# Invariance of $\phi^{4}$ measure under nonlinear wave and Schrödinger
equations on the plane

Nikolay Barashkov, Petri<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics and Statistics, University
of Helsinki

###### Abstract

We show probabilistic existence and uniqueness for the Wick-ordered cubic
nonlinear wave equation in a weighted Besov space over $\mathbb{R}^{2}$. To
achieve this, we show that a weak limit of $\phi^{4}$ measures on increasing
tori is invariant under the equation. We review and slightly simplify the
periodic theory and the construction of the weak limit measure, and then use
finite speed of propagation to reduce the infinite-volume case to the previous
setup. Our argument also gives a weak (Albeverio–Cruzeiro) invariance result
on the nonlinear Schrödinger equation in the same setting.

MSC (2020): 35L71, 60H30 (Primary); 35Q55, 60H15, 81T08
Keywords: nonlinear wave equation, nonlinear Schrödinger equation, invariant
measure, stochastic quantization, $\phi^{4}$ measure, weighted Besov space

##  1 Introduction

Since Jean Bourgain’s work in the 1990s, invariant measures have been an
important tool in probabilistic solution theory of dispersive PDEs. Bourgain
initially considered the nonlinear Schrödinger equation

$i\partial_{t}u(x,t)+\Delta u(x,t)=\pm\lambda\,{:\\!u^{3}\\!:}$ (NLS)

on one-dimensional torus $\mathbb{T}$ [8]. He proved almost sure well-
posedness in the Sobolev space $H^{1/2-\varepsilon}$ when the initial data is
distributed according to a so-called $\phi^{4}$ measure. Later on in [9], he
extended the result to $\mathbb{T}^{2}$. In two or more dimensions the
$\phi^{4}$ measure is supported on distributions, and it then becomes
necessary to renormalize the nonlinearity $u^{3}$ by Wick ordering, denoted by
${:\\!u^{3}\\!:}$.

Our main subject is the defocusing massive nonlinear wave equation

$\partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)=-\lambda\,{:\\!u^{3}\\!:}$ (NLW)

on two-dimensional spatial domain $\mathbb{R}^{2}$. This equation was
previously solved on periodic domain by Oh and Thomann [46]. The main result
of this article can be stated as follows:

######  Theorem 1.1 (Global existence and uniqueness).

Let $\mu$ be the product of infinite-volume $\phi^{4}$ and white-noise
measures, and $\varepsilon>0$. For $\mu$-almost all initial data and any
$T>0$, the nonlinear wave equation (NLW) has a unique global solution in the
weighted space $L^{2}([0,T];\;H^{-2\varepsilon}(\rho))$.

Our approach is to first construct solutions in periodic domains
$[{-L},{L}]^{2}$, and then approximate infinite-volume solutions with them.
The high-level proof strategy in a periodic domain goes back to Bourgain:

  1. 1.

Define a probability distribution on the initial data.

  2. 2.

Prove deterministic well-posedness for time interval $[0,\tau]$ when the
initial data belongs to some set $A$ of large probability. The small time
$\tau$ depends on the size of $A$.

  3. 3.

Prove that the probability measure is invariant in time under the equation.

  4. 4.

Intersect the sets of initial and final values, which have same probability by
invariance. By iteration, the probability of blow-up by time $T=n\tau$ is
bounded by $n(1-\mathbb{P}(A))$.

  5. 5.

Use stochastic estimates to show that an increase of $\mathbb{P}(A)$ cancels
the corresponding increase of iterations $n$; thus the probability of blow-up
can be made arbitrarily small.

This argument reduces the global solution theory into understanding the
invariance and large deviations of a suitable probability measure. To show
invariance, we use finite-dimensional approximation. Liouville’s theorem
states that the Gibbs measure associated to the Hamiltonian formulation of
(NLW) is invariant. These approximate measures converge in total variation to
the untruncated, periodic-domain measures.

The extension to infinite volume relies on two insights. The first is the
existence of uniform bounds for the measures in a polynomially weighted space
[38]. This yields a convergent subsequence of measures as $L\to\infty$. The
second step is to use the finite speed of propagation of (NLW) to reduce all
statements about measurable events to the periodic case.

###  1.1 The $\phi^{4}$ measure

What is the natural candidate for the invariant measure? As mentioned above,
Fourier-truncated versions of these equations converve the Hamiltonian $H$,
with which we can define the Gibbs measure proportional to $\exp(-\beta H)$.
The parameter $\beta>0$ is called the inverse temperature. For truncated and
periodic (NLW), the Gibbs measure has density222In the following, we set
$\beta=\lambda=1$ as they are not too relevant for our present topic.

$\exp\left(-\beta\int_{\mathbb{T}^{d}}\frac{\lambda\,{:\\!u^{4}\\!:}}{4}+\frac{m^{2}{\left|u\right|}^{2}+{\left|\nabla
u\right|}^{2}+{\left|\partial_{t}u\right|}^{2}}{2}\,\mathrm{d}x\right)$ (1.1)

with respect to Lebesgue measure on the Fourier coefficients. The second term
yields a Gaussian factor that can be utilized in removing the truncation.

The continuum versions of these Gibbs measures are studied in constructive
quantum field theory [23]. Stochastic quantization (see e.g. [50]) is a
rigorous PDE approach for their study. In this approach the $\phi^{4}_{d}$
measure is regarded as an invariant measure for a nonlinear heat equation with
white noise forcing. These equations are singular and cannot be solved
classically.

The periodic $\phi^{4}_{2}$ equation was solved by Da Prato and Debussche
[19]. The limit measure is absolutely continuous with respect to a Gaussian
measure. Existence of infinite-volume solutions for the 2D equation was later
shown by Mourrat and Weber in a polynomially weighted space [38]; see also
[37]. We will rely heavily on these ideas in Section 3.

The local well-posedness theory for the more singular 3D case came in three
approaches in mid-2010s: Hairer’s regularity structures [29]; Gubinelli,
Imkeller and Perkowski’s [25] paracontrolled distributions; and Kupiainen’s
renormalization group approach [33]. The bounds of Mourrat and Weber were then
exploited by Albeverio and Kusuoka [3] and Gubinelli and Hofmanová [24] to
give a self-contained construction of the $\phi^{4}_{3}$ measure.

In dimensions $d\geq 4$, the $\phi^{4}_{d}$ measures collapse to trivial
Gaussian measures. The final case $d=4$ was proved recently by Aizenman and
Duminil-Copin; see their article [1] for discussion.

The $\phi^{4}$ measure is expected to be invariant under three PDEs that share
essentially the same Hamiltonian: (NLS), (NLW), and the cubic stochastic
nonlinear heat equation. As shown in [13, Figure 1], the periodic-domain
invariance theory is almost done, with only the three-dimensional Schrödinger
missing.

This theory, and hence the global well-posedness of the equations, is much
less developed in the infinite volume. For wave and Schrödinger equations the
previous results are limited to one dimension [10] or radial setting [56].

The largest complication is that the infinite-volume $\phi^{4}$ measures are
only defined as weak limits of approximating sequences, and in particular they
are no longer absolutely continuous with respect to Gaussian measure. This
means that total variation convergence is no longer available and we have to
prove local well-posedness for non-Gaussian initial data. Depending on the
coupling constant $\lambda$, the accumulation point of the approximating
sequence might not be unique.

However, we are able to show that our invariant distribution can still be
coupled to a Gaussian and the perturbation term enjoys better analytic bounds.
A similar fact was exploited by Bringmann and collaborators in [11, 12, 13],
in situations where the singularity of the measure arises in finite volume due
to short scale divergencies.

For the nonlinear Schrödinger equation the situation is even more complicated,
as there is no finite speed of propagation. This means that we cannot reduce
the problem to the periodic setup. However, by giving up some
differentiability and thus uniqueness, we can still prove a weaker form of
invariance. This sense of invariance was initially developed for Euler and
Navier–Stokes equations by Albeverio and Cruzeiro [2], and was explored in the
case of periodic 2D NLS in [45].

As this manuscript was being prepared, Oh, Tolomeo, Wang, and Zheng published
their preprint [47] where similar ideas appear. They prove Theorem 1.1 for a
more challenging equation, (NLW) with additive stochastic forcing. Their
approach is based on an optimal transport argument developed in [39], whereas
our globalization argument depends more heavily on finite speed of
propagation. There are also slight differences in the use of stochastic
quantization, and we also consider the Schrödinger equation.

###  1.2 Previous literature and extensions

Let us take a moment here to review the history of this question. As mentioned
above, the general strategy to prove probabilistic well-posedness was
developed by Bourgain [8] in context of one-dimensional periodic (NLS). This
was in response to earlier work of Lebowitz, Rose, and Speer [34] in late
1980s.

Invariant measures for the one-dimensional wave equation were considered by
Zhidkov [57] and McKean and Vaninsky [36]. Radially symmetric (NLW) on a
three-dimensional ball was considered by Burq and Tzvetkov [17], and extended
by Xu to infinite volume [56]. Recently progress has been made in three
dimensions, culminating in the proof of invariance of periodic $\Phi^{4}_{3}$
under the wave equation [11, 12, 13].

NLW has also been considered with random data not sampled from the invariant
measure [30, 32]. Related to the invariance of Gibbs measures is the program
for showing quasi-invariance of Gaussian measures under Hamiltonian PDEs [55];
in this notion the distribution of solutions at any given time remains
absolutely continuous with respect to the initial measure. For the wave
equation this was carried out in [28, 49].

Another related development is the solution theory for (NLW) with additive
white noise forcing. This was achieved for the 2D cubic wave equation in [26]
and extended to global well-posedness in [27, 54]. The preprint [47] of Oh,
Tolomeo, Wang, and Zheng considers this case. If the equation also includes
dispersion, the invariant measure is moreover ergodic [53].

The nonlinearity can be replaced by a general polynomial or exponential term
as in [40, 47]. It is also possible to let the solution take values in a
manifold instead of $\mathbb{R}$. The invariant measures for these wave maps
equations [14, 16] are known as nonlinear sigma models in the physics
literature. In one dimension they can be interpreted as Brownian paths on a
manifold.

For (NLS) in one dimension it is possible to consider both focusing and
defocusing nonlinearities, due to the presence of an $L^{2}$ conservation law.
Restricting to a ball in $L^{2}$ leads to normalizable measure if the
nonlinearity is subquintic. In the quintic case the measure is normalizable
only if and only if the coupling is suffiently weak; remarkably, this
threshold is known exactly [43].

In two dimensions the defocusing case can still be investigated, as was done
by Bourgain [9] for the cubic case and later for general polynomial
nonlinearities by Deng, Nahmod, and Yue [20]. For the focusing NLS the $L^{2}$
cutoff does not lead to normalizable measure anymore [15]. Quasi-invariance
has also been investigated for the NLS [41, 44, 48].

In [42, 51] invariant measures of the Zakharov–Yukawa system have been
studied. This is a system of coupled wave and Schrödinger equations with
nonpositive Hamiltonian and an $L^{2}$ conservation law. Due to these
properties it behaves similarly to the defocusing NLS.

The activity described above has mostly taken place on the torus. In infinite
volume we mention the early result of Bourgain on one-dimensional NLS [10], as
well as the work of Cacciafesta and Suzzoni on the NLS and other Hamiltonian
equations [18]. These are in addition to the aforementioned papers [47, 56] on
two- and three-dimensional NLW.

Let us conclude this review with a comment on possible extensions of our work
and open problems. Our method extends in a straightforward way to more general
polynomial nonlinearities and to vector-valued models.

######  Example 1.2.

The mass term $m^{2}>0$ in (NLW) and (NLS) is used to avoid problems with the
zero Fourier mode. Still, negative mass is interesting to consider. One can
modify a negative-mass equation

$\partial_{tt}u(x,t)-(m^{2}+\Delta)u(x,t)=-\lambda\,{:\\!u^{3}\\!:}$

by adding $2m^{2}\,u(x,t)$ to both sides of the equation. Then the
nonlinearity will be of form $-\lambda\,{:\\!u^{3}\\!:}+2m^{2}u$, which is
still dominated by the cubic term.

For the weak invariance we also expect the extension to long-range models
(with fractional Laplacian) to be straightforward, provided the resulting
measures are not too singular. If one can find a suitable replacement for
finite speed of propagation in the wave case, we also expect strong invariance
to follow in a straighforward fashion.

It would be interesting to consider the strong invariance of $P(\phi)_{1}$
theories under 1-dimensional (NLS). The $\phi^{4}$ case has been solved by
Bourgain [10], but his argument does not apply to higher-order polynomial
nonlinearities. Of course the corresponding 2D problem in the full space is
also very interesting, as well as the case of cosine and exponential
nonlinearities.

Given the recent preprint [13] on invariance of three-dimensional periodic
(NLW), it is intriguing to ask about the extension to $\mathbb{R}^{3}$. While
the measure-theoretic part of our argument is dimension-independent, the
analytic estimates would require significant changes to account for the more
singular behaviour.

###  1.3 Outline of the article

Our argument consists of mainly putting together existing pieces, hence the
majority of this article presents the techniques at a more pedestrian pace.

All the estimates happen in Besov spaces, a generalization of Sobolev spaces.
Multiplicative estimates take a particularly nice form in these spaces, and
there are many (sometimes compact) embeddings between the spaces. We collect
the main results in Section 2.

We then collect the stochastic estimates for the measures in Section 3. There
we outline the proof of existence and bounds for the $\phi^{4}$ measure over
polynomially weighted $\mathbb{R}^{2}$ space. In particular, we show the
tightness of the sequence on increasing tori. Some of the calculations are
deferred to appendices.

We solve (NLW) on a periodic domain in Section 4. This argument is originally
due to Oh and Thomann [46], but we slightly simplify the argument by using
only Besov spaces. We also present explicitly the Bourgain-style global
existence argument omitted in the cited article.

The main result in this article is presented in Section 5. We use a measure-
theoretic argument to reduce the full flow to the periodic case, and thus
prove invariance of the infinite-volume $\phi^{4}$ measure.

In Section 6, we finally consider (NLS) on $\mathbb{R}^{2}$. We prove
invariance in Albeverio–Cruzeiro sense with some weaker estimates on the
solutions.

###  1.4 Notation

The weighted Besov spaces $B^{s}_{p,r}(\rho)$ are defined in Section 2 below.
We abbreviate $H^{s}(\rho)\coloneqq B^{s}_{2,2}(\rho)$ and
$\mathcal{C}^{s}(\rho)\coloneqq B^{s}_{\infty,\infty}(\rho)$. Here $\rho$ is a
polynomially decaying weight, the parameter of which may change from section
to section.

We also denote by $B^{s}_{p,r}(A)$ the flat weight on a set
$A\subset\mathbb{R}^{2}$, and abuse notation by writing
$B^{s}_{p,r}(\Lambda_{L})$ for the space of periodic functions. The periodic
domain is denoted by $\Lambda_{L}\coloneqq{[{-L},{L}]}^{2}$.

We use $\mu$ for either the $\phi^{4}_{2}$ measure, or the product of
$\phi^{4}_{2}$ and white noise measures; we use $\mu_{L}$ and $\mu_{L,N}$ for
the bounded-domain and bounded-domain Fourier-truncated versions respectively.
The product measure form is used in Sections 4 and 5. The $\phi^{4}_{2}$
measure is decomposed into a Gaussian free field $Z$ coupled with a more
regular part $\phi$. The full product measure is supported on a space
${\mathcal{H}}^{s}(\rho)\coloneqq H^{s}(\rho)\times H^{s-1}(\rho)$.

Capital $\Phi_{t}$ is reserved for the flow of (NLW). Subscripts similar to
those above are used for bounded and Fourier-truncated equations. We define
linear solution operators ${\mathcal{C}}_{t}$ and ${\mathcal{S}}_{t}$ for
(NLW) and ${\mathcal{T}}_{t}$ for (NLS) in the corresponding sections.

The Littlewood–Paley blocks $\Delta_{k}$ defined below are supported on dyadic
sets. We define the projection $P_{N}$ as a sharp Fourier cutoff to
$B(0,2^{N})$.

We use the general notation of $A\lesssim B$ if $A\leq cB$ for some
independent constant $c$, and $A\simeq B$ for $A\lesssim B\lesssim A$.
Positive constants $c,C$ may vary from line to line. The small constant
$\varepsilon>0$ appears mainly in regularity of the spaces and may change
between sections (but only finitely many times). We use $\delta$ to signify
other small parameters.

###  1.5 Acknowledgements

NB was supported by the ERC Advanced Grant 741487 “Quantum Fields and
Probability”. PL was supported by the Academy of Finland project 339982
“Quantum Fields and Probability”. PL would like to thank Kalle Koskinen,
Jaakko Sinko, and Aleksis Vuoksenmaa for useful conversations. Both authors
would like to thank Leonardo Tolomeo for helpful comments on the preprint.

##  2 Besov spaces

Besov spaces are a generalization of Sobolev spaces that support some useful
multiplication estimates and embeddings. We collect in this section the
necessary resuls, but largely omit the proofs. An excellent introduction to
the topic is in the article of Mourrat and Weber [38]. Some results are also
collected in the appendix of [24]. The textbook of Bahouri, Chemin, and
Danchin [4] treats the unweighted case.

######  Remark 2.1.

There are two conventions of weighted $L^{p}$ spaces in common use. [38] and
[24] respectively define

${\left\|f\right\|}_{L^{p}_{w}}^{p}\coloneqq\int
f(x)^{p}w(x)\,\mathrm{d}x\quad\text{and}\quad{\left\|f\right\|}_{L^{p}(w)}^{p}\coloneqq\int
f(x)^{p}w(x)^{p}\,\mathrm{d}x.$

We use the latter convention since it lets us apply a weight also when
$p=\infty$. For $p<\infty$ the conventions are interchangeable, and the
statements and their proofs require only minor changes.

######  Definition 2.2 (Littlewood–Paley blocks).

We fix $\Delta_{k}$ to be Fourier multipliers that restrict the support of
$\hat{u}$ to a partition of unity. More precisely, for $k\geq 0$ they are
smoothed indicators of the annuli $B(0,2^{k}\,8/3)\setminus B(0,2^{k}\,3/4)$,
and for $k=-1$ of the ball $B(0,3/4)$. The precise choice is irrelevant.

