On the stochastic nonlinear Schr\"odinger equations at critical regularities

We consider the Cauchy problem for the defocusing stochastic nonlinear Schr\"odinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by the first author with B\'enyi and Pocovnicu (2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.

1. Introduction 1.1. Stochastic nonlinear Schrödinger equations. We consider the Cauchy problem for the stochastic nonlinear Schrödinger equation (SNLS) with an additive noise: where ξ(t, x) denotes a 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 present a concise argument in establishing global well-posedness of (1.1) in the so-called mass-critical and energy-critical cases. Let us first go over the notion of the scaling-critical regularity for the (deterministic) defocusing nonlinear Schrödinger equation (NLS): i∂ t u + ∆u = |u| p−1 u, (1.2) namely, (1.1) with φ ≡ 0. The equation (1.2) is known to enjoy the following dilation symmetry: for λ > 0. If u is a solution to (1.2), then the scaled function u λ is also a solution to (1.2) with the rescaled initial data. This dilation symmetry induces the following scaling-critical Sobolev regularity: such that the homogeneousḢ s crit (R d )-norm is invariant under the dilation symmetry. This critical regularity s crit provides a threshold regularity for well-posedness and ill-posedness of (1.2). Indeed, when s ≥ max(s crit , 0), the Cauchy problem (1.2) is known to be locally well-posed in H s (R d ) [19,22,36,6]. 1 On the other hand, it is known that NLS (1.2) is ill-posed in the scaling supercritical regime: s < s crit . See [9,26,28].
Moreover, the following space-time bound on a global solution u to (1.2) holds: with (i) k = 0 in the mass-critical case and (ii) k = 1 in the energy-critical case. This bound in particular implies that global-in-time solutions scatter, i.e. they asymptotically behave like linear solutions as t → ±∞.
Let us now turn our attention to SNLS (1.1). We say that u is a solution to (1.1) if it satisfies the following Duhamel formulation (= mild formulation): (1.4) where S(t) = e it∆ denotes the linear Schrödinger propagator. The last term on the righthand side of (1.4) is called the stochastic convolution, which we denote by Ψ. Fix a probability space (Ω, F, P ) endowed with a filtration {F t } t≥0 and let W denote the L 2 (R d )cylindrical Wiener process associated with the filtration {F t } t≥0 ; see (2.3) below for a precise definition. Then, the stochastic convolution Ψ is defined by (1.5) See Section 2 for the precise meaning of the definition (1.5); in particular see (2.4). 1 When p is not an odd integer, we may need to impose an extra assumption due to the non-smoothness of the nonlinearity.
Our main goal is to construct global-in-time dynamics for (1.4) in the mass-critical and energy-critical cases. This means that we take (i) p = 1 + 4 d in the mass-critical case and (ii) p = 1 + 4 d−2 in the energy-critical case. Furthermore, we take the stochastic convolution Ψ in (1.5) to be at the corresponding critical regularity. Suppose that φ ∈ HS (L 2 ; H s ), namely, φ is a Hilbert-Schmidt operator from L 2 (R d ) to H s (R d ). Then, it is known that Ψ ∈ C(R + ; H s (R d )) almost surely; see [12]. Therefore, we will impose that (i) φ ∈ HS (L 2 ; L 2 ) in the mass-critical case and (ii) φ ∈ HS (L 2 ; H 1 ) in the energy-critical case.
Previously, de Bouard and Debussche [14] studied SNLS (1.1) in the energy-subcritical setting: s crit < 1, 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 well-posedness of (1.1) in H 1 (R d ) with φ ∈ HS (L 2 ; H 1 ) in the energy-subcritical case: 1 < p < 1 + 4 d−2 when d ≥ 3 and 1 < p < ∞ when d = 1, 2. We point out that when s ≥ max(s crit , 0), a slight modification of the argument in [14] with the regularity properties of the stochastic convolution (see Lemma 2.2 below) yields local well-posedness 2 of (1.1) in H s (R d ), provided that φ ∈ HS (L 2 ; H s ). See Lemma 2.3 for the statements in the mass-critical and energy-critical cases. We also mention recent papers [30,8] on local well-posedness of (1.1) with additive noises rougher than the critical regularities, i.e. φ ∈ HS (L 2 ; H s ) with s < s crit .
