Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

Abstract

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus ${\mathbb {T}}^3$. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on ${\mathbb {T}}^3$ by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing $\Phi ^4_3$-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

1 Introduction

1.1 Singular stochastic nonlinear wave equation

In this paper, we study the following Cauchy problem for the stochastic nonlinear wave equation (SNLW) with a cubic nonlinearity on the three dimensional torus ${\mathbb {T}}^3=({\mathbb {R}}/(2\pi {\mathbb {Z}}))^3$, driven by an additive noise:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u + (1 - \Delta ) u + u^3 = \phi \xi \\ (u, \partial _tu) |_{t = 0} = (u_0, u_1), \end{array}\right. } \quad (x, t) \in {\mathbb {T}}^3\times {\mathbb {R}}, \end{aligned}$$

(1.1)

where $\xi (x, t)$ denotes a (Gaussian) space-time white noise on ${\mathbb {T}}^3\times {\mathbb {R}}$ with the space-time covariance given by

$$\begin{aligned} {\mathbb {E}}\big [ \xi (x_1, t_1) \xi (x_2, t_2) \big ] = \delta (x_1 - x_2) \delta (t_1 - t_2) \end{aligned}$$

and $\phi$ is a bounded operator on $L^2({\mathbb {T}}^3)$. Our main goal is to present a concise proof of local well-posedness of (1.1), when $\phi$ is the Bessel potential of order $\alpha$:

$$\begin{aligned} \phi = \langle \nabla \rangle ^{-\alpha } = (1- \Delta )^{-\frac{\alpha }{2}} \end{aligned}$$

(1.2)

for any $\alpha > 0$. Namely, we consider (1.1) with an “almost” space-time white noise.

Given $\alpha \in {\mathbb {R}}$, let $\phi = \phi _\alpha$ be as in (1.2). Then, a standard computation shows that the stochastic convolution:

(1.3)

belongs almost surely to $C({\mathbb {R}}; W^{s, \infty }({\mathbb {T}}^3))$ for any $s < \alpha - \frac{1}{2}$. See Lemma 3.1 below. Here, we adopted Hairer’s convention to denote stochastic terms by trees; the vertex in corresponds to the random noise $\phi \xi = \langle \nabla \rangle ^{-\alpha } \xi$, while the edge denotes the Duhamel integral operator:

$$\begin{aligned} {\mathcal {I}}= (\partial _t^2 + (1 - \Delta ))^{-1}, \end{aligned}$$

(1.4)

corresponding to the forward fundamental solution to the linear wave equation. Note that when $\alpha > \frac{1}{2}$, the stochastic convolution is a function of positive (spatial) regularity $\alpha - \frac{1}{2}-\varepsilon$.¹ Then, by proceeding with the first order expansion:

and studying the equation for the residual term , we can show that (1.1) is locally well-posed, when $\alpha > \frac{1}{2}$. See [13, 58] in the case of the deterministic cubic nonlinear wave equation (NLW):

$$\begin{aligned} \partial _t^2 u + (1 - \Delta ) u + u^3 = 0 \end{aligned}$$

(1.5)

with random initial data. Furthermore, by controlling the growth of the ${\mathcal {H}}^1$-norm of the residual term v via a Gronwall-type argument, we can prove global well-posedness of (1.1), when $\alpha > \frac{1}{2}$.² See [13].

When $\alpha \le \frac{1}{2}$, solutions to (1.1) are expected to be merely distributions of negative regularity $\alpha -\frac{1}{2} -\varepsilon$, inheriting the regularity of the stochastic convolution, and thus we need to consider the renormalized version of (1.1), which formally reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u + (1 - \Delta ) u + u^3 - \infty \cdot u = \langle \nabla \rangle ^{-\alpha } \xi \\ (u, \partial _tu) |_{t = 0} = (u_0, u_1), \end{array}\right. } \end{aligned}$$

(1.6)

where the formal expression $u^3- \infty \cdot u$ denotes the renormalization of the cubic power $u^3$. In the range $\frac{1}{4} < \alpha \le \frac{1}{2}$, a straightforward computation with the second order expansion:

yields local well-posedness of the renormalized SNLW (1.6) (in the sense of Theorem 1.1 below). Here, the second order process is defined by

where denotes the renormalized version of . See [51] for this argument in the context of the deterministic renormalized cubic NLW (1.5) with random initial data.

We state our main result.

Theorem 1.1

Let $0 < \alpha \le \frac{1}{2}$. Given $s > \frac{1}{2}$, let $(u_0, u_1) \in {\mathcal {H}}^{s}({\mathbb {T}}^3) = H^s({\mathbb {T}}^3)\times H^{s-1}({\mathbb {T}}^3)$. Then, there exists a unique local-in-time solution to the renormalized cubic SNLW (1.6) with $(u, \partial _tu)|_{t = 0} = (u_0, u_1)$.

More precisely, given $N \in {\mathbb {N}}$, let $\xi _N = \pi _N \xi$, where $\pi _N$ is the frequency projector onto the spatial frequencies $\{|n|\le N\}$ defined in (1.13) below. Then, there exists a sequence of time-dependent constants $\{\sigma _N(t)\}_{N\in {\mathbb {N}}}$ tending to $\infty$ (see (1.16) below) such that, given small $\varepsilon = \varepsilon (s) > 0$, the solution $u_N$ to the following truncated renormalized SNLW:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u_N + (1- \Delta ) u_N + u_N^3 - 3\sigma _N u_N = \langle \nabla \rangle ^{-\alpha } \xi _N\\ (u_N, \partial _tu_N)|_{t = 0} = (u_0, u_1) \end{array}\right. } \end{aligned}$$

(1.7)

converges to a non-trivial³ stochastic process $u \in C([-T, T]; H^{\alpha -\frac{1}{2} -\varepsilon } ({\mathbb {T}}^3))$ almost surely, where $T = T(\omega )$ is an almost surely positive stopping time.

Stochastic nonlinear wave equations have been studied extensively in various settings; see [15, Chapter 13] for the references therein. In particular, over the last few years, we have witnessed a rapid progress in the theoretical understanding of nonlinear wave equations with singular stochastic forcing and/or rough random initial data; see [12, 19, 20, 25,26,27, 45, 47,48,49,50,51, 53,54,57, 66]. In [26], Gubinelli, Koch, and the first author studied the quadratic SNLW on ${\mathbb {T}}^3$:

$$\begin{aligned} \partial _t^2 u + (1 - \Delta ) u + u^2 = \xi . \end{aligned}$$

(1.8)

By adapting the paracontrolled calculus [24], originally introduced by Gubinelli, Imkeller, and Perkowski in the study of stochastic parabolic PDEs, to the dispersive setting, the authors of [26] reduced (1.8) into a system of two unknowns. This system was then shown to be locally well-posed by exploiting the following two ingredients: (i) multilinear dispersive smoothing coming from a multilinear interaction of random waves (see also [12, 45]) and (ii) novel random operators (the so-called paracontrolled operators) which incorporate the paracontrolled structure in their definition. These random operators are used to replace commutators which are standard in the parabolic paracontrolled approach [14, 40].

More recently, Okamoto, Tolomeo, and the first author [48] and Bringmann [12] independently studied the following SNLW with a cubic Hartree-type nonlinearity:⁴

$$\begin{aligned} \partial _t^2 u + (1 - \Delta ) u + (V*u^2) u = \xi , \end{aligned}$$

(1.9)

where V is the kernel of the Bessel potential $\langle \nabla \rangle ^{-\beta }$ of order $\beta > 0$.⁵ In [48], the authors proved local well-posedness for $\beta > 1$ by viewing the nonlinearity as the nested bilinear interactions and utilizing the paracontrolled operators introduced in [26]. In [12], Bringmann went much further and proved local well-posedness of (1.9) for any $\beta > 0$. The main strategy in [12] is to extend the paracontrolled approach in [26] to the cubic setting. The main task is then to study regularity properties of various random operators and random distributions. This was done by an intricate combination of deterministic analysis, stochastic analysis, counting arguments, the random matrix/tensor approach by Bourgain [9, 10] and Deng, Nahmod, and Yue [18], and the physical space approach via the (bilinear) Strichartz estimates due to Klainerman and Tataru [36], analogous to the random data Cauchy theory for the nonlinear Schrödinger equations on ${\mathbb {R}}^d$ as in [2,3,4].

From the scaling point of view, the cubic SNLW (1.6) with a slightly smoothed space-time white noise (i.e. small $\alpha >0$) is essentially the same as the Hartree SNLW (1.9) with small $\beta > 0$. Hence, Theorem 1.1 is expected to hold in view of Bringmann’s recent result [12]. The main point of this paper is that we present a concise proof of Theorem 1.1without using the paracontrolled calculus. In the next subsection, we outline our strategy.

Due to the time reversibility of the equation, we only consider positive times in the remaining part of the paper.

Remark 1.2

The equations (1.1) and (1.6) indeed correspond to the stochastic nonlinear Klein–Gordon equations. The same results with inessential modifications also hold for the stochastic nonlinear wave equation, where we replace the linear part in (1.1) and (1.6) by $\partial _t^2 u- \Delta u$. In the following, we simply refer to (1.1) and (1.6) as the stochastic nonlinear wave equations.

Remark 1.3

Our argument also applies to the deterministic (renormalized) cubic NLW on ${\mathbb {T}}^3$ with random initial data of the form:

$$\begin{aligned} (u_0^\omega , u_1^\omega ) = \bigg (\sum _{n \in {\mathbb {Z}}^3} \frac{g_n(\omega )}{\langle n \rangle ^{1+\alpha }}e^{in\cdot x}, \sum _{n \in {\mathbb {Z}}^3} \frac{h_n(\omega )}{\langle n \rangle ^{\alpha }}e^{in\cdot x}\bigg ), \end{aligned}$$

where the series $\{ g_n \}_{n \in {\mathbb {Z}}^3}$ and $\{ h_n \}_{n \in {\mathbb {Z}}^3}$ are two families of independent standard complex-valued Gaussian random variables conditioned that $g_n=\overline{g_{-n}}$, $h_n=\overline{h_{-n}}$, $n \in {\mathbb {Z}}^3$. In particular, Theorem 1.1 provides an improvement of the main result (almost sure local well-posedness) in [51] from $\alpha > \frac{1}{4}$ to $\alpha > 0$.

Remark 1.4

• (i) The first part of the statement in Theorem 1.1 is merely a formal statement in view of the divergent behavior $\sigma _N (t) \rightarrow \infty$ for $t\ne 0$. In the next subsection, we provide a precise meaning to what it means to be a solution to (1.6) and also make the uniqueness statement more precise. See Remark 1.9.

• (ii) In the case of the defocusing cubic SNLW with damping:

  $$\begin{aligned} \partial _t^2 u + \partial _tu + (1 - \Delta ) u + u^3 = \langle \nabla \rangle ^{-\alpha } \xi , \end{aligned}$$

  a combination of our argument with that in [47] yield the following triviality result. Consider the following truncated (unrenormalized) SNLW with damping:

  $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u_N + \partial _tu_N + (1- \Delta ) u_N + u_N^3 = \langle \nabla \rangle ^{-\alpha } \xi _N\\ (u_N, \partial _tu_N)|_{t = 0} = (u_0, u_1), \end{array}\right. } \end{aligned}$$

  where $\xi _N = \pi _N \xi$. As we remove the regularization (i.e. take $N\rightarrow \infty$), the solution $u_N$ converges in probability to the trivial function $u_\infty \equiv 0$ for any (smooth) initial data $(u_0, u_1)$. See [47] for details.

Remark 1.5

• (i) In our proof, we use the Fourier restriction norm method (i.e. the $X^{s, b}$-spaces defined in (2.8)), following [12, 57]. While it may be possible to give a proof of Theorem 1.1 based only on the physical-side spaces (such as the Strichartz spaces) as in [25,26,27], we do not pursue this direction since our main goal is to present a concise proof of Theorem 1.1 by adapting various estimates in [12] to our current setting. Note that the use of the physical-side spaces would allow us to take the initial data $(u_0, u_1)$ in the critical space ${\mathcal {H}}^\frac{1}{2}({\mathbb {T}}^3)$ (for the cubic NLW on ${\mathbb {T}}^3$). See for example [25]. One may equally use the Fourier restriction norm method adapted to the space of functions of bounded p-variation and its pre-dual, introduced and developed by Tataru, Koch, and their collaborators [28, 31, 37], which would also allow us to take the initial data $(u_0, u_1)$ in the critical space ${\mathcal {H}}^\frac{1}{2}({\mathbb {T}}^3)$. See for example [3, 46] in the context of the nonlinear Schrödinger equations with random initial data. Since our main focus is to handle rough noises (and not about rough deterministic initial data), we do not pursue this direction.

• (ii) On ${\mathbb {T}}^3$, the Bessel potential $\phi _\alpha = \langle \nabla \rangle ^{-\alpha }$ is Hilbert–Schmidt from $L^2({\mathbb {T}}^3)$ to $H^s({\mathbb {T}}^3)$ for $s < \alpha - \frac{3}{2}$. It would be of interest to extend Theorem 1.1 to a general Hilbert–Schmidt operator $\phi$, say from $L^2({\mathbb {T}}^3)$ to $H^{\alpha - \frac{3}{2}}({\mathbb {T}}^3)$ as in [16, 44, 52].⁶ Note that our argument uses the independence of the Fourier coefficients of the stochastic convolution but that such independence will be lost for a general Hilbert–Schmidt operator $\phi$.

Remark 1.6

• (i) When $\alpha = 0$, SNLW (1.6) with damping

  $$\begin{aligned} \partial _t^2 u + \partial _tu + (1 - \Delta ) u + u^3 - \infty \cdot u = \xi \end{aligned}$$

  (1.10)

  corresponds to the so-called canonical stochastic quantization equation⁷ for the Gibbs measure given by the $\Phi ^4_3$-measure on u and the white noise measure on $\partial _tu$. See [60]. In this case (i.e. when $\alpha = 0$), our approach and the more sophisticated approach of Bringmann [12] for (1.9) with $\beta > 0$ completely break down. This is a very challenging problem, for which one would certainly need to use the paracontrolled approach in [12, 26, 48] and combine with the techniques in [18].

• (ii) As mentioned above, when $\alpha > \frac{1}{2}$, the globalization argument by Burq and Tzvetkov [13] yields global well-posedness of SNLW (1.1) with $\phi$ as in (1.2). When $\alpha = 0$, we expect that (a suitable adaptation of) Bourgain’s invariant measure argument would yield almost sure global well-posedness once we could prove local well-posedness of (1.10) (but this is a very challenging problem). It would be of interest to investigate the issue of global well-posedness of (1.6) for $0 < \alpha \le \frac{1}{2}$. See [27, 66] for the global well-posedness results on SNLW with an additive space-time white noise in the two-dimensional case.

1.2 Outline of the proof

Let us now describe the strategy to prove Theorem 1.1. Let W denote a cylindrical Wiener process on $L^2({\mathbb {T}}^3)$:⁸

$$\begin{aligned} W(t) = \sum _{n \in {\mathbb {Z}}^3} B_n (t) e_n, \end{aligned}$$

where $e_n(x) = e^{ i n \cdot x}$ and $\{ B_n \}_{n \in {\mathbb {Z}}^3}$ is defined by $B_n(t) = \langle \xi , {\mathbf {1}}_{[0, t]} \cdot e_n \rangle _{ x, t}$. Here, $\langle \cdot , \cdot \rangle _{x, t}$ denotes the duality pairing on ${\mathbb {T}}^3\times {\mathbb {R}}$. As a result, we see that $\{ B_n \}_{n \in {\mathbb {Z}}^3}$ is a family of mutually independent complex-valued Brownian motions conditioned so that $B_{-n} = \overline{B_n}$, $n \in {\mathbb {Z}}^3$. In particular, $B_0$ is a standard real-valued Brownian motion. Note that we have, for any $n \in {\mathbb {Z}}^2$,

$$\begin{aligned} \text {Var}(B_n(t)) = {\mathbb {E}}\big [ \langle \xi , {\mathbf {1}}_{[0, t]} \cdot e_n \rangle _{x, t}\overline{\langle \xi , {\mathbf {1}}_{[0, t]} \cdot e_n \rangle _{x, t}} \big ] = \Vert {\mathbf {1}}_{[0, t]} \cdot e_n\Vert _{L^2_{x, t}}^2 = t. \end{aligned}$$

With this notation, we can formally write the stochastic convolution in (1.3) as

(1.11)

where $\langle \nabla \rangle = \sqrt{1-\Delta }$ and $\langle n \rangle = \sqrt{1 + |n|^2}$. We indeed construct the stochastic convolution in (1.11) as the limit of the truncated stochastic convolution defined by

(1.12)

for $N \in {\mathbb {N}}$, where $\pi _N$ denotes the (spatial) frequency projector defined by

$$\begin{aligned} \pi _N f = \sum _{ |n| \le N} \widehat{f} (n) \, e_n. \end{aligned}$$

(1.13)

A standard computation shows that the sequence is almost surely Cauchy in⁹$C([0,T];W^{\alpha - \frac{1}{2} - ,\infty }({\mathbb {T}}^3))$ and thus converges almost surely to some limit, which we denote by  , in the same space. See Lemma 3.1 below.

We then define the Wick powers and by

(1.14)

and the second order process by

(1.15)

where ${\mathcal {I}}$ denotes the Duhamel integral operator in (1.4). Here, $\sigma _N(t)$ is defined by¹⁰

(1.16)

We point out that a standard argument shows that and converge almost surely to in $C([0,T];W^{2\alpha -1 - ,\infty }({\mathbb {T}}^3))$ and to in $C([0,T];W^{3\alpha -\frac{3}{2} - ,\infty }({\mathbb {T}}^3))$, respectively, but that we do not need these regularity properties of the Wick powers and in this paper.

As for the second order process in (1.15), if we proceed with a “parabolic thinking”,¹¹ then we expect that has regularity¹²$3\alpha - \frac{1}{2} - = (3\alpha - \frac{3}{2} -) + 1$, which is negative for $\alpha \le \frac{1}{6}$. In the dispersive setting, however, we can exhibit multilinear smoothing by exploiting multilinear dispersion coming from an interaction of (random) waves. In fact, by adapting the argument in [12] to our current problem, we can show an extra $\sim \frac{1}{2}$-smoothing for , uniformly in $N \in {\mathbb {N}}$, and for the limit and thus they have positive regularity. See Lemma 3.1. As in [12, 26], such multilinear smoothing plays a fundamental role in our analysis.

Let us now start with the truncated renormalized SNLW (1.7) and obtain the limiting formulation of our problem. By proceeding with the second order expansion:

(1.17)

we rewrite (1.7) as

(1.18)

where we used (1.14). The main problem in studying singular stochastic PDEs lies in making sense of various products. In this formal discussion, let us apply the following “rules”:

• A product of functions of regularities $s_1$ and $s_2$ is defined if $s_1 + s_2 > 0$. When $s_1 > 0$ and $s_1 \ge s_2$, the resulting product has regularity $s_2$.

• A product of stochastic objects (not depending on the unknown) is always well defined, possibly with a renormalization. The product of stochastic objects of regularities $s_1$ and $s_2$ has regularity $\min ( s_1, s_2, s_1 + s_2)$.

We postulate that the unknown v has regularity $\frac{1}{2}+$,¹³ which is subcritical with respect to the standard scaling heuristics for the three-dimensional cubic NLW. In order to close the Picard iteration argument, we need all the terms on the right-hand side of (1.18) to have regularity $-\frac{1}{2}+$. With the aforementioned regularities of the stochastic terms , , and and applying the rules above, we can handle the products on the right-hand side of (1.18), giving regularity $-\frac{1}{2}+$, except for the following terms (for small $\alpha >0$):

(1.19)

As for the first term , we first use stochastic analysis to make sense of with regularity $\alpha -\frac{1}{2} -$, uniformly in $N \in {\mathbb {N}}$, (see Lemma 3.3) and then interpret the product as

Note that the right-hand side is well defined since the sum of the regularities is positive: $(\alpha -\frac{1}{2} -) + (\frac{1}{2} +) > 0$. The last product in (1.19) makes sense but the resulting regularity is $2\alpha - 1-$, smaller than the required regularity $-\frac{1}{2}+$, when $\alpha$ is close to 0. As for the second term in (1.19), it depends on the unknown $v_N$ and thus the product does not make sense (at this point) since the sum of regularities is negative (when $\alpha > 0$ is small).

As we see below, by studying the last two terms in (1.19) under the Duhamel integral operator ${\mathcal {I}}$, we can indeed give a meaning to them and exhibit extra $(\frac{1}{2}+)$-smoothing with the resulting regularity $\frac{1}{2}+$ (under ${\mathcal {I}}$), which allows us to close the argument. By writing (1.18) with initial data $(u_0, u_1)$ in the Duhamel formulation, we have

(1.20)

where $S(t) (u_0, u_1) = \cos (t\langle \nabla \rangle ) u_0 + \frac{\sin (t\langle \nabla \rangle )}{\langle \nabla \rangle }u_1$ denotes the (deterministic) linear solution. Here, denotes the random operator defined by

(1.21)

and (as the notation suggests), the last term in (1.20) is defined by

(1.22)

(without a renormalization). By exploiting random multilinear dispersion, we show that

• the random operator maps functions of regularity $\frac{1}{2}+$ to those of regularity $\frac{1}{2}+$ (measured in the $X^{s, b}$-spaces) with the operator norm uniformly bounded in $N \in {\mathbb {N}}$ and converges to some limit, denoted by , as $N \rightarrow \infty$. We study the random operator via the random matrix approach [9, 10, 12, 18, 59].¹⁴ See Lemma 3.5.

• the third order process has regularity $\frac{1}{2} +$ (measured in the $X^{s, b}$-spaces) with the norm uniformly bounded in $N \in {\mathbb {N}}$ and converges to some limit, denoted by  , as $N \rightarrow \infty$. See Lemma 3.4.

We deduce these claims as corollaries to Bringmann’s work [12]. In [12], the smoothing coming from the potential $V = \langle \nabla \rangle ^{-\beta }$ in the Hartree nonlinearity $(V*u^2)u$ played an important role. In our problem, this is replaced by the smoothing $\langle \nabla \rangle ^{-\alpha }$ on the noise and we reduce our problem to that in [12], essentially by the following simple observation:

$$\begin{aligned} \prod _{j = 1}^k \langle n_j \rangle ^{-\gamma } \lesssim \langle n_1+ \cdots + n_k \rangle ^{-\gamma } \end{aligned}$$

(1.23)

for any $\gamma \ge 0$.

