Stochastic nonlinear Schr\"odinger equations on tori

We consider the stochastic nonlinear Schr\"odinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.

1. Introduction 1.1.Stochastic nonlinear Schrödinger equations.In this paper, we study the following Cauchy problem associated to a stochastic nonlinear Schrödinger equation (SNLS) of the form: where k, d ≥ 1 are integers, T d := R d /Z d , and u : [0, ∞) × T d → C is the unknown stochastic process.The term F (u, φξ) is a stochastic forcing and in this paper we treat the following cases: the additive noise, i.e.
where the right-hand side of (1.3) is understood as an Itô product 1 .Here, ξ is a space-time white noise, i.e. a Gaussian stochastic process with correlation function E[ξ(t, x)ξ(s, y)] = δ(t − s)δ(x − y), where δ denotes the Dirac delta function.We recall that the white noise is very rough: the spatial regularity of ξ is less than − d 2 .Since the linear Schrödinger equation does not provide any smoothing properties, we consider instead a spatially smoothed out version φξ, where φ is a linear operator from L 2 (T d ) into H s (T d ), on which we make certain assumptions, depending on whether we are working with (1.2) or (1.3).
Our main goal in this paper is to prove local well-posedness of SNLS with either additive or multiplicative noise in the Sobolev space H s (T d ), for any subcritical non-negative regularity s (see below for the meaning of "subcritical").In this work, solutions to (1.1) are understood as solutions to the mild formulation where S(t) := e −it∆ is the linear Schrödinger propagator.The term Ψ(t) is a stochastic convolution corresponding to the stochastic forcing F (u, φξ), see (1.11) and (1.12) below.Our local-in-time argument uses the Fourier restriction norm method introduced by Bourgain [6] and the periodic Strichartz estimates proved by Bourgain and Demeter [5].In establishing local well-posedness for the multiplicative SNLS, we also have to combine these tools with the truncation method used by de Bouard and Debussche [18,17,19].Moreover, by proving probabilistic a priori bounds on the mass and energy of solutions, we establish global well-posedness in (i) L 2 (T) for cubic nonlinearities (i.e.k = 1) when d = 1, and (ii) H 1 (T d ) for all defocusing energy-subcritical nonlinearities -see Theorem 1.5 and the preceding discussion for more details.Previously, de Bouard and Debussche [17,18] studied SNLS on R d .They considered noise φξ that is white in time but correlated in space, where φ is a smoothing operator from L 2 (R d ) to H s (R d ).They proved global existence and uniqueness of mild solutions in (i) L 2 (R) for the one-dimensional cubic SNLS and (ii) H 1 (R d ) for defocusing energysubcritical SNLS.Other works related to SNLS on R d include the works by Barbu, Röckner, and Zhang [1,2] and by Hornung [24].
On the R d setting, the arguments given in [17,18] use fixed point arguments in the space C t H 1 x ∩L p t W 1,q x ([0, T ]×R d ), for some T > 0 and some suitable p, q ≥ 1. 2 In particular, they use the (deterministic) Strichartz estimates: where the pair (p, q) is admissible, i.e. 2 p + d q = d 2 , 2 ≤ p, q, ≤ ∞, and (p, q, d) = (2, ∞, 2).On T d , Bourgain and Demeter [5] proved the ℓ 2 -decoupling conjecture, and as a corollary, the following periodic Strichartz estimates: Here, P ≤N is the Littlewood-Paley projection onto frequencies {n ∈ Z d : |n| ≤ N }, p ≥ 2(d+2) d , and ε > 0 is an arbitrarily small quantity 3 .However, such Strichartz estimates are not strong enough for a fixed point argument in mixed Lebesgue spaces for the deterministic NLS on T d .To overcome this problem, we shall employ the Fourier restriction norm method by means of X s,b -spaces defined via the norms (1.7) The indices s, b ∈ R measure the spatial and temporal regularities of functions u ∈ X s,b , and F t,x denotes Fourier transform of functions defined on R × T d .This harmonic analytic 2. Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm: , where n := 1 + |n| 2 .When r = 2, we have H s (T d ) = W s,2 (T d ).