######  Definition 2.3 (Weighted Besov space).

We define the space $B^{s}_{p,r}(w)$ as the completion of
$C_{c}^{\infty}(\mathbb{R}^{d})$ with respect to the norm

${\left\|f\right\|}_{B^{s}_{p,r}(w)}\coloneqq{\left\|2^{ks}\,{\left\|w(x)[\Delta_{k}f](x)\right\|}_{L^{p}}\right\|}_{\ell^{r}}$

where the $L^{p}$ norm is taken over $x\in\mathbb{R}^{d}$ and the $\ell^{r}$
norm over $k\geq-1$. We abbreviate

$H^{s}(w)\coloneqq
B^{s}_{2,2}(w)\quad\text{and}\quad\mathcal{C}^{s}(w)\coloneqq
B^{s}_{\infty,\infty}(w).$

In particular, the space $H^{s}$ coincides with the usual (fractional-order)
Sobolev space, where the norm is defined as the $L^{2}$ norm of function
multiplied by ${\left\langle{\nabla}\right\rangle}^{s}$. One can also replace
the ball $B(0,3/4)$ with annuli for all $k\in-\mathbb{N}$ to define
_homogeneous_ Besov spaces: some of the following results translate to
homogeneous spaces, but the zero Fourier mode is then ignored and the
necessary embeddings are not available.

We will use throughout the article a nonhomogeneous polynomial weight

$\rho(x)\coloneqq{\left\langle{x}\right\rangle}^{-\alpha}\coloneqq(1+{\left|x\right|}^{2})^{-\alpha/2}$

for $\alpha\geq 0$ sufficiently large. What “sufficiently large” means may
vary from section to section, but the final choice is finite. In some sections
we also use the unweighted space ($\alpha=0$); this is indicated by omitting
$\rho$.

The following multiplicative inequality shows that products of distributions
and smooth enough functions are well-defined distributions. A recurring
‘trick’ in the following sections is to decompose stochastic objects into
distributional and more regular parts. There are also analogues of the usual
$L^{p}$ duality and interpolation.

######  Theorem 2.4 (Multiplicative inequality).

Let $s_{1}<s_{2}$ be non-zero such that $s_{1}+s_{2}>0$, and let
$1/p=1/p_{1}+1/p_{2}$. Then

${\left\|fg\right\|}_{B^{s_{1}}_{p,r}(\rho_{1}\rho_{2})}\lesssim{\left\|f\right\|}_{B^{s_{1}}_{p_{1},r}(\rho_{1})}{\left\|g\right\|}_{B^{s_{2}}_{p_{2},r}(\rho_{2})}.$

######  Theorem 2.5 (Duality).

Let $(p,p^{\prime})$ and $(r,r^{\prime})$ be Hölder conjugate pairs, and
$\rho_{1}$ and $\rho_{2}$ polynomial weights. Then

${\left\|fg\right\|}_{L^{1}(\rho_{1}\rho_{2})}\lesssim{\left\|f\right\|}_{B^{s}_{p,r}(\rho_{1})}{\left\|f\right\|}_{B^{-s}_{p^{\prime},r^{\prime}}(\rho_{2})}$

######  Theorem 2.6 (Interpolation).

Fix $\theta\in(0,1)$, $s=\theta s_{1}+(1-\theta)s_{2}$, and

$\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}},\quad\frac{1}{r}=\frac{\theta}{r_{1}}+\frac{1-\theta}{r_{2}},\quad\alpha=\theta\beta+(1-\theta)\gamma.$

Then

${\left\|f\right\|}_{B^{s}_{p,r}(\rho^{\alpha})}\leq{\left\|f\right\|}_{B^{s_{1}}_{p_{1},r_{1}}(\rho^{\beta})}^{\theta}{\left\|f\right\|}_{B^{s_{2}}_{p_{2},r_{2}}(\rho^{\gamma})}^{1-\theta}.$

We shall use the following three embedding results. The first lets us trade
smoothness for $L^{p}$ and $\ell^{r}$ regularity. The second plays a crucial
role in the weak convergence argument by letting us pass to a convergent
subsequence in a compact space.

######  Theorem 2.7 (Besov embeddings).

Let $s\in\mathbb{R}$, $1\leq q\leq p\leq\infty$, and

$s^{\prime}\geq s+d\left(\frac{1}{q}-\frac{1}{p}\right).$

Then

${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s^{\prime}}_{q,r}(\rho)}.$

The usual chain $\ell^{\infty}\subset\ell^{r}\subset\ell^{1}$ applies. For any
$\varepsilon>0$ we also have

${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+\varepsilon}_{p,\infty}(\rho)}.$

Finally,
${\left\|f\right\|}_{B^{k}_{p,\infty}(\rho)}\lesssim{\left\|f\right\|}_{W^{k,p}(\rho)}\lesssim{\left\|f\right\|}_{B^{k}_{p,1}(\rho)}$
for all $k\in\mathbb{N}$; in particular, for $k=0$ that corresponds to
$L^{p}$.

######  Theorem 2.8 (Compact embedding).

Let $\rho_{2}$ and $\rho_{1}$ be polynomial weights with respective parameters
$\alpha_{2}>\alpha_{1}>d$, and $s_{2}<s_{1}$. The space
$B^{s_{1}}_{p,r}(\rho_{1})$ then embeds compactly into the less regular space
$B^{s_{2}}_{p,r}(\rho_{2})$.

For the finite-volume results, we also need periodic Besov spaces. The same
multiplicative inequalities work also in this case, and furthermore the
following lemma shows that we can move between periodic and polynomial-weight
spaces easily. We use the Mourrat–Weber [38] definition of these spaces.

######  Definition 2.9 (Periodic Besov space).

Given the set $\Lambda_{L}\coloneqq{[{-L},{L}]}^{d}$, we define the space
$B^{s}_{p,r}(\Lambda_{L})$ as the completion of $2L$-periodic
$C^{\infty}(\mathbb{R}^{d})$ functions with respect to the Besov norm. The
inner $L^{p}$ norm is unweighted but over $\Lambda_{L}$.

######  Lemma 2.10 (Embedding into polynomial-weight space).

Let $\rho$ be a polynomial weight with parameter $\alpha$, and let $f\in
C^{\infty}(\mathbb{R}^{d})$ be $2L$-periodic. Then

${\left\|\rho\right\|}_{L^{1}}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\lesssim
L^{\alpha}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}.$

###### Proof.

The second inequality follows from

${\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\leq\left(\sup_{x\in\Lambda_{L}}\rho(x)^{-1}\right){\left\|f\right\|}_{B^{s}_{p,r}(\rho\mathbf{1}_{\Lambda_{L}})}\lesssim_{\rho}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}.$

Similarly, if we denote by $\Lambda_{L}^{j}$ the translates $\Lambda_{L}+j2L$,
the first inequality is

${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\leq\sum_{j\in\mathbb{Z}}{\left\|f\right\|}_{B^{s}_{p,r}(\rho\mathbf{1}_{\Lambda_{L}^{j}})}\leq{\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\sum_{j\in\mathbb{Z}}\sup_{x\in\Lambda_{L}^{j}}\rho(x).$

We must still justify that $f$ belongs to the polynomial-weight space.

Let $F$ be the restriction of $f$ to $\Lambda_{L}$, and let $F_{n}$ consist of
$2n+1$ repeats of $F$ along each axis, thus supported on
$[{-(2n+1)L},{(2n+1)L}]^{d}$. Then $F_{n}$ and $f$ coincide in any compact set
of $\mathbb{R}^{d}$ for large enough $n$.

However, $F_{n}$ might not belong to $B^{s}_{p,r}(\rho)$ because we did not
assume $F$ to vanish at the boundary. To fix that issue, we need to multiply
$F_{n}$ by the smoothed indicator of some slightly smaller set; call the
result $\tilde{F}_{n}$. Then the difference $\tilde{F}_{n+1}-\tilde{F}_{n}$
consists of two disjoint $C_{c}^{\infty}$ parts, and these parts are only
translated as $n$ increases.

With any $g\in C_{c}^{\infty}(\mathbb{R}^{d})$, we can approximate

${\left\|g\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|g\right\|}_{L^{p}(\rho)}+{\left\|D^{\lceil
s\rceil}g\right\|}_{L^{p}(\rho)}.$

These norms are clearly local, so it is easier to see that
${\|\tilde{F}_{n+1}-\tilde{F}_{n}\|}_{B^{s}_{p,r}(\rho)}$ is an $f$-dependent
constant multiplied by $\sup_{x\in\Lambda_{L}^{n}}\rho(x)$. We can then use
triangle inequality to estimate

${\left\|\tilde{F}_{n+m}-\tilde{F}_{n}\right\|}_{B^{s}_{p,r}(\rho)}\lesssim
C_{f}\sum_{j=n}^{\infty}\sup_{x\in\Lambda_{L}^{j}}\rho(x),$

which proves that the approximating sequence $(\tilde{F}_{n})$ is Cauchy in
$B^{s}_{p,r}(\rho)$. ∎

##  3 Stochastic quantization

In this section we construct the $\phi^{4}$ measure in the infinite domain
$\mathbb{R}^{2}$ equipped with a suitable weight. This construction is well-
known in the literature of stochastic quantization, and we only outline the
results we will need.

We define the stochastic objects both on the periodic space
$\Lambda_{L}\coloneqq{[{-L},{L}]}^{2}$ and the full space $\mathbb{R}^{2}$.
The basic building block, Gaussian free field, is straightforwardly defined in
both cases, whereas for the $\phi^{4}_{2}$ we need to take a weak limit as
$L\to\infty$.

######  Remark 3.1.

In this section we denote by $\mu$ the $\phi^{4}$ measure, whereas in Sections
4 and 5 we need to consider the product of $\phi^{4}$ and white noise
measures. We go with this abuse of notation since the $\phi^{4}$ part is
always the “interesting” measure. However, it is important to keep in mind
that the NLW solution will also depend on the white noise.

###  3.1 Gaussian free field

######  Definition 3.2 (Gaussian free field).

The massive GFF $\nu_{L}$ is the Gaussian measure on
$\mathcal{S}^{\prime}(\Lambda_{L})$ with covariance

$\int\langle f,\phi\rangle\langle
g,\phi\rangle\,\mathrm{d}\nu_{L}(\phi)=\langle
f,(m^{2}-\Delta)^{-1}g\rangle_{L^{2}(\Lambda_{L})}.$

Similarly we can introduce the infinite-volume GFF $\nu$ supported on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ and with covariance

$\int\langle f,\phi\rangle\langle
g,\phi\rangle\,\mathrm{d}\nu_{L}(\phi)=\langle
f,(m^{2}-\Delta)^{-1}g\rangle_{L^{2}(\mathbb{R}^{2})}.$

Note that we can view $\nu_{L}$ as a measure on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ by periodic extension. The following
proposition is proved in [38, Theorem 5.1].

######  Theorem 3.3 (Uniform bounds for GFF).

$\nu_{L}$ and $\nu$ have samples almost surely in
$\mathcal{C}^{-\varepsilon}(\rho)$, and for all $p<\infty$ the expectations
are bounded (uniformly in $L$):

$\sup_{L}\int{\left\|\psi\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}\,\mathrm{d}\nu_{L}(\psi)<\infty,\quad\int{\left\|\psi\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}\,\mathrm{d}\nu(\psi)<\infty.$

We will denote random variables sampled from $\nu_{L},\nu$ by $Z_{L},Z$. With
some abuse of notation we will write the projections $Z_{L,N}=P_{N}Z_{L}$ and
$Z_{N}=P_{N}Z$. We can sample from the GFF by realizing it as

$Z_{L}=\frac{1}{L}\sum_{n\in
L^{-1}\mathbb{Z}^{2}}\frac{g_{n}e_{n}}{(m^{2}+|n|^{2})^{1/2}},$ (3.1)

where $g_{n}$ are complex Gaussians with variance $1$ and
$e_{n}(x)\coloneqq\exp(2\pi nx)$. We require $g_{-n}=\overline{g_{n}}$ to make
the field real, but otherwise $g_{n}$ are independent. For the full-space case
we can write

$Z=\int_{\mathbb{R}^{2}}\frac{\xi(y)e_{y}}{(m^{2}+|y|^{2})^{1/2}}\,\mathrm{d}y$
(3.2)

where $\xi$ is a white noise as defined below.

######  Definition 3.4 (White noise).

The white noise $\xi$ is a Gaussian process on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ with covariance

$\mathbb{E}\,[{\left\langle\xi,\psi\right\rangle}_{L^{2}(\mathbb{R}^{2})}{\left\langle\xi,\phi\right\rangle}_{L^{2}(\mathbb{R}^{2})}]={\left\langle\psi,\phi\right\rangle}_{L^{2}(\mathbb{R}^{2})}.$

The measure $\nu_{L}$ does not have samples of positive regularity. This means
that taking powers of distributions sampled from $\nu_{L}$ does not make
sense. Yet the Gaussian structure of the randomness allows us to still define
powers of the field by so-called Wick ordering.

######  Definition 3.5 (Wick ordering, periodic space).

Let $a_{N,L}=\mathbb{E}\,[Z_{L,N}(0)^{2}]$. We then define

$\displaystyle{:\\!Z^{4}_{L,N}\\!:}_{L}$
$\displaystyle=Z^{4}_{L,N}-6a_{N,L}Z^{2}_{L,N}+3a^{2}_{N,L},$
$\displaystyle{:\\!Z^{3}_{L,N}\\!:}_{L}$
$\displaystyle=Z^{3}_{L,N}-3a_{N,L}Z_{L,N},$
$\displaystyle{:\\!Z^{2}_{L,N}\\!:}_{L}$ $\displaystyle=Z^{2}_{L,N}-a_{N,L}.$

As $N\to\infty$, the constants $a_{N,L}$ diverge logarithmically, and the
counterterms cancel the divergence of $Z^{k}_{L,N}$. Further Wick powers can
be defined using Hermite polynomials. Wick-ordered polynomials are defined by
Wick-ordering each monomial term separately. We remark that
$\mathbb{E}\,[Z_{L,N}(x)^{2}]$ does not depend on the choice of $x$ since the
Gaussian free field is translation-invariant.

It will be useful to define the Wick powers with a renormalization constant
that is independent of $L$. For this purpose we will use the expectation of
the full-space GFF.

######  Definition 3.6 (Wick ordering, full space).

We denote by $a_{N}=\mathbb{E}\,[Z_{N}(0)^{2}]$, and define

$\displaystyle{:\\!Z^{4}_{L,N}\\!:}$
$\displaystyle=Z^{4}_{L,N}-6a_{N}Z^{2}_{N,L}+3a^{2}_{N},$
$\displaystyle{:\\!Z^{3}_{L,N}\\!:}$
$\displaystyle=Z^{3}_{L,N}-3a_{N}Z_{L,N},$
$\displaystyle{:\\!Z^{2}_{L,N}\\!:}$ $\displaystyle=Z^{2}_{L,N}-a_{N}.$

The difference between these two renormalizations is a polynomial of strictly
lower degree; for the fourth Wick powers it is

${:\\!Z^{4}_{L,N}\\!:}_{L}-{:\\!Z^{4}_{L,N}\\!:}=-6(a_{N,L}-a_{N})Z^{2}_{L,N}+3(a_{N,L}^{2}-a_{N}^{2}).$
(3.3)

The next lemma asserts that the difference of renormalization constants goes
to zero as $N,L\to\infty$. This lets us always take Wick ordering with respect
to the full-space GFF.

######  Lemma 3.7.

For $a_{N,L}=\mathbb{E}\,[Z_{L,N}(0)^{2}]$ and
$a_{N}=\mathbb{E}\,[Z_{N}(0)^{2}]$ we have

$|a_{N,L}-a_{N}|\lesssim\frac{1}{N}+\frac{1}{L}.$

For every $L<\infty$ we can thus define
$r_{L}\coloneqq\lim_{N\to\infty}(a_{N,L}-a_{N})\lesssim\frac{1}{L}$.

###### Proof.

The first renormalization constant can be written as

$\mathbb{E}\,{\left|Z_{L,N}(0)^{2}\right|}=\frac{1}{L^{2}}\sum_{\begin{subarray}{c}n\in
L^{-1}\mathbb{Z}^{2}\\\ {\left|n\right|}\leq
N\end{subarray}}\frac{1}{m^{2}+{\left|n\right|}^{2}}=\sum_{\begin{subarray}{c}n\in
L^{-1}\mathbb{Z}^{2}\\\ {\left|n\right|}\leq
N\end{subarray}}\int_{P(n)}\frac{1}{m^{2}+{\left|n\right|}^{2}}\,\mathrm{d}x,$

where $P(n)$ is the rectangle $n+[{0},{1/L}]^{2}$. By covariance of the
continuum white noise, the second renormalization constant is

$\mathbb{E}\,{\left|Z_{L}(0)^{2}\right|}=\int_{{\left|x\right|}\leq
N}\frac{1}{m^{2}+{\left|x\right|}^{2}}\,\mathrm{d}x.$

The difference of these integrals is estimated in two parts. Some of the
rectangles $P(n)$ extend outside the ball $B(0,N)$; this introduces an error
of order

$\sum_{\begin{subarray}{c}n\in L^{-1}\mathbb{Z}^{2}\\\ P(n)\not\subset
B(0,N)\end{subarray}}\frac{1}{m^{2}+{\left|n\right|}^{2}}\lesssim
N\cdot\frac{1}{N^{2}}.$

Meanwhile for $x\in P(n)\cap B(0,N)$ we can estimate

$\displaystyle{\left|\frac{1}{m^{2}+{\left|n\right|}^{2}}-\frac{1}{m^{2}+{\left|x\right|}^{2}}\right|}$
$\displaystyle={\left|\frac{{\left|x\right|}^{2}-{\left|n\right|}^{2}}{(m^{2}+{\left|n\right|}^{2})(m^{2}+{\left|x\right|}^{2})}\right|}$
$\displaystyle\leq\frac{1/L}{{(m^{2}+{\left|n\right|}^{2})}^{2}}.$

This implies that the difference of integrals inside $B(0,N)$ is bounded by

$\frac{1}{L^{3}}\sum_{\begin{subarray}{c}n\in L^{-1}\mathbb{Z}^{2}\\\
{\left|n\right|}\leq
N\end{subarray}}\frac{1}{{(m^{2}+{\left|n\right|}^{2})}^{2}}\lesssim\frac{1}{L}.\qed$

We can now state that all Wick powers of the Gaussian free field are well-
defined. This result also translates into regular enough perturbations of the
GFF, in particular the $\phi^{4}$ measure defined below.

