Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

Abstract

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic $\Phi ^{4}_{3}$-model. This result is the hyperbolic counterpart to seminal works on the parabolic $\Phi ^{4}_{3}$-model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018).

The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates.

The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate.

Appendices

Appendix A: The nonlinear stochastic heat equation with sharp frequency-cutoffs

In Sect. 6.1, we discussed the frequency-truncated stochastic nonlinear heat equation (6.6), i.e.,

$$ \textstyle\begin{cases} \big(\partial _{s} + 1 - \Delta \big) \Phi _{\leq N}^{\cos} = - P_{ \leq N} \Big( :\hspace {-0.5ex}\big(P_{\leq N} \Phi _{\leq N}^{\cos} \big)^{3} \hspace {-0.5ex}:+ \gamma _{\leq N} \Phi ^{\cos}_{\leq N} \Big)+ \sqrt{2} \mathrm{d}W ^{\cos} \\ \quad (s,x) \in (s_{0},\infty ) \times \mathbb{T}^{3}, \\ \Phi ^{\cos}_{\leq N} \big|_{s=s_{0}}= \phi ^{\cos}. \end{cases} $$

(A.1)

In the proof of Proposition 3.5 at the end of Sect. 6.1, we claimed that the local well-posedness of (A.1) essentially follows from the previous literature [27, 62, 65]. Due to the sharp frequency-cutoffs in (A.1), however, some modifications are necessary.

The sharp frequency-projections $(P_{\leq N})_{N}$, which are based on cubes, are uniformly bounded on $L_{x}^{p}$ for every $1< p<\infty$ but unbounded on $L_{x}^{\infty}$. However, for every $\delta >0$, $(P_{\leq N})_{N}$ is uniformly bounded as a map from $\mathcal {C}_{x}^{\delta}$ to $L_{x}^{\infty}$. More generally, for every $\alpha \in \mathbb{R}$ and $\delta >0$, $(P_{\leq N})_{N}$ is uniformly bounded as a map from $\mathcal {C}_{x}^{\alpha +\delta}$ to $\mathcal {C}_{x}^{\alpha}$. Due to the smoothing properties of heat equations and the sub-criticality of (A.1), the resulting $\delta$-loss is acceptable, and most of the argument in [27] applies verbatim to (A.1).

The main technical difficulty concerns commutator terms, which do not exhibit any gains under the sharp frequency-cutoffs. To illustrate this, let $N\geq 1$, let $n=(N,0,0) \in \mathbb{Z}^{3}$, and $m=(1,0,0)\in \mathbb{Z}^{3}$. Then, it holds that

$$ \big[ P_{\leq N}, e^{i\langle m,x\rangle}\big] e^{i\langle n, x \rangle} = P_{\leq N}\big( e^{i \langle m, x \rangle} e^{i\langle n, x \rangle} \big) - e^{i \langle m, x \rangle} P_{\leq N}\big( e^{i \langle n, x \rangle} \big) = - e^{i \langle m+n , x \rangle}. $$

(A.2)

In particular, (A.2) exhibits no gain in $N$. Due to the missing commutator estimates, a few steps in [27] do not directly carry over to (A.1). For example, the decomposition in [27, (3.3)] has no direct counterpart in our setting. In the rest of this section, we show how the main estimate [27, Proposition 3.8] can be extended to our setting, but do not discuss any other (more minor) modifications.

1.1 A.1 Preparations

Let $0<\epsilon \ll \delta \ll 1$ be parameters. In (6.3), we defined the heat propagator $e^{-s (1-\Delta )}$ as the Fourier multiplier with symbol $n\mapsto e^{- s \langle n \rangle ^{2}}$. For any initial time $s_{0} \in \mathbb{R}$, we define the associated Duhamel integral as

$$ \operatorname{Duh}_{s_{0}} \big[ F \big](s) := \int _{s_{0}}^{s} \mathrm {d}s^{\prime }e^{-(s-s^{ \prime})(1-\Delta )} F(s^{\prime}). $$

We now recall the following Schauder-type estimate.

Lemma 1

Schauder-type estimate

For any $\alpha \in \mathbb{R}$, $\theta \in (0,1)$, $s_{0} \in [-1,0]$, and $F\colon [s_{0},0]\times \mathbb{T}^{3} \rightarrow \mathbb{R}$, it holds that

$$ \sup _{s\in [s_{0},1]} |s-s_{0}|^{\theta }\, \big\| \operatorname{Duh}_{s_{0}} \big[ F \big] \big\| _{\mathcal {C}_{x}^{\alpha +2-2\delta}} \lesssim \sup _{s \in [s_{0},1]} |s-s_{0}|^{\theta }\, \big\| F \big\| _{\mathcal {C}_{x}^{\alpha}}. $$

(A.3)

Furthermore, if $F(s_{0})=0$, we also have that

$$ \sup _{s\in [s_{0},1]} |s-s_{0}|^{\theta }\, \big\| \operatorname{Duh}_{s_{0}} \big[ \partial _{s} F \big] \big\| _{\mathcal {C}_{x}^{\alpha -2\delta}} \lesssim \sup _{s\in [s_{0},1]} |s-s_{0}|^{\theta }\, \big\| F \big\| _{ \mathcal {C}_{x}^{\alpha}}. $$

(A.4)

Except for a boundary term in $\operatorname{Duh}_{s_{0}}[\partial _{s} F]$, the $\delta$-loss in spatial derivatives can be used to obtain better weights in $|s-s_{0}|$.