3.More recently, Killip and Vişan [25] removed the arbitrarily small loss of ε derivatives in (1.6) when p > 2(d+2)   d .However, we do not need this scale-invariant improvement in our results.method was introduced by Bourgain [6] for the deterministic nonlinear Schrödinger equation (NLS): i∂ t u − ∆u ± |u| 2k u = 0 . (1.8) 1.2.Main results.We now state more precisely the problems considered here.Let (Ω, A, {A t } t≥0 , P) be a filtrated probability space.Let W be the L 2 (T d )-cylindrical Wiener process given by W (t, x, ω) := β n (t, ω)e n (x), (1.9) where {β n } n∈Z d is a family of independent complex-valued Brownian motions associated with the filtration {A t } t≥0 and e n (x) := exp(2πin • x), n ∈ Z d .The space-time white noise ξ is given by the (distributional) time derivative of W , i.e. ξ = ∂W ∂t .Since the spatial regularity of W is too low (more precisely, for each fixed t ≥ 0, ) almost surely for any ε > 0), we consider a smoothed out version φW as follows.Recall that a bounded linear operator φ from a separable Hilbert space H to a Hilbert space where {h n } n∈Z d is an orthonormal basis of H (recall that • L 2 (H;K) does not depend on the choice of {h n } n∈Z d ).Throughout this work, we assume φ ∈ L 2 (L 2 (T d ); H s (T d )) for appropriate s ≥ 0. In this case, φW is a Wiener process with sample paths in H s (T d ) and its time derivative φξ corresponds to a noise which is white in time and correlated in space (with correlation function depending on φ).We can now define the stochastic convolution Ψ(t) from (1.4) for (i) the additive noise (1.2): and (ii) the multiplicative noise (1.3): We are now ready to state our first result.
Then for any u 0 ∈ H s (T d ), there exist a stopping time T = T ( u 0 H s , Ψ) that is almost surely positive, and a unique adapted solving SNLS with additive noise on [0, T ] almost surely, for some ε > 0.
Here, X s,b ([0, T ]) is a time restricted version of the X s,b -space, see (2.5) below.The proof of this result relies on a fixed point argument for (1.4) in a closed subset of X s,b ([0, T ]).We are required to use b = 1  2 − ε in order to capture the temporal regularity of Ψ.Since X s,b ([0, T ]) does not embed into C([0, T ]; H s ) when b < 1 2 , we need to prove the continuity in time of solutions a posteriori.Our local well-posedness result above (as well as Theorem 1.6 below) covers all non-negative subcritical regularities.Remark 1.2.We point out that s crit is negative only for the one-dimensional cubic NLS, i.e. (d, k) = (1, 1) for which s crit = − 1 2 .Below L 2 (T), the deterministic cubic NLS on T was shown to be ill-posed.Indeed, Christ, Colliander and Tao [12] and Molinet [31] showed that the solution map u 0 ∈ H s (T) → u(t) ∈ H s (T) is discontinuous whenever s < 0.More recently, Guo and Oh [20] showed an even stronger ill-posedness result, in the sense that for any u 0 ∈ H s (T), s ∈ (− 1 8 , 0), there is no distributional solution u that is also a limit of smooth solutions in C([−T, T ]; H s (T)).In the (super)critical regime, i.e. for s ≤ − 1 2 = s crit , Oh [34] and Oh and Wang [35] showed a norm inflation phenomenon at any u 0 ∈ H s (T): for any ε > 0 and u 0 ∈ H s (T), there exists a solution u ε to NLS such that u ε (0) − u 0 H s (T) < ε and u ε (t) H s (T) > ε −1 for some t ∈ (0, ε).
Remark 1.3.Although we present our results for SNLS on the standard torus T d = R d /Z d , our arguments hold on any torus . This is because the periodic Strichartz estimates (1.6) of Bourgain and Demeter [5] hold for irrational tori (T d α is irrational if there is no γ ∈ Q d such that γ • α = 0).Prior to [5], Strichartz estimates were harder to establish on irrational tori -see [21] and references therein.
Remark 1.4.The deterministic NLS is locally well-posed in the critical space H s crit (T d ), for almost all pairs (d, k), except for the cases (1, 2), (2, 1), (3,1) which are still open -see [7,22,23,38].In these papers, the authors employ the critical spaces X s , Y s based on the spaces U 2 , V 2 of Koch and Tataru [28].We point out that Brownian motions belong almost surely to V p , for p > 2, but not V 2 (hence neither to U 2 ).Consequently, the spaces X s , Y s are not suitable for obtaining local well-posedness of SNLS.Now let us recall the following conservation laws for the deterministic NLS: where the sign ± in (1.14) matches that in (1.1) and (1.4).Recall that SNLS (1.1) with the + sign is called defocusing (and focusing for the − sign).We say that SNLS is energysubcritical if s crit < 1 (i.e. for any k ≥ 1 when d = 1, 2 and for k = 1 when d = 3).For solutions of SNLS these quantities are no longer necessarily conserved.However, Itô's lemma allows us to bound these in a probabilistic manner similarly to de Bouard and Debussche [18,17].Therefore, we obtain the following: Then the H s -valued solutions of Theorem 1.1 extend globally in time almost surely in the following cases: (i) the (focusing or defocusing) one-dimensional cubic SNLS for all s ≥ 0; (ii) the defocusing energy-subcritical SNLS for all s ≥ 1.