######  Lemma 3.8 (Moments of GFF powers).

Let $Z_{L}$ be sampled from $\nu_{L}$ and let $Z_{L,N}=P_{N}Z_{L}$. Then for
any $p<\infty$ and $i=1,2,\ldots$ we have

$\sup_{N}\mathbb{E}\left[{\left\|{:\\!Z^{i}_{L,N}\\!:}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\Lambda_{L})}\right]<\infty\quad$

Furthermore ${:\\!Z^{i}_{L,N}\\!:}$ converges almost surely in
$\mathcal{C}^{-\varepsilon}(\mathbb{T}^{2})$ and in any
$L^{p}(\nu_{L},C^{-\varepsilon}(\rho))$ to a well-defined limit. We denote
this limit by ${:\\!Z_{L}^{i}\\!:}$ and we have

$\quad\sup_{L}\mathbb{E}\left[{\left\|{:\\!Z^{i}_{L}\\!:}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}\right]<\infty.$

###### Proof.

The proof of the first statement is in [19, Lemma 3.2], and the polynomial-
weight case is [38, Theorem 5.1]. ∎

######  Lemma 3.9 (Wick powers of perturbations).

Let $\phi\in L^{p}(\nu_{L},B_{p,p}^{2\varepsilon})$, where $\varepsilon>0$ and
$p$ is large enough. Then

${:\\!(Z_{L}+\phi)^{j}\\!:}=\sum_{i=0}^{j}{:\\!Z^{i}_{L}\\!:}\;\phi^{j-i}.$

###### Proof.

It follows from properties of Hermite polynomials that

${:\\!(Z_{L,N}+\phi)^{j}\\!:}=\sum_{i=0}^{j}\binom{j}{i}{:\\!Z^{i}_{L,N}\\!:}\;\phi^{j-i}.$

Each term is well-defined as an element of
$B^{-\varepsilon}_{p/j,p/j}(\rho^{j+1})$ by Theorem 2.4; moreover the
multiplication is a continuous operation. The claim then follows by passing
$N\to\infty$. ∎

The following result lets us compute covariances of Wick powers by passing to
a Green’s function. For the proof, see e.g. [52, Theorem I.3].

As an application, we see that we can approximate the third Wick power by
continuous maps. We use this lemma to prove that sequences of periodic
solutions satisfy the PDEs also in the limit. The proof is somewhat technical,
and we leave it to Appendix A.

######  Theorem 3.10 (Wick’s theorem).

If $X$ and $Y$ are Gaussian, then

$\mathbb{E}\,[{:\\!X^{n}\\!:}\;{:\\!Y^{n}\\!:}]=n!\left(\mathbb{E}\,[XY]\right)^{n}.$

######  Lemma 3.11 (Approximation of Wick powers).

For every $\delta>0$ and $s>0$, there exists a continuous map
$f^{\delta}\colon H^{-s}(\rho)\to L^{2}(\rho)$ such that the following holds.
Let $Z$ (respectively $Z_{L}$) be sampled from the (periodic) Gaussian free
field and $\phi\in L^{p}(\mathbb{P},B_{p,p}^{\varepsilon}(\rho))$. Then

$\lim_{\delta\to
0}\mathbb{E}\,{\left\|f^{\delta}(Z+\phi)-{:\\!(Z+\phi)^{3}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}=0,$

and respectively with $Z$ replaced by $Z_{L}$.

###  3.2 Coupling of the $\phi_{4}$ measure and the GFF

We now turn to study the $\phi_{2}^{4}$ measure that will be the invariant
measure for our PDEs. We can define $\phi^{4}$ directly only in the periodic
case; we need to take a weak limit to get to infinite volume.

Let us first recall the definition and some basic results of weak convergence
of probability measures. These can be found in most probability textbooks; see
for example [31, Definition 13.12].

######  Theorem 3.12 (Weak convergence).

A sequence of probability measures $(\mu_{L})$ taking values in $\mathcal{X}$
is said to _converge weakly_ to $\mu$ if

$\lim_{L\to\infty}\int_{\Omega}f(\phi)\,\mathrm{d}\mu_{L}(\phi)=\int_{\Omega}f(\phi)\,\mathrm{d}\mu(\phi)\quad\text{for
all }f\in C_{b}(\mathcal{X};\mathbb{R}).$

If $\mathcal{X}$ is a Polish space, then the weak limit is unique.

######  Definition 3.13 (Tightness).

A family $(\mu_{L})_{L\in\mathbb{N}}$ of probability measures on $\mathcal{X}$
is called _tight_ if for any $\varepsilon>0$ there exists a compact set
$K_{\varepsilon}$ such that

$\sup_{L\in\mathbb{N}}\mu_{L}(\mathcal{X}\setminus K)<\varepsilon.$

######  Lemma 3.14 (Prokhorov’s theorem; [31, Theorem 13.29]).

Suppose that the sequence $(\mu_{L})$ is tight and defined on a metric space
$\mathcal{X}$. Then there is a subsequence $(\mu_{L_{k}})$ that converges
weakly to a limit measure $\mu$ on $\mathcal{X}$.

We will consider a sequence of $\phi^{4}_{2,L}$ measures over increasingly
large tori and show that it is tight over a polynomially weighted Besov space.
This will give us a weak limiting measure $\phi_{2}^{4}$.

######  Definition 3.15.

In finite volume $\Lambda_{L}$, the $\phi^{4}_{2,L}$ measure is given by

$\mathrm{d}\mu_{L}(\psi)\coloneqq
Z_{L}^{-1}\exp\left(-\lambda\int_{\Lambda_{L}}{:\\!\psi^{4}(x)\\!:}\,\mathrm{d}x\right)\,\mathrm{d}\nu_{L}(\psi),$

where $Z_{L}^{-1}$ is a normalization constant.

By Section 3.1 the Wick power ${:\\!\psi^{4}\\!:}$ makes sense as a
distribution $\nu_{L}$-almost surely, and one can show that the exponential
belongs to $L^{p}(\nu_{L})$ for any $p<\infty$ and $L<\infty$. For our
purposes, it is easier to consider $\mu_{L}$ as the invariant measure of the
stochastic quantization equation (see [38])

$\partial_{t}u+(m^{2}-\Delta)u+{:\\!u^{3}\\!:}=\xi,\quad u\in
C(\mathbb{R}_{+},H^{-\delta}(\Lambda_{L})).$ (SQE)

Here $\xi$ is space-time white noise, the Gaussian process valued in
$\mathcal{S}^{\prime}(\mathbb{R}\times\mathbb{R}^{2})$ with covariance

$\mathbb{E}\,[{\left\langle\xi,f\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})}{\left\langle\xi,g\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})}]={\left\langle
f,g\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})},$

where $f,g\in\mathcal{S}(\mathbb{R}\times\mathbb{R}^{2})$.

We will use (SQE) to control the $\phi_{2,L}^{4}$ measure in the limit
$L\to\infty$. That $\mu_{L}$ is indeed a stationary measure for this equation
was shown by Da Prato and Debussche [19]. We begin by decomposing the solution
as $u=Z+\phi$, where $Z$ is the Gaussian part that solves the stationary
equation

$\left\\{\begin{aligned} \partial_{t}Z+(m^{2}-\Delta)Z&=\xi,\\\
Z(0)&=Z_{0}\sim\text{GFF},\end{aligned}\right.$ (3.4)

and $\phi$ solves

$\left\\{\begin{aligned}
\partial_{t}\phi+(m^{2}-\Delta)\phi+{:\\!(Z+\phi)^{3}\\!:}&=0,\\\
\phi(0)&=u(0)-Z(0).\end{aligned}\right.$ (3.5)

We can take $(u,Z)$ to be jointly stationary solutions to (SQE) and (3.4) so
that $Z(t)\sim\text{GFF}$; see the beginning of Section 4.3 in [24]. In
particular the Wick power ${:\\!Z^{i}\\!:}$ is a well-defined random
distribution. Now multiplying (3.5) by $\rho\phi$ and integrating we obtain

$\partial_{t}\|\rho^{1/2}\phi\|^{2}_{L^{2}}+m^{2}\|\rho^{1/2}\phi\|^{2}_{L^{2}}+\|\rho^{1/2}\nabla\phi\|_{L^{2}}^{2}+\|\rho^{1/4}\phi\|^{4}_{L^{4}}=-G(Z,\phi),$
(3.6)

where

$\displaystyle G(Z,\phi)$
$\displaystyle=3\int\rho\;{:\\!Z^{3}\\!:}\;\phi\,\mathrm{d}x+3\int\rho\;{:\\!Z^{2}\\!:}\;\phi^{2}\,\mathrm{d}x$
$\displaystyle\qquad+\int\rho
Z\phi^{3}\,\mathrm{d}x+\int([\nabla,\rho]\phi)\nabla\phi\,\mathrm{d}x.$ (3.7)

Here $[\nabla,\rho]\coloneqq\nabla(\rho\phi)-\rho\nabla\phi=(\nabla\rho)\phi$
is the commutator of the weight and the gradient. In Appendix B we show that

${\left|G(Z,\phi)\right|}\leq
Q(Z)+\frac{1}{2}({\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}}),$
(3.8)

where

$\sup_{L}\mathbb{E}\,[|Q(Z)|^{p}]<\infty\quad\text{for any }p<\infty.$

By moving common terms to the left-hand side of (3.6), we get the estimate

$\partial_{t}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+\frac{1}{2}(m^{2}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}})\leq
Q(Z).$ (3.9)

The time derivative term vanishes in expectation since $\phi=u-Z$ was assumed
to be stationary. Therefore we are left with

$\frac{1}{2}\mathbb{E}\,[(m^{2}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}})]\leq\mathbb{E}\,Q(Z),$
(3.10)

which proves

$\sup_{L}\mathbb{E}\,{\|\rho^{1/2}\phi\|}^{2}_{H^{1}}<\infty,$

and thus tightness in $H^{1-\varepsilon}(\rho^{1/2+\varepsilon})$. We still
strengthen this in Section 3.3.

###  3.3 Wick powers of $\phi_{2}^{4}$

The bounds on the $\phi^{4}$ samples can be improved to exponential tails,
which then implies $L^{p}$ expectations for all $p$. We defer the proof of
this result to Appendix C.

######  Theorem 3.16 (Exponential tails).

There exists $\delta>0$ such that

$\int_{\Omega}\exp\left(\delta{\left\|W_{L}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{2}\right)\mathrm{d}\mu_{L}(W_{L})\lesssim
1,$

and the bound is uniform in $L$. The bound also holds in the limit $\mu$.

Since the nonlinearity in (NLW) is cubic, we will need the first three Wick
powers of the $\phi^{4}$ field. We construct and estimate the Wick powers of
$\phi_{2,L}^{4}$ uniformly in $L$, and thus in the $L\to\infty$ limit.

######  Theorem 3.17 (Wick powers of $\phi^{4}$).

Let $W_{L}=Z_{L}+\phi_{L}$ be sampled from $\phi_{2,L}^{4}$. Then
${:\\!W^{j}\\!:}$ is a well-defined random distribution for $j\leq 3$, and for
any $\varepsilon>0$ and $p<\infty$ we have

$\sup_{L}\mathbb{E}\,{\|{:\\!W_{L}^{j}\\!:}\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}<\infty.$

Furthermore, if $W$ is sampled from the full-space $\phi_{2}^{4}$ measure,
then

$\mathbb{E}\,{\|{:\\!W^{j}\\!:}\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}<\infty.$

###### Proof.

We only do the proof in the most difficult case $j=3$. The other cases are
analogous. Recall that we have

${:\\!W_{L}^{3}\\!:}=\sum_{i=0}^{3}\binom{3}{i}{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}.$

Now for $q=4/\varepsilon$ we can estimate

$\displaystyle{\left\|{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}$
$\displaystyle\lesssim{\left\|{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}\right\|}_{B_{q,q}^{-\varepsilon/2}(\rho^{2})}$
$\displaystyle\lesssim{\left\|{:\\!Z_{L}^{i}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon/2}(\rho)}{\left\|\phi_{L}^{3-i}\right\|}_{B_{q,q}^{\varepsilon/2}(\rho)}$
$\displaystyle\leq{\left\|{:\\!Z_{L}^{i}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon/2}(\rho)}{\left\|\phi_{L}\right\|}^{3-i}_{B_{3q,3q}^{\varepsilon/2}(\rho^{1/3})}.$
(3.11)

The Gaussian part is bounded by Lemma 3.8. For the perturbation part, Theorem
3.16 implies that
$\mathbb{E}\,{\left\|W_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}<\infty$,
and we can decompose

$\sup_{L}\mathbb{E}\,{\left\|\phi_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}}\lesssim\sup_{L}\,(\mathbb{E}\,{\left\|W_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}(\rho)}+\mathbb{E}\,{\left\|Z_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}(\rho)})<\infty.$

This estimate provides $L^{p}$ regularity, whereas the estimate
$\mathbb{E}\,{\left\|\phi\right\|}^{2}_{H^{1}(\rho^{1/3})}<\infty$ from
Section 3.2 provides differentiability. We can interpolate between these two
through

$\displaystyle{\left\|\phi_{L}\right\|}_{B_{3q,3q}^{\varepsilon/2}(\rho^{1/3})}$
$\displaystyle\lesssim{\left\|\phi_{L}\right\|}^{\theta}_{\mathcal{C}^{-\varepsilon/12}(\rho^{1/3})}{\left\|\phi_{L}\right\|}^{1-\theta}_{H^{1}(\rho^{1/3})}$
$\displaystyle\lesssim{\left\|\phi_{L}\right\|}^{2\theta}_{\mathcal{C}^{-\varepsilon/12}(\rho^{1/3})}+{\left\|\phi_{L}\right\|}^{2-2\theta}_{H^{1}(\rho^{1/3})},$

where $\theta=1-\varepsilon/12$. As we substitute this back into (3.3), we
find that the final expectation is bounded. ∎

From Theorem 3.17 we can bootstrap a stronger statement for the coupling. The
perturbation $\phi$ is two derivatives more regular than $Z$, instead of just
one derivative as showed earlier.

######  Corollary 3.18 (Strong bound for regular part).

We can find random variables $Z_{L},\phi_{L}$ such that $Z_{L}\sim\nu_{L}$,
$Z_{L}+\phi_{L}\sim\mu_{L}$, and

$\sup_{L}\mathbb{E}\,\|\phi_{L}\|^{p}_{H^{2-\varepsilon}(\rho)}\lesssim 1.$

###### Proof.

Recall that from the stochastic quantization equation (SQE) we have

$\phi_{L}(t)=\int^{t}_{0}e^{-(t-s)\Delta}{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\,\mathrm{d}s+e^{-t\Delta}\phi_{L}(0).$

So provided $p$ is large enough that $|t-s|^{(1-\varepsilon/2)p/(p-1)}\in
L^{1}$, we can use the smoothing effect of the heat operator ([38, Proposition
5]) to estimate

$\displaystyle\mathbb{E}\,\|\phi_{L}(t)\|^{p}_{H^{2-\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\mathbb{E}\,{\left\|\int^{t}_{0}e^{-(t-s)\Delta}{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\,\mathrm{d}s\right\|}^{p}_{H^{2-\varepsilon}(\rho)}+\mathbb{E}\,{\left\|e^{-t\Delta}\phi_{L}(0)\right\|}^{p}_{H^{2-\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\mathbb{E}\left[\int^{t}_{0}\frac{{\left\|{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\right\|}_{H^{-\varepsilon/2}(\rho)}}{{\left|t-s\right|}^{1-\varepsilon/2}}\,\mathrm{d}s\right]^{p}+\frac{\mathbb{E}\,{\left\|\phi_{L}(0)\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}}{t^{1-\varepsilon/2}}$
$\displaystyle\lesssim\;$
$\displaystyle\int^{t}_{0}\mathbb{E}\,{\left\|{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}\,\mathrm{d}s+\frac{\mathbb{E}\,{\left\|\phi_{L}(0)\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}}{t^{1-\varepsilon/2}}.$

Since $Z_{L}$ and $\phi_{L}$ are both stationary, we may choose $t$ as we
like. The integrand is then uniformly bounded by Theorem 3.17. ∎

In total we have obtained that
$\sup_{L}\mathbb{E}\,{\left\|\phi\right\|}^{p}_{H^{2-\varepsilon}(\rho)}<\infty$.
By the same compactness argument as above, $\operatorname{Law}(Z,\phi)$ is
tight on $H^{-\varepsilon}(\rho)\times
H^{2-\varepsilon}(\rho^{1+\varepsilon})$. In particular
$\mu_{L}=\operatorname{Law}(Z+\phi)$ is tight on
$H^{-\varepsilon}(\rho^{1+\varepsilon})$ and has a weakly converging
subsequence. We have thus proved the following:

######  Theorem 3.19 ($\phi^{4}_{2}$ as a weak limit).

Let $\rho$ be a sufficiently integrable polynomial weight. The measure
$\mu_{L}$ can be represented as

$\mu_{L}=\operatorname{Law}(Z_{L}+\phi_{L})$

where $Z_{L}$ is a GFF on $\Lambda_{L}$, and $\phi_{L}$ satisfies
$\sup_{L}\mathbb{E}\,\|\phi_{L}\|^{p}_{H^{2}(\rho)}<\infty$. Identifying
$Z_{L}+\phi_{L}$ with its periodic extension on $\mathbb{R}^{2}$ we have that
$\mu_{L}=\operatorname{Law}(Z_{L}+\phi_{L})$ is tight on
$H^{-\varepsilon}(\rho^{1+\varepsilon})$ and any limiting point $\mu$
satisfies

$\mu=\operatorname{Law}(Z+\phi)$

where $Z$ is a Gaussian free field on $\mathbb{R}^{2}$ and
$\mathbb{E}\,{\left\|\phi\right\|}^{p}_{H^{2}(\rho)}<\infty$.

###### Proof.

We know that the limit of $\operatorname{Law}(Z_{L})$ as $L\to\infty$ is a
Gaussian free field on $\mathbb{R}^{2}$; this follows for instance from the
convergence of the covariances. It remains to show that

$\mathbb{E}\,\|\phi\|^{p}_{H^{2-\varepsilon}(\rho)}<\infty$

but since $\|\phi\|^{2}_{H^{2}(\rho)}$ is lower semicontinuous on
$H^{2-\varepsilon}(\rho^{1+\varepsilon})$ we have by weak convergence

$\mathbb{E}\,[\|\phi\|^{p}_{H^{2}(\rho)}]\leq\liminf_{L\to\infty}\mathbb{E}\,[\|\phi_{L}\|^{p}_{H^{2}(\rho)}]<\infty.\qed$

######  Remark 3.20.

We were careful to state the preceding theorem for “any limiting point $\mu$”.
When the coupling parameter $\lambda$ is large enough, there exist
subsequences of $(\phi^{4}_{2,L})$ that converge to different weak limits.
This is one of the main complications in our study.

##  4 Invariance of periodic NLW

Let us now move on to solving the nonlinear wave equation. We fix a bounded
domain $\Lambda_{L}={[{-L},{L}]}^{2}$ and consider

$\left\\{\begin{aligned}
\partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)&=-\lambda{:\\!u^{3}\\!:}(x,t),\\\
u(x,0)&=u_{0}(x),\\\
\partial_{t}u(x,0)&=u_{0}^{\prime}(x)\end{aligned}\right.$ (4.1)

on $\Lambda_{L}\times\mathbb{R}_{+}$. The initial value $u_{0}$ will be
sampled from a $\phi^{4}$ measure and the initial time derivative
$u_{0}^{\prime}$ from a white noise measure; as remarked at the beginning of
Section 3, we denote by $\mu$ now the product $(\phi^{4},\text{WN})$ measure.
The Wick ordering will always be taken with respect to the full space
covariance, even if we start from periodic initial data.

