Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schr\"odinger equation on $\mathbb{R}^3$

We consider the cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2]. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

(1.1) The cubic NLS (1.1) has been studied extensively from both the theoretical and applied points of view. In this paper, we continue our study in [1,2] and further investigate the probabilistic well-posedness of (1.1) with random and rough initial data.
Recall that the equation (1.1) is scaling critical inḢ 1 2 (R 3 ) in the sense that the scaling symmetry: u(t, x) → λu(λ 2 t, λx) preserves the homogeneousḢ 1 2 -norm (when it is applied to functions only of x). It is known that the Cauchy problem (1.1) is locally well-posed in H s (R 3 ) for s ≥ 1 2 [9]. See also [12,25,28,17] for partial global well-posedness results. On the other hand, (1.1) is known to be ill-posed in H s (R 3 ) for s < 1 2 [11]. In [2], we studied the probabilistic well-posedness property of (1.1) below the scaling critical regularity s crit = 1 2 under a suitable randomization of initial data; see (1.2) below. In particular, we proved that (1.1) is almost surely locally well-posed in H s (R 3 ), s > 1 4 . Our main goal in this paper is to introduce an iterative procedure to improve this regularity threshold for almost sure local well-posedness. Furthermore, we study a critical regularity associated with this iterative procedure. By introducing a modified iterative approach, we then prove almost sure local well-posedness of (1.1) in an almost optimal range with respect to the original iterative procedure (Theorem 1.7). Beyond the concrete results in this paper, we believe that the iterative procedures based on (modified) partial power series expansions are themselves of interest for further development in the well-posedness theory of dispersive equations with random initial data and/or random forcing.
Such a probabilistic construction of solutions to dispersive PDEs first appeared in the works by McKean [33] and Bourgain [4] in the study of invariant Gibbs measures for the cubic NLS on T d , d = 1, 2. In particular, they established almost sure local well-posedness with respect to particular random initial data, basically corresponding to the Brownian motion. (These local-in-time solutions were then extended globally in time by invariance of the Gibbs measures. In the following, however, we restrict our attention to local-in-time solutions.) In [7], Burq-Tzvetkov further elaborated this idea and considered a randomization for any rough initial condition via the Fourier series expansion. More precisely, they studied the cubic nonlinear wave equation (NLW) on a three dimensional compact Riemannian manifold below the scaling critical regularity. By introducing a randomization via the multiplication of the Fourier coefficients by independent random variables, they established almost sure local well-posedness below the critical regularity with respect to the randomization. Such randomization via the Fourier series expansion is natural on compact domains [14,34] and more generally in situations where the associated elliptic operators have discrete spectra [45,43]. See [2] for more references therein. 1 Our discussion in this paper can be easily adapted to the cubic NLS on R d (and to other nonlinear dispersive PDEs). For the sake of concreteness, however, we only consider the d = 3 case in the following. See also the comment after Theorem 1.2 for our particular interest in the three-dimensional problem.
Then, given a function φ on R 3 , we have We define the Wiener randomization φ ω of φ by where {g n } is a sequence of independent mean-zero complex-valued random variables on a probability space (Ω, F, P ). In the following, we assume that the probability distribution µ n of g n satisfies the following exponential moment bound: for all κ ∈ R 2 and n ∈ Z 3 . This condition is satisfied by the standard complex-valued Gaussian random variables and the uniform distribution on the circle in the complex plane. We now recall the almost sure local well-posedness result from [2] which is of interest to us. 2 Theorem A. Let 1 4 < s < 1 2 . Given φ ∈ H s (R 3 ), let φ ω be its Wiener randomization defined in (1.2). Then, the Cauchy problem (1.1) is almost surely locally well-posed with respect to the random initial data φ ω . Moreover, the solution u lies in the class:  2 (R 3 )) is defined in Section 2 below. Recall that while the Wiener randomization (1.2) does not improve differentiability, it improves integrability (see Lemma 4 in [1]). See [7,40,26] for the corresponding statements in the context of the random Fourier series. The main idea for proving Theorem A is to exploit this gain of integrability. More precisely, let denote the random linear solution with φ ω as initial data and write Then, we see that v := u − z 1 satisfies the following equation: i∂ t v + ∆v = |v + z 1 | 2 (v + z 1 ) v| t=0 = 0, (1.5) where z 1 is viewed as a given (random) source term. The main point is that the gain of space-time integrability of the random linear solution z 1 (Lemma 2.1) makes this problem subcritical 3 and hence we can solve it by a standard fixed point argument. Over the last several years, there have been many results on probabilistic well-posedness of nonlinear dispersive PDEs, using this change of viewpoint. 4 See, for example, [33,4,7,8,14,34,31,1,2,41,37,24,32,6,29,36]. In [2], we studied the Duhamel formulation for (1.5): by carrying out case-by-case analysis on terms of the form vvv, vvz 1 , vz 1 z 1 , etc. in X In [2], we already proved (1.8) when σ = 1 2 and s > 1 4 , giving the regularity restriction in Theorem A. See Section 3 for the proof of Proposition 1.1 for a general value of σ. In the proof of Theorem A, in order to carry out the case-by-case analysis for (1.6), we need to have z 3 ∈ X 1 2 loc , (where we have a deterministic well-posedness for (1.1)). In view of Proposition 1.1, this imposes the regularity restriction s > 1 4 . Note, however, that even when s ≤ 1 4 , z 3 is a well defined space-time function of spatial regularity 2s− < 1 4 . This motivates us to consider the following second order expansion: and remove the second order interaction z 1 z 1 z 1 . Indeed, the residual term v := u − z 1 − z 3 satisfies the following equation: (1.10) where N (u) = |u| 2 u. In terms of the Duhamel formulation, we have Then, by studying the fixed point problem (1.11) for v, we have the following improved almost sure local well-posedness of (1.1).