We now move onto the problem with multiplicative noise, i.e.SNLS with (1.3).For this case, we need a stronger assumption on φ.By a slight abuse of notation, for a bounded linear operator φ from L 2 (T d ) to a Banach space B, we say that φ 4. In fact, such operators are known as nuclear operators of order 2 and their introduction goes back to the work of A. Grothendieck on nuclear locally convex spaces.
For s ∈ R and r ≥ 1, we also define the Fourier-Lebesgue space FL s,r (T d ) via the norm Clearly, when r = 2 we have FL s,r (T d ) = H s (T d ) and for s 1 ≤ s 2 and r 1 ≤ r 2 we have FL s 2 ,r 1 (T d ) ⊂ FL s 1 ,r 2 (T d ).We now state our local well-posedness result for the multiplicative SNLS.
for some r ∈ 1, d d−s when s > 0 and r = 1 when s = 0. Let F (u, φ) = u • φξ.Then for any u 0 ∈ H s (T d ), there exist a stopping time T that is almost surely positive, and a unique adapted process We point out that an extra condition in the multiplicative case was also used by de Bouard and Debussche [18] in their study of SNLS in H 1 (R d ), namely they required that φ is a γ-radonifying operator from L 2 (R d ) into W 1,α (R d ) for some appropriate α, as compared to the requirement that φ is Hilbert-Schmidt from L 2 (R d ) into H s (R d ) in the additive case.
In the multiplicative case, the stochastic convolution depends on the solution u and this forces us to work in the space in (1.16).In order to control the nonlinearity in this space, we use a truncation method which has been used for SNLS on R d by de Bouard and Debussche [18,17].Moreover, we combine this method with the use of X s,b -spaces in a similar manner as in [19], where the same authors studied the stochastic KdV equation with low regularity initial data on R.This introduces some technical difficulties which did not appear when using the more classical Strichartz spaces as those used in [18,17].
Next, we prove global well-posedness of SNLS (1.1) with multiplicative noise.Similarly to the additive case, the main ingredient is the probabilistic a priori bound on the mass and energy of a local solution u.However, we further need to obtain uniform control on the X s,b -norms for solutions to truncated versions of (1.4).
Theorem 1.8 (Global well-posedness for multiplicative SNLS).Let s ≥ 0. Given φ with the same assumptions as in Theorem 1.6, let F (u, φ) = u • φξ and u 0 ∈ H s (T d ).Then the H s -valued solutions of Theorem 1.6 extend globally in time in the following cases: (i) the (focusing or defocusing) one-dimensional cubic SNLS for all s ≥ 0; (ii) the defocusing energy-subcritical SNLS for all s ≥ 1.
Before concluding this introduction let us state two remarks.Remark 1.9.We point out that Theorem 1.1 and Theorem 1.6 are almost optimal for handling the regularity of initial data since the deterministic NLS is ill-posed for s < s crit (see Remark 1.2).In terms of the regularity of the noise, at least in the additive noise case, it is possible to consider rougher noise by employing the Da Prato-Debussche trick, namely by writing a solution u to (1.4) as u = v + Ψ and considering the equation for the residual part v.In general, this procedure allows one to treat rougher noise, see for example [3,4,14].where they treat NLS with rough random initial data.In the periodic setting however, the argument gets more complicated (see for example [3,4] on R d versus [14,32] on T d ).The actual implementation of the aforementioned trick requires cumbersome caseby-case analysis where the number of cases grows exponentially in k.Even for the cubic case on T d the analysis is involved, whereas on R d one can use bilinear Strichartz estimates which are not available on T d .Remark 1.10.In the multiplicative noise case, there are well-posedness results on a general compact Riemannian manifold M without boundaries.In [9], Brzeźniak and Milllet use the Strichartz estimates of [10] and the standard space-time Lebesgue spaces (i.e.without the Fourier restriction norm method).For M = T d , Theorem 1.6 improves the result in [9] since it requires less regularity on the noise and initial data.In [8], Brzeźniak, Hornung, and Weiss construct martingale solutions in H 1 (M ) for the multiplicative SNLS with energysubcritical defocusing nonlinearities and mass-subcritical focusing nonlinearities.