Thanks to the finite speed of propagation (formulated more precisely in
Theorem 4.1), boundary effects are visible only outside the ball $B(0,L-t)$.

By solving the equation in Fourier space, we can write the mild solution as

$u(x,t)={\mathcal{C}}_{t}u_{0}(x)+{\mathcal{S}}_{t}u_{0}^{\prime}(x)+\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!u^{3}\\!:}](x)\,\mathrm{d}s,$
(4.2)

where we use the cosine and sine operators

${\mathcal{C}}_{t}=\cos((m^{2}-\Delta)^{1/2}t),\quad{\mathcal{S}}_{t}=\frac{\sin((m^{2}-\Delta)^{1/2}t)}{(m^{2}-\Delta)^{1/2}}.$
(4.3)

These are defined as Fourier multiplier operators. We see that
${\mathcal{C}}_{t}$ preserves the $H^{s}(\Lambda_{L})$ regularity of its
argument whereas ${\mathcal{S}}_{t}$ increases it by one.

We split the solution into nonlinear and linear parts $u=v+w$. Here $w$ solves
the linear equation with the given initial data, leaving $v$ to solve the
coupled equation

$\partial_{t}v(x,t)+(m^{2}-\Delta)v(x,t)=\lambda\,{:\\!(v+w)^{3}\\!:}$ (4.4)

with zero initial data. We will see that $v$ has one degree higher regularity
than $w$, and its growth is controlled by $w$. The final solution will exist
in $L^{p}([0,\tau];B^{-\varepsilon}_{p,p}(\Lambda_{L}))$ up to some short,
stochastic time $\tau$.

The result in this section was already proved by Oh and Thomann [46], and
stated without proof by Bourgain in 1999. The argument presented below
replaces the more specific Fourier restriction norm by a general Besov norm,
and includes the details on convergence of solutions.

###  4.1 Linear part

It is a basic property of the wave equation that all wave packets travel at a
fixed speed. This property applies to the Duhamel formulation (4.2) as well.
The solution operators are then also bounded in weighted spaces since the
weight does not change too much within a ball.

######  Lemma 4.1 (Finite speed of propagation).

If $(u_{0},u_{0}^{\prime})$ and $(v_{0},v_{0}^{\prime})$ coincide on $B(0,R)$,
then the corresponding linear wave equation solutions $u(t)$ and $v(t)$
coincide on $B(0,R-t)$.

###### Sketch of proof.

By linearity, we only need to show that

${\mathcal{C}}_{t}w_{0}(x)+{\mathcal{S}}_{t}w_{0}^{\prime}(x)$

vanishes on $B(0,R-t)$ if both $w_{0}$ and $w_{0}^{\prime}$ vanish inside
$B(0,R)$. By density, we can further assume $w_{0}$ and $w_{0}^{\prime}$ to be
smooth and compactly supported. The idea is to consider the local energy

$E(s)\coloneqq\int_{B(0,R-s)}m^{2}u(x,s)^{2}+{\left|\nabla
u(x,s)\right|}^{2}+{\left|\partial_{t}u(x,s)\right|}^{2}\,\mathrm{d}x.$

By a calculation with some vector analysis, we see that $\partial_{s}E(s)\leq
0$ up to time $t$, and $E(0)=0$ by assumption. See [22, Section 2.4.3] for
details. ∎

######  Lemma 4.2 (Boundedness of solution operators).

Let $t\leq T$. Then for $s\in\mathbb{R}$ and $f\in H^{s}(\rho)$ we have

${\left\|{\mathcal{C}}_{t}f\right\|}_{H^{s}(\rho)}\lesssim{\left\|f\right\|}_{H^{s}(\rho)}\quad\text{and}\quad{\left\|{\mathcal{S}}_{t}f\right\|}_{H^{s}(\rho)}\lesssim{\left\|f\right\|}_{H^{s-1}(\rho)}.$

###### Proof.

Denote by $K$ the convolution kernel of ${\mathcal{C}}_{t}$. As noted in Lemma
4.1, $K$ is supported in $B(0,T)$. It suffices to study the case $s=0$ as
${\mathcal{C}}_{t}$ commutes with derivatives. We see that ${\mathcal{C}}_{t}$
is bounded on $L^{2}$ with flat weight, since it is a Fourier multiplier with
bounded symbol. Then

$\displaystyle{\left\|{\mathcal{C}}_{t}f\right\|}_{L^{2}(\rho)}^{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}\rho^{2}(x)\left[\int_{\mathbb{R}^{2}}K(x-y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle\lesssim\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}^{2}}K(x-y)\rho(x-y)^{-1}\rho(y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle\lesssim\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}^{2}}K(x-y)\rho(y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle={\left\|{\mathcal{C}}_{t}(\rho f)\right\|}_{L^{2}}^{2}$
$\displaystyle\lesssim{\left\|\rho f\right\|}_{L^{2}}^{2},$

We could estimate $\rho(x-y)^{-1}\leq C(T)$ since $K$ vanishes outside
$|x-y|\leq T$. For ${\mathcal{S}}_{t}$ the proof is identical, except that we
gain a derivative of regularity. ∎

In probabilistic terms, the linear part looks almost like the coupled
$\phi^{4}$ measure: there is an invariant Gaussian free field part and a more
regular term with conserved norm. This yields a moment bound, which we then
use to control the norm of the nonlinearity in the next section.

######  Lemma 4.3 (Distribution of linear part).

As we substitute $u_{0}=Z_{L}+\phi_{L}$, the linear part becomes

$w(\cdot,t)=\bigg{[}{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\bigg{]}+{\mathcal{C}}_{t}\phi_{L}.$

Here the term in brackets is GFF, whereas ${\mathcal{C}}_{t}\phi\in
H^{2-\varepsilon}(\rho)$ almost surely.

###### Proof.

The latter part follows from preservation of Sobolev regularity by the cosine
operator. To prove the first part, we need to compute the covariance. For any
test functions $\varphi$, $\psi$ we have

$\displaystyle\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}$
$\displaystyle=\;$
$\displaystyle\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z\right\rangle}\bigg{]}+\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}$

by independence of $Z$ and $u_{0}^{\prime}$. Because ${\mathcal{C}}_{t}$ is a
self-adjoint operator, the first term becomes

$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z\right\rangle}\bigg{]}={\left\langle{\mathcal{C}}_{t}\varphi,\frac{{\mathcal{C}}_{t}\psi}{m^{2}-\Delta}\right\rangle}={\left\langle\varphi,\frac{\cos(m^{2}-\Delta)^{2}}{m^{2}-\Delta}\psi\right\rangle}.$

For the second term we have white noise covariance instead:

$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}={\left\langle{\mathcal{S}}_{t}\varphi,{\mathcal{S}}_{t}\psi\right\rangle}={\left\langle\varphi,\frac{\sin(m^{2}-\Delta)^{2}}{m^{2}-\Delta}\psi\right\rangle}.$

Now the trigonometric identity $\sin^{2}+\cos^{2}=1$ implies

$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}={\left\langle\varphi,\frac{1}{m^{2}-\Delta}\psi\right\rangle}.\qed$

######  Lemma 4.4 (Moment bounds for linear part).

Let $L$ belong to the convergent subsequence of Theorem 3.19. Let $w_{L}$ be
the $L$-periodic linear part started from data sampled from $L$-periodic
$(\phi^{4}_{2},\mathrm{WN})$ measure. For $j=1,2,3$ and $T>0$ we have the
moment bounds

$\mathbb{E}\,{\|{:\\!w_{L}^{j}\\!:}\|}_{L^{p}([0,T];\,B^{-\varepsilon}_{p,p}(\rho))}^{p}\lesssim_{p,T,\varepsilon}1.$

This also implies that

$\mathbb{E}\,{\|{:\\!w_{L}^{j}\\!:}\|}_{L^{p}([0,T];\,\mathcal{C}^{-2\varepsilon}(\rho))}^{p}\lesssim_{p,T,\varepsilon}1$

for $p$ sufficiently large. Both estimates are uniform in $L$.

###### Proof.

By Lemma 4.3 we can decompose $w_{L}=w_{\mathrm{st}}+\psi$ where
$\psi(t)={\mathcal{C}}_{t}\phi_{L}$ and $w_{\mathrm{st}}$ has distribution
$\nu_{L}$ for every time. We thus have by stationarity

$\int_{0}^{T}\mathbb{E}{\|{:\\!w_{\mathrm{st}}^{j}\\!:}\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}\,\mathrm{d}t=T\,\mathbb{E}{\left\|{:\\!(Z(0))^{j}\\!:}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}.$

When $j=3$, we can expand the binomial and estimate

$\mathbb{E}{\left\|{:\\!w_{\mathrm{st}}^{3}\\!:}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\psi\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|w_{\mathrm{st}}\psi^{2}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|\psi^{3}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}$

at a fixed time. The middle terms can be expanded with Theorem 2.4 like in

$\displaystyle{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\phi\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}$
$\displaystyle\lesssim{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\right\|}_{B^{-\varepsilon}_{2p,p}(\rho^{1/2})}^{p}{\left\|\phi\right\|}_{B^{2\varepsilon}_{2p,p}(\rho^{1/2})}^{p}$
$\displaystyle\lesssim
1+{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\right\|}_{B^{-\varepsilon/2}_{2p,2p}(\rho^{1/2})}^{2p}{\left\|\psi\right\|}_{B^{2-\varepsilon}_{2p,2p}(\rho^{1/2})}^{2p}.$

The other cases $j=1$ and $j=2$ are analogous. Now we can deduce the claim
since the norm of $w_{\mathrm{st}}$ is in $L^{p}([0,T])$ by stationarity, and
$\psi$ is in $C([0,T],B^{\varepsilon}_{p,p})$ by Corollary 3.18 uniformly for
a sequence of $L\to\infty$.

The second claim follows from the embedding stated in Theorem 2.7 by choosing
$p\geq 2/\varepsilon$. ∎

We can now show that powers of the linear parts converge as $L\to\infty$. We
will use this result as we pass to the full space in Section 5.

######  Lemma 4.5 (Convergence of linear parts).

Let $1\leq p<\infty$. As $L\to\infty$, ${:\\!w^{i}_{L}\\!:}$ converges in
probability to ${:\\!w^{i}\\!:}$ in
$L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\rho^{3}))$, where $w$ is a linear
solution started from initial data $Z+\phi$.

###### Proof.

The convergence of ${\mathcal{C}}_{t}\phi_{L}$ to ${\mathcal{C}}_{t}\phi$ in
$H^{2-\varepsilon}(\rho)$ follows from continuity of ${\mathcal{C}}_{t}$ in
$H^{2-\varepsilon}(\rho)$. We need to show that
${:\\!w^{i}_{{\mathrm{st}},L}\\!:}\to{:\\!w_{\mathrm{st}}^{i}\\!:}$ in
$L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\rho^{3}))$. Then continuity of Besov
product from $\mathcal{C}^{-\varepsilon}\times H^{2-\varepsilon}$ to
$\mathcal{C}^{-\varepsilon}$ implies convergence of
$(w_{{\mathrm{st}}_{L}}+\phi_{L})^{3}$.

We have that $w_{{\mathrm{st}},L}\to w_{\mathrm{st}}$ in
$C([0,T],H^{-\varepsilon}(\rho))$ by continuity of the linear operators. Now
with $f^{\delta}$ as in Lemma 3.11 we have

$\displaystyle{:\\!w^{i}_{{\mathrm{st}},L}\\!:}-{:\\!w^{i}_{\mathrm{st}}\\!:}=\;$
$\displaystyle[{:\\!w^{i}_{{\mathrm{st}},L}\\!:}-f^{\delta}(w_{{\mathrm{st}},L})]$
$\displaystyle\quad+[f^{\delta}(w_{{\mathrm{st}},L})-f^{\delta}(w_{\mathrm{st}})]+[f^{\delta}(w_{\mathrm{st}})-{:\\!w^{i}_{\mathrm{st}}\\!:}].$

The middle term goes to $0$ since $f^{\delta}$ is continuous from
$H^{-\varepsilon}(\rho)$ to $\mathcal{C}^{-\varepsilon}(\rho^{3})$, and for
the first and last term we have by stationarity

$\mathbb{E}\bigg{[}\int_{0}^{T}{\left\|f^{\delta}(w_{\mathrm{st}})-{:\\!w^{i}_{\mathrm{st}}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{3})}^{p}\,\mathrm{d}s\bigg{]}=T{\left\|f^{\delta}(w_{\mathrm{st}}(0))-{:\\!w^{i}_{\mathrm{st}}(0)\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{3})}.$

This goes to $0$ uniformly in $L$ by Lemma 3.11. ∎

###  4.2 Fixed-point iteration

We now use the standard fixed-point argument for

$v(x,t)=-\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}](x)\,\mathrm{d}s.$
(4.5)

As usual, we need to check boundedness and contractivity of the solution
operator. We do the iteration in the periodic Besov space
$L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L}))$. The weight must be
“flat” because we need it to be the same on both sides of the multiplicative
estimates.

This argument is completely deterministic. We control the growth of $v$ by
assuming bounds on the linear part $w$; these bounds will be verified by
stochastic estimates in Section 4.3.

######  Lemma 4.6 (Boundedness).

Let
$M=\max_{j=1,2,3}{\left\|{:\\!w^{j}\\!:}\right\|}_{L^{4}([0,1];\,\mathcal{C}^{-\varepsilon}(\rho))}$.
The operator

$(\mathcal{F}v)(x,t)\coloneqq-\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}](x)\,\mathrm{d}s$

maps a ball of radius $R$ into a ball of radius
$C_{\Lambda}\lambda\tau^{1/2}M(1+R^{3})$ in the periodic space
$L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L}))$.

###### Proof.

We can commute the Fourier multiplier and apply Jensen’s inequality in

$\displaystyle{\left\|\mathcal{F}v\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}$
$\displaystyle=\lambda\sup_{0\leq
t\leq\tau}\left[\int_{\mathbb{R}^{2}}{\left|{\left\langle{\nabla}\right\rangle}^{1-\varepsilon}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}\,\mathrm{d}s\right|}^{2}\,\mathrm{d}x\right]^{1/2}$
$\displaystyle\leq\lambda\tau^{1/2}\sup_{0\leq
t\leq\tau}\left[\int_{\mathbb{R}^{2}}\int_{0}^{t}{\left|{\left\langle{\nabla}\right\rangle}^{1-\varepsilon}{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}\right|}^{2}\,\mathrm{d}s\,\mathrm{d}x\right]^{1/2}$
$\displaystyle=\lambda\tau^{1/2}\left[\int_{0}^{\tau}\int_{\mathbb{R}^{2}}{\left|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}{:\\!(v+w)^{3}\\!:}\right|}^{2}\,\mathrm{d}s\,\mathrm{d}x\right]^{1/2}$
$\displaystyle=\lambda\tau^{1/2}{\left\|{:\\!(v+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}.$

In the second-to-last step we used the increase in Besov regularity from
${\mathcal{S}}_{t}$. We can now expand the binomial power by triangle
inequality and estimate each term separately. First,
${\left\|{:\\!w^{3}\\!:}\right\|}_{L^{2}H^{-\varepsilon}(\Lambda_{L})}\lesssim
L^{c}M$ by assumption and Jensen. The second term is estimated as

${\left\|{:\\!w^{2}\\!:}v\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}\lesssim{\left\|{:\\!w^{2}\\!:}\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})},$

and for the third one we use Theorem 2.4 twice:

${\left\|wv^{2}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}\lesssim{\left\|w\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v^{2}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}\lesssim
L^{c}M{\left\|v\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}^{2}.$

We also perform the a similar multiplicative estimate for the $v^{3}$ term.
Thus we have estimated

$\displaystyle{\left\|{:\\!(v+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim\;$ $\displaystyle
L^{c}M\left[1+{\left\|v\right\|}_{L^{4}_{\tau}\mathcal{H}^{2\varepsilon}(\Lambda_{L})}+{\left\|v\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}^{2}+{\left\|v\right\|}_{L^{12}_{\tau}B^{3\varepsilon}_{6,6}(\Lambda_{L})}^{3}\right],$

which yields the required bound after embedding $H^{1-\varepsilon}$ into
$B^{3\varepsilon}_{6,6}$ by Theorem 2.7. With the estimates above, this is
possible for $\varepsilon<1/12$. ∎

######  Lemma 4.7 (Contraction).

In the setting of Lemma 4.6, we also have

${\left\|\mathcal{F}v-\mathcal{F}\tilde{v}\right\|}_{L^{\infty}([0,\tau];\,H^{1-\varepsilon})}\lesssim
C_{\Lambda}\lambda\tau^{1/2}M(1+R^{2}){\left\|v-\tilde{v}\right\|}_{L^{\infty}([0,\tau];\,H^{1-\varepsilon})}.$

###### Proof.

We can begin as in Lemma 4.6 to get the upper bound

$\lambda\tau^{1/2}{\left\|{:\\!(v+w)^{3}\\!:}-{:\\!(\tilde{v}+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}.$

When we again expand the binomials, we get three terms to estimate (since the
$w^{3}$ terms cancel each other). First,

$\displaystyle{\left\|{:\\!w^{2}\\!:}(v-\tilde{v})\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim{\left\|{:\\!w^{2}\\!:}\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v-\tilde{v}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim
M{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}.$

In the second term we additionally need to expand

$\displaystyle{\left\|v^{2}-\tilde{v}^{2}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim{\left\|v-\tilde{v}\right\|}_{L^{4}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}{\left\|v+\tilde{v}\right\|}_{L^{4}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}$
$\displaystyle\lesssim
2R{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}.$

In the final term, the corresponding expansion is

$\displaystyle{\left\|v^{3}-\tilde{v}^{3}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}}$
$\displaystyle=\;$
$\displaystyle{\left\|(v-\tilde{v})(v^{2}+v\tilde{v}+\tilde{v}^{2})\right\|}_{L^{4}_{\tau}H^{2\varepsilon}}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|v-\tilde{v}\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}}\left({\left\|v\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}^{2}+{\left\|v\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}{\left\|\tilde{v}\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}+{\left\|\tilde{v}\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}^{2}\right)$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}R^{2}.$

All together, we get the claimed inequality for $\varepsilon$ small. ∎

######  Theorem 4.8.

The nonlinear equation (4.5) has a unique solution

$v\in L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L})).$

The norm of $v$ depends only on $M$ as given in Lemma 4.6, and the solution
time $\tau$ depends on both $M$ and the period length of $\Lambda_{L}$.