, let φ ω be its Wiener randomization defined in (1.2). Then, the cubic NLS (1.1) on R 3 is almost surely locally well-posed with respect to the random initial data φ ω . Moreover, the solution u lies in the class: for T = T (φ, ω) > 0 almost surely.
By taking σ = 1 2 , Theorem 1.2 states that (1.1) is almost surely locally well-posed in H s (R 3 ), provided s > 1 5 . This in particular improves the almost sure local well-posedness in Theorem A. On the other hand, by taking σ = 1, Theorem 1.2 allows us to construct In particular, we can take initial data below the scaling critical regularity s crit = 1 2 , while we construct the residual part v in X 1 ([0, T ]). This opens up the possibility of studying the global-in-time behavior of v, using the (non-conserved) energy of v: with random initial data below the scaling critical regularity. We remark that by inspecting the argument in [2], a modification of (the proof of) Theorem A yields almost sure local well-posedness of (1.1) in the class: only for s > 1 2 . (This restriction on s can be easily seen by setting σ = 1 in Proposition 1.1.) In particular, Theorem A does not allow us to take initial data below the scaling critical regularity s crit = 1 2 in studying the global-in-time behavior of v ∈ X 1 ([0, T ]). As our focus in this paper is the local-in-time analysis, we do not pursue further this issue on almost sure global well-posedness of (1.1) below the scaling critical regularity in this paper. We, however, point out two recent results [29,36] on almost sure global well-posedness below the energy space for the defocusing energy-critical NLS in higher dimensions.
The main strategy for proving Theorem 1.2 is to study the fixed point problem (1.11) by carrying out case-by-case analysis on w 1 w 2 w 3 , for w i = v, z 1 , or z 3 , i = 1, 2, 3, but not all w i equal to z 1 (1.12) in N σ ([0, T ]), where the dual norm is defined by .
Note that the number of cases has increased from the case-by-case analysis in the proof of Theorem A, where we had w i = v or z 1 . One of the main ingredients is the smoothing on z 3 stated in Proposition 1.1 above. Note, however, that in order to exploit this smoothing, we need to measure z 3 in the X 2s− ([0, T ])-norm, which imposes a certain rigidity on the space-time integrability. 6 In order to prove Theorem 1.2, we also need to exploit a gain of integrability on z 3 . In Lemma 3.3, we use the dispersive estimate (see (3.10) below) and the gain of integrability on each z 1 of the three factors in (1.7) and show that z 3 also enjoys a gain of integrability by giving up some differentiability.
, let φ ω be its Wiener randomization. Given v 0 ∈ H 1 2 (R 3 ), we can also consider (1.1) with the random initial data of the form u ω 0 = v 0 + φ ω : Then, by slightly modifying the proofs, we see that the analogues of Theorem A and 1.2 (with σ = 1 2 ) also hold for (1.13). Namely, (1.13) is almost surely locally well-posed, provided s > 1 5 . This amounts to considering the following Cauchy problems: In this case, the critical nature of the problem appears through the vvv interaction in the case-by-case analysis (1.12) due to the deterministic (non-zero) initial data v 0 at the critical regularity. The required modification is straightforward and thus we omit details. See Proposition 6.3 in [2] and Lemma 6.2 in [36]. We point out that the discussion in the next subsection also applies to (1.13). Remark 1.4. (i) Let I(u 1 , u 2 , u 3 ) denote the trilinear operator defined by (1.14) Then, for σ > 3s − 1, there is no finite constant C > 0 such that In particular, this shows that when s ≤ 1 2 , there is no deterministic smoothing for I, i.e. (1.15) does not hold for σ > s. (ii) The proof of Proposition 1.1 only exploits "the randomization at the linear level". Given s ∈ R, let R s denote the class of functions defined by R s = u on R × R 3 : i∂ t u + ∆u = 0 and u ∈ L q t,loc W s,r x (R 3 ) for any 2 ≤ q, r < ∞ . Then, it follows from the proof of Proposition 1.1 that the estimate (1.15) "basically holds" 7 for any u 1 , u 2 , u 3 ∈ R s under the same regularity assumption: σ < 2s and 0 ≤ s < 1. In other words, the proof of Proposition 1.1 only uses the fact that the random linear solution z 1 = S(t)φ ω belongs to R s almost surely; see the probabilistic Strichartz estimate (Lemma 2.1). The multilinear random structure of z 3 in terms of the random linear solution z 1 yields further cancellation. See, for example, Lemma 3.6 in [14]. Such extra cancellation seems to improve only space-time integrability and we do not know how to use it to improve the regularity threshold (i.e. differentiability) at this point. A similar comment applies to the unbalanced higher order terms ζ 2k−1 (including z 5 below) studied in Proposition 1.6 below.
