Charles-Edouard Bréhier
email: brehier@math.univ-lyon1.fr

David Cohen
email: david.cohen@chalmers.se.

ANALYSIS OF A SPLITTING SCHEME FOR A CLASS OF NONLINEAR STOCHASTIC SCHRÖDINGER EQUATIONS

Keywords: 65C30, 65C50, 65J08, 60H15, 60M15, 60-08, 35Q55, Keywords, Stochastic partial differential equations, Stochastic Schrödinger equations, Splitting integrators, Strong convergence, Geometric numerical integration, Trace formulas

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Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

Introduction

Deterministic Schrödinger equations are widely used within physics, plasma physics or nonlinear optics, see for instance [START_REF] Sulem | The nonlinear Schrödinger equation[END_REF][START_REF] Cohen | Nonlinear Fiber Optics[END_REF][START_REF] Brezzi | The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation[END_REF][START_REF] Illner | Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems[END_REF]. In certain physical situations it may be appropriate to incorporate some randomness into the model. One possibility is to add a driving random force and obtain a stochastic partial differential equation (SPDE) of the form i Bu Bt px, tq " ∆upx, tq `F px, uq `ξpx, tq, considered for x P T d , the d-dimensional torus, with periodic boundary conditions. The nonlinearity F and the white noise ξ are described in details below. See Equation [START_REF] Cohen | Nonlinear Fiber Optics[END_REF] for the formulation of this problem as a stochastic evolution equation. See for example [START_REF] Konotop | Nonlinear random waves[END_REF][START_REF] De Bouard | Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations[END_REF][START_REF] De Bouard | On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation[END_REF][START_REF] De Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF][START_REF] De Bouard | Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation[END_REF] and references therein for further details and applications. The nonlinearities we shall consider encompass for instance the case of an external potential or of a nonlocal interaction cubic nonlinearity. Such long-range interaction is defined as the convolution of an interaction kernel with the density function |u| 2 and is a smooth version of the Schrödinger-Poisson equation. Such nonlinearities are used in modeling deterministic problems arising in quantum physics, chemistry, materials sciences, and biology [START_REF] Bao | Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT[END_REF][START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. However, the case of power-law nonlinearities cannot be treated by the techniques employed in this paper.

Let us now review the relevant literature on temporal discretizations of stochastic Schrödinger equations driven by an Itô noise. In [START_REF] De Bouard | Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation[END_REF], a Crank-Nicolson scheme is studied for the stochastic Schrödinger equation with regular coefficients. First order of convergence, resp. rate one half is obtained in the case of additive noise, resp. multiplicative Itô noise. In addition, convergence in probability as well as almost-surely are studied for the case of a power-law nonlinearity. Observe that the numerical scheme from [START_REF] De Bouard | Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation[END_REF] is implicit. The references [START_REF] Debussche | Numerical simulation of focusing stochastic nonlinear Schrödinger equations[END_REF][START_REF] Barton-Smith | Numerical study of two-dimensional stochastic NLS equations[END_REF] present thorough numerical simulations and numerically study the effect of noise in the stochastic Schrödinger equation with a power-law nonlinearity. The work [START_REF] Grecksch | Approximation of stochastic nonlinear equations of Schrödinger type by the splitting method[END_REF] provides a strong convergence analysis of a splitting strategy to the variational solution of Schrödinger's equation with regular coefficients. The recent article [START_REF] Anton | Exponential integrators for stochastic Schrödinger equations driven by Itô noise[END_REF] proves strong convergence of an exponential integrator for stochastic Schrödinger equations with regular coefficients. In addition, longtime behaviors of the numerical solutions of a linear model is investigated. The paper [START_REF] Harms | Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg-de Vries equations[END_REF] provides a convergence rate of the weak error under noise discretizations of some Schrödinger's equations. Finally, the work [START_REF] Hong | Convergence analysis of a symplectic semi-discretization for stochastic NLS equation with quadratic potential[END_REF] shows convergence in probability of a stochastic (implicit) symplectic scheme for stochastic nonlinear Schrödinger equations with quadratic potential and an additive noise.

In the present work, we shall analyze a splitting strategy for an efficient time integration of a class of nonlinear stochastic Schrödinger equations. In a nutshell, the main idea 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 equation [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF].

Despite the fact that splitting schemes are widely used for an efficient time integration of deterministic Schrödinger-type equations, see for instance [START_REF] Blanes | Splitting methods for the time-dependent Schrödinger equation[END_REF][START_REF] Besse | Order estimates in time of splitting methods for the nonlinear Schrödinger equation[END_REF][START_REF] Ch | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF][START_REF] Dujardin | Qualitative behavior of splitting methods for the linear Schrödinger equation in molecular dynamics[END_REF][START_REF] Gauckler | Splitting integrators for nonlinear Schrödinger equations over long times[END_REF][START_REF] Liu | Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations[END_REF][START_REF] Auzinger | Convergence of a Strang splitting finite element discretization for the Schrödinger-Poisson equation[END_REF], we are not aware of a numerical analysis of such integrators approximating mild solutions of nonlinear stochastic Schrödinger equations driven by an additive Itô noise. In the present publication we prove ' bounds for the exponential moments of the mass of the exact and numerical solutions (Theorem 10); ' a kind of longtime stability, a so called trace formula for the mass, of the numerical solutions (Proposition 5); ' preservation of symplecticity for the exact and numerical solutions (Proposition 8); ' strong convergence estimates (with order) of the splitting scheme, as well as orders of convergence in probability and almost surely (Theorem 14 and Corollary 16).

Observe that, since the nonlinearity in the class of stochastic Schrödinger equation considered here may not be globally Lipschitz, we employ the exponential moments estimates mentioned above to obtain strong rates of convergence, see Propositions 12 and 13. In these propositions, we consider moments of the error multiplied by an exponential discounting factor, and obtain the expected rate of convergence for this quantity. To the best of our knowledge, this quantity has not been considered elsewhere in the literature. Combining those estimates with the above exponential moment bounds to remove the exponential factor, we can then prove Theorem 14. Note finally, that the choice of a splitting strategy is crucial in obtaining exponential moment bounds for the numerical solution.

We begin the exposition by introducing some notations, present our main assumptions and provide several moment bound estimates for the exact solution to the considered SPDE. We then present the splitting scheme and study some geometric properties of the exact and numerical solutions in Section 3. The main results of this publication are presented in Section 4. In particular, exponential moments in the L 2 norm of the exact and numerical solutions are given, as well as several convergence results. More involved and technical proofs of results needed for convergence estimates are provided in Section 5. Various numerical experiments illustrating the main properties of the splitting scheme when applied to stochastic Schrödinger equations driven by Itô noise are given in Section 6. The paper ends with an appendix containing proofs of auxiliary results.

We use C to denote a generic constant, independent of the time-step size of the numerical scheme, which may differ from one place to another.

Setting

In this work, we consider the following class of stochastic nonlinear Schrödinger equations [START_REF] Cohen | Nonlinear Fiber Optics[END_REF] i duptq " ∆uptq dt `F puptqq dt `αdW Q ptq,

up0q " u 0 ,
where the unknown `uptq ˘tě0 is a stochastic process with values in the Hilbert space L 2 " L 2 pT d q of square integrable complex-valued functions defined on the d-dimensional torus T d . Details concerning the regularity and growth properties of the nonlinearity F and the covariance operator Q are provided below. In addition, α ą 0 is a real parameter measuring the size of the noise W Q . The initial condition u 0 P L 2 is deterministic, however the results below can be adapted to random initial conditions, satisfying appropriate integrability conditions, using a standard conditioning argument. The space L 2 is equipped with the norm ∥¨∥ L 2 , where for all u, v P L 2 ,

∥u∥ 2 L 2 " xu, uy, xu, vy " ż T d ūpxqvpxq dx.
The Sobolev spaces H 1 " H 1 pT d q and H 2 " H 2 pT d q are Hilbert spaces, and the associated norms are denoted by ∥¨∥ H 1 and ∥¨∥ H 2 . The notation H 0 " L 2 will also be used below. For σ P t0, 1, 2u, let also ∥¨∥ C σ denote the norm in the Banach space C σ " C σ pT d q of functions of class C σ defined in T d . Solutions of (1) are understood in the mild sense:

( In addition, for σ P t1, 2u, there exists C σ P p0, 8q such that for all u P H σ and all t ě 0,

`Sptq ´I˘u L 2 ď C σ t σ 2 ∥u∥ H σ .
The Wiener process W Q , with covariance operator Q, in the SPDE (1) is defined by

W Q ptq " ÿ kPN γ k β k ptqe k ,
where `ek ˘kPN is a complete orthonormal system of L 2 , `βk ˘kPN is a sequence of independent real-valued standard Wiener processes on a stochastic basis pΩ, F, P, pFptqq tě0 q, and `γk ˘kPN is a sequence of complex numbers such that ÿ

kPN |γ k | 2 ă 8. The linear operators Q and Q 1 2
are defined by Qe k " γ 2 k e k and Q 1 2 e k " γ k e k , for all k P N. For a linear operator Ψ from H σ to H σ , and any complete orthonormal system pε k q kPN of H σ , we define

∥Ψ∥ 2 L σ 2 " ÿ kPN ∥Ψε k ∥ 2 H σ .
This definition is independent of the choice of the orthonormal system.

With this notation, Q

1 2 2 L σ 2 " ÿ kPN |γ k | 2 ∥e k ∥ 2
H σ (whenever the sum is finite). We now set the assumptions on the spatial Sobolev regularity of the noise as well as on the nonlinearity in the stochastic Schrödinger equation (1) required to prove well-posedness for the SPDE (1), to prove H 1 -regularity of the solution, and to show strong convergence of order 1{2 of the proposed splitting integrator in Section 4.

