Almost conservation laws for stochastic nonlinear Schr\"odinger equations

In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the $I$-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schr\"odinger equation (SNLS) on $\mathbb R^3$ with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the $I$-method with Ito's lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.


Stochastic nonlinear Schrödinger equation.
We consider the Cauchy problem for the stochastic nonlinear Schrödinger equation (SNLS) with an additive noise: where ξ(t, x) denotes a (Gaussian) space-time white noise on R × R d and φ is a bounded operator on L 2 (R d ). In this paper, we restrict our attention to the defocusing case. Our main goal is to establish global well-posedness of (1.1) in the energy-subcritical case with a rough noise, namely, with a noise not belonging to the energy space H 1 (R d ). Here, the energy-subcriticality refers to the following range of p: (i) 1 < p < 1 + 4 d−2 for d ≥ 3 and (ii) 1 < p < ∞ for d = 1, 2. In terms of the scaling-critical regularity s crit defined by the energy-subcriticality is equivalent to the condition s crit < 1. We say that u is a solution to (1.1) if it satisfies the following Duhamel formulation (= mild formulation): where S(t) = e it∆ denotes the linear Schrödinger propagator. The last term on the righthand side represents the effect of the stochastic forcing and is called the stochastic convolution, which we denote by Ψ: (1.2) See Subsection 2.3 for the precise meaning of the definition (1.2); see (2.6) and (2.7). In the following, we assume that φ ∈ HS(L 2 ; H s ) for appropriate values of s ≥ 0, namely, φ is taken to be a Hilbert-Schmidt operator from L 2 (R d ) to H s (R d ). It is easy to see that φ ∈ HS(L 2 ; H s ) implies Ψ ∈ C(R; H s (R d )) almost surely; see [14]. Our main interest is to study (1.1) when φ ∈ HS(L 2 ; H s ) for s < 1 such that the stochastic convolution does not belong to the energy space H 1 (R d ).
A standard contraction argument with the Strichartz estimates (see (2.3) below) yields local well-posedness of (1.3) in H s (R d ) when s ≥ max(s crit , 0); see [18,24,35,5]. 1 On the other hand, (1.3) is known to be ill-posed in the scaling supercritical regime: s < s crit . See [6,27,29]. In the energy-subcritical case, global well-posedness of (1.3) in H 1 (R d ) easily follows from iterating the local-in-time argument in view of the following conservation laws for (1.3): Mass: M (u(t)) =ˆR d |u(t, x)| 2 dx, providing a global-in-time a priori control on the H 1 -norm of a solution to (1.3).
There are analogues of these well-posedness results in the context of SNLS (1.1). In [15], de Bouard and Debussche studied (1.1) in the energy-subcritical setting, assuming that φ ∈ HS(L 2 ; H 1 ). By using the Strichartz estimates, they showed that the stochastic convolution Ψ almost surely belongs to a right Strichartz space, which allowed them to prove local wellposedness of (1.1) in H 1 (R d ). When s ≥ max(s crit , 0), a slight modification of the argument in [15] and the improved space-time regularity of the stochastic convolution (see Lemma 2.2 below) yields local well-posedness of (1.1) in H s (R d ), provided that φ ∈ HS(L 2 ; H s ). In the energy-subcritical case, one can adapt the globalization argument for the deterministic NLS (1.3), based on the conservation laws (1.4), to the stochastic setting with a sufficiently regular noise. More precisely, assuming φ ∈ HS(L 2 ; H 1 ), de Bouard and Debussche [15] proved global well-posedness of (1.1) in H 1 (R d ) by applying Ito's lemma to the mass M (u) and the energy E(u) in (1.4) and establishing an a priori H 1 -bound of solutions to (1.1). In this paper, we also consider the energy-subcritical case but we treat a rougher noise: φ ∈ HS(L 2 ; H s ) for s < 1.