Remark 1.7

In the following, we also set

(1.24)

By carrying out analysis analogous to (but more involved than) that for studied in Lemma 3.3 below, we can show that forms a Cauchy sequence in $C([0,T];W^{\alpha - \frac{1}{2} - ,\infty }({\mathbb {T}}^3))$ almost surely, thus converging to some limit . In this paper, however, we proceed with space-time analysis as in [12]. Namely, we study in the $X^{s, b}$-spaces and show that it converges to some limit denoted by . See Lemma 3.4.

Putting everything together, we can take $N \rightarrow \infty$ in (1.20) and obtain the following limiting equation for :

(1.25)

By the Fourier restriction norm method with the Strichartz estimates, we can then prove local well-posedness of (1.25) in the deterministic manner. Namely, given the following enhanced data set

(1.26)

of appropriate regularities (depicted by stochastic analysis), there exists a unique local-in-time solution v to (1.25), continuously depending on the enhanced data set $\Xi$. See Proposition 3.7 for a precise statement.

This local well-posedness result together with the convergence of and then yields the convergence of in (1.17) to the limiting process

where v is the solution to (1.25).

Remark 1.8

In terms of regularity counting, the sum of the regularities in is positive. In the parabolic setting, one may then proceed with a product estimate. In the current dispersive setting, however, integrability of functions plays an important role and thus we need to proceed with care. See Lemmas 2.7 and 3.6.

Remark 1.9

(i) By the use of stochastic analysis, the stochastic terms , , , , , and in the enhanced data set are defined as the unique limits of their truncated versions. Furthermore, by deterministic analysis, we prove that a solution v to (1.25) is pathwise unique in an appropriate class. Therefore, under the decomposition , the uniqueness of u refers to (a) the uniqueness of and as the limits of and and (b) the uniqueness of v as a solution to (1.25).

(ii) In this paper, we work with the frequency projector $\pi _N$ with a sharp cutoff function on the frequency side. It is also possible to work with smooth mollifiers $\eta _{\delta }(x) = \delta ^{-3}\eta (\delta ^{-1}x)$, where $\eta \in C^\infty ({\mathbb {R}}^3 ; [0, 1])$ is a smooth, non-negative, even function with $\int \eta dx = 1$ and ${{\,\mathrm{supp}\,}}\eta \subset (-\pi , \pi ]^3\simeq {\mathbb {T}}^3$. In this case, working with

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u_\delta + (1- \Delta ) u_\delta + u_\delta ^3 - 3\sigma _\delta u_\delta = \langle \nabla \rangle ^{-\alpha } \eta _\delta * \xi \\ (u_\delta , \partial _tu_\delta )|_{t = 0} = (u_0, u_1), \end{array}\right. } \end{aligned}$$

(1.27)

we can show that a solution $u_\delta$ to (1.27) converges in probability to some limit u in $C([-T_\omega , T_\omega ]; H^{\alpha -\frac{1}{2} -\varepsilon } ({\mathbb {T}}^3))$ as $\delta \rightarrow 0$. Furthermore, the limit $u_\delta$ is independent of the choice of a mollification kernel $\eta$ and agrees with the limiting process u constructed in Theorem 1.1. This is the second meaning of the uniqueness of the limiting process u.

Remark 1.10

• (i) From the “scaling” point of view, our problem for $0 < \alpha \ll 1$ is more difficult than the quadratic SNLW (1.8) considered in [26], where the paracontrolled calculus played an essential role. On the other hand, for the proof of Theorem 1.1, we do not need to use the paracontrolled ansatz for the remainder terms thanks to the smoothing on the noise and the use of space-time estimates, which allows us to place v in the subcritical regularity $\frac{1}{2} +$.

  Our approach to (1.6) and Bringmann’s approach in [12] crucially exploit various multilinear smoothing, gaining $\sim \frac{1}{2}$-derivative. When $\alpha = 0$ (or $\beta = 0$ in the Hartree SNLW (1.9)), such multilinear smoothing seems to give (at best) $\frac{1}{2}$-smoothing and thus the arguments in this paper and in [12] break down in the $\alpha = 0$ case.

• (ii) In [26], Gubinelli, Koch, and the first author studied the quadratic SNLW on ${\mathbb {T}}^3$ with an additive space-time white noise (i.e. $\alpha = 0$):

  $$\begin{aligned} \partial _t^2 u + (1 - \Delta ) u + u^2 = \xi . \end{aligned}$$

  (1.28)

  With the Wick renormalization and the second order expansion , where , the remainder term satisfies

  

  (1.29)

  As observed in [26], the main issue in studying (1.29) comes from the regularity $\frac{1}{2} -$ of v, which is inherited from the regularity $-\frac{1}{2} -$ of . As a result, the product in (1.29) is not well defined since the sum of the regularities of and v is negative. As in (1.21), it is tempting to directly define the random operator , using the random matrix estimates. However, there is an issue in handling the “high $\times$ high $\rightarrow$ low” interaction and thus the random matrix approach alone is not sufficient to close the argument. In [26], this issue was overcome by a paracontrolled ansatz and an iteration of the Duhamel formulation. We point out that the use of the paracontrolled ansatz in [26] led to the following paracontrolled operator , which avoids the undesirable high $\times$ high $\rightarrow$ low interaction. Instead of the paracontrolled calculus, one may use the random averaging operator from [17] together with an iteration of the Duhamel formulation. We, however, point out that due to the problematic high $\times$ high interaction, the random averaging operator as introduced in [17] alone (without iterating the Duhamel formulation) does not seem to be sufficient to study the quadratic SNLW (1.28).

Organization of the paper In Sect. 2, we go over the basic definitions and lemmas from deterministic and stochastic analysis. In Sect. 3, we first state the almost sure regularity and convergence properties of (the truncated versions of) the stochastic objects in the enhanced data set $\Xi$ in (1.26). Then, we present the proof of our main result (Theorem 1.1). In Sect. 4, we establish the almost sure regularity and convergence properties of the stochastic objects in the enhanced data set. In Appendix A, we recall the counting lemmas from [12] which play a crucial role in Sect. 4. In Appendices B and C, we provide the basic definitions and lemmas on multiple stochastic integrals and (random) tensors, respectively.

2 Notations and basic lemmas

We write $A \lesssim B$ to denote an estimate of the form $A \le CB$. Similarly, we write $A \sim B$ to denote $A \lesssim B$ and $B \lesssim A$ and use $A \ll B$ when we have $A \le c B$ for small $c > 0$. We also use $a+$ (and $a-$) to mean $a + \varepsilon$ (and $a-\varepsilon$, respectively) for arbitrarily small $\varepsilon >0$.

When we work with space-time function spaces, we use short-hand notations such as $C_T H^s_x = C([0, T]; H^s({\mathbb {T}}^3))$.

When there is no confusion, we simply use $\widehat{u}$ or ${\mathcal {F}}(u)$ to denote the spatial, temporal, or space-time Fourier transform of u, depending on the context. We also use ${\mathcal {F}}_x$, ${\mathcal {F}}_t$, and ${\mathcal {F}}_{x, t}$ to denote the spatial, temporal, and space-time Fourier transforms, respectively.

We use the following short-hand notation: $n_{ij} = n_i + n_j$, etc. For example, $n_{123} = n_1 + n_2 + n_3$.

2.1 Sobolev spaces and Besov spaces

Let $s \in {\mathbb {R}}$ and $1 \le p \le \infty$. We define the $L^2$-based Sobolev space $H^s({\mathbb {T}}^3)$ by the norm:

$$\begin{aligned} \Vert f \Vert _{H^s} = \Vert \langle n \rangle ^s \widehat{f} (n) \Vert _{\ell ^2_n} \end{aligned}$$

and set ${\mathcal {H}}^s({\mathbb {T}}^3)$ to be

$$\begin{aligned} {\mathcal {H}}^s({\mathbb {T}}^3) = H^s({\mathbb {T}}^3)\times H^{s-1}({\mathbb {T}}^3). \end{aligned}$$

We also define the $L^p$-based Sobolev space $W^{s, p}({\mathbb {T}}^3)$ by the norm:

$$\begin{aligned} \Vert f \Vert _{W^{s, p}} = \big \Vert {\mathcal {F}}^{-1} (\langle n \rangle ^s \widehat{f}(n))\big \Vert _{L^p}. \end{aligned}$$

When $p = 2$, we have $H^s({\mathbb {T}}^3) = W^{s, 2}({\mathbb {T}}^3)$.

Let $\phi :{\mathbb {R}}\rightarrow [0, 1]$ be a smooth bump function supported on $\big [-\frac{8}{5}, \frac{8}{5}\big ]$ and $\phi \equiv 1$ on $\big [-\frac{5}{4}, \frac{5}{4}\big ]$. For $\xi \in {\mathbb {R}}^3$, we set $\phi _0(\xi ) = \phi (|\xi |)$ and

$$\begin{aligned} \phi _{j}(\xi ) = \phi \big (\tfrac{|\xi |}{2^j}\big )-\phi \big (\tfrac{|\xi |}{2^{j-1}}\big ) \end{aligned}$$

for $j \in {\mathbb {N}}$. Note that we have

$$\begin{aligned} \sum _{j \in {\mathbb {N}}_0} \phi _j(\xi ) = 1 \end{aligned}$$

(2.1)

for any $\xi \in {\mathbb {R}}^3$. Then, for $j \in {\mathbb {N}}_0 := {\mathbb {N}}\cup \{0\}$, we define the Littlewood-Paley projector ${\mathbf {P}}_j$ as the Fourier multiplier operator with a symbol $\phi _j$. Thanks to (2.1), we have

$$\begin{aligned} f = \sum _{j = 0}^\infty {\mathbf {P}}_j f. \end{aligned}$$

(2.2)

Next, we recall the following paraproduct decomposition due to Bony [6]. See [1, 24] for further details. Let f and g be functions on ${\mathbb {T}}^3$ of regularities $s_1$ and $s_2$, respectively. Using (2.2), we write the product fg as

(2.3)

The first term (and the third term ) is called the paraproduct of g by f (the paraproduct of f by g, respectively) and it is always well defined as a distribution of regularity $\min (s_2, s_1+ s_2)$. On the other hand, the resonant product is well defined in general only if $s_1 + s_2 > 0$.

We briefly recall the basic properties of the Besov spaces $B^s_{p, q}({\mathbb {T}}^3)$ defined by the norm:

$$\begin{aligned} \Vert u \Vert _{B^s_{p,q}} = \Big \Vert 2^{s j} \Vert {\mathbf {P}}_{j} u \Vert _{L^p_x} \Big \Vert _{\ell ^q_j({\mathbb {N}}_0)}. \end{aligned}$$

Note that $H^s({\mathbb {T}}^3) = B^s_{2,2}({\mathbb {T}}^3)$.

Lemma 2.1

• (i) (paraproduct and resonant product estimates) Let $s_1, s_2 \in {\mathbb {R}}$ and $1 \le p, p_1, p_2, q \le \infty$ such that $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$. Then, we have

  

  (2.4)

  When $s_1 < 0$, we have

  

  (2.5)

  When $s_1 + s_2 > 0$, we have

  

  (2.6)

• (ii) Let $s_1 < s_2$ and $1\le p, q \le \infty$. Then, we have

  $$\begin{aligned} \Vert u \Vert _{B^{s_1}_{p,q}}&\lesssim \Vert u \Vert _{W^{s_2, p}}. \end{aligned}$$

  (2.7)

The product estimates (2.4), (2.5), and (2.6) follow easily from the definition (2.3) of the paraproduct and the resonant product. See [1, 39] for details of the proofs in the non-periodic case (which can be easily extended to the current periodic setting). The embedding (2.7) follows from the $\ell ^{q}$-summability of $\big \{2^{(s_1 - s_2)j}\big \}_{j \in {\mathbb {N}}_0}$ for $s_1 < s_2$ and the uniform boundedness of the Littlewood-Paley projector ${\mathbf {P}}_j$.

We also recall the following product estimate from [25].

Lemma 2.2

Let $0\le s\le 1$. Let $1<p,q,r<\infty$ such that $s \ge 3\big (\frac{1}{p}+\frac{1}{q}-\frac{1}{r}\big )$. Then, we have

$$\begin{aligned} \Vert \langle \nabla \rangle ^{-s}(fg)\Vert _{L^r({\mathbb {T}}^3)} \lesssim \Vert \langle \nabla \rangle ^{-s} f\Vert _{L^{p}({\mathbb {T}}^3)} \Vert \langle \nabla \rangle ^{s} g\Vert _{L^{q}({\mathbb {T}}^3)} . \end{aligned}$$

Note that while Lemma 2.2 was shown only for $s = 3\big (\frac{1}{p}+\frac{1}{q}-\frac{1}{r}\big )$ in [25], the general case $s \ge 3\big (\frac{1}{p}+\frac{1}{q}-\frac{1}{r}\big )$ follows the embedding $L^{r_1}({\mathbb {T}}^3) \subset L^{r_2}({\mathbb {T}}^3)$, $r_1 \ge r_2$.

2.2 Fourier restriction norm method and Strichartz estimates

We first recall the so-called $X^{s, b}$-spaces, also known as the hyperbolic Sobolev spaces, due to Klainerman-Machedon [34] and Bourgain [7], defined by the norm:

$$\begin{aligned} \Vert u\Vert _{X^{s, b} ( {\mathbb {T}}^3 \times {\mathbb {R}})} = \Vert \langle n \rangle ^s \langle |\tau |- \langle n \rangle \rangle ^b \widehat{u}(n, \tau )\Vert _{\ell ^2_n L^2_\tau ( {\mathbb {Z}}^3\times {\mathbb {R}})}. \end{aligned}$$

(2.8)

For $b > \frac{1}{2}$, we have $X^{s, b} \subset C({\mathbb {R}}; H^s({\mathbb {T}}^3))$. Given an interval $I \subset {\mathbb {R}}$, we define the local-in-time version $X^{s, b}(I)$ as a restriction norm:

$$\begin{aligned} \Vert u \Vert _{X^{s, b}(I)} = \inf \big \{ \Vert v\Vert _{X^{s, b}({\mathbb {T}}^3 \times {\mathbb {R}})}: \, v|_I = u\big \}. \end{aligned}$$

(2.9)

When $I = [0, T]$, we set $X^{s, b}_T = X^{s, b}(I)$.

Next, we recall the Strichartz estimates for the linear wave/Klein–Gordon equation. Given $0 \le s \le 1$, we say that a pair (q, r) is s-admissible if $2 < q \le \infty$, $2 \le r < \infty$,

$$\begin{aligned} \frac{1}{q} + \frac{3}{r} = \frac{3}{2} - s \quad \text {and} \quad \frac{1}{q} + \frac{1}{r} \le \frac{1}{2}. \end{aligned}$$

Then, we have the following Strichartz estimates.

Lemma 2.3

Given $0 \le s \le 1$, let (q, r) be s-admissible. Then, we have

$$\begin{aligned} \Vert S(t)(\phi _0, \phi _1) \Vert _{L^q_TL^r_x({\mathbb {T}}^3)} \lesssim \Vert (\phi _0 ,\phi _1) \Vert _{{\mathcal {H}}^{s}({\mathbb {T}}^3)} \end{aligned}$$

(2.10)

for any $0 < T \le 1$.

See Ginibre–Velo [23], Lindblad–Sogge [38], and Keel–Tao [32] for the Strichartz estimates on ${\mathbb {R}}^d$. See also [33]. The Strichartz estimates (2.10) on ${\mathbb {T}}^3$ in Lemma 2.3 follows from those on ${\mathbb {R}}^3$ and the finite speed of propagation.

When $b > \frac{1}{2}$, the $X^{s, b}$-spaces enjoy the transference principle. In particular, as a corollary to Lemma 2.3, we obtain the following space-time estimate. See [35, 64] for the proof.

Lemma 2.4

Let $0 < T \le 1$. Given $0 \le s \le 1$, let (q, r) be s-admissible. Then, for $b > \frac{1}{2}$, we have

$$\begin{aligned} \Vert u \Vert _{L^q_T L^r_x} \lesssim \Vert u\Vert _{X_T^{s, b}}. \end{aligned}$$

We also state the nonhomogeneous linear estimate. See [22].

Lemma 2.5

Let $- \frac{1}{2} < b' \le 0 \le b \le b'+1$. Then, for $0 < T \le 1$, we have

$$\begin{aligned} \Vert {\mathcal {I}}(F)\Vert _{X^{s, b}_T}= \bigg \Vert \int _0^t \frac{\sin ((t-t')\langle \nabla \rangle )}{\langle \nabla \rangle } F(t') dt'\bigg \Vert _{X^{s, b}_T} \lesssim T^{1-b+b'} \Vert F\Vert _{X^{s-1, b'}_T}. \end{aligned}$$

In the following, we briefly go over the main trilinear estimate for the basic local well-posedness of the cubic NLW (1.5) in ${\mathcal {H}}^{\frac{1}{2}+\varepsilon }({\mathbb {T}}^3)$.

Lemma 2.6

Fix small $\delta _1, \delta _2 > 0$ with $4\delta _2 \le \delta _1$. Then, we have

$$\begin{aligned} \Vert {\mathcal {I}}(u_1u_2u_3)\Vert _{X^{\frac{1}{2} + \delta _1 , \frac{1}{2} + \delta _2}_T} \lesssim T^{\delta _2}\prod _{j = 1}^3 \Vert u_j\Vert _{X_T^{\frac{1}{2} +\delta _1, \frac{1}{2}+\delta _2}} \end{aligned}$$

(2.11)

for any $0 < T\le 1$.

Proof

Recall that $(q, r) = (4, 4)$ is $\frac{1}{2}$-admissible. Then, in view of Lemma 2.4, interpolating

$$\begin{aligned} \Vert u \Vert _{L^4_{T, x}} \lesssim \Vert u\Vert _{X_T^{\frac{1}{2} , \frac{1}{2}+\delta _0}} \quad \text {and} \quad \Vert u \Vert _{L^2_{T, x}} = \Vert u\Vert _{X_T^{0, 0}} \end{aligned}$$

(2.12)

with small $\delta _0 > 0$, we obtain

$$\begin{aligned} \Vert u \Vert _{L^\frac{4}{1+2\delta _1}_{T, x}} \lesssim \Vert u\Vert _{X_T^{\frac{1}{2} -\delta _1, \frac{1}{2}-\frac{1}{2}\delta _1}}. \end{aligned}$$

(2.13)

Moreover, noting that $\big (\frac{12}{3-2\delta _1},\frac{12}{3-2\delta _1}\big )$ is $\big (\frac{1}{2} + \frac{2}{3} \delta _1\big )$-admissible, we obtain from Lemma 2.4 that

$$\begin{aligned} \Vert u \Vert _{L^\frac{12}{3-2\delta _1}_{T, x}} \le C_{\delta _1, \delta _2} \Vert u\Vert _{X_T^{\frac{1}{2} +\frac{2}{3} \delta _1, \frac{1}{2}+\delta _2}} \end{aligned}$$

(2.14)

for any $\delta _2 > 0$.

Hence, from Lemma 2.5, duality, Hölder’s inequality, (2.13), and (2.14), we obtain

$$\begin{aligned} \Vert {\mathcal {I}}(u_1u_2u_3)\Vert _{X^{\frac{1}{2} + \delta _1 , \frac{1}{2} + \delta _2}_T}&\lesssim T^{\delta _2} \Vert u_1u_2u_3\Vert _{X^{-\frac{1}{2} + \delta _1, - \frac{1}{2} + 2\delta _2}}\\&= T^{\delta _2} \sup _{\Vert w\Vert _{X^{\frac{1}{2} - \delta _1, \frac{1}{2} - 2\delta _2}}= 1} \bigg |\int _0^T \int _{{\mathbb {T}}^3} u_1u_2u_3 w dx dt\bigg |\\&\le T^{\delta _2} \sup _{\Vert w\Vert _{X^{\frac{1}{2} - \delta _1, \frac{1}{2} - 2\delta _2}}= 1} \bigg ( \prod _{j = 1}^3 \Vert u_j \Vert _{L^\frac{12}{3-2\delta _1}_{T, x}}\bigg ) \Vert w \Vert _{L^\frac{4}{1+2\delta _1}_{T, x}} \\&\lesssim T^{\delta _2} \prod _{j = 1}^3 \Vert u_j\Vert _{X_T^{\frac{1}{2} +\frac{2}{3} \delta _1, \frac{1}{2}+\delta _2}}, \end{aligned}$$

provided that $0 < 4\delta _2 \le \delta _1 \ll 1$. This proves (2.11). $\square$

We conclude this part by establishing the following trilinear estimate, which will be used to control the term in (1.25). See Proposition 8.6 in [12] for an analogous trilinear estimate.

Lemma 2.7

Let $\delta _1, \delta _2 > 0$ be sufficiently small such that $8 \delta _2 \le \delta _1$. Then, we have

$$\begin{aligned} \Vert u_1 u_2 u_3 \Vert _{X^{-\frac{1}{2} +\delta _1, -\frac{1}{2}+2\delta _2}_T} \lesssim \Vert u_1\Vert _{L^\infty _T W^{ - \frac{1}{2} + 2\delta _1, \infty }_x} \Vert u_2\Vert _{X^{\frac{1}{2} + \delta _1, \frac{1}{2} + \delta _2 }_T} \Vert u_3 \Vert _{X^{\frac{1}{2} + \delta _1 , \frac{1}{2} + \delta _2}_T} \end{aligned}$$

(2.15)

for any $0 < T \le 1$.

Proof

By applying the Littlewood-Paley decompositions, we have

$$\begin{aligned}&\text {LHS of } (2.15) \\&\quad \le \sum _{j_1, j_{23}, j_{123} = 0}^\infty \big \Vert {\mathbf {P}}_{j_{123}}\big ({\mathbf {P}}_{j_1} u_1 {\mathbf {P}}_{j_{23}}(u_2 u_3)\big ) \big \Vert _{X^{-\frac{1}{2} +\delta _1, -\frac{1}{2}+2\delta _2}_T}. \end{aligned}$$

For simplicity of notation, we set $N_1 = 2^{j_1}$, $N_{23} = 2^{j_{23}}$, and $N_{123} = 2^{j_{123}}$, denoting the dyadic frequency sizes of $n_1$ (for $u_1$), $n_{23}$ (for $u_2u_3$), and $n_{123}$ (for $u_1 u_2 u_3$), respectively. We set $v_k = {\mathbf {P}}_{j_k} u_k$. In view of $n_{123} = n_1 + n_{23}$, we separately estimate the contributions from (i)  $N_{123} \sim \max (N_1, N_{23})$ and (ii) $N_{123} \ll \max (N_1, N_{23})$.

Case 1: $N_{123} \sim \max (N_1, N_{23})$.

