# The Probabilistic Scaling Paradigm

Yu Deng1 1 Department of Mathematics, University of Southern California, Los
Angeles, CA 90089, USA<EMAIL_ADDRESS>,  Andrea R. Nahmod2 2 Department of
Mathematics, University of Massachusetts, Amherst MA 01003
<EMAIL_ADDRESS>and  Haitian Yue3 3 Institute of Mathematical Sciences,
ShanghaiTech University, Shanghai, 201210, China
<EMAIL_ADDRESS>Dedicated to Carlos E. Kenig

###### Abstract.

In this note we further discuss the probabilistic scaling introduced by the
authors in [21, 22]. In particular we do a case study comparing the stochastic
heat equation, the nonlinear wave equation and the nonlinear Schrödinger
equation.

Keywords: Random data theory, probabilistic scaling, NLS, NLW, stochastic heat
equation, probabilistic well posedness; Gibbs measure.

Mathematics Subject Classification (2020): 35Q60, 35R60 (15A69 15B52 37E20
60H30).

1Y. D. is funded in part by NSF-DMS-2246908 and a Sloan Fellowship.

2A.N. is funded in part by NSF-DMS-2052740, NSF-DMS-2101381 and the Simons
Foundation Collaboration Grant on Wave Turbulence (Nahmod’s Award ID 651469).

3H.Y. is funded in part by the Shanghai Technology Innovation Action Plan
(No.22JC1402400) and a Chinese overseas high-level young talents program
(2022).

###### Contents

  1. 1 Introduction
  2. 2 The scaling heuristics
    1. 2.1 Choice of noise and initial data
      1. 2.1.1 The parabolic case
      2. 2.1.2 The hyperbolic case
      3. 2.1.3 The dispersive case
    2. 2.2 The criticality threshold
      1. 2.2.1 The parabolic case
      2. 2.2.2 The hyperbolic case
      3. 2.2.3 The dispersive case
  3. 3 Discussions
    1. 3.1 Possible discrepancies
      1. 3.1.1 The role of high-high-to-low interactions
      2. 3.1.2 The role of dispersion relations and counting estimates
    2. 3.2 The comparison with existing results and with Gibbs measures
    3. 3.3 Long time solutions and connection to wave turbulence theory
    4. 3.4 Other geometries

##  1\. Introduction

In recent years we have been interested in the general question of
understanding and describing how randomness affects the behavior of solutions
to PDEs, particularly the study of propagation of randomness. Randomness may
come into the problem in various ways but two common ones are: from the
equation such as in stochastic problems with additive or multiplicative noise;
from random initial data which obeys some canonical law of distribution (e.g.
Gaussian law).

Some111One may also ask about how their distribution evolve over time; at
least how do the ensemble averages evolve. Such questions arise in wave
turbulence; see [11]; [17, 18, 19, 20]; [13, 14] and references therein.
fundamental questions are: what is the optimal regime where the solution
exists and is unique almost surely, at least locally in time; can one describe
the solution in terms of the random structure of the initial data at least for
short times; if there are (formally invariant) Gibbs measures, can we justify
their invariance. In these works [21, 22, 23] and in subsequent joint work
with B. Bringmann [10] we provided some answers to these questions in the
context of nonlinear Schrödinger equations (NLS) and nonlinear wave equations
(NLW).

In this paper we reprise and extend a cornerstone idea that lies behind the
works above. We consider general evolution PDEs of parabolic, hyperbolic and
dispersive type, in the probabilistic setting. We are interested in the local
theory of these equations in different regularity spaces, and in particular
the threshold between probabilistic almost-sure well-posedness and ill-
posedness.

We will discuss the _scaling heuristics_ for each type of equations that
predicts a _critical_ regularity exponent. This is a guiding principle for the
study of probabilistic local well-posedness, and in many (but not all cases
this indeed matches the actual local well-posedness and ill-posedness results.
In fact, in the parabolic and deterministic settings, it is not uncommon that
the scaling does not match optimal local well-posedness
threshold222Heuristically, one can check the following example [46]:
$\dot{x}=\sigma(x)\ \dot{w}$. At least when $w$ is fractional Brownian motion,
there is no canonical stochastic definition of iterated integrals below the
threshold $\gamma=\frac{1}{4}$, where $\gamma$ refers the regularity of $w$.
However, the regime $\gamma>0$ is in principle subcritical in the parabolic
setting., but the scaling criticality still acts as a good guiding principle.

In the _parabolic_ setting, the notion of scaling has been systematically
studied in earlier works [30, 31]. By using the theory of regularity
structures [30], it has been proved that, under suitable additional
restrictions and with suitable renormalization, any nonlinear parabolic SPDE
that is _above_ the scaling threshold must be almost-surely locally well-
posed.

In the _hyperbolic_ and _dispersive_ settings, the situation is much more
complicated. In [21, 22], the authors introduced the notion of _probabilistic
scaling_ for nonlinear Schrödinger equations, and proved the analog result to
[30], namely that any nonlinear Schrödinger equation (of at least cubic
nonlinearity) that is _above_ the scaling threshold must be almost-surely
locally well-posed.

However, unlike parabolic equations, the behavior of dispersive equations,
especially that of _resonances_ , is much richer and contains more
possibilities. For this reason, results similar to [22] have not been obtained
for general dispersion relations.

In this note, we will describe a general framework of the heuristic
(probabilistic) scaling argument that treats the cases of parabolic,
hyperbolic and dispersive equations in a unified manner. In addition, we will
describe the cases in which this scaling heuristics is known or expected to be
precise, as well as possible reasons that may cause it to deviate from the
actual threshold. Finally, we conclude by connecting this scaling heuristics
to recent advancements in wave turbulence theory.

##  2\. The scaling heuristics

In this section we describe the unified scaling argument that covers
parabolic, hyperbolic and dispersive equations. For the purpose of
demonstration, we will fix the domain of our equation to be the $d$
dimensional torus $\mathbb{T}^{d}$, and consider the following three model
equations:

  * •

The stochastic heat equation:

$(\partial_{t}-\Delta)u+\mathcal{N}_{p}(u)=\xi,$ (2.1)

where $\xi$ is some spacetime Gaussian noise on
$\mathbb{R}\times\mathbb{T}^{d}$;

  * •

The semilinear wave equation:

$(\partial_{t}^{2}-\Delta)u+\mathcal{N}_{p}(u)=0,$ (2.2)

with initial data being a Gaussian random field on $\mathbb{T}^{d}$.

  * •

The semilinear Schrödinger equation:

$(i\partial_{t}-\Delta)u+\mathcal{N}_{p}(u)=0,$ (2.3)

with initial data being a Gaussian random field on $\mathbb{T}^{d}$.