###### Proof.

It only remains to choose $R$ and $\tau$ such that

$\begin{cases}C_{\Lambda}\lambda\tau^{1/2}M(1+R^{3})\leq R,\\\
C_{\Lambda}\lambda\tau^{1/2}M(1+R^{2})\leq\frac{1}{2}.\end{cases}$

We can select $R=\max\\{1,M\\}$ and $\tau=C_{\Lambda}^{-2}(4\lambda
R^{3})^{-2}$. ∎

###  4.3 Extension to global time

The analysis of previous sections also applies to the truncated equation

$\left\\{\begin{aligned} \partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)&=-\lambda
P_{N}{:\\!P_{N}u^{3}\\!:},\\\ u(x,0)&=P_{N}u_{0}(x),\\\
\partial_{t}u(x,0)&=P_{N}u_{0}^{\prime}(x)\end{aligned}\right.$ (4.6)

where $P_{N}$ truncates the Fourier series to terms with frequency at most
$2^{N}$ in absolute value.333Recall that we define the Besov space with a
full-space Fourier transform; the Fourier transform is a linear combination of
Dirac deltas in this case. The estimates are only changed by a constant factor
since the projection operators $P_{N}$ are bounded uniformly in Besov norm,
and the linear operators ${\mathcal{C}}_{t}$ and ${\mathcal{S}}_{t}$ do not
change the Fourier support.

The reason to pass to (4.6) is that the state space now consists of finitely
many Fourier modes. Because the equation is still Hamiltonian, a theorem of
Liouville automatically implies invariance of the corresponding Gibbs measure.

######  Definition 4.9 (Truncated Gibbs measure).

The measure $\mu_{L,N}$ is supported on the subset of
${\mathcal{H}}^{-\varepsilon}(\rho)$ that contains $2L$-periodic functions
Fourier-truncated to ${[{-2^{N}},{2^{N}}]}^{2}$, and is given by the density

$f(u_{0},u_{0}^{\prime})=\exp\left(-\lambda\int{:\\!P_{N}u^{4}_{0}\\!:}\,\mathrm{d}x\right)$

with respect to the periodic, truncated (Gaussian free field, white noise)
measure.

######  Theorem 4.10 (Local-in-time invariance).

The flow of (4.6) is well-defined up to time $\tau$, and the Fourier-truncated
measure $\mu_{L,N}$ is invariant under the flow.

###### Proof.

Existence of solution follows from the previous sections, and invariance of
measure from Liouville’s theorem. ∎

The invariance of measure allows us to probabilistically extend the solution
to arbitrary time. If the local time $\tau$ would be a conserved quantity, we
could simply restart the flow from $u(\tau)$ and get a solution up to time
$2\tau$. This is the case for $L^{2}$ solutions of (NLS). Such a conservation
law does not exist here, but the growth of Besov norm can be controlled in a
high-probability set. This is because the solutions at time $\tau$ are
distributed identically to the initial data.

######  Definition 4.11 (Bounded-moment set).

Let us recall that we denote by ${\mathcal{H}}^{-\varepsilon}(\rho)$ the space
of all initial data $H^{-\varepsilon}(\rho)\times H^{-1-\varepsilon}(\rho)$.
We define

$B_{M}\coloneqq\left\\{(u_{0},u_{0}^{\prime})\in{\mathcal{H}}^{-\varepsilon}(\rho)\colon{\|{:\\!w^{j}\\!:}\|}_{L^{4}([0,1];\;\mathcal{C}^{-\varepsilon/2}(\rho))}\leq
M\text{ for }j=1,2,3\right\\},$

where $w$ is the $L$-periodic linear solution to (4.1) with data
$(u_{0},u_{0}^{\prime})$.

Let us denote the truncated nonlinear flow by $\Phi_{N,t}$, and let $\tau$ be
the solution time from Theorem 4.8 with $M$ as above. Then the set $B_{M}$
contains initial data that lead to a well-defined solution in $[0,\tau]$.
Correspondingly, $\Phi_{N,\tau}^{-1}B_{M}$ contains such data that the
solution exists in $[-\tau,0]$; by intersection with $B_{M}$, we thus get a
solution in $[0,2\tau]$. By iterating this $m$ times, we get a set of initial
data that supports the flow up to time $T=m\tau$.

By overlapping these solution intervals, we can guarantee uniqueness of the
solution. The dependency on period length $L$ is contained in the solution
time $\tau$ as chosen in Theorem 4.8, and thus the next growth bound is
uniform in $L$. However, the number of intersections $m$ depends on $L$ and
moment bound $M$ via $\tau$.

######  Lemma 4.12 (Growth bound).

Let us define

$\mathcal{B}_{M,L,N}\coloneqq
B_{M}\cap\Phi_{N,\tau/2}^{-1}B_{M}\cap\cdots\cap\Phi_{N,\tau/2}^{-2m}B_{M}.$

For $\mu_{L,N}$-almost all $(u_{0},u_{0}^{\prime})\in\mathcal{B}_{M,L,N}$,
there exists a unique solution $u$ to Eq. (4.6) up to time $T=2m\tau$, and

${\left\|{:\\!u^{j}\\!:}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/2}M^{j}$

for $j=1,2,3$. The constant is independent of $N$ and $L$.

###### Proof.

Although the definition of $B_{M}$ uses the non-truncated linear equation, we
may pass to the truncated equation by
${\mathcal{C}}_{t}(P_{N}u_{0})=P_{N}({\mathcal{C}}_{t}u_{0})$ and boundedness
of $P_{N}$ on the periodic space.

That the solution is unique follows from the fact that the local solution
intervals overlap and each local solution is unique. It thus remains to verify
the moment bounds.

For $j=1$ the claim follows immediately from writing $u=v+w$; the nonlinear
part $v$ has $L^{\infty}([k\tau,(k+1)\tau],H^{1-\varepsilon}(\Lambda_{L}))$
norm bounded by $M$ in Theorem 4.8, whereas the linear part satisfies

${\left\|w\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/4}M$

by Hölder and the definition of $B_{M}$. For $j=2$ we are to estimate

${\left\|{:\\!w^{2}\\!:}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}+2{\left\|vw\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}+{\left\|v^{2}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}.$

Here the only relevant difference is estimating

${\left\|vw\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim{\left\|v\right\|}_{L^{4}([0,T];\,H^{2\varepsilon}(\Lambda_{L}))}{\left\|w\right\|}_{L^{4}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}$

with Besov multiplication and Hölder. Thanks to regularity of $v$, we have

${\left\|v^{2}\right\|}_{L^{2}([0,T];\,H^{-2\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/2}{\left\|v\right\|}_{L^{\infty}([0,T];\;H^{1-\varepsilon}(\Lambda_{L}))}^{2}\leq
T^{1/2}M^{2}.$

The case $j=3$ follows similarly. ∎

Moreover, this set of initial data has high probability. Here we use the
finite-dimensional invariance to bound the probabilities.

######  Lemma 4.13 (Data has high probability).

Given $k\in\mathbb{N}$, there exists $M_{k}$ such that
$\mu_{L,N}(\mathcal{B}_{M_{k},L,N})\geq 1-2^{-k}$. The value of $M_{k}$ may
depend on $L$ but not $N$.

###### Proof.

We may first use the triangle inequality and union bound to estimate the
probability of the complement (non-existence of local solution) as

$\displaystyle\mathbb{P}\left(\max_{\begin{subarray}{c}j=1,2,3\\\
k=0,\ldots,m\end{subarray}}{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}>M\right)$
$\displaystyle\leq\;$
$\displaystyle\sum_{j=1}^{3}\sum_{k=0}^{m}\mathbb{P}\left({\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}>M\right).$

This in turn is bounded with Markov’s inequality and invariance:

$\sum_{k=0}^{m}\frac{\mathbb{E}\,{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}^{p}}{M^{p}}\lesssim
m\frac{\mathbb{E}\,{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{p}([0,1];\;\mathcal{C}^{-\varepsilon}(\rho))}^{p}}{M^{p}}.$

The expectation is bounded by Lemma 4.4 for any large $p$; this estimate is
uniform in $N$. Now we substitute $m=T/\tau$ and $\tau=C_{L}M^{-C}$ from
Theorem 4.8. By choosing $p$ large enough, our final estimate is

$\mathbb{P}\left(w_{N}\notin\mathcal{D}_{M,N}\right)\lesssim_{L,p}TM^{C-p},$

which vanishes as $M$ is chosen to be large. ∎

###  4.4 Invariance of non-truncated measure

Let us use Lemma 4.13 to rename the sets of initial data defined above:

######  Definition 4.14 (High-probability set of data).

We define the set $\mathcal{D}_{k,L,N}$ to equal $\mathcal{B}_{M_{k},L,N}$
where $M_{k}$ is chosen with Lemma 4.13 such that
$\mu_{L,N}(\mathcal{D}_{k,L,N})\geq 1-2^{-k}$.

We can now take a limit of these sets and get a high-probability set of
initial data without any Fourier truncation. We follow here the argument of
Burq and Tzvetkov [17, Section 6]. We need to drop the regularity a bit in
order to use the compact embedding (Theorem 2.8).

To define the limit measure $\mu_{L}$, we remove the truncation in Definition
4.9. By construction, the measures $\mu_{L,N}$ and $\mu_{L}$ are all
absolutely continuous with respect to the non-truncated (GFF, WN) measure, and
$\mu_{L,N}\to\mu_{L}$ in total variation.

######  Theorem 4.15 (Limiting set of initial data).

We define a subset of ${\mathcal{H}}^{-\varepsilon}(\rho)$ by

$\mathcal{D}_{k,L}\coloneqq\left\\{(u_{0},u_{0}^{\prime})=\lim_{m\to\infty}(u_{0,N_{m}},u_{0,N_{m}}^{\prime})\in\mathcal{D}_{k,L,N_{m}}\text{
for some }N_{m}\to\infty\right\\},$

where the limit is taken in ${\mathcal{H}}^{-2\varepsilon}(\rho)$. Then
$\mu_{L}(\mathcal{D}_{k,L})\geq 1-2^{-k}$. The linear parts of solutions
started from $(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$ satisfy the same
moment bound as in Lemma 4.13.

###### Proof.

It follows from the definition that

$\limsup_{N\to\infty}\mathcal{D}_{k,L,N}\subset\mathcal{D}_{k,L},$

and then Fatou’s lemma implies

$\displaystyle\mu_{L}(\mathcal{D}_{k,L})$
$\displaystyle\geq\mu_{L}\left(\limsup_{N\to\infty}D_{k,L,N}\right)$
$\displaystyle\geq\limsup_{N\to\infty}\mu_{L}\left(D_{k,L,N}\right)$
$\displaystyle=\limsup_{N\to\infty}\mu_{L,N}\left(D_{k,L,N}\right)$
$\displaystyle\geq 1-2^{-k}.$

Here the equality holds by convergence of $\mu_{L,N}\to\mu_{L}$ in total
variation. The bound for moments follows from continuity of
${\mathcal{S}}_{t}$ and ${\mathcal{C}}_{t}$ and the matching bound for
$\mathcal{D}_{k,L,N}$, uniform in $N$. ∎

To show invariance of the limiting measure as $N\to\infty$, we need to
approximate full solutions by Fourier-truncated solutions. The next lemma
gives convergence in a qualitative sense.

######  Lemma 4.16 (Limit solves NLW).

If $u_{m}\in L^{p}([0,T];\;H^{-\varepsilon}(\rho))$ are solutions to the
truncated equation (4.6) with data $(u_{0,N_{m}},u_{0,N_{m}}^{\prime})$ as in
Theorem 4.15, then $u(x,t)\coloneqq\lim_{m\to\infty}u_{m}(x,t)$ solves the
non-truncated equation (4.1) with the limiting initial data
$(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$.

###### Proof.

As the solution operators
$({\mathcal{C}}_{t},{\mathcal{S}}_{t})\colon{\mathcal{H}}^{-\varepsilon}(\Lambda_{L})\to
H^{-\varepsilon}(\Lambda_{L})$ are continuous, the linear part converges:

$w(t)={\mathcal{C}}_{t}u_{0}+{\mathcal{S}}_{t}u_{0}^{\prime}=\lim_{m\to\infty}\left({\mathcal{C}}_{t}u_{0,N_{m}}+{\mathcal{S}}_{t}u_{0,N_{m}}^{\prime}\right).$

Let us then consider the integral part in (4.2). We need to show that

$\lim_{m\to\infty}\int_{0}^{t}{\mathcal{S}}_{t-s}\left(P_{N_{m}}{:\\!P_{N_{m}}u_{m}^{3}\\!:}-{:\\!u^{3}\\!:}\right)\\!(x,s)\,\mathrm{d}s=0.$

We write the inner term as
$P_{N_{m}}({:\\!P_{N_{m}}u_{m}^{3}\\!:}-{:\\!u^{3}\\!:})-(1-P_{N_{m}}){:\\!u^{3}\\!:}$
and note that

$\int^{T}_{0}{\left\|(1-P_{N_{m}}){:\\!u^{3}\\!:}\right\|}_{H^{-2\varepsilon}}\,\mathrm{d}s\leq\int^{T}_{0}2^{-N_{m}\varepsilon}{\left\|P_{>N_{m}}{:\\!u^{3}\\!:}\right\|}_{H^{-\varepsilon}}\,\mathrm{d}s$

goes to $0$ as $N_{m}\to\infty$ by the definition of Besov norm. Since
$P_{N_{m}}$ is bounded (uniformly in $N_{m}$) it is sufficient to show that

${:\\!P_{N_{m}}u_{N_{m}}^{3}\\!:}\to{:\\!u^{3}\\!:}\quad\text{in
}L^{1}([0,T];\;H^{-2\varepsilon}(\Lambda_{L})).$

We write $u_{N_{m}}=w_{N_{m}}+v_{N_{m}}$ where
${\left\|v_{N_{m}}\right\|}_{H^{1-\varepsilon}}\lesssim M^{3}$ by
construction. This means that we can extract a subsequence of $v_{N_{m}}$ with
a limit $v=u-w$.

Assume that ${:\\!P_{N_{m}}w^{j}_{N_{m}}\\!:}\to{:\\!w^{j}\\!:}$ for $j=1,2,3$
in $L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\Lambda_{L}))$. Then
${:\\!(P_{N_{m}}u_{N_{m}})^{3}\\!:}\to{:\\!u^{3}\\!:}$ follows from

${:\\!(P_{N_{m}}u)^{3}\\!:}=\sum_{j=0}^{3}{:\\!(P_{N}w)^{j}\\!:}\,v^{3-j}$

and continuity of the products.

We observe that $(u_{0},u_{0}^{\prime})\to{:\\!w^{j}\\!:}$ is a measurable map
from $B_{p,p}^{-\varepsilon}\times B_{p,p}^{-1-\varepsilon}$ to
$L^{p}([0,T],\mathcal{B}^{-\varepsilon}_{p,p})$. Lusin’s theorem implies for
any $\delta>0$ there exists a set $A_{\delta}$ such that
$\nu_{L}(A_{\delta})<\delta$ and ${:\\!w^{j}\\!:}$ depends continuously on
$(u_{0},u_{0}^{\prime})$. Furthermore
${:\\!(P_{N}w)^{j}\\!:}\to{:\\!w^{j}\\!:}$ almost surely, at least up to
subsequence. By Egorov’s theorem we can find a set $\bar{A}_{\delta}$ such
that ${:\\!(P_{N}w)^{j}\\!:}\to{:\\!w^{j}\\!:}$ uniformly and
$\nu_{L}(\bar{A}_{\delta})<\delta$. Then on the complement of
$\tilde{A}\coloneqq A_{\delta}\cup\bar{A}_{\delta}$ we have that (note that
$w$ depends on the initial data $u_{0}$)

$\displaystyle\lim_{m\to\infty}{\left\|{:\\!(P_{N_{m}}w_{N_{m}})^{j}\\!:}-{:\\!w^{j}\\!:}\right\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$
$\displaystyle\leq\;$
$\displaystyle\lim_{m\to\infty}\sup_{u_{0}\in\bar{A}_{\delta}}{\left\|{:\\!(P_{N_{m}}w)^{j}\\!:}-{:\\!w^{j}\\!:}\right\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$
$\displaystyle\quad+{\|{:\\!w_{N_{m}}^{j}\\!:}-{:\\!w^{j}\\!:}\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$

Now the first term goes to $0$ by uniform convergence and the second by
continuity. This implies that on the complement of $\tilde{A}_{\delta}$ the
equation holds. Thus it holds also on $\cup_{\delta>0}\tilde{A}_{\delta}^{c}$,
which is a set of probability $1$. ∎

To quantify the convergence, we derive pointwise bounds in the bounded set
$\mathcal{D}_{k,L}$. These pointwise results follow from Fourier projections
in Besov spaces, although we need to drop the regularity of our target space
by $\varepsilon$. Again, this change is irrelevant since $\varepsilon$ is
arbitrarily small.

######  Theorem 4.17 (Invariance of finite-volume measure).

We have

$\lim_{N\to\infty}\sup_{A}{\left|\mu_{L}(A)-\mu_{L,N}(A)\right|}=0,$

where the supremum is taken over all measurable subsets of
${\mathcal{H}}^{-2\varepsilon}(\Lambda_{L})$. This implies that
$\mu_{L}(\Phi_{t}A)=\mu_{L}(A)$ for all $t\in{[{0},{T}]}$.

###### Proof.

The first statement extends total variation convergence to the product
measure, which holds since the first marginal converges in total variation. We
may assume $A$ to be a subset of $\mathcal{D}_{k,L}$ for $k$ large, since the
complement of $\mathcal{D}_{k,L}$ has probability $2^{-k}$. By the same
reasoning, we may also intersect with the set of probability $1-2^{-k}$ to be
defined below. Moreover, we can bound the difference
${\left|\mu_{L}(\Phi_{t}A)-\mu_{L}(A)\right|}$ by

${\left|\mu_{L,N}(\Phi_{t}^{N}P_{N}A)-\mu_{L,N}(P_{N}A)\right|}+\sum_{s\in\\{0,t\\}}{\left|\mu_{L}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}^{N}P_{N}A)\right|}.$

Here the first term vanishes by finite-dimensional invariance from Theorem
4.10.

The second term can be split as

${\left|\mu_{L}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}A)\right|}+{\left|\mu_{L,N}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}^{N}P_{N}A)\right|},$

where the first part vanishes uniformly in $A$ due to convergence in total
variation. The latter part is the measure of a symmetric difference, which can
be bounded by the measure of $B(0,R)\times H^{-1-2\varepsilon}(\Lambda_{L})$.
Here the radius of $B(0,R)\subset H^{-2\varepsilon}(\Lambda_{L})$ must satisfy

$R=\sup_{u_{0}\in
A}{\left\|\Phi_{s}u_{0}-\Phi_{s}^{N}P_{N}u_{0}\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}.$

This pointwise bound is crude but suffices since we assume
$(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$. In particular, we do not need to
care about estimating the time derivatives, as the product measure of the
difference is already small by the first factor.