Assumption 1. One has

Q 1 2 2 L 1 2 " ÿ kPN |γ k | 2 ∥e k ∥ 2 H 1 ă 8.
The nonlinearity F satisfies F puq " V rusu for all u P L 2 , where V :

u P L 2 Þ Ñ V rus P R is a real-valued mapping. Furthermore, it is assumed that V ru 1 s " V ru 2 s if |u 1 | " |u 2 | (i. e.

the potential V is a function of the modulus).

In addition to the above, assume that the mapping F is locally Lipschitz continuous with at most cubic growth: there exists C F P p0, 8q and K F P p0, 8q such that for all u 1 , u 2 P L 2 , one has

(3) ∥F pu 2 q ´F pu 1 q∥ L 2 ď ´CF `KF p∥u 1 ∥ 2 L 2 `∥u 2 ∥ 2 L 2 q ¯∥u 2 ´u1 ∥ L 2 .
Finally, there exists C 1 P p0, 8q and a polynomial mapping P 1 , such that for all u P H 1 , one has

(4) ∥F puq∥ H 1 ď C 1 ∥u∥ H 1 ´1 `∥u∥ 2 L 2 |Impx∇u, ∇F puqyq| ď C 1 ∥∇u∥ 2 L 2 `P1 ´∥u∥ 2 L 2 ¯.
Note that assuming that V rus is real-valued implies that the equality Impxu, F puqyq " 0 holds for all u P L 2 .

The value of K F appearing in the right-hand side of (3) plays a crucial role in the convergence analysis below.

Let us recall the definition of the stochastic integral in the mild form (2) and the associated Itô isometry property. If for all t ě 0, Ψptq is a linear operator from H σ to H σ , the stochas-

tic integral ż T 0 Ψptq dW Q ptq is understood as ÿ kPN γ k ż T 0
Ψptqe k dβ k ptq, and the Itô isometry formula is given by

E « ż T 0 Ψptq dW Q ptq 2 H σ ff " ż T 0 ΨptqQ 1 2 2 L σ 2 dt.
Under Assumption 1, the stochastic convolution ´i ż t 0 Spt ´sq dW Q psq is thus well-defined and takes values in H 1 . It solves the linear stochastic Schrödinger equation driven by additive noise i duptq " ∆uptq dt `dW Q ptq, up0q " 0.

Most of the analysis can be performed when Assumption 1 is satisfied, in particular we will prove below that (1) admits a unique global solution, and that the splitting scheme has a strong convergence order 1{2. To get strong convergence order 1 of the proposed splitting integrator for the semilinear problem (1), we need further assumptions.

Assumption 2. On top of Assumption (1), let us assume that one has

Q 1 2 2 L 2 2 " ÿ kPN |γ k | 2 ∥e k ∥ 2 H 2 ă 8.
Furthermore, let us assume that the nonlinearity F is twice differentiable, and there exists C P p0, 8q such that for all u, h, k P L 2 , one has

(5) F 1 puq.h L 2 ď Cp1 `∥u∥ 2 L 2 q ∥h∥ L 2 F 2 puq.ph, kq L 2 ď Cp1 `∥u∥ L 2 q ∥h∥ L 2 ∥k∥ L 2 .
Finally, let us assume that there exists C 2 P p0, 8q and a polynomial mapping P 2 , such that for all u P H 2 , one has

(6) ∥F puq∥ H 2 ď C σ ∥u∥ H 2 ´1 `∥u∥ 2 L 2 |Impx∇ 2 u, ∇ 2 F puqyq| ď C 2 ∇ 2 u 2 L 2 `P2 ´∥u∥ 2 L 2 , ∥∇u∥ 2 L 2 ¯.
We next give examples of nonlinearities verifying Assumption 1 or 2. First, the conditions in Assumption 1 or 2 are satisfied in the case of a linear mapping F puq " V u, where the external potential function V : T d Ñ R is a real-valued mapping of class C σ , with σ " 1 (resp. σ " 2) to satisfy Assumption 1 (resp. Assumption 2). In that case, the mapping F is globally Lipschitz continuous, and (3) holds with C F " ∥V ∥ C 0 and K F " 0. Second, the conditions in Assumption 1 or 2 also hold for the following class of nonlocal interaction cubic nonlinearities. Note that K F ą 0 in this case. Proposition 2. Let σ P t1, 2u and let V : T d Ñ R be a real-valued mapping of class C σ . For every u P L 2 , set

V rus " V ‹ |u| 2 " ż V p¨´xq|upxq| 2 dx,
where ‹ denotes the convolution operator.

Then Assumption 1 (resp. Assumption 2) is satisfied for the nonlinearity F puq " V rusu " `V ‹ |u| 2 ˘u when σ " 1 (resp. when σ " 2).

Proof. Observe that for any u P L 2 , the mapping V rus is of class C σ , with

∇ σ V rus " ∇ σ V ‹ |u| 2 . It thus follows that ∥V rus∥ C σ ď ∥V ∥ C σ ∥u∥ 2
L 2 for all u P L 2 . First, assume that σ " 1. Let us check that (3) holds. Let u 1 , u 2 P L 2 , then one has

∥F pu 2 q ´F pu 1 q∥ L 2 ď ∥V ru 2 spu 2 ´u1 q∥ L 2 `∥pV ru 2 s ´V ru 1 squ 1 ∥ L 2 ď ∥V ru 2 s∥ C 0 ∥u 2 ´u1 ∥ L 2 `∥V ru 2 s ´V ru 1 s∥ C 0 ∥u 1 ∥ L 2 ď ∥V ∥ C 0 ´∥u 2 ∥ 2 L 2 `∥u 1 `u2 ∥ L 2 ∥u 1 ∥ L 2 ¯∥u 2 ´u1 ∥ L 2 ď 3 2 ∥V ∥ C 0 ´∥u 1 ∥ 2 L 2 `∥u 2 ∥ 2 L 2 ¯∥u 2 ´u1 ∥ L 2 .
Thus (3) holds with C F " 0 and K F " 3 2 ∥V ∥ C 0 . The conditions in (4) follow from straightforward computations.

Second, assume that σ " 2. The conditions in (5) follow from writing, for all u, h, k P L 2 ,

F 1 puq.h " V rush `2`V ‹ Repūhq ˘u F 2 puq.ph, kq " 2 `V ‹ Repūhq ˘k `2`V ‹ Rep kuq ˘h `2`V ‹ Rep hkq ˘u.
The conditions in (6) follow from straightforward computations. This concludes the proof of the Proposition. □

Note that the conditions in Assumption 1 or 2 are not satisfied in the standard cubic nonlinear Schrödinger case, where V rus " ˘|u| 2 , or for other (non-trivial) power-law nonlinearities.

To conclude this section, let us state a well-posedness result for the stochastic Schrödinger equation (1) in terms of mild solutions (2), and several moment bound estimates. Note that additional bounds for the exponential moments in L 2 of the exact solution are given in Section 4.

Proposition 3. Let Assumption 1 be satisfied.

For any initial condition u 0 P L 2 , there exists a unique mild solution `uptq ˘tě0 of the stochastic Schrödinger equation [START_REF] Cohen | Nonlinear Fiber Optics[END_REF], which satisfies (2) for all t ě 0. In addition, for every T P p0, 8q, σ P t0, 1, 2u, u 0 P H σ , and p P r1, 8q, there exists C p pT, α, Q, u 0 q P p0, 8q such that one has a moment bound in H σ sup 0ďtďT

Er∥uptq∥ 2p

H σ s ď C p pT, α, Q, u 0 q, with σ " 1, resp. σ " 2, when Assumption 1, resp. Assumption 2, is satisfied. Finally one has the following temporal regularity estimate: for all t 1 , t 2 P r0, T s,

E " ∥upt 2 q ´upt 1 q∥ 2p L 2 ı ď C p pT, α, Q, u 0 q|t 2 ´t1 | p .
The proof uses standard arguments and is postponed to the appendix.

Splitting scheme

In this section we define a splitting integrator for the stochastic Schrödinger equation (1) and show some geometric properties of this time integrator. The main idea of splitting schemes is to decompose the original problem, equation (1) in our case, into subsystems that can be solved explicitly (or efficiently numerically). Splitting schemes are widely used for time discretization of deterministic cubic Schrödinger equations, see, e.g. the key early reference [START_REF] Hardin | Applications of the split-step fourier method to the numerical solution of nonlinear and variable coefficient wave equations[END_REF].

The definition of the splitting scheme studied in this article relies on the flow associated with the differential equation i 9

u " F puq " V rusu. For all u P L 2 and t P R, define

Φ t puq " e ´itV rus u. Since V rus P R, one has |Φ t puq| " |u| for all t ě 0, which gives V rΦ t puqs " V rus using Assumption 1 or 2. It is then straightforward to check that `Φt ˘tPR is the flow associated with the differential equation i 9

u " F puq. Indeed, for all u P L 2 and all t P R, one has

i d dt Φ t puq " F pΦ t puqq .
Observe that the flow of the above ODE preserves the L 2 -norm: ∥Φ t puq∥ L 2 " ∥u∥ L 2 for all t ě 0 and all u P L 2 . The splitting scheme for the stochastic Schrödinger equation ( 1) considered in this article is then defined by the explicit recursion

(8) u n`1 " Spτ q `Φτ puq ´iαδW Q n ˘,
where τ denotes the time-step size, and δW Q n " W Q ppn `1qτ q ´W Q pnτ q are Wiener increments. Recall that Spτ q " e ´iτ ∆ . Without loss of generality, it is assumed that τ P p0, 1q. The scheme is obtained using a splitting strategy: at each time step, first one may write ũn " Φ τ pu n q, i. e. the equation i 9

u " F puq with initial condition u n is solved exactly, second one has u n`1 " Spτ qũ n ´iαSpτ qδW Q n , which comes from applying an exponential Euler scheme to the stochastic differential equation idu " ∆u dt `αdW Q ptq. Observe that bounds for the exponential moments in L 2 of the numerical solution are given in Section 4.