Organization of the paper.In Section 2, we provide some preliminaries for the Fourier restriction norm method and prove the multilinear estimates necessary for the local wellposedness results.In Section 3, we prove some properties of the stochastic convolutions Ψ and Ψ[u] given respectively by (1.11) and (1.12).We prove Theorems 1.1 and 1.6 in Section 4. Finally, in Section 5 we prove the global results Theorems 1.5 and 1.8.
Notations.Given A, B ∈ R, we use the notation A B to mean A ≤ CB for some constant C ∈ (0, ∞) and write A ∼ B to mean A B and B A. We sometimes emphasize any dependencies of the implicit constant as subscripts on , , and ∼; e.g.A p B means A ≤ CB for some constant C = C(p) ∈ (0, ∞) that depends on the parameter p.We denote by A ∧ B and A ∨ B the minimum and maximum between the two quantities respectively.Also, ⌈A⌉ denotes the smallest integer greater or equal to A, while ⌊A⌋ denotes the largest integer less than or equal to A.
Given a function g : U → C, where U is either T d or R, our convention of the Fourier transform of g is given by For the sake of convenience, we shall omit the 2π from our writing since it does not play any role in our arguments.
For c ∈ R, we sometimes write c+ to denote c + ε for sufficiently small ε > 0, and write c− for the analogous meaning.For example, the statement 'u ∈ X s, 1 2 − ' should be read as 'u ∈ X s, 1 2 −ε for sufficiently small ε > 0'.For the sake of readability, in the proofs we sometimes omit the underlying domain T d from various norms, e.g.we write The authors were supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh.

Fourier restriction norm method
Let s, b ∈ R. The Fourier restriction norm space X s,b adapted to the Schrödinger equation on T d is the space of tempered distributions u on R × T d such that the norm is finite.Equivalently, the X s,b -norm can be written in its interaction representation form: where S(t) = e −it∆ is the linear Schrödinger propagator.We now state some facts on X s,b -spaces.The interested reader can find the proof of these and further properties in [37].Firstly, we have the following continuous embeddings We have the duality relation (2.4) Lemma 2.1 (Transference principle, [37, Lemma 2.9]).Let Y be a Banach space of func- Given a time interval I ⊆ R, one defines the time restricted space X s,b (I) via the norm We note that for s ≥ 0 and 0 ≤ b < 1 2 , we have see for example [19,Lemma 2.1] for a proof (for X s,b spaces adapted to the KdV equation).
Lemma 2.2 (Linear estimates, [37, Proposition 2.12]).Let s ∈ R and suppose η is smooth and compactly supported.Then, we have By localizing in time, we can gain a smallness factor, as per lemma below.
We now give the proofs of the multilinear estimates necessary to control the nonlinearity |u| 2k u.Recall the L 4 -Strichartz estimate due to Bourgain [6] (see also [37, Proposition 2.13]): , and b ′ ≤ 5 8 .Then, for any time interval I ⊂ R, we have (2.10) Proof.By the triangle inequality it suffices to prove (2.10) for s = 0. We claim that for any factors u 1 , u 2 , u 3 , v. Indeed, this follows immediately from Hölder inequality and (2.9) for each of the four factors (hence the restrictions b, 1 − b ′ ≥ 3 8 ).Thus, the globalin-time version of (2.10), i.e.I = R, follows by the duality relation (2.4).For an arbitrary time interval I, if ũj is an extension of u j , j = 1, 2, 3, then ũ1 ũ2 ũ3 is an extension of u 1 u 2 u 3 .We use the previous step to get and then we take infimum over all extensions ũj 's and (2.10) follows.
Due to the scaling and Galilean symmetries of the linear Schrödinger equation, the periodic Strichartz estimate (1.6) of Bourgain and Demeter [5] is equivalent with ) By the transference principle (Lemma 2.1), we get for any b > 1 2 .By interpolating (2.12) with (which follows immediately from Sobolev inequalities, (2.1), and the H s (T d )-isometry of S(−t)), we can lower the time regularity from b = 1 2 + δ to b = 1 2 − δ, for sufficiently small δ > 0. Thus, we also have Lemma 2.4 only treats the cubic nonlinearity when d = 1.We now prove the following general multilinear estimates to treat other cases.The proof borrows techniques from [21].