By the uniform bounds on $\mathcal{D}_{k,L}$, we know that the full solution
$u$ and the Fourier-truncated solution $u_{N}$ are well-defined up to time
$T$. We split the difference $(u-u_{N})(t)$ further into three parts in Lemmas
4.18, 4.19, and 4.20 below. Thanks to the uniform moment bounds given in
Theorem 4.15, we get

$\displaystyle R$ $\displaystyle=\sup_{u_{0}\in
A}{\left\|\Phi_{s}u_{0}-\Phi_{s}^{N}P_{N}u_{0}\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim 2^{-cN}(M^{3}+\exp(CM^{3}))+H_{N}\exp(CM^{3}).$

This bound is uniform in $A$ and the constant does not depend on $N$. The
first term vanishes as we take $N\to\infty$. Inside the set of probability
$1-2^{-k}$ from Lemma 4.20 we have

$0\leq H_{N}\leq\kappa_{N}\xrightarrow[N\to\infty]{}0,$

so the second term goes to $0$ as well. This means that the ball vanishes in
the limit, and correspondingly $\mu_{L}(B(0,R)\times
H^{-1-2\varepsilon}(\Lambda_{L}))\to 0$. ∎

The first two estimates are deterministic and use Besov space properties;
however, the low frequencies of nonlinear parts may have more complicated
interactions. The probabilistic terms are due to presence of the GFF. Again,
the moment bounds for GFF allow us to control the nonlinear part $v$ in sets
of arbitrarily high probability.

######  Lemma 4.18 (Fourier approximation, linear part).

The linear parts satisfy

${\left\|w(t)-w_{N}(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\leq
C_{k}2^{-\varepsilon N}$

with probability $1-2^{-k}$.

###### Proof.

Since the operators defining $w$ and $w_{N}$ are Fourier multipliers, the
difference can be written as a projection:

${\left\|w(t)-w_{N}(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}={\left\|P_{>N}w(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}.$

It now follows from the definition of Besov space that

${\left\|P_{>N}w(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\lesssim
2^{-\varepsilon N}{\left\|w(t)\right\|}_{H^{-\varepsilon}(\Lambda_{L})}.$

The bounds in Lemma 4.4 are $L^{p}$ in time and thus inapplicable here, but by
stationarity it is enough to consider $w(0)$. Since the moments of $\phi^{4}$
are bounded by Lemma 3.17, we can choose $C_{k}$ such that
$\mathbb{P}({\left\|w(t)\right\|}_{H^{-\varepsilon}(\Lambda_{L})}>C_{k})<2^{-k}$.
∎

######  Lemma 4.19 (Fourier approximation, high-frequency nonlinearity).

We have for all $t\leq T$ the estimate

${\left\|\int_{0}^{t}P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\,\mathrm{d}s\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\lesssim
2^{-\varepsilon N}M^{3}.$

###### Proof.

By Jensen

${\left\|\int_{0}^{t}P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\,\mathrm{d}s\right\|}_{H^{-2\varepsilon}}^{2}\leq
t\int_{0}^{t}{\left\|P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-2\varepsilon}}^{2}\,\mathrm{d}s.$

Now we can again use the boundedness and pay a little regularity to get

${\left\|P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-2\varepsilon}}^{2}\lesssim
2^{-2\varepsilon
N}{\left\|{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-1-\varepsilon}}^{2}.$

Now the estimate follows from Lemma 4.12. By continuity, the lemma holds also
in the limit. ∎

######  Lemma 4.20 (Fourier approximation, low-frequency nonlinearity).

There exist random variables
$H_{N}\colon{\mathcal{H}}^{-\varepsilon}(\Lambda_{L})\to\mathbb{R}_{+}$ such
that on $\mathcal{D}_{k,L}$ we have

$\left\|\int{\mathcal{S}}_{t-s}\big{[}{:\\!u^{3}\\!:}(s)-P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\big{]}\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}(\Lambda_{L})}\\!\\!\lesssim\exp(CM^{3})(2^{-\varepsilon
N}+H_{N}),$

and for any $k\in\mathbb{N}$ there exists a sequence $\kappa_{N}\rightarrow 0$
such that

$\nu(H_{N}\geq\kappa_{N})\leq 2^{-k}.$

###### Proof.

Let us rewrite the left-hand side as $\|v-v_{N}\|_{H^{1-2\varepsilon}}$, where

$v_{N}=\int_{0}^{t}{\mathcal{S}}_{t-s}P_{N}{:\\!(P_{N}u_{N}(s))^{3}\\!:}\,\mathrm{d}s,\quad\text{and}\quad
v=\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s.$

By Lemma 4.12 we have

$\|v_{N}(t)\|_{H^{1-\varepsilon}}=\left\|\int_{0}^{t}S_{t-s}P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\,\mathrm{d}s\right\|_{H^{1-\varepsilon}}\lesssim
M^{3},$

and similarly for the limit $v$. We can split

$\displaystyle\|v-v_{N}\|_{H^{1-2\varepsilon}}$ $\displaystyle=$
$\displaystyle\left\|\int_{0}^{t}S_{t-s}\big{[}{:\\!u^{3}\\!:}(s)-P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\big{]}\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\|{:\\!u^{3}\\!:}(s)-{:\\!(P_{N}u_{N})^{3}\\!:}(s)\|_{H^{-2\varepsilon}}\,\mathrm{d}s+\left\|\int_{0}^{t}P_{>N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{t}\|{:\\!u^{3}\\!:}(s)-{:\\!(P_{N}u_{N})^{3}\\!:}(s)\|_{H^{-2\varepsilon}}\,\mathrm{d}s+2^{-\varepsilon
N}M^{3},$

where the second term is bounded as in Lemma 4.19. We expand the remaining
integrand to get

${:\\!u^{3}\\!:}-{:\\!(P_{N}u_{N})^{3}\\!:}=\sum_{j=0}^{3}\binom{3}{j}({:\\!w^{j}\\!:}v^{3-j}-{:\\!w^{j}_{N}\\!:}(P_{N}v_{N})^{3-j}),$

and further split it into

$\displaystyle{:\\!w^{j}\\!:}v^{3-j}-{:\\!w^{j}_{N}\\!:}(P_{N}v_{N})^{3-j}$
$\displaystyle=({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(P_{N}v_{N})^{3-j}$
$\displaystyle\quad{}+{:\\!w^{j}\\!:}(v^{3-j}-(P_{N}v_{N})^{3-j}).$

Now the first term is bounded by

$\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(P_{N}v_{N})^{3-j}\|_{H^{-2\varepsilon}}\lesssim\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})\|_{H^{-2\varepsilon}}M^{3},$

and in the nontrivial cases $j=1,2$ we can bound the second term by

$\displaystyle\|{:\\!w^{j}\\!:}(v^{3-j}-(P_{N}v_{N})^{3-j})\|_{H^{-2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\|{:\\!w^{j}\\!:}\|_{\mathcal{C}^{-\varepsilon}}\|v^{3-j}-(P_{N}v_{N})^{3-j}\|_{H^{2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\|{:\\!w^{j}\\!:}\|_{\mathcal{C}^{-\varepsilon}}\|v-P_{N}v_{N}\|_{H^{1-2\varepsilon}}M^{3}.$

By again separating the high-frequency part, the final factor equals

$\|v-P_{N}v_{N}\|_{H^{1-2\varepsilon}}\lesssim\|v-v_{N}\|_{H^{1-2\varepsilon}}+2^{-\varepsilon
N}M^{3}.$

Now setting

$H_{N}\coloneqq\sup_{j\leq
3}\int^{T}_{0}\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(s)\|_{\mathcal{C}^{-\varepsilon}}\,\mathrm{d}s,$

we finally obtain

$\displaystyle\|(v-v_{N})(t)\|_{H^{1-2\varepsilon}}$ $\displaystyle\lesssim\;$
$\displaystyle M^{3}\bigg{[}2^{-\varepsilon
N}+H_{N}+\int_{0}^{t}\sum_{j=1}^{2}\|{:\\!w^{j}\\!:}(s)\|_{\mathcal{C}^{-\varepsilon}}\|(v-v_{N})(s)\|_{H^{1-2\varepsilon}}\,\mathrm{d}s\bigg{]}.$

Then Grönwall’s lemma gives that

$\displaystyle\|(v-v_{N})(t)\|_{H^{1-2\varepsilon}}$ $\displaystyle\lesssim
M^{3}(2^{-\varepsilon
N}+H_{N})\exp\left(\int^{T}_{0}\sum_{j=1}^{2}\|{:\\!w^{j}\\!:}(s)\|_{C^{-\varepsilon}}\mathrm{\,\mathrm{d}s}\right)$
$\displaystyle\lesssim\exp(CM^{3})(2^{-\varepsilon N}+H_{N}).\qed$

##  5 Global invariance of NLW

In any bounded region the behaviour of (NLW) only depends on the light cone,
and we are free to use the periodic solution theory. Within this bounded
region, it is impossible to distinguish between flows of different period
length (as long as the period is sufficiently long). We use this property to
pass the period length to limit.

In Section 5.1 we show that all statements about measurable events can be
reduced back to the periodic case. We then show in Section 5.2 that the
unperiodic flow can be approximated by periodic solutions started from
periodic data. The main results are finally proved in Section 5.3.

Let us first reference some more properties of weak convergence, in addition
to those defined in Section 3.2. We use these implicitly in the following.

######  Lemma 5.1 (Weak limits in product spaces; [7, Theorem 2.8]).

Assume that $\mathcal{X}\times\mathcal{X}^{\prime}$ is separable. Then
$(\mu_{L}\times\mu_{L}^{\prime})$ converges weakly to $\mu\times\mu^{\prime}$
if and only if $(\mu_{L})$ and $(\mu_{L}^{\prime})$ converge weakly to $\mu$
and $\mu^{\prime}$ respectively.

######  Lemma 5.2 (Skorokhod’s theorem; [7, Theorem 6.7]).

Suppose that $(\mu_{L})$ converge weakly to $\mu$ supported on a separable
space. Then there exist a common probability space $\tilde{\mathbb{P}}$ and
random variables $X_{L}$, $X$ such that $\text{Law}(X_{L})=\mu_{L}$,
$\text{Law}(X)=\mu$, and $X_{L}\to X$ almost surely.

###  5.1 Reduction to bounded domain

The Borel $\sigma$-algebra of $\mathbb{R}^{2}$ can be generated by compact
sets, or even just closed balls. We will show below an analogous result for
the Borel $\sigma$-algebra of ${\mathcal{H}}^{-2\varepsilon}(\rho)$: the
$\sigma$-algebra is generated by restrictions of distributions to bounded
domains.

######  Theorem 5.3 ($\sigma$-algebra generated by bounded-domain functions).

Let $s,s^{\prime}\in\mathbb{R}$, and let $\mathcal{A}^{s}$ be the family of
sets that where inclusion only depends on restrictions to compact domains:

$\mathcal{A}^{s}\coloneqq\left\\{A\subset H^{s}(\rho)\colon\exists\text{
compact }D\text{ s.t. }f\in A\Longleftrightarrow f|_{D}\in A\quad\forall f\in
H^{s}(\rho)\right\\}.$

That is, $\mathbf{1}_{A}(f)=g_{A}(f|_{D})$ for some
$g_{A}\colon{\mathcal{H}}^{-2\varepsilon}(D)\to\\{0,1\\}$. Then

  1. 1.

the closed ball $\bar{B}=\bar{B}(f,R)\subset H^{s}(\rho)$ can be constructed
with $\sigma$-closed operations from sets in $\mathcal{A}^{s}$;

  2. 2.

the Borel $\sigma$-algebra of $H^{s}(\rho)$ is a sub-$\sigma$-algebra of
$\sigma(\mathcal{A}^{s})$;

  3. 3.

the Borel $\sigma$-algebra of $H^{s}(\rho)\times H^{s^{\prime}}(\rho)$ is a
sub-$\sigma$-algebra of
$\sigma(\mathcal{A}^{s})\times\sigma(\mathcal{A}^{s^{\prime}})$.

###### Proof.

By the definitions of Borel $\sigma$-algebra and product $\sigma$-algebra, it
is enough to verify the first statement for arbitrary $\bar{B}=\bar{B}(f,R)$
and $s\in\mathbb{R}$. We can write

$\displaystyle\bar{B}$ $\displaystyle=\left\\{g\in
H^{s}(\rho)\colon\int_{\mathbb{R}^{2}}\rho(x)^{2}{\left|(1-\Delta)^{s/2}(f-g)\right|}^{2}(x)\,\mathrm{d}x\leq
R^{2}\right\\}$ $\displaystyle=\left\\{g\in
H^{s}(\rho)\colon\int_{\mathbb{R}^{2}}\rho(x)^{2}{\left|\int_{\mathbb{R}^{2}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right|}^{2}\,\mathrm{d}x\leq
R^{2}\right\\}$ $\displaystyle=\limsup_{N\to\infty}\bigg{\\{}g\in
H^{s}(\rho)\colon\sum_{\ell,m,n=1}^{N}\int_{A_{\ell}}\\!\rho(x)^{2}\left[\int_{A_{m}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right]$
$\displaystyle\hskip
150.00023pt\left[\int_{A_{n}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right]\,\mathrm{d}x\leq
R^{2}\bigg{\\}}.$

Here we denote by $K_{s}$ the convolution kernel of $(1-\Delta)^{s/2}$ and by
$(A_{j})_{j\in\mathbb{N}}$ some partitioning of $\mathbb{R}^{2}$, e.g. by unit
squares. For finite $N$, the set thus depends on $K_{s}$, $f$, and $g$ only
inside the compact set $\cup_{m,n=1}^{N}\overline{(A_{m}+A_{n})}$. Since
$\limsup$ is a $\sigma$-closed operation, this proves that $\bar{B}$ can be
constructed from sets in $\mathcal{A}^{s}$. ∎

We need to combine this result with the definition of weak limit. If we a
priori assume the weak limit to exist, we can show uniqueness by testing
against a much smaller class of test functions. This strategy of adapting the
test functions to the specific model is very common; see the book of Ethier
and Kurtz [21, Section 3.4] for details.

######  Corollary 5.4 (Reduction to bounded domains).

Let $\mathcal{F}$ be the set of bounded Lipschitz functions
$\varphi\colon{\mathcal{H}}^{-2\varepsilon}(\rho)\to\mathbb{R}$ that depend
only on the restriction of argument to some compact domain: for any
$\varphi\in\mathcal{F}$, there exists a compact $D\subset\mathbb{R}^{2}$ such
that $\varphi(f)=\varphi(f|_{D})$ for all
$f\in{\mathcal{H}}^{-2\varepsilon}(\rho)$.

Assume that the sequence $(\mu_{L})$ has a weak limit $\mu^{*}$, and let $\mu$
be a Borel probability measure on ${\mathcal{H}}^{-2\varepsilon}(\rho)$. If

$\lim_{L\to\infty}\int_{\Omega}\varphi(f)\,\mathrm{d}\mu_{L}(f)=\int_{\Omega}\varphi(f)\,\mathrm{d}\mu(f)$

for all $\varphi\in\mathcal{F}$, then the weak limit $\mu^{*}$ equals $\mu$.

###### Proof.

Fix two distinct points $(f,f^{\prime})$ and $(g,g^{\prime})$ in
${\mathcal{H}}^{-2\varepsilon}(\rho)$. By the general theory of distributions,
there exist $\alpha,\beta\in C_{c}^{\infty}(\mathbb{R}^{2})$ such that
${\left\langle\alpha,f-g\right\rangle}\neq 0$ or
${\left\langle\beta,f^{\prime}-g^{\prime}\right\rangle}\neq 0$. Then

$\eta(f,f^{\prime})\coloneqq{\left|{\left\langle\alpha,f\right\rangle}\right|}+{\left|1+{\left\langle\alpha,f\right\rangle}\right|}+{\left|{\left\langle\beta,f^{\prime}\right\rangle}\right|}+{\left|1+{\left\langle\beta,f^{\prime}\right\rangle}\right|}$

depends only on the compact domain
$\operatorname{Supp}\alpha\cup\operatorname{Supp}\beta$, is continuous, takes
only non-negative values, and $h(f,f^{\prime})\neq h(g,g^{\prime})$.
Continuity follows from linearity and Theorem 2.5. Furthermore

$\tilde{\eta}(f,f^{\prime})\coloneqq\frac{\eta(f,f^{\prime})}{1+\eta(f,f^{\prime})}$

is bounded and composed of Lipschitz functions; thus it belongs to
$\mathcal{F}$.

As $(f,f^{\prime})$ and $(g,g^{\prime})$ were arbitrary, we have shown
$\mathcal{F}$ to separate points. Then [21, Theorem 3.4.5] implies that
$\mathcal{F}$ is a separating family, meaning that

$\int_{\Omega}\varphi(f)\,\mathrm{d}\mu^{*}(f)=\int_{\Omega}\varphi(f)\,\mathrm{d}\mu(f)\quad\text{for
all }\varphi\in\mathcal{F}$

implies $\mu^{*}=\mu$. The claim follows from the uniqueness of weak limit. ∎

###  5.2 Invariance of measure

Assuming that the period is large enough, the solution in a bounded domain $D$
is independent of the choice of periodization. However, the initial data
sampled from $\mu_{L}$ still depends on the period length. In this section we
quantify the convergence of solutions.

Let us first construct a probabilistic solution set associated with the region
$D\subset\mathbb{R}^{2}$. This argument is analogous to Theorem 4.13, but with
a twist: by Theorem 4.8 the growth bound in $D$ is independent of the
periodization, but the local solution time $\tau$ is not. However, at discrete
times $\\{k\tau\\}$ we can use the invariance of measure; this property is
qualitative and holds for all period lengths.

######  Theorem 5.5 (Limiting set of data, polynomial weight).

Fix a bounded domain $D\subset\mathbb{R}^{2}$. There exists a set
$\mathcal{E}_{k}\subset{\mathcal{H}}^{-2\varepsilon}(\rho)$ and $M_{k}>0$ such
that $\mu(\mathcal{E}_{k})>1-2^{-k}$ and for almost all
$(u_{0},u_{0}^{\prime})\in\mathcal{E}_{k}$ the flow $\Phi_{t}$ of (NLW) is
bounded by $CTM_{k}^{c}$ in $L^{2}([0,T]\;H^{-2\varepsilon}(D))$.

###### Proof.

Let us consider periodic initial data $(u_{0,L},u^{\prime}_{0,L})$ sampled
from $\mu_{L}$, and define the probabilistic data sets

$\mathcal{E}_{L,M}(D)\coloneqq\left\\{{\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}\leq
M\right\\}.$

We extend this definition as $\mathcal{E}_{\infty,M}$ for non-periodic data
sampled from $\mu$. By Skorokhod’s lemma we may assume that $u_{0,L}\to u_{0}$
almost surely. To estimate the probability of this limit, we first use Fatou’s
lemma:

$\mathbb{P}\left(\mathcal{E}_{\infty,M}\right)=\tilde{\mathbb{P}}\left(\mathcal{E}_{\infty,M}\right)\geq\tilde{\mathbb{P}}\left(\limsup_{L\to\infty}\mathcal{E}_{L,M}\right)\geq\limsup_{L\to\infty}\tilde{\mathbb{P}}\left(\mathcal{E}_{L,M}\right).$

Let us now bound the norm of the flow. We have
$D+B(0,T)\subset\Lambda_{R},\Lambda_{L}$ for $R$ sufficiently large and $L\geq
R$; thus the restriction norm is not able to distinguish between $\Phi_{R,t}$
and $\Phi_{L,t}$. By moving between these two flows, we can extract the
solution time $\tau$ to depend on $R$ (fixed) and not $L$ (divergent).

