The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

Abstract

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

1. Introduction

We consider the following Cauchy problem of the stochastic nonlinear fractional Schrödinger equation:

$$\left\{\begin{array}{cc}idu-({(-\Delta )}^{s}u+\lambda {\left|u\right|}^{2\sigma }u)dt=g\left(u\right)\circ d\mathcal{W},\hfill & x\in {\mathbb{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(1)

where $u_0\left(x\right)\in H^{\alpha }$ is non-radial, $s\in (0,1)\backslash \left\{\frac{1}{2}\right\}$, $\lambda =\pm 1$, $\sigma >0$, and ∘ denotes the Stratonovitch product. The variable $g\left(u\right)$ is a nonlinear complex-valued function. And the assumptions related to $g\left(u\right)$ are given later. The fractional Laplacian operator ${(-\Delta )}^s$ is defined as

$${(-\Delta )}^{s}u={\mathcal{F}}^{-1}\left({\left|\xi \right|}^{2s}\mathcal{F}\left(u\right)\right).$$

To state the noise term precisely, we introduce a probability space $(\mathsf{\Omega },\mathbb{F},\mathbb{P})$, which depends on a filtration ${\left({\mathbb{F}}_t\right)}_{t\ge 0}$ and a sequence of independent real-valued Brownian motions ${\left({\mathcal{B}}_{\mathcal{k}}\right)}_{\mathcal{k}\in \mathbb{N}}$. The random process

$$\mathcal{W}(t,x)={\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\mathcal{B}}_{\mathcal{k}}\left(t\right)\varphi e_{\mathcal{k}}\left(x\right),\mathcal{k}\in \mathbb{N}$$

is a Winner process in the space of square-integrable functions, where linear operator $\varphi$ is bounded on $L^2$, and ${\left(e_{\mathcal{k}}\right)}_{\mathcal{k}\in \mathbb{N}}$ is an Hilbertian basis on $L^2$. The equivalent Itô equation for Problem (1) is presented by

$$\left\{\begin{array}{cc}idu-({(-\Delta )}^{s}u+\lambda {\left|u\right|}^{2\sigma }u)dt=g\left(u\right)d\mathcal{W}\hfill & -\frac{i}{2}{g}^{\prime }\left(u\right)g\left(u\right)F_{\varphi }dt,\hfill \\ & x\in {\mathbb{R}}^n,t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(2)

where

$$F_{\varphi }={\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\left(\varphi e_{\mathcal{k}}\left(x\right)\right)}^2.$$

The mild solution to Problem (2) can be formulated as the following:

$$\begin{array}{cc}\hfill u\left(t\right)& =S\left(t\right)u_0-i\lambda {\int }_0^{t}S(t-t^{\prime }){\left|u\right|}^{2\sigma }ud{t}^{\prime }+i{\int }_0^{t}S(t-t^{\prime })g\left(u\right)d\mathcal{W}\left(t^{\prime }\right)\hfill \\ & -\frac{1}{2}{\int }_0^{t}S(t-t^{\prime })g^{\prime }\left(u\right)g\left(u\right)F_{\varphi }d{t}^{\prime }.\hfill \end{array}$$

(3)

The last term of (3),

$$\mathsf{\Psi }\left(t\right):=i{\int }_0^{t}S(t-t^{\prime })g\left(u\right)d\mathcal{W}\left(t^{\prime }\right)$$

is called stochastic convolution, where $S\left(t\right)$ denotes the linear fractional Schrödinger propagator.

In recent years, fractional differential equations have been widely investigated for applications in physics and other fields (see [1,2,3,4,5,6]). For example, Laskin [5,6] introduced a deterministic fractional Schrödinger equation:

$$\begin{array}{c}\hfill i{\partial }_{t}u-({(-\Delta )}^{s}u+\lambda {\left|u\right|}^{2\sigma }u)=0.\end{array}$$

(4)

For more detailed results of Equation (4), one can see [7,8,9,10,11,12,13,14]. In order to describe the results of Equation (4) more clearly, we first recall the notion of scaling-critical for Equation (4), which is based on the dilation symmetry

$$u(t,x)⟶u_{\mu }(t,x)={\mu }^{-\frac{s}{\sigma }}u\left({\mu }^{-2s}t,{\mu }^{-1}x\right)$$

for $\mu >0$. It is clear that if u is a solution to Equation (4), then $u_{\mu }$ is also a solution to Equation (4) with the corresponding rescaled initial value. In addition, the scaling-critical Sobolev regularity

$$s_{crit}=\frac{n}{2}-\frac{s}{\sigma }$$

is obtained by using the dilation symmetry, which makes the homogeneous ${\dot{H}}^{s_{crit}}\left({\mathbb{R}}^n\right)$-norm invariant. And $s_{crit}$ works as a threshold law for the well-posedness and ill-posedness of Equation (4). Local well-posedness in $H^{\alpha }$ for the Cauchy problem of Equation (4) has been investigated in the sub-critical case ($\alpha \ge max(0,s_{crit})$) (see [10,13]). For the critical case $\alpha =s_{crit}$, Dinh [10] discussed the local existence and uniqueness of solutions to the Cauchy problem. Furthermore, Hong and Sire [13] showed ill-posedness for the Cauchy problem in scaling the super-critical regime ($\alpha <s_{crit}$). In this paper, we consider the well-posedness for Problem (1) in the sub-critical case.

Although the propagation of waves is often described by deterministic models, spatial and temporal fluctuation of parameters of the medium have to be taken into account in some circumstances. This often occurs through a random potential or by describing the propagation of dispersive waves in non-homogeneous or random media. So it is natural to introduce stochastic Schrödinger equations. In [15,16], de Bouard and Debussche investigated the Cauchy problem of the following equation:

$$\begin{array}{cc}i{u}_t-(\Delta u+\lambda {\left|u\right|}^{2\sigma }u)=u\circ d\mathcal{W},\hfill & x\in {\mathbb{R}}^n,t>0.\hfill \end{array}$$

(5)

From deterministic Strichartz estimates, de Bouard and Debussche established that the stochastic convolution almost surely belongs to a right Strichartz space, which allowed them to show local well-posedness in $L^2$ and $H^1$, respectively. In [15], the mass conversation for the deterministic equation and Itô’s lemma were used to construct a global solution in $L^2$ almost surely. The authors also proved that the global well-posedness in $H^1$ follows from an a priori $H^1$ -bound based on the conservation of the energy for the deterministic classical Schrödinger equation and Itô’s lemma in [16]. Furthermore, Barbu et al. [17,18] discussed the effects of a finite dimensional Winner process on the local and global well-posedness. They introduced the scaling transformation $u=e^{-i\mathcal{W}}y$ such that they could apply the fixed-point argument related to the deterministic equation and Strichartz estimates of the evolution operator $A\left(s\right):=i(\Delta +b(s)\nabla +c(s\left)\right)$. However, the drawback is this method only works on the finite dimensional Wiener process. For nonlinear multiplicative noise, Ondreját [19] studied the pathwise uniqueness and norm continuity of the local solution to the stochastic wave equation. Brźniak et al. [20] proved the existence and uniqueness of the global solution on a compact n-dimensional Riemannian manifold. Furthermore, Fabian [21] considered local well-posedness for the classical Schrödinger equation with a nonlinear perturbation term in $L^2$ by utilizing deterministic and stochastic Strichartz estimates; then, the global solution was given based on the uniform bound of u. Yang and Chen [22] showed the existence of martingale solutions for the stochastic nonlinear fractional Schrödinger equation on a bounded interval. We refer readers to references [23,24,25,26,27,28,29,30]. for more details.

Inspired by [10,15,16,19,20,21,22], we consider the Cauchy problem of the nonlinear fractional Schrödinger equation with a nonlinear noise term in this paper. Compared to the known results about Equation (5), we overcome two difficulties in this work: one is that the noise term is nonlinear, and the another one is the loss of derivatives. Under more suitable assumptions regarding the noise term, we established estimates related to the stochastic convolution, which allows us to discuss the local well-posedness. Furthermore, effects of the loss of derivatives can be compensated for by using the Sobolev embedding. Based on deterministic non-radial Strichartz estimates and the estimates associated with the stochastic convolution, we prove local existence and uniqueness of the mild solution (see Theorem 1). In order to study the existence of the global solution, our main tool is Itô’s lemma. More precisely, we investigate the generalizations of the mass and energy in random settings via Itô’s formula. Then, with the help of stopping time arguments, we show the global well-posedness in $H^{\alpha }$ almost surely (see Theorem 2). The new contribution in this work is that we show the existence of a non-radial global solution under fractional-order derivatives and a nonlinear noise term.