Remark 4.

Alternatively, solving exactly the stochastic differential equation idu " ∆u dt ὰdW Q ptq yields the following numerical scheme for the SPDE (1) [START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF] u n`1 " Spτ qΦ τ puq ´iα

ż pn`1qτ nτ Sppn `1qτ ´tq dW Q ptq.
Generalizing the results obtained below to this numerical scheme is straightforward.

The error analysis for the splitting scheme (8) presented in the next section will make use of the following additional assumption. Assumption 3. There exists C P p0, 8q such that for all t P r0, 1s and u P L 2 one has

∥Φ t puq ´u∥ L 2 ď C|t| ´1 `∥u∥ 3 L 2 ¯.
Note that Assumption 3 is satisfied for the two examples of nonlinearities described in Section 2. Indeed, ∥Φ t puq ´u∥

L 2 ď t ∥V rus∥ C 0 ∥u∥ L 2 , with ∥V rus∥ C 0 " ∥V ∥ C 0 in the external potential case (V rus " V ) and ∥V rus∥ C 0 ď ∥V ∥ C 0 ∥u∥ 2 L 2 in the nonlocal interaction case (V rus " V ‹ |u| 2 ).
We now present some geometric properties of the splitting scheme (8).

3.1.

Trace formula for the mass. It is well known that, under periodic boundary conditions for instance, the mass, or L 2 -norm or density

M puq :" ∥u∥ 2 L 2 " ż |u| 2 dx
of the deterministic Schrödinger equation i Bu Bt ´∆u ´V rusu " 0, where V rus " V (external potential) or V rus " V ‹ |u| 2 (nonlocal interaction) or V rus " |u| 2 (cubic), is a conserved quantity. In the stochastic case under consideration, one immediately gets a trace formula for the mass of the exact solution of equation ( 1) as well as for the numerical solution given by the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF].

Proposition 5. Consider the stochastic Schrödinger equation (1) with a trace class covariance operator Q and an initial value satisfying ErM pu 0 qs ă 8. We assume that the nonlinearity in (1) is such that F puq " V rusu, where V rus is real-valued and a function of the modulus |u|. Furthermore, we assume that an exact global solution exists and that the differential equation in the splitting scheme can be solved exactly. This is the case for instance when one considers an external potential, a nonlocal interaction, a cubic or power-law nonlinearity.

Then, the exact solution (2) satisfies a trace formula for the mass:

E rM puptqqs :" E " ∥uptq∥ 2 L 2
ı " E rM pu 0 qs `tα 2 TrpQq for all time t.

Furthermore, the numerical solution [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF] to the nonlinear stochastic Schrödinger equation (1) satisfies the exact same trace formula for the mass:

E rM pu n qs " E rM pu 0 qs `tn α 2 TrpQq for all time t n " nτ.

Observe that the above result for the exact solution is already available in the literature in different settings, for instance in [START_REF] De Bouard | The stochastic nonlinear Schrödinger equation in H 1[END_REF][START_REF] Anton | Exponential integrators for stochastic Schrödinger equations driven by Itô noise[END_REF]. However, to the best of our knowledge, the result for the numerical solution is one of the first results in the literature on a longtime qualitative behavior of the numerical solution to nonlinear SPDEs driven by Itô noise. Such a longtime behavior is not satisfied for classical time integrators like the (semi-implicit) Euler-Maruyama schemes, see the numerical experiments below.

Proof. We apply Itô's formula to the mass M puptqq and get (10)

M puptqq " M pup0qq `ż t 0 xM 1 pupsqq, ´iαdW psqy `ż t 0 xM 1 pupsqq, ´i∆upsq ´iV rusupsqy ds `ż t 0 1 2 α 2 Tr " M 2 pupsqq ´Q1{2 ¯´Q 1{2 ¯˚ı ds.
An integration by part and the hypothesis on the potential V show that the third term on the right-hand side is zero. Taking expectation now gives E rM puptqqs " E rM pup0qqs `tα 2 TrpQq which concludes the proof of the trace formula for the mass of the exact solution.

We next show that the above trace formula is also satisfied for the numerical solution given by the splitting integrator [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. Using the definition of the numerical scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF], properties of the Wiener increments δW Q n , as well as the isometry property of Spτ q, one gets

E rM pu n`1 qs " E " ∥Spτ qΦ τ pu n q∥ 2 L 2 ı `α2 E " δW Q n 2 L 2 ı " E " ∥Φ τ pu n q∥ 2 L 2 ı `τ α 2 TrpQq.
The definition of the flow Φ τ yields E rM pu n`1 qs " E rM pu n qs `τ α 2 TrpQq and a recursion completes the proof of the proposition. □ Remark 6. The same trace formula for the mass holds for the numerical solution given by the time integrator [START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF]. Indeed, using the definition of the numerical scheme (9), Itô's isometry, as well as the isometry property of the operator Spτ q, one gets

E rM pu n`1 qs " E " ∥Φ τ pu n q∥ 2 L 2 ı `τ α 2 TrpQq.
Employing the definition of the flow Φ τ followed by a recursion shows the trace formula for the mass of the splitting scheme [START_REF] Blanes | A concise introduction to geometric numerical integration[END_REF].

Remark 7. It may also be possible to study the longtime behavior of the exact and numerical solutions along the expected value of the Hamiltonian of (1) with α ‰ 0. However, in general, the drift in the expected Hamiltonian will depend on the solution u, see for example [START_REF] Debussche | Numerical simulation of focusing stochastic nonlinear Schrödinger equations[END_REF]Equation p11q] for the cubic case. In particular, the evolution of this quantity will not be linear in time. Such a trace formula for the energy will thus unfortunately not be as simple as the one for the mass. Very recent studies have been carried on for (mainly) the Crank-Nicolson scheme in the preprint [START_REF] Millet | Behavior of solutions to the 1D focusing stochastic L 2 -critical and supercritical nonlinear Schrödinger equation with space-time white noise[END_REF]. In particular, it is observed that this numerical scheme does not verify an exact trace formula for the mass, see also the numerical experiments below. We leave the question of investigating such trace formula for the Hamiltonian of the splitting scheme for future work.

3.2. Stochastic symplecticity. Symplectic schemes are known to have excellent longtime properties when applied to Hamiltonian (partial) differential equations, see for instance [START_REF] Leimkuhler | Simulating Hamiltonian dynamics[END_REF][START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Cohen | Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems[END_REF][START_REF] Cohen | Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations[END_REF][START_REF] Cohen | One-stage exponential integrators for nonlinear Schrödinger equations over long times[END_REF][START_REF] Cohen | Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions[END_REF][START_REF] Gauckler | Numerical long-time energy conservation for the nonlinear Schrödinger equation[END_REF] and references therein. These particular integrators have thus naturally come into the realm of stochastic (partial) differential equations, see for example [START_REF] Bou-Rabee | Stochastic variational integrators[END_REF][START_REF] Burrage | Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise[END_REF][START_REF] Anton | Symplectic schemes for stochastic Hamiltonian systems preserving Hamiltonian functions[END_REF][START_REF] Milstein | Symplectic integration of Hamiltonian systems with additive noise[END_REF][START_REF] Chen | Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation[END_REF][START_REF] Hong | Convergence analysis of a symplectic semi-discretization for stochastic NLS equation with quadratic potential[END_REF][START_REF] Cohen | Exponential integrators for stochastic Maxwell's equations driven by Itô noise[END_REF]] and references therein.

The next result shows that the exact flow of the SPDE (1) as well as the proposed splitting scheme (8) are stochastic symplectic. Proposition 8. Consider the stochastic Schrödinger equation [START_REF] Cohen | Nonlinear Fiber Optics[END_REF] and assume that a global solution exists. Under the same assumptions as in the previous proposition, the exact flow of this SPDE is stochastic symplectic in the sense that it preserves the symplectic form

ωptq " ż T d dp ^dq dx a.s.,
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 the torus. Here, p and q denote the real and imaginary parts of u.

Furthermore, the splitting scheme (8) applied to the stochastic Schrödinger equation (1) is stochastic symplectic in the sense that it possesses the discrete symplectic structure: ωn`1 " ωn a.s., for the symplectic form ωn :" ż

T d
dp n ^dq n dx, where p n , resp. q n denoting the real and imaginary parts of u n , and d denotes differentials in the phase space.

Proof. The symplecticity of the phase flow of the stochastic Schrödinger equation ( 1) can be shown using similar arguments as in [33, Theorem 3.1] for a stochastic cubic Schrödinger equation with quadratic potential, see also [START_REF] Chen | Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation[END_REF].

In order to show that the numerical solution is stochastic symplectic as well, we use the same argument as in the proof of [START_REF] Cohen | Exponential integrators for stochastic Maxwell's equations driven by Itô noise[END_REF]Prop. 4.3]. Taking the differential of the numerical solution yields du n`1 " d `Spτ q `Φτ pu n q ´iαδW Q n ˘˘" d pSpτ qΦ τ pu n qq " du n , where in the last equality we have used the fact that the composition of exact flows is symplectic. This concludes the proof. □ Remark 9. The exact same proof shows that the splitting scheme (9) possesses a discrete symplectic structure.