By Hölder’s inequality and the $L^4$-Strichartz estimate (2.12), we have

$$\begin{aligned}&\big \Vert {\mathbf {P}}_{j_{123}} \big (v_1 {\mathbf {P}}_{j_{23}} (u_2 u_3)\big ) \big \Vert _{X^{-\frac{1}{2} +\delta _1, -\frac{1}{2}+2\delta _2}_T} \lesssim N_{123}^{-\frac{1}{2} + \delta _1} \Vert v_1 {\mathbf {P}}_{j_{23}}(u_2 u_3) \Vert _{L^2_{T, x}}\\&\quad \lesssim N_{123}^{- \delta _1} \Vert u_1 \Vert _{L^\infty _T W^{-\frac{1}{2} + 2\delta _1, \infty }_x} \prod _{j= 2}^3\Vert u_j \Vert _{L^4_{T, x}}\\&\quad \lesssim N_{123}^{- \delta _1} \Vert u_1\Vert _{L^\infty _T W^{ - \frac{1}{2} +2 \delta _1, \infty }_x} \prod _{j = 2}^3\Vert u_j\Vert _{X^{\frac{1}{2} , \frac{1}{2} + \delta _2}_T}. \end{aligned}$$

This is summable in dyadic $N_1, N_{23}, N_{123}\ge 1$, yielding (2.15) in this case.

Case 2: $N_{123} \ll \max (N_1, N_{23})$.

In this case, we further apply the Littlewood-Paley decompositions for $u_2$ and $u_3$ and write

$$\begin{aligned} u_2 u_3 = \sum _{j_2, j_3= 0}^\infty ({\mathbf {P}}_{j_2} u_2)({\mathbf {P}}_{j_3} u_3). \end{aligned}$$

Without loss of generality, assume $N_3 \ge N_2$, where $N_k = 2^{j_k}$, $k = 2, 3$. Then, we have

$$\begin{aligned} N_{123} \lesssim N_1 \sim N_{23} \lesssim N_3. \end{aligned}$$

(2.16)

By duality and (2.13) (with $\delta _1 = 4\delta _2$), we have

$$\begin{aligned} \Vert {\mathbf {P}}_j u \Vert _{X^{0, - \frac{1}{2} + 2\delta _2}_T}= & {} \sup _{\Vert v \Vert _{X^{0, \frac{1}{2} - 2\delta _2}}= 1}\bigg | \int _0^T\int _{{\mathbb {T}}^3} ({\mathbf {P}}_j u) \big (({\mathbf {P}}_{j-1} + {\mathbf {P}}_j + {\mathbf {P}}_{j+1})v\big ) dx dt \bigg | \nonumber \\\lesssim & {} 2^{(\frac{1}{2} - 4\delta _2)j} \Vert {\mathbf {P}}_j u \Vert _{L^\frac{4}{3-8 \delta _2}_{T, x}}. \end{aligned}$$

(2.17)

Then, from (2.17), (2.14), and (2.16) with $8 \delta _2 \le \delta _1$, we have

$$\begin{aligned}&\big \Vert {\mathbf {P}}_{j_{123}} \big (v_1 {\mathbf {P}}_{j_{23}} (v_2 v_3)\big ) \big \Vert _{X^{-\frac{1}{2} +\delta _1, -\frac{1}{2}+2\delta _2}_T} \lesssim N_{123}^{-\frac{1}{2} + \delta _1} N_{123}^{\frac{1}{2} - 4\delta _2} \Vert v_1 {\mathbf {P}}_{j_{23}} (v_2 v_3) \Vert _{L^\frac{4}{3-8 \delta _2}_{T, x}}\\&\quad \lesssim N_{123}^{\delta _1 - 4\delta _2} N_1^{\frac{1}{2} - 2\delta _1} \Vert v_1 \Vert _{L^\infty _T W^{-\frac{1}{2} + 2 \delta _1, \infty }_x} \Vert v_2 \Vert _{L^\frac{4}{1 - 8 \delta _2}_{T, x}} \Vert v_3 \Vert _{L^2_{T, x}}\\&\quad \lesssim N_{123}^{ \delta _1 - 4\delta _2} N_1^{\frac{1}{2} - 2\delta _1} N_3^{-\frac{1}{2} - \delta _1}\\&\qquad \times \Vert u_1 \Vert _{L^\infty _T W^{-\frac{1}{2} + 2 \delta _1, \infty }_x} \Vert u_2 \Vert _{X^{\frac{1}{2} + 8 \delta _2 , \frac{1}{2} + \delta _2}_T} \Vert u_3 \Vert _{X^{\frac{1}{2} + \delta _1 , 0}_T}\\&\quad \lesssim N_{123}^{ - 4\delta _2} N_1^{- \delta _1} N_3^{ - \delta _1} \Vert u_1 \Vert _{L^\infty _T W^{-\frac{1}{2} + 2 \delta _1, \infty }_x} \prod _{j = 2}^3 \Vert u_j \Vert _{X^{\frac{1}{2} + \delta _1 , \frac{1}{2} + \delta _2}_T}. \end{aligned}$$

This is summable in dyadic $N_1, N_2, N_3, N_{23}, N_{123}\ge 1$, yielding (2.15) in this case. $\square$

2.3 On discrete convolutions

Next, we recall the following basic lemma on a discrete convolution.

Lemma 2.8

• (i) Let $d \ge 1$ and $\alpha , \beta \in {\mathbb {R}}$ satisfy

  $$\begin{aligned} \alpha + \beta > d \quad \text {and}\quad \alpha , \beta < d. \end{aligned}$$

  Then, we have

  $$\begin{aligned} \sum _{n = n_1 + n_2} \frac{1}{\langle n_1 \rangle ^\alpha \langle n_2 \rangle ^\beta } \lesssim \langle n \rangle ^{d - \alpha - \beta } \end{aligned}$$

  for any $n \in {\mathbb {Z}}^d$.

• (ii) Let $d \ge 1$ and $\alpha , \beta \in {\mathbb {R}}$ satisfy $\alpha + \beta > d$. Then, we have

  $$\begin{aligned} \sum _{\begin{array}{c} n = n_1 + n_2\\ |n_1|\sim |n_2| \end{array}} \frac{1}{\langle n_1 \rangle ^\alpha \langle n_2 \rangle ^\beta } \lesssim \langle n \rangle ^{d - \alpha - \beta }\end{aligned}$$

  for any $n \in {\mathbb {Z}}^d$.

Namely, in the resonant case (ii), we do not have the restriction $\alpha , \beta < d$. Lemma 2.8 follows from elementary computations. See, for example, Lemmas 4.1 and 4.2 in [41] for the proof.

2.4 Tools from stochastic analysis

We conclude this section by recalling useful lemmas from stochastic analysis. See [5, 43, 62] for basic definitions. See also Appendix B for basic definitions and properties for multiple stochastic integrals.

Let $(H, B, \mu )$ be an abstract Wiener space. Namely, $\mu$ is a Gaussian measure on a separable Banach space B with $H \subset B$ as its Cameron-Martin space. Given a complete orthonormal system $\{e_j \}_{ j \in {\mathbb {N}}} \subset B^*$ of $H^* = H$, we define a polynomial chaos of order k to be an element of the form $\prod _{j = 1}^\infty H_{k_j}(\langle x, e_j \rangle )$, where $x \in B$, $k_j \ne 0$ for only finitely many j’s, $k= \sum _{j = 1}^\infty k_j$, $H_{k_j}$ is the Hermite polynomial of degree $k_j$, and denotes the B–$B^*$ duality pairing. We then denote the closure of polynomial chaoses of order k under $L^2(B, \mu )$ by ${\mathcal {H}}_k$. The elements in ${\mathcal {H}}_k$ are called homogeneous Wiener chaoses of order k. We also set

$$\begin{aligned} {\mathcal {H}}_{\le k} = \bigoplus _{j = 0}^k {\mathcal {H}}_j \end{aligned}$$

for $k \in {\mathbb {N}}$.

Let $L = \Delta -x \cdot \nabla$ be the Ornstein-Uhlenbeck operator.¹⁵ Then, it is known that any element in ${\mathcal {H}}_k$ is an eigenfunction of L with eigenvalue $-k$. Then, as a consequence of the hypercontractivity of the Ornstein-Uhlenbeck semigroup $U(t) = e^{tL}$ due to Nelson [42], we have the following Wiener chaos estimate [63, Theorem I.22]. See also [65, Proposition 2.4].

Lemma 2.9

Let $k \in {\mathbb {N}}$. Then, we have

$$\begin{aligned} \Vert X \Vert _{L^p(\Omega )} \le (p-1)^\frac{k}{2} \Vert X\Vert _{L^2(\Omega )} \end{aligned}$$

for any $p \ge 2$ and any $X \in {\mathcal {H}}_{\le k}$.

The following lemma will be used in studying regularities of stochastic objects. We say that a stochastic process $X:{\mathbb {R}}_+ \rightarrow {\mathcal {D}}'({\mathbb {T}}^d)$ is spatially homogeneous if $\{X(\cdot , t)\}_{t\in {\mathbb {R}}_+}$ and $\{X(x_0 +\cdot \,, t)\}_{t\in {\mathbb {R}}_+}$ have the same law for any $x_0 \in {\mathbb {T}}^d$. Given $h \in {\mathbb {R}}$, we define the difference operator $\delta _h$ by setting

$$\begin{aligned} \delta _h X(t) = X(t+h) - X(t). \end{aligned}$$

Lemma 2.10

Let $\{ X_N \}_{N \in {\mathbb {N}}}$ and X be spatially homogeneous stochastic processes $:{\mathbb {R}}_+ \rightarrow {\mathcal {D}}'({\mathbb {T}}^d)$. Suppose that there exists $k \in {\mathbb {N}}$ such that $X_N(t)$ and X(t) belong to ${\mathcal {H}}_{\le k}$ for each $t \in {\mathbb {R}}_+$.

• (i) Let $t \in {\mathbb {R}}_+$. If there exists $s_0 \in {\mathbb {R}}$ such that

  $$\begin{aligned} {\mathbb {E}}\big [ |\widehat{X}(n, t)|^2\big ]\lesssim \langle n \rangle ^{ - d - 2s_0} \end{aligned}$$

  (2.18)

  for any $n \in {\mathbb {Z}}^d$, then we have $X(t) \in W^{s, \infty }({\mathbb {T}}^d)$, $s < s_0$, almost surely.

• (ii) Suppose that $X_N$, $N \in {\mathbb {N}}$, satisfies (2.18). Furthermore, if there exists $\gamma > 0$ such that

  $$\begin{aligned} {\mathbb {E}}\big [ |\widehat{X}_N(n, t) - \widehat{X}_M(n, t)|^2\big ]\lesssim N^{-\gamma } \langle n \rangle ^{ - d - 2s_0} \end{aligned}$$

  for any $n \in {\mathbb {Z}}^d$ and $M \ge N \ge 1$, then $X_N(t)$ is a Cauchy sequence in $W^{s, \infty }({\mathbb {T}}^d)$, $s < s_0$, almost surely, thus converging to some limit in $W^{s, \infty }({\mathbb {T}}^d)$.

• (iii) Let $T > 0$ and suppose that (i) holds on [0, T]. If there exists $\sigma \in (0, 1)$ such that

  $$\begin{aligned} {\mathbb {E}}\big [ |\delta _h \widehat{X}(n, t)|^2\big ] \lesssim \langle n \rangle ^{ - d - 2s_0+ \sigma } |h|^\sigma \end{aligned}$$

  for any $n \in {\mathbb {Z}}^d$, $t \in [0, T]$, and $h \in [-1, 1]$,¹⁶ then we have $X \in C([0, T]; W^{s, \infty }({\mathbb {T}}^d))$, $s < s_0 - \frac{\sigma }{2}$, almost surely.

• (iv) Let $T > 0$ and suppose that (ii) holds on [0, T]. Furthermore, if there exists $\gamma > 0$ such that

  $$\begin{aligned} {\mathbb {E}}\big [ |\delta _h \widehat{X}_N(n, t) - \delta _h \widehat{X}_M(n, t)|^2\big ] \lesssim N^{-\gamma }\langle n \rangle ^{ - d - 2s_0+ \sigma } |h|^\sigma \end{aligned}$$

for any $n \in {\mathbb {Z}}^d$, $t \in [0, T]$, $h \in [-1, 1]$, and $M\ge N \ge 1$, then $X_N$ is a Cauchy sequence in $C([0, T]; W^{s, \infty }({\mathbb {T}}^d))$, $s < s_0 - \frac{\sigma }{2}$, almost surely, thus converging to some process in $C([0, T]; W^{s, \infty }({\mathbb {T}}^d))$.

Lemma 2.10 follows from a straightforward application of the Wiener chaos estimate (Lemma 2.9). For the proof, see Proposition 3.6 in [41] and Appendix in [50]. As compared to Proposition 3.6 in [41], we made small adjustments. In studying the time regularity, we made the following modifications: $\langle n \rangle ^{ - d - 2s_0+ 2\sigma }\mapsto \langle n \rangle ^{ - d - 2s_0+ \sigma }$ and $s< s_0 - \sigma \mapsto s < s_0 - \frac{\sigma }{2}$ so that it is suitable for studying the wave equation. Moreover, while the result in [41] is stated in terms of the Besov-Hölder space ${\mathcal {C}}^s({\mathbb {T}}^d) = B^s_{\infty , \infty }({\mathbb {T}}^d)$, Lemma 2.10 handles the $L^\infty$-based Sobolev space $W^{s, \infty }({\mathbb {T}}^3)$. Note that the required modification of the proof is straightforward since $W^{s, \infty }({\mathbb {T}}^d)$ and $B^s_{\infty , \infty }({\mathbb {T}}^d)$ differ only logarithmically:

$$\begin{aligned} \Vert f \Vert _{W^{s, \infty }} \le \sum _{j = 0}^\infty \Vert {\mathbf {P}}_j f \Vert _{W^{s, \infty }} \lesssim \Vert f\Vert _{B^{s+\varepsilon }_{\infty , \infty }} \end{aligned}$$

(2.19)

for any $\varepsilon > 0$. For the proof of the almost sure convergence claims, see [50].

3 Local well-posedness of SNLW, $\alpha > 0$

In this section, we present the proof of local well-posedness of (1.25) (Theorem 1.1). In Sect. 3.1, we first state the regularity and convergence properties of the stochastic objects in the enhanced data set $\Xi$ in (1.26). In Sect. 3.2, we then present a deterministic local well-posedness result by viewing elements in the enhanced data set as given (deterministic) distributions and a given (deterministic) operator with prescribed regularity properties.

3.1 On the stochastic terms

In this subsection, we state the regularity and convergence properties of the stochastic objects in (1.26) whose proofs are presented in Sect. 4.

Lemma 3.1

Let $\alpha >0$ and $T > 0$.

• (i) For any $s< \alpha - \frac{1}{2}$, defined in (1.12) is a Cauchy sequence in $C([0,T];W^{s,\infty }({\mathbb {T}}^3))$, almost surely. In particular, denoting the limit by (formally given by (1.11)), we have

  

  for any $\varepsilon >0$, almost surely.

• (ii) Let $0 < \alpha \le \frac{1}{2}$. Then, for any $s< \alpha$, defined in (1.15) is a Cauchy sequence in $C([0,T];W^{s,\infty }({\mathbb {T}}^3))$ almost surely. In particular, denoting the limit by , we have

  

  for any $\varepsilon >0$, almost surely.

Remark 3.2

• (i) As mentioned in Sect. 1, a parabolic thinking gives regularity $3\alpha - \frac{1}{2}-$ for  . Lemma 3.1 (ii) states that, when $\alpha > 0$ is small, we indeed gain about $\frac{1}{2}$-regularity by exploiting multilinear dispersion as in the quadratic case studied in [26]. We point out that our proof is based on an adaptation of Bringmann’s analysis on the corresponding term in the Hartree case [12] and thus the regularities we obtain in Lemma 3.1 (ii) as well as Lemmas 3.3, 3.4, and 3.5 may not be sharp (especially for large $\alpha >0$; see, for example, a crude bound (4.9)). They are, however, sufficient for our purpose.

• (ii) In this section, we only state almost sure convergence but the same argument also yields convergence in $L^p(\Omega )$ with an exponential tail estimate (as in [12, 27, 48]). Our goal is, however, to prove local well-posedness and thus the almost sure convergence suffices for our purpose.

Lemma 3.3

Let $0 < \alpha \le \frac{1}{2}$ and $T >0$. Let and be as in (1.12) and (1.15). Then, for any $s< \alpha - \frac{1}{2}$, is a Cauchy sequence in $C([0,T];W^{s,\infty }({\mathbb {T}}^3))$ almost surely. In particular, denoting the limit by , we have

for any $\varepsilon >0$, almost surely.

Lemma 3.4

Let $\alpha > 0$, $T > 0$, and $b > \frac{1}{2}$ be sufficiently close to $\frac{1}{2}$.

• (i) For any $s < \alpha + \frac{1}{2}$, defined in (1.22) is a Cauchy sequence in $X^{s, b}([0, T])$. In particular, denoting the limit by , we have

  

  for any $\varepsilon > 0$, almost surely.

• (ii) For any $s < \alpha + \frac{1}{2}$, defined in (1.24) is a Cauchy sequence in $X^{s, b}([0, T])$. In particular, denoting the limit by , we have

  

  for any $\varepsilon > 0$, almost surely.

Given Banach spaces $B_1$ and $B_2$, we use ${\mathcal {L}}(B_1; B_2)$ to denote the space of bounded linear operators from $B_1$ to $B_2$. We also set

$$\begin{aligned} {\mathcal {L}}^{s_1, s_2, b}_{T_0} = \bigcap _{0< T < T_0} {\mathcal {L}}\big (X^{s_1, b}([0, T]); X^{s_2, b}([0, T])\big ) \end{aligned}$$

(3.1)

endowed with the norm given by

$$\begin{aligned} \Vert S\Vert _{{\mathcal {L}}^{s_1, s_2, b}_{T_0}} = \sup _{0< T < T_0} T^{-\theta } \Vert S\Vert _{{\mathcal {L}}(X^{s_1, b}_T; X^{s_2, b}_T)} \end{aligned}$$

(3.2)

for some small $\theta >0$.

Lemma 3.5

Let $\alpha > 0$ and $T_0>0$. Then, given sufficiently small $\delta _1, \delta _2 > 0$, the sequence of the random operators defined in (1.21) is a Cauchy sequence in the class ${\mathcal {L}}^{\frac{1}{2} + \delta _1, \frac{1}{2} +\delta _1, \frac{1}{2} +\delta _2}_{T_0}$, almost surely. In particular, denoting the limit by , we have

almost surely.

The following trilinear estimate is an immediate consequence of Lemma 2.7.

Lemma 3.6

Let $\alpha > 0$. Let $\delta _1, \delta _2, \varepsilon > 0$ be sufficiently small such that $2\delta _1 + \varepsilon \le \alpha$. Then, we have

for any $0 < T \le 1$.

3.2 Proof of Theorem 1.1

In this section, we prove the following proposition. Theorem 1.1 then follows from this proposition and Lemmas 3.1 - 3.5.

Proposition 3.7

Let $\alpha > 0$, $s > \frac{1}{2}$, and $T_0 >0$. Then, there exists small $\varepsilon = \varepsilon (\alpha , s)$, $\delta _1 = \delta _1(\alpha , s)$, $\delta _2 = \delta _2(\alpha , s) >0$ such that if

• is a distribution-valued function belonging to $C([0, T_0]; W^{\alpha -\frac{1}{2} - \varepsilon , \infty }({\mathbb {T}}^3))$,

• is a distribution-valued function belonging to $C([0, T_0]; W^{\alpha - \varepsilon , \infty }({\mathbb {T}}^3))$,

• is a distribution-valued function belonging to $C([0, T_0]; W^{\alpha -\frac{1}{2} - \varepsilon , \infty }({\mathbb {T}}^3))$,

• is a function belonging to $X^{\alpha + \frac{1}{2} -\varepsilon , \frac{1}{2} +\delta _2}([0, T_0])$,

• is a function belonging to $X^{\alpha + \frac{1}{2} -\varepsilon , \frac{1}{2} +\delta _2}([0, T_0])$,

• the operator belongs to the class ${\mathcal {L}}^{\frac{1}{2} + \delta _1, \frac{1}{2} +\delta _1, \frac{1}{2} +\delta _2}_{T_0}$ defined in (3.1),

then the Eq. (1.25) is locally well-posed in ${\mathcal {H}}^{s}({\mathbb {T}}^3)$. More precisely, given any $(u_0, u_1)\in {\mathcal {H}}^{s}({\mathbb {T}}^3)$, there exist $0 < T \le T_0$ and a unique solution v to the cubic SNLW (1.25) on [0, T] in the class

$$\begin{aligned} X^{\frac{1}{2} +\delta _1, \frac{1}{2} +\delta _2}([0, T]) \subset C([0, T]; H^{\frac{1}{2} +\delta _1}({\mathbb {T}}^3)). \end{aligned}$$

Furthermore, the solution v depends continuously on the enhanced data set

(3.3)

in the class

$$\begin{aligned} {\mathcal {X}}^{s, \alpha , \varepsilon }_T&= {\mathcal {H}}^{s}({\mathbb {T}}^3) \times C([0,T]; W^{\alpha -\frac{1}{2} - \varepsilon , \infty }({\mathbb {T}}^3))\\&\quad \times C([0,T]; W^{\alpha - \varepsilon , \infty }({\mathbb {T}}^3)) \times C([0,T]; W^{\alpha - \frac{1}{2} - \varepsilon , \infty }({\mathbb {T}}^3))\\&\quad \times X^{\alpha + \frac{1}{2} -\varepsilon , \frac{1}{2} +\delta _2}([0, T]) \times X^{\alpha + \frac{1}{2} -\varepsilon , \frac{1}{2} +\delta _2}([0, T])\\&\quad \times {\mathcal {L}}\big (X^{\frac{1}{2} + \delta _1, \frac{1}{2} +\delta _2}([0, T]); X^{\frac{1}{2} +\delta _2, \frac{1}{2} +\delta _2}([0, T])\big ). \end{aligned}$$

Proof

Given $\alpha > 0$ and $s > \frac{1}{2}$, fix small $\varepsilon > 0$ such that $\varepsilon < \min (\alpha , s - \frac{1}{2} )$. Given an enhanced data set $\Xi$ as in (3.3), we set

and

where ${\mathcal {L}}^{\frac{1}{2} + \delta _1, \frac{1}{2} + \delta _1, \frac{1}{2} + \delta _2}_{T_0}$ is as in (3.2). In the following, we assume that

$$\begin{aligned} \Vert \Xi (\xi ) \Vert _{{\mathcal {Y}}^{\alpha , \varepsilon }_{T_0}} \le K \end{aligned}$$

(3.4)

for some $K \ge 1$.