1.3. Partial power series expansion and the associated critical regularity. In this subsection, we discuss possible improvements over Theorem 1.2 by considering further expansions. For this purpose, we fix σ = 1 2 in the following. By examining the proof of Theorem 1.2, we see that the regularity restriction s > 1 5 comes from the following third order term: (1.17) Namely, we have (j 1 , j 2 , j 3 ) = (1, 1, 3) up to permutations. In Lemma 4.1, we show that given 0 < s < 1 2 , we have z 5 ∈ X 5 2 s− loc . In particular, we have z 5 ∈ X 1 2 loc , provided s > 1 5 , yielding the regularity threshold in Theorem 1.2.
A natural next step is to remove this non-desirable third order interaction In view of the deterministic non-smoothing discussed above, the constant C in (1.15) must depend on uj (0) in this case; given T > 0, uj L q t ([0,T ];W s,r x ) < ∞ for uj ∈ R s . In particular, we have uj L q t ([0,T ];W s,r x ) ≤ Cq,r,T (uj(0)) uj (0) H s (1.16) and the constant C in (1.15) depends on Cq,r,T (uj (0)) for relevant values of (q, r). See the proof of Proposition 1.1.
in the case-by-case analysis in (1.12) by considering the following third order expansion: In this case, the residual term v := u − z 1 − z 3 − z 5 satisfies the following equation: We expect that (1.19) is almost surely locally well-posed for s > 2 11 , which would be an improvement over Theorem 1.2. The proof will be once again based on case-by-case analysis: . Note the increasing number of combinations. In the following, however, we do not discuss details of this particular improvement over Theorem 1.2. Instead, we consider further iterative steps and a possible limitation of this procedure. By drawing an analogy to the previous steps, we expect that the worst contribution comes from the following fourth order terms: (1.20) There are basically two contributions to (1.20): (j 1 , j 2 , j 3 ) = (1, 3, 3) or (1, 1, 5) up to permutations. In Lemma 4.2, we show that the contribution from (j 1 , j 2 , j 3 ) = (1, 1, 5) is worse, being responsible for the expected regularity restriction s > 4 11 . In order to remove this term, we can consider the following fourth order expansion: power series expansion of a solution to (1.1) in terms of the random initial data can be expressed as a summation of certain multilinear operators over ternary trees. See, for example, [10,35]. Here, the term "order" in our context corresponds to "generation (of the associated trees) +1" with the convention that the trivial tree consisting only of the root node is of the zeroth generation. For example, the third order term z5 in (1.17) appears as the summation over all the multilinear operators associated to the ternary trees of the third generation. In terms of the graphical representation in [35], we have z5 " = " + + where " " denotes the random linear solution z1 = S(t)φ ω and " " denotes the trilinear Duhamel integral operator I(u1, u2, u3) defined in (1.14) with its three children as its arguments u1, u2, and u3.
as in the previous steps and try to solve the following equation for the residual term We can obviously iterate this argument and consider the following kth order expansion: In this case, we need to consider the following equation for the residual term v : and hope to construct a solution v ∈ X 1 2 ([0, T ]) by carrying out the following case-by-case analysis: it is not of the form z j 1 z j 2 z j 3 with j 1 + j 2 + j 3 ∈ {3, 5, . . . , 2k − 1} (1.23) in N We now consider a "critical" regularity s * < 1 2 with respect to this iterative procedure for proving almost sure local well-posedness of (1.1). We simply define the critical regularity s * < 1 2 for this problem to be the infimum of the values of s < 1 2 such that given any φ ∈ H s (R 3 ), the above iterative procedure 9 shows that (1.1) is almost surely locally well-posed with respect to the Wiener randomization φ ω of φ. This is an empirical notion of criticality; unlike the scaling criticality, we can not a priori compute this critical regularity s * . Moreover, our discussion will be based on the estimates on the stochastic multilinear terms (Proposition 1.1 and Proposition 1.6 below). In the following, we discuss a (possible) lower bound on s * , presenting a limitation to our iterative procedure based on partial power series expansions.
Within the framework of the iterative procedure discussed above, a necessary condition for carrying out the case-by-case analysis (1.23) to study (1.22) in X  (k + 1)st order term z 2k+1 = z 2(k+1)−1 defined in a recursive manner: . By the nature of this iterative procedure, we may assume that the lower order terms z 2ℓ−1 , ℓ ∈ {1, 3, . . . , k}, belong to X s ℓ ([0, T ]) for some s ℓ < 1 2 but not in X 1 2 ([0, T ]), since if any of the lower order terms, say z 2ℓ−1 for some ℓ ∈ {1, 3, . . . , k}, were in X 1 2 ([0, T ]), then we would have stopped the iterative procedure at the (ℓ − 1)th step. As in the previous steps, we expect that z 2k+1 is responsible for a regularity restriction at the kth step of this iterative approach.
In anticipating the alternative expansion (1.31) below, let us study the regularity property of the unbalanced kth order term ζ 2k−1 : (1.26) For k = 2, 3, 4, we have (with appropriate restrictions on the range of s) 10 In fact, we study the kth order term ζ 2k−1 in Proposition 1.6. 11 By associating the (2k + 1)-linear terms appearing in the summation in (1.24) with ternary trees of the kth generation as in [35], the summands in (1.25) corresponds to the "unbalanced" trees of the kth generation, where two of the three children of the root node are terminal.
