hfill \mathbb{E}\left\{\eta {\int }_0^T{(\alpha z_m\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}⟶\mathbb{E}\left\{\eta {\int }_0^T{(\alpha z\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}.\end{array}$$

Notice that the term $E_6=\mathbb{E}\left\{\eta {\int }_0^T{(\beta \left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right\}$ is independent of m. Lastly, we consider $E_7=\mathbb{E}\left\{\eta {\int }_0^T{(\hslash \left(t\right)e_i,h\left(t\right)\mathrm{d}{W}_m\left(t\right))}_{L^2\left(I\right)}\right\}$. The definition of $W_m$ as the m-dimensional truncation of W and Lemma 4 imply that

$$W_m\to W,m\to \infty ,\mathrm{in}\mathrm{U}.$$

By virtue of assumption (A3), $\mathbb{E}{\int }_0^T{\parallel h{\left(t\right)}^{*}(\hslash \left(t\right)e_i)\parallel }_Q^{2}dt<\infty .$ We deduce the convergence of the stochastic integral as follows:

$$\mathbb{E}\left\{\eta {\int }_0^T{(\hslash \left(t\right)e_i,h\left(t\right)\mathrm{d}{W}_m\left(t\right))}_{L^2\left(I\right)}\right\}⟶\mathbb{E}\left\{\eta {\int }_0^T{(\hslash \left(t\right)e_i,h\left(t\right)\mathrm{d}W\left(t\right))}_{L^2\left(I\right)}\right\}.$$

Taking the limit in (21), we conclude that for every $\hslash \in C_T^1[0,T]$, every $i\in \mathbb{N}$, and every $\eta \in L^{\infty }\left(\mathsf{\Omega }\right)$,

$$\begin{array}{cc}\hfill 0=& \mathbb{E}\left[\eta {(\varphi ,\hslash \left(0\right)e_i)}_{L^2\left(I\right)}\right]+\mathbb{E}\left[\eta {\int }_0^T{(z\left(t\right),{\hslash }^{\prime }\left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right]\hfill \\ \hfill & -\mathbb{E}\left[\eta {\int }_0^T{⟨{(-\Delta )}^{\gamma }z\left(t\right),e_i⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hslash \left(t\right)\mathrm{d}t\right]\hfill \\ \hfill & -\mathbb{E}\left[\eta {\int }_0^T{\left(\alpha \right|z\left(t\right)|}^{\theta }z\left(t\right),\hslash \left(t\right)e_i{)}_{L^2\left(I\right)}\mathrm{d}t\right]+\mathbb{E}\left[\eta {\int }_0^T{(\alpha z\left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right]\hfill \\ \hfill & +\mathbb{E}\left[\eta {\int }_0^T{(\beta \left(t\right),\hslash \left(t\right)e_i)}_{L^2\left(I\right)}\mathrm{d}t\right]+\mathbb{E}\left[\eta {\int }_0^T{(\hslash \left(t\right)e_i,h\left(t\right)\mathrm{d}W\left(t\right))}_{L^2\left(I\right)}\right].\hfill \end{array}$$

As ${\left\{e_i\right\}}_{i\ge 1}$ is complete in $B_0\left(I\right)$, by linearity and density, this identity holds with $e_i$ replaced by any $ð\in B_0\left(I\right)$. Therefore, if we denote by $X\left(\omega \right)$ the quantity

$$\begin{array}{cc}\hfill X:=& -{\left(\varphi ,\hslash \left(0\right)ð\right)}_{L^2\left(I\right)}-{\int }_0^T{\left(z\left(t\right),{\hslash }^{\prime }\left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & +{\int }_0^T{⟨{(-\Delta )}^{\gamma }z\left(t\right),ð⟩}_{B_0^{*},B_0}\hslash \left(t\right)\mathrm{d}t+{\int }_0^T\left({\alpha \left|z\left(t\right)\right|}^{\theta }z\left(t\right),\hslash \left(t\right)ð\right)\mathrm{d}t\hfill \\ \hfill & -{\int }_0^T\left(\alpha z\left(t\right),\hslash \left(t\right)ð\right)\mathrm{d}t-{\int }_0^T{\left(\beta \left(t\right),\hslash \left(t\right)ð\right)}_{L^2\left(I\right)}\mathrm{d}t\hfill \\ \hfill & -{\int }_0^T{\left(\hslash \left(t\right)ð,h\left(t\right)\mathrm{d}W\left(t\right)\right)}_{L^2\left(I\right)},\hfill \end{array}$$