Here in (2.1)–(2.3), $\mathcal{N}_{p}(u)$ is a $p$-th order polynomial of $u$
without derivative (the exact form does not matter in local theory), and the
precise form of noise or initial data will be specified in Section 2.1 below.

###  2.1. Choice of noise and initial data

####  2.1.1. The parabolic case

We start with the heat equation (2.1). A canonical choice of the noise would
be a fractional derivative (or antiderivative) of spacetime white noise,
namely

$\xi(t,x)=\sum_{k\in\mathbb{Z}^{d}}\langle k\rangle^{-\alpha}e^{ik\cdot
x}\xi_{k}(t),$ (2.4)

for some $\alpha\in\mathbb{R}$, where $\xi_{k}=\mathrm{d}B_{k}/\mathrm{d}t$
are derivatives of independent one-dimensional Brownian motions. In terms of
regularity, we know that

$\xi\in C_{t}^{0}C_{x}^{\alpha-1-(d/2)-\varepsilon}$

almost surely333Here and also in the following parts of this paper we write
$C^{\gamma}$ for the Besov space $B_{\infty,\infty}^{\gamma}$.. Note that the
linear noise component of the solution $u$ to (2.1) is given by

$u_{\mathrm{lin}}:=\int_{0}^{t}e^{(s-t)\Delta}\eta(s)\,\mathrm{d}s;$ (2.5)

since the parabolic regularity gains two derivatives, it is easy to see that

$u_{\mathrm{lin}}\in C_{t}^{0}C_{x}^{\alpha+1-(d/2)-\varepsilon}$ (2.6)

almost surely, which is the space in which the solution $u$ is constructed
provided the initial data $u(0)$ has matching regularity (see for example
[30]). Ignoring the arbitrary small constant $\varepsilon$, we may call the
value $\alpha+1-(d/2)$ the _regularity level_ of the solution to (2.1) with
noise (2.4).

####  2.1.2. The hyperbolic case

We now turn to the wave equation (2.2), for which the randomness comes from
the initial data. We fix the initial data
$(u_{0},u_{1})=(u(0),\partial_{t}u(0))$ as

$u_{0}=\sum_{k\in\mathbb{Z}^{d}}\langle k\rangle^{-\alpha-1}e^{ik\cdot
x}g_{k},\quad u_{1}=\sum_{k\in\mathbb{Z}^{d}}\langle
k\rangle^{-\alpha}e^{ik\cdot x}g_{k}^{\prime},$ (2.7)

where $g_{k}$ and $g_{k}^{\prime}$ are i.i.d. Gaussians, for some
$\alpha\in\mathbb{R}$. In terms of regularity, we have that

$u_{0}\in C_{x}^{\alpha+1-(d/2)-\varepsilon},\quad u_{1}\in
C_{x}^{\alpha-(d/2)-\varepsilon}$

almost surely, so the linear component of the solution

$u_{\mathrm{lin}}:=\cos(t|\nabla|)u_{0}+\frac{\sin(t|\nabla|)}{|\nabla|}u_{1}$
(2.8)

satisfies that

$u_{\mathrm{lin}}\in C_{x}^{\alpha+1-(d/2)-\varepsilon}$ (2.9)

almost surely. In fact, by the properties of the i.i.d. Gaussians, it is easy
to see that $u_{\mathrm{lin}}$ is also in $H_{x}^{\alpha+1-(d/2)-\varepsilon}$
almost surely. Similar to above, we call the value $\alpha+1-(d/2)$ the
_regularity level_ of the solution to (2.2) with data (2.7).

####  2.1.3. The dispersive case

The case of the Schrödinger equation (2.3) is similar to (2.2), but we only
need to specify the initial data $u(0)$. Suppose

$u(0)=u_{0}=\sum_{k\in\mathbb{Z}^{d}}\langle k\rangle^{-\alpha-1}e^{ik\cdot
x}g_{k},$ (2.10)

where $g_{k}$ are i.i.d. Gaussians, then in the same way as (2.2), we have

$u_{0}\in C_{x}^{\alpha+1-(d/2)-\varepsilon},$

and hence the linear component of the solution

$u_{\mathrm{lin}}:=e^{-it\Delta}u_{0}$ (2.11)

satisfies that

$u_{\mathrm{lin}}\in C_{x}^{\alpha+1-(d/2)-\varepsilon}$ (2.12)

almost surely. Similarly as in the hyperbolic case $u_{\mathrm{lin}}$ is also
in $H_{x}^{\alpha+1-(d/2)-\varepsilon}$ almost surely. Again we call the value
$\alpha+1-(d/2)$ the _regularity level_ of the solution to (2.3) with data
(2.10).

###  2.2. The criticality threshold

Note that the noise (2.4), as well as the initial data (2.7) and (2.10), are
essentially homogeneous when restricted to high frequency/small physical
scales; in other words, rescaling the noise (2.4) or the initial data (2.7) or
(2.10) just corresponds to restricting to frequency $\sim N$ for some $N\gg
1$.

In general, we can view the criticality of a given equation in a given space
$C^{s}$ (or $H^{s}$) in the following aspect: suppose one starts with some
initial data $f$, which is rescaled to frequency $\sim N\gg 1$ and normalized
to have $C^{s}$ norm $\sim 1$, and computes the second order iterations of the
nonlinearity with this $f$. Then, the equation is _subrcritical_ if the result
of iteration carries a _negative_ power of $N$ when measured in $C^{s}$ norm,
is _critical_ if the result of iteration is _independent_ of $N$ when measured
in $C^{s}$ norm, and is _supercritical_ if the result of iteration carries a
_positive_ power of $N$ when measured in $C^{s}$ norm. Due to the above
scaling covariant observation, it is easy to see that the above definition is
equivalent to the usual definition of scaling criticality using rescaling.

The advantage of such definition of criticality is that it is well adapted to
the probabilistic setting, since the functions involved are not rescaling of
Schwartz functions, but of Fourier-randomized Schwartz functions.