Proof

The estimates essentially follow from [27, Proposition 2.7]. For the second estimate, we also use the identity

$$ \operatorname{Duh}_{s_{0}}\big[ F \big] = F - \operatorname{Duh}_{s_{0}} \big[ (1- \Delta ) F \big]. $$

 □

Furthermore, we define the following bilinear para-products.⁴⁰

Definition 2

Bilinear para-products

For any $f,g \colon \mathbb{T}^{3} \rightarrow \mathbb{R}$, we define

1.2 A.2 Main estimate

We now prove the analogue of the main estimate [27, Proposition 3.8] with sharp frequency-cutoffs. In order to precisely state the estimate, we use the Wick-ordered square

Proposition 3

For all $A\geq 1$, there exists an $A$-certain event $E_{A} \in \mathcal{E}$ on which the following estimate holds: For all $\theta \in (0,1)$, $N \geq 1$, $s_{0}\in [-1,0)$, and $\Psi \colon [s_{0},0] \times \mathbb{T}^{3} \rightarrow \mathbb{R}$, we have that

Remark 4

We make a few comments on Proposition A.3.

• (i) If instead of the nonlinear remainder $\Psi$ the low-frequency component is given by the cubic stochastic object

  

  the term can be treated as an explicit stochastic object and estimated more easily.

• (ii) The regularity condition on $\Psi$ on the right-hand side of (A.5) is far from optimal, but sufficient for our applications to (A.1). It is the result of a simple (but crude) Sobolev-type embedding.

• (iii) The local well-posedness theory of (A.1) also requires a minor variant of (A.5), where the Duhamel integral is estimated at a lower spatial regularity but with better pre-factors of $|s-s_{0}|$.

Proof

In the following, we denote the frequencies in the first and second factor of

by $(n_{1},n_{2})$ and $(n_{3},n_{4})$, respectively. We also denote the frequency of $\Psi$ by $n_{5}$. Inserting the definitions of the stochastic objects, Duhamel integral, and para-products, it follows that

where

(A.7)

In order to proceed with our estimates, we decompose $\mathcal {G}_{\leq N}$ into terms with zero, one, and two resonances. To this end, we define the fourth-order chaos by

(A.8)

the second-order chaos by

(A.9)

and the zeroth-order chaos (or constant) by

$$ \begin{aligned} &\mathcal {G}^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5}) \\ =& 18 \cdot 2^{4/2} e^{i\langle n_{5}, x \rangle}\\ \times & \sum _{n_{1},n_{2},n_{3},n_{4} \in \mathbb{Z}^{3}} \sum _{ \substack{N_{12},N_{345}\colon \\ N_{12} \sim N_{345}, \\ N_{345}\leq N }} \sum _{\substack{N_{34} \colon \\ N_{34} \gg N_{5} }} \bigg[ \mathbf{1}\{ n_{13}=n_{24}=0 \} 1_{N_{12}}(n_{34}) \\ \times & 1_{N_{345}}(n_{345}) 1_{N_{34}}(n_{34}) 1_{N_{5}}(n_{5})\Big( \prod _{3\leq j \leq 4} 1_{\leq N}(n_{j}) \Big) e^{-(s-s^{\prime}) \langle n_{345} \rangle ^{2}} \\ \times & \Big( \int _{- \infty}^{s^{\prime}} \int _{-\infty}^{s^{\prime}} \mathrm {d}s_{3} \mathrm {d}s_{4} \prod _{3\leq j \leq 4} e^{-(s+s^{\prime}-2s_{j}) \langle n_{j} \rangle ^{2}} \Big) \bigg]. \end{aligned} $$

(A.10)

We now treat the contribution of each chaos separately.

Step 1: Contribution of $\mathcal {G}^{(4)}$. For expository purposes, we decompose the argument into three sub-steps.

Step 1.a: In the first step, we define a dyadic decomposition of $\mathcal {G}^{(4)}$. We define

$$ \mathcal {G}^{(4)}(s,s^{\prime},x;n_{5},N_{\ast}) = \mathcal {G}^{(4)}(s,s^{\prime},x;n_{5},N_{1},N_{2},N_{3},N_{4},N_{5},N_{12},N_{34},N_{345}) $$

by inserting dyadic cut-offs in (A.8). More precisely, we remove the sum over (but keep the restrictions on) $N_{12}$, $N_{34}$, and $N_{345}$ and replace

$$ \prod _{1\leq j \leq 4} 1_{\leq N}(n_{j}) \qquad \text{with} \qquad \prod _{1\leq j \leq 4} 1_{N_{j}}(n_{j}) . $$

Due to our previous frequency-restrictions and symmetry, we can always restrict ourselves to the case

$$ N_{1} \geq N_{2}, \, N_{3} \geq N_{4}, \, N_{12} \sim N_{34} \sim N_{345} \gg N_{5}. $$