First, we have the following bound for any $\tau>0$:

$\displaystyle{\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle={\left\|\Phi_{R,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle=\sum_{k=0}^{T/\tau}{\left\|\Phi_{R,t}u_{0,L}\right\|}_{L^{2}({[{k\tau},{(k+1)\tau}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle=\sum_{k=0}^{T/\tau}{\left\|\Phi_{R,t}\Phi_{L,k\tau}u_{0,L}\right\|}_{L^{2}({[{0,\tau}]};\;H^{-2\varepsilon}(D))}^{2}.$

Let us show that the $R$-periodic flow is valid. Assume that

${\left\|{:\\!(\Phi_{\text{lin}}\Phi_{L,k\tau}u_{0,L})^{j}\\!:}\right\|}_{L^{2}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{2}\leq
M$

for $j=1,2,3$ and all $k\leq T/\tau$, where $\tau$ is chosen as per Theorem
4.8 for the domain $\Lambda_{R}$ (fixed) and moment bound $M$. Then the local
growth bound in Lemma 4.12 gives that $\Phi_{R}$ is valid up to time $\tau$
and

${\left\|\Phi_{R,t}\Phi_{L,k\tau}u_{0,L}\right\|}_{L^{2}({[{0,\tau}]};\;H^{-2\varepsilon}(D))}\leq
2M.$

Correspondingly the full sum satisfies

${\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}\lesssim\frac{TM^{2}}{\tau}\lesssim
TM^{2+c}.$

The assumption made above is probabilistic, and as before our estimate for the
sum holds with probability at least

$1-\mathbb{P}\left(\max_{\begin{subarray}{c}j=1,2,3\\\
k=0,\ldots,T/\tau\end{subarray}}{\left\|{:\\!(\Phi_{\text{lin}}\Phi_{L,k\tau}u_{0,L})^{j}\\!:}\right\|}_{L^{2}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{2}>M\right).$

It is here that we use the invariance of $\mu_{L}$ under $\Phi_{L}$. As in
Lemma 4.13, we bound the probability from below by

$1-CTM^{c}\frac{\mathbb{E}\,{\left\|{:\\!(\Phi_{\text{lin}}u_{0,L})^{j}\\!:}\right\|}_{L^{p}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{p}}{M^{p}}.$

The expectation is bounded by Lemma 4.4 uniformly in $L$. Again we choose $p$
large enough to make the power on $M$ negative and then pass to $M$ large
enough. ∎

As we approximate full-space solutions by periodic ones, we also need to
estimate the error made. The following result is analogous to Lemma 4.20. In
contrast to Theorem 4.17, now we also need to estimate the time derivatives.
However, the estimate for time derivatives is bootstrapped from the one for
the $\phi^{4}$ component.

######  Lemma 5.6 (Stability, $\phi^{4}$ component).

Assume that

$\max_{j=1,2,3}{\left\|{:\\!w_{L}^{j}\\!:}\right\|}_{L^{4}({[0,1]};\mathcal{C}^{-2\varepsilon}(\rho))}\leq
M$

holds for all $L$ and in the limit. Fix a bounded domain
$D\subset\mathbb{R}^{2}$, and define the spatial cutoff $\chi_{t}$ as a
mollified indicator of $D$ and $\chi_{t-s}$ as that of $D+B(0,s+1)$. Then the
random solutions satisfy

${\left\|\chi_{t}(u_{L}-u)(t)\right\|}_{H^{-2\varepsilon}(\rho)}\lesssim{\left\|\chi_{0}(u_{0,L},u_{0,L}^{\prime})-\chi_{0}(u_{0},u_{0}^{\prime})\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+\exp(CM^{3})H_{L},$

where $H_{L}$ are random variables satisfying: for any $k\in\mathbb{N}$ there
exist $\kappa_{L}\to 0$ such that $\tilde{\mathbb{P}}(H_{L}\geq\kappa_{L})\leq
2^{-k}$.

###### Proof.

The first term on the right-hand side comes from continuity of the linear
solution operators (Lemma 4.2). For the nonlinear term, our goal is to show

$\chi_{t}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s=\lim_{L\to\infty}\chi_{t}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u_{L}(s)^{3}\\!:}\,\mathrm{d}s.$

We can first apply finite speed of propagation to ${\mathcal{S}}_{t-s}$ to
reduce

$\displaystyle\int_{0}^{t}{\left\|\chi_{t}{\mathcal{S}}_{t-s}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\int_{0}^{t}{\left\|\chi_{t-s}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}.$

We then perform the same manipulations as in Lemma 4.20, only replacing the
Fourier cutoff $N$ by the period length $L$ and keeping track of the weights.
Thanks to the bounded domain, we can always re-introduce $\rho$ such as in

$\displaystyle{\left\|{:\\!w^{2}(s)\\!:}(v-v_{L})(s)\chi_{t-s}\right\|}_{H^{-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|\chi^{\prime}{:\\!w^{2}(s)\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(1)}{\left\|\chi_{t-s}(v-v_{L})(s)\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|\chi^{\prime}\right\|}_{\mathcal{C}^{1}(\rho^{-1})}{\left\|{:\\!w^{2}(s)\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}{\left\|\chi_{t-s}(v-v_{L})(s)\right\|}_{H^{1-2\varepsilon}(\rho)},$

where $\chi^{\prime}$ is a smooth cutoff of some larger domain. We also define
the stochastic term

$H_{L}\coloneqq\sup_{j\leq
3}\int_{0}^{t}{\left\|({:\\!w^{j}\\!:}-{:\\!w_{L}^{j}\\!:})(s)\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}\,\mathrm{d}s$

as a variation of that in Lemma 4.20. Thus we have bounded

$\displaystyle{\left\|\chi_{t}(v-v_{L})(t)\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$ $\displaystyle
M^{3}\left[H_{L}+\int_{0}^{t}\sum_{j=1}^{2}{\left\|{:\\!w^{j}\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}{\left\|\chi_{t-s}(v-v_{L})(t-s)\right\|}_{H^{1-2\varepsilon}(\rho)}\right],$

and again Grönwall gives

${\left\|\chi_{t}(v-v_{L}(t))\right\|}_{H^{1-2\varepsilon}(\rho)}\lesssim\exp(CM^{3})H_{L}.\qed$

######  Lemma 5.7 (Stability, time derivative).

Fix $k,\ell\in\mathbb{N}$, and assume that

$\lim_{L\to\infty}{\left\|\chi(u_{L}-u)(t)\right\|}_{H^{-2\varepsilon}(\rho)}=0,$

where $\chi$ is any cutoff. Then we have

${\left\|\chi\partial_{t}(u_{L}-u)(t)\right\|}_{H^{-1-2\varepsilon}(\rho)}\leq
2^{-\ell}$

in a set of probability at least $1-2^{-k}$.

###### Proof.

Let us first note that by passing to the mild formulation we have

$\lim_{s\to 0}\frac{u(t+s)-u(t)}{s}=\lim_{s\to
0}\left[\frac{1}{s}\int_{t}^{t+s}{\mathcal{S}}_{t-r}{:\\!u(r)^{3}\\!:}\,\mathrm{d}r+\frac{{\mathcal{C}}_{t+s}-{\mathcal{C}}_{t}}{s}u_{0}+\frac{{\mathcal{S}}_{t+s}-{\mathcal{S}}_{t}}{s}u_{0}^{\prime}\right].$

The first term converges almost surely to ${:\\!u(t)^{3}\\!:}$. The limit of
the last two terms is a bounded linear operator $\mathcal{L}$ from
${\mathcal{H}}^{-2\varepsilon}(\rho)$ to $H^{-1-2\varepsilon}(\rho)$, as can
be seen by considering the Fourier multiplier symbols.

We estimate the difference of Wick powers with Lemma 3.11:

${\left\|{:\\!u_{L}(t)^{3}\\!:}-{:\\!u(t)^{3}\\!:}\right\|}_{H^{-2\varepsilon}(\rho)}\leq
J_{L}+{\left\|f^{\delta}(u_{L}(t))-f^{\delta}(u(t))\right\|}_{H^{-2\varepsilon}(\rho)}+J_{\infty},$

where

$J_{L}\coloneqq{\left\|u_{L}(t)^{3}-f^{\delta}(u_{L}(t))\right\|}_{\mathcal{H}^{-2\varepsilon}(\rho)}$

and $J_{\infty}$ is defined analogously. By Lemma 3.11 the expectations of
$J_{L}$ and $J_{\infty}$ vanish as $\delta\to 0$; we get two $2^{-\ell}/3$
terms with probability $1-2^{-k}$ by fixing $\delta$ small enough. We then
bound the middle term by $2^{-\ell}/3$ by continuity of $f^{\delta}$, choosing
$L$ to be large enough. ∎

###  5.3 Invariance proof finished

######  Theorem 5.8 (Global invariance).

We have $\mu\circ\Phi_{t}=\mu$ for all $0\leq t\leq T$.

###### Proof.

We know a priori that the pushforward measure $\mu\circ\Phi_{t}$ exists as
$\Phi_{t}$ is a measurable map. (By Theorem 5.3, we only need to check
restrictions to bounded domains. There $\Phi_{t}$ is almost surely defined as
a composition of small-time periodic flows.)

By the weak limit and finite-volume invariance, we also have that for all
bounded and continuous
$f\colon{\mathcal{H}}^{-2\varepsilon}(\rho)\to\mathbb{R}$,

$\int_{{\mathcal{H}}}\\!f(u_{0},u_{0}^{\prime})\,\mathrm{d}\mu=\lim_{L\to\infty}\int_{{\mathcal{H}}}f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))\,\mathrm{d}\mu_{L}.$

Since the weak limit is unique, we only need to show that

$\lim_{L\to\infty}\int_{{\mathcal{H}}}f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))\,\mathrm{d}\mu_{L}=\int_{{\mathcal{H}}}f(\Phi_{t}(u_{0},u_{0}^{\prime}))\,\mathrm{d}\mu.$

Corollary 5.4 lets us assume that $f$ is Lipschitz and depends on the
restriction of its arguments to some $\Lambda_{R}$. We can further pass to a
common probability space by Lemma 5.2.

Let $\mathcal{F}_{L,M}$ be the intersection of those data sets such that both
the full and $L$-periodic flow have moment bound $M$ in $\Lambda_{R}$ by
Theorem 5.5, that $H_{L}\leq 2^{-\ell}$ in Lemma 5.6, and that the $2^{-\ell}$
bound in Lemma 5.7 holds. This set has probability at most $C2^{-k}$ when $M$
and $L$ are large enough. Let us recall that $M$ only depends on $R$ and not
$L$. We can then estimate

$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))-f(\Phi_{t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))\right|}$
$\displaystyle\leq\;$
$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|\mathbf{1}_{\mathcal{F}_{L,M}}[f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))-f(\Phi_{t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))]\right|}+2^{-k}{\left\|f\right\|}_{\infty}$
$\displaystyle=\;$
$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|\mathbf{1}_{\mathcal{F}_{L,M}}[f(\Phi_{L,t}(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime}))-f(\Phi_{L,t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))]\right|}+2^{-k}{\left\|f\right\|}_{\infty}$
$\displaystyle\leq\;$
$\displaystyle\lim_{L\to\infty}\text{Lip}_{f}\tilde{\mathbb{E}}\,\mathbf{1}_{\mathcal{F}_{L,M}}{\left\|\chi[\Phi_{L,t}(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime})-\Phi_{L,t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime})]\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+2^{-k}{\left\|f\right\|}_{\infty}.$

Here we used respectively boundedness, that $f$ cannot distinguish between
$\Phi_{t}$ and $\Phi_{L,t}$ once $L\geq R+T$, and Lipschitz continuity. The
spatial cutoff $\chi$ depends on $f$ through $R$.

The two components of ${\mathcal{H}}^{-2\varepsilon}(\rho)$ are now estimated
with Lemmas 5.6 and 5.7: The first component is bounded by

$\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left\|\chi[(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime})-(\tilde{u}_{0},\tilde{u}_{0}^{\prime})]\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+\exp(CM^{3})2^{-\ell}.$

The initial data converges almost surely and uniform integrability allows us
to commute the limit and expectation. By increasing $\ell$ and $L$ as
necessary, we can make this term arbitrarily small. This then allows us to
bound the second component. ∎

We can rephrase the invariance as a global existence and uniqueness result as
in the introduction. The same result holds for the second and third powers of
the solution since the corresponding $\phi^{4}$ moments are finite.

###### Proof of Theorem 1.1.

We have now showed that a limit exists in any compact region of
$\mathbb{R}^{2}$, and that the product measure $\mu$ is invariant under the
flow. The latter implies that

$\mathbb{E}\,{\left\|\Phi_{t}u_{0}\right\|}_{L^{2}([0,T];\;H^{-2\varepsilon}(\rho))}^{2}=\int_{0}^{T}\mathbb{E}\,{\left\|u_{0}\right\|}_{H^{-2\varepsilon}(\rho)}^{2}\,\mathrm{d}t\lesssim
T,$

meaning that the norm is almost surely finite. We still need to check that the
limit really solves (NLW).

The convergence of linear parts follows from Lemma 4.5. We only need to show
that

$v(t)=\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s.$

Equivalently, we can show that

$\lim_{L\to\infty}{\left\|v-v_{L}\right\|}_{H^{1-2\varepsilon}(\rho)}\lesssim\lim_{L\to\infty}\int_{0}^{t}{\left\|{\mathcal{S}}_{s-t}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}\,\mathrm{d}s$

vanishes as $L\to\infty$. Since ${\mathcal{S}}_{t-s}$ is bounded by Lemma 4.2
and ${:\\!u^{3}\\!:}$ is assumed to belong to $H^{-2\varepsilon}(\rho)$, we
can use dominated convergence. Then we are left with estimating

$\lim_{L\to\infty}{\left\|\sum_{j=0}^{3}\binom{3}{j}\left[{:\\!w(s)^{j}\\!:}v(s)^{3-j}-{:\\!w_{L}(s)^{j}\\!:}v_{L}(s)^{3-j}\right]\right\|}_{H^{-2\varepsilon}(\rho)}.$

As $w_{L}$ converges in $L^{p}([0,T];H^{-2\varepsilon}(\rho))$ and $v_{L}$
converges in $C([0,T];H^{1-2\varepsilon}(\rho))$, and the product is a
continuous operator, we can pass $w_{L}$ and $v_{L}=u_{L}-w_{L}$ to the limit.
This shows that $v$ satisfies the mild formulation. ∎

##  6 Weak invariance of NLS

The situation is more complicated for the nonlinear Schrödinger equation

$\left\\{\begin{aligned}
i\partial_{t}u_{L}+(m^{2}-\Delta)u_{L}&=-\lambda{:\\!u_{L}^{3}\\!:},\\\
u_{L}(0)&\sim\phi_{2,L}^{4},\end{aligned}\right.$ (6.1)

on $\Lambda_{L}\times\mathbb{R}$. Invariance of the periodic $\phi^{4}_{2}$
measure under this equation was shown already by Bourgain [9] – see also the
more pedagogic review [45] – but our preceding extension argument is broken
for two reasons.

The linear solution operator

${\mathcal{T}}_{t}u\coloneqq\exp(it\Delta)u\coloneqq\mathcal{F}^{-1}\left[\exp(-it|\xi|^{2})\hat{u}(\xi)\right]$
(6.2)

does not increase the regularity of its argument. Therefore the mild solution

$u(t)={\mathcal{T}}_{t}u(0)+\lambda\int_{0}^{t}[{\mathcal{T}}_{t-s}{:\\!u(s)^{3}\\!:}](x)\,\mathrm{d}s$
(6.3)

is not amenable to the fixpoint argument of Section 4. Moreover, NLS does not
possess finite speed of propagation: wave packets propagate at a speed
proportional to their frequency squared. This means that the argument in
Section 5 is not applicable either.

However, if we can accept some loss of regularity, we can still use the
previous tightness argument. That allows us to approximate full-space
solutions by (a subsequence of) periodic solutions. This sense of invariance
was introduced by Albeverio and Cruzeiro [2] in the context of Navier–Stokes
equations.

######  Remark 6.1.

Unlike with the wave equation, here the initial data is complex-valued. Our
construction of the $\phi^{4}$ measure holds also in this setting and all
results from Section 3 carry over.

Compactness is given by a version of the usual embedding theorem for Hölder-
continuous functions:

######  Lemma 6.2 (Arzelà–Ascoli theorem).

The Hölder space $C^{\alpha}([0,T];H^{s}(\rho))$ is defined by the norm

${\left\|f\right\|}_{C^{\alpha}([0,T];\;H^{s}(\rho))}\coloneqq\sup_{0\leq
s\neq t\leq
T}\frac{{\left\|f(t)-f(s)\right\|}_{H^{s}(\rho)}}{{\left|t-s\right|}^{\alpha}}.$

Then the embedding

$C^{2\varepsilon}([0,T];\;H^{s}(\rho))\hookrightarrow
C^{\varepsilon}([0,T];\;H^{s-\varepsilon}(\rho^{1+\varepsilon}))$

is compact.

We first collect a lemma needed for the tightness proof. By giving up some
differentiability, we can get a dispersive estimate that is independent of
time.