In the deterministic setting, Colliander, Keel, Staffilani, Takaoka, and Tao [8] introduced the so-called I-method (also known as the method of almost conservation laws) and proved global well-posedness of the energy-subcritical defocusing cubic NLS ((1.3) with p = 3) on R d , d = 2, 3, below the energy space. Since then, the I-method has been applied to a wide class of dispersive models in establishing global well-posedness below the energy spaces (or more generally below regularities associated with conservation laws), where there is no a priori bound on relevant norms (for iterating a local-in-time argument) directly given by a conservation law. Our strategy for proving global well-posedness of SNLS (1.1) when φ ∈ HS(L 2 ; H s ), s < 1, is to implement the I-method in the stochastic PDE setting. This will provide a general framework for establishing global well-posedness of stochastic dispersive equations with additive noises below energy spaces.
1.2. Main result. For the sake of concreteness, we consider SNLS (1.1) in the threedimensional cubic case (d = 3 and p = 3): We point out, however, that our implementation of the I-method in the stochastic PDE setting is sufficiently general and can be easily adapted to other dispersive models with rough additive stochastic forcing. We now state our main result.
Suppose that φ ∈ HS(L 2 ; H s ) for some s > 5 6 . Then, the defocusing stochastic cubic NLS (1.5) on R 3 is globally well-posed in H s (R 3 ).
Note that the regularity range s > 5 6 is exactly the same as that in the deterministic case [8]. In view of the global well-posedness result by de Bouard and Debussche [15], we only consider 5 6 < s < 1 in the following. Let us first go over the main idea of the I-method argument in [8] applied to the deterministic cubic NLS on R 3 , i.e. (1.5) with φ = 0. Fix u 0 ∈ H s (R 3 ) for some 5 6 < s ≤ 1. Then, the standard Strichartz theory yields local well-posedness of (1.3) with u| t=0 = u 0 in the subcritical sense, namely, time of local existence depends only on the H s -norm of the initial data u 0 . Hence, once we obtain an a priori control of the H s -norm of the solution, we can iterate the local-in-time argument and prove global existence. When s = 1, the conservation of the mass and energy in (1.4) provides a global-in-time a priori control of the H 1 -norm of the solution. When 5 6 < s < 1, the conservation of the energy E(u) is no longer available (since E(u) = ∞ in general), while the mass M (u) is still finite and conserved. Therefore, the main goal is to control the growth of the homogeneous Sobolev H s -norm of the solution.
Unlike the s = 1 case, we do not aim to obtain a global-in-time boundedness of thė H s -norm of the solution. Instead, the goal is to show that, given any large target time T ≫ 1, theḢ s -norm of the solution remains finite on the time interval [0, T ], with a bound depending on T . The main idea of the I-method is to introduce a smoothing operator I = I N , known as the I-operator, mapping H s (R 3 ) into H 1 (R 3 ). Here, the I-operator depends on a parameter N = N (T ) ≫ 1 (to be chosen later) such that I N acts essentially as the identity operator on low frequencies {|ξ| N } and as a fractional integration operator of order 1−s on high frequencies {|ξ| ≫ N }; see Section 3 for the precise definition. Thanks to the smoothing of the I-operator, the modified energy: is finite for u ∈ H s (R 3 ). Moreover, the modified energy E(I N u) controls u 2Ḣ s . See (3.1) below. Hence, the main task is reduced to controlling the growth of the modified energy E(I N u).
While the energy E(u) is conserved for (smooth) solutions to NLS (1.3), the modified energy E(I N u) is no longer conserved since I N u does not satisfy the original equation. Instead, I N u satisfies the following I-NLS: (1.6) where N (u) = |u| 2 u denotes the cubic nonlinearity. The commutator term is the source of non-conservation of the modified energy E(I N u). A direct computation shows See (5.1). Thanks to the commutator structure, it is possible to obtain a good estimate (with a decay in the large parameter N ) for ∂ t E(I N u) on each local time interval (See Proposition 4.1 in [8]). Then, by using a scaling argument (with a parameter λ = λ(T ) ≫ 1, depending on the target time T ), we (i) first reduce the situation to the small data setting, (ii) then iterate the local-in-time argument with a good bound on ∂ t E(I N u λ ) on the scaled solution u λ , and (iii) choose N = N (T ) ≫ 1 sufficiently large such that the scaled target time λ 2 T is (at most) the doubling time for the modified energy E(I N u λ ). This yields the regularity restriction s > 5 6 in [8]. Let us turn to the case of the stochastic NLS (1.5). In proceeding with the I-method, we need to estimate the growth of the modified energy E(I N u). In this stochastic setting, we have two sources for non-conservation of E(I N u). The first one is the commutator term [I N , N ](u) in (1.7) as in the deterministic case described above. This term can be handled almost in the same manner as in [8] but some care must be taken due to a weaker regularity in time (b < 1 2 ). See Proposition 4.1 below. The second source for non-conservation of E(I N u) is the stochastic forcing. In particular, in estimating the growth of the modified energy E(I N u), we need to apply Ito's lemma to E(I N u), which introduces several correction terms.