(A.11)

In the following, we also write

$$ N_{\max}:= \max \big( N_{1},N_{2},N_{3},N_{4},N_{5} , N_{345} \big). $$

Step 1.b: In the second step, we prove for all $p\geq 2$ that

$$ \begin{aligned} &\mathbb{E} \bigg[ \sup _{-1\leq s^{\prime }\leq s \leq 0} \Big\| P_{N_{0}} \mathcal {G}^{(4)}(s,s^{\prime},x,n_{5},N_{\ast}) \Big\| _{L_{x}^{\infty}}^{p} \bigg]^{1/p} \\ \lesssim &\, p^{2} \exp \Big(-1/8 |s-s^{\prime}| N_{345}^{2}\Big) N_{0}^{1/2} N_{345}^{2} \min \big( 1, N_{0} N_{5}^{-1}\big) N_{\mathrm{max}}^{-1/2+ \epsilon}. \end{aligned} $$

(A.12)

Using the reduction arguments in Sect. 5.7 (or Sobolev embedding in the $s$ and $s^{\prime}$-variables), it suffices to treat fixed $-1\leq s^{\prime }\leq s \leq 0$. For any $p\geq 2$, we obtain from Gaussian hypercontractivity that

$$\begin{aligned} &\mathbb{E}\Big[ \Big\| P_{N_{0}} \mathcal {G}^{(4)}(s,s^{\prime},x,n_{5},N_{\ast}) \Big\| _{L_{x}^{\infty}}^{p} \Big]^{2/p} \\ \lesssim & \, p^{2} N_{\mathrm{max}}^{\epsilon} \mathbb{E}\Big[ \Big\| P_{N_{0}} \mathcal {G}^{(4)}(s,s^{\prime},x,n_{5},N_{5}) \Big\| _{L_{x}^{2}}^{2} \Big] \\ \lesssim &\, p^{2} N_{\mathrm{max}}^{\epsilon} \sum _{ \substack{n_{0},n_{1},n_{2},n_{3},n_{4},n_{5} \in \mathbb{Z}^{3} \colon \\ n_{0}=n_{12345}}} \bigg[ \Big( \prod _{j=0}^{5} 1_{N_{j}}(n_{j}) \Big) \\ &\quad \times 1_{N_{12}}(n_{12}) 1_{N_{34}}(n_{34}) 1_{N_{345}}(n_{j}) \exp \Big(-2(s-s^{\prime}) \langle n_{345} \rangle ^{2}\Big) \\ &\quad \times \int _{-\infty}^{s} \int _{-\infty}^{s} \mathrm {d}s_{1} \mathrm {d}s_{2} \int _{-\infty}^{s^{\prime}} \int _{-\infty}^{s^{\prime}} \mathrm {d}s_{3} \mathrm {d}s_{4} \Big( \prod _{j=1}^{2} e^{-(s-s_{j}) \langle n_{j} \rangle ^{2}} \Big) \Big( \prod _{j=3}^{4} e^{-(s^{\prime}-s_{j}) \langle n_{j} \rangle ^{2}} \Big) \bigg] \\ \lesssim &\, p^{2} N_{\mathrm{max}}^{\epsilon} \Big( \prod _{j=1}^{4} N_{j}^{-2} \Big) \exp \Big(-1/2 (s-s^{\prime}) N_{345}^{2}\Big) \\ &\quad \times \sum _{ \substack{n_{0},n_{1},n_{2},n_{3},n_{4},n_{5} \in \mathbb{Z}^{3} \colon \\ n_{0}=n_{12345}}} \bigg[ \Big( \prod _{j=0}^{5} 1_{N_{j}}(n_{j}) \Big) 1_{N_{12}}(n_{12}) 1_{N_{34}}(n_{34}) 1_{N_{345}}(n_{j}) \bigg]. \end{aligned}$$

(A.13)

By viewing $n_{0}$, $n_{2}$, $n_{345}$, and $n_{4}$ as the free variables and recalling (A.11), we obtain that

$$\begin{aligned} \text{(A.13)} &\lesssim p^{2} N_{\mathrm{max}} \Big( \prod _{j=1}^{4} N_{j}^{-2} \Big) (N_{0} N_{2} N_{345} N_{4} )^{3} \exp \Big(-1/2 (s-s^{ \prime}) N_{345}^{2}\Big) \\ &\lesssim N_{\mathrm{max}}^{\epsilon }N_{0} N_{345}^{4} \times \Big( N_{0}^{2} N_{1}^{-1} N_{3}^{-1} N_{345}^{-1} \Big) \times \exp \Big(-1/2 (s-s^{ \prime}) N_{345}^{2}\Big). \end{aligned}$$

(A.14)

In order to prove (A.12), it only remains to bound the second factor in (A.14). Under our frequency-restrictions in (A.11), it holds that $\max (N_{0},N_{5}) \lesssim N_{345}$ and $N_{345} \lesssim \min (N_{1},N_{3})$. As a result,

$$\begin{aligned} N_{0}^{2} N_{1}^{-1} N_{3}^{-1} N_{345}^{-1} &\lesssim \big( N_{0}^{2} N_{345}^{-2} \big) \times \big( N_{1}^{-1} N_{3}^{-1} N_{345} \big) \\ &\lesssim \min (1,N_{0}^{2} N_{5}^{-2}) \max (N_{1},N_{3})^{-1} \\ &\sim \min (1,N_{0}^{2} N_{5}^{-2}) N_{\mathrm{max}}^{-1}. \end{aligned}$$

This yields the desired estimate in (A.12).

