# Stochastic Burgers Equation via Energy Solutions
from Non-Stationary Particle Systems

Kevin Yang Stanford University

###### Abstract.

The primary objective of the current article is to provide a first true result
in establishing the stochastic Burgers equation as a continuum model for the
dynamics of the density fluctuations for asymmetric particle systems both with
a possibly unbounded total number of particles and also outside stationarity.
More broadly, we provide the first application of the energy solution theory
developed by Goncalves-Jara in [11], for example, to particle systems outside
any stationary measure, and as an application we treat particle systems for
which the more popular approach via microscopic Cole-Hopf transformation fails
quite seriously; this includes a large family of classical zero-range
processes. The approach consists of constructing appropriate "stationary
approximations" to non-stationary particle systems, propagating these
approximations both globally in-time and uniformly in the continuum limit, and
a stochastic analytic argument at the level of continuum SPDEs . As a
consequence, we obtain the "crossover" fluctuations, showing the transition
from Gaussian continuum limits to the stochastic Burgers equation as was done
in [11], but now for non-stationary particle systems.

September 2, 2024

##  1\. Introduction

The purpose of the current article is to extend the universality of the
Kardar-Parisi-Zhang stochastic PDE to a wider class of particle systems
through the following avatar of the KPZ equation, which is often referred to
as the _stochastic Burgers equation_ :

$\displaystyle\partial_{T}\mathscr{Y}_{T}\ =\
\alpha_{1}\partial_{x}^{2}\mathscr{Y}_{T}\ +\
\alpha_{2}\partial_{x}\mathscr{Y}_{T}^{2}\ +\
\alpha_{3}\partial_{x}\dot{\mathscr{W}}_{T};$ (1.1)

the constants $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}_{>0}$ are
structural constants determining the physics of the model, and
$\dot{\mathscr{W}}_{T}$ is a space-time white noise. The solution to the KPZ
equation is provided by taking a primitive
$\mathscr{H}_{T}=\partial_{x}^{-1}\mathscr{Y}_{T}$ of the stochastic Burgers
equation.

To provide historical background, in [2] the authors provided both a solution
theory for the KPZ equation via a Cole-Hopf linearization along with a
particle system known as ASEP from which the KPZ equation could be realized as
a continuum model, thereby justifying their solution theory with a physical
example. We emphasize the nontrivial nature of this result, as the KPZ
equation and stochastic Burgers equation are both singular SPDEs given the
irregularity of the space-time white noise. In the seminal article [13], and
extended in [14], Hairer introduced an _intrinsic_ solution theory for the KPZ
equation which was well-suited to physical approximation through the theory of
regularity structures. However, this pursuit does not currently admit an
extension to lattice models such as interacting particle systems given its
heavy analytic flavor.

The strategy behind establishing the continuum model for the interface
associated to ASEP in [2] was to implement the Cole-Hopf linearization at the
level of the lattice model. As discussed in [4], this admits an interpretation
of a self-duality for ASEP, and the success of this strategy is intimately
connected to the solvability properties of ASEP. We emphasize that such a
strategy has succeeded for a limited number of particle systems outside ASEP,
though outside [6] and [18], these particle systems were perturbations of ASEP
which may also be realized as "solvable" in some sense. In [6], the article
was concerned with a non-simple exclusion process and introduced classical
hydrodynamic analysis as an additional ingredient to establish KPZ continuum
limits; these systems were limited to have maximal jump-length bounded above
by 3, however. In [18], this latter restriction was entirely removed via a
novel dynamical variation and improvement of the classical one-block estimate;
this dynamical one-block strategy also significantly improved another
structural assumption of the model in [6] as well. In particular, the strategy
towards establishing universality of the KPZ equation consisting of
implementing a Cole-Hopf linearization at the level of the lattice model has
been recently revived in a new robust analytic nature rather than algebraic
fashion; these recent advances have indicated significant success for
analyzing lattice models that are both in statistical non-equilibrium and far
from solvable.