Given the enhanced data set $\Xi$ in (3.3), define a map $\Gamma _\Xi$ by

Fix $0 < T \le T_0$. From Lemmas 2.5 and 2.4 with (3.4), we have

(3.5)

and

(3.6)

for some $\theta > 0$. Similarly, we have

(3.7)

From Lemma 2.5 and Lemma 2.2 with (3.4), we have

(3.8)

provided that $\delta _1 + \varepsilon \le \alpha$. From (3.2) and (3.4), we have

(3.9)

Hence, by applying Lemmas 2.3 and 2.5, then Lemma 2.6, (3.5), Lemma 3.6, (3.6), (3.8), (3.9), (3.7), and Lemma 3.4 with (3.4), we have

$$\begin{aligned} \Vert \Gamma _\Xi ( v)\Vert _{X^{\frac{1}{2} +\delta _1, \frac{1}{2} +\delta _2}_T}&\lesssim \Vert (u_0, u_1)\Vert _{{\mathcal {H}}^s} + T^\theta \Big ( \Vert v\Vert _{X^{\frac{1}{2} +\delta _1, \frac{1}{2} +\delta _2}_T}^3 + K^3 \Big ) + K. \end{aligned}$$

An analogous computation yields a difference estimate on $\Gamma _\Xi ( v_1) - \Gamma _\Xi ( v_2)$. Therefore, Proposition 3.7 follows from a standard contraction argument. $\square$

4 Regularities of the stochastic terms

In this section, we present the proof of Lemmas 3.1 - 3.5, which are basic tools in applying Proposition 3.7 to finally prove Theorem 1.1. In view of the local well-posedness result in [51], we assume that $0 < \alpha \le \frac{1}{4}$ in the following. Without loss of generality, we assume that $T \le 1$. The main tools in this section are the counting estimates from [12, Section 4] and the random matrix estimate (see Lemma C.3 below) from [18], which capture the multilinear dispersive effect of the wave equation. For readers’ convenience, we collect the relevant counting estimates in Appendix A and the relevant definitions and estimates for random matrices and tensors in Appendix C. We show in details how to reduce the relevant stochastic estimates to some basic counting and (random) matrix/tensor estimates studied in [12, Section 4] and [18].

In the remaining part of this section, we assume $0< T < T_0 \le 1$.

4.1 Basic stochastic terms

We first present the proof of Lemma 3.1.

Proof

(i) Let $t \ge 0$. From (1.16), we have

(4.1)

for any $n \in {\mathbb {Z}}^3$ and $N \ge 1$. Also, by the mean value theorem and an interpolation argument as in [26], we have

for any $\theta \in [0, 1]$, $n \in {\mathbb {Z}}^3$, and $0 \le t_2 \le t_1 \le T$ with $t_1-t_2 \le 1$, uniformly in $N \in {\mathbb {N}}$. Hence, from Lemma 2.10, we conclude that for any $\varepsilon > 0$, almost surely. Moreover, a slight modification of the argument, using Lemma 2.10, yields that is almost surely a Cauchy sequence in $C([0, T]; W^{\alpha -\frac{1}{2} - \varepsilon , \infty }({\mathbb {T}}^3))$, thus converging to some limit . Since the required modification is exactly the same as in [26], we omit the details here.

Remark 4.1

In the remaining part of this section, we establish uniform (in N) regularity bounds on the truncated stochastic terms (such as ) but may omit the convergence part of the argument. Furthermore, as for studied in Lemma 3.3, we only establish a uniform (in N) regularity bound on for each fixed $0 < t \le T \le 1$. A slight modification as above yields continuity in time but we omit details.

(ii) It is possible to prove this part by proceeding as in [26, 45] (i.e. without the use of the $X^{s, b}$-spaces). In the following, however, we follow Bringmann’s approach [12], adapted to the stochastic PDE setting. More precisely, we show that given any $\delta _1> 0$ and sufficiently small $\delta _2 > 0$, the sequence is a Cauchy sequence in $X^{\alpha - 1 - \delta _1, - \frac{1}{2} + \delta _2}([0, T])$, almost surely, and thus converges almost surely to in the same space, where is the almost sure limit of in $C([0,T];W^{3\alpha -\frac{3}{2} - ,\infty }({\mathbb {T}}^3))$ discussed in Sect. 1.

Our first goal is to prove the following bound; given any $\delta _1> 0$ and sufficiently small $\delta _2 > 0$, there exists $\theta > 0$ such that

(4.2)

for any $p \ge 1$ and $0 < T \le 1$, uniformly in $N \in {\mathbb {N}}$.

Let us first compute the space-time Fourier transform of (with a time cutoff function). From (1.14) with (1.12), we can write the spatial Fourier transform as the following multiple Wiener–Ito integral (as in [41]):

(4.3)

We emphasize that the renormalization in (1.14) is embedded in the definition of the multiple Wiener–Ito integral.

We now compute the space-time Fourier transform of   , where ${\mathbf {1}}_{[0, T]}$ denotes the sharp cutoff function on the time interval [0, T]. From (4.3) and the stochastic Fubini theorem ([15, Theorem 4.33]; see also Lemma B.2), we have

(4.4)

where $F_{n_1, n_2, n_3} (t_1, t_2, t_3, \tau )$ is defined by

$$\begin{aligned} F_{n_1, n_2, n_3} (t_1, t_2, t_3, \tau ) = \int _{0}^T e^{- i t \tau } \prod _{j = 1}^3 \frac{\sin ((t - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} {\mathbf {1}}_{[0,t]}(t_j) dt . \end{aligned}$$

(4.5)

Note that $F_{n_1, n_2, n_3} (t_1, t_2, t_3, \tau )$ is symmetric in $t_1, t_2, t_3$.

Given dyadic $N_j \ge 1$, $j = 1, 2, 3$, let us denote by $A^N_{N_1, N_2, N_3}$ the contribution to from $|n_j|\sim N_j$, $j = 1, 2, 3$, in (4.4). We first compute the $X^{s-1, b }$-norm of $A^N_{N_1, N_2, N_3}$ with $b = -\frac{1}{2} - \delta$ for $\delta > 0$. We then interpolate it with the trivial $X^{0, 0}$-bound. Recall the trivial bound:

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{X^{s, b}}&= \Vert \langle n \rangle ^{s} \langle |\tau | - \langle n \rangle \rangle ^b \, \widehat{u}(n, \tau )\Vert _{\ell ^2_n L^2_\tau }\\&\le \sum _{\varepsilon _0 \in \{-1, 1\}} \Vert \langle n \rangle ^{s} \langle \tau + \varepsilon _0 \langle n \rangle \rangle ^b \, \widehat{u}(n, \tau )\Vert _{\ell ^2_n L^2_\tau }\\&= \sum _{\varepsilon _0 \in \{-1, 1\}} \Vert \langle n \rangle ^{s} \langle \tau \rangle ^b \, \widehat{u}(n, \tau - \varepsilon _0 \langle n \rangle )\Vert _{\ell ^2_n L^2_\tau } \end{aligned} \end{aligned}$$

(4.6)

for any $s, b \in {\mathbb {R}}$. Then, defining $\kappa ({\bar{n}}) = \kappa _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3}(n_1, n_2, n_3)$ by

$$\begin{aligned} \kappa ({\bar{n}}) = \varepsilon _0 \langle n_{123} \rangle + \varepsilon _1\langle n_1 \rangle + \varepsilon _2\langle n_2 \rangle + \varepsilon _3\langle n_3 \rangle , \end{aligned}$$

(4.7)

with $\varepsilon _j \in \{-1, 1\}$ for $j = 0,1,2,3$, it follows from (4.6), (4.4), Fubini’s theorem, Ito’s isometry, and expanding the sine functions in (4.5) in terms of the complex exponentials that

$$\begin{aligned} \Big \Vert \Vert&A^N_{N_1, N_2, N_3} \Vert _{X^{s - 1, -\frac{1}{2} - \delta }_T}\Big \Vert _{L^2(\Omega )}^2 \nonumber \\&\lesssim \sum _{\varepsilon _0\in \{-1, 1\}} \sum _{n \in {\mathbb {Z}}^3}\int _{\mathbb {R}}\langle n \rangle ^{2(s-1)} \langle \tau \rangle ^{-1 - 2\delta } \nonumber \\&\quad \times \Bigg \{\sum _{\begin{array}{c} n = n_1 + n_2 + n_3\\ |n_j|\le N\\ |n_j|\sim N_j \end{array}} \int _{[0, T]^3} |F_{n_1, n_2, n_3} (t_1, t_2, t_3, \tau -\varepsilon _0\langle n \rangle ) |^2 dt_3 dt_2 dt_1\Bigg \} d\tau \nonumber \\&\lesssim \sum _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}} \sum _{n \in {\mathbb {Z}}^3}\int _{\mathbb {R}}\langle n \rangle ^{2(s-1)} \langle \tau \rangle ^{-1 - 2\delta } \Bigg \{\sum _{\begin{array}{c} n = n_1 + n_2 + n_3\\ |n_j|\le N\\ |n_j|\sim N_j \end{array}} \prod _{j = 1}^3 \frac{1}{\langle n_j \rangle ^{2(1+\alpha )}} \nonumber \\&\quad \times \int _{[0, T]^3} \bigg | \int _{\max (t_1, t_2, t_3)}^T e^{- i t (\tau - \kappa ({\bar{n}}))} dt\bigg |^2 dt_3 dt_2 dt_1\Bigg \} d\tau \nonumber \\&\lesssim \sum _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}} \sum _{n \in {\mathbb {Z}}^3}\sum _{\begin{array}{c} n = n_1 + n_2 + n_3\\ |n_j|\sim N_j \end{array}} \frac{\langle n \rangle ^{2(s-1)} }{\prod _{j =1}^3 \langle n_j \rangle ^{2(1+\alpha )}} \int _{\mathbb {R}}\frac{1}{\langle \tau \rangle ^{1 + 2\delta } \langle \tau - \kappa ({\bar{n}}) \rangle ^2} d\tau \nonumber \\&\lesssim \sum _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}} \sum _{n \in {\mathbb {Z}}^3}\sum _{\begin{array}{c} n = n_1 + n_2 + n_3\\ |n_j|\sim N_j \end{array}} \frac{\langle n \rangle ^{2(s-1)} }{\prod _{j =1}^3 \langle n_j \rangle ^{2(1+\alpha )}} \langle \kappa ({\bar{n}}) \rangle ^{-1 - 2\delta } \nonumber \\&\lesssim \sum _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}} \sup _{m \in {\mathbb {Z}}} \sum _{n \in {\mathbb {Z}}^3}\sum _{\begin{array}{c} n = n_1 + n_2 + n_3\\ |n_j|\sim N_j \end{array}} \frac{\langle n \rangle ^{2(s-1)} }{\prod _{j =1}^3 \langle n_j \rangle ^{2(1+\alpha )}} \cdot {\mathbf {1}}_{\{|\kappa ({\bar{n}}) - m|\le 1\}} \end{aligned}$$

(4.8)

for any $\delta > 0$, uniformly in dyadic $N_j \ge 1$, $j = 1, 2, 3$. By noting

$$\begin{aligned} \prod _{j = 1}^3 \langle n_j \rangle ^{- 2\alpha }\lesssim \langle n_{12} \rangle ^{-2\alpha }, \end{aligned}$$

(4.9)

we can reduce the right-hand side of (4.8) to the setting of the Hartree nonlinearity studied in [12]. In particular, from (4.8) with (4.9) and the cubic sum estimate (Lemma A.1), we obtain

$$\begin{aligned} \Big \Vert \Vert A^N_{N_1, N_2, N_3} \Vert _{X^{s - 1, -\frac{1}{2} - \delta }_T}\Big \Vert _{L^2(\Omega )} \lesssim N_{\max }^{s - \alpha }, \end{aligned}$$

(4.10)

where $N_{\max } = \max (N_1, N_2, N_3)$. This provides an estimate for $s < \alpha$ and $b = - \frac{1}{2} - \delta < -\frac{1}{2}$.

On the other hand, using (4.4), we have

$$\begin{aligned} \begin{aligned} \Big \Vert \Vert&A^N_{N_1, N_2, N_3} \Vert _{X^{0, 0}_T}\Big \Vert _{L^2(\Omega )}^2 = \Big \Vert \Vert A^N_{N_1, N_2, N_3} \Vert _{L^2_{T, x}}\Big \Vert _{L^2(\Omega )}^2\\&\lesssim T^\theta \sum _{\begin{array}{c} n_1 , n_2 , n_3\in {\mathbb {Z}}^3 \\ |n_j|\sim N_j \end{array}} \prod _{j =1}^3 \langle n_j \rangle ^{-2(1+\alpha )}\\&\lesssim T^\theta N_{\max }^{3 - 6\alpha } \end{aligned} \end{aligned}$$

(4.11)

for some $\theta > 0$. Hence, it follows from interpolating (4.10) and (4.11) and then applying the Wiener chaos estimate (Lemma 2.9) that given $s < \alpha$, there exist small $\delta _2 > 0$ and $\varepsilon > 0$ such that

$$\begin{aligned} \Big \Vert \Vert A^N_{N_1, N_2, N_3} \Vert _{X^{s - 1, -\frac{1}{2} + \delta _2}_T}\Big \Vert _{L^p(\Omega )} \lesssim p^\frac{3}{2} T^\theta N_{\max }^{- \varepsilon } \end{aligned}$$

for any $p \ge 1$, uniformly in dyadic $N_j \ge 1$, $j = 1, 2, 3$. By summing over dyadic blocks $N_j\ge 1$, $j = 1, 2, 3$, we obtain the bound (4.2) (with $b = -\frac{1}{2} + \delta _2 > -\frac{1}{2}$).

As for the convergence of to in $X^{\alpha - 1 - \delta _1, - \frac{1}{2} + \delta _2}([0, T])$, we can simply repeat the computation above to estimate the difference for $M \ge N \ge 1$. Fix $s < \alpha$. Then, in (4.8), we replace the restriction $|n_j| \le N$ in the summation of $n_j$, $j = 1, 2, 3$, by $N \le \max (|n_1|, |n_2|, |n_3|) \le M$, which allows us to gain a small negative power of N. As a result, in place of (4.10), we obtain

$$\begin{aligned} \Big \Vert \Vert A^M_{N_1, N_2, N_3} - A^N_{N_1, N_2, N_3} \Vert _{X^{s - 1, -\frac{1}{2} - \delta }_T}\Big \Vert _{L^2(\Omega )} \lesssim N^{-\varepsilon } N_{\max }^{s - \alpha +\varepsilon } \end{aligned}$$

for any small $\varepsilon > 0$ and $M \ge N \ge 1$. Then, the interpolation argument with (4.11) as above yields that given $s < \alpha$, there exist small $\delta _2>0$ and $\varepsilon > 0$ such that

(4.12)

for any $p \ge 1$ and $M \ge N \ge 1$. Then, by applying Chebyshev’s inequality and the Borel–Cantelli lemma, we conclude the almost sure convergence of . See [51].

Finally, fix $s< \alpha$. Given $N \in {\mathbb {N}}$, let . Then, we have

(4.13)

for $t \in [0, T]$, Note that from (4.4), we have $\widehat{H}_N(n, t) \in {\mathcal {H}}_3$ and, furthermore, by the independence of $\{B_n\}_{n \in {\mathbb {Z}}^3}$ (modulo $B_{-n} = \overline{B}_n$), we have

$$\begin{aligned} {\mathbb {E}}\big [\widehat{H}_N(n, t_1)\widehat{H}_N(m, t_2)\big ] = {\mathbf {1}}_{n+m = 0} \, {\mathbb {E}}\big [ \widehat{H}_N(n, t_1) \overline{\widehat{H}_N(n, t_2)}\big ] \end{aligned}$$

(4.14)

for any $t_1, t_2 \in {\mathbb {R}}$. Then, by (4.13), Sobolev’s inequality (with finite $r\gg 1$ such that $r\delta _0 > 3$ for some small $\delta _0 > 0$), Minkowski’s integral inequality, the Wiener chaos estimate (Lemma 2.9) with (4.14), Hausdorff–Young’s inequality (in time), we have, for any $p \ge \max (q, r) \gg 1$,

Now, by the triangle inequality: $\langle \tau \rangle ^{\delta _0}\lesssim \langle |\tau | - \langle n \rangle \rangle ^{\delta _0} \langle n \rangle ^{\delta _0}$, Hölder’s inequality (in $\tau$), followed by the nonhomogeneous linear estimate (Lemma 2.5) and (4.12) (with $p = 2$, $M = \infty$, and s replaced by $s + 2\delta _0 < \alpha$), we obtain

by choosing $\delta _0 > 0$ sufficiently small. Then, the regularity and convergence claim for follows from applying Chebyshev’s inequality and the Borel–Cantelli lemma as before. $\square$

Remark 4.2

Given a function $f \in L^2(({\mathbb {Z}}^3\times {\mathbb {R}}_+)^k)$, define the multiple stochastic integral $I_k[f]$ by

$$\begin{aligned} I_k [f] = \sum _{n_1,\dots ,n_k \in {\mathbb {Z}}^3} \int _{[0,\infty )^k} f( n_1, t_1, \dots , n_k, t_k) d B_{n_1} (t_1) \cdots d B_{n_k} (t_k). \end{aligned}$$

See Appendix B for the basic definitions and properties of multiple stochastic integrals. In terms of multiple stochastic integrals, we can express (4.3) as

where $f_{n, t}$ is defined by

$$\begin{aligned} f_{n, t}(n_1,t_1, n_2, t_2,n_3,t_3) = {\mathbf {1}}_{n = n_{123}}\cdot \bigg ( \prod _{j = 1}^3 \frac{\sin ((t - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j| \le N}\cdot {\mathbf {1}}_{[0,t]}(t_j)\bigg ) \end{aligned}$$

for $(n_1,t_1, n_2, t_2,n_3,t_3) \in ({\mathbb {Z}}^3 \times {\mathbb {R}}) ^3$. Then, by Fubini’s theorem for multiple stochastic integrals (Lemma B.2), we have

where ${\mathcal {F}}_t$ denotes the Fourier transform in time. With this notation, it follows from Lemma B.1 that we can write the second moment of the $X^{s, b}$-norm of $A^N_{N_1, N_2, N_3}$, appearing in (4.8) and (4.11), in a concise manner:

$$\begin{aligned}&\Big \Vert \Vert A^N_{N_1, N_2, N_3} \Vert _{X^{s , b}_T}\Big \Vert _{L^2(\Omega )}^2\\&= 3! \sum _{n \in {\mathbb {Z}}^3}\int _{\mathbb {R}}\langle n \rangle ^{2s} \langle |\tau |- \langle n \rangle \rangle ^{2b} \big \Vert {\mathcal {F}}_t({\mathbf {1}}_{[0, T]}f_{n, \cdot }^{{\bar{N}}})(\tau )\big \Vert _{\ell ^2_{n_1, n_2, n_3} L^2_{t_1, t_2, t_3} }^2d\tau , \end{aligned}$$

where $f_{n, t}^{{\bar{N}}}$ is given by

$$\begin{aligned} f_{n, t}^{{\bar{N}}} = f_{n, t}\cdot \prod _{j = 1}^3 {\mathbf {1}}_{|n_j|\sim N_j}. \end{aligned}$$

In the following, for conciseness of the presentation, we express various stochastic objects as multiple stochastic integrals on $({\mathbb {Z}}^3\times {\mathbb {R}}_+)^k$ and carry out analysis. For this purpose, we set

$$\begin{aligned} z_j = (n_j, t_j) \in {\mathbb {Z}}^3 \times {\mathbb {R}}_+ \end{aligned}$$

(4.15)

and use the following short-hand notation:

$$\begin{aligned} \Vert f(z_j) \Vert _{L^p_{z_j}} = \Vert f(n_j, t_j)\Vert _{\ell ^p_{n_j} L^p_{t_j}}. \end{aligned}$$

(4.16)

Note, however, that one may also carry out equivalent analysis at the level of multiple Wiener–Ito integrals as in the proof of Lemma 3.1 presented above.

Next, we briefly discuss the proof of Lemma 3.3.

Proof of Lemma 3.3

By the paraproduct decomposition (2.3), we have

In view of Lemma 2.1 with (2.19), the paraproducts and belong to $C([0,T];W^{\alpha - \frac{1}{2} - \varepsilon ,\infty }({\mathbb {T}}^3))$ for any $\varepsilon >0$, almost surely. Hence, it remains to study the resonant product . We only study the regularity of the resonant product for a fixed time since the continuity in time and the convergence follow from a systematic modification. In the following, we show

(4.17)

for any $n \in {\mathbb {Z}}^3$ and $N \ge 1$. Note the bound (4.17) together with Lemma 2.10 shows that the resonant product is smoother and has (spatial) regularity $2\alpha - \frac{1}{2} - = (\alpha - ) + \big (\alpha - \frac{1}{2}-\big )$.

As in [41], by decomposing into components in the homogeneous Wiener chaoses ${\mathcal {H}}_k$, $k = 2, 4$, we have

where and . See, for example, [43, Proposition 1.1.2] and Lemma B.4 on the product formula for multiple Wiener–Ito integrals (and it also follows from Ito’s lemma as explained in [41]). From the orthogonality of ${\mathcal {H}}_4$ and ${\mathcal {H}}_2$, we have

Hence, it suffices to prove (4.17) for , $j = 2, 4$.

From a slight modification¹⁷ of (4.8) with Lemma A.2, we have

(4.18)

for any $n \in {\mathbb {Z}}^3$ and $N \ge 1$. Then, from Jensen’s inequality (see (B.2)),¹⁸ (4.1), (4.18), and Lemma 2.8, we have

(4.19)

for any $n \in {\mathbb {Z}}^3$ and $N \ge 1$, where $|n_1|\sim |n- n_1|$ signifies the resonant product . This yields (4.17) for .