Step 1.c: In the final step, we control the contribution of $\mathcal {G}^{(4)}$ to the left-hand side of (A.5). Due to Lemma A.1, it suffices to prove for all frequency-scales that

$$\begin{aligned} &\sup _{s\in [s_{0},0]} |s-s_{0}|^{\theta }\Big\| P_{N_{0}} \sum _{n_{5} \in \mathbb{Z}^{3}} \int _{s_{0}}^{s} \mathrm {d}s^{\prime }\widehat{P_{N_{5}} \Psi}(s^{ \prime},n_{5}) \mathcal {G}^{(4)}(s,s^{\prime},x;n_{5},N_{\ast}) \Big\| _{\mathcal {C}_{x}^{-1/2-4 \delta}} \\ \lesssim & \, A N_{\mathrm{max}}^{-\epsilon} \sup _{s\in [s_{0},0]} |s-s_{0}|^{ \theta }\big\| \Psi (s) \big\| _{H_{x}^{1-\delta}}. \end{aligned}$$

(A.15)

By using (A.12), we obtain (on an $A$-certain event) that, for all $-1\leq s^{\prime }\leq s \leq 0$ and all frequency-scales,

$$\begin{aligned} &\Big\| P_{N_{0}} \mathcal {G}^{(4)}(s,s^{\prime},x;n_{5},N_{\ast}) \Big\| _{L_{x}^{ \infty}} \\ &\quad \leq A \exp \Big(-1/8 |s-s^{\prime}| N_{345}^{2} \Big) N_{0}^{1/2} N_{345}^{2} \min (1,N_{0} N_{5}^{-1}) N_{\mathrm{max}}^{-1/2+\epsilon}. \end{aligned}$$

(A.16)

As a result,

$$\begin{aligned} &\Big\| P_{N_{0}} \sum _{n_{5} \in \mathbb{Z}^{3}} \int _{s_{0}}^{s} \mathrm {d}s^{ \prime }\widehat{P_{N_{5}} \Psi}(s^{\prime},n_{5}) \mathcal {G}^{(4)}(s,s^{ \prime},x;n_{5},N_{\ast}) \Big\| _{\mathcal {C}_{x}^{-1/2-4\delta}} \\ \lesssim & \, \int _{s_{0}}^{s} \mathrm {d}s^{\prime }\big\| P_{N_{0}} \mathcal {G}^{(4)}(s,s^{ \prime},x;n_{5},N_{\ast}) \big\| _{\mathcal {C}_{x}^{-1/2-4\delta}} \sum _{n_{5}} \big| \widehat{P_{N_{5}} \Psi}(s^{\prime},n_{5}) \big| \\ \lesssim & \, A N_{0}^{-4\delta} \min \big(1,N_{0} N_{5}^{-1}\big) N_{345}^{2} N_{5}^{1/2+\delta} N_{\mathrm{max}}^{-1/2+\epsilon} \\ \times & \Big( \int _{s_{0}}^{s} \mathrm {d}s^{\prime }\exp \Big(-1/8 |s-s^{ \prime}| N_{345}^{2} \Big) |s^{\prime}-s_{0}|^{-\theta} \Big) \sup _{s^{ \prime }\in [s_{0},0]} |s^{\prime }-s_{0}|^{\theta }\big\| \Psi (s^{ \prime}) \big\| _{H_{x}^{1-\delta}}. \end{aligned}$$

From a direct calculation, we obtain that

$$\begin{aligned} &A N_{0}^{-4\delta} \min \big(1,N_{0} N_{5}^{-1}\big) N_{345}^{2} N_{5}^{1/2+ \delta} N_{\mathrm{max}}^{-1/2+\epsilon} \\ &\quad {}\times \Big( \int _{s_{0}}^{s} \mathrm {d}s^{ \prime }\exp \Big(-1/8 |s-s^{\prime}| N_{345}^{2} \Big) |s^{\prime}-s_{0}|^{- \theta} \Big) \\ \lesssim & \, A N_{345}^{2\epsilon} N_{5}^{1/2-3\delta} N_{ \mathrm{max}}^{-1/2+\epsilon} \Big( \int _{s_{0}}^{s} \mathrm {d}s^{\prime }|s-s^{ \prime}|^{-(1-\epsilon )} |s^{\prime}-s_{0}|^{-\theta} \Big) \\ \lesssim & \, A N_{\mathrm{max}}^{3(\epsilon -\delta )} |s-s_{0}|^{- \theta}. \end{aligned}$$

This yields the desired estimate (A.15).

