Charles-Edouard Br

David Cohen
email: david.cohen@chalmers.se

STRONG RATES OF CONVERGENCE OF A SPLITTING SCHEME FOR SCHR ÖDINGER EQUATIONS WITH NONLOCAL INTERACTION CUBIC NONLINEARITY AND WHITE NOISE DISPERSION

Keywords: AMS Classification, 65C30, 65J08, 60H15, 60H35, 60-08, 35Q55 Keywords, Stochastic partial differential equations, Stochastic Schrödinger equations, White noise dispersion, Nonlocal interaction cubic nonlinearity, Locally Lipschitz nonlinearity, Geometric numerical integration, Splitting integrators, Strong convergence rates

Strong rates of convergence of a splitting scheme for Schrödinger equations with nonlocal interaction cubic nonlinearity and white noise dispersion

Introduction

We consider the time discretization by a splitting scheme for the following class of nonlinear Schrödinger equations with white noise dispersion iduptq `∆uptq ˝dβptq `V ruptqsuptq dt " 0 up0q " u 0 , [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] where the unknown u " upt, ¨q, with t ě 0, is a complex valued random process defined on R d , ∆u "

d ÿ j"1 B 2 u Bx 2 j
denotes the Laplacian in R d , and β " βptq is a real-valued standard Brownian motion.

The nonlinearity Ψ 0 puq " V rusu in the Stochastic Partial Differential Equation (SPDE) (1) is a nonlocal interaction cubic nonlinearity, V rus " V ‹ |u| 2 " ż V p¨´xq|upxq| 2 dx, where ‹ denotes the convolution operator and the real-valued mapping V : R d Ñ R is at least continuous and bounded, more precise regularity conditions are imposed below. Such long-range interaction is a smooth version of the nonlinearity in the (deterministic) Schrödinger-Poisson equation, see for instance [START_REF] Ch | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF]. Splitting schemes for Schrödinger equations driven by additive space-time noise with this type of nonlinearities were recently studied in [START_REF] Bréhier | Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations[END_REF]. Observe that, the case of power-law nonlinearities cannot be treated by the techniques employed in the present publication. The SPDE [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] is understood in the Stratonovich sense, using the ˝symbol for the Stratonovich product.

Theoretical results on well-posedness of the SPDE (1) are relatively scarce and given mostly for the case of a power-law nonlinearity (|u| 2σ u for σ a positive real number) in place of the nonlocal interaction nonlinearities considered in this article. For instance, it has been shown that SPDEs of the type (1) with power-law nonlinearities have solutions in H 1 for dimension d " 1 and σ " 2, [7, Theorem 2.2], and for σ ă 2{d in any dimension, [START_REF] Debussche | 1d quintic nonlinear schrdinger equation with white noise dispersion[END_REF]Theorem 2.3].

To the best of our knowledge, no strong convergence rates are know for a time discretization of the SPDE [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] with the considered type of locally Lipschitz nonlinearity. However, strong convergence results have been proved in the case of a globally Lipschitz nonlinearity in place of the above nonlinearity. In addition, rates of convergence in probability for a pure cubic nonlinearity in place of the above nonlocal interaction cubic nonlinearity have also been obtained. We now review these known convergence results. The work [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF] studies a Lie-Trotter splitting integrator. The mean-square order of convergence of this explicit numerical method is proven to be at least 1{2 for a (truncated) Lipschitz nonlinearity [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF]Sect. 5 and 6]. Furthermore, [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF] conjectures that this splitting scheme should have strong order one, and supports this conjecture numerically. Sharp order estimates for the same splitting scheme (but applied to a more general problem) were recently presented in the preprint [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF]. The authors of [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] study a semi-implicit Crank-Nicolson scheme. In particular, they show that this time integrator has mean-square order of convergence one for a truncated problem and order of convergence in probability one in the case of a cubic nonlinearity. For the same problem, the same convergence rates are obtained for a multi-symplectic integrator in [START_REF] Cui | Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion[END_REF] and for an explicit exponential scheme in [START_REF] Cohen | Exponential integrators for nonlinear Schrödinger equations with white noise dispersion[END_REF]. We conclude this list of references with the recent work [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] which considers a randomized exponential integrator for time discretization of related (non-random) nonlinear modulated Schrödinger equations.

In the present publication, we consider an explicit splitting integrator for an efficient time discretization of the nonlinear stochastic Schrödinger equation [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF]. In essence, the main principle of splitting integrators is to decompose the vector field of the original differential equation in several parts, such that the arising subsystems are exactly (or easily) integrated. We refer interested readers to [START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF][START_REF] Mclachlan | Splitting methods[END_REF] for details on splitting schemes for ordinary (partial) differential equations. The splitting scheme considered in this publication is given by

u n`1 " e ipβptn`1q´βptnqq∆ `eiτV runs u n ˘,
where τ denotes the time-step size, t n " nτ , and u n « upt n q, see Equation ( 14) below for details.

The main result of this paper is a strong convergence result for the explicit and easy to implement splitting integrator for the time discretization of (1) defined above, see Section 4 for a precise statement. Note that the nonlocal interaction cubic nonlinearity is only locally Lipschitz continuous in the Sobolev spaces H m pR d , Cq, where m " 0, 1, 2, . . ., and not globally Lipschitz as in the above references, see Section 2 for details. Theorem 8 states that the splitting scheme converges with order 1 in the L p pΩ, H m pR d , Cqq sense, for all p P r1, 8q. One also obtains convergence with order 1 in probability and in the almost sure sense. A crucial property for showing these results is the fact that the splitting scheme exactly preserves the L 2 -norm as does the exact solution to [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF], see Proposition 3 and Proposition 4. To the best of our knowledge, this is the first strong convergence result obtained for a time discretization scheme applied to the nonlinear Schrödinger equation with white noise dispersion with a non-globally Lipschitz continuous nonlinearity.

In order to show such convergence results, we begin the exposition by introducing some notations and recalling useful results in Section 2. Section 3 then provides various properties of the exact solution to the SPDE [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF]. After that, we present the splitting scheme and analyse its strong convergence in Section 4. The proof of the main convergence result is given in Section 5. Several numerical experiments illustrating the main properties of the proposed numerical scheme are presented in Section 6.

Throughout this article, we denote by C a generic constant that may vary from line to line. Furthermore, we set N " t1, 2, . . .u and N 0 " t0, 1, . . .u. Finally, the initial value u 0 of the SPDE (1) is assumed to be non-random for ease of presentation. The results of this paper can be extended to the case of random u 0 (independent of the given Brownian motion and with appropriate moment bounds).