From Ito’s lemma (see also the product formula, Lemma B.4), (1.12), and (4.3) with (4.15), we have

where $g_{n, t, t'}$ is defined by

$$\begin{aligned} \begin{aligned} g_{n, t, t'}(z_2, z_3)&= \sum _{\begin{array}{c} |n_1|\le N\\ |n_1| \sim |n_{123}| \end{array}} {\mathbf {1}}_{n = n_{23}}\cdot {\mathbf {1}}_{|n_2|\le N}\cdot {\mathbf {1}}_{|n_3|\le N} \int _{0}^{t'} \frac{\sin ((t - t')\langle n_{123} \rangle )}{\langle n_{123} \rangle } \\&\qquad \times \bigg (\prod _{j = 1}^3 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{[0,t']}(t_j) \bigg ) \frac{\sin ((t - t_1)\langle n_1 \rangle )}{\langle n_1 \rangle ^{1+\alpha }} dt_1. \end{aligned} \end{aligned}$$

(4.20)

Note that $g_{n, t, t'}(z_2, z_3)$ is symmetric (in $z_2$ and $z_3$). From Fubini’s theorem (Lemma B.2), we have

(4.21)

We now apply Lemma B.1 to compute the second moment of(4.21). Then, with $\kappa ({\bar{n}})$ as in (4.7), it follows from expanding the sine functions in (4.20) in terms of the complex exponentials and switching the order of integration in $t'$ and $t_1$ that

Under the condition $|n_1| \sim |n_{123}|$ and $n = n_2 + n_3$, we have $|n_1|\gtrsim |n|$. Then, by applying the basic resonant estimate (Lemma A.3) and Lemma 2.8, we obtain

(4.22)

This computation with Lemma 2.10 shows that is even smoother and has (spatial) regularity $4\alpha -$.

Therefore, putting (4.19) and (4.22) together, we obtain the desired bound (4.17). $\square$

4.2 Quintic stochastic term

In this subsection, we present the proof of Lemma 3.4 (i) on the quintic stochastic process defined in (1.22). In view of Lemma 2.5, we prove the following bound; given any $\varepsilon >0$ and sufficiently small $\delta _2 >0$, there exists $\theta >0$ such that

(4.23)

for any $p \ge 1$ and $0 < T \le 1$, uniformly in $N \in {\mathbb {N}}$.

We start by computing the space-time Fourier transform of with a time cutoff. As shown in (1.22), the quintic stochastic objects is a convolution of in (1.15) and in (1.14):

(4.24)

Using Lemma B.2, we can write and as multiple stochastic integrals:

(4.25)

where $f_{n, t, t'}$ and $g_{n, t}$ are defined by

$$\begin{aligned} \begin{aligned} f_{n, t, t'}(z_1, z_2, z_3)&= {\mathbf {1}}_{n = n_{123}} \cdot \frac{\sin ((t - t')\langle n_{123} \rangle )}{\langle n_{123} \rangle } \\&\quad \times \bigg (\prod _{j = 1}^3 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,t']}(t_j) \bigg ), \\ g_{n, t}(z_1, z_2)&= {\mathbf {1}}_{n = n_{12}} \cdot \bigg (\prod _{j = 1}^2 \frac{\sin ((t - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,t]}(t_j)\bigg ). \end{aligned} \end{aligned}$$

(4.26)

By the product formula (Lemma B.4) to (4.24), we can decompose into the components in the homogeneous Wiener chaoses ${\mathcal {H}}_k$, $k = 1, 3, 5$:

(4.27)

where , , and . By taking the Fourier transforms in time, the relation (4.27) still holds. Then, by using the orthogonality of ${\mathcal {H}}_5$, ${\mathcal {H}}_3$, and ${\mathcal {H}}_1$, we have

Hence, it suffices to prove (4.23) for each , $j = 1, 3, 5$.

Case (i): Non-resonant term . From (4.25) and (4.26), we have

where $f^{(5)}_{n, t}$ is defined by

$$\begin{aligned} f^{(5)}_{n, t}(z_1, z_2, z_3, z_4, z_5)= & {} {\mathbf {1}}_{n = n_{12345}} \cdot \int _0^t \frac{\sin ((t - t')\langle n_{123} \rangle )}{\langle n_{123} \rangle } \nonumber \\&\quad \times \bigg (\prod _{j = 1}^3 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,t']}(t_j) \bigg ) dt' \nonumber \\&\quad \times \bigg (\prod _{j = 4}^5 \frac{\sin ((t - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,t]}(t_j)\bigg ). \nonumber \\ \end{aligned}$$

(4.28)

Let ${{\,\mathrm{\mathtt {Sym}}\,}}(f^{(5)}_{n, t})$ be the symmetrization of $f^{(5)}_{n, t}$ defined in (B.1). Then, from Lemma B.1 (ii), we have

Then, by taking the temporal Fourier transform and applying Fubini’s theorem (Lemma B.2), we have

Then, by (4.6), Fubini’s theorem, and Lemma B.1 (iii) with (4.15) and (4.16), we have

(4.29)

where ${\bar{z}} = (z_1, \dots , z_5)$.

By expanding the sine functions in (4.28) in terms of the complex exponentials, we have

$$\begin{aligned} \begin{aligned} f^{(5)}_{n, t} (z_1, z_2, z_3, z_4, z_5)&= c\cdot {\mathbf {1}}_{n = n_{12345}} \sum _{ {\mathcal {E}}} \widehat{\varepsilon }\cdot \frac{e^{i t\kappa _1({\bar{n}} )}}{\langle n_{123} \rangle } \int _{\max (t_1, t_2, t_3)}^t e^{- i t' \kappa _2({\bar{n}} )} dt' \\&\quad \times \bigg (\prod _{j = 1}^5 \frac{1}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N}\bigg ) \bigg (\prod _{j =4}^5 {\mathbf {1}}_{[0,t]}(t_j)\bigg ) F_1(z_1, \dots , z_5), \end{aligned} \end{aligned}$$

(4.30)

where $F_1(z_1, \dots , z_5)$ is independent of t and $t'$ with $|F_1|\le 1$. Here, ${\mathcal {E}}$, $\widehat{\varepsilon }$, $\kappa _1({\bar{n}})$, and $\kappa _2({\bar{n}})$ are defined by

$$\begin{aligned} \begin{aligned} {\mathcal {E}}&= \big \{\varepsilon _1, \dots , \varepsilon _5, \varepsilon _{123} \in \{-1, 1\}\big \}, \quad \widehat{\varepsilon }= \varepsilon _{123} \prod _{j= 1}^5 \varepsilon _j, \\ \kappa _1({\bar{n}})&= \varepsilon _{123} \langle n_{123} \rangle + \varepsilon _{4} \langle n_{4} \rangle + \varepsilon _{5} \langle n_{5} \rangle , \\ \kappa _2({\bar{n}})&= \varepsilon _{123} \langle n_{123} \rangle - \varepsilon _{1} \langle n_{1} \rangle - \varepsilon _{2} \langle n_{2} \rangle - \varepsilon _{3} \langle n_{3} \rangle . \end{aligned} \end{aligned}$$

(4.31)

By integrating in $t'$, we have

$$\begin{aligned} \int _{\max (t_1, t_2, t_3)}^t e^{- i t' \kappa _2({\bar{n}} )} dt' = \frac{e^{-it \kappa _2({\bar{n}})} -e^{-it_{123}^* \kappa _2({\bar{n}})}}{-i \kappa _2({\bar{n}})}, \end{aligned}$$

(4.32)

where $t_{123}^* = \max (t_1, t_2, t_3)$. Then, from (4.30) and (4.32), we have

$$\begin{aligned} \begin{aligned} \big |{\mathcal {F}}_t({\mathbf {1}}_{[0,T]}f^{(5)}_{n, \cdot }({\bar{z}}) )(\tau - \varepsilon _0 \langle n \rangle )\big |&\lesssim {\mathbf {1}}_{n = n_{12345}} \frac{1}{\langle \kappa _2({\bar{n}}) \rangle \langle \min (|\tau - \kappa _3({\bar{n}})|, |\tau - \kappa _4({\bar{n}})|) \rangle }\\&\quad \times \frac{1}{\langle n_{123} \rangle } \bigg (\prod _{j = 1}^5 \frac{1}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,T]}(t_j)\bigg ), \end{aligned} \end{aligned}$$

(4.33)

where $\kappa _3({\bar{n}})$ and $\kappa _4({\bar{n}})$ are defined by

$$\begin{aligned} \begin{aligned} \kappa _3({\bar{n}})&= \varepsilon _0 \langle n_{12345} \rangle + \varepsilon _{123} \langle n_{123} \rangle + \varepsilon _{4} \langle n_{4} \rangle + \varepsilon _{5} \langle n_{5} \rangle , \\ \kappa _4({\bar{n}})&= \varepsilon _0 \langle n_{12345} \rangle +\sum _{j= 1}^5 \varepsilon _{j} \langle n_{j} \rangle . \end{aligned} \end{aligned}$$

(4.34)

Given dyadic $N_j \ge 1$, $j = 1,2,3,4,5$, we denote by $B^N_{N_1,\cdots ,N_5}$ the contribution to from $|n_j| \sim N_j$ in (4.33). Let ${\mathcal {E}}_0 = {\mathcal {E}}\cup \{\varepsilon _0 \in \{-1, 1\}\}$ and $N_{\max } = \max (N_1,\dots ,N_5)$. Then, from (4.29), Jensen’s inequality (B.2), and (4.33) with (1.23), we have

$$\begin{aligned}&\Big \Vert \big \Vert {\mathbf {1}}_{[0,T]} B^N_{N_1,\cdots ,N_5} \big \Vert _{X^{s -1, -\frac{1}{2} - \delta }_T}\Big \Vert _{L^2(\Omega )}^2\nonumber \\&\lesssim T^\theta \sum _{ {\mathcal {E}}_0} \sum _{n \in {\mathbb {Z}}^3} \sum _{\begin{array}{c} n = n_{12345}\\ |n_j|\sim N_j \end{array}} \frac{\langle n \rangle ^{2(s-1)}}{\langle n_{123} \rangle ^2} \frac{1}{\langle \kappa _2({\bar{n}}) \rangle ^2} \bigg (\prod _{j = 1}^5 \frac{1}{\langle n_j \rangle ^{2+2\alpha }}\bigg )\nonumber \\&\qquad \times \int _{\mathbb {R}}\frac{1}{\langle \tau \rangle ^{1 + 2\delta }\langle \min (|\tau - \kappa _3({\bar{n}})|, |\tau - \kappa _4({\bar{n}})|) \rangle ^2}d\tau \nonumber \\&\lesssim T^\theta \sum _{ {\mathcal {E}}_0} \sup _{m,m' \in {\mathbb {Z}}} \sum _{\begin{array}{c} n_1, \dots , n_5 \in {\mathbb {Z}}^3\\ |n_j|\sim N_j \end{array}} \frac{\langle n_{12345} \rangle ^{2(s - \alpha + \frac{1}{2} \varepsilon -1)} }{ \langle n_{1234} \rangle ^{\frac{1}{2}\varepsilon }\langle n_{12} \rangle ^{\frac{1}{2}\varepsilon } \langle n_{123} \rangle ^{2} \prod _{j=1}^5 \langle n_j \rangle ^{2}} \nonumber \\&\qquad \times {\mathbf {1}}_{\{ |\kappa _2({\bar{n}}) -m|\le 1\}} \Big ({\mathbf {1}}_{\{ |\kappa _3({\bar{n}}) -m'|\le 1\}}+{\mathbf {1}}_{\{ |\kappa _4({\bar{n}}) -m'|\le 1\}}\Big ) \end{aligned}$$

(4.35)

for some $\theta > 0$, provided that $\delta > 0$. In the last step, we used the following bound:

$$\begin{aligned} \int _{\mathbb {R}}&\frac{1}{\langle \tau \rangle ^{1 + 2\delta }\langle \min (|\tau - \kappa _3({\bar{n}})|, |\tau - \kappa _4({\bar{n}})|) \rangle ^2}d\tau \\&\le \int _{\mathbb {R}}\frac{1}{\langle \tau \rangle ^{1 + 2\delta }\langle \tau - \kappa _3({\bar{n}}) \rangle ^2}d\tau + \int _{\mathbb {R}}\frac{1}{\langle \tau \rangle ^{1 + 2\delta }\langle \tau - \kappa _4({\bar{n}}) \rangle ^2}d\tau \\&\lesssim \langle \kappa _3({\bar{n}} ) \rangle ^{-1-2\delta } + \langle \kappa _4({\bar{n}} ) \rangle ^{-1-2\delta }\\&\lesssim \sum _{m'\in {\mathbb {Z}}} \frac{1}{\langle m' \rangle ^{1+2\delta }} \Big ({\mathbf {1}}_{\{ |\kappa _3({\bar{n}}) -m'|\le 1\}}+{\mathbf {1}}_{\{ |\kappa _4({\bar{n}}) -m'|\le 1\}}\Big ) \end{aligned}$$

for $\delta > 0$. Then, by applying Lemma A.4 to (4.35), we obtain

$$\begin{aligned} \Big \Vert \big \Vert&{\mathbf {1}}_{[0,T]} B^N_{N_1,\cdots ,N_5} \big \Vert _{X^{\alpha -\frac{1}{2}- \varepsilon , -\frac{1}{2} -\delta }_T}\Big \Vert _{L^2(\Omega )}^2 \lesssim T^\theta N_{\max }^{- \delta _0} \end{aligned}$$

(4.36)

for some $\delta _0>0$, provided that $\varepsilon , \delta > 0$. Using (4.29) and (4.33), a crude bound shows

$$\begin{aligned} \bigg \Vert \Big \Vert {\mathbf {1}}_{[0,T]} B^N_{N_1,\cdots ,N_5} \Big \Vert _{X^{0,0} } \bigg \Vert _{L^2 (\Omega )}^2 \lesssim T^\theta N_{\max }^{K} \end{aligned}$$

(4.37)

for some (possibly large) $K > 0$. By interpolating (4.36) and (4.37), applying the Winner chaos estimate (Lemma 2.9), and then summing over dyadic $N_j$, $j = 1, \dots ,5$, we obtain

for some $\theta >0$, uniformly in $N \in {\mathbb {N}}$. Proceeding as in the end of the proof of Lemma 3.1 (ii) on , a slight modification of the argument above yields convergence of to . Since the required modification is straightforward, we omit details. A similar comment applies to and studied below.

Case (ii): Single-resonance term . In view of the product formula (Lemma B.4)¹⁹ and Definition B.3 together with (4.25) and (4.26), we have

where $f^{(3)}_{n, t}$ is defined by

$$\begin{aligned} f^{(3)}_{n, t}(z_1, z_2, z_4)&= \sum _{n_3 \in {\mathbb {Z}}^3} {\mathbf {1}}_{n = n_{124}} \cdot \bigg (\prod _{j = 1}^4 {\mathbf {1}}_{|n_j|\le N}\bigg ) \frac{\sin ((t - t_j)\langle n_4 \rangle )}{\langle n_4 \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{[0,t]}(t_4)\\&\quad \times \int _0^t \frac{\sin ((t - t')\langle n_{123} \rangle )}{\langle n_{123} \rangle } \bigg (\prod _{j = 1}^2 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{[0,t']}(t_j) \bigg )\\&\quad \times \bigg (\int _0^{t'} \frac{\sin ((t - t_3)\langle n_3 \rangle ) \sin ((t' - t_3)\langle n_3 \rangle )}{\langle n_3 \rangle ^{2+2\alpha }} dt_3\bigg ) dt'. \end{aligned}$$

By the Wiener chaos estimate (Lemma 2.9) and Hölder’s inequality, we have

(4.38)

for small $\delta _2> 0$. Hence, (4.23) follows once we prove

(4.39)

for $\varepsilon >0$, uniformly in $N \in {\mathbb {N}}$.

With the symmetrization ${{\,\mathrm{\mathtt {Sym}}\,}}(f^{(3)}_{n, t})$ defined in (B.1), it follows from Lemma B.1 and Jensen’s inequality (B.2) that

(4.40)

where ${\mathfrak {I}}^{(3)} (z_1, z_2, t_4)$ is defined by

$$\begin{aligned} \begin{aligned} {\mathfrak {I}}^{(3)} (z_1, z_2, t_4)&= \sum _{|n_3|\le N} \frac{1}{\langle n_{123} \rangle \langle n_3 \rangle ^{2+2\alpha }} \int _{\max (t_1,t_2)}^t \sin ((t-t') \langle n_{123} \rangle )\\&\quad \times \bigg ( \prod _{j=1}^2 \sin ((t'-t_j) \langle n_{j} \rangle ) \bigg ) \\&\quad \times \int _0^{t'} \sin ((t -t_3) \langle n_{3} \rangle ) \sin ((t'-t_3) \langle n_{3} \rangle ) dt_3 dt' . \end{aligned} \end{aligned}$$

(4.41)

By switching the order of the integrals in (4.41) (with $a = \max (t_1,t_2)$):

$$\begin{aligned} \int _{a}^t \int _0^{t'} f dt_3 dt' = \int _0^{a} \int _{a}^t f dt' dt_3 + \int _{a}^t \int _{t_3}^t f dt' dt_3 \end{aligned}$$

and integrating in $t'$ first, we have

$$\begin{aligned} |{\mathfrak {I}}^{(3)} (z_1, z_2, t_4)| \lesssim \sum _{\varepsilon _1, \varepsilon _2, \varepsilon _3, \varepsilon _{123} \in \{-1, 1\}} \sum _{|n_3|\le N} \frac{1}{\langle n_{123} \rangle \langle n_3 \rangle ^{2+2\alpha }\langle \kappa _2({\bar{n}}) \rangle }, \end{aligned}$$

(4.42)

where $\kappa _2({\bar{n}})$ is as in (4.31). Hence, from (4.40), (4.42), and Lemma A.3, we obtain

By applying Lemma 2.8 iteratively, we then obtain

provided that $\delta _1 > 0$. This yields (4.39).

Case (iii): Double-resonance term . As in Case (ii), from the product formula (Lemma B.4) and Definition B.3 together with (4.25) and (4.26), we have

where $f^{(1)}_{n, t}$ is defined by

$$\begin{aligned} f^{(1)}_{n, t}(z_1)= & {} \sum _{n_2, n_3 \in {\mathbb {Z}}^3} {\mathbf {1}}_{n = n_{1}} \cdot \bigg (\prod _{j = 1}^3 {\mathbf {1}}_{|n_j|\le N}\bigg ) \\&\quad \times \int _0^t \frac{\sin ((t - t')\langle n_{123} \rangle )}{\langle n_{123} \rangle } \frac{\sin ((t' - t_1)\langle n_1 \rangle )}{\langle n_1 \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{[0,t']}(t_1) \\&\quad \times \bigg (\int _0^{t'} \int _0^{t'} \prod _{j = 2}^3 \frac{\sin ((t - t_j)\langle n_j \rangle ) \sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{2+2\alpha }} dt_2 dt_3\bigg ) dt'. \end{aligned}$$

Arguing as in (4.38), it suffices to show

(4.43)

for $\varepsilon >0$, uniformly in $N \in {\mathbb {N}}$.

With the symmetrization ${{\,\mathrm{\mathtt {Sym}}\,}}(f^{(1)}_{n, t})$ defined in (B.1), it follows from Lemma B.1 and Jensen’s inequality (B.2) that

(4.44)

where ${\mathfrak {I}}^{(1)} (z_1)$ is defined by

$$\begin{aligned} \begin{aligned} {\mathfrak {I}}^{(1)} (z_1)&= \sum _{|n_2|, |n_3|\le N} \frac{1}{\langle n_{123} \rangle \langle n_2 \rangle ^{2+2\alpha } \langle n_3 \rangle ^{2+2\alpha }}\\&\qquad \times \int _{t_1}^t \sin ((t-t') \langle n_{123} \rangle \sin ((t'-t_1) \langle n_{1} \rangle ) \\&\qquad \times \int _0^{t'} \int _0^{t'} \prod _{j = 2}^3 \sin ((t -t_j) \langle n_{j} \rangle ) \sin ((t'-t_j) \langle n_{j} \rangle ) dt_2 dt_3 dt' . \end{aligned} \end{aligned}$$

(4.45)

By switching the order of the integrals in (4.45) and integrating in $t'$ first, we have

$$\begin{aligned} |{\mathfrak {I}}^{(1)} (z_1)| \lesssim \sum _{|n_2|, |n_3|\le N} \frac{1}{\langle n_{123} \rangle \langle n_2 \rangle ^{2+2\alpha } \langle n_3 \rangle ^{2+2\alpha }\langle \kappa _2({\bar{n}}) \rangle }, \end{aligned}$$

(4.46)

where $\kappa _2({\bar{n}})$ is as in (4.31). Hence, from (4.44) and (4.46), we obtain

Now, apply the dyadic decompositions $|n_j|\sim N_j$, $j = 1, 2, 3$. By noting that $\langle n_{12} \rangle ^\alpha \lesssim N_1^\alpha N_2^\alpha$ and that $|\kappa _2({\bar{n}}) - m |\le 1$ implies $|m| \lesssim N_{\max } = \max (N_1, N_2, N_3)$, it follows from Lemma A.5 that

provided that $\varepsilon > 0$, where $\gamma =\gamma (\varepsilon , \alpha ) > 0$ is sufficiently small. This yields (4.43).

This concludes the proof of Lemma 3.4 (i).

4.3 Septic stochastic term

In this subsection, we present the proof of Lemma 3.4 (ii) on the septic stochastic term defined in (1.24). Proceeding as in (4.38), it suffices to show

(4.47)

for $\varepsilon > 0$, uniformly in $N \in {\mathbb {N}}$. As in the previous subsections, we decompose into the components in the homogeneous Wiener chaoses ${\mathcal {H}}_k$, $k = 1, 3, 5, 7$:

(4.48)

where . From the orthogonality of ${\mathcal {H}}_k$, we have

Hence, it suffices to prove (4.47) for , $j = 0,1,2,3$.