then $\mathbb{E}\left(\eta X\right)=0$ for all $\eta \in L^{\infty }\left(\mathsf{\Omega }\right)$, which implies that $X=0$ $\mathbb{P}$-a.s. Thus, z satisfies (8). Moreover, $z\left(0\right)=\varphi$ follows from the Galerkin initial condition and the convergence of $P_m\varphi$ to $\varphi$. Therefore, z is a weak solution of (1).

Step 4. Uniqueness.

Let $z_1$ and $z_2$ be two weak solutions of (1) with the same initial datum $\varphi$, and set $y:=z_1-z_2.$ Then, y satisfies

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}+{(-\Delta )}^{\gamma }y=\alpha y-\alpha \left(|z_1{|}^{\theta }{z}_1-{\left|z_2\right|}^{\theta }{z}_2\right),y\left(0\right)=0,$$

in the weak sense. Taking the $L^2\left(I\right)$ inner product with $y\left(t\right)$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel y\left(t\right)\parallel }_{L^2\left(I\right)}^2+{⟨{(-\Delta )}^{\gamma }y\left(t\right),y\left(t\right)⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}\hfill \\ \hfill =& {(\alpha y\left(t\right),y\left(t\right))}_{L^2\left(I\right)}-{\left(\alpha \left(\right|z_1\left(t\right){|}^{\theta }{z}_1\left(t\right)-\left|z_2\left(t\right){|}^{\theta }{z}_2\left(t\right)\right),y\left(t\right)\right)}_{L^2\left(I\right)}.\hfill \end{array}$$

By Lemma 1, ${⟨{(-\Delta )}^{\gamma }y,y⟩}_{{\left\{B_0\left(I\right)\right\}}^{*},B_0\left(I\right)}={\displaystyle \frac{1}{2}}C(n,\gamma ){\parallel y\parallel }_{B_0\left(I\right)}^2\ge 0.$ Since the map $r\mapsto {\left|r\right|}^{\theta }r$ is monotone on $\mathbb{R}$ and $\alpha \left(x\right)\ge {\alpha }_0>0$, a.e. on I,

$${\left(\alpha \left(\right|z_1{|}^{\theta }{z}_1-|z_2{|}^{\theta }{z}_2),z_1-z_2\right)}_{L^2\left(I\right)}\ge 0.$$

Therefore, ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\parallel y\left(t\right)\parallel }_{L^2\left(I\right)}^2\le {2\parallel \alpha \parallel }_{L^{\infty }\left(I\right)}{\parallel y\left(t\right)\parallel }_{L^2\left(I\right)}^2.$ As $y\left(0\right)=0$, Gronwall’s inequality yields

$${\parallel y\left(t\right)\parallel }_{L^2\left(I\right)}^2=0,\forall t\in [0,T],\mathbb{P}-\mathrm{a}.\mathrm{s}.$$

Hence, $z_1=z_2$ almost surely on $[0,T]\times I$.

Combining Steps 1–4, we conclude that problem (1) admits a unique weak solution $z\in L_a^2(0,T;B_0\left(I\right))\cap L_a^{\theta +2}(0,T;L^{\theta +2}\left(I\right))\cap L_a^{\infty }(0,T;L^2\left(I\right)).$ This completes the proof. □

4. Conclusions

In this study, we investigate the well-posedness of the stochastic fractional Ginzburg–Landau equation. First, we introduced a framework for weak solutions within adapted solution spaces. The existence of these solutions was established using a Galerkin-type approximation and the Dominated Convergence Theorem, supported by detailed energy estimates and compactness arguments. Furthermore, the uniqueness of the weak solution was confirmed through the application of the monotonicity of nonlinear terms and Gronwall’s inequality, ensuring the stability of the system.

The results presented in this work contribute to the theoretical understanding of nonlocal stochastic partial differential equations (SPDEs). Future work could extend these results to investigate the long-term asymptotic behavior, such as the existence of random attractors, or explore optimal control problems for this fractional stochastic system.