In the deterministic case [8], one iteratively applies the local-in-time argument and estimate energy increment on each local time interval. A naive adaptation of this argument to the stochastic setting would lead to iterative applications of Ito's lemma to estimate the growth of the modified energy E(I N u). In controlling an expression of the form we need to apply Burkholder-Davis-Gundy inequality, which introduces a multiplicative constant C > 1. See Lemma 5.1 below. Namely, if we were to apply Ito's lemma iteratively on each time interval of local existence, then this would lead to an exponential growth of the constant in front of the modified energy. This causes an iteration argument to break down.
We instead apply Ito's lemma only once on the global time interval [0, λ 2 T ]. At the same time, we estimate the contribution from the commutator term iteratively on each local time interval. Note that this latter task requires a small data assumption, which we handle by introducing a suitable stopping time and iteratively verifying such a small data assumption. See Section 6.
As in the deterministic setting, we employ a scaling argument to reduce the problem to the small data regime. In the stochastic setting, we need to proceed with care in applying a scaling to the noise φξ since we need to apply Ito's lemma after scaling. Namely, we need to express the scaled noise as φ λ ξ λ , where ξ λ is another space-time white noise (defined by the white noise scaling; see (3.16) below) such that Ito calculus can be applied. This forces us to study the scaled Hilbert-Schmidt operator φ λ . In the application of Ito's lemma, there are correction terms due to I N φ λ besides the commutator term [I N , N ](u λ ). In order to carry out an iterative procedure, we need to make sure that the contribution from the correction terms involving I N φ λ is negligible as compared to that from the commutator term. See Subsection 3.3 and Section 6. As a result, the regularity restriction s > 5 6 comes from the commutator term as in the deterministic case.
We conclude this introduction by several remarks.
Remark 1.2. In this paper, we implement the I-method in the stochastic PDE setting.
There is a recent work [22] by Gubinelli, Koch, Tolomeo, and the third author, establishing global well-posedness of the (renormalized) defocusing stochastic cubic nonlinear wave equation on the two-dimensional torus T 2 , forced by space-time white noise. The I-method was also employed in [22]. We point out that our argument in this paper is a genuine extension of the I-method to the stochastic setting, which can be applied to a wide class of stochastic dispersive equations. On the other hand, in [22], the I-method was applied to the residual term v = u − Ψ wave in the Da Prato-Debussche trick [13], where Ψ wave denotes the stochastic convolution in the wave setting. Furthermore, the I-method argument in [22] is pathwise, namely, entirely deterministic once we take the pathwise regularity of Ψ wave (and its Wick powers) from [21].
In a recent paper [30], the third author and Okamoto studied SNLS (1.1) in the mass-critical case (p = 1 + 4 d ) and the energy-critical case (p = 1 + 4 d−2 , d ≥ 3). By adapting the recent deterministic mass-critical and energy-critical global theory, they proved global well-posedness of (1.1) in the critical spaces. In particular, when d = 2 and p = 3, this yields global well-posedness the two-dimensional defocusing stochastic cubic NLS in L 2 (R 2 ). This is the reason why we only considered the three-dimensional case in Theorem 1.1, since our I-method argument would yield global well-posedness only for s > 4 7 in the two-dimensional cubic case (just as in the deterministic case [8]), which is subsumed by the aforementioned global well-posedness result in [30]. Remark 1.4. In an application of the I-method, it is possible to introduce a correction term (away from a nearly resonant part) and improve the regularity range. See [11]. It would be of interest to implement such an argument to the stochastic PDE setting since a computation of a correction term would involve Ito's lemma.