Setting and useful results

We denote the classical Lebesgue space of complex functions by L 2 " L 2 pR d , Cq, endowed with its real vector space structure, and with the inner product

ÿ |α|ďm B α v 2 L 2 ,
where α " pα 1 , . . . , α d q P N d 0 is a multi-index and |α| "

d ÿ i"1 α i . Note that, if m 1 ď m 2 , one has H m1 Ă H m2 and v H m 1 ď v H m 2 for all v P H m2 . If α and γ are two multi-indices, it is said that γ ď α if γ i ď α i for all i " 1, . . . , d. If γ ď α, we also introduce the notation α ´γ " pα i ´γi q 1ďiďd and ˆα γ ˙" d ź i"1 ˆαi γ i ˙. The Banach space C m " C m pR d , Cq of complex-valued functions of class C m is equipped with the norm }v} C m " sup |α|ďm sup xPR d |B α vpxq|.
Let us recall a version of the Leibniz rule. See for instance [9, Theorem 1, Sect. 5.2.3] for a proof in the case of smooth compactly supported functions v P C 8 c , and [START_REF] Bressan | Lecture notes on functional analysis[END_REF]Theorem 8.25] for the argument to extend the result to v P C m . Lemma 1 (Leibniz rule). For all m P N, there exists C m P p0, 8q such that for all u P H m and v P C m , one has uv P H m , and

uv H m ď C m u H m v C m .
In addition, Leibniz rule holds: for all α P N d 0 with |α| ď m, one has

B α puvq " ÿ γďα ˆα γ ˙Bγ uB α´γ v.
Let β " `βptq ˘tě0 be a standard real-valued Brownian Motion defined on a filtered probability space pΩ, F, P, tF t u tě0 q satisfying the usual conditions. For all t, s ě 0, define the operator Spt, sq as follows:

(2)

Spt, sq " e ipβptq´βpsqq∆ .

In Fourier variables, one has the expression Spt, sqv pζq " exp ´´i |ζ| 2 pβptq ´βpsqq ¯p vpζq, for all t, s ě 0, all ζ P R d and any v P H m , m P N 0 .

The operators Spt, sq for t ě s play an important role in this work: if v is a F s -measurable random function with values in H m , then t Þ Ñ v s ptq " Spt, sqv is the solution of the stochastic linear Schrödinger equation idv s ptq `∆v s ptq ˝dβptq " 0, t ě s, with v s psq " v.

Two properties of the operators Spt, sq will be used repeatedly in this article. First, for all m P N 0 , all t, s ě 0 and all v P H m , one has the isometry property

(3) Spt, sqv H m " v H m .
Second, for all m P N 0 , all t, s ě 0 and all v P H m`2 , one has

(4) Spt, sqv ´v H m ď |βptq ´βpsq| v H m`2 .
Let us now study properties of the nonlinearity in the SPDE (1) defined by Ψ 0 puq " V rusu " `V ˚|u| 2 ˘u.

If V P C m , the mapping Ψ 0 : H m Ñ H m is well-defined and is locally Lipschitz continuous. More precisely, one has the following result. Lemma 2. Let m P N 0 and assume that V P C m . There exists C m pV q P p0, 8q such that the following properties hold.

First, for all u P H m , Ψ 0 puq P H m and one has

(5) Ψ 0 puq H m ď C m pV q}u} 2 L 2 }u} H m . In addition, Ψ 0 is locally Lipschitz continuous in H m : for all u 1 , u 2 P H m , one has (6) Ψ 0 pu 2 q ´Ψ0 pu 1 q H m ď C m pV q ´ u 1 2 H m ` u 2 2 H m ¯ u 2 ´u1 H m .
Finally, Ψ 0 is twice differentiable, and its first and second order derivatives satisfy the following result: for all u, h, k P H m , one has

(7) Ψ 1 0 puq.h H m ď C m pV q u L 2 u H m h H m and (8) Ψ 2 0 puq.ph, kq H m ď C m pV q u L 2 h H m k H m .
Proof of Lemma 2. Let m P N 0 be fixed.

Using the definition of Ψ 0 , Leibniz rule (Lemma 1) and the property

V rus C m " V ˚|u| 2 C m ď V C m u 2 L 2 , the proof of (5) is straightforward: for all u P H m , one has Ψ 0 puq H m " V rusu H m ď C m V rus C m u H m ď C m V C m u 2 L 2 u H m .
To prove [START_REF] De Bouard | The nonlinear Schrödinger equation with white noise dispersion[END_REF] and [START_REF] Debussche | 1d quintic nonlinear schrdinger equation with white noise dispersion[END_REF], note that the expressions for the derivatives are given by Ψ 1 0 puq.h " V rush `2V ˚pRepūhqq u Ψ 2 0 puq.ph, kq " 4V ˚`Rep khq ˘u `2V ˚pRepūkqq h `2V ˚pRepūhqq k. Using Leibniz rule (Lemma 1) again, one obtains

Ψ 1 0 puq.h H m ď V rus C m h H m `2 V ˚pRepūhqq C m u H m ď V C m u 2 L 2 h H m `2 V C m u L 2 h L 2 u H m , and
Ψ 2 0 puq.ph, kq H m ď 4 V ˚`Rep khq ˘ C m u H m `2 V ˚pRepūkqq C m h H m `2 V ˚pRepūhqq C m k H m ď 4 V C m p}u} H m }h} L 2 }k} L 2 `}u} L 2 }h} H m }k} L 2 `}u} L 2 }h} L 2 }k} H m q .
Finally, in order to prove [START_REF] Cui | Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion[END_REF], it suffices to write Ψ 0 pu 2 q ´Ψ0 pu 1 q " ż 1 0 Ψ 1 0 `p1 ´ξqu 1 `ξu 2 ˘.pu 2 ´u1 q dξ, and to use [START_REF] De Bouard | The nonlinear Schrödinger equation with white noise dispersion[END_REF]. One then obtains

Ψ 0 pu 2 q ´Ψ0 pu 1 q H m ď C m pV qp u 1 2 H m ` u 2 2
H m q u 2 ´u1 H m . This concludes the proof of Lemma 2.

Properties of the exact solution

In this section, we provide a well-posedness result and some properties of the exact solution uptq of the nonlinear Schrödinger equation with white noise dispersion [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF].

Proposition 3. Assume that V P C 0 .
For any (non-random) initial condition u 0 P L 2 , there exists a unique mild solution puptqq tě0 of the Schrödinger with white noise dispersion (1) in L 2 , which means that for all t ě 0 one has uptq " Spt, 0qu 0 `i ż t 0 Spt, rq pV ruprqsuprqq dr, [START_REF] Evans | Partial differential equations[END_REF] where `Spt, sq ˘těsě0 is defined by [START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF].