To better understand the main results of this paper, we first introduce some notations.

(1)

$\lceil \alpha \rceil$ is the smallest positive integer greater than or equal to $\alpha$.

(2)

Throughout this paper, we call $(r,p)$ is an admissible pair if it satisfies

$$(r,p)\in {[2,\infty ]}^2,{\displaystyle \frac{2}{r}}={\displaystyle \frac{n}{2}}-{\displaystyle \frac{n}{p}},(r,p,n)\ne (2,\infty ,2).$$

(6)

In addition, for $(r,p)\in {[1,\infty ]}^2$, we set

$${\gamma }_{r,p}={\displaystyle \frac{n}{2}}-{\displaystyle \frac{n}{p}}-{\displaystyle \frac{2s}{r}}.$$

(7)

The inequality ${\gamma }_{r,p}>0$ holds true for all admissible pairs except for the case of $(r,p)=(\infty ,2)$.

(3)

Let $\tilde{g}:[0,\infty )⟶R$ be a function of class $C^1$ such that

$$g\left(u\right)=\tilde{g}\left({\left|u\right|}^2\right)u.$$

Furthermore, there exists $g^{\prime }\left(u\right)g\left(u\right)={\left(\tilde{g}\left({\left|u\right|}^2\right)\right)}^{2}u$ and $g^{\prime }\left(u\right)g\left(u\right)\overline{u}={\left|g\left(u\right)\right|}^2$. For any admissible pair $(r,p)$, there exists a probability space

$$V:=L^{\rho }(\mathsf{\Omega };C(0,\tau ;H^{\gamma })\cap L^r(0,\tau ;W^{\beta ,p})),$$

where $\rho \ge r$, $\gamma =max\{s,\alpha \}$ and $\beta =\alpha -{\gamma }_{r,p}$. We assume that $g\left(u\right):V\to V$ is a continuous mapping. For any $u_1$, $u_2\in V$, it satisfies

$$\begin{array}{ccc}& & \left(i\right)g\left(0\right)=0,\hfill \\ & & \left(ii\right)\parallel g\left(u_1\right)-g\left(u_2\right){\parallel }_V\le C_1{\parallel u_1-u_2\parallel }_V,\hfill \\ & & \left(iii\right)\parallel g^{\prime }\left(u_1\right)g\left(u\right)-g^{\prime }\left(u_2\right)g\left(u_2\right){\parallel }_V\le C_2{\parallel u_1-u_2\parallel }_V,\hfill \end{array}$$

where $C_1$ and $C_2$ are two positive constants, and $\gamma =max\{s,\alpha \}$.

The main results of this paper can be stated as the following:

Theorem 1.

Assume that, $0<\sigma <\frac{2s}{n-2\alpha }.$ and $\varphi \in R(L^2;W^{\alpha ,\infty })$. Let $\sigma \ge \frac{\lceil \alpha \rceil }{2}$. If $u_0$ is a ${\mathbb{F}}_0$ -measurable random variable in $H^{\alpha }$, then Problem (1) has a unique solution

$$u\in C(0,\tau ;H^{\alpha })\cap L^r(0,\tau ;W^{\beta ,p})\mathbb{P}-a.s.,$$

where $0<\tau <{\tau }^{*}\left(u_0\right)$, $\beta =\alpha -{\gamma }_{r,p}$, and

$$r>max(2\sigma ,6).$$

(8)

Moreover, there is an alternative

$$\begin{array}{cc}\hfill & \left(i\right)either{\tau }^{*}\left(u_0\right)=\infty almostsurely,\hfill \\ \hfill & \left(ii\right)or{\tau }^{*}\left(u_0\right)<\infty and\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,t;H^{\alpha }))}=\infty \mathrm{if}{\tau }^{*}\left(u_0\right)<\infty .\hfill \end{array}$$

Theorem 2.

Let $n\le 4$. Given $s\ge \alpha$ and $\varphi \in R(L^2;W^{s,\infty })$. If $\sigma <\frac{2s}{n}$ or $\lambda =-1$, then the solution u of Problem (1) given by Theorem 1 is almost surely global.

This paper is organized as follows. Section 2 introduces some notations and inequalities. The proof of local well-posedness for Problem (1) is given in Section 3. In Section 4, we present the proof of the existence of a global solution to Problem (1).

2. Preliminaries

In this section, we introduce some notations and estimates.

We first introduce some notations.

(1)

Given $\beta \in \mathbb{R}$ and $1\le p\le \infty$, the non-homogeneous Sobolev space is defined by

$$W^{\beta ,p}:=\{u\in W^{\beta ,p}{,\parallel u\parallel }_{W^{\beta ,p}}:=\parallel {\mathsf{\Lambda }}^{\beta }u{\parallel }_{L^p}<\infty \},\mathsf{\Lambda }:=\sqrt{1+(-▵)}.$$

And the definition of the homogeneous Sobolev space is given by

$${\dot{W}}^{\beta ,p}:=\{u\in {\dot{W}}^{\beta ,p}{,\parallel u\parallel }_{{\dot{W}}^{\beta ,p}}:={\parallel |-\nabla |}^{\beta }u{\parallel }_{L^p}<\infty \}.$$

(2)

For $r,p,\rho \ge 1$, the space of random function $u(t,x,\omega )$ in $L^r(0,T;W^{\beta ,p})$ is denoted by $L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))$ such that

$${\parallel u\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))}:={\left(\mathbb{E}(\parallel u{\parallel }_{L^r(0,T;W^{\beta ,p})}^{\rho })\right)}^{\frac{1}{\rho }}<\infty .$$

(3)

We review the definition of space $R(L^2;W^{\alpha ,\infty })$, which is a space of Hilbert–Schmidt operator $\varphi$ from $L^2$ into $W^{\alpha ,\infty }$. The norm of operator $\varphi$ in $R(L^2;W^{\alpha ,\infty })$ is defined as

$${\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}=\parallel {\displaystyle \sum _{\mathcal{k}=1}^{\infty }}\varphi e_{\mathcal{k}}{\parallel }_{W^{\alpha ,\infty }},$$

where ${\left(e_{\mathcal{k}}\right)}_{\mathcal{k}\in \mathbb{N}}$ is the orthonormal basis on $L^2$.

We next introduce some necessary estimates.

In the deterministic case, the Cauchy problem of deterministic fractional Schrödinger equation

$$\left\{\begin{array}{cc}i{\partial }_{t}u-{(-\Delta )}^{s}u=F(x,t),\hfill & (x,t)\in {\mathbb{R}}^n\times {\mathbb{R}}^{+},\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(9)

exists a solution

$$u(x,t)=S\left(t\right)u_0-i{\int }_0^{t}S(t-s)F\left(s\right)ds,$$

where $S\left(t\right):=e^{-it{(-\Delta )}^s}$ satisfies the following lemma.

Lemma 1

(Deterministic non-radial Strichartz estimates [13]).  Given $u_0\in L^2$ and $F\in L^r(\mathbb{R};L^p)$, then for all $(r,p)$ and $(a,b)$ admissible pairs, the group $S\left(t\right)=e^{-it{(-\Delta )}^s}$ ($t\in \mathbb{R}$) satisfies

$$\parallel S\left(t\right)u_0{\parallel }_{L^r(\mathbb{R};L^p)}\le {C\parallel |-\nabla |}^{{\gamma }_{r,p}}{u}_0{\parallel }_{L^2}$$

and

$$\parallel {\int }_0^t{S(t-s)F\left(s\right)ds\parallel }_{L^r(\mathbb{R};L^p)}\le {C\parallel |-\nabla |}^{{\gamma }_{r,p}-{\gamma }_{a^{\prime },b^{\prime }}-2s}{F\parallel }_{L^{a^{\prime }}(\mathbb{R};L^{b^{\prime }})},$$

where $\frac{1}{a}+\frac{1}{a^{\prime }}=\frac{1}{b}+\frac{1}{b^{\prime }}=1$.

We recall the following lemmas that are needed for the proof of local well-posedness.

Lemma 2

(Nonlinear estimate [10]).  Let $f\left(u\right)={\left|u\right|}^{2\sigma }u$ with $\sigma >0$. Given $1<p,q<\infty$ and $1<m\le \infty$. Assume that

$${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{q}}+{\displaystyle \frac{2\sigma }{m}}$$

holds.