Remark 1.5. We mentioned that our implementation of the I-method in the stochastic PDE setting is sufficiently general and is applicable to other dispersive equations forced by additive noise. This is conditional to an assumption that a commutator term can be treated with a weaker temporal regularity b < 1 2 . In the case of SNLS, this can be achieved by a simple interpolation argument, at a slight loss of spatial regularity. See Section 4. See also [7] for an analogous argument in the periodic case. In this regard, it is of interest to study the stochastic KdV equation in negative Sobolev spaces since crucial estimates for KdV require the temporal regularity to be b = 1 2 . See [3,25,10].
This paper is organized as follows. In Section 2, we go over the preliminary materials from deterministic and stochastic analysis. We then reduce a proof of Theorem 1.1 to controlling the homogeneousḢ s -norm of a solution (Remark 2.4). In Section 3, we introduce the Ioperator and go over local well-posedness of I-SNLS (3.2). Then, we discuss the scaling properties of I-SNLS in Subsection 3.3. In Section 4, we briefly go over the nonlinear estimates, indicating required modifications from [8]. In Section 5, we apply Ito calculus to bound the modified energy in terms of a term involving the commutator [I N , N ]. Lastly, we put all the ingredients together and present a proof of Theorem 1.1 in Section 6.

Preliminaries
In this section, we first introduce notations and function spaces along with the relevant linear estimates. We also go over preliminary lemmas from stochastic analysis. We then discuss a reduction of the proof of Theorem 1.1; see Remark 2.4.

2.1.
Notations. For simplicity, we drop 2π in dealing with the Fourier transforms. We first recall the Fourier restriction norm spaces . The X s,b -space is defined by the norm: where · = (1 + | · | 2 ) 1 2 . When b > 1 2 , we have the following embedding: Given a time interval J ⊂ R, we also define the local-in-time version X s,b (J) in an analogous manner.
When we work with space-time function spaces, we use short-hand notations such as . We write A B to denote an estimate of the form A ≤ CB. Similarly, we write A ∼ B to denote A B and B A and use A ≪ B when we have A ≤ cB for small c > 0. We may use subscripts to denote dependence on external parameters; for example, A p,q B means A ≤ C(p, q)B, where the constant C(p, q) depends on parameters p and q. We also use a+ (and a−) to mean a + ε (and a − ε, respectively) for arbitrarily small ε > 0. As it is common in probability theory, we use A ∧ B to denote min(A, B).
In view of the time reversibility of the problem, we only consider positive times in the following.

2.3.
Tools from stochastic analysis. Lastly, we go over basic tools from stochastic analysis and then provide some reduction for the proof of Theorem 1.1.
We first recall the regularity properties of the stochastic convolution Ψ defined in (1.2). Given two separable Hilbert spaces H and K, we denote by HS(H; K) the space of Hilbert-Schmidt operators φ from H to K, endowed with the norm: where {f n } n∈N is an orthonormal basis of H. Recall that the Hilbert-Schmidt norm of φ is independent of the choice of an orthonormal basis of H.
