Stochastic nonlinear Schrödinger equations on tori

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1 Stochastic nonlinear Schrödinger equations on tori Kelvin Cheung 1 Razvan Mosincat Received: 8 March 18 The Author(s) 18 Abstract We consider the stochastic nonlinear Schrödinger 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 (T). As for other power-type nonlinearities, namely (i) (super)quintic when d = 1 and (ii) (super)cubic when d, 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. Keywords Stochastic nonlinear Schrödinger equations Well-posedness Stochastic NLS Mathematics Subject Classification 6H15 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: { i t u u ± u k u = F(u,φξ) u t= = u H s (T d (t, x) (, ) T d, (1.1) ) B Razvan Mosincat r.o.mosincat@sms.ed.ac.uk Kelvin Cheung K.K.Cheung-3@sms.ed.ac.uk 1 School of Mathematics and Computer Sciences, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh EH14 4AS, UK School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK

2 where k, d 1 are integers, T d := R d /Z d, and u :, ) 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. and the (linear) multiplicative noise, i.e. F(u,φξ)= φξ (1.) F(u,φξ)= u φξ, (1.3) 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 Eξ(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. 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 (T d ) into H s (T d ), on which we make certain assumptions, depending on whether we are working with (1.) 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 u(t) = S(t)u ± i S(t t )( u k u)(t ) dt i (t), t, (1.4) where S(t) := e it is the linear Schrödinger propagator. The term (t) is a stochastic convolution corresponding to the stochastic forcing F(u,φξ), see(1.11) and (1.1) below. Our local-in-time argument uses the Fourier restriction norm method introduced by Bourgain 6 and Klainerman and Machedon 6, as well as 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 Moreover, by proving probabilistic a priori bounds on the mass and energy of solutions, we establish global well-posedness in (i) L (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 15,16 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 (R d ) to H s (R d ). They proved global existence and uniqueness of mild solutions in (i) L (R) for the one-dimensional cubic SNLS and (ii) H 1 (R d ) for 1 The multiplicative noise given by the Stratonovich product u φξ with real-valued ξ is relevant in physical applications, as it conserves the mass of u (i.e. t u(t) L x (T d is constant) almost surely. Our analysis ) can handle either the Itô or the Stratonovich product, and we choose to work with the former for the sake of simpler exposition.

3 defocusing energy-subcritical SNLS. Other works related to SNLS on R d include the works by Barbu, Röckner, and Zhang 1, and by Hornung 3. More recently, by using the dispersive estimate for the linear Schrödinger operator, Oh, Pocovnicu and Wang 3 treat additive SNLS equation with rougher noise than that considered in 16(seealsoRemark1.9). On the R d setting, the arguments given in 15,16 use fixed point arguments in the space C t Hx 1 L t p Wx 1,q (, T R d ),forsomet > and some suitable p, q 1. In particular, they use the (deterministic) Strichartz estimates: S(t) f L p t L q x (R R d ) C p,q f L x (R d ), (1.5) where the pair (p, q) is admissible, i.e. p + q d = d, p, q,, and (p, q, d) = (,, ). OnT d, Bourgain and Demeter 5 proved the l -decoupling conjecture, and as a corollary, the following periodic Strichartz estimates: S(t)P N f L p t,x (,T T d ) C p,t,ε N d d+ p +ε f L x (T d ). (1.6) Here, P N is the Littlewood-Paley projection onto frequencies {n Z d : n N}, p (d+) d, and ε>isanarbitrarily 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 u X s,b := n s τ n b F t,x (u)(τ, n) L τ l n (R Zd ). (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 method was introduced by Bourgain 6 for the deterministic nonlinear Schrödinger equation (NLS): i t u u ± u k u =. (1.8) Independently, Klainerman and Machedon 6 used the same method in the context of nonlinear wave equations. 1. Main results We now state more precisely the problems considered here. Let (, A, {A t } t, P) be a filtrated probability space. Let W be the L (T d )-cylindrical Wiener process given by Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm: u W s,r (T d ) := s u L r (T d ) = F 1 ( n s û(n)) l r n (Z d ), where n := 1 + n.whenr =, we have H s (T d ) = W s, (T d ). 3 More recently, Killip and Vişan 4 removed the arbitrarily small loss of ε derivatives in (1.6) when p > (d+) d. However, we do not need this scale-invariant improvement in our results.

4 W (t, x,ω):= n Z d 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 and e n (x) := exp(πin 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, W (t) H d ε (T d ) almost surely for any ε>), 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 K is Hilbert-Schmidt if φ L (H;K ) := n Z d φh n K <, (1.1) where {h n } n Z d is an orthonormal basis of H (recall that L (H;K ) does not depend on the choice of {h n } n Z d ). Throughout this work, we assume φ L (L (T d ); H s (T d )) for appropriate s. 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 (t) from (1.4) for (i) the additive noise (1.): (t) := and (ii) the multiplicative noise (1.3): (t) := u(t) := S(t t )φ dw(t ) (1.11) S(t t )u(t )φ dw(t ). (1.1) For the deterministic nonlinear Schrödinger equation (NLS), i.e. F in(1.1), there is the so-called scaling-critical regularity and is given by s crit := d 1 k. (1.13) See 35, Section 3.1. In this paper, we consider the Cauchy problem (1.1) (with either (1.)or(1.3)) posed in H s (T d ) with s > s crit, i.e. we consider the scaling-subcritical Cauchy problem. For the one-dimensional cubic problems (i.e. when (d, k) = (1, 1)), we also require s. We are now ready to state our first result. Theorem 1.1 (Local well-posedness for additive SNLS) Given s > s crit non-negative, let φ L (L (T d ); H s (T d )) and F(u,φ) = φξ. Then for any u H s (T d ), there exist a stopping time T that is almost surely positive, and a unique solution u C(, T ; H s (T d )) X s, 1 ε (, T ) to SNLS with additive noise, for some ε>. Here, X s,b (, T ) is a time restricted version of the X s,b -space, see (.5) below. The proof of this result relies on a fixed point argument for (1.4) in a closed subset of