Convergence results

In this section, we study various types of convergence (strong, in probability and almostsurely) of the splitting scheme (8) when applied to the stochastic Schrödinger equation [START_REF] Cohen | Nonlinear Fiber Optics[END_REF]. In order to do this, we first show bounds for the exponential moments in the L 2 norm of the exact and numerical solutions as well as two auxiliary results. The proofs of these results are given in Section 5 for the reader's convenience.

Theorem 10. Let us apply the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF] to the stochastic Schrödinger equation (1) with a trace class covariance operator Q and deterministic initial value u 0 P L 2 . Assume that the nonlinearity in (1) satisfies F puq " V rusu, where V rus " V rūs is real-valued, and that the exact and numerical solutions are well defined on the interval r0, T s. One then has the following bounds for the exponential moments: there exists κ ą 0 and τ ‹ such that if µα 2 T ă κ TrpQq , then one has:

sup 0ďtďT E " exp ´µ ∥uptq∥ 2 L 2 ¯ı ď Cpµ, T, α, Q, u 0 q ă 8
for the exact solution and

sup τ Pp0,τ ‹ q sup 0ďnτ ďT E " exp ´µ ∥u n ∥ 2 L 2 ¯ı ď Cpµ, T, α, Q, u 0 q ă 8
for the numerical solution.

In the proof of this theorem, the lower bound κ ě e ´1 2 is obtained, note that it does not depend on the nonlinearity. Furthermore, observe that the condition µα 2 T ă κ TrpQq gets more restrictive when α and T increase.

It is immediate to deduce the following moment estimates for the exact and numerical solutions from Theorem 10.

Corollary 11.

Under the assumptions of the previous theorem, for any p P r1, 8q and T P p0, 8q, one has the following moment estimates for the L 2 norm of the exact and numerical solutions: for any u 0 P L 2 , there exists C p pT, α, Q, u 0 q P p0, 8q such that

sup 0ďtďT Er∥uptq∥ 2p L 2 s ď C p pT, α, Q, u 0 q and sup τ Pp0,τ ‹ q sup 0ďnτ ďT Er∥u n ∥ 2p L 2 s ď C p pT, α, Q, u 0 q.
Observe that one could show this statement for τ P p0, 1q directly without using the exponential moments estimates given in Theorem 10.

In order to show the main convergence result of this article, we will make use of the following two propositions. Each one of these propositions are used to show strong convergence order 1{2, resp. 1, of the numerical solution given by the splitting scheme. Proposition 12. Consider the time discretization of the stochastic Schrödinger equation (1) by the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. Let Assumptions 1 and 3 be satisfied. Assume that u 0 P H 1 .

Let T P p0, 8q. For every q P r1, 8q, there exists C q pT, u 0 q P p0, 8q (which depends on F , Q and on α), such that for every τ P p0, τ ‹ q, one has

sup 0ďnτ ďT E " expp´qK F S n q ∥u n ´upt n q∥ q L 2 ‰ ď C q pT, u 0 qτ q 2 ,
where K F is given in (3) (see Assumption 1), and

S n " τ n´1 ÿ k"0 ´∥upkτ q∥ 2 L 2 `∥u k ∥ 2 L 2
¯.

Proposition 13. Consider the time discretization of the stochastic Schrödinger equation (1)

by the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. Let Assumptions 2 and 3 be satisfied. Assume that u 0 P H 2 . Let T P p0, 8q. For every q P r1, 8q, there exists C q pT, u 0 q P p0, 8q (which depends on F , Q and on α), such that for every τ P p0, τ ‹ q, one has

sup 0ďnτ ďT E " expp´qK F S n q ∥u n ´upt n q∥ q L 2 ‰ ď C q pT, u 0 qτ q ,
where K F is given in (3) (see Assumption 1) and

S n " τ n´1 ÿ k"0 ´∥upkτ q∥ 2 L 2 `∥u k ∥ 2 L 2
¯.

The proofs of the technical results, Theorem 10 and Propositions 12 and 13, are postponed to Section 5.

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

Theorem 14. Let uptq denote the exact solution to the stochastic Schrödinger equation (1) and u n the numerical solution given by the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. Let Assumption 3 be satisfied. Let also σ " 1, resp. σ " 2, if Assumption 1, resp. Assumption 2, is satisfied. Assume that

u 0 P H σ . Recall the notation S n " τ n´1 ÿ k"0 ´∥upkτ q∥ 2 L 2 `∥u k ∥ 2 L 2 ¯.
Let T P p0, 8q. Assume that µ P p0, 8q and τ 0 P p0, τ ‹ q are chosen such that

(11) sup τ Pp0,τ 0 q sup 0ďnτ ďT E rexppµS n qs " CpT, u 0 , α, Q, τ 0 , µq ă 8.
Then, for all r P p0, 8q and all µ P p0, µq, there exists C r pT, u 0 , α, Q, τ 0 , µq ă 8 such that for all τ P p0, τ 0 q one has

(12) sup 0ďnτ ďT pE r∥u n ´upt n q∥ r L 2 sq 1 r ď C r pT, u 0 , α, Q, τ 0 , µqτ σ 2 minp1, µ rK F q .
As a consequence of this theorem, the convergence is polynomial in L r pΩq, for all r P r1, 8q. The rate of convergence of the splitting scheme depends on r in [START_REF] Brezzi | The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation[END_REF], and vanishes when r Ñ 8. Note that for sufficiently small r ą 0, one has minp1, µ rK F q " 1, thus the convergence rate of the splitting scheme is σ 2 when r is sufficiently small. Observe also that a sufficient condition for condition [START_REF] Bou-Rabee | Stochastic variational integrators[END_REF] to be verified is that

µ ă κ α 2 T 2 TrpQq ,
where κ ą 0 is some positive constant (see Remark 15 below). Thus the value of minp1, µ rK F q depends on the quantity α 2 T 2 K F (considering that TrpQq is fixed and that the size of the noise is given by α). The larger this quantity, the more restrictive the condition to have minp1, µ rK F q " 1 becomes. In the external potential case V rus " V u, one has K F " 0, thus there is no restrictions and the order of convergence is σ 2 in L r pΩq for all r P r1, 8q. Remark 15. Owing to Theorem 10 concerning exponential moments of the exact and numerical solutions, the set of parameters µ, τ 0 such that [START_REF] Bou-Rabee | Stochastic variational integrators[END_REF]

holds is non-empty. Indeed, recalling that S n " τ n´1 ÿ k"0 ´∥upkτ q∥ 2 L 2 `∥u k ∥ 2 L 2 ¯and using Cauchy-Schwarz inequality, one has E rexppµS n qs " E « n´1 ź k"0 ´exppµτ ∥upt k q∥ 2 L 2 q exppµτ ∥u k ∥ 2 L 2 q ¯ff ď n´1 ź k"0 ´E " expp2nτ µ ∥upt k q∥ 2 L 2 q ı E " expp2nτ µ ∥u k ∥ 2 L 2 q ı¯1 2n ď sup 0ďkďn E " expp2T µ ∥upt k q∥ 2 L 2 q ı sup 0ďkďn E " expp2T µ ∥u k ∥ 2 L 2 q ı ď Cpµ, T, α, Q, u 0 q, if µ ă κ α 2 T 2
TrpQq and τ ă τ ‹ , where κ and τ ‹ are given in Theorem 10. The value of µ obtained by the argument above (as well as the values of κ " e ´1 2 and τ ‹ ) may not be optimal.

Proof of Theorem 14. Set e n " ∥u n ´upt n q∥ L 2 . For every R P p0, 8q, let χ n,R " 1 SnďR . Then Ere r n s " Ere r n χ n,R s `Ere r n p1 ´χn,R qs. For a given µ P p0, µq, let p P p1, 8q such that µ " p1 ´1 p qµ.

On the one hand, applying the Cauchy-Schwarz and Markov inequalities yields This concludes the proof of the theorem. □

E
To conclude this section, let us state results concerning convergence in probability, with order of convergence equal to σ 2 , and almost sure convergence with order of convergence σ 2 ´ε for all ε P p0, 1 2 q, with σ P t1, 2u. Corollary 16. Consider the stochastic Schrödinger equation (1) on the time interval r0, T s with solution denoted by uptq. Let u n be the numerical solution given by the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF] with time-step size τ . Under the assumptions of Theorem 14, one has convergence in probability of order σ

2 lim CÑ8 P ´∥u N ´upT q∥ L 2 ě Cτ σ 2 ¯" 0,
where T " N τ .

Moreover, consider the sequence of time-step sizes given by τ M " T 2 M , M P N. Then, for every ε P p0, σ 2 q, there exists an almost surely finite random variable C ε , such that for all M P N one has

∥u 2 M ´upT q∥ L 2 ď C ε ˆT 2 M ˙σ 2 ´ε .
Proof. Let r be chosen sufficiently small, such that applying Theorem 14 yields

E r∥u N ´upT q∥ r L 2 s ď Cpr, T qτ rσ 2 .
Then the convergence in probability result is a straightforward consequence of Markov's inequality:

P ´∥u N ´upT q∥ L 2 ě Cτ σ 2 ¯" P ´∥u N ´upT q∥ r L 2 ě C r τ rσ 2 ¯ď E r∥u N ´upT q∥ r L 2 s C r τ rσ 2 " Cpr, T q C r Ñ CÑ8 0.
To get the almost sure convergence result, it suffices to observe that (again by applying Theorem 14)

8 ÿ m"0 E r∥u 2 m ´upT q∥ r L 2 s τ rp σ 2 ´εq m ă 8, thus ∥u 2 M ´upT q∥ r L 2 τ rp σ 2 ´εq M Ñ M Ñ8
0 almost surely. □

Proofs of technical results

This section is devoted to giving the proofs to Theorem 10 and Propositions 12 and 13.