Case (i): Non-resonant septic term We first study the non-resonant term . From (1.12) and (4.25) with (4.26) and (4.15), we have

(4.49)

where $f^{(7)}_{n, t}$ is defined by

$$\begin{aligned} \begin{aligned} f^{(7)}_{n, t}(z_1, \dots , z_7)&= {\mathbf {1}}_{n = n_{1234567}}\cdot \bigg (\prod _{j = 1}^7 {\mathbf {1}}_{|n_j|\le N}\bigg )\\&\qquad \times \int _0^t \frac{\sin ((t-t') \langle n_{123} \rangle )}{\langle n_{123} \rangle } \bigg (\prod _{j = 1}^3 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} {\mathbf {1}}_{[t_j,t]} (t')\bigg ) dt' \\&\qquad \times \int _0^t \frac{\sin ((t-t'') \langle n_{456} \rangle )}{\langle n_{456} \rangle } \bigg (\prod _{j = 4}^6 \frac{\sin ((t'' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} {\mathbf {1}}_{[t_j,t]} (t'')\bigg ) dt'' \\&\qquad \times \frac{\sin ( (t-t_7) \langle n_7 \rangle )}{\langle n_7 \rangle ^{1+\alpha }} {\mathbf {1}}_{[0,t]} (t_7). \end{aligned} \end{aligned}$$

(4.50)

By defining the amplitude $\Phi$ by

$$\begin{aligned} \Phi (t, z_1, z_2, z_3) = \int _{\max (t_1,t_2,t_3)}^t \frac{\sin ((t-t') \langle n_{123} \rangle )}{\langle n_{123} \rangle } \prod _{j= 1}^3 \frac{\sin ((t' - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} dt', \end{aligned}$$

(4.51)

we have

$$\begin{aligned} f^{(7)}_{n, t}(z_1, \dots , z_7)&= \Phi (t, z_1, z_2, z_3) \Phi (t, z_4, z_5, z_6) \frac{\sin ( (t-t_7) \langle n_7 \rangle )}{\langle n_7 \rangle ^{1+\alpha }}. \end{aligned}$$

Let $\kappa _2({\bar{n}})$ be as in (4.31). Then, from (4.51), we have

$$\begin{aligned} \sup _{t \in [0,T]} | \Phi (t, z_1, z_2, z_3)| \lesssim K (n_1,n_2,n_3)\prod _{j = 1}^3 \langle n_j \rangle ^{-\alpha } , \end{aligned}$$

where $K (n_1,n_2,n_3)$ is defined by

$$\begin{aligned} K (n_1,n_2,n_3) = \frac{1}{\langle n_{123} \rangle \langle \kappa _2({\bar{n}}) \rangle } \prod _{j= 1}^3 \frac{1}{\langle n_j \rangle } . \end{aligned}$$

(4.52)

Note that from Lemma A.1, we have

$$\begin{aligned} \sum _{\begin{array}{c} n_1, n_2, n_3\in {\mathbb {Z}}^3\\ |n_j|\sim N_j \end{array}} K^2 (n_1,n_2,n_3) \lesssim \max (N_1, N_2, N_3)^\gamma \end{aligned}$$

(4.53)

for any $\gamma > 0$. In view of (4.52) and (4.31), $K (n_1,n_2,n_3)$ depends on $\varepsilon _{123}, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}$. In the following, however, we drop the dependence on $\varepsilon _{123}, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}$ since (4.53) uniformly in $\varepsilon _{123}, \varepsilon _1, \varepsilon _2, \varepsilon _3 \in \{-1, 1\}$. The same comment applies to (4.54) below.

With the symmetrization ${{\,\mathrm{\mathtt {Sym}}\,}}(f^{(7)}_{n, t})$ defined in (B.1), it follows from Lemma B.1, Jensen’s inequality (B.2), and Lemma 2.8 (to sum over $n_7$) that

for some $\theta > 0$, provided that $\delta _1 > 0$. By applying the dyadic decomposition $|n_j| \sim N_j$, $j = 1, \dots , 7$, and then applying (4.53), we then obtain

as long as $\gamma < 2\alpha$. This proves (4.47).

Case (ii): General septic terms As we saw in the previous subsections, all other terms in (4.48) come from the contractions of the product of . In order to fully describe these terms, we recall the notion of a pairing from [12, Definition 4.30] to describe the structure of the contractions.

Definition 4.3

(pairing) Let $J\ge 1$. We call a relation ${\mathcal {P}} \subset \{1,\dots ,J\}^2$ a pairing if

• (i) ${\mathcal {P}}$ is reflexive, i.e. $(j,j) \notin {\mathcal {P}}$ for all $1 \le j \le J$,

• (ii) ${\mathcal {P}}$ is symmetric, i.e. $(i,j) \in {\mathcal {P}}$ if and only if $(j,i) \in {\mathcal {P}}$,

• (iii) ${\mathcal {P}}$ is univalent, i.e. for each $1 \le i \le J$, $(i,j) \in {\mathcal {P}}$ for at most one $1 \le j \le J$.

If $(i,j) \in {\mathcal {P}}$, the tuple (i, j) is called a pair. If $1 \le j \le J$ is contained in a pair, we say that j is paired. With a slight abuse of notation, we also write $j \in {\mathcal {P}}$ if j is paired. If j is not paired, we also say that j is unpaired and write $j \notin {\mathcal {P}}$. Furthermore, given a partition ${\mathcal {A}} = \{A_\ell \}_{\ell = 1}^L$ of $\{1, \cdots , J\}$, we say that ${\mathcal {P}}$ respects ${\mathcal {A}}$ if $i,j\in A_\ell$ for some $1 \le \ell \le L$ implies that $(i,j)\notin {\mathcal {P}}$. Namely, ${\mathcal {P}}$ does not pair elements of the same set $A_\ell \in {\mathcal {A}}$. We say that $(n_1, \dots , n_J) \in ({\mathbb {Z}}^3)^J$ is admissible if $(i,j) \in {\mathcal {P}}$ implies that $n_i + n_j = 0$.

In order to represent , $k = 1, 3, 5$, as multiple stochastic integrals as in (4.49), we start with (4.50) and perform a contraction over the variables $z_j = (n_j, t_j)$, namely, we consider a (non-trivial)²⁰ pairing on $\{1, \dots , 7\}$. Then, by integrating in $t'$ and $t''$ first in (4.50) after a contraction, a computation analogous to that in Case (i) yields

(4.54)

where K is as in (4.52) and the non-resonant frequency $n_{\text {nr}}$ is defined by

$$\begin{aligned} n_{\text {nr}} = \sum _{j \notin {\mathcal {P}}} n_j. \end{aligned}$$

(4.55)

Here, $\Pi _k$ denotes the collection of pairings ${\mathcal {P}}$ on $\{1, \dots , 7\}$ such that (i) ${\mathcal {P}}$ respects the partition ${\mathcal {A}}= \big \{\{1,2,3\}, \{4,5,6\}, \{7\}\big \}$ and (ii) $|{\mathcal {P}}| = 7-k$ (when we view ${\mathcal {P}}$ as a subset of $\{1, \dots , 7\}$). Note that the estimate on discussed in Case (i) is a special case of (4.54) with ${\mathcal {P}} = \varnothing$. By applying Lemma A.6 (with (1.23)), we then obtain

provided that $\varepsilon >0$. This concludes the proof of Lemma 3.4 (ii).

4.4 Random operator

In this subsection, we present the proof of Lemma 3.5 on the random operator defined in (1.21).

In view of (3.1) and (3.2) in the definition of ${\mathcal {L}}^{s_1, s_2, b}_{T_0}$, (1.21), and the nonhomogeneous linear estimate (Lemma 2.5), it suffices to show the following bound:

(4.56)

for some small $\delta _1, \delta _2 >0$ and any $p \ge 1$, uniformly in $N \in {\mathbb {N}}$. From (2.9), we see that (4.56) follows once we prove

(4.57)

Furthermore, by inserting a sharp time-cutoff function on [0, 1], we may drop the supremum in T and reduce the bound (4.57) to proving

(4.58)

As in the proof of Lemma 3.1 (ii), we first prove

(4.59)

namely with $b = - \frac{1}{2} - \delta < - \frac{1}{2}$ on the $X^{s, b}$-norm of   for $\delta >0$. In fact, we prove a frequency-localized version of (4.59) (see (4.72) below) and interpolate it with a trivial $X^{0, 0}$ estimate (see (4.73) below), as in the proof of Lemma 3.1 (ii) and Lemma 3.4 (i), to establish (4.58) with $b = - \frac{1}{2} + 2\delta _2 > - \frac{1}{2}$

We start by computing the space-time Fourier transform of  . From (4.25) and (4.26), we have

where $g_{n - n_3, t}(z_1, z_2)$ is as in (4.26). Now, write $v = v_1 + v_{-1}$, where

$$\begin{aligned} \widehat{v}_{1}(n, \tau ) = {\mathbf {1}}_{[0, \infty )}(\tau )\cdot \widehat{v}(n, \tau ) \quad \text {and} \quad \widehat{v}_{-1}(n, \tau ) = {\mathbf {1}}_{(-\infty , 0)}(\tau )\cdot \widehat{v}(n, \tau ). \end{aligned}$$

Then, by noting $|\widehat{v}(n, \tau )|^2 = |\widehat{v}_{1}(n, \tau )|^2+ |\widehat{v}_{-1}(n, \tau )|^2$, we have

$$\begin{aligned} \begin{aligned} \Vert v\Vert _{X^{s, b}}^2&= \sum _{\varepsilon _3 \in \{-1, 1\}}\Vert v_{\varepsilon _3}\Vert _{X^{s, b}}^2\\&= \sum _{\varepsilon _3 \in \{-1, 1\}} \big \Vert \langle n \rangle ^{s} \langle \tau \rangle ^b \, \widehat{v}_{\varepsilon _3}(n, \tau + \varepsilon _3 \langle n \rangle )\big \Vert _{\ell ^2_n L^2_\tau }^2. \end{aligned} \end{aligned}$$

(4.60)

With this in mind, we write

(4.61)

where $\varepsilon _0, \varepsilon _3 \in \{-1, 1\}$ and the kernel $H = H^{\varepsilon _0, \varepsilon _3}$ is given by

$$\begin{aligned} H(n,n_3,\tau ,\tau _3)&= \langle n \rangle ^{-\frac{1}{2} + \delta _1} \langle n_3 \rangle ^{-\frac{1}{2} - \delta _1} \frac{1}{\sqrt{2\pi }} \int _0^1 e^{-i t (\tau - \tau _3 - \varepsilon _0\langle n \rangle - \varepsilon _3\langle n_3 \rangle )} I_2[g_{n - n_3, t}] dt. \end{aligned}$$

By Fubini’s theorem (Lemma B.2), we can write H as

$$\begin{aligned} \begin{aligned} H(n,n_3,\tau ,\tau _3)&= \langle n \rangle ^{-\frac{1}{2} + \delta _1} \langle n_3 \rangle ^{-\frac{1}{2} - \delta _1} I_2[h_{n, n_3, \tau , \tau _3}], \end{aligned} \end{aligned}$$

(4.62)

where $h_{n, n_3, \tau , \tau _3}$ is given by

$$\begin{aligned} \begin{aligned} h_{n, n_3, \tau , \tau _3}(z_1, z_2)&= {\mathbf {1}}_{n- n_3 = n_{12}} \cdot \frac{1}{\sqrt{2\pi }} \int _0^1 e^{-i t (\tau - \tau _3 - \varepsilon _0\langle n \rangle - \varepsilon _3\langle n_3 \rangle )} \\&\quad \times \bigg (\prod _{j = 1}^2 \frac{\sin ((t - t_j)\langle n_j \rangle )}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{|n_j|\le N} \cdot {\mathbf {1}}_{[0,t]}(t_j)\bigg ) dt. \end{aligned} \end{aligned}$$

(4.63)

Then, by (4.6), (4.61), Cauchy–Schwarz’s inequality, and (4.60), we have

as long as $\delta , \delta _2 > 0$, where, in the last step, we used Minkowski’s integral inequality followed by Hölder’s inequality (in $\tau$ and $\tau _3$). Here, we viewed $H(n,n_3,\tau ,\tau _3)$ (for fixed $\tau , \tau _3 \in {\mathbb {R}}$) as an infinite dimensional matrix operator mapping from $\ell ^2_{n_3}$ into $\ell ^2_n$. Hence, the estimate (4.59) is reduced to proving

$$\begin{aligned} \sup _{\varepsilon _0, \varepsilon _3} \sup _{\tau ,\tau _3 \in {\mathbb {R}}} \Big \Vert \Vert H(n,n_3,\tau ,\tau _3) \Vert _{\ell ^2_{n_3} \rightarrow \ell ^2_n} \Big \Vert _{L^p(\Omega )} \lesssim p. \end{aligned}$$

(4.64)

As mentioned above, we instead establish a frequency-localized version of (4.64):

$$\begin{aligned} \sup _{\varepsilon _0, \varepsilon _3} \sup _{\tau ,\tau _3 \in {\mathbb {R}}} \Big \Vert \Vert H_{N_1, N_2, N_3} (n,n_3,\tau ,\tau _3) \Vert _{\ell ^2_{n_3} \rightarrow \ell ^2_n} \Big \Vert _{L^p(\Omega )} \lesssim p N_{\max }^{-\delta _0}, \end{aligned}$$

(4.65)

for some small $\delta _0 > 0$, uniformly in dyadic $N_1, N_2, N_3\ge 1$, where $N_{\max } = \max (N_1, N_2, N_3)$ and $H_{N_1, N_2, N_3}$ is defined by (4.62) and (4.63) with extra frequency localizations ${\mathbf {1}}_{|n_j|\sim N_j}$, $j = 1, 2, 3$. Namely, we have

$$\begin{aligned} \begin{aligned} H_{N_1, N_2, N_3}(n,n_3,\tau ,\tau _3)&= \langle n \rangle ^{-\frac{1}{2} + \delta _1} \langle n_3 \rangle ^{-\frac{1}{2} - \delta _1} I_2\big [h^{N_1, N_2, N_3}_{n, n_3, \tau , \tau _3}\big ], \end{aligned} \end{aligned}$$

(4.66)

where $h^{N_1, N_2, N_3}_{n, n_3, \tau , \tau _3}$ is given by

$$\begin{aligned} h^{N_1, N_2, N_3}_{n, n_3, \tau , \tau _3}(z_1, z_2)= & {} \sum _{\varepsilon _1, \varepsilon _2\in \{-1, 1\}}c_{\varepsilon _1, \varepsilon _2} {\mathbf {1}}_{n- n_3 = n_{12}}\cdot {\mathbf {1}}_{|n_3|\sim N_3}\cdot \frac{1}{\sqrt{2\pi }} \int _0^1 e^{-i t (\tau - \tau _3 - \kappa ({\bar{n}}) )} \nonumber \\&\quad \times \bigg (\prod _{j = 1}^2 \frac{e^{-it_j \varepsilon _j \langle n_j \rangle }}{\langle n_j \rangle ^{1+\alpha }} \cdot {\mathbf {1}}_{\begin{array}{c} |n_j|\sim N_j\\ |n_j|\le N \end{array}} \cdot {\mathbf {1}}_{[0,t]}(t_j)\bigg ) dt \end{aligned}$$

(4.67)

with $\kappa ({\bar{n}})$ as in (4.7).

For $m \in {\mathbb {Z}}$, define the tensor ${\mathfrak {h}}^m$ by

$$\begin{aligned} \begin{aligned} {\mathfrak {h}}^m_{nn_1n_2n_3}&= c_{\varepsilon _1, \varepsilon _2} {\mathbf {1}}_{n = n_{123} } \cdot {\mathbf {1}}_{|n_3|\sim N_3} \bigg (\prod _{j=1}^2{\mathbf {1}}_{\begin{array}{c} |n_j|\sim N_j\\ |n_j|\le N \end{array}}\bigg )\\&\quad \times {\mathbf {1}}_{\{|\kappa ({\bar{n}}) - m|\le 1\}} \frac{ \langle n \rangle ^{-\frac{1}{2} + \delta _1}}{\langle n_1 \rangle ^{1+ \alpha } \langle n_2 \rangle ^{1+ \alpha } \langle n_3 \rangle ^{\frac{1}{2} + \delta _1}}. \end{aligned} \end{aligned}$$

(4.68)

Then, from (4.66), (4.67), and (4.68), we have

$$\begin{aligned} \begin{aligned} H_{N_1, N_2, N_3}(n,n_3,\tau ,\tau _3)&= \sum _{\varepsilon _1, \varepsilon _2 \in \{-1, 1\}}\sum _{m \in {\mathbb {Z}}} H^m(n,n_3, \tau ,\tau _3)\\ : \!&=\sum _{\varepsilon _1, \varepsilon _2 \in \{-1, 1\}} \sum _{m \in {\mathbb {Z}}} I_2\big [ {\mathfrak {h}}^m_{nn_1n_2n_3} {\mathfrak {H}}^m_{n_3, \tau , \tau _3} \big ], \end{aligned} \end{aligned}$$

(4.69)

where ${\mathfrak {H}}^m_{n_3, \tau , \tau _3}$ is given by

$$\begin{aligned}&{\mathfrak {H}}^m_{n_3, \tau , \tau _3} (z_1, z_2)\\&= \frac{1}{\sqrt{2\pi }} \int _0^1 {\mathbf {1}}_{\{|\kappa ({\bar{n}}) - m| \le 1\}} e^{-i t (\tau - \tau _3 - \kappa ({\bar{n}}) )} \Big (\prod _{j = 1}^2 e^{-it_j \varepsilon _j \langle n_j \rangle } \cdot {\mathbf {1}}_{[0,t]}(t_j)\Big ) dt. \end{aligned}$$

Performing t-integration, we have

$$\begin{aligned} \Vert {\mathfrak {H}}^m_{n_3, \tau , \tau _3} (z_1, z_2) \Vert _{ \ell ^{\infty }_{n_1,n_2} L^2_{t_1, t_2}([0, 1]^2)} \lesssim \langle \tau - \tau _3 - m \rangle ^{-1}. \end{aligned}$$

(4.70)

Then from Lemma C.3, (4.70), and Lemma C.2 (with (1.23)), there exists $\delta _3 > 0$ such that

$$\begin{aligned}&\Big \Vert \Vert H^m (n,n_3,\tau ,\tau _3) \Vert _{\ell ^2_{n_3} \rightarrow \ell ^2_n} \Big \Vert _{L^p(\Omega )}\nonumber \\&\lesssim p N_{\max }^{\varepsilon } \langle \tau - \tau _3 - m \rangle ^{-1} \nonumber \\&\quad \times \max \Big ( \Vert {\mathfrak {h}}^m \Vert _{n_1n_2n_3 \rightarrow n}, \Vert {\mathfrak {h}}^m \Vert _{n_3 \rightarrow nn_1n_2}, \Vert {\mathfrak {h}}^m \Vert _{n_1n_3 \rightarrow nn_2}, \Vert {\mathfrak {h}}^m \Vert _{n_2n_3 \rightarrow nn_1} \Big ) \nonumber \\&\lesssim p N_{\max }^{\varepsilon -\delta _3} \langle \tau - \tau _3 - m \rangle ^{-1} . \end{aligned}$$

(4.71)

for any $\varepsilon >0$, provided that $\delta _1 < \alpha$, which is needed to apply Lemma C.2. Hence, by noting that the condition $|\kappa ({\bar{n}}) - m|\le 1$ implies $|m| \lesssim N_{\max }$ and summing over $m \in {\mathbb {Z}}$, the bound (4.65) follows from (4.69) and (4.71) (by taking $\varepsilon >0$ sufficiently small), which in turn implies

(4.72)

for some $\delta _0 > 0$, where $v_{N_3} = {\mathcal {F}}_x^{-1} ({\mathbf {1}}_{|n|\sim N_3} \widehat{v}(n))$ and

Namely, the frequencies $n_1$, $n_2$, and $n_3$ are localized to the dyadic blocks $\{|n_j|\sim N_j\}$, $j = 1, 2, 3$.

On the other hand, a crude bound shows

(4.73)

for some (possibly large) $K > 0$. By interpolating (4.72) and (4.73) and then summing over dyadic $N_j$, $j = 1, \dots ,3$, we obtain (4.58) for some small $\delta _2 > 0$.

Lastly, as for the convergence of to , we can simply repeat the computation above to estimate the difference for $M \ge N \ge 1$. In considering the difference of the tensors ${\mathfrak {h}}^m$ in (4.68), we then obtain a new restriction $\max (|n_1|, |n_2|) \gtrsim N$, which allows us to gain a small negative power of N. As a result, we obtain

for some small $\varepsilon , \delta _0' > 0$, Then, interpolating this with (4.73) and summing over dyadic blocks, we then obtain

for any $p \ge 1$ and $M \ge N \ge 1$. Then, by applying Chebyshev’s inequality, summing over $N \in {\mathbb {N}}$, and applying the Borel–Cantelli lemma, we conclude the almost sure convergence of  . This concludes the proof of Lemma 3.5.

Appendices

Appendix A: Counting estimates

In this section, we state the counting estimates used in Sect. 4 to study the regularities of the stochastic terms. These lemmas are taken from Bringmann [12]. Note that some statements are given in a slightly simplified form. The same comment applies to Lemma C.2.

Lemma A.1

(Proposition 4.20 in [12]) Let $0 < s \le \frac{1}{2}$ and $0 \le \beta \le \frac{1}{2}$. Given $\varepsilon _j \in \{-1, 1 \}$ for $j = 0,1,2,3$, let $\kappa ({\bar{n}}) = \kappa _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3}(n_1, n_2, n_3)$ be as in (4.7). Then, we have

$$\begin{aligned} \sup _{m \in {\mathbb {Z}}} \sum _{\begin{array}{c} n_1,n_2,n_3 \in {\mathbb {Z}}^3\\ |n_j|\sim N_j \end{array}}&\langle n_{123} \rangle ^{2(s-1)} \frac{{\mathbf {1}}_{\{|\kappa ({\bar{n}})- m| \le 1\}}}{\langle n_{12} \rangle ^{2\beta } \prod _{j=1}^3 \langle n_j \rangle ^{2} } \lesssim N_{\max }^{2(s- \beta )}, \end{aligned}$$

uniformly in dyadic $N_1,N_2,N_3 \ge 1$ and $\varepsilon _j \in \{-1, 1 \}$ for $j = 0,1,2,3$, where $N_{\max } = \max (N_1,N_2,N_3)$.

Lemma A.2

(Lemma 4.22 (i) in [12]) Given $\varepsilon _j \in \{-1, 1\}$ for $j = 0,1,2,3$, let $\kappa ({\bar{n}}) = \kappa _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3}(n_1, n_2, n_3)$ be as in (4.7). Then, we have

$$\begin{aligned} \sup _{m \in {\mathbb {Z}}} \sup _{n \in {\mathbb {Z}}^3} \,&\#\Big \{(n_1, n_2, n_3) \in {\mathbb {Z}}^3: |n_j|\sim N_j, \, j = 1, 2, 3, \, n \,{=}\, n_{123}, \, \{|\kappa ({\bar{n}})- m| \,{\le }\, 1\}\Big \} \\&\lesssim {{\,\mathrm{med}\,}}(N_1, N_2, N_3)^3 \min (N_1, N_2, N_3)^2, \end{aligned}$$

uniformly in dyadic $N_1,N_2,N_3 \ge 1$ and $\varepsilon _j \in \{-1, 1 \}$ for $j = 0,1,2,3$.

Next, we recall the basic resonant estimate.

Lemma A.3

(Lemma 4.25 in [12]) Given $\varepsilon _j \in \{- 1, 1\}$ for $j = 0,1,2,3$, let $\kappa ({\bar{n}}) = \kappa _{\varepsilon _0, \varepsilon _1, \varepsilon _2, \varepsilon _3}(n_1, n_2, n_3)$ be as in (4.7). Then, we have

$$\begin{aligned} \sum _{m \in {\mathbb {Z}}} \sum _{\begin{array}{c} n_1 \in {\mathbb {Z}}^3\\ |n_1|\sim N_1 \end{array}} \frac{{\mathbf {1}}_{\{|\kappa ({\bar{n}})- m| \le 1\}}}{ \langle m \rangle \langle n_{123} \rangle \langle n_1 \rangle ^{2} } \lesssim \frac{\log (2+N_1)}{\langle n_{23} \rangle }, \end{aligned}$$

uniformly in dyadic $N_1 \ge 1$ and $\varepsilon _j \in \{-1, 1 \}$ for $j = 0,1,2,3$.