If $0\le \lceil \alpha \rceil <2\sigma$, there exists $C=C(n,\sigma ,\alpha ,p,q,m)$ such that for $u,v\in L^m\cap W^{\alpha ,p}$, we have

$${\parallel f\left(u\right)-f\left(v\right)\parallel }_{W^{\alpha ,p}}\le C{(\parallel u\parallel }_{L^m}^{2\sigma }+\parallel v{\parallel }_{L^m}^{2\sigma }){\parallel u-v\parallel }_{W^{\alpha ,q}}+C{(\parallel u\parallel }_{L^m}^{2\sigma -1}+\parallel v{\parallel }_{L^m}^{2\sigma -1}){(\parallel u\parallel }_{W^{\alpha ,q}}+\parallel v{\parallel }_{W^{\alpha ,q}}){\parallel u-v\parallel }_{L^m}.$$

In order to estimate the stochastic integral, we introduce the following lemmas.

Lemma 3

(Bürkholder inequality [31]).  Given $g\in L^2(0,\infty ;L^p)$. Assume that ${\int }_0^{t}g\left(s\right)d\mathcal{B}\left(s\right)$ $(t\ge 0)$ is an $L^p$ -valued square-integrable martingale with continuous modification and zero mean. If $\rho \in [2,\infty )$, then there exists a constant C such that

$$\mathbb{E}(\underset{0\le t\le T}{sup}\parallel {\int }_0^t{g\left(s\right)d\mathcal{B}\left(s\right)\parallel }_{L^p}^{\rho })\le C\mathbb{E}\left[\right({\int }_0^T{\parallel g\left(s\right)\parallel }_{L^p}^{2}ds{)}^{\frac{\rho }{2}}],$$

where C is independent of T and g.

3. Local Well-Posedness

3.1. Estimations

In this subsection, we introduce some estimations that are important for proving Theorem 1. First, we need to estimate the stochastic convolution

$$\mathsf{\Psi }\left(t\right)=i{\int }_0^{t}S(t-t^{\prime })g\left(u\right)d\mathcal{W}\left(s\right)$$

in $L^{\rho }(\mathsf{\Omega };C(0,\tau ;H^{\alpha })\cap L^r(0,\tau ;W^{\beta ,p}))$.

Lemma 4.

Let the conditions for r and q be the same as in Theorem 1. If $T>0$, $\rho \ge r$, and $u\in L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))$, it holds that $\mathsf{\Psi }\left(t\right)\in L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))$ and

$$\begin{array}{c}\hfill {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))}\le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}{T}^{\frac{1}{2}-\frac{2}{r}-\frac{1}{\rho }}{\parallel u\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}.\end{array}$$

Proof. 

In an application of the Hölder’s inequality and $\rho \ge r$, we have

$$\begin{array}{cc}\hfill & {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))}^{\rho }\hfill \\ & \le T^{\frac{\rho }{r}-1}\mathbb{E}\left({\int }_0^T{\parallel {\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^{t}S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)d{\mathcal{B}}_{\mathcal{k}}\parallel }_{L^p}^{\rho }dt\right)\hfill \\ & \le T^{\frac{\rho }{r}-1}\mathbb{E}\left(\underset{0\le t\le T}{sup}{\parallel {\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^{t}S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)d{\mathcal{B}}_{\mathcal{k}}\parallel }_{L^p}^{\rho }\right).\hfill \end{array}$$

(10)

Employing the Bürkholder inequality yields

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{0\le t\le T}{sup}{\parallel {\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^{t}S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)d{\mathcal{B}}_{\mathcal{k}}\parallel }_{L^p}^{\rho }\right)\hfill \\ & \le C\mathbb{E}\left({\left({\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^T{\parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\parallel }_{L^p}^{2}d{t}^{\prime }\right)}^{\frac{\rho }{2}}\right).\hfill \end{array}$$

(11)

According to the decay estimate in [13]:

$${\parallel S\left(t\right)f\parallel }_{L^p}\le c{\left|t\right|}^{-\frac{n}{2}(1-\frac{2}{p})}{\parallel |-\nabla |}^{{\gamma }_{r,p}}{f\parallel }_{L^{p^{\prime }}},$$

we have

$$\begin{array}{cc}\hfill \parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right){\parallel }_{L^p}& \le c{|t-t^{\prime }|}^{-\frac{n}{2}(1-\frac{2}{p})}\parallel {\mathsf{\Lambda }}^{\alpha }\left(g\left(u\right)\right)\varphi e_{\mathcal{k}}{)\parallel }_{L^{p^{\prime }}},\hfill \end{array}$$

where $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. Application of the Hölder’s inequality and $\frac{1}{p}-\frac{\beta }{n}\le \frac{1}{p^{\prime }}-\frac{\alpha }{n}$ implies that

$$\begin{array}{cc}\hfill & \parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right){\parallel }_{L^p}\hfill \\ & \le C{|t-t^{\prime }|}^{-\frac{2}{r}}(\parallel {\mathsf{\Lambda }}^{\alpha }g\left(u\right){\parallel }_{L^{p^{\prime }}}\parallel \varphi e_{\mathcal{k}}{\parallel }_{L^{\infty }}+\parallel g\left(u\right){\parallel }_{L^{p^{\prime }}}\parallel {\mathsf{\Lambda }}^{\alpha }\varphi e_{\mathcal{k}}{\parallel }_{L^{\infty }})\hfill \\ & \le C{|t-t^{\prime }|}^{-\frac{2}{r}}\parallel g\left(u\right){\parallel }_{W^{\beta ,p}}{\parallel \varphi e_{\mathcal{k}}\parallel }_{W^{\alpha ,\infty }}.\hfill \end{array}$$

(12)

Combining (11) and (12), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\left({\left({\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^T{\parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\beta }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\parallel }_{L^p}^{2}d{t}^{\prime }\right)}^{\frac{\rho }{2}}\right)\hfill \\ & \le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}^{\rho }{T}^{\rho (\frac{1}{2}-\frac{3}{r})}{\mathbb{E}(\parallel u\parallel }_{L^r(0,\tau ;W^{\beta ,p})}^{\rho }).\hfill \end{array}$$

(13)

Then, substituting (13) into (10), for $\rho \ge r>max(2\sigma ,6)$, we have

$$\begin{array}{cc}\hfill & {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\alpha ,p}))}^{\rho }\le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}^{\rho }{T}^{\rho (\frac{1}{2}-\frac{2}{r}-\frac{1}{\rho })}{\parallel u\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}^{\rho }\hfill \end{array}$$

which complements the proof of the lemma. □

The estimate about $\mathsf{\Psi }\left(t\right)$ on $L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))$ is presented as follows:

Lemma 5.

Suppose that the conditions for r and q are the same as in Theorem 1. Let $T>0$ and $\rho \ge r$. If $u\in L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))$, we have $\mathsf{\Psi }\left(t\right)\in L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))$ and

$$\begin{array}{c}\hfill {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}\le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}{T}^{\frac{1}{2}-\frac{1}{r}}{\parallel u\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T;W^{\beta ,p}))}.\end{array}$$

(14)

Proof. 

Applying the Bürkholder inequality, we get

$$\begin{array}{cc}\hfill & {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}^{\rho }\hfill \\ & \le C\mathbb{E}\left({\left({\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^T{\parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\alpha }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\parallel }_{L^2}^{2}d{t}^{\prime }\right)}^{\frac{\rho }{2}}\right).\hfill \end{array}$$

(15)

Furthermore, employing the fact that linear operator $S\left(t\right)$ is isometrically isomorphic on $L^2$ and the Hölder’s inequality leads to

$$\begin{array}{cc}\hfill & \mathbb{E}\left({\left({\displaystyle \sum _{\mathcal{k}=0}^{\infty }}{\int }_0^T{\parallel S(t-t^{\prime }){\mathsf{\Lambda }}^{\alpha }\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\parallel }_{L^2}^{2}d{t}^{\prime }\right)}^{\frac{\rho }{2}}\right)\hfill \\ & \le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}^{\rho }\mathbb{E}\left({\left({\int }_0^T{\parallel g\left(u\right)\parallel }_{H^{\alpha }}^{2}d{t}^{\prime }\right)}^{\frac{\rho }{2}}\right).\hfill \end{array}$$

(16)

Thanks to $\frac{1}{p}-\frac{\beta }{n}\le \frac{1}{2}-\frac{\alpha }{n}$, according to the Sobolev embedding theorem, we obtain $W^{\beta ,p}\subset H^{\alpha }$. Then, substituting (16) into (15), we get

$$\begin{array}{cc}\hfill & {\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}^{\rho }\le {C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}^{\rho }{T}^{\rho (\frac{1}{2}-\frac{1}{r})}{\mathbb{E}(\parallel u\parallel }_{L^r(0,T;W^{\beta ,p})}^{\rho }).\hfill \end{array}$$

We derive the conclusion (14). □

Moreover, the following lemmas are necessary for later processing.