5 X s,b (, T ). We are required to use b = 1 ε in order to capture the temporal regularity of. Since X s,b (, T ) does not embed into C(, T ; H s ) when b < 1, 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. 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.BelowL (T), the deterministic cubic NLS on T was shown to be ill-posed. Indeed, Christ, Colliander and Tao 11 and Molinet 9 showed that the solution map u H s (T) u(t) H s (T) is discontinuous whenever s <. More recently, Guo and Oh 19 showed an even stronger ill-posedness result, in the sense that for any u H s (T), s ( 1 8, ), 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 = s crit, Kishimoto 5 showed a norm inflation phenomenon at any u H s (T): for any ε>and u H s (T), there exists a solution u ε to NLS such that u ε () u H s (T) <εand u ε (t) H s (T) >ε 1 for some t (,ε). See also 31,33. Regarding the one-dimensional cubic SNLS on T, we point out that recently Forlano, Oh and Wang 18 studied a renormalized (Wick ordered, see also 1) additive SNLS with a weaker assumption than that of Theorem 1.1 above. While we assume that φ L (L (T); L (T)), the work of 18 assumesthatφ is γ -radonifying from L (T) into the Fourier-Lebesgue space F L s,p (T) with s > and 1 < p <. In particular, this allows them to handle almost space-time white noise, namely φ = x α with α>arbitrarily small. 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 T d α = d j=1 R/α j Z, where α = (α 1,...,α d ), ) d. 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 Qd such that γ α = ). Prior to 5, Strichartz estimates were harder to establish on irrational tori see 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, ), (, 1), (3, 1) which are still open see 7,1,,36. In these papers, the authors employ the critical spaces X s, Y s based on the spaces U, V of Koch and Tataru 7. We point out that Brownian motions belong almost surely to V p,forp >, but not V (hence neither to U ). 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: M(u(t)) := 1 u(t, x) dx (1.14) T d E(u(t)) := 1 x u(t, x) ± 1 u(t, x) k+ dx, (1.15) T d k + T d where the sign ± in (1.15) 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

6 energy-subcritical if s crit < 1 (i.e. for any k 1 when d = 1, 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 15,16. Therefore, we obtain the following: Theorem 1.5 (Global well-posedness for additive SNLS) Let s. Given φ L (L (T d ); H s (T d )), letf(u,φ) = φξ and u H s (T d ). 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 ; (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 (T d ) to a Banach space B, we say that φ L (L (T d ); B) if 4 φ L (L (T d );B) := n Z d φe n B <. For s R and r 1, we also define the Fourier-Lebesgue space F L s,r (T d ) via the norm f F L s,r (T d ) := n s f (n) l r n (Z d ). Clearly, when r = wehavef L s,r (T d ) = H s (T d ) and for s 1 s and r 1 r we have F L s,r 1 (T d ) F L s 1,r (T d ). We now state our local well-posedness result for the multiplicative SNLS. Theorem 1.6 (Local well-posedness for multiplicative SNLS) Given s > s crit nonnegative, let φ L (L (T d ); H s (T d )). Ifs d, we further impose that φ L (L (T d ); F L s,r (T d )) (1.16) for some r ) d 1, d s when s > and r = 1 when s =. Let F(u,φ)= u φξ. Then for any u H s (T d ), there exist a stopping time T that is almost surely positive, and a unique solution u C(, T ; H s (T d )) X s, 1 ε (, T ) to SNLS with multiplicative noise, for some ε>. Remark 1.7 If φξ is a spatially homogeneous noise, i.e. φ is translation invariant, then the extra assumption (1.16) is superfluous. Indeed, if φe n (m) =, for all m, n Z d, m = n and φ L (L (T d ); H s (T d )), then φ L (L (T d ); F L s,r (T d )) for any r 1. 4 In fact, such operators are known as nuclear operators of order and their introduction goes back to the work of A. Grothendieck on nuclear locally convex spaces.

7 We point out that an extra condition in the multiplicative case was also used by de Bouard and Debussche 16 in their study of SNLS in H 1 (R d ), namely they required that φ is a γ -radonifying operator from L (R d ) into W 1,α (R d ) for some appropriate α, as compared to the requirement that φ is Hilbert-Schmidt from L (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 L ( ; C(, T ; H s (T d )) X s, 1 ε (, T ) ). 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 15,16. Moreover, we combine this method with the use of X s,b -spaces in a similar manner as in 17, 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 15,16. 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. Given φ with the same assumptions as in Theorem 1.6,letF(u,φ)= u φξ and u 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 ; (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 Theorems 1.1 and 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.). 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) asu = v + and considering the equation for the residual part v. In general, this procedure allows one to treat rougher noise, see for example 3,4,1 where they treat NLS with rough random initial data and more recently 3 where they handled supercritical noise for the additive SNLS on R d.in the periodic setting however, the argument gets more complicated (see for example 3,4 onr d vs. 1,3 ont d ). The actual implementation of the aforementioned trick requires cumbersome case-by-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 1.1 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 1 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.

8 In 8, Brzeźniak, Hornung, and Weiss construct martingale solutions in H 1 (M) for the multiplicative SNLS with energy-subcritical defocusing nonlinearities and masssubcritical focusing nonlinearities. Organization of the paper In Sect., we provide some preliminaries for the Fourier restriction norm method and prove the multilinear estimates necessary for the local well-posedness results. In Sect. 3, we prove some properties of the stochastic convolutions and u given respectively by (1.11) and (1.1). We prove Theorems 1.1 and 1.6 in Sect. 4. Finally, in Sect. 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 CBfor some constant C (, ) 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) (, ) 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 ĝ(ξ) = U e πiξ x g(x) dx, where ξ is either an element of Z d (if U = T d ) or an element of R (if U = R). For the sake of convenience, we shall omit the π 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>, and write c for the analogous meaning. For example, the statement u X s, 1 should be read as u X s, 1 ε for sufficiently small ε>. For the sake of readability, in the proofs we sometimes omit the underlying domain T d from various norms, e.g. we write f H s instead of f H s (T d ) and φ L (L ;H s ) instead of φ L (L (T d );H s (T d )). 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 u X s,b := n s τ n b F t,x (u)(τ, n) l n L τ (Zd R) is finite. Equivalently, the X s,b -norm can be written in its interaction representation form: u X s,b = n s τ b F t,x (S( t)u(t)) (n,τ), (.1) l n L τ (Zd R)