####  2.2.1. The parabolic case

Start with (2.1). Since we are interested in local theory, for simplicity we
will periodize the time, so the noise

$\xi\sim\sum_{(k,\ell)\in\mathbb{Z}^{d+1}}\langle
k\rangle^{-\alpha}e^{i(k\cdot x+\ell\cdot t)}g_{k,\ell},$

where $g_{k,\ell}$ are i.i.d. Gaussians, and hence

$u_{\mathrm{lin}}\sim\sum_{(k,\ell)\in\mathbb{Z}^{d+1}}\langle
k\rangle^{-\alpha}(1+|k|^{2}+|\ell|)^{-1}e^{i(k\cdot x+\ell\cdot
t)}g_{k,\ell}.$

Plugging into (2.1) and assuming (say) $\mathcal{N}_{p}(u)=u^{p}$, we obtain
that the second iteration

$u_{\mathrm{iter}}\sim\sum_{(k,\ell)\in\mathbb{Z}^{d+1}}X_{k,\ell}e^{i(k\cdot
x+\ell\cdot t)},$

where the random variables

$X_{k,\ell}:=(1+|k|^{2}+|\ell|)^{-1}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{p}=k\\\
\ell_{1}+\cdots+\ell_{p}=\ell\end{subarray}}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-\alpha}(1+|k_{q}|^{2}+|\ell_{q}|)^{-1}g_{k_{q},\ell_{q}}.$

Here, assuming all the $(k_{q},\ell_{q})$ involved are different, one may
invoke the square root cancellation444This square root cancellation is derived
from a classic large deviation property for the sum of independent Gaussian
variables, as referenced in Lemma 4.4 in [22]. for sums of independent
Gaussians to obtain that with high probability,

$|X_{k,\ell}|\lesssim(1+|k|^{2}+|\ell|)^{-1}\bigg{(}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{p}=k\\\
\ell_{1}+\cdots+\ell_{p}=\ell\end{subarray}}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-2\alpha}(1+|k_{q}|^{2}+|\ell_{q}|)^{-2}\bigg{)}^{1/2}.$ (2.13)

Now, assume we are dealing with _high-high-to-high_ interactions, then each
$|k_{q}|\sim|k|\sim N$, and $|\ell_{q}|\sim|\ell|\sim N^{2}$, hence by the
heuristic dimensional counting argument of
$(k_{q},\ell_{q})\in\mathbb{Z}^{d+1}$ for $q=1,...,p$ in the above sum, the
parabolic scaling being $d+2$ when $(k_{q},\ell_{q})\in\mathbb{Z}^{d+1}$, the
inequality (2.13) implies

$|X_{k,\ell}|\lesssim N^{-p\alpha-2p-2+(d+2)(p-1)/2}.$

From this it is easy to see that

$\|u_{\mathrm{iter}}\|_{C_{t}^{0}C_{x}^{s}}\lesssim
N^{-(p-1)\alpha-2+(d-2)(p-1)/2},$

where $s=\alpha+1-d/2$ is the regularity level of the solution. We thus see
that the equation is

$\left\\{\begin{aligned}
&\mathrm{subcritical},&&\mathrm{if}\quad\alpha>\frac{d}{2}-1-\frac{2}{p-1}\quad\mathrm{or}\quad
s>\frac{-2}{p-1},\\\
&\mathrm{critical},&&\mathrm{if}\quad\alpha=\frac{d}{2}-1-\frac{2}{p-1}\quad\mathrm{or}\quad
s=\frac{-2}{p-1},\\\
&\mathrm{supercritical},&&\mathrm{if}\quad\alpha<\frac{d}{2}-1-\frac{2}{p-1}\quad\mathrm{or}\quad
s<\frac{-2}{p-1}.\end{aligned}\right.$

In this way we get the critical threshold of (2.1) is $C^{s}$ where
$s=-2/(p-1)$, which agrees with the standard parabolic scaling as described in
Hairer’s work [30].

####  2.2.2. The hyperbolic case

In the hyperbolic case (2.2) we argue similarly, but also need to take into
account the time oscillation. For simplicity we shall replace
$u_{1}=\partial_{t}u(0)$ by $0$, which does not affect the scaling argument;
then the linear component

$u_{\mathrm{lin}}=\sum_{k\in\mathbb{Z}^{d}}\cos(t|k|)\langle
k\rangle^{-\alpha-1}e^{ik\cdot x}g_{k}.$

By using $\sin z=(e^{iz}-e^{-iz}/2)$, plugging the linear component into (2.2)
and assuming (say) $\mathcal{N}_{p}(u)=u^{p}$, we obtain that the second
iteration

$u_{\mathrm{iter}}\sim\sum_{k\in\mathbb{Z}^{d}}X_{k}e^{i(k\cdot
x\pm_{0}|k|t)},$

where the random variables

$X_{k}:=\langle
k\rangle^{-1}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{p}=k\end{subarray}}\frac{e^{it\Omega-1}}{\Omega}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-\alpha-1}g_{k_{q}},$

where

$\Omega=\mp_{0}|k|\pm_{1}|k_{1}|+\cdots\pm_{p}|k_{p}|.$

Here, assuming all the $k_{q}$ involved are different, one may invoke the
square root cancellation for sums of independent Gaussians to obtain that with
high probability,

$|X_{k}|\lesssim\langle
k\rangle^{-1}\bigg{(}\sum_{\begin{subarray}{c}k_{1}+\cdots+k_{p}=k\end{subarray}}\langle\Omega\rangle^{-1}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-2\alpha-2}\bigg{)}^{1/2}.$ (2.14)

Now, assuming we are dealing with _high-high-to-high_ interactions, then each
$|k_{q}|\sim|k|\sim N$, hence by by omitting a loss of $N^{\varepsilon}$ and
the lattice counting with fixed $\Omega$ (Lemma 3.1), except for the case when
$d\in\\{1,2\\}$ and $p=2$, we have

$|X_{k}|\lesssim N^{-p(\alpha+1)+d(p-1)/2-3/2}.$

From this it is easy to see that

$\|u_{\mathrm{iter}}\|_{C_{t}^{0}H_{x}^{s}}\lesssim
N^{-(p-1)\alpha+(d-2)(p-1)/2-3/2},$

where $s=\alpha+1-d/2$ is the regularity level of the solution. Also we hold
the same bound in $C_{t}^{0}C_{x}^{s}$ by Khintchine’s inequality (see the
similar estimates in Section 7.2 of [10]). We thus see that the equation is

$\left\\{\begin{aligned}
&\mathrm{subcritical},&&\mathrm{if}\quad\alpha>\frac{d}{2}-1-\frac{3}{2(p-1)}\quad\mathrm{or}\quad
s>\frac{-3}{2(p-1)},\\\
&\mathrm{critical},&&\mathrm{if}\quad\alpha=\frac{d}{2}-1-\frac{3}{2(p-1)}\quad\mathrm{or}\quad
s=\frac{-3}{2(p-1)},\\\
&\mathrm{supercritical},&&\mathrm{if}\quad\alpha<\frac{d}{2}-1-\frac{3}{2(p-1)}\quad\mathrm{or}\quad
s<\frac{-3}{2(p-1)}.\end{aligned}\right.$

In this way we get the critical threshold of (2.2) is $C^{s}$ where
$s=-\frac{3}{2(p-1)}$ except for the case when $d=1,2$ and $p=2$.

In the case $d=1$ and $p=2$, by Lemma 3.1 we have $|X_{k}|\lesssim
N^{-p(\alpha+1)+d(p-1)/2-1}.$ And hence it is easy to see that the critical
threshold is $C^{s}$ where $s=-1$. In the case $d=2$ and $p=2$, by Lemma 3.1
we have $|X_{k}|\lesssim N^{-p(\alpha+1)+d(p-1)/2-1}.$ And hence it is easy to
see that the critical threshold is $C^{s}$ where $s=-\frac{5}{4}$.