Lemma 6

([15]). Let $u_1$, $u_2\in Y_T$, and $u_3\in L^r(\mathsf{\Omega };L^1(0,t_1^R;H^{\alpha }))$. Then, we have

$$\begin{array}{c}\hfill \parallel ({\theta }_R\left(u_1\right)-{\theta }_R\left(u_2\right))u_3{\parallel }_{L^r(\mathsf{\Omega };L^1(0,t_1^R;H^{\alpha }))}\le c\parallel u_1-u_2{\parallel }_{Y_T}{\parallel u_3\parallel }_{L^r(\mathsf{\Omega };L^1(0,t_1^R;H^{\alpha }))},\end{array}$$

for some positive constant c.

Lemma 7.

Suppose that the stopping time τ satisfies $\tau \le {\tau }^{*}\left(u_0\right)$ and $\mathsf{\Psi }\left(t\right)\in L^r$ $(\mathsf{\Omega };L^r(0,\tau ;W^{\beta ,p}))$. Given a solution to Problem (1), $u\in L^{\infty }(0,\tau ;H^{\alpha })$ almost surely. Then, there exists

$$\begin{array}{c}\hfill h\left(\omega \right)=C_0{(1+\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,\tau ;H^{\alpha }))}^{2\sigma +1}+{\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^r(\mathsf{\Omega };L^r(0,\tau ;W^{\beta ,p}))}),\end{array}$$

such that

$$\begin{array}{c}\hfill {\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}\le C\left({\tau }^{*}\left(u_0\right)\right)h{\left(\omega \right)}^{\frac{2\sigma }{\lambda }+1}.\end{array}$$

Proof. 

For $0<T_1\le \tau$, according to the Young’s inequality and Lemmas 4–5, we obtain

$$\begin{array}{cc}\hfill & {\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}\hfill \\ \hfill & \le C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}+C{T}_1^{1-\frac{2\sigma }{r}}(\frac{1}{2\sigma +1}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,T_1;H^{\alpha }))}^{2\sigma +1}+\frac{2\sigma }{2\sigma +1}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}^{2\sigma +1})\hfill \\ & +{C\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}^2{T}_1^{1-\frac{1}{r}}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}+{\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}.\hfill \end{array}$$

Let $T_1\le T_2$ and $C{T}_2^{1-\frac{1}{r}}{\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,q})}^2\le {\displaystyle \frac{1}{2}}$. If there exists a large enough

$$C_0:=C(\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })},{\tau }^{*}\left(u_0\right),r,\sigma )$$

then

$${\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}\le h\left(\omega \right)+C{T}_1^{\lambda }{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}^{2\sigma +1}$$

where $\lambda =1-\frac{2\sigma }{r}$ and

$$\begin{array}{c}\hfill h\left(\omega \right)=C_0{(1+\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,\tau ;H^{\alpha }))}^{2\sigma +1}+{\parallel \mathsf{\Psi }\left(t\right)\parallel }_{L^r(\mathsf{\Omega };L^r(0,\tau ;W^{\beta ,p}))}).\end{array}$$

If we choose $T_1=inf\{\tau ,T^{\prime }\}$ with

$$T^{\prime }=2^{\frac{-1}{\lambda }}h{\left(\omega \right)}^{\frac{-2\sigma }{\lambda }},$$

it follows that

$${\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}\le 2h\left(\omega \right).$$

We divide the interval $[0,\tau ]$ into many sub-intervals: namely,

$$[0,\tau ]={\displaystyle \sum _{j=0}^{\frac{\tau \bigwedge {\tau }^{*}\left(u_0\right)}{T_1}}}[j{T}_1,(j+1)T_1].$$

Then, we have

$${\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(j{T}_1,(j+1)T_1;W^{\beta ,p}))}\le 2h\left(\omega \right).$$

Hence, we arrive at

$$\begin{array}{cc}\hfill & {\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_1;W^{\beta ,p}))}\hfill \\ & \le \frac{\tau \wedge {\tau }^{*}\left(u_0\right)}{T_1}h\left(\omega \right)\hfill \\ & \le C\left({\tau }^{*}\left(u_0\right)\right)h{\left(\omega \right)}^{\frac{2\sigma }{\lambda }+1}.\hfill \end{array}$$

(17)

□

3.2. Proof of Theorem 1

Now, we give the proof of Theorem 1. Denote

$$X_t=C(0,t;H^{\alpha })\cap L^r(0,t;W^{\beta ,p}).$$

We consider $\theta \in C^{\infty }$ as a cut-off function on ${\mathbb{R}}^{+}$, and

$$\theta \left(u\right)=\theta \left(\frac{{\parallel u\parallel }_{Y_t}}{R}\right):=\left\{\begin{array}{cc}1,\hfill & r\in [0,R],\hfill \\ 0,\hfill & r\in [2R,\infty ),\hfill \end{array}\right.$$

(18)

where $R\in {\mathbb{N}}^{+}$ and $Y_t:=L^r(\mathsf{\Omega };X_t)$.

Since the nonlinear term of Problem (1) is local Lipschitz, we introduce the following Cauchy problem of a truncated equation

$$\left\{\begin{array}{cc}id{u}_R-({(-\Delta )}^s{u}_R+\lambda \theta \left(u_R\right){\left|u_R\right|}^{2\sigma }{u}_R)dt=\theta \left(u_R\right)g\left(u_R\right)\circ d\mathcal{W},\hfill & x\in {\mathbb{R}}^n,t>0,\hfill \\ u_R(x,0)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(19)

The mild solution to Problem (19) is

$$\begin{array}{cc}\hfill u_R\left(t\right)=& S\left(t\right)u_0-i\lambda {\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right){\left|u_R\right|}^{2\sigma }ud{t}^{\prime }\hfill \\ & -i{\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right)g\left(u_R\right)d\mathcal{W}\left(t^{\prime }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right)g^{\prime }\left(u_R\right)g\left(u_R\right)F_{\varphi }d{t}^{\prime }.\hfill \end{array}$$

(20)

Define a mapping as following

$$\begin{array}{cc}\hfill {\psi }_R{u}_R=& S\left(t\right)u_0-i\lambda {\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right){\left|u_R\right|}^{2\sigma }ud{t}^{\prime }\hfill \\ & -i{\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right)g\left(u_R\right)d\mathcal{W}\left(t^{\prime }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^{t}S(t-t^{\prime })\theta \left(u_R\right)g^{\prime }\left(u_R\right)g\left(u_R\right)F_{\varphi }d{t}^{\prime }.\hfill \end{array}.$$

Step 1: We show that for a fixed $R>0$, there exists $T>0$ depending only on R such that Problem (8) has a unique solution $u_R$ that satisfies $u_R\in X_{t^{\prime }}$ almost surely.

We consider the space

$$E(T_R,R):=\{u_R\in Y_{T_R}:\parallel u_R{\parallel }_{Y_{T_R}}\le R\}$$

with the following metric:

$$d(u_R,v_R):=\parallel u_R-v_R{\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,T_R;H^{\alpha }))}+{\parallel u_R-v_R\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}.$$

We need to prove that ${\psi }_R$ is a contraction on $E(T_R,R)$ if $T_R$ depends on R.