To simplify notation, we let Q α " α 2 Q, where we recall that Q is the covariance operator of the noise in the SPDE (1). ¯eλTrpQαqT , where we recall that λ " 1 2T TrpQαq " 1 2α 2 T TrpQq . This concludes the proof of the exponential moment estimates for the exact solution of the stochastic Schrödinger equation [START_REF] Cohen | Nonlinear Fiber Optics[END_REF].

Let us now prove the exponential moment estimates for the numerical solution [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF]. Let p, q ą 1 such that 1{p `1{q " 1, and set λ " 1 2T pTrpQαq . Define r n " λ expp´n N q for n " 1, . . . , N , where N τ " T , and introduce the filtration F n " σtδW Qα k ; k ď n ´1u. Note that u n is F n -measurable. Let also τ ‹ P p0, ppp ´1qq.

Using the definition of the scheme (8) and Hölder's inequality, one has

E " exp `rn`1 ∥u n`1 ∥ 2 L 2 ˘| F n ‰ ď Erexp ´rn`1 ∥u n ∥ 2 L 2 ¯s `Erexpp2pr n`1 ImpxΦ τ pu n q, δW Qα n yqq | F n s ˘1 p ´Erexp ´qr n`1 δW Qα n 2 L 2 ¯s¯1 q .
On the one hand, since δW Qα n is a centered Gaussian random variable and by definition of r n , one has

E " exp ´qr n`1 δW Qα n 2 L 2 ¯ı ď ´1 ´2qr n`1 E " δW Qα n 2 L 2 ı¯´1 2 ď p1 ´2qλτ TrpQ α qq ´1 2 ,
under the condition that τ ă 1 2qλTrpQαq " p 2 q . This condition thus holds when τ ă τ ‹ . On the other hand, conditional on F n , the random variable xΦ τ pu n q, δW Qα n y is also Gaussian and centered, thus

E " exp `2pr n`1 ImpxΦ τ pu n q, δW Qα n yq ˘| F n ‰ ď exp `2p 2 r 2 n`1 VarrxΦ τ pu n q, δW Qα n ys ď exp ´2p 2 λr n`1 τ TrpQ α q ∥u n ∥ 2 L 2 ¯.
Gathering these estimates and taking expectation yield

E " exp ´rn`1 ∥u n`1 ∥ 2 L 2 ¯ı ď E " exp ´rn`1 p1 `2pλτ TrpQ α qq ∥u n ∥ 2 L 2 ¯ı p1 ´2qλτ TrpQ α qq ´1 2q .
Having chosen λ " 1 2pT TrpQαq , one then gets r n`1 p1 `2pλτ TrpQ α qq " r n e ´τ T p1 `τ T q ď r n . A recursion on n then gives the following estimate

sup 0ďnτ ďT E " exp ´rn ∥u n ∥ 2 L 2 ¯ı ď exppλ ∥u 0 ∥ 2 L 2 qp1 ´2qλτ TrpQ α qq ´N 2q ď Cpλ, u 0 q ă 8,
for τ ă τ ‹ , where the quantity Cpλ, u 0 q does not depend on τ .

We are now in position to conclude the proof of exponential moments estimates for the numerical solution. Let µ such that µ ă e ´1 2T TrpQαq . Note that r N " λe ´1, thus there exists p ą 1 such that µ ď r N ď r n for all n P t0, . . . , N u. This then implies that

sup 0ďnτ ďT E " exp ´µ ∥u n ∥ 2 L 2 ¯ı ď Cpµ, T, Q, u 0 q ă 8,
for all τ P p0, τ ‹ q. This concludes the proof of Theorem 10. □ 5.2. Proofs of Propositions 12 and 13. Before we start with these proofs, it is convenient to introduce some auxiliary notation and provide the steps that are common for both proofs.

Define wptq " ´αi ż t 0 Spt ´sq dW Q psq for all t ě 0 and w n " ´αi n´1 ÿ k"0 Spτ q n´k δW Q k for all n ě 0. Introduce also vptq " uptq ´wptq and v n " u n ´wn . Let t k " kτ . Recall that

S n " τ n´1 ÿ k"0 ´∥upkτ q∥ 2 L 2 `∥u k ∥ 2 L 2 ¯.
Define ϵ n " ∥vpt n q ´vn ∥ L 2 and e n " ∥upt n q ´un ∥ L 2 . Then the error between the numerical and exact solution reads e n ď ϵ n `∥w n ´wpt n q∥ L 2 .

Let us first deal with the error term ∥w n ´wpt n q∥ L 2 for the stochastic convolution: employing the Itô isometry formula, with σ " 1 (resp. σ " 2) if Assumption 1 (resp. Assumption 2) is satisfied, one has

E " ∥w n ´wpt n q∥ 2 L 2 ı " α 2 E » - n´1 ÿ k"0 ż t k`1 t k ´Spτ q n´k ´Spt n ´tq ¯dW Q ptq 2 L 2 fi fl " α 2 n´1 ÿ k"0 ż t k`1 t k ´Spτ q n´k ´Spt n ´tq ¯Q 1 2 2 L 0 2 dt ď α 2 n´1 ÿ k"0 ż t k`1 t k |t ´tk | σ dt Q 1 2 2 L σ 2 ď CpT, α, Qqτ σ ,
using properties of the semigroup S. Since the distribution of w n ´wpt n q is Gaussian, for every q P r1, 8q, there exists C q pT, Qq P p0, 8q such that one has [START_REF] Burrage | Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise[END_REF] Er∥w n ´wpt n q∥ q L 2 s ď C q pT, α, Qqτ qσ 2 . It remains to treat the error term ϵ n " ∥v n ´vpt n q∥ L 2 . Using the mild formulation (2) of the solution upt n q and the definition of the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF] for u n , one obtains ( 14)

v n`1 ´vpt n`1 q " ˆSpτ qv n ´i ż t n
" Spτ q pv n ´vpt n qq `E1 n `E2 n `E3 n `E4 n , where E 1 n " i ż t n`1 tn pSpt n`1 ´tq ´Spτ qq F puptqq dt E 2 n " i ż t n`1
tn Spτ q pF puptqq ´F pupt n qqq dt

E 3 n " iτ Spτ q pF pupt n qq ´F pu n qq E 4 n " i ż t n`1
tn Spτ q pF pu n q ´F pΦ t´tn pu n qqq dt.

For the first term, using properties of the semigroup S (see Lemma 1), for σ P t1, 2u, one has

E 1 n L 2 ď Cτ σ 2 ż t n`1 tn ∥F puptqq∥ H σ dt.
The treatment of the second term E 2 n is different for the two propositions, details are provided below.

For the third term, recall that ∥u n ´upt n q∥ L 2 ď ϵ n `∥w n ´wpt n q∥ L 2 . Using (3), one obtains

E 3 n L 2 ď τ ´CF `KF p∥upt n q∥ 2 L 2 `∥u n ∥ 2 L 2 q ¯∥u n ´upt n q∥ L 2 ď τ ´CF `KF p∥upt n q∥ 2 L 2 `∥u n ∥ 2 L 2 q ¯ϵn `τ ´CF `KF p∥upt n q∥ 2 L 2 `∥u n ∥ 2 L 2 q ¯∥w n ´wpt n q∥ L 2 .
For the fourth term, using (3), the equality ∥Φ t´tn pu n q∥ L 2 " ∥u n ∥ L 2 , and Assumption 3, one obtains

E 4 n L 2 ď C ż t n`1 tn ´1 `∥u n ∥ 2 L 2 `∥Φ t´tn pu n q∥ 2 L 2 ¯∥u n ´Φt´tn pu n q∥ L 2 dt ď C ´1 `2 ∥u n ∥ 5 L 2 ¯ż t n`1 tn |t ´tn | dt ď Cτ 2 ´1 `∥u n ∥ 5 L 2 ¯.
At this stage, it is necessary to treat separately the proofs for Proposition 12 and 13.