####  2.2.3. The dispersive case

In the dispersive case (2.3), then the linear component

$u_{\mathrm{lin}}=\sum_{k\in\mathbb{Z}^{d}}\langle
k\rangle^{-\alpha-1}e^{i(k\cdot x+|k|^{2}t)}g_{k}.$

Plugging into (2.3) and assuming (say) $\mathcal{N}_{p}(u)=u^{\pm_{1}}\cdots
u^{\pm_{p}}$ (denote that $u^{+}:=u$ and $u^{-}:=\overline{u}$), we obtain
that the second iteration

$u_{\mathrm{iter}}\sim\sum_{k\in\mathbb{Z}^{d}}X_{k}e^{i(k\cdot
x\pm_{0}|k|^{2}t)},$

where the random variables

$X_{k}:=\sum_{\begin{subarray}{c}\pm_{1}k_{1}+\cdots\pm_{p}k_{p}=k\end{subarray}}\frac{e^{it\Omega-1}}{\Omega}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-\alpha-1}g_{k_{q}},$

where

$\Omega=\mp_{0}|k|^{2}\pm_{1}|k_{1}|^{2}+\cdots\pm_{p}|k_{p}|^{2}.$

Here, assuming there are no _parings_ 555$k_{i}\neq k_{j}$ if the
corresponding signs $\pm_{i}$ and $\pm_{j}$ are the opposite. in the $p$-th
order power $\mathcal{N}_{p}$, one may invoke the square root cancellation for
sums of independent Gaussians to obtain that with high probability,

$|X_{k}|\lesssim\bigg{(}\sum_{\begin{subarray}{c}\pm_{1}k_{1}+\cdots\pm_{p}k_{p}=k\end{subarray}}\langle\Omega\rangle^{-1}\prod_{q=1}^{p}\langle
k_{q}\rangle^{-2\alpha-2}\bigg{)}^{1/2}.$ (2.15)

Now, assuming we are dealing with _high-high-to-high_ interactions, then each
$|k_{q}|\sim|k|\sim N$. From this it is easy to see that

$\|u_{\mathrm{iter}}\|_{C_{t}^{0}H_{x}^{s}}\lesssim
N^{-(p-1)(\alpha+1)-d/2}\times\\\ \left(\\#\\{k,k_{q}\in\mathbb{Z}^{d}\text{
for all }q:|k|\sim N,|k_{q}|\sim N,\text{ for all
}q,\pm_{1}k_{1}+\cdots\pm_{p}k_{p}=k,\Omega\text{ is
fixed}\\}\right)^{\frac{1}{2}}$ (2.16)

where $s=\alpha+1-d/2$ is the regularity level of the solution. Hence by
omitting a loss of $N^{\varepsilon}$ and the lattice counting with fixed
$\Omega$ (Lemma 3.2), we have

$\|u_{\mathrm{iter}}\|_{C_{t}^{0}H_{x}^{s}}\lesssim
N^{-(p-1)\alpha+(d-2)(p-1)/2-1}.$

Also we hold the same bound in $C_{t}^{0}C_{x}^{s}$ by Khintchine’s
inequality. We thus see that the equation is

$\left\\{\begin{aligned}
&\mathrm{subcritical},&&\mathrm{if}\quad\alpha>\frac{d}{2}-1-\frac{1}{p-1}\quad\mathrm{or}\quad
s>\frac{-1}{p-1},\\\
&\mathrm{critical},&&\mathrm{if}\quad\alpha=\frac{d}{2}-1-\frac{1}{p-1}\quad\mathrm{or}\quad
s=\frac{-1}{p-1},\\\
&\mathrm{supercritical},&&\mathrm{if}\quad\alpha<\frac{d}{2}-1-\frac{1}{p-1}\quad\mathrm{or}\quad
s<\frac{-1}{p-1}.\end{aligned}\right.$

In this way we get the critical threshold of (2.3) is $C^{s}$ where
$s=-\frac{1}{p-1}$.

######  Remark 2.1.

The estimate in (2.16) heavily relies on the square root cancellation for sums
of independent Gaussians. If we replace the Gaussians in the data (2.10)
simply by $1$ and restrict the data to frequency $\sim N$, then the data would
be a smooth data whose Fourier modes are supported around $N$. With such
smooth deterministic data, we lose the square root gain in (2.16), and by the
similar calculations as above we get another critical threshold
$s=\frac{d}{2}-\frac{2}{p-1}$ of (2.2) in $H^{s}$, which coincides the usual
deterministic scaling for (2.3). Similarly in the hyperbolic case with such
smooth deterministic data, we can also obtain the usual deterministic scaling
critical $s$ for (2.2).

##  3\. Discussions

###  3.1. Possible discrepancies

We should point out that the scaling heuristics in Section 2 provide only a
_guiding principle_ to the probabilistic well-posedness problem of the
corresponding dynamics and _should not be understood_ that the actual
threshold between a.s. local well-posedness and ill-posedness is always given
by scaling criticality. In most cases this is indeed true, including for
example general NLS equations of cubic or higher nonlinearity [22], but in
some cases, especially concerning low dimensions and/or low degree
nonlinearity, discrepancies may occur between the scaling prediction and well-
posedness result666For example [41, 36] focus on such discrepancies..

Such discrepency is not uncommon in other settings involving the notion of
scaling, in fact it is well-known that:

  * •

In the deterministic setting, the local well-posedness threshold for NLW
equations generally does not equal that predicted by scaling criticality due
to the Lorentzian symmetry; similar phenomena happen for NLS equations
involving negative regularity data.

  * •

In the case of stochastic heat equations, the local well-posedness threshold
again deviates from the parabolic scaling prediction, if the noise involved is
rougher than the spacetime white noise (for a more comprehensive description
see [37] and references therein).

In our case, the discrepancy is mainly caused by two mechanisms: (1) the high-
high-to-low interaction, which is the same reason for the discrepancy in the
stochastic heat equations, and (2) the anomalies occurring in various counting
estimates depending on the specific dispersion relation, which is related to
the discrepancy in deterministic problems for NLW. We now describe these
mechanisms in more detail.

####  3.1.1. The role of high-high-to-low interactions

As indicated in Section 2.2, the probabilistic scaling criticality is
determined through heuristic calculations of the _high-high-to-high_
interactions, which are also taken into account when obtaining the critical
exponent for the usual deterministic scaling. As indicated in Section 3.1, the
_high-high-to-low_ interactions may also play a role in the local theory of
the random data problem for the dispersive and wave equations in some special
cases which we will discuss below.