In the energy-subcritical case, assuming φ ∈ HS (L 2 ; H 1 ), global well-posedness of (1.1) in H 1 (R d ) follows from an a priori H 1 -bound of solutions to (1.1) based on the conservation of the energy E(u) for the deterministic NLS and Ito's lemma; see [14]. See also Lemma 2.4. In a recent paper [7], Cheung, Li, and the first author adapted the I-method [10] to the stochastic PDE setting and established global well-posedness of energy-subcritical SNLS below H 1 (R d ). In the mass-subcritical case, global well-posedness in L 2 (R d ) also follows from an a priori L 2 -bound based on the conservation of the mass M (u) for the deterministic NLS and Ito's lemma.
We extend these global well-posedness results to the mass-critical and energy-critical settings.
In the following, we only consider deterministic initial data u 0 . This assumption is, however, not essential and we may also take random initial data (measurable with respect to the filtration F 0 at time 0).
In the mass-critical case (and the energy-critical case, respectively), the a priori L 2bound (and the a priori H 1 -bound, respectively) does not suffice for global well-posedness (even in the case of the deterministic NLS (1.2)). The main idea for proving Theorem 1.1 is to adapt the probabilistic perturbation argument introduced by the authors [4,29] in studying global-in-time behavior of solutions to the defocusing energy-critical cubic NLS with random initial data below the energy space. Namely, by letting v = u − Ψ, where Ψ is the stochastic convolution defined in (1.5), we study the equation satisfied by v: where N (u) = |u| p−1 u. Write the nonlinearity as Then, the regularity properties of the stochastic convolution (see Lemma 2.2 below) and the fact that their space-time norms can be made small on short time intervals allow us to view the second term on the right-hand side as a perturbative term. By invoking the perturbation lemma (Lemmas 3.2 and 4.3), we then compare the solution v to (1.6) with a solution to the deterministic NLS (1.2) on short time intervals as in [4,29]. See also [35,24] for similar arguments in the deterministic case. In the energy-critical case, we rely on the Lipschitz continuity of ∇N (u) in the perturbation argument, which imposes the assumption d ≤ 6 in Theorem 1.1.

Remark 1.2.
We remark that solutions constructed in this paper are adapted to the given filtration {F t } t≥0 . For example, adaptedness of a solution v to (1.6) directly follows from the local-in-time construction of the solution via the Picard iteration. Namely, we consider the map Γ defined by Then, we define the jth Picard iterate P j by setting for j ∈ N. Since the stochastic convolution Ψ is adapted to the filtration {F t } t≥0 , it is easy to see from (1.7) that P j is adapted for each j ∈ N. Furthermore, the local well-posedness of (1.6) by a contraction mapping principle (see Lemmas 3.1 and 4.1 below) shows that the sequence {P j } j∈N converges, in appropriate functions spaces, to a limit v = lim j→∞ P j , which is a solution to (the mild formulation of) (1.6). By invoking the closure property of measurability under a limit, we conclude that the solution v to (1.6) is also adapted to the filtration {F t } t≥0 . The same comment applies to Lemma 2.3 below.
Remark 1.3. (i) In the focusing case, i.e. with −|u| p−1 u in (1.1), de Bouard and Debussche [13] proved under appropriate conditions that, starting with any initial data, finite-time blowup occurs with positive probability.
(ii) In the mass-subcritical and energy-critical cases, SNLS with a multiplicative noise has been studied in [1,2,3]. In recent preprints, Fan and Xu [18] and Zhang [39] proved global well-posedness of SNLS with a multiplicative noise in the mass-critical and energy-critical setting.