By solving the recursive relation (1.28), we have and thus we have α 2 = 2, α 3 = 5 2 , and α 4 = 11 4 . In particular, Proposition 1.6 agrees with (1.27). Moreover, since α k is increasing and lim k→∞ α k = 3, the regularity restriction s < α −1 k−1 in Proposition 1.6 does not cause any issue since our main focus is to study the probabilistic local well-posedness of (1.1) in the range of s that is not covered by Theorem 1.2. Namely, we may assume s ≤ 1 5 in the following. In view of Propositions 1.1 and 1.6, one obvious lower bound for the critical regularity s * for this iterative procedure is given by s 0 := 0 since there is no gain of regularity when s = 0 (even in moving from z 1 to z 3 ). On the other hand, in order to prove almost sure local wellposedness of (1.1) by carrying out the case-by-case analysis (1.23) for the equation (1.22), we need to show that the (k + 1)st order term ζ 2k+1 belongs to X 1 2 ([0, T ]). This gives rise to a regularity restriction s k := 1 2α k+1 at the kth step of the iterative procedure. By taking k → ∞, we obtain another "lower" bound 12 s ∞ := 1 6 on this critical regularity s * . As mentioned above, the case-by-case analysis (1.23) for general k ∈ N may be a combinatorially overwhelming task due to (i) the number of the increasing combinations in (1.23) and (ii) the random multilinear terms z j , j ∈ {3, 5, . . . , 2k − 1}, themselves having nontrivial combinatorial structures which makes it difficult to establish nonlinear estimates; see (1.24). In the following, we instead consider an alternative iterative procedure based on the following expansion: in place of (1.21). This expansion allows us to prove the following almost sure local wellposedness of (1.1) for s > s ∞ = 1 6 .
, let φ ω be its Wiener randomization defined in (1.2). Then, the cubic NLS (1.1) on R 3 is almost surely locally well-posed with respect to the random initial data φ ω . Moreover, by letting k ∈ N such that 1 2α k+1 < s ≤ 1 2α k , the solution u lies in the class: In view of Proposition 1.6, Theorem 1.7 proves almost sure local well-posedness of (1.1) in an almost "optimal" 13 regularity range s > s ∞ = 1 6 with respect to the original iterative procedure based on the partial power series expansion (1.21). The proof of Theorem 1.7 is analogous to that of Theorem 1.2. Given 1 6 < s < 1 2 , fix k ∈ N such that 1 2α k+1 < s ≤ 1 2α k . 14 Write a solution u as in (1.31). Note that as s gets closer and closer to the critical value s ∞ = 1 6 , the expansion (1.31) gets arbitrarily long. In view of (1.26), the residual term v := u − k ℓ=1 ζ 2ℓ−1 satisfies the following equation: (1.32) Hence, we need to carry out the following case-by-case analysis: . While Theorem 1.7 yields almost sure local well-posedness in an almost optimal range with respect to the original iterative procedure based on the partial power series expansion (1.21), the required analysis is much simpler than that required for the original iterative procedure based on the expansion (1.21). First, note that while the case-by-case analysis (1.23) involves combinatorially non-trivial z 2k−1 , the case-by-case analysis (1.33) only involves the unbalanced kth order term ζ 2k−1 which has a much simpler structure than z 2k−1 . In particular, Proposition 1.6 shows that ζ 2k−1 , k ≥ 3, has a better regularity property than the second order term ζ 3 = z 3 in (1.7). In terms of space-time integrability, we show that ζ 2k−1 also enjoys gain of integrability by giving up a control on derivatives (Lemma 4.3). Finally, by inspecting the proof of Theorem 1.2 (see Lemma 4.2 and Proposition 5.1 below), we see that, except for z j 1 z j 2 z j 3 with (j 1 , j 2 , j 3 ) = (1, 1, 3) up to permutations, 15 we can bound all the terms w 1 w 2 w 3 appearing in the case-by-case analysis (1.12) in N 1 2 ([0, T ]). Hence, we can basically apply the result of the case-by-case analysis (1.12) to our problem at hand. This allows us to construct a solution v ∈ X We previously conjectured that the (k+1)st order term z 2k+1 in (1.24) would be responsible for a regularity restriction at the kth step of the original iterative procedure. Combining Proposition 1.6 and Theorem 1.7, we confirmed this claim; the regularity restriction indeed comes only from the unbalanced (k + 1)st order term ζ 2k+1 in (1.25).