9 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 35. Firstly, we have the following continuous embeddings We have the duality relation u X s,b = X s,b C(R; H s x (Td )),forb > 1, (.) X s,b X s,b,fors s and b b. (.3) sup v X s, b 1 u(t, x)v(t, x) dt dx. (.4) R T d Lemma.1 (Transference principle, 35, Lemma.9) Let Y be a Banach space of functions on R T d such that e itλ e ±it f Y f H s (T d ) for all λ R and all f H s (T d ). Then, for any b > 1, for all u X s,b. u Y u X s,b Given a time interval I R, one defines the time restricted space X s,b (I ) via the norm u X s,b (I ) := inf { ũ X s,b :ũ I = u }. (.5) We note that for s and b < 1,wehave u X s,b (I ) 1 I (t)u(t) X s,b, (.6) see for example 17, Lemma.1 for a proof (for X s,b spaces adapted to the KdV equation). Lemma. (Linear estimates, 35, Proposition.1) Let s R and suppose η is smooth and compactly supported. Then, we have η(t)s(t) f X s,b f H s (T d ), for b R; (.7) X η(t) S(t t )F(t )dt F X s,b 1, for b > 1 s,b. (.8) By localizing in time, we can gain a smallness factor, as per lemma below.

10 Lemma.3 (Time localization property, 35, Lemma.11) Let s R and 1 < b < b < 1. For any T (, 1), we have f X s,b (,T ) b,b T b b f X s,b (,T ). We now give the proofs of the multilinear estimates necessary to control the nonlinearity u k u. Recall the L 4 -Strichartz estimate due to Bourgain 6 (seealso35, Proposition.13): u L 4 t,x (R T) u X, 3 8. (.9) Lemma.4 Let d = 1, s, b 3 8, and b 5 8. Then, for any time interval I R, we have 3 u 1 u u 3 X s,b 1 (I ) u j X s,b (I ). (.1) Proof By the triangle inequality it suffices to prove (.1) fors =. We claim that j=1 u 1 u u 3 v dxdt R T d 3 u j X,b v X,1 b j=1 for any factors u 1, u, u 3,v. Indeed, this follows immediately from Hölder inequality and (.9) for each of the four factors (hence the restrictions b, 1 b 3 8 ). Thus, the global-in-time version of (.1), i.e. I = R, follows by the duality relation (.4). For an arbitrary time interval I,ifũ j is an extension of u j, j = 1,, 3, then ũ 1 ũ ũ 3 is an extension of u 1 u u 3. We use the previous step to get u 1 u u 3 X s,b 1 (I ) ũ1 ũ ũ 3 X s,b 1 3 ũ j X,b j=1 and then we take infimum over all extensions ũ j s and (.1) 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 S(t)P Q f L p t,x (I T d ) I Q 1 d+ pd + f L x (T d ), (.11) for any d 1, p (d+) d, I R finite time interval, and Q R d dyadic cube. Here, P Q denotes the frequency projection onto Q, i.e. P Q f (n) = 1 Q (n) f (n). By the transference principle (Lemma.1), we get P Q u L p t,x (I T d ) I Q 1 d+ pd + u X,b (I ), (.1)

11 for any b > 1. By interpolating (.1) with P Q u L p t,x (I T d ) Q 1 1 p u X, 1 1p (I ), (.13) (which follows immediately from Sobolev inequalities, (.1), and the H s (T d )- isometry of S( t)), we can lower the time regularity from b = 1 + δ to b = 1 δ, for sufficiently small δ>.thus,wealsohave P Q u L p t,x (I T d ) I,δ Q 1 d+ pd +o(δ) u X, 1 δ (I ) (.14) Lemma.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. Lemma.5 Let d, k 1 such that dk and let I R be a finite time interval. Then for any s > s c, there exist b = 1 and b = 1 + such that u 1 u u k u k+1 X s,b 1 (I ) I k+1 j=1 u j X s,b (I ). (.15) Proof In view of (.6), we can assume that u j (t) = 1 I (t)u j (t) and thus by the duality relation (.4), it suffices to show ( s (u 1 u u k+1 ) ) v dxdt v R T d X,1 b k+1 j=1 u j X s,b. (.16) We use Littlewood-Paley decomposition: we estimate the left-hand side of (.16) when v = P N v, u j = P N j u j for some dyadic numbers N, N j Z,1 j k + 1. Then the claim follows by triangle inequality and performing the summation. (.17) N 1 N N N 1 N N N 1 N k+1 N k+1 N k Notice that without loss of generality, we may assume that N 1 N N k+1, in which case we also have N N 1, and that the factors v and u j are real-valued and non-negative. Let ε := s s c, and we distinguish two cases. Case 1: N 1 N. By Hölder inequality, N s u 1 u u k+1 v dxdt N s R T d 1 u 1 q Lt,x N s k+1 u q Lt,x u j p Lt,x v L r t,x, j=3 (.18)

12 with p, q, r chosen such that k 1 p + q + r 1 = 1. We take p, q such that d d+ p = s crit and d d+ q = 1 s crit, or equivalently p = k(d + ) and q = 4k(d+) dk+. These give the Hölder exponent r = (d+) d.by(.14) and (.1), we get N s j u j q Lt,x N ε + j u j X s,b, j = 1, (.19) u j p Lt,x N ε+ j u j X s,b, 3 j k + 1, (.) v L r t,x N + v X,1 b. (.1) By choosing δ, δ ε in b := 1 δ and in 1 b = 1 δ, respectively, we get k+1 RHS of (.18) N 4 ε v X,1 b j=1 N ε 4 j u j X s,b. (.) The factors N ε 4, N ε 4 j guarantee that we can perform (.17). Case : N 1 N. Then, we necessarily have N 1 N or else the left hand side of (.16) vanishes. By Hölder inequality, N s R T d u 1 u u k+1 v dxdt N s 1 u 1 L q t,x k+1 j= u j L p t,x v L r t,x, (.3) with k p + q 1 + r 1 = 1. As in Case 1, we would like to have p such that d d+ p = s crit, or equivalently p = k(d +). However, the best we can do with the Strichartz estimate for the remaining factors is to choose q = r = (d+) d, so that we have N1 s u 1 q Lt,x N1 + 1 X s,b, (.4) u j p Lt,x N ε+ j u j X s,b, j k + 1, (.5) v L r t,x N1 + X,1 b. (.6) Notice that we can overcome the loss of derivative N1 s 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, i.e. {ξ 1 : ξ 1 N 1 } l Q l, {ξ : ξ N} j R j. There are approximately ( N1 N ) d-many cubes needed, and so u 1 = l P Ql u 1 =: l u 1,l, v = j P R j v =: j v j