In addition, one has conservation of the L 2 -norm: for all t ě 0, one has almost surely [START_REF] Hairer | Geometric numerical integration[END_REF] uptq

L 2 " u 0 L 2 .
Furthermore, the SPDE (1) is a stochastic Hamiltonian system, in the sense of [6, Section 2], and thus its solution preserves the stochastic symplectic structure

ω " ż R d dp ^dq dx almost surely,
where the overbar on ω is a reminder that the two-form dp ^dq (with differentials made with respect to the initial value) is integrated over R d . Here, pptq " Repuptqq and qptq " Impuptqq denote the real and imaginary parts of uptq.

Moreover, one can bound the solution in H m in the following sense. Let m P N and assume that V P C m . There exists C m pV q P p0, 8q such that if u 0 P H m , then almost surely uptq P H m for all t ě 0, and [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] uptq H m ď e CmpV q u0 2 L 2 t u 0 H m . Finally, for all u 0 P H m`2 , T P p0, 8q and p P r1, 8q, there exists C p pT, u 0 H m`2 q P p0, 8q such that for all 0 ď t 1 ď t 2 ď T , one has

(12) pEr upt 2 q ´upt 1 q p H m sq 1 p ď C p pT, u 0 H m`2 qpt 2 ´t1 q 1 2 . Proof. Since Ψ 0 is locally Lipschitz continuous from H m to H m , if V P C m , for all m P N, local well-posedness of mild solutions in H m is a standard result.
To prove that solutions to (1) are global, we use a truncation argument. Let θ : r0, 8q Ñ r0, 1s be a compactly supported Lipschitz continuous function, such that θpxq " 1 for x P r0, 1s. For any R P p0, 8q, set V R puq " θpR ´1 u L 2 qV rus and F R puq " V R puqu. The mapping F R is globally Lipschitz continuous, and the SPDE idu R ptq `∆u R ptq ˝dβptq `F R pu R ptqq dt " 0, with initial condition u R p0q " u 0 , thus admits a unique global mild solution `uR ptq ˘tPr0,T s , where T is an arbitrary positive real number. Applying Itô's formula to a regularization of u R ptq as in the proof of [START_REF] De Bouard | The nonlinear Schrödinger equation with white noise dispersion[END_REF]Theorem 4.1] for instance, one checks that u R ptq L 2 " u R p0q L 2 for all t P r0, T s. Choosing R ą u 0 L 2 shows that one can define uptq " u R ptq for all t ě 0. Then uptq is the unique solution on r0, T s of the fixed point equation ( 9), i. e. uptq is the unique mild solution of (1), and one has the preservation of the L 2 -norm [START_REF] Hairer | Geometric numerical integration[END_REF].

The fact that the problem (1) is a stochastic Hamiltonian system is seen, exactly as in [6, Sect. 2], by considering its real and imaginary parts and observing that the obtained differential equations are indeed stochastic Hamiltonian systems. The preservation of the stochastic symplectic structure follows also as in [6, Sect. 2] since the potential V in ( 1) is real-valued (as opposed to a power-law nonlinearity |u| 2σ in the above reference).

Let us now prove the bound in H m , see [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF]. Using Lemma 2, one obtains

V rusu H m ď C V rus C m u H m ď C V C m u 2 L 2 u H m .
Then, using the isometry property for Spt, sq in H m (see ( 3)), the mild formulation ( 9) and the preservation of the L 2 -norm [START_REF] Hairer | Geometric numerical integration[END_REF], one then obtains

uptq H m ď u 0 H m `ż t 0 V rupsqsupsq H m ds ď u 0 H m `C ż t 0 upsq 2 L 2 upsq H m ds ď u 0 H m `C ż t 0 u 0 2 L 2 upsq H m ds.
Applying Gronwall's lemma then yields [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF].

It remains to establish the temporal regularity property [START_REF] Ch | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF]. Using the mild formulation ( 9), and the isometry property (3), one obtains [START_REF] Hairer | Geometric numerical integration[END_REF].

upt 2 q ´upt 1 q H m ď pSpt 2 , t 1 q ´Iq upt 1 q H m `ż t2 t1 V ruptqsuptq H m dt ď pSpt 2 , t 1 q ´Iq upt 1 q H m `C u 0 2 L 2 ż t2 t1 uptq H m dt, using the inequality V ruptqsuptq H m ď C uptq 2 L 2 uptq H m ď C u 0 2 L 2 uptq H m , owing to the preservation of the L 2 -norm
Finally, using (4), the fact that `Er|βpt 2 q ´βpt 1 q| p s ˘1 p ď C p |t 2 ´t1 | 1 2 , and the bound for the exact solution in the H m norm (11), one obtains, for all 0 ď t 1 ď t 2 ď T ,

pEr upt 2 q ´upt 1 q p H m sq 1 p ď C p pt 2 ´t1 q 1{2 e Cm`2pV q u0 2 L 2 t1 u 0 H m`2 `Cpt 2 ´t1 qe CmpV q u0 2 L 2 t2 u 0 3 H m ď C p pT, u 0 H m`2 qpt 2 ´t1 q 1{2 .
This yields [START_REF] Ch | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF], and concludes the proof of Proposition 3.

Numerical analysis of the splitting scheme

In this section, we propose and study an efficient time integrator for the SPDE [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF]. We state and prove some properties of the numerical solution, in particular preservation of the L 2 -norm (Proposition 4). Furthermore, we state the main strong convergence result (Theorem 8) of the paper, namely that the splitting scheme has convergence rate 1. Finally, we deduce various auxiliary results from the main theorem. 4.1. Presentation of the splitting scheme. Let T ą 0 be a fixed time horizon and an integer N ě 1. We define the step size of the numerical method by τ " T {N and denote the discrete times by t n " nτ , for n " 0, . . . , N . Without loss of generality, we assume that τ P p0, 1q.

The main idea of a splitting integrator for the SPDE (1) is based on the observation that the vector field of the original problem can be decomposed in two parts (linear and nonlinear parts respectively) that are exactly integrated.

On the one hand, the solution of the linear stochastic evolution equation iduptq `∆uptq ˝dβptq " 0, up0q " u 0 is given by uptq " Spt, 0qu 0 , where the random propagator Spt, 0q is defined by [START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF].

On the other hand, the solution of the nonlinear evolution equation iduptq `V ruptqsuptq dt " 0, up0q " u 0 is given by uptq " Φ t pu 0 q, where for all t ě 0 and all u P L 2 , one has [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF] Φ t puq " e itV rus u.

The Lie-Trotter splitting strategy yields the definition of the following time integrator for the nonlinear Schrödinger equation with white noise dispersion (1): [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF] u n`1 " Spt n`1 , t n qΦ τ pu n q.

The following notation will be used in the sequel: for all τ P p0, 1q and u P L 2 , set Ψ τ puq " Φ τ puq ´u iτ .

4.2.