Owing to $r>max(2\sigma ,6)$ and ${\gamma }_{r,p}>0$, it holds that $\alpha >\frac{n}{2}-\frac{2s}{r}$. Combining (6) and (7), we can see $\frac{1}{p}-\frac{\alpha -{\gamma }_{r,p}}{n}<0.$ Then, the Sobolev embedding implies

$$W^{\beta ,p}\subset L^{\infty }.$$

For any $u_R\in E(T_R,R)$, using Strichartz estimates, $W^{\beta ,p}\subset L^{\infty }$ and Lemmas 4–5, we get

$$\begin{array}{cc}\hfill & \parallel {\psi }_R{u}_R{\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}\hfill \\ & \le C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}+C(\mathbb{E}\parallel {\left|u_R\right|}^{2\sigma }{u}_R{\parallel }_{L^1(0,T_R;H^{\alpha })}^r{)}^{\frac{1}{r}}+{\parallel \mathsf{\Psi }(t)\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}\hfill \\ \hfill & +C(\mathbb{E}\parallel g^{\prime }(u_R)g(u_R)F_{\varphi }{{\parallel }_{L^1(0,T_R;H^{\alpha })}^r)}^{\frac{1}{r}}\hfill \\ \hfill & \le C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}+C{T}_R^{1-\frac{2\sigma }{r}}(\mathbb{E}\parallel u_R{\parallel }_{L^r(0,T;L^{\infty })}^{2\sigma r}\parallel u_R{{\parallel }_{L^{\infty }(0,T_R;H^{\alpha })}^r)}^{\frac{1}{r}}\hfill \\ \hfill & +{C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})T_R^{\frac{1}{2}-\frac{3}{r}}(\mathbb{E}\parallel u_R{{\parallel }_{L^{\infty }(0,T_R;H^{\alpha })}^r)}^{\frac{1}{r}}\hfill \\ & +{C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})T_R^{1-\frac{1}{r}}(\mathbb{E}\parallel u_R{{\parallel }_{L^r(0,T_R;W^{{\mathsf{\Lambda }}^{\beta },p})}^r)}^{\frac{1}{r}}\hfill \\ \hfill & \le C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}+{C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})T_R^{\lambda }{R}^{2\sigma +1},\hfill \end{array}$$

(21)

where $\lambda =min\{1-\frac{2\sigma }{r},\frac{1}{2}-\frac{3}{r}\}$. In a similar way, we obtain

$$\begin{array}{c}\hfill \parallel {\psi }_R{u}_R{\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}\le C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}+{C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})T_R^{\lambda }{R}^{2\sigma +1}.\end{array}$$

Fix $C\parallel u_0{\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}=\frac{R}{2}$ such that

$$2^{2\sigma +1}{C}^{2\sigma +1}{T}_R^{\lambda }{\parallel v_0\parallel }_{L^r(\mathsf{\Omega };H^{\alpha })}^{2\sigma +1}<1,$$

then, we obtain that ${\psi }_R$ is well-defined on $E(T_R,R)$ and $T_R\lesssim R^{-\frac{2\sigma }{\lambda }}$.

Taking $u_{R1}$, $u_{R2}$ $\in E(T_R,R)$, it is clear that

$$\begin{array}{cc}\hfill d({\psi }_R{u}_{R1},{\psi }_R{u}_{R2})& =\parallel {\psi }_R{u}_{R1}-{\psi }_R{u}_{R2}{\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,T_R;H^{\alpha }))}\hfill \\ & +\parallel {\psi }_R{u}_{R1}-{\psi }_R{u}_{R2}{\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}.\hfill \end{array}$$

(22)

By the Strichartz estimates, it follows that

$$\begin{array}{cc}\hfill & \parallel {\psi }_R{u}_{R1}-{\psi }_R{u}_{R2}{\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}\hfill \\ & \le C\parallel (\theta \left(u_{R1}\right){\left|u_{R1}\right|}^{2\sigma }{u}_{R1}-\theta \left(u_{R2}\right){\left|u_{R2}\right|}^{2\sigma }{u}_{R2}){\parallel }_{L^r(\mathsf{\Omega };L^1(0,T_R;H^{\alpha }))}\hfill \\ & +\parallel {\int }_0^{t}S(t-t^{\prime })\left(\theta \left(u_{R1}\right)g\left(u_{R1}\right)-\theta \left(u_{R2}\right)g\left(u_{R2}\right)\right)d\mathcal{W}\left(t^{\prime }\right){\parallel }_{L^r(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}\hfill \\ & +\frac{1}{2}{\parallel (\theta \left(u_{R1}\right)g^{\prime }\left(u_{R1}\right)g\left(u_{R1}\right)-\theta \left(u_{R2}\right)g^{\prime }\left(u_{R2}\right)g\left(u_{R2}\right))F_{\varphi }\parallel }_{L^r(\mathsf{\Omega };L^{r^{\prime }}(0,T_R;W^{{\mathsf{\Lambda }}^{\beta },p^{\prime }}))}\hfill \\ & =I+II+III.\hfill \end{array}$$

For the purpose of estimating I for $\ell =1,2$, we set

$$t_{\mathcal{l}}^R=sup\{t\in [0,T_R],\parallel u_{R\ell }{\parallel }_{X_t}\le 2R\},$$

where

$$[0,T_R]=[0,t_1^R]\cup [t_1^R,t_2^R]\cup [t_2^R,T_0].$$

Without loss of generality, we assume $t_1^R\le t_2^R$ such that

$$\begin{array}{cc}\hfill I& \le \parallel (\theta \left(u_{R1}\right)-\theta \left(u_{R2}\right)){\left|u_{R1}\right|}^{2\sigma }{u}_{R1}{\parallel }_{L^r(\mathsf{\Omega };L^1(0,t_1^R;H^{\alpha }))}\hfill \\ & +\parallel \theta \left(u_{R2}\right)({\left|u_{R1}\right|}^{2\sigma }{u}_{R1}-{\left|u_{R2}\right|}^{2\sigma }{u}_{R2}){\parallel }_{L^r(\mathsf{\Omega };L^1(0,t_1^R;H^{\alpha }))}\hfill \\ & +\parallel \theta \left(u_{R2}\right){\left|u_{R2}\right|}^{2\sigma }{u}_{R2}{\parallel }_{L^r(\mathsf{\Omega };L^1(t_1^R,t_2^R;H^{\alpha }))}\hfill \\ & =I_1+I_2+I_3.\hfill \end{array}$$

(23)

To estimate $I_1$, we recall the following lemma from [15].

According to Lemma 6, it is easy to get

$$\begin{array}{c}\hfill I_1\le C{T}_R^{1-\frac{2\sigma }{r}}{R}^{2\sigma +1}{\parallel u_{R1}-u_{R2}\parallel }_{Y_T}.\end{array}$$

(24)

For $t\in [t_1^R,t_2^R]$, there exists $\theta \left(u_{R1}\right)=0$. Thus, we can rewrite $I_3$ as

$$\begin{array}{c}\hfill I_3={\parallel (\theta \left(u_{R2}\right)-\theta \left(u_{R1}\right)){\left|u_{R2}\right|}^{2\sigma }{u}_{R2}\parallel }_{L^r(\mathsf{\Omega };L^1(t_1^R,t_2^R;H^{\alpha }))}.\end{array}$$

Using Lemma 2, it follows that

$$\begin{array}{c}\hfill I_2\le C{T}_R^{1-\frac{2\sigma }{r}}{R}^{2\sigma }{\parallel u_{R1}-u_{R2}\parallel }_{Y_{T_R}}\mathrm{.}\end{array}$$

(25)

Similarly, we obtain

$$\begin{array}{c}\hfill I_3\le C{T}_R^{1-\frac{2\sigma }{r}}{R}^{2\sigma +1}{\parallel u_{R1}-u_{R2}\parallel }_{Y_{T_R}}\end{array}$$

(26)

Then, it follows from (23)–(26) that

$$\begin{array}{c}\hfill I\le C{T}_R^{1-\frac{2\sigma }{r}}{R}^{2\sigma +1}{\parallel u_{R1}-u_{R2}\parallel }_{Y_{T_R}}.\end{array}$$

Thanks to Hölder’s inequality, one can check that

$$\begin{array}{cc}\hfill III& \le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})T_R^{1-\frac{2}{r}}{∥u_{R1}-u_{R2}∥}_{Y_{T_R}}.\hfill \end{array}$$

Combining all the above estimates, we conclude

$$\begin{array}{c}\hfill \parallel {\psi }_R{u}_{R1}-{\psi }_R{u}_{R2}{\parallel }_{L^{\rho }(\mathsf{\Omega };L^r(0,T_R;W^{\beta ,p}))}\le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }{\parallel u_{R1}-u_{R2}\parallel }_{Y_{T_R}}.\end{array}$$

Similar to the estimate of the first term on the right-hand side of (22), we arrive at

$$\begin{array}{c}\hfill \parallel {\psi }_R{u}_{R1}-{\psi }_R{u}_{R2}{\parallel }_{L^{\rho }(\mathsf{\Omega };L^{\infty }(0,T;H^{\alpha }))}\le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }{\parallel u_{R1}-u_{R2}\parallel }_{Y_{T_R}}\end{array}$$

as long as $\rho \ge r$. Thus, we have

$$d({\psi }_R{u}_{R1}-{\psi }_R{u}_{R2})\le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }d(u_{R1}-u_{R2}),$$

(27)

which means that ${\psi }_R$ is a contraction mapping on $E(T_R,R)$.

