The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

: 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 It ô ’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.

To state the noise term precisely, we introduce a probability space (Ω, F, P), which depends on a filtration (F t ) t≥0 and a sequence of independent real-valued Brownian motions (B k ) k∈N .The random process is a Winner process in the space of square-integrable functions, where linear operator ϕ is bounded on L 2 , and (e k ) k∈N is an Hilbertian basis on L 2 .The equivalent Itô equation for Problem (1) is presented by where The mild solution to Problem (2) can be formulated as the following: (3) The last term of (3), is called stochastic convolution, where S(t) denotes the linear fractional Schrödinger propagator.
And s crit works as a threshold law for the well-posedness and ill-posedness of Equation ( 4).Local well-posedness in H α for the Cauchy problem of Equation ( 4) has been investigated in the sub-critical case (α ≥ max(0, s crit )) (see [10,13]).For the critical case α = 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 (α < 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: (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 W y such that they could apply the fixed-point argument related to the deterministic equation and Strichartz estimates of the evolution operator A(s) 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 α 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) ⌈α⌉ is the smallest positive integer greater than or equal to α.
The main results of this paper can be stated as the following: Moreover, there is an alternative (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).

Preliminaries
In this section, we introduce some notations and estimates.We first introduce some notations.
(1) Given β ∈ R and 1 ≤ p ≤ ∞, the non-homogeneous Sobolev space is defined by And the definition of the homogeneous Sobolev space is given by .
(2) For r, p, ρ ≥ 1, the space of random function u(t, x, ω) in L r 0, T; W β,p is denoted by (3) We review the definition of space R L 2 ; W α,∞ , which is a space of Hilbert-Schmidt operator ϕ from L 2 into W α,∞ .The norm of operator ϕ in R L 2 ; W α,∞ is defined as where (e k ) k∈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 exists a solution where S(t) := e −it(−∆) s satisfies the following lemma.
Lemma 1 (Deterministic non-radial Strichartz estimates [13]).Given u 0 ∈ L 2 and F ∈ L r (R; L p ), then for all (r, p) and (a, b) admissible pairs, the group S(t , We recall the following lemmas that are needed for the proof of local well-posedness. Lemma 2 (Nonlinear estimate [10]).Let f (u) = |u| 2σ u with σ > 0. Given 1 < p, q < ∞ and In order to estimate the stochastic integral, we introduce the following lemmas.
where C is independent of T and g.

Estimations
In this subsection, we introduce some estimations that are important for proving Theorem 1. First, we need to estimate the stochastic convolution Lemma 4. Let the conditions for r and q be the same as in Theorem 1.If T > 0, ρ ≥ r, and u ∈ L ρ (Ω; L ∞ (0, T; H α )), it holds that Ψ(t) ∈ L ρ Ω; L r 0, T; W β,p and Proof.In an application of the Hölder's inequality and ρ ≥ r, we have Employing the Bürkholder inequality yields According to the decay estimate in [13]: we have Application of the Hölder's inequality and 1 Combining ( 11) and ( 12), we obtain Then, substituting ( 13) into (10), for ρ ≥ r > max(2σ, 6), we have which complements the proof of the lemma.□ The estimate about Ψ(t) on L ρ (Ω; L ∞ (0, T; H α )) 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 ρ ≥ r.If u ∈ L ρ (Ω; L r (0, T; W β,p )), we have Proof.Applying the Bürkholder inequality, we get Furthermore, employing the fact that linear operator S(t) is isometrically isomorphic on L 2 and the Hölder's inequality leads to Thanks to 1 p − β n ≤ 1 2 − α n , according to the Sobolev embedding theorem, we obtain W β,p ⊂ H α .Then, substituting ( 16) into (15), we get We derive the conclusion (14).□ Moreover, the following lemmas are necessary for later processing.

Proof of Theorem 1
Now, we give the proof of Theorem 1. Denote We consider θ ∈ C ∞ as a cut-off function on R + , and where R ∈ N + and Y t := L r (Ω; X t ).
Since the nonlinear term of Problem ( 1) is local Lipschitz, we introduce the following Cauchy problem of a truncated equation The mild solution to Problem ( 19) is Define a mapping as following .
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 ∈ X t ′ almost surely.
We consider the space with the following metric: We need to prove that ψ R is a contraction on E(T R , R) if T R depends on R.
Owing to r > max(2σ, 6) and γ r,p > 0, it holds that α > n 2 − 2s r .Combining ( 6) and ( 7), we can see 1  p − α−γ r,p n < 0.Then, the Sobolev embedding implies For any u R ∈ E(T R , R), using Strichartz estimates, W β,p ⊂ L ∞ and Lemmas 4-5, we get where λ = min 1 − 2σ r , 1 2 − 3 r .In a similar way, we obtain By the Strichartz estimates, it follows that For the purpose of estimating I for ℓ = 1, 2, we set Without loss of generality, we assume t R To estimate I 1 , we recall the following lemma from [15].According to Lemma 6, it is easy to get For t ∈ t R 1 , t R 2 , there exists θ(u R1 ) = 0. Thus, we can rewrite I 3 as Using Lemma 2, it follows that Similarly, we obtain Then, it follows from ( 23)-( 26) that Thanks to Hölder's inequality, one can check that almost surely.Then, it is obvious that u ∈ Y τ for all stopping times τ < τ * (u 0 ).Furthermore, the uniqueness of this local solution is consistent with the uniqueness of the solution to Equation (20) in Y τ .

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 and the energy (40)

The Stochastic Identity of M(u) and H(u)
For later use, it is convenient to give Itô's lemmas for M(u) and H(u).
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(u R ), we have Taking a sufficiently large R and t = τ, we conclude M(u(τ)) = M(u 0 ).□ Proposition 2. The assumptions about u 0 , σ, and ϕ are the same as in Theorem 1.For all stopping times τ with τ < τ * (u 0 ), there exists where u is given by Theorem 1.
Proof.Applying Itô's lemma and fractional integration by parts to Q(u), we obtain Choosing a large enough R and taking t = τ, we arrive at Thanks to the assumption n ≤ 4, we can choose σ < 2s n such that From Bürkholder and Hölder's inequalities, we have Applying Hölder's inequality to the variable s, we get According to Proposition 1, it follows that If we plug (43)-(45) back into (42), we obtain In the case of λ = 1, from σ < 2s n , the following Gagliardo-Nirenberg inequality holds (see [32]): Application of ( 47) and (40) leads to Combining ( 46) and ( 48), it follows that For s ≥ α, utilizing Gronwall's inequality and Sobolev embedding, we have which means that the boundedness of ) , E(M(u 0 )), T 0 .
In summary, we complete the proof of Theorem 2.

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 α 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 α 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.