Preliminary results
In this section, we introduce some notations and go over preliminary results. 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 {e n } n∈N is an orthonormal basis of H.
Since our focus is the mass-critical and energy-critical cases, we introduce N k (u), k = 0, 1, by Namely, k = 0 corresponds to the mass-critical case, while k = 1 corresponds to the energycritical case. The Strichartz estimates play an important role in our analysis. We say that a pair (q, r) is admissible if 2 ≤ q, r ≤ ∞, (q, r, d) = (2, ∞, 2), and Then, the following Strichartz estimates are known to hold; see [32,38,20,23].
Lemma 2.1. Let (q, r) be admissible. Then, we have For any admissible pair ( q, r), we also have where q ′ and r ′ denote the Hölder conjugates. Moreover, if the right-hand side of (2.2) is finite for some admissible pair ( q, r), Next, we provide a precise meaning to the stochastic convolution defined in (1.5). 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 associated with the filtration {F t } t≥0 . Here, the complex-valued Brownian motion means that Re β n (t) and Im β n (t) are independent (real-valued) Brownian motions. Then, the space-time white noise ξ is given by a distributional derivative (in time) of W and thus we can express the stochastic convolution Ψ as where each summand is a classical Wiener integral (with respect to the integrator dβ n ); see [27]. Then, we have the following lemma on the regularity properties of the stochastic convolution. See, for example, Proposition 5.9 in [12] for Part (i). As for Part (ii), see [30].
By the Strichartz estimates (Lemma 2.1) and Lemma 2.2 on the stochastic convolution, one can easily prove the following local well-posedness (see Lemma 2.3 below) of the masscritical and energy-critical SNLS (1.1) by essentially following the argument in [14], namely, by studying the Duhamel formulation for See also Lemmas 3.1 and 4.1 below. In the mass-critical case, the admissible pair q = r = 2(d+2) d plays an important role. In the energy-critical case, we use the following admissible pair . Then, given any u 0 ∈ L 2 (R d ), there exists an almost surely positive stopping time T = T ω (u 0 ) and a unique local-in-time solution u ∈ C([0, T ]; L 2 (R d )) to the mass-critical SNLS (1.1). Furthermore, the following blowup alternative holds; let T * = T * ω (u 0 ) be the forward maximal time of existence. Then, either (ii) (energy-critical case). Let 3 ≤ d ≤ 6, p = 1 + 4 d−2 , and φ ∈ HS (L 2 ; H 1 ). Then, given any u 0 ∈ H 1 (R d ), there exists an almost surely positive stopping time T = T ω (u 0 ) and a unique local-in-time solution u ∈ C([0, T ]; H 1 (R d )) to the energy-critical SNLS (1.1). Furthermore, the following blowup alternative holds; let T * = T * ω (u 0 ) be the forward maximal time of existence. Then, either We note that the mapping: (u 0 , Ψ) → v is continuous. See Proposition 3.5 in [14]. In the energy-critical case, the local-in-time well-posedness also holds for d > 6 (see Remark 4.2 below). As mentioned earlier, the perturbation argument requires the Lipschitz continuity of ∇N and hence we need to assume d ≤ 6 in the following.
Lastly, we state the a priori bounds on the mass and energy of solutions constructed in Lemma 2.3.
where u is the solution to the mass-critical SNLS (1.1) with u| t=0 = u 0 and T * = T * ω (u 0 ) is the forward maximal time of existence.
(ii) (energy-critical case). Assume the hypotheses in Lemma 2.3 (ii). Then, given where u is the solution to the defocusing energy-critical SNLS (1.1) with u| t=0 = u 0 and For Part (ii), we need to assume that the equation is defocusing. These a priori bounds follow from Ito's lemma and the Burkholder-Davis-Gundy inequality. In order to justify an application of Ito's lemma, one needs to go through a certain approximation argument. See, for example, Proposition 3.2 in [14]. In our mass-critical and energy-critical settings, however, such an approximation argument is more involved and hence we present a sketch of the argument in Appendix A.