We conclude this introduction with several remarks. 13 Once again, this is based on the estimates in Propositions 1.1 and 1.6. In particular, the "optimality" of the regularity threshold in Theorem 1.7 is with respect to Propositions 1.1 and 1.6. If one can improves the bounds in Propositions 1.1 and 1.6, then one can lower the regularity threshold in Theorem 1.7. 14 The lower bound on s guarantees that ζ 2k+1 ∈ X . Then, by slightly modifying the proof of Theorem 1.7 based on the modified iterative approach (1.31), we can prove almost sure local wellposedness of the cubic NLS (1.13) with the random initial data v 0 + φ ω for the same range of s. See Remark 1.3 Remark 1.9. The ill-posedness result in [11] show that the solution map is not continuous for (1.1) when s < 1 2 . In proving Theorem A, we studied the perturbed NLS (1.5) for v = u − z 1 . In particular, the proof shows that we can factorize the solution map for (1.1) as 16 where the first map can be viewed as a universal lift map and the second map Ψ is the solution map to (1.5), which is in fact continuous in z 1 ∈ S s 1 ([0, T ]). 17 In the case of Theorem 1.2 (with σ = 1 2 ), we have the following factorization of the solution map for (1.1): where the second map Ψ is the solution map to (1.10), which is continuous from ). In the case of Theorem 1.7, we create k stochastic objects in the first step, where k = k(s) ∈ N: (1.36) Once again, the second map Ψ (which is the solution map to (1.32)) is continuous from ). We point out an analogy of these factorizations (1.34), (1.35), and (1.36) of the ill-posed solution map into (i) the first step, involving stochastic analysis and (ii) the second step, which is a deterministic continuous map to similar factorizations in the rough path theory [18] and more recent studies on stochastic parabolic PDEs [20,22].
In the proof of Theorem 1.7, we could consider the following expansion of an infinite order: (1.37) 16 Similarly, we can factorize the solution map for (1.13) as such that the second map Ψ is continuous in (v0, z1) ∈ H This would allow us to present a single argument that works for all 1 6 < s < 1 3 . In this case, the residual part In particular, we would need to worry about the convergence issue of infinite series and hence there seems to be no simplification in considering the infinite order expansion (1.37). Another strategy would be to treat ζ ∞ := ∞ ℓ=1 ζ 2ℓ−1 as one stochastic object and write It follows from (1.26) that ζ ∞ satisfies the following equation: (1.38) Noting that we can rewrite (1.38) as (1.39) (1.40) The equations (1.39) and (1.40) do not particularly appear to be in such a friendly format, allowing a simplification over the case-by-case analysis (1.33) for the equation (1.32).
In a recent work [38], the second author (with Tzvetkov and Y. Wang) proved invariance of the white noise for the (renormalized) cubic fourth order NLS on the circle. One novelty of this work is that we introduced an infinite sequence {z res 2ℓ−1 } ℓ∈N of stochastic (2ℓ − 1)linear objects (depending only on the random initial data) and considered the following expansion: For this problem, it turned out that z res ∞ := ∞ ℓ=1 z res 2ℓ−1 satisfies a particularly simple equation. 18 Then by treating z res ∞ as one stochastic object, we wrote a solution u as u = 18 In fact, the series z res ∞ = ∞ ℓ=1 z res 2ℓ−1 corresponds to the power series expansion of the resonant cubic fourth order NLS. We point out that z res ∞ does not belong to Wiener homogeneous chaoses of any finite order. z res ∞ + v, which led to the following factorization: for s < − 1 2 , where φ ω denotes the Gaussian white noise on the circle. Remark 1.10. In [42], the third author (with Y. Wang) recently studied probabilistic local well-posedness of NLS on R d within the framework of the L p -based Sobolev spaces, using the dispersive estimate. In the context of the cubic NLS (1.1) on R 3 , their result yields an almost sure unique local-in-time solution u for the randomized initial data φ ω ∈ H s (R 3 ), provided s ≥ 0. Their result, however, shows that the solution u only belongs to C([0, T ]; L 4 (R 3 )) almost surely. Note that our solution u constructed in Theorems A, 1.2, and 1.7 lies in the class u ∈ C([0, T ]; H s (R 3 )) almost surely. This paper is organized as follows. In Section 2, we recall probabilistic and deterministic lemmas along with the definitions of the basic function spaces. In Section 3, we study the regularity property of the second order term z 3 in (1.7). In Section 4, we further investigate the regularity properties of the higher order terms z 5 and z 7 and the unbalanced higher order terms ζ 2k−1 . Note that the analysis on z 5 and z 7 contains part of the case-by-case analysis (1.12) needed for proving Theorem 1.2. In Section 5, we then carry out the rest of the case-by-case analysis (1.12) and prove Theorem 1.2. In Section 6, we briefly describe the proof of Theorem 1.7 by indicating how the analysis in the previous sections can lead to the proof. In Appendix A, we prove the deterministic non-smoothing of the Duhamel integral operator discussed in Remark 1.4.

Notations:
We use a+ (and a−) to denote a + ε (and a − ε, respectively) for arbitrarily small ε ≪ 1, where an implicit constant is allowed to depend on ε > 0 (and it usually diverges as ε → 0).

Strichartz estimates and function spaces
It follows from (2.1) and Sobolev's inequality that for p ≥ 10 3 . We will use the following admissible pairs in this paper: (∞, 2), 5, 30 11 , 10 3 10 3 , (2+, 6−). In particular, by Sobolev's inequality, we have One of the important key ingredients for probabilistic well-posedness is the probabilistic Strichartz estimates. Such probabilistic estimates were first exploited by McKean [33] and Bourgain [4]. In the following, we state the probabilistic Strichartz estimates under the Wiener randomization (1.2). See [1,37] for the proofs.
We also need the following lemma on the control of the size of H s -norm of φ ω .

Function spaces and their properties.