13 are decompositions into finitely many terms. Since ξ 1 ξ N for ξ 1 supp(û 1 ), ξ supp( v) on the convolution hyperplane, there exists a constant K such that if dist(q l, Q j )>KN, then the integral in (.16) vanishes. Hence the summation (.17) is replaced by. (.7) N 1 N N N 1 Also, in place of (.4) (.5), we now have N k+1 N k+1 N k l, j j l N1 s u 1,l q Lt,x N + 1,l X s,b, (.8) u i p Lt,x Ni ε+ u i X s,b, i k + 1, (.9) v j q Lt,x N + j X,1 b, (.3) Therefore, by Cauchy-Schwarz inequality and Plancherel identity, LHS of (.16) N N N k+1 i=1 N 1 l, j N 1 N l j N ε+ k+1 N ε+ u 1,l X s,b v j X,1 b u i X s,b N 1 N 1 N u 1,l X s,b l 1 i= N N N k+1 N ε+ u 1 X s,b v X,1 b u i X s,b u i X s,b v X,1 b i= v j X,1 b j 1 k+1 i= u i X s,b and the proof is complete. 3 The stochastic convolution In this section, we prove some X s,b -estimates on the stochastic convolution (t) given either by (1.11) or(1.1). We first record the following Burkholder-Davis-Gundy inequality, which is a consequence of 8, Theorem 1.1. Lemma 3.1 (Burkholder Davis Gundy inequality) Let H, K be separable Hilbert spaces, T >, and W is an H-valued Wiener process on, T. Suppose that {ψ(t)} t,t is an adapted process taking values in L (H; K ). Then for p 1, E sup ψ(t ) dw(t p ( T ) p E ψ(t ) ) p L (H;K ) dt. t,t K

14 In addition, we prove that (t) is pathwise continuous in both cases. To this end, we employ the factorization method of Da Prato 13, Lemma.7, i.e. we make use of the following lemma and (3.3) below. Lemma 3. Let H be a Hilbert space, T >, α (, 1), and σ> ( 1 α, ). Suppose that f L σ (, T ; H). Then the function F(t) := S(t t )(t t ) α 1 f (t ) dt, t, T (3.1) belongs to C(, T ; H). Moreover, sup F(t) H σ,t f L σ (,T ;H). (3.) t,t We make use of the above lemma in conjunction with the following fact: μ (t t ) α 1 (t μ) α dt = π sin(πα), (3.3) for all <α<1 and all μ<t. This can be seen via considerations with Euler-Beta functions, see 13. We now treat the additive and multiplicative cases separately below in Sects. 3.1 and 3. 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 (t) = e n (φe j )(n) e i(t t ) n dβ j (t ). (3.4) n Z d j Z d We first prove the following X s,b -estimate on : Lemma 3.3 Let s, b < 1,T >, and σ, ). Assume that φ L (L (T d ); H s (T d )). Then for given by (3.4) we have E σ X s,b T σ (,T ) (1 + T ) σ φ σ L (L (T d );H s (T d )). (3.5) Proof Since 1,T (t)1,t (t ) = 1,T (t) = 1 whenever t t T,wehave 1,T (t) (t)(x) = n Z d e n j Z d φe j (n)1,t (t)e it n 1,T (t )e it n dβ j (t )

15 By (.6), we have (t) X s,b (,T ) 1,T (t) (t) X s,b = n s τ b ( F t,x S( t)1,t (t) (t) ) (τ, n) L τ l n = n s τ b F t gn (t) (τ), (3.6) L τ l n where g n (t) := 1,T (t) 1,T (t )e it n φe j (n)dβ j (t ). j Z d By the stochastic Fubini theorem (see 14, Theorem 4.33), we have F t g n (t)(τ) = e itτ g n (t)dt R = 1,T (t )e it n φe j (n) 1,T (t)e itτ dt dβ j (t ). j Z d t Since 1,T (t)e itτ dt min{t, τ 1 }, (3.7) t by Burkholder-Davis-Gundy inequality (Lemma 3.1), we get E F t g n (t)(τ) σ T φe j (n) j Z d t 1,T (t)e itτ dt dt T φe j (n) min{t, τ } j Z d σ. σ (3.8) By (3.6), (3.8), and Minkowski inequality, we get L σ ( ;X s,b (,T )) n s τ b ( E Fg n (τ) σ ) σ dτ n Z d T 1 n s φe j (n) n, j Z d T 1 φ L (L ;H s ) (T τ <1 1 τ b min{t, τ } dτ dτ + τ b dτ)1. τ 1 1

16 This completes the proof of Lemma 3.3. We now prove that 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, T >, and σ <. Assume that φ L (L (T d ); H s (T d )). Then ( ) belongs to C(, T ; H s (T d )) almost surely and E sup t,t (t) σ H s (T d ) T φ σ L (L (T d );H s (T d )). (3.9) Proof We fix α (, 1 ) and we write the stochastic convolution as follows: (t) = sin(πα) π = sin(πα) π μ (t t ) α 1 (t μ) α dt S(t μ)φ dw(μ) S(t t )(t t ) α 1 S(t μ)(t μ) α φ dw(μ) dt, (3.1) where we used the stochastic Fubini theorem 14, Theorem 4.33 and the group property of S( ). By Lemma 3. and (3.1) it suffices to show that the process f (t ) := S(t μ)(t μ) α φ dw(μ) satisfies T E f (t ) σ Hx s dt C ( ) T,σ, φ L (L ;H s ) <, (3.11) for some σ> 1 α. By Burkholder-Davis-Gundy inequality (Lemma 3.1), for any σ and any t, T, we get E f (t ) ( ) σ t σ Hx s S(t μ)(t μ) α φ L (L ;H s ) dμ = φ σ L (L ;H s ) (t μ) α S(t μ)φe j H s dμ j Z d ( T 1 α 1 α ) σ, σ