Lemma 2.5.Let d, k ≥ 1 such that dk ≥ 2 and let I ⊂ R be a finite time interval.Then for any s > s c , there exist b = 1  2 − and b ′ = 1 2 + such that Proof.In view of (2.6), we can assume that u j (t) = ½ I (t)u j (t) and thus by the duality relation (2.4), it suffices to show We use Littlewood-Paley decomposition: we estimate the left-hand side of (2.16) when v = P N v, u j = P N j u j for some dyadic numbers N, N j ∈ 2 Z , 1 ≤ j ≤ 2k + 1.Then the claim follows by triangle inequality and performing the summation (2.17) Notice that without loss of generality, we may assume that N 1 ≥ N 2 ≥ . . .≥ N 2k+1 , in which case we also have N N 1 , and that the factors v and u j are real-valued and non-negative.
Case 2: N 1 ≫ N 2 .Then, we necessarily have N 1 ∼ N or else the left hand side of (2.16) vanishes.By Hölder inequality, with 2k p + 1 q + 1 r = 1.As in Case 1, we would like to have p such that d 2 − d+2 p = s crit , or equivalently p = k(d + 2).However, the best we can do with the Strichartz estimate for the remaining factors is to choose q = r = 2(d+2) d , so that we have Notice that we can overcome the loss of derivative N s 1 only up to a logarithmic factor.We need a slightly refined analysis.
We cover the dyadic frequency annuli of u 1 and of v with dyadic cubes of side-length N 2 , i.e.
There are approximately N 1 N 2 d -many cubes needed, and so 16) vanishes.Hence the summation (2.17) is replaced by (2.27) Also, in place of (2.24)-(2.25),we now have Therefore, by Cauchy-Schwartz inequality and Plancherel identity, LHS of (2.16) and the proof is complete.

The stochastic convolution
In this section, we prove some X s,b -estimates on the stochastic convolution Ψ(t) given either by (1.11) or (1.12).We first record the following Burkholder-Davis-Gundy inequality, which is a consequence of [30, Theorem 1.1].Lemma 3.1 (Burkholder-Davis-Gundy inequality).Let H, K be separable Hilbert spaces, T > 0, and W is an H-valued Wiener process on [0, T ].Suppose that {ψ(t)} t∈[0,T ] is an adapted process taking values in L 2 (H; K).Then for p ≥ 1, .
In addition, we prove that Ψ(t) is pathwise continuous in both cases.To this end, we employ the factorization method of Da Prato [15, Lemma 2.7], i.e. we make use of the following lemma and (3.3) below.Lemma 3.2.Let H be a Hilbert space, T > 0, α ∈ (0, 1), and σ > 1 α , ∞ .Suppose that f ∈ L σ ([0, T ]; H).Then the function We make use of the above lemma in conjunction with the following fact: for all 0 < α < 1 and all 0 ≤ µ < t.This can be seen via considerations with Euler-Beta functions, see [15].
We now treat the additive and multiplicative cases separately below in Subsection 3.1 and 3.2 respectively.The arguments for the two cases are similar, albeit with some extra technicalities in the multiplicative case.
3.1.The additive stochastic convolution.By Fourier expansion, the stochastic convolution (1.11) for the additive noise problem can be written as We first prove the following X s,b -estimate on Ψ: . Then for Ψ given by (3.4) we have By (2.6), we have where By the stochastic Fubini theorem (see [16,Theorem 4.33]), we have by Burkholder-Davis-Gundy inequality (Lemma 3.1), we get .
This completes the proof of Lemma 3.3.
We now prove that Ψ has a continuous version taking values in H s (T d ).This is the content of the next lemma.Lemma 3.4 (Continuity of the additive noise).Let s ≥ 0, T > 0, and Proof.We fix α ∈ 0, 1 2 and we write the stochastic convolution as follows: where we used the stochastic Fubini theorem [16,Theorem 4.33] and the group property of S(•).By Lemma 3.2 and (3.10) it suffices to show that the process for some σ > 1 α .By Burkholder-Davis-Gundy inequality (Lemma 3.1), for any σ ≥ 2 and any t ′ ∈ [0, T ], we get , where in the last step we used 2α ∈ (0, 1) and the H s (T d )-isometry property of S(t ′ − µ).

3.2.
The multiplicative stochastic convolution.The multiplicative stochastic convolution Ψ = Ψ[u] from (1.12) can be written as Recall that if s > d 2 , then we have access to the algebra property of H s (T d ): which is an easy consequence of the Cauchy-Schwartz inequality.This simple fact is useful for our analysis in the multiplicative case.On the other hand, (3.13) is not available to us for regularities below d 2 , but we use the following inequalities.