Mass-critical case
In this section, we prove global well-posedness of the defocusing mass-critical SNLS (1.1) (Theorem 1.1 (i)). In Subsection 3.1, we first study the following defocusing mass-critical NLS with a deterministic perturbation: where N 0 is as in (2.1) and f is a given deterministic function, satisfying certain regularity conditions. By applying the perturbation lemma, we prove global existence for (3.1), assuming an a priori L 2 -bound of a solution v to (3.1). See Proposition 3.3. In Subsection 3.2, we then present the proof of Theorem 1.1 (i) by writing (1.1) in the form (3.1) (with f = Ψ) and verifying the hypotheses in Proposition 3.3.

3.1.
Mass-critical NLS with a perturbation. By the standard Strichartz theory, we have the following local well-posedness of the perturbed NLS (3.1).
Proof. We show that the map Γ defined by provided that η > 0 is sufficiently small. Noting that the Hölder conjugate of 2(d+2) and for any v ∈ B 2η . This shows that v ∈ C(I; L 2 (R d )).
Next, we recall the long-time stability result in the mass-critical setting. See [35] for the proof.
Lemma 3.2 (Mass-critical perturbation lemma). Let I be a compact interval. Suppose that v ∈ C(I; L 2 (R d )) satisfies the following perturbed NLS: for some u 0 ∈ L 2 (R d ), some t 0 ∈ I, and some ε < ε 0 , then there exists a solution u ∈ C(I; L 2 (R d )) to the defocusing mass-critical NLS: In the remaining part of this subsection, we consider long time existence of solutions to the perturbed NLS (3.1) under several assumptions. Given T > 0, we assume that there exist C, θ > 0 such that f (ii) Given a solution v to (3.1), the following a priori L 2 -bound holds: for some R ≥ 1.

Proof of Theorem 1.1 (i).
We are now ready to present a proof of Theorem 1.1 (i). Given a local-in-time solution u to (1.1), let v = u − Ψ. Then, v satisfies (3.13) Theorem 1.1 (i) follows from applying Proposition 3.3 to (3.13) with f = Ψ, once we verify the hypotheses (i) and (ii). Fix T > 0. From Lemma 2.4 and Markov's inequality, we have the following almost sure a priori bound: for a solution u to (1.1) with p = 1 + 4 d . Then, from (3.14) and Lemma 2.2 (i), we obtain sup Then, Markov's inequality yields

Energy-critical case
In this section, we prove global well-posedness of the defocusing energy-critical SNLS (1.1) (Theorem 1.1 (ii)). The idea is to follow the argument for the mass-critical case presented in Section 3. Namely, we study the following defocusing energy-critical NLS with a deterministic perturbation: where N 1 is as in (2.1) and f is a given deterministic function, satisfying certain regularity conditions.
Let q d and r d be as in (2.5) and set ρ d := 2d 2 (d−2) 2 for d ≥ 3. A direct calculation shows that 4.1. Energy-critical NLS with a perturbation. We first go over the local theory for the perturbed NLS (4.1) in the energy-critical case.
for some η ≤ η 0 and some time interval Proof. We show that the map Γ defined by is a contraction on B 2η ⊂ L q d (I; W 1,r d (R d )) of radius 2η > 0 centered at the origin, provided that η > 0 is sufficiently small. It follows from Lemma 2.1 and (4.2) with (4.3) that there exists small η 0 > 0 such that for v ∈ B 2η and 0 < η ≤ η 0 . Recall that ∇N 1 is Lipschitz continuous when 3 ≤ d ≤ 6 and we have See, for example, Case 4 in the proof of Proposition 4.1 in [29]. Then, proceeding as above with (4.4), we have . This shows that v ∈ C(I; H 1 (R d )).