Proof of Proposition 12. Assume that σ " 1. For the second error term E 2 n , using the assumption on F and Cauchy-Schwarz inequality, one has

E 2 n 2 L 2 ď C ż t n`1 tn ´1 `∥uptq∥ 2 L 2 `∥upt n q∥ 2 L 2 ¯2 dt ż t n`1 tn ∥uptq ´upt n q∥ 2 L 2 dt.
Gathering all the estimates, and using the isometry property ∥Spτ q pv n ´vpt n qq∥ L 2 " ∥v n ´vpt n q∥ L 2 " ϵ n , from ( 14) one obtains

ϵ n`1 ď `1 `CF τ `KF τ Γ n qϵ n `Rn , where we define Γ n " ∥upt n q∥ 2 L 2 `∥u n ∥ 2 L 2 and R n " E 1 n L 2 ` E 2 n L 2 ` E 4 n L 2 `KF τ Γ n ∥w n ´wpt n q∥ L 2 .
Using a discrete Gronwall inequality and the equality ϵ 0 " 0, one gets for all n P t0, . . . , N u

exp ˜´C F nτ ´KF τ n´1 ÿ k"0 Γ k ¸ϵn ď n´1 ÿ k"0 R k . Rewriting τ n´1 ÿ k"0 Γ k " S n and ∥u n ´upt n q∥ L 2 ď ϵ n `∥w n ´wpt n q∥ L 2 ,
applying Minkowskii's inequality yields for q P r1, 8q

E " exp p´qK F S n q ∥u n ´upt n q∥ q L 2 ‰ 1 q ď e C F T n´1 ÿ k"0 `E " R q k ‰˘1 q `eC F T `E " ∥w n ´wpt n q∥ q L 2 ‰˘1 q .
We now estimate each of the terms above. Let us first recall that

R k " E 1 k L 2 ` E 2 k L 2 ` E 4 k L 2 `KF τ Γ k ∥w k ´wpt k q∥ L 2 .
Using the triangle inequality, followed by Cauchy-Schwarz's inequality, the assumption on the nonlinearity F as well as moment estimates in the L 2 and H 1 norms for the exact solution (Corollary 11 and Proposition 3), one obtains

E " E 1 k q L 2 ‰ 1{q ď Cτ 1{2 ż t k`1 t k E " ∥F puptqq∥ q H 1 ‰ 1{q dt ď Cτ 1{2 ż t k`1 t k E " ∥uptq∥ 2q H 1 ı 1{p2qq E " p1 `∥uptq∥ 2 L 2 q 2q ı 1{p2qq dt ď Cτ 1{2 ż t k`1 t k dt ď Cτ 3{2 .
For the second term, we use Cauchy-Schwarz's inequality and moment bounds and regularity properties of the exact solution from Proposition 3 to get

E " E 2 k q L 2 ‰ 1{q ď C ˜ż t k`1 t k E " ´1 `∥uptq∥ 2 L 2 `∥upt k q∥ 2 L 2 ¯2q ȷ 1{q dt ¸1{2 ˆż t k`1 t k E " ∥uptq ´upt k q∥ 2q L 2 ı 1{q dt ˙1{2 ď Cτ 1{2 ˆż t k`1 t k |t ´tk | dt ˙1{2 ď Cτ 3{2 .
Similarly, using the Cauchy-Schwarz's inequality and the moment estimates in the L 2 norm for the numerical solution (Corollary 11), we obtain

E " E 4 k q L 2 ‰ 1{q ď Cτ 2 E " ´1 `2 ∥u n ∥ 2 L 2 ¯2q ȷ 1{p2qq E " ∥u n ∥ 10q L 2 ı 1{p2qq ď Cτ 2 .
Thanks to the bounds for the moments in the L 2 norm given by Corollary 11, as well as to the error estimate [START_REF] Burrage | Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise[END_REF] for the stochastic convolution proved above, we obtain the estimate

E " pK F τ Γ k ∥w k ´wpt k q∥q q L 2 ‰ 1{q ď Cτ E " Γ 2q k ı 1{p2qq E " ∥w k ´wpt k q∥ 2q L 2 ı 1{p2qq ď Cτ E " ∥w k ´wpt k q∥ 2q L 2 ı 1{p2qq ď Cτ τ 1{2 ď Cτ 3{2 .
With all these estimates at hand, we arrive at

n´1 ÿ k"0 `E " R q k ‰˘1 q ď C q pT, u 0 , α, Qqτ 1 2 .
Finally, we obtain

E " exp p´qK F S n q ∥u n ´upt n q∥ q L 2 ‰ 1 q ď e C F T n´1 ÿ k"0 `E " R q k ‰˘1 q `eC F T `E " ∥w n ´wpt n q∥ q L 2 ‰˘1 q ď C q pT, u 0 , α, Qqτ 1 2 `Cq pT, α, Qqτ 1 2 , using (13)
in the last step. This concludes the proof of Proposition 12.

□

We now turn to the proof of the second auxiliary result.

Proof of Proposition 13. Assume that σ " 2. As explained above, one requires to substantially modify the treatment of the error term E 2 n . As will be clear below, some changes in the analysis of the error ϵ n are required too.

Using a second-order Taylor expansion of the nonlinearity F and equation ( 5) (assumption on F 2 ), one obtains the decomposition

E 2 n " E 2,1 n `E2,2 n where E 2,1 n " i ż t n`1 tn Spτ qF 1 pupt n qq. `uptq ´upt n q ˘dt E 2,2 n L 2 ď C ż t n`1 tn p1 `∥upt n q∥ L 2 `∥uptq∥ L 2 q ∥uptq ´upt n q∥ 2 L 2 dt.
In addition, using the mild formulation of the exact solution (2), one has the decomposition

E 2,1 n " E 2,1,1 n `E2,1,2 n `E2,1,3 n
, where

E 2,1,1 n " i ż t n`1 tn Spτ qF 1 pupt n qq. pSpt ´tn q ´Iqq upt n q dt E 2,1,2 n " i ż t n`1 tn Spτ qF 1 pupt n qq. ˆż t tn Spt ´sqF pupsqq ds ˙dt E 2,1,3 n " iα ż t n`1 tn Spτ qF 1 pupt n qq. ˆż t tn Spt ´sq dW Q psq ˙dt.
Owing to Lemma 1 and to equation ( 5) in Assumption 2, the first and second terms above are treated as follows: one has

E 2,1,1 n L 2 ď Cτ 2 ´1 `∥upt n q∥ 2 L 2 ¯∥upt n q∥ H 2 , and E 2,1,2 n L 2 ď C ´1 `∥upt n q∥ 2 L 2 ¯τ ż t n`1 tn ∥F pupsqq∥ L 2 ds.
Using the stochastic Fubini Theorem, the third term is written as

E 2,1,3 n " iα ż t n`1 tn Spτ qF 1 pupt n qq. `ż t tn Spt ´sq dW Q psq ˘dt " iα ż t n`1 tn ˆSpτ qF 1 pupt n qq. ż t n`1 s Spt ´sq dt ˙dW Q psq " iα ż t n`1 tn Θ n psq dW Q psq,
where we have defined the quantity Θ n psq " Spτ qF 1 pupt n qq.

ż t n`1 s Spt ´sq dt.
Applying Itô's formula, one gets

(15) E " E 2,1,3 n 2 L 2 ı " α 2 ż t n`1 tn E " Θ n psqQ 1 2 2 L 0 2 ȷ ds ď Cτ 3
using again (5) from Assumption 2 and the moment estimates in the L 2 norm of the exact solution from Corollary 11. However the estimate [START_REF] Cohen | Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems[END_REF] is not sufficient to directly obtain the required error estimate for ϵ n as in the proof of Proposition 12. Improving this estimate requires to exploit the

orthogonality property E " xE 2,1,3 n , E 2,1,3 m y ı " 0 if n ‰ m,
and to modify the approach used above to deal with the error.

Starting from [START_REF] Chen | Symplectic Runge-Kutta semidiscretization for stochastic Schrödinger equation[END_REF], one obtains for all n ě 0

v n ´vpt n q " n´1 ÿ k"0 Spτ q n´k´1 `E1 k `E2 k `E3 k `E4 k ˘.
Recalling the decomposition

E 2,1 k " E 2,1,1 k `E2,1,2 k `E2,1,3
k and using the above bounds for the term E 3 k then yields

ϵ n ď τ n´1 ÿ k"0 pC F `KF Γ k q ϵ k `τ n´1 ÿ k"0 pC F `KF Γ k q ∥wpt k q ´wk ∥ L 2 `n´1 ÿ k"0 ` E 1 k L 2 ` E 4 k L 2 n´1 ÿ k"0 E 2 k ´E2,1,3 k L 2 ` n´1 ÿ k"0 Spτ q n´1´k E 2,1,3 k L 2
.

Applying the Gronwall inequality to get an almost sure inequality, then using the Cauchy-Schwarz and Minkowskii's inequalities, one obtains for all n ě 0 and all q P r1, 8q

e ´CF nτ `Ere ´qK F Sn ϵ q n s ˘1{q ď τ n´1 ÿ k"0 ´E " pC F `KF Γ k q 2q ı¯1 {p2qq ´E " ∥wpt k q ´wk ∥ 2q L 2 ı¯1 {p2qq `n´1 ÿ k"0 ´`E " E 1 k q L 2 ‰˘1 {q ``E " E 4 k q L 2 ‰˘1 {q n´1 ÿ k"0 ´E " E 2 k ´E2,1,3 k q L 2 ı¯1 {q `¨E » - n´1 ÿ k"0 Spτ q n´1´k E 2,1,3 k 2q L 2 fi fl '1{p2qq ,
where in the last term, we have used the inclusion L 2q pΩq Ă L q pΩq. Using the same arguments as in the proof of Proposition 12 (in particular moment estimates of Proposition 3 and Corollary 11, and the error estimate [START_REF] Burrage | Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise[END_REF] for the stochastic convolution), the treatment of the error terms in the right-hand side is straightforward, except for the last one which requires more details that we now present. First, note that using the Burkholder-Davis-Gundy inequality one has

¨E » - n´1 ÿ k"0 Spτ q n´1´k E 2,1,3 k 2q L 2 fi fl '1{p2qq ď C q ¨E » - n´1 ÿ k"0 Spτ q n´1´k E 2,1,3 k 2 L 2 fi fl '1{2 ,
thus it is sufficient to deal with the case q " 1. Second, one has

E » - n´1 ÿ k"0 Spτ q n´1´k E 2,1,3 k 2 L 2 fi fl " n´1 ÿ k"0 E " E 2,1,3 k 2 L 2 ȷ ď Cτ 2 ,
using the orthogonality property E " xE 2,1,3 n , E 2,1,3 m y ı " 0 if n ‰ m mentioned above, and the estimate [START_REF] Cohen | Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems[END_REF].