Step 2: Contribution of $\mathcal {G}^{(2)}$. Since the argument is similar to the treatment of $\mathcal {G}^{(4)}$, we omit the details.

Step 3: Contribution of $\mathcal {G}^{(0)}$. We emphasize that the contribution of $\mathcal {G}_{\leq N}^{(0)}$ has to be renormalized, i.e., has to cancel with $\gamma _{\leq N}$. We first perform the sum over the frequencies $n_{1}$ and $n_{2}$ and perform the integrals in $s_{3}$ and $s_{4}$, which yields

$$ \begin{aligned} &\mathcal {G}^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5}) \\ =& 18 \cdot 1_{N_{5}}(n_{5}) \, e^{i\langle n_{5}, x \rangle} \sum _{n_{3},n_{4} \in \mathbb{Z}^{3}} \sum _{ \substack{N_{12},N_{345} \colon \\ N_{12} \sim N_{345}, \\ N_{345} \leq N}} \sum _{\substack{N_{34} \colon N_{34} \gg N_{5} }} \bigg[ 1_{N_{12}}(n_{34}) 1_{N_{345}}(n_{345}) \\ \times & 1_{N_{34}}(n_{34}) 1_{\leq N}(n_{345}) \Big( \prod _{3\leq j \leq 4} 1_{\leq N}(n_{j}) \Big) e^{-(s-s^{\prime}) \langle n_{345} \rangle ^{2}} \Big( \prod _{3 \leq j \leq 4} \frac{e^{-(s-s^{\prime}) \langle n_{j} \rangle ^{2}}}{\langle n_{j} \rangle ^{2}} \Big) \bigg]. \end{aligned} $$

(A.17)

We now isolate the main term in (A.17), which is given by

$$ \begin{aligned} &\widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5}) \\ =& 18 \cdot 1_{N_{5}}(n_{5}) e^{i\langle n_{5}, x \rangle} \\ &{}\times \sum _{n_{3},n_{4} \in \mathbb{Z}^{3}} \bigg[ 1_{\leq N}(n_{345}) \Big( \prod _{3\leq j \leq 5} 1_{ \leq N}(n_{j}) \Big) e^{-(s-s^{\prime}) \langle n_{345} \rangle ^{2}} \Big( \prod _{3\leq j \leq 4} \frac{e^{-(s-s^{\prime}) \langle n_{j} \rangle ^{2}}}{\langle n_{j} \rangle ^{2}} \Big) \bigg]. \end{aligned} $$

(A.18)

In the error term $\mathcal {G}^{(0)}_{\leq N}-\widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}$, one can gain a factor of $N_{5} N_{345}^{-1}$, which makes the estimate rather easy. As a result, we focus only on the main term $\widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}$. Using symmetry in $n_{3}$, $n_{4}$, and $n_{345}$, it follows that

$$ \begin{aligned} &\widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5}) \\ =& - 6 \cdot 1_{N_{5}}(n_{5}) e^{i\langle n_{5} ,x \rangle} \\ &{}\times \partial _{s} \bigg( \sum _{n_{3},n_{4} \in \mathbb{Z}^{3}} \bigg[ 1_{\leq N}(n_{345}) \frac{e^{-(s-s^{\prime}) \langle n_{345} \rangle ^{2}}}{\langle n_{345} \rangle ^{2}} \Big( \prod _{3\leq j \leq 4} 1_{\leq N}(n_{j}) \frac{e^{-(s-s^{\prime}) \langle n_{j} \rangle ^{2}}}{\langle n_{j} \rangle ^{2}} \Big) \bigg] \bigg). \end{aligned} $$

(A.19)

With a slight abuse of notation,⁴¹ we define

$$ \begin{aligned} &\Gamma _{\leq N}(n_{5},s;N_{5}) \\ &\quad := 6 \cdot 1_{N_{5}}(n_{5}) \sum _{n_{3},n_{4} \in \mathbb{Z}^{3}} \bigg[ 1_{\leq N}(n_{345}) \frac{e^{-s \langle n_{345} \rangle ^{2}}}{\langle n_{345} \rangle ^{2}} \Big( \prod _{3\leq j \leq 4} 1_{\leq N}(n_{j}) \frac{e^{-s \langle n_{j} \rangle ^{2}}}{\langle n_{j} \rangle ^{2}} \Big) \bigg] \bigg). \end{aligned} $$

(A.20)

Equipped with this notation, (A.19) reads

$$ \widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5})= - \partial _{s} \Gamma _{ \leq N}(n_{5},s-s^{\prime};N_{5}) \, e^{i\langle n_{5}, x \rangle}. $$

(A.21)

It follows that

$$\begin{aligned} &\sum _{n_{5} \in \mathbb{Z}^{3}} \int _{s_{0}}^{s} \mathrm {d}s^{\prime } \widehat{P_{N_{5}} \Psi}(s^{\prime},n_{5}) \widetilde{\mathcal {G}} \vphantom{\mathcal {G}} ^{(0)}_{\leq N}(s,s^{\prime},x;n_{5},N_{5}) - \gamma _{\leq N} P_{N_{5}} \Psi (s) \\ =& \, - \partial _{s} \sum _{n_{5} \in \mathbb{Z}^{3}} \int _{s_{0}}^{s} \mathrm {d}s^{\prime }\widehat{P_{N_{5}}\Psi}(s^{\prime},n_{5}) \Gamma _{ \leq N}(n_{5},s-s^{\prime};N_{5}) e^{i \langle n_{5} , x \rangle} \end{aligned}$$