Following similar calculations as in Section 2.2 (see (2.13) for the heat
equation, (2.14) for the wave equation and (2.15) for the Schrödinger
equation) and assuming we are dealing with _high-high-to-low_ interactions
(each $|k_{q}|\sim N$ and $|k|\sim 1$) then up to an $N^{\varepsilon}$ loss
and using the lattice counting lemmas in Section 3.1.2, we can obtain that

$\left\\{\begin{aligned}
&\text{heat:}&&\|u_{\mathrm{iter}}\|_{C_{t}^{0}C_{x}^{s}}\lesssim
N^{-sp-(d+2)/2},\\\
&\text{wave:}&&\|u_{\mathrm{iter}}\|_{C_{t}^{0}C_{x}^{s}}\lesssim\begin{cases}N^{-sp-d/2}\quad&\text{when
}\mathcal{N}_{p}(u)=u^{2},\\\
N^{-sp-(d+1)/2}\quad&\text{otherwise,}\end{cases}\\\
&\text{Schr\"{o}dinger:}&&\|u_{\mathrm{iter}}\|_{C_{t}^{0}C_{x}^{s}}\lesssim\begin{cases}N^{-sp-(d+1)/2}\quad&\text{when
}\mathcal{N}_{p}(u)=|u|^{2},\\\
N^{-sp-(d+2)/2}\quad&\text{otherwise}.\end{cases},\end{aligned}\right.$

where $s=\alpha+1-d/2$. As a result, we can see the critical threshold arising
from the high-high-to-low interactions is $C^{s}$ where,

$\left\\{\begin{aligned} &\text{heat:}&&s=-\frac{d+2}{2p},\\\
&\text{wave:}&&s=\begin{cases}-\frac{d}{2p}\quad&\text{when
}\mathcal{N}_{p}(u)=u^{2},\\\
-\frac{d+1}{2p}\quad&\text{otherwise,}\end{cases}\\\
&\text{Schr\"{o}dinger:}&&s=\begin{cases}-\frac{d+1}{2p}\quad&\text{when
}\mathcal{N}_{p}(u)=|u|^{2},\\\
-\frac{d+2}{2p}\quad&\text{otherwise}.\end{cases}\end{aligned}\right.$

Let us compare the _probabilistic scaling_ critical threshold $s_{pr}$ and the
critical threshold $s$ above which we denote now by $s_{hhl}$ for the _high-
high-to-low interactions_ in the context of the stochastic heat equation
(2.1), the semilinear wave equation (2.2) and the semilinear Schrödinger
equation (2.3). As shown in Table 1, for (2.1) we obtain $s_{pr}\geq s_{hhl}$
when $d>2$ and $p\geq\frac{d+2}{d-2}$; for (2.2) we obtain $s_{pr}\geq
s_{hhl}$ when $d>2$, $p\geq\max(\frac{d+1}{d-2},3)$ and when $d\geq 5$, $p=2$;
for (2.3) we obtain $s_{pr}\geq s_{hhl}$ when $d\geq 3$,
$\mathcal{N}_{p}(u)=|u|^{2}$ and when $p\geq\frac{d+2}{d}$,
$\mathcal{N}_{p}(u)\neq|u|^{2}$.

|  $\mathcal{N}_{p}(u)$ |  $s_{pr}$ |  $s_{hhl}$
---|---|---|---
Heat |  all $p$ |  $-\frac{2}{p-1}$ |  $-\frac{d+2}{2p}$
Wave |  $p=2$ |  $-\frac{3}{2}$111Note that, for $d\in\\{1,2\\}$, the actual threshold is higher than this value $-3/2$, which has to do with specific counting estimates for wave equations; see Lemma 3.1. This same discrepancy between scaling critical and well-posedness result also happens for the deterministic wave equation in 2D. |  $-\frac{d}{4}$
|  $p\geq 3$ |  $-\frac{3}{2(p-1)}$ |  $-\frac{d+1}{2p}$
Schrödinger |  $p=2$222Here $p=2$ only means the case $\mathcal{N}_{p}(u)=|u|^{2}$. When $\mathcal{N}_{p}(u)=u^{2}$ or $\overline{u}^{2}$, $s_{hhl}$ still follows $-\frac{d+2}{2p}=-\frac{d+2}{4}$ |  $-1$ |  $-\frac{d+1}{4}$
|  $p\geq 3$ |  $-\frac{1}{p-1}$ |  $-\frac{d+2}{2p}$
Table 1. Probabilistic scaling criticality v.s. high-high-to-low criticality

####  3.1.2. The role of dispersion relations and counting estimates

The scaling estimates in Sections 2.2 and 3.1.1 crucially rely on the
following counting estimates which are intimately related to the dispersion at
hand. Although in Lemma 3.1 and Lemma 3.2 we only state upper bounds, in most
cases these are optimal. For the sake of brevity, we do not repeat the proofs
in this note, but the reader may find these in [10, 16, 21, 22]. Note in
particular the less favorable counting estimate in (3.1) when $A\ll N$, which
corresponds to high-high-to-low interactions discussed above, and the
different counting estimates (3.2) and (3.4), which has to do with the
anomalous scaling for wave equation in 2D.

######  Lemma 3.1 (The counting lemma for wave equations).

Given dyadic numbers $A,N$ and $a\in\mathbb{Z}^{d}$ satisfying $|a|\sim A$, we
have the following lattice point counting bounds

  1. (1)

when $d\geq 3$

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}^{d}:|n|\sim N,|\langle
a+n\rangle-\langle n\rangle-m|\leq 1\\}\lesssim\min(A,N)^{-1}N^{d}.$ (3.1)

  2. (2)

when $d\geq 3$

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}^{d}:|n|\sim N,|\langle
a+n\rangle+\langle n\rangle-m|\leq 1\\}\lesssim N^{d-1}.$ (3.2)

  3. (3)

when $d=2$

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}^{2}:|n|\sim N,|\langle
a+n\rangle-\langle n\rangle-m|\leq 1\\}\lesssim N^{\frac{3}{2}}.$ (3.3)

  4. (4)

when $d=1$

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}:|n|\sim N,|\langle
a+n\rangle-\langle n\rangle-m|\leq 1\\}\lesssim N.$ (3.4)

######  Lemma 3.2 (The counting lemma for Schrödinger equations).

Given dyadic numbers $N$ and $a\in\mathbb{Z}^{d}$, we have the following
lattice point counting bounds

  1. (1)

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}^{d}:|n|\sim
N,\big{|}|a+n|^{2}-|n|^{2}-m\big{|}\leq 1\\}\lesssim N^{d-1}.$ (3.5)

  2. (2)

$\sup_{m\in\mathbb{Z}}\\#\\{n\in\mathbb{Z}^{d}:|n|\sim
N,\big{|}|a+n|^{2}+|n|^{2}-m\big{|}\leq 1\\}\lesssim N^{d-2+\varepsilon}.$
(3.6)

###  3.2. The comparison with existing results and with Gibbs measures

The Gibbs measure associated to the stochastic heat equation with space-time
white noise (2.1), the nonlinear wave equation (2.2) and the nonlinear
Schrödinger equation (2.3) is known to exist for $d=1,2$ and for all power
nonlinearity $\mathcal{N}_{p}(u)$ with odd $p\geq 3$ (see [7, 40] for
details). When $d=3$, the Gibbs measure is also known to exist but only when
$p=3$; this is so-called $\Phi^{4}_{3}$ measure (see [4, 5, 26] and references
therein for details). However, one has the marginal triviality of the
$\Phi^{4}_{d}$-measure for $d\geq 4$ (see the works of Aizenman [1], Fröhlich
[25], and Aizenman-Duminil-Copin [2]). Loosely speaking, this implies that the
$\Phi^{4}_{d}$-measure in dimension $d\geq 4$ essentially yields a Gaussian
measure, for any renormalization of the potential energy term giving rise to a
well defined measure.