Finally, recalling that ∥u n ´upt n q∥ L 2 ď ϵ n `∥w n ´wpt n q∥ L 2 , gathering all these estimates and using the bounds on the error in the stochastic convolution [START_REF] Burrage | Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise[END_REF], we obtain

E " exp p´qK F S n q ∥u n ´upt n q∥ q L 2 ‰ 1 q ď C q pT, u 0 , Qqτ.
This concludes the proof of Proposition 13. □

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 (8) (denoted by Split below) with the following time integrators ' the classical Euler-Maruyama scheme (denoted EM) u n`1 " u n ´iτ ∆u n ´iτ F pu n q ´iαδW Q n . ' the classical semi-implicit Euler-Maruyama scheme (denoted sEM)

u n`1 " u n ´iτ ∆u n`1 ´iτ F pu n q ´iαδW Q n .
' the stochastic exponential integrator from [START_REF] Anton | Exponential integrators for stochastic Schrödinger equations driven by Itô noise[END_REF] (denoted sEXP)

u n`1 " Spτ q `un ´iτ F pu n q ´iαδW Q n ˘.
' the Crank-Nicolson-Euler-Maruyama (denoted CN) u n`1 " u n ´iτ ∆u n`1{2 ´iτ F pu n q ´iαδW Q n , where u n`1{2 " 1 2 pu n `un`1 q. This is a slight modification of the Crank-Nicolson from [START_REF] De Bouard | Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation[END_REF].

6.1. Trace formulas for the mass. We consider the stochastic Schrödinger equation (1) on the interval r0, 2πs with periodic boundary condition, the coefficient α " 1, and a covariance operator with pγ k q kPZ " ´1 1`k 2 ¯kPZ and pe k pxqq kPZ " ´1 ? 2π e ikx ¯kPZ . We consider the initial value u 0 " 2 2´cospxq and the following nonlinearities: V pxqu "

3

5´4 cospxq u (external potential), `V ‹ |u| 2 ˘u with V pxq " cospxq (nonlocal interaction), F puq " `|u| 2 u (cubic). We refer to [START_REF] De Bouard | The stochastic nonlinear Schrödinger equation in H 1[END_REF]Theorem 3.4] for a result on global existence of solutions to the cubic case. We use a pseudo-spectral method with N x " 2 8 modes and the above time integrators with time-step size τ " 0.1.

Figure 1 displays the evolution of the expected value of the mass on the time intervals r0, 1s (external potential) and r0, 25s (other cases). The expected values are approximated using M " 75000 samples. The exact trace formulas for the splitting scheme, shown in Proposition 5, can be observed. The growth rates of the other schemes are qualitatively different than this linear rate of the exact solution: observe for instance the exponential drift of EM in the first plot, the fact that sEXP seems to overestimate the linear drift and the fact that sEM underestimates it. The CN scheme performs relatively well, except in the cubic case (not displayed), where it should use a much smaller step-size in order not to explode. Trace formulas for mass of the splitting scheme (Split), the Euler-Maruyama scheme (EM), the semi-implicit Euler-Maruyama scheme (sEM), the exponential integrator (sEXP), and the Crank-Nicolson (CN) schemes.

Strong convergence.

In this subsection, we illustrate the strong convergence of the splitting scheme [START_REF] Besse | Energy preserving methods for nonlinear Schrödinger equations[END_REF] as stated in Theorem 14.

To do this, we consider the stochastic Schrödinger equation (1) on the interval r0, 2πs with periodic boundary condition, and a covariance operator with pγ k q kPZ " ´1 1`k 2 ¯kPZ .

We consider the external potential V pxq " 3 5´4 cospxq and nonlocal interaction given by the potential V pxq " cospxq. We take the initial value u 0 " 2 2´cospxq (external potential) and u 0 " 1 1`sinpxq 2 (nonlocal interaction). Additional parameters are: coefficient in front of the noise α " 1.5, time interval r0, 1s, 250 samples used to approximate the expectations. We use a pseudo-spectral method with N x " 2 10 modes and the above time integrators. Strong errors, measured with r " 1 at the end point, are presented in Figure 2. or this numerical experiment, the splitting and exponential integrators give very close results. For clarity, only some of the values for the exponential integrator are displayed. An order 1{2 of convergence for the splitting scheme is observed. Note that, the strong order of convergence of the other time integrators are not known in the case of the nonlocal interaction potential.

In order to illustrate the higher order of convergence for the splitting scheme, we consider a smoother noise with covariance operator with pγ k q kPZ " ´1 1`k 4 ¯kPZ (the other parameters for the simulation are as above). Results are presented in Figure 3, where a strong order of convergence 1 is observed for the proposed time integrator, in agreement with Theorem 14.

Convergence in probability.

In this subsection we numerically demonstrate the order of convergence in probability for the splitting scheme (8). This order has been shown to be 1{2 in Corollary 16 above. Numerically, we investigate the order in probability by using the equation ( 16)

max nPt1,2,...,N u ∥u n ´uref pt n q∥ L 2 ě Cτ δ ,
where u ref denotes a reference solution computed using the splitting scheme with step-size τ ref " 2 ´16 . We then study the proportion of samples, P , fulfilling equation ( 16) for given C and δ and observe whether P Ñ 0 for the given δ as τ Ñ 0 and C increases. We simulate 50 samples of the splitting scheme applied to the SPDE (1) with the initial value u 0 " 2 2´cospxq , the nonlocal interaction and the same noise as in the previous subsection (non-smooth case). In addition, we take the following parameters: t P r0, 1s, N x " 2 8 Fourier modes and τ " 2 n where n " ´6, ´7, . . . , ´14. We then estimate the proportion P of samples fulfilling [START_REF] Cohen | Exponential integrators for stochastic Maxwell's equations driven by Itô noise[END_REF] for each given τ , δ " 0.4, 0.5, 0.6, and C " 10 c for c " 1, 2, 3. The results are presented in Figure 4. In this figure, one sees how the proportion of samples P quickly goes to zero for δ ď 1{2 and an increasing C. Furthermore, this property does not hold for δ ą 1{2. This numerical experiment thus confirms that the order of convergence in probability of the splitting scheme is 1{2, as stated in Corollary 16.

6.4. Computational times. In this numerical experiment, we compare the computational costs of the above time integrators (expect the classical Euler-Maruyama scheme). To do this, we consider the SPDE (1) with the above nonlocal interaction potential for times t P r0, 2s. We discretize this SPDE using N x " 2 10 Fourier modes in space. We run 100 samples for each numerical scheme. For each scheme and each sample, we run several time steps and compare the L 2 error at the final time with a reference solution provided for the same sample by the same scheme for a very small time-step τ " 2 ´13 . Figure 5 displays the total computational time for all the samples, for each numerical scheme and each time-step, as a function of the averaged final error. One observes better performance for the splitting scheme. This appendix provides the proofs of properties of the exact solution to (1).

Global well-posedness. Let Assumption 1 be satisfied. Since the nonlinearity F is only locally Lipschitz continuous, a truncation argument is used to prove global well-posedness. Let us stress that the key property of (1) used in the argument below is the fact that V rus is real-valued.

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 dt `F R pu R ptqq dt `αdW Q ptq, with initial condition u R p0q " u 0 , thus admits a unique global solution `uR ptq ˘tPr0,T s . Since the mapping V R is real-valued, the trace formula holds, see [START_REF] Blanes | Splitting methods for the time-dependent Schrödinger equation[END_REF] where one observes that CpT, Q, u 0 q does not depend on R, using the trace formula above for the term in the integral u R psq 2 L 2 . Setting the truncation argument is then straightforward. Let τ R " inftt ě 0; u R ptq L 2 ą Ru. If R 1 , R 2 ě R, then u R 1 ptq " u R 2 ptq for all t ď τ R , by construction of F R . This allows us to define uptq solving (1) for all t P r0, τ q, where τ " lim RÑ8 τ R . Finally, τ " 8 almost surely, indeed for every T P p0, 8q,

Ppτ ď T q " lim RÑ8 Ppτ R ď T q " lim RÑ8 Pp sup 0ďtďT u R ptq 2 L 2 ě R 2 q ď lim RÑ8
CpT, Q, u 0 q R 2 " 0, using the moment estimate above. This concludes the proof of the global well-posedness of (1). Moment estimates in H 1 . Next, let us prove the moment bounds for the exact solution to (1). We provide details only for the moment estimates in the H 1 norm (under Assumption 1)

sup 0ďtďT Er∥∇uptq∥ 2p L 2 s ď C p pT, Q, u 0 q.
Indeed, the moment estimates for the L 2 norm, namely

sup 0ďtďT
Er∥uptq∥ 2p L 2 s ď C p pT, Q, u 0 q, can either be obtained using similar arguments, or be deduced from the exponential moment estimates for which a detailed proof is provided above. Likewise, the proof of moment estimates Temporal regularity. It remains to deal with the temporal regularity estimate. Using the mild formulation (2), for any 0 ď t 1 ă t 2 ď T , one has upt 2 q ´upt 1 q " Spt 1 q `Spt 2 ´t1 q ´I˘u 0 ´i ż

sup

Figure 1 .

 1 Figure1. Trace formulas for mass of the splitting scheme (Split), the Euler-Maruyama scheme (EM), the semi-implicit Euler-Maruyama scheme (sEM), the exponential integrator (sEXP), and the Crank-Nicolson (CN) schemes.

Figure 2 .Figure 3 .

 23 Figure 2. Strong errors for the stochastic Schrödinger equations.

Figure 4 .