The next two lemmas (and Lemma A.3 above) are used for estimating the quintic stochastic term.

Lemma A.4

Let $s \le \frac{1}{2} - \eta$ and $\beta > 0$ for some $\eta > 0$. Given $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 0, \dots , 5$, let $\kappa _2({\bar{n}})$, $\kappa _3({\bar{n}})$, and $\kappa _4({\bar{n}})$ be as in (4.31) and (4.34). Then, we have

$$\begin{aligned} \begin{aligned}&\sup _{m, m' \in {\mathbb {Z}}} \sum _{\begin{array}{c} n_1, \dots , n_5 \in {\mathbb {Z}}^3\\ |n_j|\sim N_j \end{array}} \frac{ \langle n_{12345} \rangle ^{2(s-1)}}{ \langle n_{1234} \rangle ^{2\beta } \langle n_{12} \rangle ^{2\beta } \langle n_{123} \rangle ^{2} \prod _{j=1}^5 \langle n_j \rangle ^{2}} \\&\qquad \times {\mathbf {1}}_{\{ |\kappa _2({\bar{n}}) -m|\le 1\}} \Big ({\mathbf {1}}_{\{ |\kappa _3({\bar{n}}) -m'|\le 1\}}+{\mathbf {1}}_{\{ |\kappa _4({\bar{n}}) -m'|\le 1\}}\Big ) \\&\quad \lesssim \max (N_1, N_2, N_3, N_4)^{-2\beta + \varepsilon } N_5^{-\eta } \end{aligned} \end{aligned}$$

(A.1)

for any $\varepsilon > 0$, uniformly in dyadic $N_1,\dots ,N_5 \ge 1$ and $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 0, \dots , 5$, where $N_{\max } = \max (N_1,\dots ,N_5)$.

Lemma A.4 is essentially Lemma 4.27 in [12], where the condition $|\kappa _4({\bar{n}}) -m'|\le 1$ in (A.1) is replaced by $|\kappa _4({\bar{n}}) + \varepsilon _{123}\langle n_{123} \rangle -m'|\le 1$. We point out that this modification does not make any difference in the proof. In our notation, the first step of the proof of Lemma 4.27 in [12] is to sum over $n_5$, using [12, Lemma 4.17], for which the conditions $|\kappa _4({\bar{n}}) -m'|\le 1$ in (A.1) and $|\kappa _4({\bar{n}}) + \varepsilon _{123}\langle n_{123} \rangle -m'|\le 1$ do not make any difference since the extra term $\varepsilon _{123}\langle n_{123} \rangle$ is fixed in summing over $n_5$.

Lemma A.5

(Lemma 4.29 in [12]) Let $\beta > 0$. Given $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 1, 2, 3$, let $\kappa _2({\bar{n}})$ be as in (4.31). Then, we have

$$\begin{aligned} \sup _{m \in {\mathbb {Z}}^3} \sup _{|n_1| \sim N_1} \sum _{\begin{array}{c} |n_2|\sim N_2\\ |n_3|\sim N_3 \end{array}} \frac{{\mathbf {1}}_{\{|\kappa _2({\bar{n}}) - m |\le 1\}}}{\langle n_{123} \rangle \langle n_{12} \rangle ^\beta \langle n_2 \rangle ^{2} \langle n_3 \rangle ^{2}} \lesssim \max (N_1,N_2)^{-\beta + \varepsilon } \end{aligned}$$

for any $\varepsilon >0$, uniformly in dyadic $N_1,N_2,N_3 \ge 1$ and $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 1,2, 3$.

Lastly, we state the septic counting estimate. See Definition 4.3 in Sect.  4.3 for the definition of a paring.

Lemma A.6

(Lemma 4.31 in [12])

Let $\frac{1}{2}< s < 1$ and $\beta > 0$. Given $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 1, 2, 3$, let $\kappa _2({\bar{n}})$ be as in (4.31) and set

$$\begin{aligned} {\mathcal {K}} (n_1,n_2,n_3) = \sum _{m \in {\mathbb {Z}}} \frac{{\mathbf {1}}_{\{|\kappa _2({\bar{n}}) -m|\le 1\}}}{\langle m \rangle \langle n_{123} \rangle \langle n_{12} \rangle ^\beta } \prod _{j=1}^3 \frac{1}{\langle n_j \rangle }. \end{aligned}$$

Let ${\mathcal {P}}$ be a pairing on $\{1, \cdots , 7\}$ which respects the partition $\big \{\{1,2,3\}, \{4, 5, 6\}, \{7\}\big \}$. Then, we have

$$\begin{aligned}&\sum _{\{n_j\}_{j \notin {\mathcal {P}}}} \langle n_{nr } \rangle ^{2(s-1)} \bigg ( \sum _{\{n_j\}_{j \in {\mathcal {P}}}} {\mathbf {1}}_{|n_{1234567}|\sim N_{1234567}} \cdot {\mathbf {1}}_{|n_{1237}|\sim N_{1237}} \cdot {\mathbf {1}}_{|n_{456}|\sim N_{456}} \cdot {\mathbf {1}}_{|n_7|\sim N_7}\\&\qquad \times {\mathbf {1}}_{\begin{array}{c} (n_1, \dots , n_7) \\ admissible \end{array}} \cdot \frac{{\mathcal {K}} (n_1,n_2,n_3) {\mathcal {K}} (n_4,n_5,n_6)}{ \langle n_7 \rangle }\bigg )^2\\&\quad \lesssim N_{\max }^{2 s- 1 + \varepsilon } \end{aligned}$$

for any $\varepsilon > 0$, uniformly in dyadic $N_{1234567}, N_{1237}, N_{456}, N_7 \ge 1$ and $\varepsilon _{123}, \varepsilon _j \in \{- 1, 1\}$ for $j = 1,2, 3$, where $N_{\max } = \max (N_1, \cdots , N_7)$ and $n_{nr }$ is as in (4.55).

Appendix B: Multiple stochastic integrals

In this section, we go over the basic definitions and properties of multiple stochastic integrals. See [43] and also [12, Section 4] for further discussion.

Let $\lambda$ be the measure on $Z: = {\mathbb {Z}}^3 \times {\mathbb {R}}_+$ defined by

$$\begin{aligned} d \lambda = d n d t, \end{aligned}$$

where dn is the counting measure on ${\mathbb {Z}}^3$. Given $k \in {\mathbb {N}}$, we set $\lambda _k = \bigotimes _{j =1}^k \lambda$ and $L^2(Z^k) = L^2( ({\mathbb {Z}}^3 \times {\mathbb {R}}_+)^k, \lambda _k )$. Given a function $f \in L^2(Z^k)$, we can adapt the discussion in [43, Section 1.1] (in particular, [43, Example 1.1.2]) to the complex-valued setting and define the multiple stochastic integral $I_k[f]$ by

$$\begin{aligned} I_k [f] = \sum _{n_1,\dots ,n_k \in {\mathbb {Z}}^3} \int _{[0,\infty )^k} f( n_1, t_1, \dots , n_k, t_k) d B_{n_1} (t_1) \cdots d B_{n_k} (t_k). \end{aligned}$$

Given a function $f \in L^2(Z^k)$, we define its symmetrization ${{\,\mathrm{\mathtt {Sym}}\,}}(f)$ by

$$\begin{aligned} {{\,\mathrm{\mathtt {Sym}}\,}}(f)(z_1, \dots , z_k) = \frac{1}{k!} \sum _{\sigma \in S_k} f( z_{\sigma (1)}, \dots ,z_{\sigma (k)} ), \end{aligned}$$

(B.1)

where $z_j = (n_j, t_j)$ as in (4.15) and $S_k$ denotes the symmetric group on $\{1, \dots , k\}$. Note that by Jensen’s equality, we have

$$\begin{aligned} |{{\,\mathrm{\mathtt {Sym}}\,}}(f)(z_1, \dots , z_k)|^p \le \frac{1}{k!} \sum _{\sigma \in S_k} |f( z_{\sigma (1)}, \dots ,z_{\sigma (k)} )|^p \end{aligned}$$

(B.2)

for any $p \ge 1$. We say that f is symmetric if ${{\,\mathrm{\mathtt {Sym}}\,}}(f) = f$. We now recall some basic properties of multiple stochastic integrals.

Lemma B.1

Let $k, \ell \in {\mathbb {N}}$. The following statements hold for any $f \in L^2(Z^k)$ and $g \in L^2(Z^\ell )$:

• (i) $I_k : L^2(Z^k) \rightarrow {\mathcal {H}}_k \subset L^2(\Omega )$ is a linear operator, where ${\mathcal {H}}_k$ denotes the kth Wiener chaos.

• (ii) $I_k[ \mathtt {Sym}(f)] = I_k[ f]$.

• (iii) Ito isometry:

  

• (iv) Furthermore, suppose that f is symmetric. Then, we have

  $$\begin{aligned} I_k[f] = k! \sum _{n_1, \cdots , n_k \in {\mathbb {Z}}^3} \int _{0}^\infty \int _{0}^{t_1} \int _{0}^{t_{k-1}} f(n_1,t_1, \dots , n_k,t_k) dB_{n_k}(t_k) \cdots dB_{n_1}(t_1), \end{aligned}$$

  where the iterated integral on the right-hand side is understood as an iterated Ito integral.

We state a version of Fubini’s theorem for multiple stochastic integrals that is convenient for our purpose. See, for example, [15, Theorem 4.33] for a version of the stochastic Fubini theorem.

Lemma B.2

Let $k \ge 1$. Given finite $T> 0$, let $f \in L^2 ( ({\mathbb {Z}}^3 \times [0, T])^k \times [0, T], d \lambda _k \otimes dt \big )$. (In particular, we assume that the temporal support (for the variables $t_1, \dots , t_k, t$) of f is contained in $[0, T]^{k+1}$ for any $(n_1, \dots , n_k)$.) Then, we have

$$\begin{aligned} \int _{0}^T I_k [f(\cdot ,t)] d t = I_k \bigg [ \int _{0}^T f(\cdot ,t) d t \bigg ] \end{aligned}$$

(B.3)

in $L^2(\Omega )$.

Proof

From Lemma B.1 (ii), we may assume that $f(z_1, \dots , z_k, t)$ is symmetric in $z_j = (n_j, t_j)$, $j = 1, \dots , k$. Let $\pmb {n} = (n_1, \dots , n_k)$ and $\pmb {t} = (t_1, \dots , t_k)$. From Minkowski’s integral inequality, Lemma B.1 (iii), and Cauchy–Schwarz’s inequality, we have

$$\begin{aligned} \begin{aligned} \Big \Vert \int _0^T I_k [(f-\varphi )(\cdot ,t)] d t \Big \Vert _{L^{2}(\Omega )}&\lesssim \int _0^T \Vert (f - \varphi )(\cdot ,t) \Vert _{\ell ^2_{\pmb {n}} (({\mathbb {Z}}^3)^k; L^2_{\pmb {t}}([0, T]^k))}dt \\&\le T^\frac{1}{2} \Vert f - \varphi \Vert _{\ell ^2_{\pmb {n}}(({\mathbb {Z}}^3)^k; L_{t, \pmb {t}}^2([0, T]^{k+1}))}. \end{aligned} \end{aligned}$$

(B.4)

On the other hand, by Lemma B.1 (iii) and Cauchy–Schwarz’s inequality, we have

$$\begin{aligned} \begin{aligned} \bigg \Vert I_k \Big [ \int _0^T (f -\varphi )(\cdot ,t) d t \Big ] \bigg \Vert _{L^2(\Omega )}&\sim \bigg \Vert \int _0^T (f -\varphi )(\cdot ,t) d t \bigg \Vert _{\ell ^2_{\pmb {n}} (({\mathbb {Z}}^3)^k; L^2_{\pmb {t}}({\mathbb {R}}_+^k))}\\&\le T^\frac{1}{2} \Vert f - \varphi \Vert _{\ell ^2_{\pmb {n}}(({\mathbb {Z}}^3)^k; L_{t, \pmb {t}}^2([0, T]^{k+1}))}. \end{aligned} \end{aligned}$$

(B.5)

Hence, it follows from (B.4), (B.5) and the density²¹ of $\ell ^2_{\pmb {n}}(({\mathbb {Z}}^3)^k;C^\infty _{t, \pmb {t}}([0, T]^{k+1}))$ in $\ell ^2_{\pmb {n}}(({\mathbb {Z}}^3)^k; L_{t, \pmb {t}}^2([0, T]^{k+1}))$ that we may assume that f is symmetric and belongs to $\ell ^2_{\pmb {n}}(({\mathbb {Z}}^3)^k;C^\infty _{t, \pmb {t}}([0, T]^{k+1}))$. Furthermore, we may assume that f has a compact support in $\pmb {n}$. Namely, there exists $K > 0$ such that if $\max (|n_1|, \dots , |n_k|) > K$, then $f(n_1, t_1, \dots , n_k, t_k, t) = 0$ for any $t_1, \dots , t_k, t \in [0, T]$. Then, together with Lemma B.1 (iv), we have

$$\begin{aligned} \begin{aligned}&\int _0^T I_k [f(\cdot ,t)] dt \\&\quad = k! \int _0^T \sum _{\begin{array}{c} n_1, \dots , n_k \in {\mathbb {Z}}^3\\ \max (|n_1|, \dots , |n_k|) \le K \end{array}}\int _{0}^{T} \int _{0}^{t_1} \cdots \int _{0}^{t_{k-1}} f(z_1, \dots ,z_k,t) d B_{n_k}(t_k) \cdots d B_{n_1}(t_1)dt \\&\quad = k! \sum _{\begin{array}{c} n_1, \dots , n_k \in {\mathbb {Z}}^3\\ \max (|n_1|, \dots , |n_k|) \le K \end{array}} \int _0^T \int _0^T \int _0^{t_1} \cdots \int _{0}^{t_{k-1}} f(z_1, \dots ,z_k,t) d B_{n_k}(t_k) \cdots d B_{n_1}(t_1)dt, \end{aligned} \end{aligned}$$

(B.6)

since the summation is over a finite set of indices $\pmb {n} = (n_1, \dots , n_k)$ and f is symmetric. Hence, it remains to justifying the t-integration with the stochastic integrals for each fixed $\pmb {n} = (n_1, \dots , n_k)$. For this reason, we suppress the dependence of f on $\pmb {n} = (n_1, \dots , n_k)$ in the following.

When $k = 1$, we can exploit the smoothness of f and have

$$\begin{aligned} \int _0^T \int _{0}^{T} f(t_1, t) d B_{n_1}(t_1) dt&= \int _0^T f(T, t) B_{n_1}(T) dt - \int _0^T \int _{0}^{T} B_{n_1}(t_1) \partial _{t_1} f (t_1, t) dt_1 dt \\&= B_{n_1}(T) \int _0^T f(T, t) dt - \int _{0}^{T} B_{n_1}(t_1) \partial _{t_1} \bigg (\int _0^T f (t_1, t)dt \bigg ) dt_1 \\&= \int _0^T \int _{0}^{T} f(t_1, t) dt d B_{n_1}(t_1), \end{aligned}$$

where, at the second equality, we used the standard Fubini’s theorem in view of the almost sure boundedness of $B_{n_1}$ on [0, T]. This proves (B.3) when $k = 1$.

For the general case, let us first consider the innermost integral in (B.6). For notational simplicity, let us suppress all the variables of f except for $t_k$ and t. Let $\Delta _m = \{ 0 \le \tau _0< \tau _1< \cdots < \tau _m \le T\}$ be a partition of [0, T] and define a step function $f_m(\cdot , t)$ by setting $f_m(\tau , t) = f(\tau _{j-1}, t)$ for $\tau _{j-1} < \tau \le \tau _j$. Then, by defining $J_m$ by

$$\begin{aligned} J_m(t) := \int _0^{t_{k-1}}f_m(t_k, t) dB_{n_{k}}(t_k) = \sum _{j = 1}^{m} ({\mathbf {1}}_{[0, t_{k-1}]} f)(\tau _{j-1}, t) \big (B_{n_k}(\tau _j) - B_{n_k}(\tau _{j-1})\big ), \end{aligned}$$

(B.7)

it follows from the definition of the Wiener integral that

$$\begin{aligned} J_m(t) \longrightarrow \int _{0}^{t_{k-1}} f(t_k,t) d B_{n_k}(t_k) \quad \text { in } L^2(\Omega ), \end{aligned}$$

(B.8)

as $m \rightarrow \infty$ (such that $|\Delta _m|\rightarrow 0$). By integrating (B.7) in t, we have

$$\begin{aligned} \int _0^T J_m(t) dt = \sum _{j = 1}^{m} \bigg (\int _0^T ({\mathbf {1}}_{[0, t_{k-1}]} f)(\tau _{j-1}, t) dt \bigg ) \big (B_{n_k}(\tau _j) - B_{n_k}(\tau _{j-1})\big ). \end{aligned}$$

(B.9)

By the definition of the Wiener integral once again, we have

$$\begin{aligned} \text {RHS of } (B.9) \longrightarrow \int _{0}^{t_{k-1}} \int _0^Tf(t_k,t)dt d B_{n_k}(t_k) \quad \text { in } L^2(\Omega ), \end{aligned}$$

(B.10)

while from Minkowski’s integral inequality, (B.8), and the bounded convergence theorem (recall that f is smooth), we have

$$\begin{aligned} \begin{aligned}&\bigg \Vert \int _0^T J_m(t) dt - \int _0^T \int _{0}^{t_{k-1}} f(t_k,t) d B_{n_k}(t_k) dt \bigg \Vert _{L^2(\Omega )}\\&\quad \le \int _0^T \Big \Vert J_m(t) - \int _{0}^{t_{k-1}} f(t_k,t) d B_{n_k}(t_k) \Big \Vert _{L^2(\Omega )} dt \longrightarrow 0, \end{aligned} \end{aligned}$$

(B.11)

as $m \rightarrow \infty$. Hence, from (B.9), (B.10), and (B.11), we conclude that

$$\begin{aligned} \int _0^T \int _{0}^{t_{k-1}} f(t_k,t) d B_{n_k}(t_k) dt = \int _{0}^{t_{k-1}} \int _0^T f(t_k,t) dt d B_{n_k}(t_k)\quad \text { in } L^2(\Omega ). \end{aligned}$$

(B.12)

Next, we consider

$$\begin{aligned} \begin{aligned}&\int _0^{t_{k-2}} \int _0^T F(t_{k-1}, t) dt dB_{n_{k-1}}(t_{k-1}) \\&\quad : = \int _0^{t_{k-2}} \int _0^T\bigg ( \int _{0}^{t_{k-1}} f(t_{k-1}, t_k,t) d B_{n_k}(t_k) \bigg )dt dB_{n_{k-1}}(t_{k-1}) . \end{aligned} \end{aligned}$$

(B.13)

Given the partition $\Delta _m$ of [0, T] as above, we define an adaptive step function $F_m(\cdot , t)$ by setting $F_m(\tau , t;\omega ) = F(\tau _{j-1}, t;\omega )$ for $\tau _{j-1} < \tau \le \tau _j$. Then, we can simply repeat the previous computation (but with Ito integrals instead of Wiener integrals) and obtain

$$\begin{aligned} \int _0^T \int _{0}^{t_{k-2}} F(t_{k-1},t) d B_{n_{k-1}}(t_{k-1}) dt = \int _{0}^{t_{k-2}} \int _0^T F(t_{k-1},t) dt d B_{n_{k-1}}(t_{k-1}) \end{aligned}$$

(B.14)

in $L^2(\Omega )$. Combining (B.13) and (B.14) with (B.12), we then obtain

$$\begin{aligned}&\int _0^T \int _0^{t_{k-2}} \int _{0}^{t_{k-1}} f(t_{k-1}, t_k,t) d B_{n_k}(t_k) dB_{n_{k-1}}(t_{k-1})dt \\&\quad = \int _0^{t_{k-2}} \int _{0}^{t_{k-1}} \int _0^T f(t_{k-1}, t_k,t)dt d B_{n_k}(t_k) dB_{n_{k-1}}(t_{k-1}) \end{aligned}$$

in $L^2(\Omega )$. By iterating this process, we conclude

$$\begin{aligned}&\int _0^T \int _0^T \int _0^{t_1} \cdots \int _{0}^{t_{k-1}} f(t_1, \dots ,t_k,t) d B_{n_k}(t_k) \cdots d B_{n_1}(t_1)dt\\&\quad = \int _0^T \int _0^{t_1} \cdots \int _{0}^{t_{k-1}} \int _0^Tf(t_1, \dots ,t_k,t)dt d B_{n_k}(t_k) \cdots d B_{n_1}(t_1) \end{aligned}$$

in $L^2(\Omega )$. Together with (B.6), this proves (B.3). $\square$

We conclude this section by stating the product formula (Lemma B.4). Before doing so, we first recall the contraction of two functions.

Definition B.3

Let $k, \ell \in {\mathbb {N}}$. Given an integer $0 \le r \le \min (k,l)$, we define the contraction $f \otimes _r g$ of r indices of $f\in L^2(Z^k)$ and $g\in L^2(Z^\ell )$ by

$$\begin{aligned} (f \otimes _r g)(z_1, \dots , z_{k + \ell - 2 r})&= \sum _{m_1, \dots , m_r \in {\mathbb {Z}}^3} \int _{{\mathbb {R}}_+^r} f(z_1, \dots , z_{k-r}, \zeta _1, \dots , \zeta _r)\\&\quad \times g(z_{k+1-r}, \dots , z_{k+\ell -2r}, \widetilde{\zeta }_1, \dots , \widetilde{\zeta }_r ) ds_1 \cdots ds_r, \end{aligned}$$

where $\zeta _j = (m_j, s_j)$ and $\widetilde{\zeta }_j = (-m_j, s_j)$.

Note that even if f and g are symmetric, their contraction $f\otimes _r g$ is not symmetric in general. We now state the product formula. See [43, Proposition 1.1.3].