Properties of the numerical solution. This subsection lists useful properties of the numerical solution given by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF].

Conservation of the L 2 -norm. The splitting scheme exactly preserves the L 2 -norm as does the exact solution to the SPDE (1), see equation [START_REF] Hairer | Geometric numerical integration[END_REF] in Proposition 3. This conservation property plays a crucial role in the error analysis presented below. Proposition 4. Let u 0 P L 2 , τ P p0, 1q and let `un ˘nPN0 be given by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF], one then has conservation of the L 2 -norm: for all n P N, one has almost surely [START_REF] Mclachlan | Splitting methods[END_REF] u n L 2 " u 0 L 2 .

Proof. Using the isometry property (3), then the definition (13) of the flow Φ τ , a direct computation from the definition of the scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF] gives for all n P N 0

u n`1 L 2 " Spt n`1 , t n qΦ τ pu n q L 2 " Φ τ pu n q L 2 " u n L 2 .
A straightforward recursion argument concludes the proof.

Bounds for the numerical solution in H m . Proposition 5 below states almost sure upper bounds for the numerical solution u n H m , for all n P N 0 and m P N. Proposition 5. Let m P N and assume that V P C m . There exists C m pV q P p0, 8q, such that for any initial condition u 0 P H m , the numerical solution u n defined by the splitting scheme (14) satisfies the following upper bound: for all n P N 0 , one has almost surely

(16) u n H m ď e CmpV qtn u0 2m L 2 u 0 H m .
The proof of Proposition 5 requires the following auxiliary result. Lemma 6. Let m P N and assume that V P C m .

There exists C m pV q P p0, 8q such that for all τ P p0, 1q and all u P H m , one has

Φ τ puq H m ď ´1 `Cm pV qτ p1 ` u 2m L 2 q ¯ u H m .
Proof of Lemma 6. Using the definition [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF], one has the identity Φ τ puq " e iτ V rus u, with V rus " V ˚|u| 2 P C m and e iτ V rus P C m , since V P C m .

The following expression holds: for all u P H m one has Φ τ puq ´u " θ τ pV rusqu, where θ τ pyq " e iτ y ´1 for all y P R. Applying the inequality from Lemma 1, one has, for all u P H m ,

Φ τ puq ´u H m ď C m θ τ pV rusq C m u H m .
It remains to study the behavior of θ τ pV rusq C m . The auxiliary function θ τ satisfies the following properties: for all y P R, all τ P p0, 1q and all k P N,

|θ τ pyq| ď τ |y| |θ pkq τ pyq| ď τ k .
Using the Faà di Bruno formula, one obtains the bounds

B γ θ τ pV rusq C 0 ď C # τ V rus C 0 , γ " 0, C |γ| τ ´1 ` V rus |γ| C |γ| ¯, γ ‰ 0. Finally, using the inequality V rus |γ| C |γ| " B γ `V ˚|u| 2 ˘ |γ| C 0 ď C m pV q u 2m L 2
if 1 ď |γ| ď m then concludes the proof of Lemma 6.

We are now in position to provide the proof of Proposition 5.

Proof of Proposition 5. Using the definition (14) of the splitting scheme, the isometry property (3) of the random propagator Spt n`1 , t n q, and Lemma 6, one gets

u n`1 H m " Spt n`1 , t n qΦ τ pu n q H m " Φ τ pu n q H m ď ´1 `Cm pV q u n 2m L 2 τ ¯ u n H m .
Using the preservation property (15) of the L 2 -norm by the splitting integrator, see Proposition 4, one then obtains the following estimate: for all n P N 0

u n`1 H m ď ´1 `Cm pV q u 0 2m L 2 τ ¯ u n H m .
Finally, a straightforward recursion argument yields the following bound: for all n P N 0 , one has

u n H m ď e CmpV qtn u0 2m L 2 u 0 H m .
All the estimates above hold in an almost sure sense. This concludes the proof of Proposition 5.

Numerical preservation of the stochastic symplectic structure. As seen in Proposition 3, the exact solution to the SPDE (1) preserves a stochastic symplectic structure. The next result states that the same geometric structure is also preserved by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF]. Proposition 7. Consider the numerical discretization of the Schrödinger equation with white noise dispersion (1) by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF]. Then, the splitting scheme preserves the stochastic symplectic structure ωn`1 " ωn for n " 0, . . . , N ´1 almost surely, where ωn " ż R d

dp n ^dq n dx and p n , resp. q n , are the real, resp. imaginary parts of u n .

Proof. The splitting integrator ( 14) is obtained by solving exactly sequentially the following differential equations: iduptq `V ruptqsuptq dt " 0 and iduptq `∆uptq ˝dβptq " 0. Considering the real and imaginary parts of these differential equations and using the fact that V is real-valued, one gets dpptq " ´V rppptq, qptqqsqptq dt, dqptq " V rppptq, qptqqspptq dt and dpptq " ´∆qptq ˝dβptq, dqptq " ∆pptq ˝dβptq. The above problems are infinite-dimensional stochastic Hamiltonian systems in the sense of [6, Eq. ( 6)]. It thus follows, as in [START_REF] Cui | Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion[END_REF]Prop. 3.3], that the splitting scheme preserves the stochastic symplectic structures of each of these Hamiltonian systems, as it is obtained as composition of symplectic maps, and hence the statement.

Convergence results.

We are now in position to state the main result of this article.

Theorem 8. Let `uptq ˘tě0 , resp. `un ˘nPN0 , be the solutions of the stochastic Schrödinger equation (1), resp. of the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF], with (non-random) initial condition u 0 .

Let m P N 0 and assume that V P C m`4 . For all p P r1, 8q, all T P p0, 8q and all u 0 P H m`4 , there exists C m,p pT, u 0 H m`4 q P p0, 8q such that, for all τ P p0, 1q, one has

(17) sup 0ďnďN pE r u n ´upt n q p H m sq 1 p ď C m,p pT, u 0 H m`4 qτ.
The proof of Theorem 8 is postponed to Section 5.

Note that contrary to previous works in the literature, [START_REF] Marty | On a splitting scheme for the nonlinear Schrödinger equation in a random medium[END_REF][START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF][START_REF] Cohen | Exponential integrators for nonlinear Schrödinger equations with white noise dispersion[END_REF][START_REF] Cui | Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion[END_REF], concerning the analysis of numerical schemes for stochastic Schrödinger equations with white noise dispersion with a globally Lipschitz continuous nonlinearity, in Theorem 8 we consider the moments of arbitrary order p P r1, 8q, instead of only p " 2 (mean-square error). We also consider the error in the H m norm, for arbitrary m P N 0 . We could use the same strategy of proof as in those references when p " 2, however we need to use a different strategy when p ‰ 2 and directly consider the general case p P r1, 8q.