Another alternative to establishing continuum models for particles of KPZ-type
is a theory of energy solutions introduced by Goncalves and Jara; see [9],
[10], and [11], for example. This strategy is a nonlinear martingale problem
formulation of the stochastic Burgers equation; the upshot is that the
relevant random space-time fields arising from interacting particle systems
fit nicely into such a framework. Indeed, for lattice models with spin spaces
larger than those of Ising models, the latter of which is the spin space for
the aforementioned exclusion processes, the approach taken via energy
solutions has successfully established the stochastic Burgers equation
continuum limit; see [11] for the example known as the zero-range process.
Even with the aforementioned recent successes of the Cole-Hopf linearization
approach, analyzing lattice models with larger spin spaces such as
$\mathbb{Z}_{\geqslant 0}$ is still a remarkable challenge to successfully
perform via the Cole-Hopf linearization, and the reasons are, at this point,
purely algebraic and without a clear physical meaning.

One extreme disadvantage to the approach via energy solutions is that it
currently works _only_ when the particle system both admits and begins in a
well-understood stationary measure, whereas most interesting physical
information exists far outside such a regime; see [1] and [5], for example.

The physical relevance of this article is to further the universality of the
KPZ equation and/or stochastic Burgers equation, as said in the beginning of
this introduction. The mathematical relevance behind this article is to
provide a first effort towards introducing the theory of energy solutions to
particle systems much outside stationarity, and to study particle systems with
potentially unbounded spins. We do this for a family of zero-range processes
to fix ideas, provided our specific interest that this particle system allows
for a potentially unbounded number of particles at any given site, but the
methods employed are not entirely model-specific and extend to other models
such as the classical ASEP as studied in this context within [2] and
misanthrope processes.

##  2\. Main Results

###  2.1. Continuum Equations

This article begins with some proper treatment of the continuum SPDEs of
primary interest. Before we proceed, we declare that all relevant SPDEs will
be defined and analyzed on the torus $\mathbb{T}^{1}=[0,1)$ realized with
periodic boundary conditions; we comment on this in Remark 2.8.

######  Definition 2.1.

We define the solution to the following additive stochastic heat equation
(ASHE)

$\displaystyle\partial_{T}\mathscr{H}_{T}\ =\
\alpha_{1}\partial_{x}^{2}\mathscr{H}_{T}\ +\
\alpha_{2}\dot{\mathscr{W}}_{T},$ (2.1)

for $\alpha_{1},\alpha_{2}\in\mathbb{R}_{>0}$, via the following mild
formulation:

$\displaystyle\mathscr{H}_{T}\ $ $\displaystyle=\
\mathrm{e}^{\alpha_{1}T\partial_{x}^{2}}\mathscr{H}_{0}\ +\
\int_{0}^{T}\mathrm{e}^{\alpha_{1}(T-s)\partial_{x}^{2}}\dot{\mathscr{W}}_{s}\
\mathrm{d}s,$ (2.2)

where $\mathrm{e}^{\alpha_{1}t\partial_{x}^{2}}$ denotes the semigroup
operator corresponding to the operator $\alpha_{1}\partial_{x}^{2}$, and the
integral against the white noise an Ito stochastic integral.

We further define the solution to the following $\partial$-additive stochastic
heat equation ($\partial$-ASHE)

$\displaystyle\partial_{T}\mathscr{Y}_{T}\ =\
\alpha_{1}\partial_{x}^{2}\mathscr{Y}_{T}\ +\
\alpha_{2}\partial_{x}\dot{\mathscr{W}}_{T}$ (2.3)

via differentiation in the analytically weak-sense
$\mathscr{Y}_{T}=\partial_{x}\mathscr{H}_{T}$.

######  Definition 2.2.

We define the energy solution to the following stochastic Burgers equation
(SBE)

$\displaystyle\partial_{T}\mathscr{Y}_{T}\ =\
\frac{\alpha}{2}\partial_{x}^{2}\mathscr{Y}_{T}\ -\
\frac{\alpha^{\prime}}{2}\partial_{x}\mathscr{Y}_{T}^{2}\ +\
\alpha^{\frac{1}{2}}\dot{\mathscr{W}}_{T}$ (2.4)

to be the random field defined through the following action on
$J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$:

$\displaystyle\langle J,\mathscr{Y}_{T}\rangle\ $
$\displaystyle\overset{\bullet}{=}\
-\langle\partial_{x}J,\log\mathscr{Z}_{T}\rangle,$ (2.5)

where $\mathscr{Z}_{T}$ solves the following multiplicative-noise stochastic
heat equation:

$\displaystyle\partial_{T}\mathscr{Z}_{T,X}\ =\
\frac{\alpha}{2}\partial_{X}^{2}\mathscr{Z}_{T,X}\ +\
\lambda\alpha^{\frac{1}{2}}\mathscr{Z}_{T,X}\dot{\mathscr{W}}_{T,X}.$ (2.6)

We briefly note that $\mathscr{Z}_{T,X}$ is a linear SPDE which admits a
solution via classical Ito calculus; moreover, per the comparison principle of
Mueller in [17], any such solution with positive initial data remains to be
positive with probability 1 under the multiplicative-noise SHE evolution.
Indeed, this is certainly the situation given that the initial data for the
multiplicative-noise SHE by exponentiating the initial condition
$\partial_{x}^{-1}\mathscr{Y}_{T}$.