In this subsection, we go over the basic definitions and properties of the functions spaces used for the Fourier restriction norm method (i.e. analysis involving the X s,b -spaces introduced in [3]) adapted to the space of functions of bounded p-variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators [30,21,23]. We refer readers to [21,23] for the proofs of the basic properties. See also [2]. Let Z be the set of finite partitions −∞ < t 0 < t 1 < · · · < t K ≤ ∞ of the real line. By convention, we set u(t K ) := 0 if t K = ∞. We use 1 I to denote the sharp characteristic function of a set I ⊂ R.
(i) We define a U p -atom to be a step function a : with the norm where the infimum is taken over all possible representations for u.
By convention, we impose that the limits lim t→±∞ u(t) exist in L 2 (R 3 ).
(iii) Let V p rc be the closed subspace of V p of all right-continuous functions u ∈ V p with lim t→−∞ u(t) = 0.
(iv) We define U p ∆ := S(t)U p (and V p ∆ := S(t)V p , respectively) to be the space of all functions u : rc,∆ is defined in an analogous manner. Recall the following inclusion relation; for 1 ≤ p < q < ∞, ). The space V p is the classical space of functions of bounded p-variation and the space U p appears as the pre-dual of V p ′ with 1 p + 1 p ′ = 1. Their duality relation and the atomic structure of the U p -space turned out to be very effective in studying dispersive PDEs in critical settings.
We are now ready to define the solution spaces.
with respect to the X s -norm defined by (ii) Let s ∈ R. We define Y s (R) to be the space of all functions u ∈ C(R; H s (R 3 )) such that the map t → P N u lies in V 2 rc,∆ H s for any N ∈ 2 N 0 and u Y s (R) < ∞, where the Y s -norm is defined by

18Á. BÉNYI, T. OH, AND O. POCOVNICU
Recall the following embeddings: Given an interval I ⊂ R, we define the local-in-time versions X s (I) and Y s (I) of these spaces as restriction norms. For example, we define the X s (I)-norm by We also define the norm for the nonhomogeneous term on an interval I = [t 0 , t 1 ): We conclude this section by presenting some basic estimates involving these function spaces. See [21,23,2] for the proofs.
for any T > 0 and u 1 , Proof. From the bilinear refinement of the Strichartz estimate [5,13] and the transference principle, we have for all u 1 , u 2 ∈ Y 0 . See [2] for the proof of (2.5). On the other hand, it follows from Hölder's and Sobolev's inequalities that Then, the estimate (2.4) follows from interpolating (2.5) and (2.6).

On the second order term z 3
In this and the next sections, we study the regularity properties of the various stochastic terms that appear in the iterative procedures. Given φ ∈ H s (R 3 ), let φ ω be the Wiener randomization of φ defined in (1.2) and set In this section, we study the regularity properties of the second order term: We first present the proof of Proposition 1.1. We follow closely the argument in [2].
for some almost surely finite constant C(ω, φ H s ) > 0 and θ > 0, where w Y 0 T ≤ 1. In the following, we drop the complex conjugate when it does not play any role.
Define A s 3 (T ) by Then, by applying the dyadic decomposition, it suffices to prove for all N 1 , . . . , N 4 ∈ 2 N 0 with N 3 ≥ N 2 ≥ N 1 . Once we prove (3.4), the desired estimate (3.2) follows from summing (3.4) over dyadic blocks and applying Lemmas 2.1 and 2.2. Recall our shorthand notation: z 1,N j = P N j z 1 and w N 4 = P N 4 w.
(ii) Given N ≫ 1 and small ℓ > 0, consider the following deterministic initial condition φ whose Fourier transform is given by (1, 0, 0), and e 2 = (0, 1, 0). By taking ℓ > 0 sufficiently small, we have supp φ ⊂ N e 1 + Q and thus we can neglect the effect of the randomization in (1.2) since all the three terms on the right-hand side will be multiplied by a common random number g N e 1 . Without loss of generality, we assume that g N e 1 = 1 in the following.
We estimate from below the contribution to 1 Q N,ℓ (ξ) z 3 (t, ξ), where Q N,ℓ = N e 1 + 100ℓ(e 1 + e 2 ) + ℓQ. 19 In the remaining part of this paper, we repeatedly apply this argument when there is a frequency separation. We shall simply refer to it as the "bilinear Strichartz estimate" argument.
Lemma 3.1. There exists c > 0 such that for all a, b, ξ ∈ R 3 .
By applying Lemma 3.1 to (3.6) with (3.8) and (3.9), we obtain Therefore, for any σ > 0, we have  On the one hand, Proposition 1.1 shows that z 3 controls almost 2s derivatives. On the other hand, we need to measure z 3 in the X 2s− -norm, which controls only the admissible space-time Lebesgue norms (with 2s− derivatives). The following lemma breaks this rigidity by giving up a control on derivatives. In particular, it allows us to control a wider range of space-time Lebesgue norms of z 3 . The main idea is to use the dispersive estimate for the linear Schrödinger operator: (3.10) 20 Here, the constant depends on u(0) in the sense of (1.16).
This allows us to reduce the analysis to a product of the random linear solution z 1 = S(t)φ ω and apply Lemma 2.1.
, let φ ω be its Wiener randomization defined in (1.2). Then, for any finite q, r ≥ 1, we have

11)
for any T > 0 and N ∈ 2 N 0 . Note that the right-hand side of (3.11) is almost surely finite thanks to the probabilistic Strichartz estimate (Lemma 2.1).