Remark 9. If the initial condition u 0 and the potential V are less regular than in Theorem 8, it is possible to obtain the following result: assume that u 0 P H m`2 and that V P C m`2 , then

sup 0ďnďN pE r u n ´upt n q p H m sq 1 p ď C m,p pT, u 0 H m`2 qτ 1 2 .
As immediate consequences of the main result of this article we obtain the following corollaries.

Corollary 10. Under the assumptions of Theorem 8, one obtains the following error estimate: for all ε P p0, 1q, there exists C m,p,ε pT, u 0 H m`4 q P p0, 8q such that, for all τ P p0, 1q, one has

(18) ˆE " sup 0ďnďN u n ´upt n q p H m ˙1 p ď C m,p,ε pT, u 0 H m`4 qτ 1´ε .
Proof. The second error estimate (18) follows from the first error estimate (17) by an elementary argument. Let ε P p0, 1q and p P r1, 8q, and choose q ą maxpp, ε ´1q, then using (17) one obtains

E " sup 0ďnďN u n ´upt n q q H m  ď N ÿ n"0 E r u n ´upt n q q H m s ď T τ
`Cm,q pT, u 0 H m`4 qτ q q ď T C m,q pT, u 0 H m`4 q q τ qp1´1 q q ď C m,q pT, u 0 H m`4 qτ qp1´εq , where we recall that τ " T {N . Finally one obtains (18) as follows:

ˆE " sup 0ďnďN u n ´upt n q p H m ˙1 p ď ˆE " sup 0ďnďN u n ´upt n q q H m ˙1 q ď C m,p,ε pT, u 0 H m`4 qτ 1´ε .
The argument described above gives a slight reduction in the order of convergence from 1 to 1 ´ε, with arbitrarily small ε ą 0. It may be possible to obtain (18) with ε " 0 using refined arguments in the analysis of the error. To keep the presentation simple, this is not performed in the sequel.

The fact that the first error estimate (17) holds with arbitrarily large p is important and allows us to choose arbitrarily small ε. If one applies the argument detailed above only when p " 2 for instance, one obtains an order of convergence 1 2 in (18). Corollary 11. Consider the stochastic Schrödinger equation (1) on the time interval r0, T s with solution denoted by puptqq tPr0,T s . Let u n be the numerical solution given by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF] with time-step size τ . Under the assumptions of Theorem 8, one has convergence in probability of order one lim CÑ8 sup τ Pp0,1q P p u N ´upT q H m ě Cτ q " 0, where we recall that T " N τ . Moreover, consider the sequence of time-step sizes given by τ L " T 2 L , L P N. Then, for every ε P p0, 1q, there exists an almost surely finite random variable C ε , such that for all L P N one has

u 2 L ´upT q H m ď C ε ˆT 2 L ˙1´ε .
Proof. The result on convergence in probability is a straightforward consequence of Markov's inequality followed by Theorem 8:

P p u N ´upT q H m ě Cτ q ď E r u N ´upT q H m s Cτ " C m,1 pT, u 0 H m`4 q C Ñ CÑ8 0.
To get the result on almost sure convergence, it suffices to observe that (again by applying Theorem 8)

8 ÿ "0 E r u 2 ´upT q H m s τ 1´ε ă 8, thus u 2 L ´upT q H m τ 1´ε L Ñ LÑ8
0 almost surely.

Error analysis: proof of Theorem 8

Before proceeding with the proof of the error estimates (17), let us state and prove an auxiliary result on the mappings Ψ 0 puq " V rusu and Ψ τ puq " Φτ puq´u iτ . Lemma 12. Let m P N 0 and assume that V P C m . There exists C m pV q P p0, 8q such that for all u P H m and all τ P p0, 1q, one has

Ψ τ puq ´Ψ0 puq H m ď C m pV qτ ´1 ` u maxp4,2mq L 2 ¯ u H m .
Proof. Let us first observe that, by definitions of the operators Ψ τ and Ψ 0 , one has

Ψ τ puq ´Ψ0 puq " Θ τ pV rusqu,
where Θ τ pyq " e iτ y ´1´iτ y iτ for all y P R.

Applying the inequality from Lemma 1, one obtains

Ψ τ puq ´Ψ0 puq H m ď C m Θ τ pV rusq C m u H m .
It remains to study the behavior of Θ τ pV rusq C m .

The auxiliary function Θ τ satisfies the following properties: for all k P N 0 , there exists C k P p0, 8q such that, for all y P R, one has

' |Θ τ pyq| ď C 0 τ |y| 2 ' |Θ 1 τ pyq| ď C 1 τ |y| ' |Θ pkq τ pyq| ď C k τ k´1 ď C k τ
for all integers k ě 2 and all τ P p0, 1q.

Using the Faà di Bruno formula, one obtains the bounds

B γ Θ τ pV rusq C 0 ď C # τ V rus 2 C 0 , γ " 0, τ ´1 ` V rus |γ| C |γ| ¯, γ ‰ 0.
Using the inequality

V rus |γ| C |γ| " B γ V ˚|u| 2 |γ| C 0 ď C m pV q u 2m L 2
if |γ| ď m then concludes the proof of Lemma 12.

We are now in position to give the proof of Theorem 8.

Proof of Theorem 8. Let us first perform a change of unknowns: for all t ě 0 and n P N 0 , set vptq " Sp0, tquptq " Spt, 0q ´1uptq , and v n " Sp0, t n qu n " Spt n , 0q ´1u n ,

where uptq is the solution of (1) whereas u n is defined by the splitting scheme [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF]. Owing to the isometry property (3) for the random propagator, one has the equality

u n ´upt n q H m " Spt n , 0q pv n ´vpt n qq H m " v n ´vpt n q H m ,
for all m P N 0 and for all n P N 0 . Thus, it is sufficient to prove estimates for the error E n " v n ´vpt n q.

Using the mild form ( 9) for uptq and the definition of the splitting scheme ( 14), for all t ě 0 and n P N 0 , one has the following expressions:

vptq " u 0 `i ż t 0
Sp0, sqΨ 0 pupsqq ds vpt n`1 q " vpt n q `i ż tn`1 tn Sp0, tqΨ 0 puptqq dt v n`1 " Sp0, t n qΦ τ pu n q " v n `iτ Sp0, t n qΨ τ pu n q.

The expressions above then give the following decomposition of the error:

E n`1 " v n`1 ´vpt n`1 q " E n ` 1 n ` 2 n ` 3 n ` 4 n ` 5
n , with local error terms defined by 1 n " iτ Sp0, t n q pΨ τ pu n q ´Ψ0 pu n qq 2 n " iτ Sp0, t n q pΨ 0 pu n q ´Ψ0 pupt n qqq It remains to prove error estimates for `Er E j n p H m s ˘1 p , for j " 1, . . . , 5 and p P r2, 8q (the case p P r1, 2q is treated using Hölder's inequality). The estimates of the error terms for j " 1, 2, 3 follow from straightforward arguments, whereas more work is required to deal with the cases j " 4 and j " 5 (in order to obtain order of convergence equal to 1, instead of the order 1{2 corresponding to the temporal Hölder regularity of the solution, see Equation ( 12)).