(A.22)

$$\begin{aligned} +& \, \sum _{n_{5} \in \mathbb{Z}^{3}} \big( \Gamma _{\leq N}(n_{5},0) - \gamma _{\leq N}\big) \widehat{P_{N_{5}} \Psi}(s,n_{5}) e^{i\langle n_{5} ,x \rangle} . \end{aligned}$$

(A.23)

For the first term (A.22), we use the second estimate in Lemma A.1, which reduces the desired estimate to

$$\begin{aligned} &\sup _{s\in [s_{0},0]} |s-s_{0}|^{\theta }\, \Big\| \sum _{n_{5} \in \mathbb{Z}^{3}} \int _{s_{0}}^{s} \mathrm {d}s^{\prime }\widehat{P_{N_{5}}\Psi}(s^{ \prime},n_{5}) \Gamma _{\leq N}(n_{5},s-s^{\prime};N_{5}) e^{i \langle n_{5} , x \rangle} \Big\| _{\mathcal {C}_{x}^{3/2-4\delta}} \\ \lesssim &\, \sup _{s\in [s_{0},0]} |s-s_{0}|^{\theta }\, \big\| \Psi (s) \big\| _{H_{x}^{1-\delta}}. \end{aligned}$$

This follows directly from the triangle inequality in $n_{5}$, inserting the definition of $\Gamma _{\leq N}$, and performing the $s^{\prime}$-integral.

For the second term (A.23), we use the first estimate in Lemma A.1, which reduces the desired estimate to

$$\begin{aligned} &\sup _{s\in [s_{0},0]} |s-s_{0}|^{\theta }\, \Big\| \sum _{n_{5} \in \mathbb{Z}^{3}} \big( \Gamma _{\leq N}(n_{5},0) - \gamma _{\leq N}\big) \widehat{P_{N_{5}} \Psi}(s,n_{5}) e^{i\langle n_{5} ,x \rangle} \Big\| _{\mathcal {C}_{x}^{-1/2-4\delta}} \\ &\quad \lesssim \, \sup _{s\in [s_{0},0]} |s-s_{0}|^{ \theta }\, \big\| \Psi (s) \big\| _{H_{x}^{1-\delta}}. \end{aligned}$$

This follows from the triangle inequality in $n_{5}$ and Lemma 7.1. □

Appendix B: Merging estimates, moment method and time integrals

We first recall the main technical tools needed from [45], namely the merging estimate (Lemma B.1) and the moment method (Proposition B.2, called trimming estimate in [45]).

Lemma 1

Merging estimates, Proposition 4.11 in [45]

Consider two tensors $h_{k_{A_{1}}}^{(1)}$ and $h_{k_{A_{2}}}^{(2)}$, where $A_{1}\cap A_{2}=C$. Let $A_{1}\Delta A_{2}=A$, define the semi-product

$$ H_{k_{A}}=\sum _{k_{C}}h_{k_{A_{1}}}^{(1)}h_{k_{A_{2}}}^{(2)}. $$

(B.1)

Then, for any partition $(X,Y)$ of $A$, let $X\cap A_{1}=X_{1}$, $Y\cap A_{1} =Y_{1}$ etc., we have

$$ \|H\|_{k_{X}\to k_{Y}}\leq \|h^{(1)}\|_{k_{X_{1}\cup C}\to k_{Y_{1}}} \cdot \|h^{(2)}\|_{k_{X_{2}}\to k_{C\cup Y_{2}}}. $$

(B.2)

Proposition 2

Moment method, Proposition 4.14 in [45]

Let $\mathcal{A}$, $\mathcal{X}$, and $\mathcal{Y}$ be disjoint finite index sets and let $h=h_{n_{\mathcal{A}} n_{\mathcal{X}} n_{\mathcal{Y}}}$ be a (deterministic) tensor. Let $N$ be a dyadic frequency-scale and assume that, on the support of $h$, $|n_{j}|\lesssim N$ for all $j\in \mathcal{A}\cup \mathcal{X} \cup \mathcal{Y}$. Furthermore, let $(\pm _{j})_{j\in \mathcal{A}} \in \big\{ +, - \big\}^{\mathcal{A}}$ be a collection of signs. Finally, define the random tensor $H=H_{n_{\mathcal{X}} n_{\mathcal{Y}}}$ by

$$ H_{n_{\mathcal{X}} n_{\mathcal{Y}}} = \sum _{n_{\mathcal{A}}} h_{n_{ \mathcal{A}} n_{\mathcal{X}} n_{\mathcal{Y}}} \operatorname{\mathcal{S}\mathcal{I}}[n_{j},\pm _{j} \colon j\in \mathcal{A}], $$