To rigorously justify the invariance of the Gibbs measures under the
corresponding dynamics, we need to first fully understand the a.s. local
wellposedness of (2.1), (2.2) and (2.3) with Gaussian data ($\alpha=0$) (2.4),
(2.7) and (2.10) respectively. The Gibbs measure is supported in $H^{s_{G}-}$
where $s_{G}=1-\frac{d}{2}$.

An interesting question arises in the context of dispersive and wave
equations. Let us focus on the NLS. In 1996, Bourgain proved in a seminal
paper [8] the a.s. global wellposedness and the invariance of the Gibbs
measure under the dynamics of the 2D cubic NLS. But since the measure is known
to exist in 2D for odd $p\geq 5$ and in 3D for $p=3$, it is natural to ask
about the a.s. global wellposedness and the invariance of the Gibbs measure in
these cases. See also [7, 34] for more works about the invariance of Gibbs
measures under nonlinear Schödinger equations. In our paper [21], we indeed
proved these facts for $d=2$ and odd $p\geq 5$. The interesting question is
why is it that the random averaging operator method introduced in [21] does
not suffice when $d=3$ and $p=3$. Note that the deterministic scaling
$s_{cr}=\frac{d}{2}-\frac{2}{p-1}$ when $d=2$ and $p=5$ is
$H^{\frac{1}{2}}(\mathbb{T}^{d})$ same as when $d=3$ and $p=3$. What tells
these two cases apart is precisely the probabilistic scaling $s_{pr}$. When
$d=2$ and $p=5$ the probabilistic scaling $s_{pr}=-\frac{1}{4}<0-=s_{G}-$, so
the the a.s. local wellposedness problem corresponding to the Gibbs measure is
probabilistic subcritical. But when $d=3$ and $p=3$ the probabilistic scaling
$s_{pr}=-\frac{1}{2}=s_{G}$, so the the corresponding a.s. local wellposedness
problem is probabilistic critical. The probabilistic scaling is thus the
correct lens through which we should view the probabilistic wellposedness
problems.

In the case of the cubic nonlinear wave equation in dimension three (the
hyperbolic $\Phi^{4}_{3}$ model), the statistiscal ensemble of its associated
Gibbs measure is probabilistical subcritical. In [10] in joint work with B.
Bringmann we prove the invariance of the Gibbs measure under the dynamics of
the three-dimensional cubic wave equation (see [9, 24, 29, 43, 42, 48] for
more about the hyperbolic $\Phi^{p+1}$ models). This result is the hyperbolic
counterpart to seminal works on the parabolic $\Phi^{4}_{3}$ model by Hairer
[30] and Hairer-Matetski [32]. See [3, 12, 15, 27, 28, 33, 37, 38, 39, 44] and
references therein for more works on the parabolic $\Phi^{p+1}$ models.

To end this section, Table 2 shows the optimal number of derivatives $\beta$
that the initial data can be rougher than the Gaussian data ($\alpha=0$)
corresponding to the Gibbs measure, when analyzing the high-high into high
(probabilistical scaling) interactions. In contrast in Table 3 we lay out the
optimal number of derivatives $\beta$ that arises when analyzing the high-high
into low interactions.

|  2D Quad. |  2D Cubic |  3D Quad. |  3D Cubic |  4D Quad. |  4D Cubic
---|---|---|---|---|---|---
Heat |  $\beta=2$ |  $\beta=1$ |  $\beta=\frac{3}{2}$ |  $\beta=\frac{1}{2}$ |  $\beta=1$ |  $\beta=0$
Wave |  $\beta=\frac{5}{4}$ |  $\beta=\frac{3}{4}$ |  $\beta=1$ |  $\beta=\frac{1}{4}$ |  $\beta=\frac{1}{2}$ |  $\beta=-\frac{1}{4}$
Schrödinger |  $\beta=1$ |  $\beta=\frac{1}{2}$ |  $\beta=\frac{1}{2}$ |  $\beta=0$ |  $\beta=0$ |  $\beta=-\frac{1}{2}$
Table 2. $\beta$ is the number of the derivatives that the initial data can be rougher than the Gaussian data $\alpha=0$ corresponding to the Gibbs measure concerning the high-high into high (probabilistical scaling) interaction. |  2D Quad. |  2D Cubic |  3D Quad. |  3D Cubic |  4D Quad. |  4D Cubic
---|---|---|---|---|---|---
Heat |  $\beta=1$ |  $\beta=\frac{2}{3}$ |  $\beta=\frac{3}{4}$ |  $\beta=\frac{1}{3}$ |  $\beta=\frac{1}{2}$ |  $\beta=0$
Wave |  $\beta=\frac{1}{2}$ |  $\beta=\frac{1}{2}$ |  $\beta=\frac{1}{4}$ |  $\beta=\frac{1}{6}$ |  $\beta=0$ |  $\beta=-\frac{1}{6}$
Schrödinger |  $\beta=\frac{3}{4}$ |  $\beta=\frac{2}{3}$ |  $\beta=\frac{1}{2}$ |  $\beta=\frac{1}{3}$ |  $\beta=\frac{1}{4}$ |  $\beta=0$
Table 3. $\beta$ is the number of the derivatives that the initial data can be
rougher than the Gaussian data ($\alpha=0$) corresponding to the Gibbs measure
concerning the high-high into low interaction.

###  3.3. Long time solutions and connection to wave turbulence theory

The same scaling heuristics in Section 2 can also be applied to more general
context beyond local well-posedness. For example, fix a dyadic scale $N$, and
assume that the initial data is supported at scale $N$ as in Section 2.2.
Suppose instead of local well-posedness (i.e. existence up to time $1$) we are
interested in existence up to time scale $T\sim N^{\beta}$ for some $\beta$,
where $\beta>0$ corresponds to long-time existence result as $N\to\infty$, and
$\beta<0$ corresponds to short-time existence result as $N\to\infty$. Then, we
may consider the following questions:

  * •

Suppose $\beta$ is fixed, what is the optimal decay rate of the initial data,
in terms of the exponent $\alpha$ in Sections 2.2.1–2.2.3, such that solution
to the given equation exists with high probability up to time $T=N^{\beta}$?