Step 2: We show that problem (19) has a unique solution on $[0,T_0]$.

Let $m\in {\mathbb{N}}^{+}$ and $u_R^m\in Y_{m{T}_R}.$ In order to extend $u_R^m$ to $[m{T}_R,(m+1)T_R]$, we introduce

$$u_R^{m+1}:=\left\{\begin{array}{cc}{u}_R^m\left(t\right),\hfill & t\in [0,m{T}_R],\hfill \\ y_R^m(t-m{T}_R),\hfill & t\in [m{T}_R,(m+1)T_R].\hfill \end{array}\right.$$

(28)

An application of the Duhamel formulation gives rise to

$$\begin{array}{cc}\hfill y_R^m\left(\tilde{t}\right)& =S\left(\tilde{t}\right)u\left(m{T}_R\right)-i\lambda {\int }_0^{\tilde{t}}S(\tilde{t}-s){\phi }_R^m\left(y_R^m\right){\left|y_R^m\right|}^{2\sigma }{y}_R^{m}d{t}^{\prime }\hfill \\ & -i{\int }_0^{\tilde{t}}S(t-t^{\prime }){\phi }_R^m\left(y_R^m\right)g\left(y_R^m\right)d\mathcal{W}\left(t^{\prime }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^{\tilde{t}}S(t-t^{\prime }){\phi }_R^m\left(y_R^m\right)g^{\prime }\left(y_R^m\right)g\left(y_R^m\right)F_{\varphi }d{t}^{\prime }\hfill \end{array}$$

where $\tilde{t}=t-m{T}_R$, $t\in [m{T}_R,(m+1)T_R]$, and ${\phi }_R^m\in C^{\infty }$ is a cut-off function given by

$${\phi }_R^m\left(y_{R}m\right)=\phi \left(\frac{\parallel y_{R}m{\parallel }_{Y_{\tilde{t}}}}{R}\right):=\left\{\begin{array}{cc}1,\hfill & r\in [0,R],\hfill \\ 0,\hfill & r\in [2R,\infty )\mathrm{.}\hfill \end{array}\right.$$

(29)

We define a map

$$\begin{array}{cc}\hfill {\psi }_y{y}_R^m\left(\tilde{t}\right)& =S\left(\tilde{t}\right)u\left(mT\right)-i\lambda {\int }_0^{\tilde{t}}S(\tilde{t}-s){\phi }_R^m\left(y_R^m\right){\left|y_R^m\right|}^{2\sigma }{y}_R^{m}d{t}^{\prime }\hfill \\ & -i{\int }_0^{t}S(t-t^{\prime }){\phi }_R^m\left(y_R^m\right)g\left(y_R^m\right)d\mathcal{W}\left(t^{\prime }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^{t}S(t-t^{\prime }){\phi }_R^m\left(y_R^m\right)g^{\prime }\left(y_R^m\right)g\left(y_R^m\right)F_{\varphi }d{t}^{\prime }.\hfill \end{array}$$

In the same spirit as the proof of Step 1, we obtain

$$\begin{array}{c}\hfill \parallel {\psi }_y{y}_R^m-{\psi }_y{\tilde{y}}_R^m{\parallel }_{Y_{\tilde{t}}}\le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }{\parallel y_R^m-{\tilde{y}}_R^m\parallel }_{Y_{\tilde{t}}}.\end{array}$$

(30)

The constant on the right-hand side of (30) is the same as the constant in (27), so we can conclude that the mapping ${\psi }_y$ is a contraction map on

$$\begin{array}{c}\hfill E(T_R,R):=\{u\in Y_{\tilde{t}}{,\parallel u\parallel }_{Y_{\tilde{t}}}\le R\}.\end{array}$$

From (28), we know that for $t\in [0,m{T}_R]$, there exists

$$u_R^{m+1}=u_R^m.$$

Then, we turn our attention to $t\in [m{T}_R,(m+1)T_R]$. It follows from

$$S\left(t_1\right)S\left(t_2\right)=S\left(t_2\right)S\left(t_1\right)=S(t_1+t_2)$$

that

$$\begin{array}{cc}\hfill u_R^{m+1}\left(t\right)& =y_R^m\left(\tilde{t}\right)\hfill \\ \hfill & =S\left(t\right)u_0+S\left(\tilde{t}\right)({\psi }_R{u}_{R}m-S\left(m{T}_R\right)u_0)+({\psi }_y{y}_{R}m-S\left(\tilde{t}\right)u\left(m{T}_R\right))\mathrm{.}\hfill \end{array}$$

(31)

Combining (18), (28), and (29), we can see that for $t\in [0,m{T}_R]$:

$$\begin{array}{c}\hfill {\theta }_R\left(u_R^m\right)={\theta }_R\left(u_R^{m+1}\right)\end{array}$$

(32)

and for $t\in [m{T}_R,(m+1)T_R]$:

$$\begin{array}{c}\hfill {\phi }_R^m\left(y_R^m\right)={\theta }_R\left(u_R^{m+1}\right).\end{array}$$

(33)

From (31)–(33), we obtain

$$\begin{array}{cc}\hfill & S\left(\tilde{t}\right)({\psi }_R{u}_{R}m-S\left(m{T}_R\right)u_0)+({\psi }_y{y}_{R}m-S\left(\tilde{t}\right))u\left(m{T}_R\right)\hfill \\ \hfill & =-i\lambda {\int }_0^{t}S(t-t^{\prime }){\theta }_R\left(u_R^{m+1}\right){\left|u_R^{m+1}\right|}^{2\sigma }{u}_R^{m+1}ds\hfill \\ \hfill & -\frac{1}{2}{\int }_0^{t}S(t-t^{\prime }){\theta }_R\left(u_R^{m+1}\right)g^{\prime }\left(u_R^{m+1}\right)g\left(u_R^{m+1}\right)F_{\varphi }ds\hfill \\ \hfill & -i{\int }_0^{t}S(t-t^{\prime }){\theta }_R\left(u_R^{m+1}\right){\left|u_R^{m+1}\right|}^{\frac{p_2}{2}}{u}_R^{m+1}d\mathcal{W}\left(t^{\prime }\right)\mathrm{.}\hfill \end{array}$$

(34)

Based on (28), (31) and (34), we can show that for $t\in [0,(m+1)T_R]$:

$$u_R^{m+1}\left(t\right)={\Gamma }_R{u}_R^{m+1}\left(t\right)$$

and

$$\begin{array}{cc}\hfill & \parallel {\psi }_R{u}_R^{m+1}-{\psi }_R{\tilde{u}}_R^{m+1}{\parallel }_{Y_{(m+1)T_R}}\le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }{\parallel u_R^{m+1}-{\tilde{u}}_R^{m+1}\parallel }_{Y_{(m+1)T_R}}.\hfill \end{array}$$

As a result, we conclude that ${\psi }_R$ is a contraction map in $Y_{(m+1)T_R}$. Taking $m=\left[\frac{T_0}{T_R}\right]+1$ gives rise to $u_R:=u_R^m$ on $[0,T_0]$.

Then, we need to present the uniqueness. Suppose that ${\overline{u}}_R(t,x)$ with $t\in [0,\overline{T})$ is another local solution to (19). It follows that

$$\begin{array}{cc}\hfill \parallel u_R-{\overline{u}}_R{\parallel }_{Y_{T_R\bigwedge \overline{T}}}& \le {C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })})R^{2\sigma +1}{T}_R^{\lambda }{\parallel u_R^{m+1}-{\tilde{u}}_R^{m+1}\parallel }_{Y_{T_R\bigwedge \overline{T}}}\hfill \\ \hfill & \lesssim \parallel u_R^{m+1}-{\tilde{u}}_R^{m+1}{\parallel }_{Y_{T_R\bigwedge \overline{T}}},\hfill \end{array}$$

which means that $u=\overline{u}$ almost surely for $t\in [0,T_R\wedge \overline{T})$. By iterating, it follows that $u=\overline{u}$ almost surely for $t\in [0,T_m)$, where $T_m=m{T}_R\wedge \overline{T}$. The assertion follows from $T_m=\overline{T}$ if m is large enough.