(B.3)

where the stochastic integrals $\operatorname{\mathcal{S}\mathcal{I}}$ are as in Sect. 2.4. Then, we have for all $\delta >0$ and all $p\geq 1$ that

$$ \Big\| \big\| H_{n_{\mathcal{X}} n_{\mathcal{Y}}} \big\| _{n_{ \mathcal{X}} \rightarrow n_{\mathcal{Y}}}\Big\| _{L^{p}_{\omega}} \lesssim _{\delta }N^{\delta }p^{\# \mathcal{A}/2} \max _{\mathcal{B}, \mathcal{C}} \big\| h_{n_{\mathcal{A}} n_{\mathcal{X}} n_{\mathcal{Y}}} \big\| _{n_{\mathcal{B}} n_{\mathcal{X}} \rightarrow n_{\mathcal{C}} n_{ \mathcal{Y}}}, $$

(B.4)

where the maximum is taken over all partitions of $\mathcal{A}$.

Remark 3

Note that Proposition B.2 is slightly different from Proposition 4.14 in [45], due to the use of the renormalized product $\mathcal {S}\mathcal {I}$ instead of products of independent Gaussians. However, this difference does not affect the proof.

In addition, we prove an almost $L^{1}$ estimate for iterated oscillatory time integrals (Proposition B.4). This is a special case of Lemma 10.2 in [40], and is used in the molecular analysis in Sect. 11.

Proposition 4

Some time integral estimates

Consider the following expressions:

$$\begin{aligned} &\mathcal {H}_{1,3,3}(t,\Omega _{0},\Omega _{1},\Omega _{2}):=\int _{0}^{t} \int _{0}^{t} \chi (t)\chi (t_{1})\chi (t_{2})e^{i(\Omega _{0}t+ \Omega _{1}t_{1}+\Omega _{2}t_{2})}\,\mathrm{d}t_{1}\mathrm{d}t_{2}, \end{aligned}$$

(B.5)

$$\begin{aligned} &\mathcal {H}_{3,3,3}(t,\Omega _{0},\Omega _{1},\Omega _{2},\Omega _{3}) \\ &\quad := \int _{0}^{t}\int _{0}^{t} \int _{0}^{t}\chi (t)\chi (t_{1})\chi (t_{2})e^{i( \Omega _{0}t+\Omega _{1}t_{1}+\Omega _{2}t_{2}+\Omega _{3}t_{3})}\, \mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}, \end{aligned}$$

(B.6)

$$\begin{aligned} &\mathcal {H}_{1,1,5}(t,\Omega _{0},\Omega _{1},\Omega _{2}):=\int _{0}^{t} \int _{0}^{t} \int _{0}^{t_{1}}\chi (t)\chi (t_{1})\chi (t_{2})e^{i( \Omega _{0}t+\Omega _{1}t_{1}+\Omega _{2}t_{2})}\,\mathrm{d}t_{2} \mathrm{d}t_{1}. \end{aligned}$$

(B.7)

Then we have the following estimates:

$$\begin{aligned} &\int _{\mathbb{R}^{3}}(\langle \Omega _{0}\rangle \langle \Omega _{1} \rangle \langle \Omega _{2}\rangle )^{8(1/2-b_{+})}|(\mathcal {F}_{t}\mathcal {H}_{1,3,3})( \xi ,\Omega _{0},\Omega _{1},\Omega _{2})|\,\mathrm{d}\Omega _{0} \mathrm{d}\Omega _{1}\mathrm{d}\Omega _{2} \\ &\quad \lesssim \langle \xi \rangle ^{4(1/2-b_{+})}, \end{aligned}$$

(B.8)

$$\begin{aligned} &\int _{\mathbb{R}^{4}}(\langle \Omega _{0}\rangle \langle \Omega _{1} \rangle \langle \Omega _{2}\rangle \langle \Omega _{3}\rangle )^{8(1/2-b_{+})}|( \mathcal {F}_{t}\mathcal {H}_{3,3,3})(\xi ,\Omega _{0},\Omega _{1},\Omega _{2}, \Omega _{3})|\,\mathrm{d}\Omega _{0}\mathrm{d}\Omega _{1}\mathrm{d} \Omega _{2}\mathrm{d}\Omega _{3} \\ &\quad \lesssim \langle \xi \rangle ^{4(1/2-b_{+})}, \end{aligned}$$

(B.9)

$$\begin{aligned} &\int _{\mathbb{R}^{3}}(\langle \Omega _{0}\rangle \langle \Omega _{1} \rangle \langle \Omega _{2}\rangle )^{8(1/2-b_{+})}|(\mathcal {F}_{t}\mathcal {H}_{1,1,5})( \xi ,\Omega _{0},\Omega _{1},\Omega _{2})|\,\mathrm{d}\Omega _{0} \mathrm{d}\Omega _{1}\mathrm{d}\Omega _{2} \\ &\quad \lesssim \langle \xi \rangle ^{4(1/2-b_{+})}. \end{aligned}$$

(B.10)

The same estimates hold for all $\Omega _{j}$ derivatives of these functions.