We now provide detailed error estimates of those five terms.

' Let us start with the first term. Using Minkowski's inequality and the isometry property (3) of the random propagator, one has

´E " E 1 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 1 k p H m ı¯1 p ď n´1 ÿ k"0 pE r iτ Sp0, t k q pΨ τ pu k q ´Ψ0 pu k qq p H m sq 1 p ď τ n´1 ÿ k"0 pE r Ψ τ pu k q ´Ψ0 pu k q p H m sq 1 p
Applying Lemma 12 and using first the preservation of the L 2 -norm property (15) for the numerical scheme (Proposition 4), second the almost sure bound ( 16) for the H m -norm of the numerical solution (Proposition 5), one finally obtains, for all n " 0, . . . , N ,

´E " E 1 n p H m ı¯1 p ď Cτ 2 n´1 ÿ k"0 ´E "´1 ` u k maxp4,2mq L 2 ¯p u k p H m ı¯1 p ď Cτ 2 n´1 ÿ k"0 ´1 ` u 0 maxp4,2mq L 2 ¯p pE r u k p H m sq 1 p ď C m,p pT, u 0 H m qτ.
' For the second term, using Minkowski's inequality and the isometry property (3) of the random propagator, one has

´E " E 2 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 2 k p H m ı¯1 p ď n´1 ÿ k"0 pE r iτ Sp0, t k q pΨ 0 pu k q ´Ψ0 pupt k qqq p H m sq 1 p ď τ n´1 ÿ k"0 pE r Ψ 0 pu k q ´Ψ0 pupt k qq p H m sq 1 p .
Using the local Lipschitz continuity property (6) of Ψ 0 (Lemma 2), then the almost sure bounds for the H m norm of the exact solution (Equation ( 11) from Proposition 3) and of the numerical solution (Equation ( 16) from Proposition 5), one obtains, for all n " 0, . . . , N ,

´E " E 2 n p H m ı¯1 p ď Cτ n´1 ÿ k"0 ´E "´ u k 2 H m ` upt k q 2 H m ¯p u k ´upt k q p H m ı¯1 p ď CpT, u 0 H m qτ n´1 ÿ k"0 pE r E k p H m sq 1 p .
' In order to estimate the third term, applying Minkowski's inequality yields

´E " E 3 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 3 k p H m ı¯1 p .
Using the inequality (4) (combined with Cauchy-Schwarz's inequality) and the local Lipschitz continuity property (6) of Ψ 0 (Lemma 2) then yields ´E "

3 k p H m ı¯1 p ď ż t k`1 t k pE r pSp0, t k q ´Sp0, tqq pΨ 0 puptqq ´Ψ0 pupt k qqq p H m sq 1 p dt ď Cτ 1 2 ż t k`1 t k ´E " Ψ 0 puptqq ´Ψ0 pupt k qq 2p H m`2 ı¯1 2p dt ď Cτ 1 2 ż t k`1 t k ˆE " ´ uptq 2 H m`2 ` upt k q 2 H m`2 ¯2p uptq ´upt k q 2p H m`2 ˙1 2p dt.
Using the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m`2 -norm of the exact solution (Proposition 3) and the temporal regularity estimate [START_REF] Ch | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF], one finally obtains ´E "

3 k p H m ı¯1 p ď CpT, u 0 H m`4 qτ 1 2 ż t k`1 t k pt ´tk q 1 2 dt ď CpT, u 0 H m`4 qτ 2 ,
and finally one has, for all n " 0, . . . , N ,

´E " E 3 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 3 k p H m ı¯1 p ď C m,p pT, u 0 H m`4 qτ.
' Let us now focus on the the fourth term. As explained above, one needs to be careful to obtain an order of convergence equal to 1. Indeed, for the fourth term, applying (4) directly (and appropriate bounds) would only give order of convergence 1{2 of the splitting scheme.

Let us define auxiliary processes: for all n P N 0 and all t P rt n , t n`1 s, set v n ptq " Sp0, tqΨ 0 pupt n qq.

Note that v n ptq " Spt n , tqv n pt n q " e ´ipβptq´βptnqq∆ v n pt n q, for all t P rt n , t n`1 s. As a consequence, the process pv n ptqq tPrtn,tn`1s is the solution of the linear stochastic evolution equation dv n ptq " ´i∆v n ptq ˝dβptq " ´i∆v n ptq dβptq ´1 2 ∆ 2 v n ptq dt, with v n pt n q " Sp0, t n qΨ 0 pupt n qq.

The local error term 4 n is rewritten as follows in terms of the auxiliary process v n : Set also

4 n " i ż tn`1 tn pSp0, t n q ´Sp0, tqq Ψ 0 pupt n qq dt " i ż tn
E 4,1 n " n´1 ÿ k"0 4,1 k and E 4,2 n " n´1 ÿ k"0 4,2
k . Using Minkowski's inequality, one then gets

´E " E 4 n p H m ı¯1 p ď ´E " E 4,1 n p H m ı¯1 p `´E " E 4,2 n p H m ı¯1 p .
On the one hand, observe that applying the stochastic Fubini theorem gives the equality Introduce the (adapted) auxiliary process v, such that vpsq " pt k`1 ´sqv k psq for all s P rt k , t k`1 s and k " 0, . . . , N ´1. Then the error term E 4,1 n is rewritten as the Itô integral

E 4,1 n " ż tn 0 ∆vpsq dβpsq,
and applying the Burkholder-Davis-Gundy and Hölder inequalities, for all p ě 2, one obtains

´E " E 4,1 n p H m ı¯1 p " ˜E « ż tn 0 ∆vpsq dβpsq p H m ff¸1 p ď C p pT q ˆż tn 0 E r ∆vpsq p H m s ds ˙1 p ď C p pT q ˆż tn 0 E " vpsq p H m`2 ‰ ds ˙1 p .
By definition of vpsq and of v k psq, one obtains

ż tn 0 E " vpsq p H m`2 ‰ ds " n´1 ÿ k"0 ż t k`1 t k E " pt k`1 ´sqv k psq p H m`2 ‰ ds ď τ p n´1 ÿ k"0 ż t k`1 t k E " Sp0, sqΨ 0 pupt k qq p H m`2 ‰ ds ď τ p`1 n´1 ÿ k"0 E " Ψ 0 pupt k qq p H m`2 ‰ ď CpT, u 0 H m`2 qτ p ,
using the isometry property (3), the inequality (5) and the exact preservation of the L 2 norm (10) as well as the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m`2 norm of the exact solution, see Proposition 3.