Remark 4.2. The restriction d ≤ 6 appears in (4.4) and (4.5), where we used the Lipschitz continuity of ∇N 1 . Following the argument in [6], we can remove this restriction and construct a solution by carrying out a contraction argument on B 2η ⊂ L q d (I; W 1,r d (R d )) equipped with the distance Indeed, a slight modification of the computation in (4.5) shows d(Γv 1 , Next, we state the long-time stability result in the energy-critical setting. See [11,33,35,25]. The following lemma is stated in terms of non-homogeneous spaces, the proof follows closely to that in the mass-critical case. for some R ≥ 1. Then, there exists ε 0 = ε 0 (R) > 0 such that if we have ≤ ε for some u 0 ∈ H 1 (R d ), some t 0 ∈ I, and some ε < ε 0 , then there exists a solution u ∈ C(I; H 1 (R d )) to the defocusing energy-critical NLS: where C 1 (R) is a non-decreasing function of R.
With Lemmas 4.1 and 4.3 in hand, we can repeat the argument in Proposition 3.3 and obtain the following proposition. The proof is essentially identical to that of Proposition 3.3 and hence we omit details. We point out that, in applying the perturbation lemma (Lemma 4.3) with e = N 1 (v + f ) − N 1 (v), we use (4.4), which imposes the restriction d ≤ 6. ( and there exist C, θ > 0 such that a solution v to (4.1), the following a priori H 1 -bound holds: Fix T > 0. As in Subsection 3.2, the hypothesis (i) in Proposition 4.4 can easily be verified from Hölder's inequality in time, Markov's inequality, and Lemma 2.2 (ii), once we note that Furthermore, the following almost sure a priori bound follows from Lemma 2.4 and Markov's inequality: for a solution u to (1.1) with p = 1 + 4 d−2 . Then, from (4.6) and Lemma 2.2 (i), we obtain sup almost surely. This shows that the hypothesis (ii) in Proposition 4.4 holds almost surely for some almost surely finite R = R(ω) ≥ 1. This proves Theorem 1.1 (ii).

Appendix A. On the application of Ito's lemma
In this appendix, we briefly discuss the derivation of the a priori bounds on the mass and the energy stated in Lemma 2.4. The argument essentially follows from that by de Bouard-Debussche [14] but we indicate certain required modifications.
A.1. Mass-critical case. We first consider the mass-critical case. Given N ∈ N, let P N denote a smooth frequency projection onto {|ξ| N } and set φ N := P N • φ. Then, consider the following truncated SNLS: where N 0 is as in (2.1). Note that P N u 0 ∈ H 1 (R d ) and φ N ∈ HS (L 2 ; H 1 ). Therefore, it follows from [14] that (A.1) is globally well-posed for each N ∈ N. Furthermore, from Proposition 3.2 in [14], we have for any t ≥ 0 and, as a consequence of (A.2) and the Burkholder-Davis-Gundy inequality (see, for example, [21, Theorem 3.28 on p. 166]), the a priori bound (2.6) holds for each u N , with the constant C 1 , independent of N ∈ N. Given T 0 > 0, let 0 < T < min(T * , T 0 ) be a given stopping time as in Lemma 2.4 (i) and u be the solution to (1.1) constructed in Lemma 2.3 (i). We now show that the solution u N to the truncated SNLS (A.1) converges to u almost surely. Then, the a priori bound (2.6) for u follows from that for u N mentioned above and the convergence of u N to u.
In the following, we suppress the spatial domain R d for simplicity of the presentation. Given R > 0, define a stopping time T 1 by setting and set T 2 := min(T, T 1 ). In view of the blowup alternative in Lemma 2.3, we have < ∞ almost surely and hence we conclude that T 2 ր T almost surely as R → ∞.