17 where in the last step we used α (, 1) and the H s (T d )-isometry property of S(t μ). Hence LHS of (3.11) = T E f (t ) σ Hx s dt φ σ L (L ;H s ) T σ (1 α)+1 <. The estimate (3.9) follows from (3.). 3. The multiplicative stochastic convolution The multiplicative stochastic convolution = u from (1.1) can be written as u(t) = e n e i(t t ) n (u(t )φe j )(n)dβ j (t ). (3.1) n Z d j Z d Recall that if s > d, then we have access to the algebra property of H s (T d ): fg H s (T d ) f H s (T d ) g H s (T d ) (3.13) which is an easy consequence of the Cauchy-Schwarz 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, but we use the following inequalities. Lemma 3.5 Let < s d and 1 r < Also, for s =, we have d d s. Then fu H s (T d ) f F L s,r (T d ) u H s (T d ). (3.14) fu L (T d ) f F L,1 (T d ) u L (T d ). (3.15) Proof Assume that < s d and let n 1 and n denote the spatial frequencies of f and u respectively. By separating the regions { n 1 n } and { n 1 n }, and then applying Young s inequality, we have fu H s (T d ) ( s f û ) (n) + ( f s u ) (n) l n f F L s,r û l p + f l 1 u H s, where p is chosen such that 1 r + 1 p = 3. By Hölder inequality, for r and q such that 1 r + 1 r = 1 and 1 q + 1 = 1 p, f l 1 n s l r f F L s,r, û l p n s l q u H s. l n

18 Since sr > d and sq > d provided that r < d s, the conclusion (3.14) follows. If s =, (3.15) follows easily from Young s inequality: d fu L (T d ) = f û l f l 1 û l = f F L,1 u L. Given φ as in Theorem 1.6, let us denote C(φ) := φ L (L (T d );F L s,r (T d )) <, (3.16) for r = when s > d,forsomer ) d 1, d s when < s d, and for r = 1 when s =. Recall that if φ is translation invariant, then it is sufficient to assume that C(φ) < with r =, for all s. We now proceed to prove the following X s,b -estimate of u. Lemma 3.6 Let s, b < 1,T >, and σ<. Suppose that φ satisfies the assumptions of Theorem 1.6. Then, for u given by (1.1) we have the estimate E u σ X s,b (T + 1) σ (,T ) C(φ) σ E u σ L (,T ;H s (T d. (3.17) )) Proof We first prove (3.17). Let g(t) := 1,T (t)s( t) (t). By the stochastic Fubini theorem 14, Theorem 4.33, F t,x (g)(τ, n) = R e itτ 1,T (t) j Z d e it n ( u(t )φe j )(n) dβ j (t ) dt = j Z d T t 1,T (t)e itτ e it n ( u(t )φe j )(n) dt dβ j (t ). Then by (.6) and the assumption b < 1, the Burkholder-Davis-Gundy inequality (Lemma 3.1), and (3.7), we have LHS of (3.17) n E s τ b σ Fg(n,τ) L τ l n E T n s τ b 1,T (t)e itτ dt ( j,n Z d R t u(t )φe j )(n) dt dτ σ (T + 1) σ T E n s ( u(t )φe j )(n) dt. j,n Z d If s > d, we apply the algebra property of H s (T d ) to get u(t )φe j l j H s φ L (L ;H s ) u(t ) H s. σ

19 If s d,wehave u(t )φe j l j H s C(φ) u(t ) H s. (3.18) and thus (3.17) follows. Next, we prove the continuity of u(t) in the same way as in Lemma 3.4, i.e. by using Lemma 3.. Lemma 3.7 (Continuity of the multiplicative noise) Let T >,s, b < 1, and σ<. Suppose that u L σ ( ; X s,b (, T ) ) and that φ satisfies the assumptions of Theorem 1.6. Then u( ) given by (3.1) belongs to C(, T ; H s (T d )) almost surely. Moreover, E sup u(t) σ H t,t s (T d ) C(φ) σ E u σ X s,b. (3.19) (,T ) Proof Applying the same factorisation procedure as in the proof of Lemma 3.4 reduces the problem to proving that the process f (t ) := (t μ) α S(t μ) u(μ)φ dw(μ) satisfies T E f (t ) σ H s x dt C (T,σ,C(φ)) < (3.) for some <α<1satisfying α> σ 1. By the Burkholder-Davis-Gundy inequality (Lemma 3.1) and Lemma 3.5, wehave E f (t ) ( σ Hx s E = E ) σ (t μ) α S(t μ)u(μ)φ L (L ;H s ) dμ (t μ) α S(t μ)u(μ)φe j H s dμ j Z d E T φe j F L s,r (t μ) α u(μ) H s dμ j Z d σ σ.

20 Then, by Fubini theorem and Minkowski inequality, we obtain T E f (t ) σ Hx s dt = f H s σ x L σ ( ;L σ t,t ) C(φ) σ (t μ) α L u(μ) H s x μ (,T ) C(φ) E σ (t μ) α u(μ) H s σ L σ ( ;L σ t,t ) σ x L σ t (,T ) L μ (,T ) ( T ) σ C(φ) σ E (T μ) ( σ 1 α) u(μ) Hx s dμ By Hölder and Sobolev inequalities and (.6), we have ( T ) 1 (T μ) ( σ 1 α) u(μ) Hx s dμ (T 1 μ) σ α L There exists α = α(σ) := σ for which we have 4 Local well-posedness 4 1+b μ (,T ) u(μ) H s x L T 1+ 1+b 4 ( σ 1 α) 1,T (μ) S( μ)u(μ) H s x L 4 1 b μ (,T ) 4 1 b μ T E f (t ) σ Hx s dt E T 1+b bσ u σ X s,b <. (,T ) 4.1 SNLS with additive noise In this subsection, we prove Theorem 1.1. Letb = b(k) = 1 be given by Lemma.4 (in the case d = k = 1) or by Lemma.5 (in the case dk ). By Lemma 3.3,for any T >, there is an event of full probability such that the stochastic convolution has finite X s,b (, T )-norm on. Now fix ω and u H s (T d ). Consider the ball. B R := { u X s,b (, T ) : u X s,b (,T ) R} where < T < 1 and R > are to be determined later. We aim to show that the operator given by u(t) = S(t)u ± i S(t t ) ( u k u ) (t )dt i (t), t,