Also, for s = 0, we have Proof.Assume that 0 < s ≤ d 2 and let n 1 and n 2 denote the spatial frequencies of f and u respectively.By separating the regions {|n 1 | |n 2 |} and {|n 1 | ≪ |n 2 |}, and then applying Young's inequality, we have where p is chosen such that 1 r + 1 p = 3 2 .By Hölder inequality, for r ′ and q such that 1 r + 1 r ′ = 1 and Given φ as in Theorem 1.6, let us denote for r = 2 when s > d 2 , for some r ∈ 1, d d−s when 0 < s ≤ d 2 , and for r = 1 when s = 0. Recall that if φ is translation invariant, then it is sufficient to assume that C(φ) < ∞ with r = 2, for all s ≥ 0. We now proceed to prove the following X s,b -estimate of Ψ[u].Lemma 3.6.Let s ≥ 0, 0 ≤ b < 1  2 , T > 0, and 2 ≤ σ < ∞.Suppose that φ satisfies the assumptions of Theorem 1.6.Then, for Ψ[u] given by (1.12) we have the estimate (3.17) Proof.We first prove (3.17 Then by (2.6) and the assumption 0 ≤ b < 1 2 , the Burkholder-Davis-Gundy inequality (Lemma 3.1), and (3.7), we have LHS of (3.17 If s > d 2 , we apply the algebra property of H s (T d ) to get and thus (3.17) follows.

Local well-posedness
4.1.SNLS with additive noise.In this subsection, we prove Theorem 1.1.Let b = b(k) = 1 2 − be given by Lemma 2.4 (in the case d = k = 1) or by Lemma 2.5 (in the case dk ≥ 2).By Lemma 3.3, for any T > 0, there is an event Ω ′ of full probability such that the stochastic convolution Ψ has finite X s,b ([0, T ])-norm on Ω ′ .Now fix ω ∈ Ω ′ and u 0 ∈ H s (T d ).Consider the ball where 0 < T < 1 and R > 0 are to be determined later.We aim to show that the operator Λ given by where Ψ is the additive stochastic convolution given by (3.4), is a contraction on B R .To this end, it remains to estimate the X s,b ([0, T ])-norm of For any δ > 0 sufficiently small (such that b + δ < 1 2 ), by Lemma 2.3 and (2.6): X s, 1 2 +δ .Let η be a smooth cut-off function, supported on [−1, T + 1], with η(t) = 1 for all t ∈ [0, T ].For any w ∈ X s,− 1 2 +δ that agrees with |u| 2k u on [0, T ], by Lemma 2.2, we obtain Then after taking the infimum over all such w, we use Lemma 2.4 or 2.5 and we get for some c > 0. Similarly, we obtain . From (4.3) and (4.4), we see that Λ is a contraction from B R to B R provided This is always possible if we choose T ≪ 1 sufficiently small.This shows the existence of a unique solution u ∈ X s,b ([0, T ]) to (1.4) on Ω ′ .Finally, we check that u ∈ C([0, T ]; H s ) on the set of full probability Ω ′′ ∩ Ω ′ , where Ω ′′ is given by Lemma 3.4, that is Ψ ∈ C([0, T ]; H s ) on Ω ′′ .By (2.6), (4.1) and Lemma 2.4 or 2.5, we also get By the embedding X s, 1 2 +δ ([0, T ]) ֒→ C([0, T ]; H s (T d )), we have D(u) ∈ C([0, T ]; H s (T d )).Since the linear term S(t)u 0 also belongs to C([0, T ]; H s (T d )), we conclude that Remark 4.1.From (4.5), we obtain the time of existence where θ = 2k δ .Note that (4.7) will be useful in our global argument.4.2.SNLS with multiplicative noise.In this subsection, we prove Theorem 1.6.Following [19], we use a truncated version of (1.4).The main idea is to apply an appropriate cut-off function on the nonlinearity to obtain a family of truncated SNLS, and then prove global well-posedness of these truncated equations.Since solutions started with the same initial data coincide up to suitable stopping times, we obtain a solution to the original SNLS in the limit.
Let η : R → [0, 1] be a smooth cut-off function such that η ≡ 1 on [0, 1] and η R and consider the equation The key ingredient for Theorem 1.6 is the following proposition.Before proving this result, we state and prove the following lemma.

.13)
We now estimate the H b (0, T ∧ τ R )-norm, for which we use the following characterization (see for example [36]): For the inhomogeneous contribution (i.e.coming from the L 2 -norm above), we have The remaining part of (4.13) needs a bit more work.Fix n ∈ Z d , then It is clear that and hence For II(n), the mean value theorem infers that Again, we split w(•, n ′ ) 2 H b (t ′ ,t) using (4.14) into the inhomogeneous contribution (the L 2norm squared part) and the homogeneous contribution (the second term of (4.14)).We control here only the homogeneous contributions for II(n) as the inhomogeneous contributions are easier.The homogeneous part of II(n) is controlled by where we used 0 Since b ∈ 0, 1 2 , by Hardy's inequality (see for example [37 . After multiplying by n 2s and summing over n ∈ Z d , we see that (4.17) is controlled by We now prove (4.11).Let τ u R and τ v R be defined as in (4.12).Assume without loss of generality that By the mean value theorem, For B, one runs through the same argument as for (4.10) but with w(t, n) replaced by We now conclude the proof of Proposition 4.2.