 4 Figure 4. Proportion of samples fulfilling (16) for the splitting scheme (N denotes the number of step-sizes).

Figure 5 .

 5 Figure 5. Computational time as a function of the averaged final error for the four numerical methods.

2 L 2 " α 2 2 ı " ∥u 0 ∥ 2 L 2 `2tα 2 R ptq 2 L 2 s ď 3 ´∥u 0 ∥ 2 L 2 ` 2 L 2 `Er u R psq 2 L 2

 22222222222222 : indeed, one obtainsd u R ptq TrpQq `2αImpxu R ptq, dW Q ptqyq.Taking expectation, one obtains the trace formula E " u R ptq 2 L TrpQq, where the right-hand side does not depend on truncation index R. Using the Burkholder-Davis-Gundy inequality, one obtainsEr sup 0ďtďT u α2 T TrpQq ¯`3α 2 ż T 0 ÿ kPN |γ k | 2 Er|xu R psq, e k y| 2 s ds ď 3 ´∥u 0 ∥ α2 T TrpQq ¯`3α 2 Q s ds ď CpT, Q, u 0 q,

  On the other hand, let q " pr for p introduced above. Applying the Cauchy-Schwarz inequality yields

	Ere r n χ n,R s " Ere r n e ´rK F Sn e rK F Sn χ n,R s ď	`E " e q n e ´qK F Sn ‰˘1 p	ˆE "	e	rK F p p´1 Sn χ n,R	ȷ˙1 ´1 p	.
	Using Proposition 12 with σ " 1, or Proposition 13 with σ " 2, and the relation q " pr, for
	the first factor one has
						`E "	e q n e ´qK F Sn ‰˘1 p ď Cτ	rσ 2 .
	For the second factor, using the exponential moment estimates and the upper bound S n ď R
	when χ n,R ‰ 0, one obtains
	ˆE "	e	rK F p p´1 Sn χ n,R	ȷ˙1 ´1 p	ď	ˆE "	e µSn ‰	exppmaxp0,	p rK F p ´1 ´µqRq ˙1´1 p
										rσ 2K F and one obtains
							Ere r n s ď Cτ
	re r n p1 ´χn,R qs ď pE re rp n sq	1 p pE r1 ´χn,R sq 1´1 p ď pEre rp n sq	1 p PpS n ą Rq 1´1 p
		ď pEre rp n sq	1 p P pexppµS n q ą exppµRqq	1´1 p ď pEre rp n sq	1 p ˆErexppµS n qs exppµRq	˙1´1 p	.
	Using moment bounds for the exact and the numerical solution (Corollary 11) and the expo-
	nential moment estimate (11) for S n , then yield (for a constant C that does not depend on
	R)
						E re r

n p1 ´χn,R qs ď Ce ´µR . ď C exp pmaxp0, rK F ´µqRq , using the identity µ " p1 ´1 p qµ.

Finally, for all R P p0, 8q, one has

Ere r n s ď C ´τ rσ 2 exp `maxp0, rK F ´µqR ˘`expp´µRq ¯.

It remains to optimize the choice of R in terms of τ . If rK F ď µ, there is no condition and passing to the limit R Ñ 8 yields Ere r n s ď Cτ rσ 2 . If rK F ą µ, the right-hand side is minimized when τ rσ 2 e rK F R " 1, i. e. e ´R " τ rσµ 2K F .

  We start the proof of Theorem 10.

	Proof. Set λ "
	By definition of the stochastic process Xptq, the above reads
	dE	" exp ´λe ´t{T ∥uptq∥ 2 L 2 dt	¯ı	ď λTrpQ α qE	" exp ´λe ´t{T ∥uptq∥ 2 L 2	¯ı
	and applying Gronwall's lemma provides the following estimate
		E	" exp ´λe ´t{T ∥uptq∥ 2 L 2 ¯ı ď exp ´λ ∥u 0 ∥ 2 L 2	¯eλTrpQαqt ,
	Finally, let µ ď e ´1 2T TrpQαq " e ´1λ. Then for all t P r0, T s,
			E	" exp ´µ ∥uptq∥ 2 L 2 ¯ı ď E	" exp ´λe ´t{T ∥uptq∥ 2 L 2

5.1. Proof of Theorem 10. 1 2T TrpQαq and define the stochastic process Xptq " e ´t{T ∥uptq∥ 2 L 2 . An application of Itô's formula gives d ´eλXptq ¯" e λXptq ˆ´λ{T Xptq dt `λe ´t{T TrpQ α q dt `λ2 2 dxXy t ˙`2λe λXptq e ´t{T xuptq, dW Qα ptqy, where the quadratic variation xXy t satisfies dxXy t ď e ´2t{T 4TrpQ α q ∥uptq∥ 2 L 2 dt ď 4TrpQ α qe ´t{T Xptq dt. Taking expectation in the first equation above and observing that Xptq ě 0 a.s, one gets dEre λXptq s dt ď λTrpQ α qEre λXptq s `Ere λXptq `2λ 2 TrpQ α q ´λ{T ˘Xptqs ď λTrpQ α qEre λXptq s by definition of λ. ¯ı ď exp ´λ ∥u 0 ∥ 2 L 2

0ďtďT

  Er∥∇uptq∥ 2pH 2 s ď C p pT, Q, u 0 q under Assumption 2 would follow from similar arguments. Let us first consider ψpuq " ∥∇u∥ 2 L 2 for all u P H 1 . Its first and second order derivatives are given by ψ 1 puq.h " 2Repx∇u, ∇hyqψ 2 puq.ph, kq " 2Repx∇h, ∇kyq for h, k P H 1 . Using Itô's formula, one gets | 2 ∥∇e k ∥ 2 L 2 .The first term in the last equality vanishes, and when taking expectation the third term also vanishes. Using the condition (4) to deal with the second term, one obtains ErP 1 p∥uptq∥ 2 L 2 qs ď CpT, Q, u 0 q due to moment bounds in the L 2 norm. Using the Gronwall Lemma then yields sup Let p ě 1, then applying Itô's formula for ψ p puq " ψpuq p yieldsImpx∇uptq, ∇F puptqqyq `Cp α 2 ÿ kPN |γ k | 2 ∥∇e k ∥ 2 L 2 Er∥∇uptq∥ ´1 `Er∥∇uptq∥ 2p L 2 s `CErP 1 p∥uptq∥ 2 L 2 q p s ūsing(4) and Young's inequality. Using Gronwall's lemma then concludes the proof of the moment bounds in the H 1 norm.

	dEr∥∇uptq∥ 2p L 2 s dt	ď C p ∥∇uptq∥	2pp´1q L 2		2pp´1q L 2	s
		ď C
	d ∥∇uptq∥ 2 L 2 " dψpuptqq
		" ψ 1 puptqq.duptq	2 `α2	ÿ
		dEr∥∇uptq∥ 2 L 2 s dt	ď C ´1 `Er∥∇uptq∥ 2 L 2 s `ErP 1 p∥uptq∥ 2 L 2 qs ¯,
	where P 1 is a polynomial mapping. Note that sup

kPN ψ 2 puptqq.pγ k e k , γ k e k q dt " 2Impx∇uptq, ∇∆uptqyq dt `2Impx∇uptq, ∇F puptqqyq dt `2αImpx∇uptq, ∇dW Q ptqyq `α2 ÿ kPN |γ k 0ďtďT 0ďtďT

Er∥∇uptq∥ 2 L 2 s ď CpT, Q, u 0 q.

.

  Spt 2 ´sq dW Q psq.Using Lemma 1, the first estimate of (4) and the moment bounds in the L 2 and H 1 norms, one obtainsSpt 1 q `Spt 2 ´t1 q ´I˘u 0 L 2 ď C|t 2 ´t1 | C|t 2 ´t1 | 2p . Spt 2 ´sq dW Q psqSince the stochastic integrals have Gaussian distribution, gathering the estimates above yieldsE " ∥upt 2 q ´upt 1 q∥ 2p L 2 ı ď C p pT, Q, u 0 q|t 2 ´t1 | p ,for all p ě 1 and t 1 , t 2 P r0, T s. This concludes the proof of Proposition 3.

					t 1	Spt 1 ´sq `Spt 2 ´t1 q	´I˘F pupsqq ds	´i ż t 2	Spt 2 ´sqF pupsqq ds
				0										t 1
				´iα	ż t 1	Spt 1 ´sq `Spt 2 ´t1 q	´I˘d	W Q psq ´iα	ż t 2
					0										t 1
															1 2 ∥u 0 ∥ H 1
	E	«	ż t 1 0	Spt 1 ´sq `Spt 2 ´t1 q	´I˘F pupsqq ds	2p L 2	ff	ď T 2p´1 |t 2 ´t1 | p	ż T 0	E	" ∥F pupsqq∥ 2p H 1	ı	ds ď C|t 2 ´t1 |	2p 2
	E ds ď Using Itô's isometry formula and Lemma 1, one has « ż t 2 t 1 Spt 2 ´sqF pupsqq ds 2p L 2 ff ď |t 2 ´t1 | 2p ż T 0 E " ∥F pupsqq∥ 2p L 2 ı
	E	«	ż t 1 0	Spt 1 ´sq `Spt 2 ´t1 q	´I˘d	W Q psq	2 L 2	ff	" t 1 `Spt 2 ´t1 q	´I˘Q 1 2	2 L 0 2	ď Ct 1 |t 2 ´t1 | Q	1 2	2 L 1 2
				E	«	ż t 2 t 1							2 ff	" |t 2 ´t1 | Q	1 2	L 0 2

We thank André Berg (Umeå University) for discussions on the implementation of Figure 4. 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.