Proof. We first consider the case r < 6. From (3.1) and (3.10), we have (3.12) When r > 6, we proceed as in (3.12) but we apply Sobolev's inequality before applying (3.10): This completes the proof of Lemma 3.3.

On the higher order terms
In this section, we study the regularity properties of the higher order terms.

4.1.
On the third order term z 5 . In this subsection, we study the third order term: The following lemma shows that the third order term z 5 enjoys a gain of extra 1 2 s derivative as compared to the second order term z 3 (Proposition 1.1). Lemma 4.1. Given 0 < s < 1 2 , let φ ω be the Wiener randomization of φ ∈ H s (R 3 ) defined in (1.2). Then, for any σ < 5 2 s, we have z 5 ∈ X σ loc , almost surely. In particular, there exists an almost surely finite constant C(ω, φ H s ) > 0 and θ > 0 such that for any T > 0.
Proof. First, note that the only possible combination for (j 1 , j 2 , j 3 ) in (4.1) is (1, 1, 3) up to permutations. The complex conjugate does not play any role in the subsequent analysis and hence we drop the complex conjugate sign and simply study By Lemma 2.5, it suffices to prove Then, by applying the dyadic decomposition, it suffices to prove for all N 1 , . . . , N 4 ∈ 2 N 0 . Once we prove (4.4), the desired estimate (4.3) follows from summing over dyadic blocks and applying Lemmas 2.1 and 2.2 and Proposition 1.1. In the following, we fix 0 < s < 1 2 . Without loss of generality, we assume that provided that σ < 3s. 1 . By Hölder's inequality and Lemma 2.6, we have LHS of (4.4) N σ 1 z 1,N 1 L 5 Hence, we obtain (4.4), provided that σ < 3s.
Putting all the cases together, we conclude that (4.4) holds, provided that σ < 5 2 s. This completes the proof of Lemma 4.1.
Putting all the cases together, we conclude that (4.9) holds when σ < 3s.
(ii) Next, we estimate ζ 7 in (4.7). This term has a similar structure to z 5 in (4.2); the only difference appears in the third factor. Hence, we can estimate ζ 7 simply by replacing the regularity 2s− (for z 3 ; see Proposition 1.1) with 5 2 s− (for z 5 ; see Lemma 4.1) in the proof of Lemma 4.1. In the following, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma 4.1. 13 4 s and 0 < s < 2 5 . 21 The modifications for Subcases (1.b) and (1.c) are straightforward under the same regularity restriction. (4.10) The modifications for Subcases (3.b), (3.c), and (3.d) are straightforward under the regularity restriction (4.10).
Putting all the cases together, we conclude that (4.8) holds under the regularity restriction (4.10). This completes the proof of Lemma 4.2.
Proof of Proposition 1.6. We proceed by induction. From (1.27), we see that the recursive relation (1.28) is satisfied when k = 2, 3, 4. In the following, by assuming ζ 2k−3 ∈ X σ loc for σ < α k−1 s almost surely, where α k−1 satisfies (1.30), we prove (1.28) and (1.29). As we mentioned above, the proof follows from the proof of Lemma 4.1 by replacing z 3 and 2s− with ζ 2k−3 and α k−1 s−, respectively. Therefore, we only indicate the necessary modifications on the powers of dyadic parameters in the proof of Lemma 4.1.
We conclude this section by proving a gain of space-time integrability for ζ 2k−1 analogous to Lemma 3.3.

Proof of Theorem 1.2
In this section, we study the fixed point problem (1.11) around the second order expansion and present the proof of Theorem 1.2. Given 1 2 ≤ σ < 1, let 2 5 σ < s < 1 2 . Given φ ∈ H s (R 3 ), let φ ω be its Wiener randomization defined in (1.2) and let z 1 and z 3 be as in (1.3) and (1.7). Define Γ by Then, we have the following nonlinear estimates.
Then, there exists θ > 0, C 1 , C 2 > 0, and an almost surely finite constant R = R(ω) > 0 such that Once we prove Proposition 5.1, Theorem 1.2 immediately follows from a standard argument and thus we omit details. See Section 5 in [2].
Proof of Proposition 5.1. Let 0 < T ≤ 1. We only prove (5.1) since (5.2) follows in a similar manner. Arguing as in the proof of Proposition 4.1 in [2], it suffices to perform a case-by-case analysis of expressions of the form: where w Y 0 T ≤ 1 and w j = v, z 1 , or z 3 , j = 1, 2, 3 but not all z 1 . Note that we have dropped the complex conjugate sign on w 2 since it does not play any essential role. Then, we need to consider the following cases: We already treated Cases (A) and (B) in Lemmas 4.1 and 4.2. In particular, Case (A) imposes the regularity restriction: As we see below, Cases (B) -(I) only impose a milder regularity restriction: σ < 3s. Given 1 2 ≤ σ ≤ 1, let 2 5 σ < s < 1 2 . Define the B s (T )-norm by where q ≫ 1 is defined in (5.4) below. Then, by applying the dyadic decomposition, we prove the following estimate: 22 for all N 1 , . . . , N 4 ∈ 2 N 0 . Once we prove (5.3), the desired estimate (5.1) follows from summing (5.3) over dyadic blocks and applying Lemmas 2.1 and 2.2 and Proposition 1.1.