We set

$$\begin{array}{c}\hfill T_R=sup\{t\in [0,T_0]{,\parallel u\parallel }_{Y_t}\le R\}\end{array}$$

(35)

and denote

$$\begin{array}{c}\hfill u=u_R\end{array}$$

(36)

on $[0,T_R]$. Moreover, we know from [15] that $T_R$ increases with R and $u_R=u_{R+1}$ on $[0,T_R]$. Thus, there exists a local solution u of Problem (1) on $[0,{\tau }^{*}\left(u_0\right)]$ for which

$$\begin{array}{c}\hfill {\tau }^{*}\left(u_0\right)=\underset{R\to \infty }{lim}{T}_R\end{array}$$

(37)

almost surely. Then, it is obvious that

$$u\in Y_{\tau }$$

for all stopping times $\tau <{\tau }^{*}\left(u_0\right)$. Furthermore, the uniqueness of this local solution is consistent with the uniqueness of the solution to Equation (20) in $Y_{\tau }$.

Step 3: We give

$$\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}\underset{s\le t}{sup}{\parallel u\left(s\right)\parallel }_{H^{\alpha }}=\infty \mathbb{P}-a.s.$$

for ${\tau }^{*}\left(u_0\right)<\infty$.

For ${\tau }^{*}\left(u_0\right)<\infty$, set

$${\tilde{T}}_R=sup\{t\in [0,{\tau }^{*}\left(u_0\right)){,\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,t;H^{\alpha }))}\le R\}.$$

Using (17), we derive

$$\begin{array}{cc}\hfill & {\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,{\tilde{T}}_R;W^{\beta ,p}))}\hfill \\ & \le C\left({\tau }^{*}\left(u_0\right)\right)h{\left(\omega \right)}^{(\frac{2\sigma }{\lambda }+1)}\hfill \\ & \le C\left({\tau }^{*}\left(u_0\right)\right)(1+R^{2\sigma +1}+{C(\parallel \varphi \parallel }_{R(L^2;W^{\alpha ,\infty })}{\left)R\right)}^{\frac{2\sigma }{\lambda }+1}.\hfill \end{array}$$

(38)

Suppose that

$${\mathbb{P}(\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,{\tau }^{*}\left(u_0\right);H^{\alpha }))}<\infty \mathrm{and}{\tau }^{*}\left(u_0\right)<\infty )>0.$$

Thus, if R is large enough, we get

$$\mathbb{P}({\tilde{T}}_R={\tau }^{*}\left(u_0\right))>0,$$

which implies that for ${\tau }^{*}\left(u_0\right)<\infty$:

$$\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,t;H^{\alpha }))}<\infty .$$

From Equation (38), it follows that

$$\begin{array}{c}\hfill \underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^r(0,t;W^{\beta ,p}))}<\infty .\end{array}$$

Thus, we arrive at

$$\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{Y_t}<\infty .$$

(39)

However, (35)–(37) lead to that, for ${\tau }^{*}\left(u_0\right)<\infty$,

$$\begin{array}{c}\hfill \underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{Y_t}=\infty ,\end{array}$$

which contradicts (39).

Hence, we get

$$\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}{\parallel u\parallel }_{L^r(\mathsf{\Omega };L^{\infty }(0,t;H^{\alpha }))}=\infty \mathrm{if}{\tau }^{*}\left(u_0\right)<\infty .$$

The proof of Theorem 1 is now complete.

4. The Existence of a Global Solution

In this section, we study the existence of the global solution. Firstly, some conclusions about invariant quantities of the deterministic nonlinear fractional Schrödinger equation are recalled. It holds that the mass

$$M\left(u\right)={\int }_{{\mathbb{R}}^n}{\left|u\right|}^{2}dx$$

and the energy

$$Q\left(u\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^n}{\left|{(-\nabla )}^{s}u\right|}^{2}dx-\frac{\lambda }{2\sigma +2}{\int }_{{\mathbb{R}}^n}{\left|u\right|}^{2\sigma +2}dx.$$

(40)

4.1. The Stochastic Identity of $M\left(u\right)$ and $H\left(u\right)$

For later use, it is convenient to give Itô’s lemmas for $M\left(u\right)$ and $H\left(u\right)$.

Proposition 1.

The assumptions about $u_0$, σ, and ϕ are the same as in Theorem 1. If u is the solution to Problem (1), then for all stopping times τ with $\tau <{\tau }^{*}\left(u_0\right)$, there exists

$$\begin{array}{c}\hfill M\left(u\left(\tau \right)\right)=M\left(u_0\right).\end{array}$$

Proof. 

Since the nonlinear term of Problem (1) is local Lipschitz, it is natural to use a truncation argument. Thus, we consider Problem (19). Applying Itô’s lemma to $M\left(u_R\right)$, we have

$$\begin{array}{cc}\hfill M\left(u_R\left(t\right)\right)& =M\left(u_0\right)+2Im\lambda {\int }_0^t{\int }_{{\mathbb{R}}^n}\theta \left(u_R\right){\left|u_R\right|}^{2\sigma +2}dxds\hfill \\ & -Re{\int }_0^t{\int }_{{\mathbb{R}}^n}\theta \left(u_R\right)\overline{u}{g}^{\prime }\left(u_R\right)g\left(u_R\right)F_{\varphi }dxds\hfill \\ & +{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^t{\int }_{{\mathbb{R}}^n}\theta \left(u_R\right){\left|g\left(u_R\right)\right|}^2{\left|\varphi e_{\mathcal{k}}\right|}^{2}dxds\hfill \\ & +2Im{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^t{\int }_{{\mathbb{R}}^n}\theta \left(u_R\right)\overline{u}g\left(u_R\right)\left(\varphi e_{\mathcal{k}}\right)dxd{\mathcal{B}}_{\mathcal{k}}.\hfill \end{array}$$

Taking a sufficiently large R and $t=\tau$, we conclude

$$M\left(u\left(\tau \right)\right)=M\left(u_0\right).$$

□

Proposition 2.

The assumptions about $u_0$, σ, and ϕ are the same as in Theorem 1. For all stopping times τ with $\tau <{\tau }^{*}\left(u_0\right)$, there exists

$$\begin{array}{cc}\hfill Q\left(u\right(\tau \left)\right)=& Q\left(u_0\right)+Im{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}{(-\nabla )}^s\stackrel{-}{u}{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)dxd{\mathcal{B}}_{\mathcal{k}}\hfill \\ & -\frac{1}{2}Re{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}\left(g^{\prime }\left(u\right)g\left(u\right)F_{\varphi }\right){(-\Delta )}^s\stackrel{-}{u}dxds\hfill \\ & +\frac{1}{2}{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}{\left|{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\right|}^{2}dxds,\hfill \end{array}$$

(41)

where u is given by Theorem 1.

Proof. 

Applying Itô’s lemma and fractional integration by parts to $Q\left(u\right)$, we obtain

$$\begin{array}{cc}\hfill Q\left(u_R\left(t\right)\right)=& Q\left(u_0\right)+Im{\displaystyle \sum _{k=0}^{\infty }}{\int }_{{\mathbb{R}}^n}{\int }_0^t{\left(-\nabla )\right.}^s{\stackrel{-}{u}}_R{(-\nabla )}^s\left(g\left(u_R\right)\varphi e_{\mathcal{k}}\right)d{\mathcal{B}}_{\mathcal{k}}dx\hfill \\ & -Im{\int }_0^t{\int }_{{\mathbb{R}}^n}\lambda (1-\theta \left(u_R\right))\times \left({\left|u_R\right|}^{2\sigma }{u}_R\right)\left({(-\Delta )}^s{\stackrel{-}{u}}_R\right)dxds\hfill \\ & -\frac{1}{2}Re{\int }_0^t{\int }_{R^n}\left(g^{\prime }\left(u_R\right)g\left(u_R\right)F_{\varphi }\right){(-\Delta )}^s{\stackrel{-}{u}}_{R}dxds\hfill \\ & +\frac{1}{2}{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^t{\int }_{R^n}{\left|{(-\nabla )}^s\left(g\left(u_R\right)\varphi e_{\mathcal{k}}\right)\right|}^{2}dxds\hfill \\ & +\frac{\lambda }{2}{\int }_0^t{\int }_{R^n}{\left|u_R\right|}^{2\sigma }{\left|g\left(u_R\right)\right|}^2{F}_{\varphi }dxds\hfill \\ & -\frac{\lambda }{2}{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^t{\int }_{R^n}{\left|u_R\right|}^{2\sigma +2}{\left|\tilde{g}\left({\left|u_R\right|}^2\right)\right|}^2{\left(\varphi e_{\mathcal{k}}\right)}^{2}dxds\mathrm{.}\hfill \end{array}$$

Choosing a large enough R and taking $t=\tau$, we arrive at

$$\begin{array}{cc}\hfill Q\left(u\right(\tau \left)\right)=& Q\left(u_0\right)+Im{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}{(-\nabla )}^s\overline{u}{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)dxd{\mathcal{B}}_{\mathcal{k}}\hfill \\ & -\frac{1}{2}Re{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}\left(g^{\prime }\left(u\right)g\left(u\right)F_{\varphi }\right){(-\Delta )}^s\overline{u}dxds\hfill \\ & +\frac{1}{2}{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^n}{\left|{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\right|}^{2}dxds.\hfill \end{array}$$

□

4.2. Proof of Theorem 2

Thanks to the assumption $n\le 4$, we can choose $\sigma <\frac{2s}{n}$ such that $\frac{\lceil \alpha \rceil }{2}\le \sigma <\frac{2s}{n}$.