Next, recall the definition of a cylindrical Wiener process W on L 2 (R d ). Let (Ω, F, P ) be a probability space endowed with a filtration {F t } t≥0 . Fix an orthonormal basis {e n } n∈N of L 2 (R d ). We define an L 2 (R d )-cylindrical Wiener process W by where {β n } n∈N is a family of mutually independent complex-valued Brownian motions 2 associated with the filtration {F t } t≥0 . Note that a space-time white noise ξ is given by a distributional derivative (in time) of W . Hence, we can express the stochastic convolution Ψ in (1.2) as The next lemma summarizes the regularity properties of the stochastic convolution. See [14] for (i) and [15,31] for (ii). As for (iii), see [16,28,7] for the proofs of the X s,bregularity of the stochastic convolution. The works [16,28,7] treat a different equation (KdV) and/or a different setting (on the circle) but the proofs can be easily adapted to our context. (ii) Given any 1 ≤ q < ∞ and finite r ≥ 2 such that r ≤ 2d d−2 when d ≥ 3, we have Ψ ∈ L q ([0, T ]; W s,r (R d )) almost surely. Once we have Lemma 2.2, we can use the Strichartz estimates (2.3) (without the X s,bspaces) to prove local well-posedness of SNLS (1.5) in H s (R 3 ) for s ≥ s crit = 1 2 , provided that φ ∈ HS(L 2 ; H s ). See [15,30]. In particular, for the subcritical range s > 1 2 , the random time δ = δ(ω) of local existence, starting from t = t 0 , satisfies for some θ > 0, where C t 0 (Ψ) > 0 denotes certain Strichartz norms of the stochastic convolution Ψ, restricted to a time interval [t 0 , t 0 + 1]. Given T > 0, it follows from Lemma 2.2 that C t 0 (Ψ) remains finite almost surely for any t 0 ∈ [0, T ]. Therefore, Theorem 1.1 follows once we show that sup t∈[0,T ] u(t) H s remains finite almost surely for any T > 0 (with a bound depending on T > 0).
Lastly, we recall the a priori mass control from [15] whose proof follows from Ito's lemma applied to the mass M (u) in (1.4) and Burkholder-Davis-Gundy inequality (see [14,Theorem 4.36]). Lemma 2.3. Assume φ ∈ HS(L 2 ; L 2 ) and u 0 ∈ L 2 (R 3 ). Let u be the solution to SNLS (1.5) with u| t=0 = u 0 and T * = T * ω (u 0 ) be the forward maximal time of existence. Then, given T > 0, there exists C 1 = C 1 (M (u 0 ), T, φ HS(L 2 ;L 2 ) ) > 0 such that for any stopping time τ with 0 < τ < min(T * , T ) almost surely, we have We then define the I-operator I = I N to be the Fourier multiplier operator with the multiplier m N : As mentioned in the introduction, I N acts as the identity operator on low frequencies {|ξ| ≤ N }, while it acts as a fractional integration operator of order 1−s on high frequencies {|ξ| ≥ 2N }. As a result, we have the following bound: 3.2. I-SNLS. By applying the I-operator to SNLS (1.5), we obtain the following I-SNLS: In this subsection, we study local well-posedness of the Cauchy problem (3.2). A similar local well-posedness result for the (deterministic) I-NLS (namely, (3.2) with φ = 0) was studied in [8,Proposition 4.2]. In order to capture the temporal regularity of the stochastic convolution (Lemma 2.2), we need to work with the X s,b -space with b < 1 2 and hence need to establish a trilinear estimate in this setting. See Lemma 3.2 below. The following proposition allows us to avoid using the L 2 -norm which is supercritical with respect to scaling (as in [8]). Proposition 3.1. Let 1 2 < s < 1, φ ∈ HS(L 2 ;Ḣ s ), and u 0 ∈Ḣ s (R 3 ). Then, there exist an almost surely positive stopping time and a unique local-in-time solution I N u ∈ C([0, δ];Ḣ 1 (R 3 )) to I-SNLS (3.2). Furthermore, if T * = T * ω denotes the forward maximal time of existence, the following blowup alternative holds: Proposition 3.1 follows from a standard contraction argument once we prove the following trilinear estimate.
As compared to Proposition 4.2 in [8], we need to work with a slightly weaker temporal regularity on the right-hand side of (3.4).
Before going over a proof of Lemma 3.2, let us briefly discuss a proof of Proposition 3.1. By writing (3.2) in the Duhamel formulation, we have where Φ = Φ I N u 0 ,I N φ and we interpreted the nonlinearity as a function of I N u: ). Fix small ε > 0 Then, by Lemmas 2.1 and 2.2 followed by Lemma 3.2, we have ∇Φ(I N u) for an almost surely finite random constant C ω > 0 and for any 0 ≤ δ ≤ 1. Similarly, we have (3.6) From (3.5) and (3.6), we conclude that Φ is almost surely a contraction on the ball of radius in ∇ −1 X 0, 1 2 −ε by choosing δ = δ ω (R) > 0 sufficiently small. Moreover, from (2.1) and Lemmas 2.1 and 2.2 with (3.5), we also conclude that I N u ∈ C([0, δ];Ḣ 1 (R 3 )). This proves Proposition 3.1. The following remark plays an important role in iteratively applying the local-in-time argument in Section 6.