  * •

Conversely, given the decay rate $\alpha$, what is the best time $T=N^{\beta}$
up to which a solution exists with high probability?

In fact, the above questions are exactly those encountered in the mathematical
theory of _wave turbulence_. The key prediction of the wave turbulence theory,
starting from random initial data (or stochastically forced) problems, is the
so-called _wave kinetic equation_ , which not only claims the existence of
solutions, but also calculates the effective dynamics of statistical
quantities of the solution. This effective dynamics occurs at a particular
time scale $T_{\mathrm{kin}}$, called the _kinetic_ or _Van Hove_ time scale.

In fact, this notion of $T_{\mathrm{kin}}$ is precisely the time scale
predicted by the heuristic calculations in Section 2.2.1–2.2.3: given the
decay of initial data quantified by the exponent $\alpha$, this
$T_{\mathrm{kin}}=N^{\beta}$ is exactly the value such that the ($C^{s}$ or)
$H^{s}$ norms of $u_{\mathrm{iter}}$ and $u_{\mathrm{lin}}$ are comparable at
time $T$. In particular, the probabilistically critical, subcritical and
supercritical problems in the sense of Section 2, precisely correspond777In
wave turbulence theory it is customary to perform another reduction so the
space scale becomes $N$ and time scale becomes $N^{2}$; in this setting the
probabilistically critical problem would correspond to
$T_{\mathrm{kin}}=N^{2}$. to those problems where $T_{\mathrm{kin}}=1$,
$T_{\mathrm{kin}}\gg 1$ and $T_{\mathrm{kin}}\ll 1$.

In the paper [22], as a consequence of the sharp subcritical a.s. local
wellposedness of (2.3) in $H^{s}(\mathbb{T}^{d})$ with $s>s_{pr}$, we also
obtained the long time existence888Thus, in particular, proving that with high
probability, there is no energy cascade between Fourier modes (i.e.
$|\widehat{u}(t,k)|^{2}\approx|\widehat{u}(0,k)|^{2}$ with negligible error
for large $N$). of solutions for well prepared smooth random data –such as
that arising in the derivation of the wave kinetic equation as we explained
above up to the time $T=N^{(p-1)(s-s_{pr})-}.$ In the case of general odd $p$,
one can show that the kinetic time scale $T_{kin}$ refered above is
$N^{2(p-1)(s-s_{pr})+2}$. After a suitable rescaling, the time $T$ obtained in
[22] reaches $N^{-\varepsilon}T_{kin}$ when $s-s_{pr}=\varepsilon/(p-1)$, for
all $p\geq 3$ odd.

The physical theory of wave turbulence started in the 1920s [45] and had
developed into a substantial field of research with a wide range of scientific
and practical applications; the rigorous mathematical theory started much
later around the 2000s, and has become particularly active in recent years,
see [13, 14, 17, 18] and references therein. It has been a major open problem
in wave turbulence theory to derive the wave kinetic equation up to the time
scale $T_{\mathrm{kin}}$; recently this has been accomplished by the first
author with Z. Hani in the work [18] for the cubic NLS, and the methods used
in [18] are closely related to those employed in [21, 22]. Note that [18]
covers the probabilistically critical case $T_{\mathrm{kin}}=1$, so it in fact
solves a _critical_ problem, which may shed some light on the Gibbs measure
invariance problem described in Section 3.2 above. Though, this latter problem
is still much harder, due to the many potential logarithmic divergences.

###  3.4. Other geometries

In the context of expanding to more general geometries, including
$\mathbb{T}^{d}$, we encounter different scenarios. In the case of compact
manifolds, the canonical randomization would be based on the spectral
expansion of the Laplacian. Here, the probabilistic scaling depends on the
_global geometry_ of the underlying manifold. This is because of the fact that
the randomized data is _not_ localized and has the same amplitude at each
point within the domain. In comparison, the _deterministic scaling_ threshold
remains independent of the geometry, because it corresponds to the data zoomed
out at a point, which is localized. On the other hand, although the gaussian
measure can be defined canonically on manifolds by expanding the
eigenfunctions of the Laplacian and this concept itself does not depend too
much on the geometry, when we treat nonlinear problems, the (probabilistic)
analysis in this case will depend on multilinear integrals involving
eigenfunctions which depend on the underlying geometry.

In the case of non-compact manifolds, for instance $\mathbb{R}^{d}$, the
canonical randomization based on Laplacian eigenfunctions (i.e., $e^{i\xi\cdot
x}$) leads to initial data with infinite $L^{2}$ mass, which is not
feasible999However, it is compatible with the wave equation due to the finite
speed of propagation; specifically, the result of [10] is expected to hold
also for the Gibbs measure on $\mathbb{R}^{3}$. We refrain from discussing
this here, but for more details, refer to [10]. for the Schrödinger equation
(2.3). There is an alternative “Wiener randomization” approach that involves
sectioning the Fourier space into unit boxes and performing randomization
within each box (refer to, for example, [47, 35, 6]). This randomization
generates localized initial data, which is not conserved by the linear flow.
In the $\mathbb{R}^{d}$ scenario, this results in a critical threshold
$s_{p}=-3/(4r)$ that is lower than that of $\mathbb{T}^{d}$.