21 where 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 (, T )-norm of D(u) := S(t t ) ( u k u ) (t ) dt. For any δ>sufficiently small (such that b + δ< 1 ), by Lemma.3 and (.6): D(u) X s,b (,T ) T δ D(u) X s,b+δ (,T ) T δ 1,T (t)d(u)(t) X s, 1 +δ. Let η be a smooth cut-off function, supported on 1, T + 1, with η(t) = 1 for all t, T. For any w X s, 1 +δ that agrees with u k u on, T, by Lemma., we obtain 1,T (t)d(u)(t) X s, 1 +δ X η(t) S(t t )w(t )dt w s, 1 +δ X s, 1 +δ Then after taking the infimum over all such w, we use Lemma.4 or.5 and we get (4.1) D(u) X s,b (,T ) T δ (uu) k u X s, 1 +δ (,T ) T δ u k+1 X s,b (,T ). (4.) It follows that u X s,b (,T ) c u H s x + ct δ u k+1 X s,b (,T ) + (t) X s,b (,T ), (4.3) for some c >. Similarly, we obtain ( ) u v X s,b (,T ) ct δ u k X s,b (,T ) + v k X s,b u v (,T ) X s,b (,T ). (4.4) Let R := c u H s x + (t) X s,b (,T ).From(4.3) and (4.4), we see that is a contraction from B R to B R provided ct δ R k+1 1 R and ct δ ( R k) 1. (4.5) This is always possible if we choose T 1 sufficiently small. This shows the existence of a unique solution u X s,b (, T ) to (1.4) on. Finally, we check that u C(, T ; H s ) on the set of full probability, where is given by Lemma 3.4, that is C(, T ; H s ) on.by(.6), (4.1) and Lemma.4 or.5, wealsoget D(u) X s, 1 1 +δ,t (t)d(u)(t) (,T ) X s, 1 +δ u k+1 X s,b (,T ). (4.6)

22 By the embedding X s, 1 +δ (, T ) C(, T ; H s (T d )), we have D(u) C(, T ; H s (T d )). Since the linear term S(t)u also belongs to C(, T ; H s (T d )), we conclude that u = u C (, T ; H s (T d ) ) on. Remark 4.1 From (4.5), we obtain the time of existence { ( ) } θ T max := max T > : T c u H s + X s,b (, T ), (4.7) where θ = k δ. Note that (4.7) will be useful in our global argument. 4. SNLS with multiplicative noise In this subsection, we prove Theorem 1.6. Following 17, 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, 1 be a smooth cut-off function such that η 1on, 1 and η outside 1,. Set η R := η ( R ) and consider the equation ( ) k+1 u i t u R u R ± η R u R X s,b (,t) R k u R = u R φξ, (4.8) with initial data u R t= = u. Its mild formulation is u R = R u R, where R is given by R u R := S(t)u ± i S(t t )η R ( u R X s,b (,t )) k+1 u R k u R (t ) dt i u R (t). The key ingredient for Theorem 1.6 is the following proposition. (4.9) Proposition 4. (Global well-posedness for (4.8)) Let s > s crit,s, and T, R >. Suppose that φ is as in Theorem 1.6. Given u H s (T d ), there exists b = 1 and a unique adapted process u R L ( ; C (, T ; H s (T d ) ) ) X s,b (, T ) solving (4.8) on, T. Before proving this result, we state and prove the following lemma.

23 Lemma 4.3 (Boundedness of cut-off) Let s,b, 1 ),R> and T >. There exist constants C 1, C (R) > such that ) } η R ( u X s,b (,t) u(t) {C min 1 u X s,b (,T ) X s,b (,T ), C (R) ; (4.1) ) ( ) η R ( u X s,b (,t) u(t) η R v X s,b (,t) v(t) C (R) u v X s,b (,T ) X s,b (,T ). (4.11) ( u X Proof We first ) prove (4.1). Let w(t, n) = F x S( t)u(t)(n), κ R (t) = η R s,b (,t) and τ R := inf { t : u X s,b (,t) R}. (4.1) Then κ R (t) = when t >τ R.By(.6) and (.1), κ R (t)u(t) X s,b (,T ) 1,T τr κ R (t)u(t) X s,b κ R (t)u(t) X s,b (,T τ R ) n Z d n s κ R (t)w(t, n) H b (,T τ R ). (4.13) We now estimate the H b (, T τ R )-norm, for which we use the following characterization (see for example 34): f H b (a 1,a ) f L (a 1,a ) + a a 1 a a 1 f (x) f (y) x y 1+b dx dy, < b < 1. (4.14) For the inhomogeneous contribution (i.e. coming from the L -norm above), we have { } n s κ R (t)w(t, n) L n Z d t (,T τ R) min u X s,b (,τ R ), u X s,b (,T ) { } min (R), u X s,b. (,T ) The remaining part of (4.13) needs a bit more work. Fix n Z d, then T τr T τr T τr + T τr κ R (t)w(t, n) κ R (t )w(t, n) t t 1+b dt dt κ R (t)(w(t, n) w(t, n)) t t 1+b dt dt (κ R (t) κ R (t ))w(t, n) t t 1+b dt dt =: I(n) + II(n).