Case (C): z 3 z 3 z 3 case. By symmetry, we assume N 1 ≤ N 2 ≤ N 3 . Note that we have N 3 ∼ N max .
• Subcase (C.2): • Subsubcase (C.2.a): 3 . By the bilinear Strichartz estimate, we have This yields (5.3), provided that σ < 4s and s < 1 2 . 22 In Case (I), we do not perform the dyadic decomposition and hence there is no need to have the factor N 0− max on the right-hand side of (5.3).
By symmetry, we assume N 1 ≥ N 2 .
provided that σ > 0 and s > 0. 1 . In this case, we have N 1 ∼ N 4 ∼ N max . By Lemmas 2.6, 2.7, and 3.3, we have 3 . In this case, we proceed as in Subsubcase (D.2.a). It suffices to note that By Hölder's inequality with (5.4) as in Subcase (C.1) and Lemmas 2.6 and 3.3, we have T , provided that σ ≥ 0 and s > 0.
By symmetry, we assume N 3 ≥ N 2 .
When N 4 ≪ N 3 , we have N 2 ∼ N 3 ∼ N max . In this case, we can repeat the computation above with the roles of z 3,N 2 and w N 4 switched. Noting that we obtain (5.3), provided that σ ≤ min(1, 6s−). In this case, we proceed with L 10 3 T,x , L 10 3 T,x , L 10 T,x , L 10 3 T,x -Hölder's inequality. It suffices to note that provided that 0 ≤ σ < 4s.

Proof of Theorem 1.7
In this section, we briefly discuss the proof of Theorem 1.7. Given 1 6 < s < 1 2 , let k ∈ N such that 1 2α k+1 < s ≤ 1 2α k , where α k is defined in (1.30). Our main goal is to study the fixed point problem (1.32). Define Γ by where ζ 2ℓ−1 , ℓ = 1, . . . , k, is as in (1.26). Then, Theorem 1.7 follows from a standard fixed point argument, once we prove the following proposition.
Proposition 6.1. Given 1 6 < s < 1 2 , let k ∈ N such that 1 2α k+1 < s < 1 α k . Then, there exists θ > 0, C 1 , C 2 > 0, and an almost surely finite constant R = R(ω) > 0 such that On the one hand, the upper bound 1 α k on the range of s in Proposition 6.1 comes from the restriction in Proposition 1.6. On the other hand, given 1 6 < s < 1 2 , we fix k ∈ N such that 1 2α k+1 < s ≤ 1 2α k and apply Proposition 6.1. Namely, the upper bound on s in Proposition 6.1 does not cause any problem in proving Theorem 1.7.
Proof. We only discuss the proof of prove (6.1) since (6.2) follows in a similar manner. As in the proof of Proposition 5.1, it suffices to perform a case-by-case analysis of expressions of the form: ˆT where w Y 0 T ≤ 1 and w j = v, ζ 2ℓ−1 , ℓ = 1, . . . , k but not of the form ζ 1 ζ 1 ζ 2ℓ−3 for ℓ ∈ {2, 3, . . . , k}. Here, we have dropped the complex conjugate sign on w 2 since it does not play any role in our analysis More concretely, we need to consider the following cases: where j 1 , j 2 , j 3 can take any value in {3, 5, . . . , 2k−1}. As we see below, the worst interaction appears in Case (A) and all the other cases can be handled as in the proof of Proposition 5.1. We first point out that, in the proof of Proposition 5.1, the regularity restriction coming from Cases (B) -(I) (with σ = 1 2 ) is s > 1 6 . Moreover, by comparing Proposition 1.1 and Lemma 3.3 with Proposition 1.6 and Lemma 4.3, we see that the unbalanced higher order term ζ 2ℓ−1 for ℓ ∈ {2, 3, . . . , k} enjoys (at least) the same regularity properties as z 3 = ζ 3 both in terms of differentiability and space-time integrability. Therefore, we can simply apply the estimates in the proofs of Lemma 4.2 for Case (B) and Proposition 5.1 for Cases (C) -(I) and conclude that the contributions from Cases (B) -(I) can be bounded, provided s > 1 6 . It remains to consider Case (A). In view of (1.25), the contribution from Case (A) is nothing but ζ 2k+1 . Hence, by applying Proposition 1.6 with σ = 1 2 , we conclude that the contribution in Case (A) can be estimated by T θ R(ω), provided that 1 2α k+1 < s < 1 α k . This completes the proof of Proposition 6.1.
Then, we have we see that the non-trivial contribution to (A.2) comes from ξ j ∈ A j , j = 1, 2, 3. Moreover, in this case, Φ(ξ) defined (3.7) satisfies |Φ(ξ)| N ℓ. In particular, by choosing such that t * |Φ(ξ)| ≪ 1, we have where v j i = u i for i = ℓ and v j ℓ = I(u ℓ , u 4 , u 5 ). This basically corresponds to the second order term appearing in the power series expansion for (1.1). In particular, we have that z 5 = I (2) (z 1 , . . . , z 1 ). By a computation similar to the proof of Lemma A.1, we can prove the following deterministic non-smoothing on the second order term. Let σ > 5s − 2. Then, there is no finite constant C > 0 such that . In particular, this shows that when s ≤ 1 2 , there is no deterministic smoothing on the second order term I (2) . A similar comment applies to the higher order terms.