Further, this is _not_ the definition of energy solutions provided in [11],
for example. However, as direct consequence of the uniqueness theorem of
Gubinelli and Perkowski in [12], the two definitions are equivalent. Moreover,
we will actually not need to work directly with the energy conditions
appearing within [11], so appealing to the preceding definition will be of no
harm in this article; in fact, it will actually serve significantly more
convenient.

We will discuss these SPDEs later in this article when necessary.

###  2.2. The Model

For simplicity, we focus on the zero-range processes; however, the methods
within this article hold for any particle system satisfying the constraints
for the particle systems in [11] and the following additional constraints:

  * •

The invariant measures are product measures indexed by the sites in the
underlying discrete torus $\mathbb{T}_{N}=\\{0,\frac{1}{N},\ldots,1\\}$.
Moreover, each marginal has uniformly bounded $2+\varepsilon$-moment for any
arbitrarily small but universal constant $\varepsilon\in\mathbb{R}_{>0}$.

  * •

The dynamic is _attractive_. Namely, the rate at which a particle jumps is a
non-decreasing function of the number of particles at the site from which it
jumps, and is a non-increasing function of the number of particles at the site
to which it jumps. This includes ASEP, the zero-range process, and the
misanthrope processes as in [7]

In this situation, the results hold for the density fluctuation fields defined
in [11].

The particle system of consideration within this article is defined via the
following data.

  * •

The state space is $\Omega_{\mathbb{T}_{N}}=\mathbb{Z}_{\geqslant
0}^{\mathbb{T}_{N}}$, the set of all particle configurations on the discrete
torus $\mathbb{T}_{N}=\\{0,\frac{1}{N},\ldots,1\\}$ realized with periodic
boundary conditions. Elements of $\Omega_{\mathbb{T}_{N}}$ will be denoted by
$\eta$, so that $\eta_{x}\in\mathbb{Z}_{\geqslant 0}$ for
$x\in\mathbb{T}_{N}$.

We continue the preceding paragraph by further defining
$\Omega_{\Lambda}=\mathbb{Z}_{\geqslant 0}^{\Lambda}$ provided any sub-lattice
$\Lambda\subseteq\mathbb{T}_{N}$. Moreover, for total clarity, a function
$\varphi:\Omega_{\mathbb{T}_{N}}\to\mathbb{R}$ is a _cylinder function_ with
support $\Lambda\subseteq\mathbb{T}_{N}$ if the following are true:

    * –

The sub-lattice $\Lambda\subseteq\mathbb{T}_{N}$ is a finite set.

    * –

The function $\varphi:\Omega_{\mathbb{T}_{N}}\to\mathbb{R}$ induces a
canonical function $\varphi^{\prime}:\Omega_{\Lambda}\to\mathbb{R}$ via the
projection map $\Pi_{\Lambda}:\Omega_{\mathbb{T}_{N}}\to\Omega_{\Lambda}$;
precisely, we stipulate $\varphi^{\prime}\circ\Pi_{\Lambda}=\varphi$ on the
set $\Omega_{\mathbb{T}_{N}}$.

  * •

The dynamics for the particle system is defined through the following action
on cylinder functions $\varphi:\Omega_{\mathbb{T}_{N}}\to\mathbb{R}$; below,
$g:\mathbb{Z}_{\geqslant 0}\to\mathbb{R}$ denotes a generic function
satisfying constraints to be determined:

$\displaystyle\mathrm{L}^{N}\ $ $\displaystyle=\
N^{2}\mathrm{L}_{\mathrm{Sym}}^{N}\ +\ N^{2}\mathrm{L}_{\wedge}^{N},$ (2.7)
$\displaystyle\mathrm{L}_{\mathrm{Sym}}^{N}\varphi(\eta)\ $ $\displaystyle=\
\sum_{x\in\mathbb{T}_{N}}g(\eta_{x})\left(\varphi(\eta^{x,x+1})-\varphi(\eta)\right)\
+\
\sum_{x\in\mathbb{Z}}g(\eta_{x+1})\left(\varphi(\eta^{x+1,x})-\varphi(\eta)\right),$
(2.8) $\displaystyle\mathrm{L}_{\wedge}^{N}\varphi(\eta)\ $ $\displaystyle=\
\gamma
N^{-\beta}\sum_{x\in\mathbb{T}_{N}}g(\eta_{x})\left(\varphi(\eta^{x,x+1})-\varphi(\eta)\right);$
(2.9)

above, $\eta^{x,y}\in\Omega_{\mathbb{T}_{N}}$ is the particle configuration
obtained by moving a particle from $x$ to $y$. Moreover,
$\beta\geqslant\frac{1}{2}$, and the function $g:\mathbb{Z}_{\geqslant
0}\to\mathbb{R}$ is assumed to satisfy the following constraints:

    * –

Assume $g(0)=0$, and that $g(k)\in\mathbb{R}_{>0}$ for all
$k\in\mathbb{Z}_{>0}$.

    * –

Assume the Lipschitz condition $|g(k+1)-g(k)|\lesssim 1$ for a universal
implied constant.

    * –

Assume that $g:\mathbb{Z}_{\geqslant 0}\to\mathbb{R}$ is monotone non-
decreasing – the dynamic is thus attractive.

Notationally, we let $\eta_{T}^{N}$ denote the state of the particle system
after evolving the initial state $\eta_{0}^{N}$ by time
$T\in\mathbb{R}_{\geqslant 0}$ under the above dynamic.

###  2.3. Density Fluctuation Field

The primary space-time random field of interest within the current article is
the following _density fluctuation field_.

######  Definition 2.3.

Provided a smooth test function
$J\in\mathscr{C}^{\infty}_{c}(\mathbb{T}^{1})$, we define the density
fluctuation field of density $\varrho\in\mathbb{R}_{\geqslant 0}$ as the
random space-time field

$\displaystyle\mathscr{Y}_{T,J}^{N}\ $ $\displaystyle=\
N^{-\frac{1}{2}}\sum_{x\in\mathbb{T}_{N}}\left(\eta_{T,x}^{N}-\varrho\right)\cdot
J\left(\frac{x}{N}-c_{N,\gamma,\beta,\varrho}T\right),\quad
T\in\mathbb{R}_{\geqslant 0};$ (2.10)

above, we have defined the constants

$\displaystyle c_{N,\gamma,\beta,\varrho}\ $
$\displaystyle\overset{\bullet}{=}\ \gamma N^{1-\beta}c_{\varrho}^{\prime},$
(2.11) $\displaystyle c_{\varrho}^{\prime}\ $
$\displaystyle\overset{\bullet}{=}\
\frac{\mathrm{d}}{\mathrm{d}\varrho}c_{\varrho},$ (2.12) $\displaystyle
c_{\varrho}\ $ $\displaystyle\overset{\bullet}{=}\
\sum_{k=0}^{\infty}\frac{\alpha_{\varrho}^{k}}{\mathfrak{Z}(\alpha_{\varrho})}\prod_{j=1}^{k-1}\frac{1}{g(j)},$
(2.13)

where $\mathfrak{Z}(\alpha_{\varrho})$ is a normalization constant and
$\alpha_{\varrho}\in\mathbb{R}_{>0}$ is defined, via monotonicity, to be the
unique positive parameter so that

$\displaystyle\sum_{k=0}^{\infty}\frac{\alpha_{\varrho}^{k}}{\mathfrak{Z}(\alpha_{\varrho})}\prod_{j=1}^{k}\frac{1}{g(j)}\cdot
k\ =\ \varrho.$ (2.14)

Provided there exists no such $\alpha_{\varrho}\in\mathbb{R}_{\geqslant 0}$ so
that $\mathfrak{Z}(\alpha_{\varrho})\neq\infty$ for this fixed
$\varrho\in\mathbb{R}_{\geqslant 0}$, we discard this density
$\varrho\in\mathbb{R}_{\geqslant 0}$. This will be the convention through the
entirety of this article.

We now present the associated _height fluctuation profile_ or _height
function_ associated to the particle system.