Proof of Proposition 4.2.Let T, R > 0 and let E T := L 2 ad Ω; X s,b ([0, T ]) be the space of adapted processes in L 2 Ω; X s,b ([0, T ]) .We solve the fixed point problem (4.9) in E T .Arguing as in the additive case, and using Lemmata 4.3 and 3.6, we have Therefore, Λ R is a contraction from E T to E T provided we choose T = T (R) sufficiently small.Thus there exists a unique solution u R ∈ E T .Note that T does not depend on u 0 H s , hence we may iterate this argument to extend u R (t) to all t ∈ [0, ∞).
Finally, to see that Then, by similar argument as in the end of Subsection 4.1, we have that where is a well-defined stopping time that is either positive or infinite almost surely.By defining we see that u is a solution of (1.4) on [0, τ * ) almost surely.

Global well-posedness
In this section, we prove Theorems 1.5 and 1.8.Recall that the mass and energy of a solution u(t) of the defocusing (1.1) are given respectively by It is well-known that these are conserved quantities for (smooth enough) solutions of the deterministic NLS equation.
For SNLS, we prove probabilistic a priori control as per Propositions 5.1 and 5.3 below.To this purpose, the idea is to compute the stochastic differentials of (5.1) and (5.2) and use the stochastic equation for u.We work with the following frequency truncated version of (1.1): where P ≤N is the Littlewood-Paley projection onto the frequency set {n ∈ Z d : |n| ≤ N }, By repeating the arguments in Section 4, one obtains local well-posedness for (5.3) with initial data P ≤N u 0 at least with the same time of existence as for the untruncated SNLS.
5.1.SNLS with additive noise.We treat the additive SNLS in this subsection.We first prove probabilistic a priori bounds on (5.1) and (5.2) of a solution u N of the truncated equation. (5.4) (5.5) The constants C 1 and C 2 are independent of N .
Proof.By applying Itô's Lemma, we have the last term being the Itô correction term.We first control (5.6).By Burkholder-Davis-Gundy inequality (Lemma 3.1), Hölder and Young inequalities, we get Hence by Young's inequality, we infer that In a straightforward way, we also have Therefore, there is some C m > 0 such that (5.9) We now wish to move the last term of (5.9) to the left-hand side.However, we do not know a priori that the moments of sup t∈[0,T ] M (u N (t)) are finite.To justify this, we note that (5.9) holds with T replaced by T R , where Now the terms that would be appearing in (5.9) are finite and hence the formal manipulation is justified.Note that T R → T almost surely as R → ∞ because u (and hence u N ) belongs in C([0, T ]; H s (T d )) almost surely.Hence by letting R → ∞ and invoking the monotone convergence theorem, one finds (5.10) Hence, by induction on m, we obtain where we note that the implicit constant is independent of N .We now turn to estimating the energy.Applying Itô's Lemma again, we find that E(u N (t)) m equals E(u N 0 ) m (5.12) We shall control here only the difficult term (5.13) as the other terms are bounded by similar lines of argument.Firstly, by Burkholder-Davis-Gundy inequality (Lemma 3.1), we deduce Then, by duality and the (dual of the) Sobolev embedding H 1 (T d ) ֒→ L 2k+2 (T d ), we have Therefore, by Hölder and Young inequalities, and similarly to the control of (5.6), we have where in the last step we used interpolation.
We also have (5.17 Gathering all the estimates, there exists C m > 0 such that Similarly to passing from (5.9) to (5.10) and by induction on m, we deduce that with constant independent of N .