Proof

First, the $\Omega _{j}$ derivatives of the ℋ functions satisfy the same estimates as themselves because taking one $\Omega _{j}$ derivative in (B.5)–(B.7) just corresponds to multiplying by $t_{j}$ or $t$, which is bounded due to the cutoff $\chi$. Thus we will only prove the bounds for the original ℋ functions. Moreover we will only prove (B.10) for $\mathcal {H}_{1,1,5}$ defined by (B.7), since the proof for the other two will be similar (and simpler as they do not involve iterated time integrals).

Now consider $\mathcal {H}:=\mathcal {H}_{1,1,5}$. Let $1=\eta _{0}(\Omega )+\eta _{\infty}(\Omega )$ be a partition of unity supported on $|\Omega |\lesssim 1$ and $|\Omega |\gtrsim 1$ respectively, then integrating by parts we have

$$\begin{aligned} &\int _{0}^{t_{1}}\chi (t_{2})e^{i\Omega _{2}t_{2}}\,\mathrm{d}t_{2} \\ &\quad = \eta _{0}(\Omega _{2})\psi (t_{1},\Omega _{2})+\eta _{\infty}(\Omega _{2}) \bigg(\frac{\chi (t_{1})e^{i\Omega _{2}t_{1}}-\chi (0)}{i\Omega _{2}}- \frac{1}{i\Omega _{2}}\int _{0}^{t_{1}}\chi '(t_{2})e^{i\Omega _{2}t_{2}} \,\mathrm{d}t_{2}\bigg), \end{aligned}$$

(B.11)

where $\psi$ is a fixed smooth function of $(t_{1},\Omega _{2})$. Let the three terms in (B.11) be $A$, $B$ and $C$, below we will only focus on the main term which is $B$. In fact the term $A$ is a Schwartz function upon localizing in $t_{1}$ (which we can always do), which is easily handled. For the term $C$, we can integrate by parts in $t_{2}$ many times, and the boundary terms all have the same form as $B$ (except $\chi$ may be replaced by $\chi '$ etc. which does not matter); the remaining bulk term will carry enough decay in $\Omega _{2}$, so upon localizing in $t_{1}$, it can be bounded as a function of $t_{1}$ and $\Omega _{2}$ in a finite high order Schwartz space, so effectively it can be treated in the same way as $A$.

Now we focus on the main term

$$ B:=\eta _{\infty}(\Omega _{2}) \frac{\chi (t_{1})e^{i\Omega _{2}t_{1}}-\chi (0)}{i\Omega _{2}}, $$

and plug it into (B.7) to get

$$ \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}}e^{i\Omega _{0}t} \int _{0}^{t}\chi (t_{1})(\chi (t_{1})e^{i\Omega _{2}t_{1}}-\chi (0))e^{i \Omega _{1}t_{1}}\,\mathrm{d}t_{1}. $$

Integrating by parts again, we obtain some remainder terms plus the main term which is

$$\begin{aligned} &e^{i\Omega _{0}t}\bigg[ \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1}+\Omega _{2})}{i(\Omega _{1}+\Omega _{2})}( \chi ^{2}(t)e^{i(\Omega _{1}+\Omega _{2})t}-\chi ^{2}(0)) \\ &\quad {}- \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1})}{i\Omega _{1}}\chi (0)(\chi (t)e^{i \Omega _{1}t}-\chi (0))\bigg]. \end{aligned}$$

(B.12)

The remainder terms can be treated in the same way as (B.11) above by integrating by parts many times, so we will only consider the main term below.

For this main term we clearly have that

$$\begin{aligned} &|\mathcal {F}_{x}\text{(B.12)}(\xi )| \\ &\quad \lesssim \sum _{\Omega \in \{ \Omega _{0},\,\Omega _{0}+\Omega _{1},\,\Omega _{0}+\Omega _{1}+ \Omega _{2}\}}\bigg(\bigg| \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1}+\Omega _{2})}{i(\Omega _{1}+\Omega _{2})} \bigg| \\ &\qquad {} +\bigg|\frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1})}{i\Omega _{1}}\bigg|\bigg)\langle \xi -\Omega \rangle ^{-10}. \end{aligned}$$

(B.13)

This easily implies (B.10), because both functions

$$ \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1}+\Omega _{2})}{i(\Omega _{1}+\Omega _{2})} \quad \textrm{and}\quad \frac{\eta _{\infty}(\Omega _{2})}{i\Omega _{2}} \frac{\eta _{\infty}(\Omega _{1})}{i\Omega _{1}} $$

are almost $L^{1}$ in $(\Omega _{1},\Omega _{2})$, and becomes $L^{1}$ when multiplied by the negative weight $(\langle \Omega _{1}\rangle \langle \Omega _{2}\rangle )^{8(1/2-b_{+})}$; the integrability in $\Omega _{0}$, as well as decay in $\xi$ in (B.10), follows because of the $\langle \xi -\Omega \rangle ^{-10}$ factor in (B.13) (so in particular we can assume $|\xi |\lesssim 1+\max _{j}|\Omega _{j}|$). Now, once (B.10) is proved for the main term, the same argument can easily be applied to the remainder terms to get the same estimate. This finishes the proof of (B.10). □