Let $u_0\in L^{2+\frac{4\sigma s}{2s-\sigma n}}(\mathsf{\Omega };L^2)\cap L^2(\mathsf{\Omega };H^{\alpha })$ and $E\left(Q\left(u_0\right)\right)$ be bounded. It is enough to prove that for ${\tau }^{*}\left(u_0\right)<\infty$, ${\parallel u\parallel }_{H^{\alpha }}$ is almost surely bounded. Denote stopping time ${\tau }^{\prime }<inf\{T_0,{\tau }^{*}\left(u_0\right)\}$, and

$${\tilde{\tau }}_R=inf\{t<{\tau }^{*}\left(u_0\right){,\parallel u\parallel }_{H^{\alpha }}\ge Ralmostsurely\}$$

for $R>0$. It is apparent for Proposition 2 that

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\left|Q\left(u\left(t\right)\right)\right|}^2\right)\hfill \\ \hfill & \le \mathbb{E}\left(\left|Q{\left(u_0\right|}^2\right)\right)+\mathbb{E}\left(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\displaystyle \sum _{\mathcal{k}=1}^{\infty }}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^n}{(-\nabla )}^s\overline{u}{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)dxd{\mathcal{B}}_{\mathcal{k}}\right|}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\mathbb{E}\left(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^n}\left(g^{\prime }\left(u\right)g\left(u\right)F_{\varphi }\right){(-\Delta )}^s\overline{u}dxds\right|}^2\right)\hfill \\ & +{\displaystyle \frac{1}{4}}\mathbb{E}\left(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^n}{\left|{(-\nabla )}^s\left(g\left(u\right)\varphi e_{\mathcal{k}}\right)\right|}^{2}dxds\right|}^2\right)\hfill \\ \hfill & =\mathbb{E}\left(\left|Q{\left(u_0\right|}^2\right)\right)+I+II+III.\hfill \end{array}$$

(42)

From Bürkholder and Hölder’s inequalities, we have

$$\begin{array}{cc}\hfill I& \le {C\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })}^2\mathbb{E}({\int }_0^t\parallel {(-\nabla )}^s\overline{u}{\parallel }_{L^2}^{4}ds).\hfill \end{array}$$

(43)

Applying Hölder’s inequality to the variable s, we get

$$\begin{array}{cc}\hfill II& \le {\displaystyle \frac{C}{4}}\parallel F_{\varphi }{\parallel }_{L^{\infty }}{T}_0\mathbb{E}({\int }_0^t{\parallel (-\nabla )}^{s}u{\parallel }_{L^2}^{4}ds).\hfill \end{array}$$

(44)

According to Proposition 1, it follows that

$$\begin{array}{cc}\hfill III& \le {C\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })}^4{T}_0\mathbb{E}({\int }_0^t{\parallel (-\nabla )}^s{u\parallel }_{L^2}^4{ds)+C\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })}^4{T}_0^2\mathbb{E}\left(M^2\left(u_0\right)\right).\hfill \end{array}$$

(45)

If we plug (43)–(45) back into (42), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\left|Q\left(u\left(t\right)\right)\right|}^2\right)\hfill \\ \hfill & \le C\left({\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })},T_0\right)\mathbb{E}({\int }_0^t\parallel {(-\nabla )}^s\overline{u}{\parallel }_{L^2}^{4}ds)+\mathbb{E}\left({\left|Q\left(u_0\right)\right|}^2\right){+C\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })}^4{T}_0^2\mathbb{E}\left(M^2\left(u_0\right)\right).\hfill \end{array}$$

(46)

In the case of $\lambda =1$, from $\sigma <\frac{2s}{n}$, the following Gagliardo–Nirenberg inequality holds (see [32]):

$${\parallel u\parallel }_{L^{2\sigma +2}}^{2\sigma +2}\le {C\parallel u\parallel }_{L^2}^{2\sigma +2-\frac{\sigma n}{s}}{\parallel (-\nabla )}^s{u\parallel }_{L^2}^{\frac{\sigma n}{s}}\le \frac{1}{4}{\parallel (-\nabla )}^s{u\parallel }_{L^2}^2+C{\parallel u\parallel }_{L^2}^{2+\frac{4\sigma s}{2s-\sigma n}}.$$

(47)

Application of (47) and (40) leads to

$${Q\left(u\right)\gtrsim \parallel (-\nabla )}^s{u\parallel }_{L^2}^2.$$

(48)

Combining (46) and (48), it follows that

$$\begin{array}{cc}\hfill \mathbb{E}(\parallel {(-\nabla )}^{s}u{\parallel }_{L^2}^2)\lesssim & C\left({\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })},T_0\right)\mathbb{E}({\int }_0^t\parallel {(-\nabla )}^s\overline{u}{\parallel }_{L^2}^{4}ds)\hfill \\ & +\mathbb{E}\left({\left|Q\left(u_0\right)\right|}^2\right)+C{\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })}^4{T}_0^2\mathbb{E}\left(M^2\left(u_0\right)\right).\hfill \end{array}$$

(49)

For $s\ge \alpha$, utilizing Gronwall’s inequality and Sobolev embedding, we have

$$\mathbb{E}(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\parallel u\parallel }_{H^{\alpha }}^2)\le \mathbb{E}(\underset{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}{sup}{\parallel u\parallel }_{H^s}^2)<\infty ,$$

which means that the boundedness of $\mathbb{E}({sup}_{t\le {\tau }^{\prime }\wedge {\tilde{\tau }}_R}\parallel u{\parallel }_{H^{\alpha }}^2)$ is independent of R. Letting $R\to \infty$, it follows that

$$\begin{array}{c}\hfill \mathbb{E}(\underset{t\le {\tau }^{\prime }}{sup}{\parallel u\parallel }_{H^{\alpha }}^2)\le C\left({\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })},\mathbb{E}\left(M\left(u_0\right)\right),T_0\right).\end{array}$$

For $\lambda =-1$, from (40), it is clear that

$${Q\left(u\right)\gtrsim \parallel (-\nabla )}^s{u\parallel }_{L^2}^2.$$

Then, according to (46), we deduce that (49) still holds. Then, applying Gronwall’s inequality and letting $R\to \infty$, we get

$$\begin{array}{c}\hfill \mathbb{E}(\underset{t\le {\tau }^{\prime }}{sup}{\parallel u\parallel }_{H^{\alpha }}^2)\le C\left({\parallel \varphi \parallel }_{R(L^2;W^{s,\infty })},\mathbb{E}\left(M\left(u_0\right)\right),T_0\right).\end{array}$$

In summary, we complete the proof of Theorem 2.

5. Conclusions

This paper is dedicated to non-radial solutions to the Cauchy problem of the nonlinear fractional Schrödinger equation with nonlinear multiplicative noise. Local well-posedness in $H^{\alpha }$ almost surely follows from the estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, the global existence of a solution in $H^{\alpha }$ almost surely follows from an a priori estimate based on the conservation of energy and Itô’s formula. Comparing to known results about Equation (5) (see [10,15,16,19,20,21,22]), there are two difficulties we have overcome in this paper: one is that the noise term is nonlinear, and the another one is the loss of derivatives. The main innovation is to show the existence of a non-radial global solution under fractional-order derivatives and a nonlinear noise term.