Lemma B.4

(product formula) Let $k, \ell \in {\mathbb {N}}$. Let $f \in L^2(Z^k)$ and $g \in L^2(Z^\ell )$ be symmetric functions. Then, we have

$$\begin{aligned}&I_k[f] \cdot I_{\ell }[g] = \sum _{r=0}^{\min (k,\ell )} r! \left( {\begin{array}{c}k\\ r\end{array}}\right) \left( {\begin{array}{c}\ell \\ r\end{array}}\right) I_{k + \ell - 2r}[f \otimes _r g]. \end{aligned}$$

Appendix C: Random tensors

In this section, we provide the basic definition and some lemmas on (random) tensors from [12, 18]. See [18, Sections 2 and 4] and [12, Section 4] for further discussion.

Definition C.1

Let A be a finite index set. We denote by $n_A$ the tuple $(n_j : j \in A)$. A tensor $h = h_{n_A}$ is a function: $({\mathbb {Z}}^3)^{A} \rightarrow {\mathbb {C}}$ with the input variables $n_A$. Note that the tensor h may also depend on $\omega \in \Omega$. The support of a tensor h is the set of $n_A$ such that $h_{n_A} \ne 0$.

Given a finite index set A, let (B, C) be a partition of A. We define the norms $\Vert \cdot \Vert _{n_A}$ and $\Vert \cdot \Vert _{n_{B} \rightarrow n_{C}}$ by

$$\begin{aligned} \Vert h \Vert _{n_A} = \Vert h\Vert _{\ell ^2_{n_A}} = \bigg (\sum _{n_A} |h_{n_A}|^2\bigg )^\frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \Vert h \Vert ^2_{n_{B} \rightarrow n_{C}} = \sup \bigg \{ \sum _{n_{C}} \Big | \sum _{n_{B}} h_{n_A} f_{n_{B}} \Big |^2 : \Vert f \Vert _{\ell ^2_{n_{B}}} =1 \bigg \}, \end{aligned}$$

(C.1)

where we used the short-hand notation $\sum _{n_Z}$ for $\sum _{n_Z \in ({\mathbb {Z}}^3)^Z}$ for a finite index set Z. Note that, by duality, we have $\Vert h \Vert _{n_{B} \rightarrow n_{C}} = \Vert h \Vert _{n_{C} \rightarrow n_{B}} = \Vert \overline{h} \Vert _{n_{B} \rightarrow n_{C}}$ for any tensor $h = h_{n_A}$. If $B = \varnothing$ or $C = \varnothing$, then we have $\Vert h \Vert _{n_{B} \rightarrow n_{C}} = \Vert h \Vert _{n_A}$.

For example, when $A = \{1, 2\}$, the norm $\Vert h \Vert _{n_{1} \rightarrow n_{2}}$ denotes the usual operator norm $\Vert h \Vert _{\ell ^2_{n_{1}} \rightarrow \ell ^2_{n_{2}}}$ for an infinite dimensional matrix operator $\{h_{n_1 n_2}\}_{n_1, n_2 \in {\mathbb {Z}}^3}$. By bounding the matrix operator norm by the Hilbert–Schmidt norm (= the Frobenius norm), we have

$$\begin{aligned} \Vert h \Vert _{\ell ^2_{n_{1}} \rightarrow \ell ^2_{n_{2}}} \le \Vert h\Vert _{\ell ^2_{n_1, n_2}} \end{aligned}$$

(C.2)

Let (B, C) be a partition of A. Then, by duality, we can write (C.1) as

$$\begin{aligned} \Vert h \Vert _{n_{B} \rightarrow n_{C}} = \sup \bigg \{ \sum _{n_{C}} \Big | \sum _{n_{B}} h_{n_A} f_{n_{B}} g_{n_C}\Big | : \Vert f \Vert _{\ell ^2_{n_{B}}} = \Vert g \Vert _{\ell ^2_{n_{C}}} =1 \bigg \}, \end{aligned}$$

from which we obtain

$$\begin{aligned} \sup _{n_A}|h_{n_A}| = \sup _{n_B, n_C}|h_{n_Bn_C}| \le \Vert h \Vert _{n_{B} \rightarrow n_{C}}. \end{aligned}$$

(C.3)

Next, we recall a key deterministic tensor bound in the study of the random cubic NLW from [12].

Lemma C.2

(Lemma 4.33 in [12])

Let $s < \frac{1}{2}+ \beta$ for some $\beta > 0$. Given $\varepsilon _j \in \{-1, 1\}$ for $j = 0,1,2,3$, let $\kappa ({\bar{n}})$ be as in (4.7). For $m \in {\mathbb {Z}}$, define the tensor $h^m$ by

$$\begin{aligned} h^m_{nn_1n_2n_3} = \bigg (\prod _{j=1}^3{\mathbf {1}}_{\begin{array}{c} |n_j|\sim N_j\\ |n_j|\le N \end{array}}\bigg ) {\mathbf {1}}_{\{|\kappa ({\bar{n}}) - m|\le 1\}} \frac{ \langle n \rangle ^{s-1}}{\langle n_{12} \rangle ^\beta \langle n_1 \rangle \langle n_2 \rangle \langle n_3 \rangle ^{\frac{1}{2}}}. \end{aligned}$$

Then, there exists $\delta _0 > 0$ such that

$$\begin{aligned}&\max \Big ( \Vert h^m \Vert _{n_1n_2n_3 \rightarrow n}, \Vert h^m \Vert _{n_3 \rightarrow nn_1n_2}, \Vert h^m \Vert _{n_1n_3 \rightarrow nn_2}, \Vert h^m \Vert _{n_2n_3 \rightarrow nn_1} \Big ) \\&\quad \lesssim \max (N_1, N_2, N_3)^{-\delta _0}, \end{aligned}$$

uniformly in $N \ge 1$, $m\in {\mathbb {Z}}$, dyadic $N_1,N_2,N_3 \ge 1$, and $\varepsilon _j \in \{-1, 1 \}$ for $j = 0,1,2,3$.

We conclude this section with the following random matrix estimate. This lemma is essentially Propositions 2.8 and 4.14 in [18]; see also Proposition 4.50 in [12]. In our stochastic PDE setting, however, we need a slightly different formulation (in particular, adapted to multiple stochastic integrals with general integrands) and thus for readers’ convenience, we present its proof.

Let A be a finite index set. As in (4.15) and (4.16), we set $z_A = (k_A,t_A)$ for $(k_A, t_A) \in ({\mathbb {Z}}^3)^A\times {\mathbb {R}}^A$ and write $f_{z_A} = f(z_A) = f(n_A, t_A)$.

Lemma C.3

Let A be a finite index set with $k = |A| \ge 1$. Let $h = h_{bcn_A}$ be a tensor such that $n_j \in {\mathbb {Z}}^3$ for each $j \in A$ and $(b,c) \in ({\mathbb {Z}}^3)^d$ for some integer $d \ge 2$. Given $N \ge 1$, assume that

$$\begin{aligned} {{\,\mathrm{supp}\,}}h \subset \big \{ |b|, |c|, |n_j| \lesssim N \text { for each }j \in A \big \}. \end{aligned}$$

(C.4)

Given a (deterministic) tensor $h_{bcn_A} \in \ell ^2_{bcn_A}$, define the tensor $H = H_{bc}$ by

$$\begin{aligned} H_{bc} = I_k \big [ h_{bcn_A} f_{z_A} \big ] \end{aligned}$$

(C.5)

for $f \in \ell ^{\infty }_{n_A}(({\mathbb {Z}}^3)^A; L^2_{t_A}({\mathbb {R}}_+^A ))$, where $I_k$ denotes the multiple stochastic integral defined in Appendix B. Then, for any $\theta > 0$, we have

$$\begin{aligned} \big \Vert \Vert H_{bc} \Vert _{b \rightarrow c} \big \Vert _{L^p(\Omega )} \lesssim p^{\frac{k}{2}} N^{\theta }\Big ( \max _{(B,C)} \Vert h \Vert _{b n_B \rightarrow c n_C}\Big ) \Vert f(n_A, t_A)\Vert _{\ell ^{\infty }_{n_A} L^2_{t_A}}, \end{aligned}$$

(C.6)

where the maximum is taken over all partitions (B, C) of A.

Remark C.4

• (i) The assumption that $h_{bcn_A} \in \ell ^2_{bcn_A}$ and $f \in \ell ^{\infty }_{n_A}(({\mathbb {Z}}^3)^A; L^2_{t_A}({\mathbb {R}}_+^A ))$ ensures that the multiple stochastic integral $I_k \big [ h_{bcn_A} f_{z_A}\big ]$ in (C.5) is well defined. Note that if for instance we have a stronger condition $f \in \ell ^2 \big ( ({\mathbb {Z}}^3)^A; L^{2}({\mathbb {R}}_+^A) \big )$, then the conclusion (C.6) trivially holds without any loss in N. We also note that even if the tensor h is random, Lemma C.3 holds with the same proof as long as h is independent of the Brownian motions $\{B_{n_A}\}$ defining multiple stochastic integrals.

• (ii) By translation invariance, we may replace the condition (C.4) in Lemma C.3 by

  $$\begin{aligned} {{\,\mathrm{supp}\,}}h \subset \big \{ |b - b_*|, |c - c_*|, |n_j - n_{j, *}| \lesssim N \text { for each }j \in A \big \} \end{aligned}$$

  for some $(b_*, c_* ) \in ({\mathbb {Z}}^3)^d$ and $n_{j, *} \in {\mathbb {Z}}^3$, $j \in A$.

Proof of Lemma C.3

We follow the proof of Proposition 4.14 in [18] and use a higher order version of Bourgain’s $TT^*$-argument [9]. Let $T : \ell ^2_c \rightarrow \ell ^2_b$ be the linear operator whose kernel is $H_{bc}$. Namely, T is defined by

$$\begin{aligned} (Tg)_b = \sum _{c} H_{bc} g_c, \quad g \in \ell ^2_c. \end{aligned}$$

(C.7)

For $j \in {\mathbb {N}}$, we define the operator $T_j$ by $T_j = (T T^*)^m$ if $j = 2m$, and $T_j = (TT^*)^m T$ if $j = 2m +1$. We claim that $T_j$ has a kernel which is given by a linear combination of terms ${\mathcal {T}}_j$ of the form

$$\begin{aligned} {\mathcal {T}}_j = {\left\{ \begin{array}{ll} I_{\ell } \big [y_{bb'}(z_D)\big ], &{} \text {when }j\text { is even}, \\ I_{\ell } \big [y_{bc}(z_D)\big ],&{} \text {when }j\text { is odd}, \end{array}\right. } \end{aligned}$$

(C.8)

for some finite index set D and $\ell = |D| \le k j$, where $y_{bb'}(z_D)$ (or $y_{bc}(z_D)$) satisfies the following bound:

$$\begin{aligned} \begin{aligned}&\Vert y_{bb'}(z_D) \Vert _{\ell ^2_{bcn_D} L^2_{t_D}} \quad \big (\text {or }\Vert y_{bc}(z_D) \Vert _{\ell ^2_{bcn_D} L^2_{t_D}}\big ) \\&\quad \lesssim \Big ( \max _{(B,C)} \Vert h \Vert _{b n_B \rightarrow c n_C} \Big )^{j-1} \Vert h_{bcn_A} \Vert _{\ell ^2_{bcn_A}} \Vert f(n_A, t_A)\Vert _{\ell ^{\infty }_{n_A} L^2_{t_A}}^j. \end{aligned} \end{aligned}$$

(C.9)

where the maximum is taken over all partitions (B, C) of A. Here, the implicit constant depends on k, $\ell$, and j. While it grows with j (and $\ell$), this does not cause an issue since for a given small $\theta > 0$ in (C.6), we fix $j = j(\theta ) \gg 1$.

Let $j = 1$. In this case, comparing (C.8) with (C.7) and (C.5) and using Lemma B.1 (ii), we have $y_{bc}(z_D) = {{\,\mathrm{\mathtt {Sym}}\,}}(h_{bc n_A} f(z_A))$ with $D = A$ and thus the bound (C.9) follows from Hölder’s inequality. Note that, in this case, it follows from Lemma B.1 (iii) that

$$\begin{aligned} \Vert y_{bc}(z_A) \Vert _{\ell ^2_{bcn_A} L^2_{t_A}} = (k!)^{-1} \big \Vert \Vert H_{bc}\Vert _{\ell ^2_{bc}}\big \Vert _{L^2(\Omega )}, \end{aligned}$$

where the right-hand side is the second moment of the Hilbert–Schmidt norm of the operator T. By taking higher powers $T_j$, we control the operator norm of T.

Now, assume that the claim with (C.8) and (C.9) hold true for $j - 1$. We assume that j is odd. The proof for even j is analogous. Noting that $T_j = T_{j-1}T$, it follows from the inductive hypothesis (C.8) with (C.5) and Lemma B.1 (ii) that the kernel for $T_j$ is given by a linear combination of terms ${\mathcal {T}}_j$ of the form

$$\begin{aligned} \begin{aligned} ({\mathcal {T}}_j)_{bc}&= \sum _{b'} ({\mathcal {T}}_{j-1})_{bb'} H_{b'c} \\&= \sum _{b'} I_{\ell } \big [ y_{bb'}(z_D) \big ] \cdot I_k \big [ h_{b'c n_A} f(z_A) \big ]\\&= \sum _{b'} I_{\ell } \big [ {{\,\mathrm{\mathtt {Sym}}\,}}(y_{bb'}(z_D)) \big ] \cdot I_k \big [ {{\,\mathrm{\mathtt {Sym}}\,}}(h_{b'c n_A} f(z_A)) \big ] . \end{aligned} \end{aligned}$$

Then, from the product formula (Lemma B.4), we have

$$\begin{aligned} ({\mathcal {T}}_j)_{bc} = \sum _{r=0}^{\min (k,\ell )} r! \left( {\begin{array}{c}k\\ r\end{array}}\right) \left( {\begin{array}{c}\ell \\ r\end{array}}\right) I_{k + \ell - 2r} \bigg [ \sum _{b'} \big ( {{\,\mathrm{\mathtt {Sym}}\,}}(y_{bb'}) \otimes _r {{\,\mathrm{\mathtt {Sym}}\,}}(h_{b'c} f) \big ) \bigg ]. \end{aligned}$$

Hence, it suffices to show that $\sum _{b'} ( {{\,\mathrm{\mathtt {Sym}}\,}}(y_{bb'}) \otimes _r {{\,\mathrm{\mathtt {Sym}}\,}}(h_{b'c} f) )$ satisfies (C.9) for each $0 \le r \le \min (k,\ell )$. For notational simplicity, we drop ${{\,\mathrm{\mathtt {Sym}}\,}}$ in ${{\,\mathrm{\mathtt {Sym}}\,}}(y_{bb'})$ and ${{\,\mathrm{\mathtt {Sym}}\,}}(h_{b'c} f)$ in the following. Note that this does not cause any issue since, in taking the $L^2(\Omega )$-norm, we can remove ${{\,\mathrm{\mathtt {Sym}}\,}}$ by Jensen’s inequality (B.2) as in Sect. 4.

Fix $0 \le r \le \min (k, \ell )$. From Definition B.3 on the contraction, we have

$$\begin{aligned} \begin{aligned}&\big ( y_{bb'} \otimes _r h_{b'c} f \big )(z_B) \\&\quad = \sum _{n_C} \int _{{\mathbb {R}}_+^r} y_{bb'}(z_{B_1}, z_{C} )\cdot (h_{b'c} f)(z_{B_2}, \widetilde{z}_C )dt_C, \end{aligned} \end{aligned}$$

(C.10)

where $\widetilde{z}_C = (- n_C, t_C)$ for given $z_C = (n_C, t_C)$. Here, $B_1$, $B_2$, and C are pairwise disjoint sets such that $|B_1| = \ell - r$, $|B_2| = k - r$, $|C| = r$, $B = B_1 \cup B_2$, and (by suitable relabeling of indices)

$$\begin{aligned} B_1 \cup C = D \quad \text {and}\quad B_2 \cup C = A. \end{aligned}$$

(C.11)

Then, from (C.10), Cauchy–Schwarz’s inequality (in $t_C$), Minkowski’s integral inequality (with $L^2_{t_B} = L^2_{t_{B_1}}L^2_{t_{B_2}}$, (C.1), and the identification in (C.11), we have

$$\begin{aligned}&\Big \Vert \sum _{b'} \big ( y_{bb'} \otimes _r h_{b'c} f \big )(n_B, t_B) \Big \Vert _{\ell ^2_{bc n_B}L^2_{t_B}}\nonumber \\&\quad \le \Big \Vert \sum _{b', n_C} \Vert y_{bb'}(n_{B_1}, t_{B_1}, n_{C}, t_{C} )\Vert _{L^2_{t_{B_1}t_{C}}}\cdot \Vert (h_{b'c} f)(n_{B_2},t_{B_2},-n_C, t_C )\Vert _{L^2_{t_{B_2}t_C}} \Big \Vert _{\ell ^2_{bc n_{B}} } \nonumber \\&\quad \le \Vert y_{bb'}(n_{B_1}, t_{B_1}, n_C, t_C ) \Vert _{\ell ^2_{bb'n_{B_1} n_C}L^2_{t_{B_1}t_C} } \Big \Vert \Vert (h_{b'c} f)(z_{B_2}, z_C)\Vert _{L^2_{t_{B_2}t_C}} \Big \Vert _{b' n_C \rightarrow cn_{B_2}} \nonumber \\&\quad = \Vert y_{bb'}(z_D ) \Vert _{\ell ^2_{bb'n_D} L^2_{t_D}} \Big \Vert h_{b'cn_A} \Vert f(z_{A} )\Vert _{L^2_{t_A}} \Big \Vert _{b' n_C \rightarrow cn_{A\setminus C}} . \end{aligned}$$

(C.12)

Moreover, from (C.1), we have

$$\begin{aligned} \begin{aligned} \Big \Vert h_{b'cn_A} \Vert f(z_{A} )\Vert _{L^2_{t_A}} \Big \Vert _{b' n_C \rightarrow cn_{A\setminus C}}&\le \Vert h_{b'c n_A } \Vert _{b' n_C \rightarrow cn_{A\setminus C}} \Vert f(n_A, t_A)\Vert _{\ell ^{\infty }_{n_A} L^2_{t_A}} \\&\le \Big (\max _{(A_1,A_2)} \Vert h_{bc n_A } \Vert _{bn_{A_1} \rightarrow cn_{A_2}} \Big ) \Vert f(n_A, t_A)\Vert _{\ell ^{\infty }_{n_A} L^2_{t_A}}, \end{aligned} \end{aligned}$$

(C.13)

where the maximum is taken over all partitions $(A_1,A_2)$ of A. Hence, from (C.12), (C.13), and the inductive hypothesis (C.9) (with $j -1$ in place of j), we obtain (C.9) for j. Therefore, by induction, the claim holds for any $j \in {\mathbb {N}}$.

We are now ready to prove (C.6). Consider the product $T_{2m} = (TT^*)^m$ for $m \ge 1$. Let us denote by ${\mathcal {R}}_{2m}$ the kernel of $T_{2m}$, which consists of terms ${\mathcal {T}}_j$, satisfying (C.8) and (C.9). Namely, we have

$$\begin{aligned} ({\mathcal {R}}_{2m})_{bb'} = \sum _{j = 1}^J I_{2k \ell _j} \big [y_{bb'}^{(j)}(z_{D^{(j)}})\big ] \end{aligned}$$

(C.14)

for some $J \ge 1$, $0 \le \ell _j \le m$, and $y_{bb'}^{(j)}$, satisfying (C.9). Note that we have ${\mathcal {R}}_{2m} \in {\mathcal {H}}_{\le 2mk}$. Then, by the standard $TT^*$ argument, (C.2), Minkowski’s integral inequality, (C.14), the Wiener chaos estimate (Lemma 2.9), Lemma B.1 (iii), and (C.9), we obtain

$$\begin{aligned} \big \Vert \Vert H_{bc} \Vert _{b \rightarrow c} \big \Vert _{L^p(\Omega )}= & {} \big \Vert \Vert H_{bc} \Vert _{b \rightarrow c}^{2m} \big \Vert _{L^{\frac{p}{2m}}(\Omega )}^{\frac{1}{2m}} = \big \Vert \Vert T \Vert _{\ell ^2_c \rightarrow \ell ^2_b }^{2m} \big \Vert _{L^{\frac{p}{2m}}(\Omega )} ^{\frac{1}{2m}} \nonumber \\= & {} \big \Vert \Vert (T T^*)^m\Vert _{\ell ^2_{b'} \rightarrow \ell ^2_{b} }\big \Vert _{L^{\frac{p}{2m}}(\Omega )} ^{\frac{1}{2m}} \le \big \Vert \Vert ({\mathcal {R}}_{2m})_{bb'} \Vert _{\ell ^2_{bb'}} \big \Vert _{L^{\frac{p}{2m}}(\Omega )} ^{\frac{1}{2m}} \nonumber \\\le & {} \big \Vert \Vert ({\mathcal {R}}_{2m})_{bb'} \Vert _{L^{\frac{p}{2m}}(\Omega )} \big \Vert _{\ell ^2_{bb'}} ^{\frac{1}{2m}} \nonumber \\\le & {} p^\frac{k}{2} \bigg ( \sum _{j = 1}^J \Big \Vert \big \Vert I_{2k \ell _j} \big [y_{bb'}^{(j)}(z_{D^{(j)}})\big ] \big \Vert _{L^{2}(\Omega )} \Big \Vert _{\ell ^2_{bb'}} \bigg )^{\frac{1}{2m}} \nonumber \\\lesssim & {} p^\frac{k}{2} \bigg ( \sum _{j = 1}^J \Vert y_{bb'}^{(j)}(z_{D^{(j)}}) \Vert _{\ell ^2_{bcn_{D^{(j)}}} L^2_{t_{D^{(j)}}}} \bigg )^{\frac{1}{2m}} \nonumber \\\lesssim & {} p^{\frac{k}{2}} \Big ( \max _{(B,C)} \Vert h \Vert _{b n_B \rightarrow c n_C} \Big )^{1 - \frac{1}{2m}} \Vert h \Vert _{\ell ^2_{bcn_A}}^{\frac{1}{2m}} \Vert f(n_A, t_A)\Vert _{\ell ^{\infty }_{n_A} L^2_{t_A}} \nonumber \\ \end{aligned}$$

(C.15)

for any $p \ge 4m$. Moreover, from (C.4) and (C.3), we have

$$\begin{aligned} \Vert h \Vert _{\ell ^2_{bcn_A}} \le N^{\frac{3}{2}(d + k)} \sup _{b,c,n_A} |h_{bcn_A}| \le N^{\frac{3}{2}(d + k)} \max _{(B,C)} \Vert h \Vert _{b n_B \rightarrow c n_C}. \end{aligned}$$

(C.16)

Therefore, by combining (C.15) and (C.16) and taking m sufficiently large, we obtain the desired bound (C.6). $\square$