On the other hand, for the second term, using Minkowski's inequality and the definition of the auxiliary processes v k , one obtains

´E " E 4,2 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 4,2 k p H m ı¯1 p ď 1 2 n´1 ÿ k"0 ż t k`1 t k ż t t k `E " v k psq p H m`4 ‰˘1 p ds dt ď Cτ 2 n´1 ÿ k"0 `E " Sp0, t k qΨ 0 pupt k qq p H m`4 ‰˘1 p ď CpT, u 0 H m`4 qτ,
using the isometry property (3), the inequality (5) and the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m`4 norm of the exact solution.

Gathering the estimates, one obtains the following estimate for the fourth error term: for all n " 0, . . . , N , p1 ´ξqΨ 2 0 pp1 ´ξqupt n q `ξuptqq.puptq ´upt n q, uptq ´upt n qq dξ " Ψ 1 0 pupt n qq. ˆpSpt, t n q ´Iq upt n q `i ż t tn Spt, sqΨ 0 pupsqq ds ˙`R n ptq, using the mild formulation [START_REF] Evans | Partial differential equations[END_REF] for the exact solution, where one has defined the auxiliary quantity

´E

R n ptq " ż 1 0
p1 ´ξqΨ 2 0 pp1 ´ξqupt n q `ξuptqq.puptq ´upt n q, uptq ´upt n qq dξ.

For all n " 0, . . . , N ´1, set It remains to obtain estimates for each of the three error terms in the right-hand side above.

piq To treat the first error terms E 5,1 n and 5,1 n , one follows the same strategy as for the error terms E 4,1 n and 4,1 n above. Let us define auxiliary processes: for all n P N 0 and t P rt n , t n`1 s, set w n ptq " Spt, t n qupt n q.

For each n P N 0 , the process pw n ptqq tPrtn,tn`1s is the solution of the linear stochastic evolution dw n ptq " i∆w n ptq ˝dβptq " i∆w n ptq dβpsq ´1 2 ∆ 2 w n ptq dt, with initial value w n pt n q " upt n q, see the definition (2) of the random propagator Spt, sq. The local error term 5,1 n is rewritten as follows in terms of the auxiliary process w n : . Using Minkowski's inequality, one then gets

´E " E 5,1 n p H m ı¯1 p ď ´E " E 5,1,1 n p H m ı¯1 p `´E " E 5,1,2 n p H m ı¯1 p .
On the one hand, observe that applying the stochastic Fubini theorem gives the equality Introduce the (adapted) auxiliary process w, such that wpsq " pt k`1 ´sqSp0, t k qΨ 1 0 pupt k qq.p∆w k psqq for all s P rt k , t k`1 s and k " 0, . . . , N ´1. Then the error term E 5,1,1 n is rewritten as the Itô integral By definition of wpsq and of w k psq, one obtains

ż tn 0 E r wpsq p H m s ds " n´1 ÿ k"0 ż t k`1 t k E " pt k`1 ´sqSp0, t k qΨ 1 0 pupt k qq.∆w k psq p H m ‰ ds ď Cτ p n´1 ÿ k"0 ż t k`1 t k E " Ψ 1 0 pupt k qq.∆Sps, t k qupt k q p H m ‰ ds ď Cτ p`1 n´1 ÿ k"0 E " upt k q 2p H m ∆upt k q p H m ı ď Cτ p`1 n´1 ÿ k"0 E " upt k q 3p H m`2 ı ď CpT, u 0 H m`2 qτ p ,
using the isometry property (3), the inequality [START_REF] De Bouard | The nonlinear Schrödinger equation with white noise dispersion[END_REF] and the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m`2 norm of the exact solution.

On the other hand, for the second term, using Minkowski's inequality and the definition of the auxiliary processes w k , one obtains

´E " E 5,1,2 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 5,1,2 k p H m ı¯1 p ď 1 2 n´1 ÿ k"0 ż t k`1 t k ż t t k ´E " Sp0, t k qΨ 1 0 pupt k qq.∆ 2 Sps, t k qupt k q p H m ı¯1 p ds dt ď Cτ 2 n´1 ÿ k"0 ´E " upt k q 2p H m ∆ 2 upt k q p H m ı¯1 p ď Cτ 2 n´1 ÿ k"0 ´E " upt k q 3p H m`4 ı¯1 p ď CpT, u 0 H m`4 qτ,
using the isometry property (3), the inequality (5) and the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m`4 norm of the exact solution.

Gathering the estimates for E 5,1,1 n and E 5,1,2 n , one finally obtains, for all n " 0, . . . , N ,

´E " E 5,1 n p H m ı¯1 p ď C m,p pT, u 0 H m`4 qτ.
piiq To deal with the error terms E 5,2 n and 5,2 n , using Minkowski's inequality, the isometry property (3) and the inequalities ( 7) and ( 5) (Lemma 2), one obtains

´E " E 5,2 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 5,2 k p H m ı¯1 p ď n´1 ÿ k"0 ż t k`1 t k ż t t k `E " Sp0, t k qΨ 1 0 pupt k qq. pSpt, sqΨ 0 pupsqqq p H m ‰˘1 p ds dt ď n´1 ÿ k"0 ż t k`1 t k ż t t k ´E " upt k q 2p H m Ψ 0 pupsqq p H m ı¯1 p ds dt ď Cτ n´1 ÿ k"0 ż t k`1 t k ´E " upt k q 2p H m upsq 2p L 2 upsq p H m ı¯1 p ds.
Finally using the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m norm of the exact solution, one obtains, for all n " 0, . . . , N

´E " E 5,2 n p H m ı¯1 p ď C m,p pT, u 0 H m qτ.
piiiq To deal with the error terms E 5,3 n and 5,3 n , using Minkowski's inequality gives

´E " E 5,3 n p H m ı¯1 p ď n´1 ÿ k"0 ´E " 5,3 k p H m ı¯1 p ď n´1 ÿ k"0 ż t k`1 t k pE r Sp0, t k qR k ptq p H m sq 1 p dt.
Using the isometry property (3), the inequality (8) (Lemma 2), one obtains

´E " E 5,3 n p H m ı¯1 p ď C n´1 ÿ k"0 ż t k`1 t k ´E " p uptq H m ` upt k q H m q p uptq ´upt k q 2p H m ı¯1 p dt ď C p pT, u 0 H m q n´1 ÿ k"0 ż t k`1 t k ´E " uptq ´upt k q 2p H m ı¯1 p dt ď C m,p pT, u 0 H m`2 qτ,
where the almost sure bound [START_REF] Hofmanová | Randomized exponential integrators for modulated nonlinear schrdinger equations[END_REF] for the H m norm of the exact solution and the temporal regularity estimate (12) (Proposition 3) have been used.