Remark 3.3. The argument above shows that there exist small η 0 , η 1 > 0 such that if, for a given interval J = [t 0 , t 0 + 1] ⊂ [0, ∞) of length 1 and ω ∈ Ω, we have then a solution I N u to I-SNLS (3.2) exists on the interval J with the bound: for some absolute constant C 0 , uniformly in N ≥ 1.
We now present a proof of Lemma 3.2. ∇Iu j X 0, 1 2 −ε u 4 X 0, 1 2 −2ε . (3.8) For j ∈ {2, 3}, we split the functions u j into high and low frequency components: As for u hi j , we claim Since N = 1, we have I ∼ |∇| s−1 . Then, by Sobolev's inequality and the transference principle (Lemma 2.1 (iii)) with an admissible pair (q, r) = 5+, 30 11  By interpolating (3.12) and (3.13), we obtain (3.11). We now estimate (3.8) by expanding u j , j = 2, 3, as u hi j + u low j . For j = 2, 3, let p j = 6 in treating u low j and p j = 5+ in treating u hi j . Then, the claimed estimate (3.8) follows from L 10 3 − t,x , L p 2 t,x , L p 3 t,x , L p 4 t,x -Hölder's inequality, Lemma 2.1 (iv), (3.10), and (3.11), where p 4 is defined by 1 3.3. Scaling property. In this subsection, we discuss the scaling properties of SNLS (1.5) and I-SNLS (3.2). Before doing so, we first recall the scaling property of the (deterministic) cubic NLS: i∂ t u + ∆u = |u| 2 u. (3.14) This equation enjoys the following scaling invariance; if u is a solution to (3.14), then the scaled function also satisfies the equation (3.14) with the scaled initial data. In the application of the I-method in the deterministic case (as in [8]), we apply this scaling first and then apply the I-operator to obtain I-NLS (1.6) (with u λ in place of u). In our current stochastic setting, when we apply the scaling, we also need to scale the noise φξ. In order to apply Ito calculus to the scaled noise, we need to make sure that the scaled noise is given by another space-time white noise ξ λ (with a scaled Hilbert-Schmidt operator φ λ ). For this purpose, we first recall the scaling property of a space-time white noise. Given a space-time white noise ξ on R × R d , it is well known that the scaled noise ξ λ defined by 4 is also a space-time white noise for any a 1 , a 2 ∈ R.
Next, let us study the scaling property of the Hilbert-Schmidt operator φ via its kernel representation. Recall from [32,Theorem VI.23] that a bounded linear operator φ on L 2 (R 3 ) is Hilbert-Schmidt if and only if it is represented as an integral operator with a kernel k ∈ L 2 (R 3 × R 3 ): x,y . More generally, we have φ HS(L 2 ;Ḣ s ) = k Ḣs x L 2 y . (3.17) With this in mind, let us evaluate φξ at ( t λ 2 , x λ ) with a factor of λ −3 . By a change of variables and (3.16) with (a 1 , a 2 ) = (2, 0), we have This motivates us to define the scaled kernel k λ by and the associated Hilbert-Schmidt operator φ λ with an integral kernel k λ . Then, it follows from (3.18) and (3.19) that Therefore, by applying the scaling (3.15) with (3.20) to SNLS (1.5) and then applying the I-operator, we obtain In the following lemma, we record the scaling property of the Hilbert-Schmidt norm of Lemma 3.4. Let d = 3, 0 < s < 1, and φ ∈ HS(L 2 ,Ḣ s ). Then, we have As a consequence, given any ε > 0, there exists θ > 0 such that for any finite p ≥ 1 and T > 0, where Ψ λ is the stochastic convolution corresponding to the scaled noise φ λ ξ λ . Furthermore, if we assume φ ∈ HS(L 2 ,Ḣ 3 4 ), then we have , (3.24) uniformly in N ≥ 1.