24 It is clear that { } I(n) min w(n) H b ((,τ R )), w(n) H b, ((,T )) and hence n Z d I(n) min { } (R), u X s,b. (,T ) For II(n), the mean value theorem infers that κr (t) κ R (t ) ( u X s,b (,t) u X s,b (,t )) 1t,tu X s,b R R 1 R n s w(, n ) H b (t,t). n Z d ( sup η (r) r R Again, we split w(, n ) using (4.14) into the inhomogeneous contribution H b (t,t) (the L -norm 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 ) 1 R T τr λ n s n Z d t = 1 T τr λ ( λ T τr R n s n Z d λ t w(t, n) t t 1+b w(λ, n ) w(λ, n ) λ λ 1+b dλ dλ dt dt ) 1 t t dt w(t, n) 1+b (4.15) w(λ, n ) w(λ, n ) λ λ 1+b dt dλ dλ, (4.16) whereweused t λ λ t T τ R to switch the integrals. Now, the integral with respect to t is equal to T τ R t b λ t b, which is bounded by T τ R t b λ t b. Thus (4.16) is controlled by

25 1 R n Z d n s T τr λ ( ) λ λ t b w(t, n) dt w(λ, n ) w(λ, n ) λ λ 1+b dλ dλ. (4.17) Since b, ) 1, by Hardy s inequality (see for example 35, Lemma A.) the t - integral is w(, n) H b (,λ ) w(, n) H b (,T τ R ). After multiplying by n s and summing over n Z d, we see that (4.17) is controlled by 1 R n,n Z d n s n s w(, n) H b (,T τ R ) w(, n) H b λ (,T τ R) 1 R u X s,b (,T τ R ) u X s,b (,T τ R ) { min 4 u X s,b (,T ), 16R}. We now prove (4.11). Let τr u and τ R v be defined as in (4.1). Assume without loss of generality that τr u τ R v. We decompose LHS of (4.11) ( ( ( )) η R u X (,t)) s,b ηr v X s,b (,t) v(t) X s,b (,T ) + ( ) ηr u X s,b (,t) (u(t) v(t)) X s,b (,T ) =: A + B. By the mean value theorem, A = ( η R ( u X s,b (,t)) ηr ( v X s,b (,t))) v(t) X s,b (,T τ v R ) 1 R v X s,b (,T τ v R ) u v X s,b (,T ) u v X s,b (,T ). For B, one runs through the same argument as for (4.1) but with w(t, n) replaced by F x S( t) ( u(t) v(t) ) (n), which yields B C(R) u v X s,b (,T ). We now conclude the proof of Proposition 4.. Proof of Proposition 4. Let T, R > and let E T := L ( ad ; X s,b (, T ) ) be the space of adapted processes in L ( ; X s,b (, T ) ). We solve the fixed point problem (4.9)inE T. Arguing as in the additive case, and using Lemmata 4.3 and 3.6,wehave

26 R u ET C 1 u H s + C (R)T δ + C 3 T b u ET ; R u R v ET C 4 (R)T δ u v ET + C 5 T b u v ET. Therefore, 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 H s, hence we may iterate this argument to extend u R (t) to all t, ). Finally, to see that u R F T := L ( ; C(, T ; H s (T d )) ), we first note that since u R E T, Lemma 3.7 infers that u R F T. Then, by similar argument as in the end of Sect. 4.1, we have that D(u R ) L ( ; X s, 1 +(, T ) ), where D(u R )(t) := S(t t ) ( u R k u R ) dt. Since L ( ; X s, 1 + (, T ) ) F T,wehaveD(u R ) F T. Also, it is clear that S(t)u F T. Hence u R F T. Proof of Theorem 1.6 Let τ R := inf { t > : u R X s,b (,t) R}. (4.18) Then, η R ( u R X s,b (,t) ) = 1 if and only if t τ R. Hence u R is a solution of (1.4) on,τ R. For any δ>, we have u R (t) = u R+δ (t) whenever t,τ R. Consequently, τ R is increasing in R. Indeed, if( τ R >τ R+δ for some ) R > and some δ>, then for t τ R+δ,τ R,wehaveη R+δ u R+δ X s,b (,t) < 1 which implies that u R (t) = u R+δ (t), a contradiction. Therefore, τ := lim R τ R (4.19) is a well-defined stopping time that is either positive or infinite almost surely. By defining u(t) := u R (t) for each t,τ R, we see that u is a solution of (1.4) on,τ ) almost surely. 5 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 SNLS (1.1) are given respectively by 1 M(u(t)) = T d u(t, x) dx, (5.1) 1 E(u(t)) = T d u(t, 1 x) + (k + 1) u(t, x) (k+1) dx. (5.) It is well-known that these are conserved quantities for (smooth enough) solutions of the deterministic NLS equation.

27 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.) and use the stochastic equation for u. We work with the following frequency truncated version of (1.1): { i t u N u N ± P N u N k u N = F(u N,φ N dw N ), u N t= = P N u =: u N (5.3) where P N is the Littlewood-Paley projection onto the frequency set {n Z d : n N}, φ N := P N φ and W N (t) := β n (t)e n. n N By repeating the arguments in Sect. 4, one obtains local well-posedness for (5.3) with initial data P N u 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.) of a solution u N of the truncated equation. Proposition 5.1 Let m N, T >, φ L (L (T d ); L (T d )), and F(u,φξ) = φξ. Suppose that u N (t) is a solution to (5.3) for t, T, for some stopping time T, T. Then there exists a constant C 1 = C 1 (m, M(u ), T, φ L (L ;L ))> such that E sup M(u N (t)) m C 1. (5.4) t T Furthermore, if (5.3) is defocusing, there exists C = C (m, E(u ), T, φ L (L ;H 1 )) > such that E sup E(u N (t)) m C. (5.5) t T The constants C 1 and C are independent of N. Proof By applying Itô s Lemma, we have M(u N (t)) m = M(u N )m + m Im j N M(u N (t )) T m 1 u N (t )φ N e j dx dβ j (t ) d (5.6)

28 + m(m 1) + m φ N j N L (L ;L ) Stoch PDE: Anal Comp M(u N (t )) T m u N (t )φ N e j dx dt. d (5.7) M(u N (t )) m 1 dt, (5.8) 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 1 E sup (5.6) t,t m E j N T M(u N (t )) (m 1) u N (t ) L φ N e j L dt { T } 1 φ N L (L ;L ) E M(u N (t)) m 1 dt φ L (L ;L )T 1 E φ L (L ;L )T 1 { sup t,t M(u N (t)) m 1 } 1 { Hence by Young s inequality, we infer that E sup (5.6) C m φ L t,t (L ;L ) T E + 1 E sup t,t } 1 M(u N (t)) m { } 1 } 1 E sup M(u N (t)) {E m 1 sup M(u N (t)) m t,t t,t sup M(u N (t)) m. t,t In a straightforward way, we also have E sup (5.7) m(m 1) φ L (L ;L ) T E t,t E sup (5.8) m φ L t,t (L ;L ) T E sup M(u N (t)) m 1 t,t sup M(u N (t)) m 1, t,t sup M(u N (t)) m 1. t,t Therefore, there is some C m > such that E sup M(u N (t)) m M(u ) m + C m T E t,t + 1 E sup M(u N (t)) m. t,t sup M(u N (t)) m 1 t,t (5.9)