######  Lemma 6.3 (Dispersive estimate).

Fix $1\leq p,q\leq\infty$, and let us endow $\mathbb{R}^{d}$ with weight
$\rho(x)=(1+{\left|x\right|}^{2})^{\alpha/2}$, where
${\left|\alpha\right|}>d$. The Schrödinger propagator ${\mathcal{T}}_{t}$ then
satisfies for all $s\in\mathbb{R}$ the estimate

${\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{p,q}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}.$

###### Proof.

Let us first assume $\alpha>d$. We will estimate the $L^{p}$ norm inside

${\left\|{\mathcal{T}}_{t}f\right\|}_{H^{s}_{p,q}(\rho)}={\left\|2^{ks}{\left\|\Delta_{k}{\mathcal{T}}_{t}f\right\|}_{L^{p}(\rho)}\right\|}_{\ell^{q}_{k}}.$

We can write
$\Delta_{k}{\mathcal{T}}_{t}=\Delta_{k}\Delta_{k}^{\prime}{\mathcal{T}}_{t}$
where $\Delta_{k}^{\prime}$ is a smooth indicator of a larger annulus, given
by multiplier symbol $\varphi(2^{-k}\,\cdot\,)$. Let $K_{k}$ be the
convolution kernel of $\Delta_{k}^{\prime}{\mathcal{T}}_{t}$; by weighted
Young’s inequality [24, Lemma A.8] we then have

$\displaystyle{\left\|K_{k}\ast(\Delta_{k}f)\right\|}_{L^{p}(w)}$
$\displaystyle\leq{\left\|K_{k}\right\|}_{L^{1}(w^{-1})}{\left\|\Delta_{k}f\right\|}_{L^{p}(w)}$
$\displaystyle\leq{\left\|1\right\|}_{L^{1}(w^{-1})}{\left\|K_{k}\right\|}_{L^{\infty}}{\left\|\Delta_{k}f\right\|}_{L^{p}(w)}.$

Since $w^{-1}$ is integrable, the first factor is finite. The second factor
satisfies the scaling relation

${\left|\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}\varphi(2^{-k}\xi)e^{-it{\left|\xi\right|}^{2}}\,\mathrm{d}\xi\right|}\leq
2^{kd}\int_{\mathbb{R}^{d}}{\left|\varphi(\eta)\right|}\,\mathrm{d}\eta,$

where $\varphi$ is compactly supported. This yields the required bound. The
case $k=-1$ (ball in Fourier space) also gives a constant factor.

For $\alpha<-d$, we proceed by duality and skew-adjointness of the operator.
By Theorem 2.5 and the above we have

$\displaystyle{\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{p,q}(\rho)}$
$\displaystyle=\sup_{g}{\left\langle{\mathcal{T}}_{t}f,g\right\rangle}$
$\displaystyle\leq\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}{\left\|{\mathcal{T}}_{-t}g\right\|}_{B^{-s-d}_{p^{\prime},q^{\prime}}(\rho^{-1})}$
$\displaystyle\lesssim\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}{\left\|g\right\|}_{B^{-s}_{p^{\prime},q^{\prime}}(\rho^{-1})}$
$\displaystyle=\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)},$

where the supremum is taken over all $g$ with
$B^{-s}_{p^{\prime},q^{\prime}}(\rho^{-1})$ norm equal to $1$. ∎

######  Remark 6.4.

We can also swap $p$ and $1$ in the weighted Young’s inequality to get

${\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{\infty,q}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+d}_{1,q}(\rho)}.$

In the unweighted case this Besov estimate follows from Theorem 2.7 and
conservation of $L^{2}$ norm. These results are reminiscent of the standard
dispersive estimate

${\left\|{\mathcal{T}}_{t}f\right\|}_{L^{\infty}(\mathbb{R}^{d})}\lesssim{\left|t\right|}^{-d/2}{\left\|f\right\|}_{L^{1}(\mathbb{R}^{d})},$

and by interpolation one can find “exchange rates” between spatial regularity
and integrability in time. Interpolation of weighted Besov spaces is explored
in [35]; see especially Theorem 4 there.

######  Theorem 6.5 (Tightness).

The sequence of periodic solutions $u_{L}$ is tight in
$C^{\alpha-\varepsilon}([0,T];\;H^{-4-2\varepsilon}(\rho^{1+\varepsilon}))$
for any $\alpha<1/2$, $\varepsilon>0$, and integrable weight $\rho$.

###### Proof.

We will show that

$\sup_{L}\mathbb{E}\,{\|u_{L}\|}_{C^{\alpha}([0,T];\;H^{-4-\varepsilon}(\rho))}^{2}<\infty.$

This implies tightness in a slightly less regular space by Lemma 6.2.

From the mild formulation of the equation we obtain that

$u_{L}(t)-u_{L}(s)=({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u_{L}(0)+\int^{t}_{s}{\mathcal{T}}_{t-l}{:\\!u_{L}(l)^{3}\\!:}\,\mathrm{d}r,$

so we will need to estimate

${\left\|\int^{t}_{s}{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}+{\left\|({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}.$

For the first term, we can use Cauchy–Schwarz to exchange the integrals:

$\displaystyle{\left\|\int_{s}^{t}{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle\leq\;$
$\displaystyle{\left|t-s\right|}^{1/2}\left[\int_{s}^{t}{\left\|{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\right\|}_{H^{-4-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}$
$\displaystyle\lesssim\;$
$\displaystyle{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|{:\\!u_{L}(r)^{3}\\!:}\right\|}_{H^{-2-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}.$

Here we used the uniform bound from Lemma 6.3. The Wick power is bounded in
expectation by Theorem 3.17, and the bound is uniform in $L$.

For the second term we use the functional derivative

$(e^{-it\Delta}-e^{-is\Delta})f=\int_{s}^{t}(-i\Delta)e^{-ir\Delta}f\,\mathrm{d}r$

and fundamental theorem of calculus to compute

$\displaystyle{\left\|({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle={\left\|\int_{s}^{t}\Delta{\mathcal{T}}_{r}u(0)\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle\leq{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|\Delta{\mathcal{T}}_{r}u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}$
$\displaystyle\leq{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|u(0)\right\|}_{H^{-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}.$

Again the expectation is bounded. All of these estimates are uniform in $L$
and hold for all $t,s\in{[{0},{T}]}$. ∎

By changing the probability space with Skorokhod’s theorem (Lemma 5.2) we can
assume that $u_{L}\to u$ almost surely. We still need to establish that $u$
satisfies the limiting equation.

######  Theorem 6.6 (Limit solves NLS).

There exists a probability space $\tilde{\mathbb{P}}$ and random variable
$\tilde{u}\in
L^{2}(\tilde{\mathbb{P}},C^{\varepsilon}([0,T];H^{-4-\varepsilon}(\rho)))$
such that

$\tilde{u}(t)={\mathcal{T}}_{t}\tilde{u}(0)+\int^{t}_{0}{\mathcal{T}}_{t-s}{:\\!\tilde{u}(s)^{3}\\!:}\,\mathrm{d}s$

and $\operatorname{Law}(\tilde{u}(t))=\mu$ for all $t\in[0,T]$.

###### Proof.

For clarity we omit the tildes in the proof. In addition to almost sure
convergence of $u_{L}$ and ${:\\!u_{L}^{3}\\!:}$, we have by tightness

$u_{L}\to u\quad\text{in}\quad
L^{2}(\tilde{\mathbb{P}};\;C^{\varepsilon}([0,T];\;H^{-4-\varepsilon}(\rho))).$

Hölder continuity implies that we also have convergence of

$u_{L}(t)\to u(t)\quad\text{in}\quad
L^{2}(\tilde{\mathbb{P}};\;H^{-4-\varepsilon}(\rho))$

for all $t\in[0,T]$.

We repeat the approximation argument of Lemma 4.5. Let $f^{\delta}(u)$
approximate ${:\\!u^{3}\\!:}$ as in Lemma 3.11. Then

$\displaystyle\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:})\,\mathrm{d}s$
$\displaystyle=\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-f^{\delta}(u(s)))\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u_{L}(s)^{3}\\!:}-f^{\delta}(u_{L}(s)))\,\mathrm{d}s.$

Here the first and last term vanish in $H^{-4-\varepsilon}(\rho)$ norm as
$\delta\to 0$. This follows directly from the approximation result and is
uniform in $L$.

To bound the second term, we use boundedness of $f^{\delta}$:

$\mathbb{E}\,{\left\|{\mathcal{T}}_{t-s}f^{\delta}(u(s))\right\|}_{H^{-2}(\rho)}\lesssim\mathbb{E}\,{\left\|f^{\delta}(u(s))\right\|}_{L^{2}(\rho)}\lesssim\mathbb{E}\,{\left\|u(s)\right\|}_{H^{-4-\varepsilon}(\rho)},$

and the same for $u_{L}$. This lets us use dominated convergence in

$\lim_{L\to\infty}\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s=0.$

Thus for any fixed $\delta>0$ and $\kappa>0$, we have for all large $L$ that

$\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s<\kappa.$

In summary, we first pass to $\delta$ small and then to $L$ large. This gives
that

$\lim_{L\to\infty}\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:})\,\mathrm{d}s=0.$

By passing to a further subsequence, we then have almost sure convergence of
these nonlinear terms. This finishes the proof. ∎

##  Appendix A Proof of Lemma 3.11

Let us state the lemma in a slightly different way. We remark that this lemma
is not used in the construction of ${:\\!(Z+\phi)^{3}\\!:}$, so the
expressions are valid.

######  Lemma A.1.

Let us fix a Fourier cutoff $\chi\in C^{\infty}_{c}(\mathbb{R}^{2})$ and
define $\chi_{\delta}(x)=\chi(\delta x)$. Let
$Z^{\delta}=\chi_{\delta}({\left\langle{\nabla}\right\rangle})Z$ and
$Z_{L}^{\delta}=\chi_{\delta}({\left\langle{\nabla}\right\rangle})Z_{L}$.
Define

$f^{\delta}(Z^{\delta})\;\coloneqq\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\;\coloneqq\;(Z^{\delta})^{3}-3a_{\delta}Z^{\delta},$

where $a_{\delta}=\mathbb{E}\,[(Z^{\delta})^{2}]$. Then if $\phi\in
L^{p}(\mathbb{P},B_{p,p}^{\varepsilon}(\rho))$, we have

$\displaystyle\lim_{\delta\to
0}\mathbb{E}\,{\left\|({:\\!(Z+\phi)^{3}\\!:}-{:\\!(Z^{\delta}+\phi)^{3}\\!:}_{\delta})\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}=0,\text{
and}$ (A.1) $\displaystyle\lim_{\delta\to
0}\sup_{L}\mathbb{E}\,{\left\|({:\\!(Z_{L}+\phi)^{3}\\!:}-{:\\!(Z_{L}^{\delta}+\phi)^{3}\\!:}_{\delta})\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}=0.$
(A.2)

Since $f^{\delta}$ restricts the Fourier support of its argument to a bounded
interval, it follows that its image is in $L^{2}$ and the map is continuous.

We first establish (A.1) in the case $\phi=0$. In this case it is enough to
prove that

$\sup_{\delta}\mathbb{E}\,{\left\|f^{\delta}(Z)\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}<\infty,$
(A.3)

and that for any fixed $\varepsilon>0$ we have

$\lim_{\delta\to
0}\mathbb{E}\,{\left\|\chi_{\varepsilon}({\left\langle{\nabla}\right\rangle})({:\\!(Z^{\delta})^{3}\\!:}_{\delta}-{:\\!Z^{3}\\!:})\right\|}^{2}_{L^{2}(\rho)}=0,$
(A.4)

which will imply convergence in tempered distributions $\mathcal{S}^{\prime}$.

Let us first verify (A.3). By Besov embedding it is enough to establish

$\sup_{\delta}\mathbb{E}\,{\left\|{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\right\|}^{p}_{B_{p,p}^{-\varepsilon/2}(\rho)}<\infty$

for $p$ large. We use the stationarity of $Z$, integrability of $\rho$, and
hypercontractivity to estimate

$\displaystyle\mathbb{E}\,{\left\|{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\right\|}^{p}_{B_{p,p}^{-\varepsilon/2}(\rho)}$
$\displaystyle=\sum_{k\geq-1}2^{-kp\varepsilon/2}\int_{\mathbb{R}^{2}}\mathbb{E}\,{\left|\Delta_{k}{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(x)\right|}^{p}\rho(x)^{p}\,\mathrm{d}x$
$\displaystyle=\sum_{k\geq-1}2^{-kp\varepsilon/2}\int_{\mathbb{R}^{2}}\mathbb{E}\,{\left|\Delta_{k}{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)\right|}^{p}\rho(x)^{p}\,\mathrm{d}x$
$\displaystyle\lesssim\sum_{k\geq-1}2^{-kp\varepsilon/2}\left(\mathbb{E}\,{\left|\Delta_{k}\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)\right|}^{2}\right)^{p/2}.$

Now we can use Wick’s theorem to get

$\displaystyle\mathbb{E}\,|\Delta_{k}\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)|^{2}$
$\displaystyle=\;$
$\displaystyle\int_{\mathbb{R}^{2+2}}\mathbb{E}\,[K_{k}(x_{1})\,{:\\!(Z^{\delta})^{3}\\!:}(x_{1})\,K_{k}(x_{2})\,{:\\!(Z^{\delta})^{3}\\!:}(x_{2})]\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}$
$\displaystyle=\;$
$\displaystyle\int_{\mathbb{R}^{2+2}}K_{k}(x_{1})K_{k}(x_{2})\,G^{\delta}(x_{1},x_{2})^{3}\,\mathrm{d}x_{1}\,\mathrm{d}x_{2},$

where $K_{k}$ is the convolution kernel of $\Delta_{k}$ and
$G^{\delta}(x_{1},x_{2})\coloneqq\mathbb{E}\,[Z^{\delta}(x_{1})Z^{\delta}(x_{2})]$.
It is known from [23, Chapter 7.2] that

${\left|G^{\delta}(x_{1},x_{2})\right|}\lesssim(C+{\left|\log({\left|x_{1}-x_{2}\right|})\right|})\exp(-m^{2}|x_{1}-x_{2}|).$

This implies that that
$\sup_{\delta>0}{\left\|G^{\delta}(x_{1},x_{2})\right\|}_{L^{p}(\mathrm{d}x_{1}\,\mathrm{d}x_{2})}<\infty$
for any $p<\infty$. In particular

$\displaystyle{\left|\int_{\mathbb{R}^{2+2}}K_{k}(x_{1})K_{k}(x_{2})G^{\delta}(x_{1},x_{2})^{3}\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\right|}$
$\displaystyle\lesssim{\left\|K_{k}(x_{1})K_{k}(x_{2})\right\|}_{L^{q}(\mathrm{d}x_{1}\,\mathrm{d}x_{2})}$
$\displaystyle\lesssim{\left\|K_{k}\right\|}^{2}_{L^{q}}$

for any $q>1$. Since $K_{k}$ has constant $L^{1}$ norm for all $k$ and its
$L^{\infty}$ norm scales as $2^{2k}$, we can take
${\left\|K_{i}(x_{1})\right\|}_{L^{q}}\lesssim 2^{k\varepsilon/3}$ by choosing
$q$ close enough to $1$. Substituting this back to the Besov norm, we get a
convergent geometric sum. This proves (A.3).

Let us then turn to (A.4). Let
$\bar{K}_{\varepsilon}=\mathcal{F}^{-1}\chi^{2}_{\varepsilon}$. We have
$\bar{K}_{\varepsilon}\in L^{\infty}$ since $\chi_{\varepsilon}^{2}$ is a
Schwartz function. Then we compute

$\displaystyle\mathbb{E}\,{\left\|\chi_{\varepsilon}({\left\langle{\nabla}\right\rangle})({:\\!(Z^{\delta})^{3}\\!:}_{\delta}-{:\\!Z^{3}\\!:})\right\|}^{2}_{L^{2}(\rho)}$
$\displaystyle=\;$
$\displaystyle\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!Z^{3}\\!:}(x)\;{:\\!Z^{3}\\!:}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle+\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(x)\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle{}-2\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!Z^{3}\\!:}(x)\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=\;$
$\displaystyle\int\rho(x)^{2}\bar{K}(x-y)G(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y+\int\rho(x)^{2}\bar{K}(x-y)G^{\delta}(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle{}-2\int\rho(x)^{2}\bar{K}(x-y)\tilde{G}^{\delta}(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y,$

where $\tilde{G}^{\delta}(x,y)=\mathbb{E}\,[Z^{\delta}(x)Z(y)]$. We will have
that the right-hand side goes to $0$ as soon as we can show that

$\lim_{\delta\to 0}G^{\delta}(x,y)=\lim_{\delta\to
0}\tilde{G}^{\delta}(x,y)=G(x,y)\quad\text{in }L^{1}(\mathbb{R}^{2}).$

Since

$\tilde{G}^{\delta}(x,y)=\int
K^{\delta}(x-z)\mathbb{E}\,[Z(z)Z(y)]\,\mathrm{d}x=\int
K^{\delta}(x-z)G(z,y)\,\mathrm{d}x,$

and $K^{\delta}$ goes to a Dirac measure, the convergence holds. The argument
for $G^{\delta}(x,y)$ is the same.

We still need to check the $\phi\neq 0$ case. Here

${:\\!(Z^{\delta}+\phi)^{3}\\!:}_{\delta}=\sum_{i=0}^{3}{:\\!(Z^{\delta})^{i}\\!:}_{\delta}\,\phi^{3-i}\longrightarrow\sum_{i=0}^{3}{:\\!(Z)^{i}\\!:}\,\phi^{3-i}={:\\!(Z+\phi)^{3}\\!:}$

where the convergence holds by convergence of
${:\\!(Z^{\delta})^{i}\\!:}_{\delta}$ in $\mathcal{C}^{-\varepsilon}(\rho)$
and continuity of the product.

For the periodic case we again start with $\phi=0$. We may pass from
${:\\!(\cdot)^{3}\\!:}_{\delta}$ to ${:\\!(\cdot)^{3}\\!:}_{\delta,L}$ where
${:\\!(\cdot)\\!:}_{\delta,L}=(\cdot)^{3}-a_{\delta,L}(\cdot)$ and
$a_{\delta,L}=\mathbb{E}Z^{2}_{\delta,L}$, since we can show in the same way
as in Lemma 3.7 that $\sup_{\delta,L}|a_{\delta,L}-a_{\delta}|<\infty$. Then
repeating the computations from the first case we arrive at having to estimate

${\left\|\int
K^{\delta}(\cdot-z)G_{L}(z,y)\,\mathrm{d}x-G_{L}(\cdot,y)\right\|}_{L^{1}}\lesssim\delta{\left\|G_{L}(\cdot,y)\right\|}_{W^{1,1}}$

and the right-hand side is known to be bounded uniformly in $L$ [23, Chapter
7.3]. The $\phi\neq 0$ case follows analogously as above.

##  Appendix B Computations for Section 3.2

In order to prove Theorem 3.19, we needed to bound the absolute value of

$3\int_{\mathbb{R}^{2}}\rho\phi{:\\!Z^{3}\\!:}\,\mathrm{d}x+3\int_{\mathbb{R}^{2}}\rho\phi^{2}{:\\!Z^{2}\\!:}\,\mathrm{d}x+\int_{\mathbb{R}^{2}}\rho\phi^{3}Z\,\mathrm{d}x+\int_{\mathbb{R}^{2}}\phi(\nabla\rho\cdot\nabla\phi)\,\mathrm{d}x$

with a term like

$Q(Z)+\frac{1}{2}\left(m^{2}{\left\|\phi\right\|}_{L^{2}(\rho^{1/2})}^{2}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}+{\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}\right),$

where $Q(Z)$ is bounded in expectation.

We bound each of the integrals in the following lemmas, selecting $Q(Z)$ to
consist of norms of Wick powers of $Z$. The norms have bounded expectation by
Lemma 3.8. In each proof we apply the multiplicative inequality (Theorem 2.4)