Gathering the estimates, one finally obtains the last required result: for all n " 0, . . . , N Applying the discrete Gronwall Lemma then gives (17) and concludes the proof of Theorem 8.

Numerical experiments

We present some numerical experiments in order to support and illustrate the above theoretical results. In addition, we shall compare the behavior of the splitting scheme (14) (denoted by Split below) with the following time integrators for the stochastic nonlinear Schrödinger equation [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] ' the stochastic exponential integrator from [START_REF] Cohen | Exponential integrators for nonlinear Schrödinger equations with white noise dispersion[END_REF] (adapted to the present nonlocal interaction cubic nonlinearity, denoted by Exp)

u n`1 " Spt n`1 , t n qu n `iτ Spt n`1 , t n qV ru n su n .

' the semi-implicit midpoint scheme (denoted Mid) i u n`1 ´un τ `χn ? τ ∆u n`1{2 `V ru n su n " 0, where u n`1{2 " 1 2 pu n `un`1 q and χ n " βptn`1q´βptnq ? τ . This is a modification of the Crank-Nicolson from [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] for the nonlinear interaction nonlinearity studied here.

Unless stated otherwise, we consider the SPDE (1) with the potential V pxq " cospxq on the one dimensional torus with periodic boundary conditions r0, 2πs. The spatial discretization is done by a pseudospectral method with M Fourier modes. The initial value is given by u 0 pxq " expp´0.5px´πq 2 q.

6.1. Evolution plots. To illustrate the interplay and the balance between the random dispersion and the nonlinearity, in Figure 1, we display the evolution of |u n | 2 along one sample of the numerical solution obtained by the splitting integrator [START_REF] Marty | Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion[END_REF]. The discretization parameters are τ " 2 ´14 and M " 2 10 and the time interval is given by r0, 0.5s. remains constant for all times, see Proposition 3. Figure 2 illustrates the corresponding behavior of the above numerical integrators along one sample path. For this numerical experiment, we consider the parameters τ " 2 ´8 and M " 2 10 Fourier modes and the time interval r0, 1s. Exact preservation of the L 2 -norm for the splitting scheme is observed, as stated in Proposition 4, whereas a small drift is observed for the exponential integrator and the midpoint scheme.

6.3. Strong convergence. We now illustrate the mean-square convergence of the splitting scheme (14) stated in Theorem 8. M " 2 10 Fourier modes are used for the spatial discretization. The meansquare errors Er u ref px, T end q ´uN pxq 2 H 1 s 1{2 at time T end " 1 are displayed in Figure 3 for various values of the time step τ " 2 ´ for " 10, . . . , 16. Here, we simulate the reference solution u ref px, tq with the splitting scheme, with a small time step τ ref " 2 ´18 . The expected values are approximated by computing averages over M s " 100 samples. In Figure 3, we observe convergence of order 1 for all time integrators. Note that, the strong order of convergence of the exponential scheme and midpoint integrator are not known in the case of the considered nonlocal interaction potential, whereas Figure 3 illustrates our main result Theorem 8 for the splitting scheme.

3 n 5 n " i ż tn` 1 tn

 351 " i ż tn`1 tn pSp0, tq ´Sp0, t n qq pΨ 0 pupt n qq ´Ψ0 puptqqq dt 4 n " i ż tn`1 tn pSp0, t n q ´Sp0, tqq Ψ 0 pupt n qq dt Sp0, t n q pΨ 0 pupt n qq ´Ψ0 puptqqq dt.For j " 1, . . . , 5, set E j n " 's inequality one obtains, for p P r1, 8q,

∆v n psq dt dβpsq " ż tn` 1 tn pt n` 1

 11 ´sq∆v n psq dβpsq.

E 5

 5 Burkholder-Davis-Gundy and Hölder's inequalities, for all p ě 2, one obtains

Figure 1 .Figure 2 .

 12 Figure 1. Space-time evolution and contour plot for the splitting integrator (14). The discretization parameters are τ " 2 ´14 and M " 2 10 Fourier modes.

6. 2 .

 2 Conservation of the L 2 -norm. It is known that the L 2 -norm of the solution to the SPDE[START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF]

Figure 3 .

 3 Figure 3. Mean-square errors as a function of the time step: Splitting scheme (˝), exponential integrator (♦), and midpoint scheme (ˆ). The dotted line has slope 1.

  L 2 . For m P N, we further denote by H m " H m pR d , Cq the Sobolev space of functions in L 2 , with weak derivatives of order 1, . . . , m in L 2 . The Fourier transform of a tempered distribution v is denoted by p v. With this notation, H m is the Sobolev space of tempered distributions v such that p1`|ζ| 2 q m{2 p v P L 2 . The Sobolev space H m is equipped with the norm defined by v

	ż	ż
	pu, vq " Re	uv dx " Re	uv dx
	R d
	as well as its norm denoted by ¨ 2 H m "

  It remains to deal with the fifth error term. Using a second-order Taylor expansion, one has, for t P rt n , t n`1 s, Ψ 0 puptqq ´Ψ0 pupt n qq " Ψ 1 0 pupt n qq. puptq ´upt n qq

	`ż 1
	0
	"	n E 4	H m p	ı¯1

p ď C m,p pT, u 0 H m`4 qτ.

'

  Sp0, t n qΨ 1 0 pupt n qq. ppSpt, t n q ´Iq upt n qq dt

					5,1 n "	´i ż tn`1
								tn
					5,2 n "	ż tn`1	Sp0, t n qΨ 1 0 pupt n qq.	ˆż t	Spt, sqΨ 0 pupsqq ds ˙dt
								tn								tn
					5,3 n "	´i ż tn`1	Sp0, t n qR n ptq dt,
								tn
		n´1
	and E 5,j n "	ÿ	5,j n , j " 1, 2, 3. Note that 5 n " 5,1 n ` 5,2 n ` 5,3 n , and E 5 n " E 5,1 n	`E5,2 n	`E5,3 n , for all
		k"0
	n " 0, . . . , N ´1, and Minkowski's inequality yields
	´E "	E 5 n	p H m	ı¯1 p	ď	´E "	E 5,1 n	p H m	ı¯1 p	`´E	"	E 5,2 n	p H m	ı¯1 p	`´E	"	E 5,3 n	p H m	ı¯1 p .

Acknowledgements

The work of CEB was partially supported by the SIMALIN project ANR-19-CE40-0016 of the French National Research Agency. The work of DC was partially supported by the Swedish Research Council (VR) (projects nr. 2018 ´04443). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N, Umeå University and at Chalmers Centre for Computational Science and Engineering.