## References

  * [1] M. Aizenman. Geometric analysis of $\Phi^{4}$ fields and Ising models. Part I and II. _Comm. Math. Phys._ 86 (1982), issue 1, 1–48.
  * [2] M. Aizenman and H. Duminil-Copin. Marginal triviality of the scaling limits of critical 4D Ising and $\Phi_{4}^{4}$ models. _Ann. of Math._ (2) 194 (2021), no. 1, 163–235.
  * [3] S. Albeverio and S. Kusuoka.  The invariant measure and the flow associated to the $\Phi^{4}_{3}$-quantum field model.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 20(4):1359–1427, 2020\.
  * [4] N. Barashkov and M. Gubinelli. A variational method for $\Phi_{3}^{4}$. _Duke Math. J._ 169 (2020), no. 17, 3339–3415.
  * [5] N. Barashkov and M. Gubinelli. The $\Phi_{3}^{4}$ measure via Girsanov’s theorem. _Electron. J. Probab._ 26 (2021), 1–29.
  * [6] Á. Bényi, T. Oh and O. Pocovnicu. On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^{d}$. _Trans. Amer. Math. Soc. Ser. B_ 2 (2015), 1–50.
  * [7] J. Bourgain. Periodic nonlinear Schrödinger equation and invariant measures. _Comm. Math. Phys._ 166 (1994), 1–26.
  * [8] J. Bourgain. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. _Comm. Math. Phys._ 176 (1996), 421–445.
  * [9] J. Bourgain.  Nonlinear Schrödinger equations.  In Hyperbolic equations and frequency interactions (Park City, UT, 1995), volume 5 of IAS/Park City Math. Ser., pages 3–157. Amer. Math. Soc., Providence, RI, 1999.
  * [10] B. Bringmann, Y. Deng, A. Nahmod and H. Yue. Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation. arXiv:2205.03893.
  * [11] T. Buckmaster, P. Germain, Z. Hani and J. Shatah. Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation. _Invent. Math._ , 225 (2021), 787–855.
  * [12] R. Catellier and K. Chouk.  Paracontrolled distributions and the 3-dimensional stochastic quantization equation.  Ann. Probab., 46(5):2621–2679, 2018.
  * [13] C. Collot and P. Germain. On the derivation of the homogeneous kinetic wave equation. arXiv:1912.10368.
  * [14] C. Collot and P. Germain. Derivation of the homogeneous kinetic wave equation: longer time scales. arXiv:2007.03508.
  * [15] G. Da Prato and A. Debussche.  Strong solutions to the stochastic quantization equations.  Ann. Probab., 31(4):1900–1916, 2003.
  * [16] K. Dean. _Private communication_.
  * [17] Y. Deng and Z. Hani. On the derivation of the wave kinetic equation for NLS. _Forum Math. Pi_ , 9 (2021), E6, 1–37.
  * [18] Y. Deng and Z. Hani. Full derivation of the wave kinetic equation. arXiv:2104.11204. To appear in Invent. Math.
  * [19] Y. Deng and Z. Hani. Propagation of chaos and the higher order statistics in the wave kinetic theory. arXiv:2110.04565.
  * [20] Y. Deng and Z. Hani. Derivation of the wave kinetic equation: full range of scaling laws. arXiv:2301.07063.
  * [21] Y. Deng, A. Nahmod and H. Yue. Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two. arXiv:1910.08492.
  * [22] Y. Deng, A. R. Nahmod, and H. Yue.  Random tensors, propagation of randomness, and nonlinear dispersive equations.  Invent. Math., (228):539–686, 2022.
  * [23] Y. Deng, A. R. Nahmod, and H. Yue.  Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three.  J. Math. Phys., 62(3):Paper No. 031514, 39, 2021.
  * [24] L. Friedlander.  An invariant measure for the equation $u_{tt}-u_{xx}+u^{3}=0$.  Comm. Math. Phys., 98(1):1–16, 1985.
  * [25] J. Fröhlich. On the triviality of $\lambda\Phi_{d}^{4}$ theories and the approach to the critical point in $d_{(-)}>4$ dimensions. _Nuclear Physics B_ 200 (1982), issue 2, 281–296.
  * [26] J. Glimm and A. Jaffe. Quantum physics, A functional integral point of view, Second edition, Springer-Verlag, New York, 1987. xxii+535 pp.
  * [27] M. Gubinelli and M. Hofmanová.  A PDE construction of the Euclidean $\phi_{3}^{4}$ quantum field theory.  Comm. Math. Phys., 384(1):1–75, 2021.
  * [28] M. Gubinelli, P. Imkeller, and N. Perkowski.  Paracontrolled distributions and singular PDEs.  Forum Math. Pi, 3:e6, 75, 2015.
  * [29] M. Gubinelli, H. Koch, and T. Oh.  Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity. _J. Eur. Math. Soc._ (2023), 10.4171/JEMS/1294.
  * [30] M. Hairer. A theory of regularity structures. _Invent. Math._ 198 (2014), no. 2, 269–504.
  * [31] M. Hairer. Introduction to regularity structures. _Braz. J. Probab. Stat._ 29 (2015), no. 2, 175–210.
  * [32] M. Hairer and K. Matetski.  Discretisations of rough stochastic PDEs.  Ann. Probab., 46(3):1651–1709, 2018.
  * [33] K. Iwata.  An infinite-dimensional stochastic differential equation with state space $C({\bf R})$.  Probab. Theory Related Fields, 74(1):141–159, 1987.
  * [34] J. Lebowitz, R. Rose and E. Speer. Statistical mechanics of the nonlinear Schrödinger equation. _J. Statist. Phys._ 50 (1988), 657–687.
  * [35] J. Lührmann and D. Mendelson. Random data Cauchy theory for nonlinear wave equations of power type on $\mathbb{R}^{3}$. _Comm. Partial Differential Equations_ 39 (2014), no. 12, 2262–2283.
  * [36] R. Liu. On the probabilistic well-posedness of the two-dimensional periodic nonlinear SchrÃ¶dinger equation with the quadratic nonlinearity $|u|^{2}$. arXiv:2205.07797.
  * [37] A. Moinat and H. Weber. Space-time localisation for the dynamic $\Phi_{3}^{4}$ model. _Comm. Pure Appl. Math._ 73 (2020), no. 12, 2519–2555.
  * [38] J.C. Mourrat and H. Weber. The dynamic $\Phi^{4}_{3}$ model comes down from infinity. _Comm. Math. Phys._ 356 (2017), no. 3, 673–753.
  * [39] J.C. Mourrat and H. Weber. Global well-posedness of the dynamic $\Phi^{4}$ model in the plane. _Ann. Probab._ 45 (2017), no. 4, 2398–2476.
  * [40] E. Nelson.  A quartic interaction in two dimensions.  In Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), pages 69–73. M.I.T. Press, Cambridge, Mass., 1966.
  * [41] T. Oh and M. Okamoto. Comparing the stochastic nonlinear wave and heat equations: a case study. arXiv:1908.03490.
  * [42] T. Oh, M. Okamoto, and L. Tolomeo.  Stochastic quantization of the $\Phi^{3}_{3}$-model.  arXiv:2108.06777, August 2021.
  * [43] T. Oh and L. Thomann.  Invariant Gibbs measures for the $2$-$d$ defocusing nonlinear wave equations.  Ann. Fac. Sci. Toulouse Math. (6), 29(1):1–26, 2020.
  * [44] G. Parisi and Y. S. Wu.  Perturbation theory without gauge fixing.  Sci. Sinica, 24(4):483–496, 1981.
  * [45] Peierls (1929) R. Peierls. Zur kinetischen theorie der wärmeleitung in kristallen. _Annalen der Physik_ 395, (1929), no. 8, 1055–1101.
  * [46] H. Weber. _Private communication_.
  * [47] T. Zhang and D. Fang. Random dada Cauchy theory for the generalized incompressible Navier-Stokes equations. _J. Math. Fluid. Mech._ 14 (2012), no. 2, 311–324.
  * [48] P. E. Zhidkov.  An invariant measure for a nonlinear wave equation.  Nonlinear Anal., 22(3):319–325, 1994.