Define the truncated stochastic convolution Ψ N by and set for j = 0, 1, . . . , J − 1. Then, it follows from the Lebesgue dominated convergence theorem (applied to (Id −P N )u 0 ) and Lemma 2.2 that almost surely as N → ∞. From the Strichartz estimates (Lemma 2.1), we have uniformly in N ≥ N 0 (ω). Thus, from (A.4) and (A.5), we conclude that In particular, we have as N → ∞. By repeating the argument above, we have ∼ η, uniformly in N ≥ N 1 (ω). Together with (A.9), this yields By successively applying the argument above to the interval I j , j = 0, 1, . . . , J − 1, we conclude that as N → ∞. Therefore, recalling that J = J(R, η) depends only on R > 0 and an absolute constant η > 0, we obtain By the almost sure convergence of u N to u in C([0, T 2 ]; L 2 (R d )), Fatou's lemma, and the uniform bound (2.6) for u N , we then have Finally, from the almost sure convergence of T 2 = T 2 (R) to T , as R → ∞, and Fatou's lemma, we conclude the bound (2.6) for u. This proves Lemma 2.4 (i).
A.2. Energy-critical case. Next, we consider the energy-critical case. In the following, we only discuss the a priori bound on the energy: since the a priori bound on the mass follows in a similar but simpler manner.
Lemma A.1. Assume the hypotheses in Lemma 2.3 (ii). Then, for any stopping time T such that 0 < T < T * almost surely, we have where u is the solution to the energy-critical SNLS (1.1) with p = 1+ 4 d−2 , N 1 is as in (2.1), and T * is the forward maximal time of existence.
Once we prove Lemma A.1, the bound (A.10) follows from the Burkholder-Davis-Gundy inequality.

Proof. A direct calculation shows that
Thus, a formal application of Ito's lemma to E(u(t)) yields (A.11). It remains to justify the application of Ito's lemma.
As in the proof of Proposition 3.3 in [14], given N ∈ N, we consider the following truncated problem: where P N and φ N are the same as those in Subsection A.1. Since the frequency truncation is harmless, the same well-posedness result as in Lemma 2.3 (ii) holds for the truncated SNLS (A.12). Moreover, by considering the corresponding Duhamel formulation for (A.12), we have u N = P 3N u N . We can therefore apply Ito's lemma (see Theorem 4.32 in [12]) to E(u N (t)) and obtain for 0 < t < T * N , where T * N is the forward maximal time of existence for the solution u N to (A.12).
Given R > 0, define a stopping T 1 by setting almost surely and hence we conclude that T 2 ր T almost surely as R → ∞.  for any interval I ⊂ [0, T 2 ]. It follows from the Lebesgue dominated convergence theorem and (A.14) that the first term on the right-hand side of (A.15) converges to 0 almost surely as N → ∞. Accordingly, proceeding as in Subsection A.1, we conclude that u N converges to u in C([0, T 2 ]; H 1 (R d )) ∩ L q d ([0, T 2 ]; W 1,r d (R d )) almost surely. In particular, there exists an almost surely finite number N 0 (ω) ∈ N such that T * N ≥ T 2 for any N ≥ N 0 (ω) and, as a result, (A.13) holds for any 0 < t < T 2 and N ≥ N 0 (ω). Moreover, from the definition of T 2 = T 2 (R), we may assume u N L q d ([0,T 2 ];W 1,r d ) ≤ R + 1 (A. 16) for any N ≥ N 0 (ω). This allows us to conclude that the third term on the right-hand side of (A.13) tends to 0 almost surely as N → ∞. Indeed, by (4.2), (4.4), (A.14), (A. 16), and the almost sure convergence of u N to u in L q d ([0, T 2 ]; W 1,r d (R d )), we have, for any 0 ≤ t ≤ T 2 ,   Let us now consider the second and fourth terms on the right-hand side of (A.13). As for the second term, we first consider the contribution from N 1 (u N )φ N e n . By Hölder's inequality with (2.1) and Sobolev's embedding: Then, by Ito's isometry along with the independence of {β n } n∈N , we obtain E n∈Nˆt 0ˆR d N 1 (u N )(t ′ )φ N e n dxdβ n (t ′ ) As for the fourth term on the right-hand side of (A.13), it follows from Hölder's and Sobolev's inequalities that