29 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,t M(u N (t)) are finite. To justify this, we note that (5.9) holds with T replaced by T R, where { } T R := sup t, T :M(u N (t)) R, R >. 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(, T ; H s (T d )) almost surely. Hence by letting R and invoking the monotone convergence theorem, one finds E sup M(u N (t)) m M(u ) m + C m T E sup M(u N (t)) m 1. (5.1) t,t t,t Hence, by induction on m, we obtain E sup M(u N (t)) m 1, (5.11) t,t 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 )m + m Im j N m Im j N + (k + 1)m + m φ N + m(m 1) j N (5.1) E(u N (t )) T m 1 u N k u N φ N e j dx dβ j (t ) (5.13) d E(u N (t )) T m 1 u N φ N e j dx dβ j (t ) (5.14) d E(u N (t )) m 1 T d u N k φ N e j dx dt (5.15) L (L ;L ) j N E(u N (t )) m 1 dt (5.16) ) E(u N (t )) ( u m N + u N k u N φe j dx dt. T d (5.17) 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

30 1 T E sup (5.13) C m E E(u N (t )) (m 1) u t,t j N T N k u N φ N e j dx dt. d Then, by duality and the (dual of the) Sobolev embedding H 1 (T d ) L k+ (T d ), we have u N k u N φ N e j dx u N k u N H φ N e j T d 1 (T d H ) 1 (T d ) u N k u N L k+ φe j k+1 H (T d ) 1 (T d ) E(u N ) k+1 k+ φe j H 1 (T d ), provided that 1 + k 1 d. Therefore, by Hölder and Young inequalities, and similarly to the control of (5.6), we have E sup (5.13) C m φ L t,t (L ;H 1 ) T E sup E(u N (t)) m 1 t,t E sup E(u N (t)) m k+ 1 t,t C m φ L (L ;H 1 ) T E E where in the last step we used interpolation. We also have sup E(u N (t)) m, t,t sup E(u N (t)) m 1 t,t E sup (5.14) C m φ L (L ;H 1 ) E sup E(u N (t)) m t,t t,t 8 E sup E(u N (t)) m t,t E sup (5.15) C m φ L(L;H t,t 1) + 18 E sup E(u N ) m t,t E sup (5.16) C m φ L(L;H t,t 1) E sup E(u N (t)) m 1, t,t E sup (5.17) C φ L(L;H t,t 1) + E sup H(u N (t)) m t,t 8 E sup H(u N (t)) m. t,t Gathering all the estimates, there exists C m > such that

31 E sup E(u N (t)) E(u ) m + C m T E sup E(u N (t)) m t,t t,t E sup E(u N (t)) m. t,t Similarly to passing from (5.9)to(5.1) and by induction on m, we deduce that E sup E(u N (t)) m 1, (5.18) t,t with constant independent of N. We now argue that the probabilistic a priori bounds in fact hold for solutions of the original SNLS. Corollary 5. For u solution to (1.1) with (1.), the estimates (5.4) and (5.5) hold with u in place of u N under the same assumptions as Proposition 5.1. Proof Let N be the mild formulation of (5.3), more precisely, ) N (v) := S(t)u N ±i S(t t )P N ( v k v (t ) dt i S(t t )φ N dw N (t ). (5.19) Then N is a contraction on a ball in X 1, 1 (, 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 ( ; C(, T ; Hx s )) for s =, 1. We only show s = 1 since the proof of s = 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 S(t)u in F T. For the noise, let N (t) denote the stochastic convolution in (5.19). Then (t) N (t) = = + n >N j Z d n N j >N S(t t )P >N φ dw(t ) + e n e i(t t ) n φe j (n)dβ j (t ) S(t t )π N P N φ dw(t ), 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 P >N φ L (L ;H 1 ) + π N P N φ L (L ;H 1 ), which tends to as N because both norms are tails of convergent series. Finally we treat the nonlinear terms ) Du(t) := S(t t ) u k u(t ) dt and D N u(t) := S(t t )P N ( u k u (t ) dt.

32 We first fix a path for which local well-posedness holds, and prove that Du D N u inx 1, 1 +. Firstly, Du D N t X u X 1, 1 S(t t )P + N ( u k u u N k u N )(t ) dt (,T ) 1, 1 + (,T ) + P >N Du X 1, 1 + (,T ) By Lemmas.,.4 and.5, wehave ( I u k X 1, 1 + u N k (,T ) II u k+1 X 1, 1 (,T ) X 1, 1 (,T ) ) u u N X 1, 1 (,T ) (5.) (5.1) In particular, (5.1) implies Du X 1, 1 + (, T ), and hence II asn.we claim that I asn as well. Indeed, N and are contractions with fixed points u N and u respectively, hence u u N (u) X 1, 1 N X (u) 1, (,T ) 1 + N (u) N (u N X ) 1, (,T ) 1 (,T ) (u) N (u) X 1, u u N X 1, (,T ) 1. (,T ) By rearranging, it suffices to show that the first term on the right-hand side above tends to as N.Now (u) N (u) X 1, 1 P >N S(t)u (,T ) X 1, 1 (,T ) X + P >N S(t t ) u k u(t ) dt 1, 1 (,T ) + >N X 1, 1. (,T ) By similar arguments as above, all the terms on the right go to as N.This proves our claim. By the embedding X 1, 1 + (, T ) C(, T ; H 1 (T d )), wehave that Du D u N (5.) C(,T ;H 1 ) almost surely as N. By the dominated convergence theorem, we have Du D N u inf 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 {, 1} be the regularity of u from Theorem 1.5. Let ε> and T > be given. We claim that there exists an event ε such that a solution