Remark 3.5. (i) It is easy to check that Lemma 3.4 remains true even if we proceed with a scaling argument in (3.18) for any a 2 ∈ R.

On the commutator estimates
In this section, we go over the commutator estimates (Proposition 4.1), corresponding to the deterministic component in our application of the I-method.
The estimates (4.1) and (4.2) are essentially the same as those appearing in the proof of Proposition 4.1 in [8]. The difference appears in the temporal regularity; on the right-hand sides of (4.1) and (4.2), we have b = 1 2 − ε, whereas the temporal regularity in [8] was b = 1 2 + ε. The desired estimates in Proposition 4.1 follow from the corresponding estimates in [8] and an interpolation argument.  (ii) Let 2 3 < s < 1. Then, the following trilinear estimate holds: for any interval J ⊂ [0, ∞), where the implicit constants are independent of N ≥ 1 and J ⊂ [0, ∞).
We now briefly discuss a proof of Proposition 4.1.

On growth of the modified energy
In this section, we use stochastic analysis to study growth of the modified energy E(I N u) associated with I-SNLS (3.2). Before doing so, we first go over the the deterministic setting. Given a smooth solution u to the cubic NLS (3.14), we can verify the conservation of the energy E(u) = 1 2´R 3 |∇u| 2 dx + 1 4´R 3 |u| 4 dx by simply differentiating in time and using the equation (3.14): In a similar manner, given a smooth solution u to the cubic NLS (3.14), the time derivative of the modified energy E(I N u) is given by Then, the fundamental theorem of calculus yields for any t 1 , t 2 ∈ R, where the right-hand side can be estimated by the commutator estimate; see [8,Proposition 4.1]. For our problem. we need to estimate growth of the modified energy E(I N u), where u is now a solution to SNLS (1.5) with a stochastic forcing. 5 As such, we need to proceed with Ito's lemma.
Furthermore, if we assume that I N φ ∈ HS(L 2 ;Ḣ 3 4 ) in addition, then there exists C > 0 such that The remaining terms are the additional terms, appearing from the application of Ito's lemma. Part (ii) follows from (5.4) and Burkholder-Davis-Gundy inequality. We need to hide some terms on the right-hand side of (5.3) to the left-hand side, which results in the factor of 2 appears in the first term on the right-hand side of (5.4).
In the deterministic setting, one can apply the commutator estimate to (5.2) on each local-in-time interval. In the current stochastic setting, however, it is not possible to apply the estimate (5.4) (and the commutator estimates (Proposition 4.1)) on each local-in-time interval since this factor of 2 on E(I N u 0 ) in (5.4) would lead to an exponential growth of the constant in iterating the local-in-time argument.
(ii) Our convention states that β n in (2.6) is a complex-valued Brownian motion. This is the reason we do not have a factor 1 2 on the last term in (5.3). (iii) In controlling the fourth and fifth terms on the right-hand side of (5.3), we need to use the HS(L 2 ;Ḣ Proof. A formal application of Ito's lemma yields (5.3). One can justify the computation by inserting several truncations and the local well-posedness argument. See [15] for details when there is no I-operator.
Let us now turn to Part (ii). By Burkholder-Davis-Gundy inequality ([23, Theorem 3.28 on p.166]), Cauchy-Schwarz inequality, and Cauchy's inequality, we estimate the third term on the right-hand side of (5.3) as ) .

(5.5)
By Burkholder-Davis-Gundy inequality and Sobolev's inequalityḢ , the fourth terms is estimated as . In this section, we present global well-posedness of SNLS (1.5) (Theorem 1.1). In the current stochastic setting, it suffices to prove the following "almost" almost sure global well-posedness. Proposition 6.1. Given 5 6 < s < 1, let u 0 ∈ H s (R 3 ) and φ ∈ HS(L 2 ; H s ). Then, given any T, ε > 0, there exists a set Ω T,ε ⊂ Ω such that Once we prove Proposition 6.1, Theorem 1.1 follows from the Borel-Cantelli lemma. See, for example, [12,1]. Hence, in the remaining part of this paper, we focus on proving Proposition 6.1.