We now argue that the probabilistic a priori bounds in fact hold for solutions of the original SNLS.Proof.Let Λ N be the mild formulation of (5.3), more precisely, Then Λ N is a contraction on a ball in X 1, 1 2 − ([0, T ]) and has a unique fixed point u N that satisfies the bounds in Proposition 5.1.Hence it suffices to show that u N in fact converges to u in F T := L 2 (Ω; C([0, T ]; H s x )) for s = 0, 1.We only show s = 1 since the proof of s = 0 is the same.To this end, we consider the mild formulations of u N and u and show that each piece of u N converges to the corresponding piece in u.Clearly, S(t)u N 0 → S(t)u 0 in F T .For the noise, let Ψ N (t) denote the stochastic convolution in (5.19).Then where π N denotes the projection onto the linear span of the orthonormal vectors {e j : |j| > N }.By Lemma 3.4, the above is controlled by which tends to 0 as N → ∞ because both norms are tails of convergent series.Finally we treat the nonlinear terms We first fix a path for which local well-posedness holds, and prove that Du By Lemmas 2.2, 2.4 and 2.5, we have In particular, (5.21) implies Du ∈ X 1, 1 2 + ([0, T ]), and hence II → 0 as N → ∞.We claim that I → 0 as N → ∞ as well.Indeed, Λ N and Λ are contractions with fixed points u N and u respectively, hence .
By rearranging, it suffices to show that the first term on the right-hand side above tends to 0 as .
By similar arguments as above, all the terms on the right go to 0 as N → ∞.This proves our claim.By the embedding almost surely as N → ∞.By the dominated convergence theorem, we have Du−D ≤N u → 0 in F T .This concludes our proof.
Finally, we conclude the proof of global well-posedness for the additive case.
Proof of Theorem 1.5.Let s ∈ {0, 1} be the regularity of u 0 from Theorem 1.5.Let ε > 0 and T > 0 be given.We claim that there exists an event Ω ε such that a solution u ∈ we have that P(Ω * ) = 1 and u exists on [0, T ], proving the theorem.Let δ ∈ (0, 1) be a small quantity chosen later.We subdivide where L > 0 is some large quantity determined later.Now by Chebyshev's inequality and Lemma 3.3, By choosing L = L(ε, T, φ) sufficiently large, we may therefore bound P(Ω c 0 ) above by ε 2 .Now let R max { u 0 H s , L} .
By local theory, there exists a unique solution u(t) to (1.1) with time of existence T max given in (4.7).In particular, we note that for ω ∈ Ω 0 , where c is as in (4.7).By choosing δ = δ(R, L) := c(R + L) −θ , we see that u(t) exists for t ∈ [0, δ] for all ω ∈ Ω 0 .Now define By the same argument, u(t) exists for t ∈ (δ, 2δ) for all ω ∈ Ω 1 .Iterating this argument, we have a chain of events Ω and u(t) exists for all t ∈ [0, (k + 1)δ] on Ω k .Setting Ω ε := Ω M −1 , u(t) exists on the full interval [0, T ] on Ω ε .It remains to check that Ω \ Ω ε remains small.By Corollary 5.2, we have for any p ∈ N. We further enlarge R if necessary by setting where have that This is smaller than ε provided we choose p = p(ε, θ) > 0 sufficiently large.Thus Ω ε satisfies our claim.
5.2.SNLS with multiplicative noise.In order to globalize solutions of SNLS, for the multiplicative noise case, we need to prove probabilistic control of the X s,b -norm of the solutions of the truncated SNLS uniformly in the truncation parameter (Lemma 5.4).This requires a priori bounds on mass and energy of solutions.From Subsection 4.2, we obtained a local solution of the multiplicative (1.1) with time of existence Under the hypotheses of Theorem 1.8, we shall prove global well-posedness by showing that τ * = ∞ almost surely.Proposition 5.3.Let T 0 > 0 and φ be as in Theorem 1.8.Suppose that u(t) is a solution for (1.1) with F (u, φξ) = u • φξ on t ∈ [0, T ] for some stopping time T ∈ [0, T 0 ∧ τ * ).Let C(φ) be as in (3.16).Then for any m ∈ N, there exists C 1 = C 1 (m, M (u 0 ), T 0 , C(φ)) > 0 such that E sup The left-hand side is bounded above by 3M, where M is maximum of the three terms of the right-hand side.In any of the three cases, we may conclude the proof via simple rearrangement arguments and Gronwall's inequality.(5.30) .
Similarly, we bound the other terms as follows: .
Arguing in the same way as for the mass of u N yields the estimate for the energy of u N .This proves the proposition for u N in place of u.The proposition then follows by letting N → ∞.
We now prove the following probabilistic a priori bound on the X s,b -norm of a solution.
Lemma 5.4.Let T, R > 0. Let u R be the unique solution of (4.8) on [0, T ].There exists Moreover, if (4.8) is defocusing, there also exists The constants C 1 and C 2 are independent of R.

Corollary 5 . 2 .
For u solution to (1.1) with (1.2), the estimates (5.4) and (5.5) hold with u in place of u N under the same assumptions as Proposition 5.1.