######  Definition 2.4.

Provided any space-time coordinate $(T,x)\in\mathbb{R}_{\geqslant
0}\times\mathbb{T}_{N}$, we define

$\displaystyle\mathscr{H}_{T,x}^{N}\ $ $\displaystyle\overset{\bullet}{=}\
N^{-\frac{1}{2}}\sum_{y<x}\left(\eta_{T,y}^{N}-\varrho\right)\ +\
N^{-\frac{1}{2}}\mathscr{H}_{T,0}^{N},$ (2.15)

where $\mathscr{H}_{T,0}^{N}$ denotes the net flux of particles across the
periodic-boundary, adopting the convention that particles moving leftwards
$1\rightsquigarrow N$ contributing positive flux. Certainly
$\mathscr{H}_{0,0}^{N}\equiv 0$.

###  2.4. Convergence

We impose _one_ assumption on the initial particle configuration measure
through its associated height function.

######  Definition 2.5.

Provided any $\varrho\in\mathbb{R}_{\geqslant 0}$, we define a sequence of
probability measures $\\{\mu_{0}^{N}\\}_{N=1}^{\infty}$ on
$\\{\Omega_{\mathbb{T}_{N}}\\}_{N=1}^{\infty}$, respectively, to be
$\varrho$-_stable_ if under these probability measures, the corresponding
initial height functions $\\{\mathscr{H}_{0,NX}^{N}\\}_{N=1}^{\infty}$
converges uniformly on $\mathbb{T}^{1}$ to a continuous function, at least
upon piecewise linear interpolation, in probability.

We remark that requiring a sequence of probability measures to be
$\varrho$-stable for some $\varrho\in\mathbb{R}_{\geqslant 0}$ is almost
entirely non-trivial. At the level of the corresponding continuum KPZ
equation, it is equivalent to having continuous initial data; outside the
narrow-wedge initial data, this is a harmless assumption. In this spirit, we
observe that under a sequence of $\varrho$-stable probability measures, the
sequence of density fluctuation fields converges upon testing against any
smooth $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ via the following summation-
by-parts formula:

$\displaystyle\mathscr{Y}_{0,J}^{N}\ $ $\displaystyle=\ -\ \
\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-11.66676pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-9.73344pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-7.4334pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-6.16672pt}}\\!\sum_{x\in\mathbb{T}_{N}}\mathscr{H}_{0,x}^{N}\cdot
N\left[J\left(\frac{x+1}{N}-c_{N,\gamma,\beta,\varrho}T\right)\ -\
J\left(\frac{x}{N}-c_{N,\gamma,\beta,\varrho}T\right)\right];$ (2.16)

the RHS clearly converges to per the definition of $\varrho$-stability
combined with the smoothness of $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$. We
conclude by preemptively emphasizing the importance of the summation-by-parts
identity (2.16) in this article.

We may now state precisely the first primary result of this article.

######  Theorem 2.6.

Suppose $\beta\in(\frac{1}{2},\infty)$, and fix $\varrho\in\mathbb{R}_{>0}$.
Provided any initial particle configuration measures
$\\{\mu_{0}^{N}\\}_{N=1}^{\infty}$ that is $\varrho$-stable, for any smooth
test function $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ the sequence
$\\{\mathscr{Y}_{T,J}^{N}\\}_{N=1}^{\infty}$ is tight in the Skorokhod space
$\mathscr{D}([0,T],\mathbb{R})$, and moreover every limit point is equal to
the solution of the $\partial$-_ASHE_ acting on
$J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$:

$\displaystyle\partial_{T}\mathscr{Y}_{T}\ $ $\displaystyle=\
\frac{1}{2}c_{\varrho^{\prime}}\partial_{x}^{2}\mathscr{Y}_{T}\ +\
\sqrt{\frac{1}{2}c_{\varrho}}\partial_{x}\dot{\mathscr{W}}_{T},$ (2.17)
$\displaystyle\mathscr{Y}_{0,\bullet}^{\infty}\ $ $\displaystyle=\
\lim_{N\to\infty}\mathscr{Y}_{0,\bullet}^{N}.$ (2.18)

Theorem 2.6 is also known as "crossover fluctuations". In particular, at the
critical value $\beta=\frac{1}{2}$, we no longer converge to $\partial$-ASHE
equations in the continuum scale. As indicated in the following result, we
obtain the stochastic Burgers equation.

######  Theorem 2.7.

Suppose $\beta=\frac{1}{2}$, and fix a density $\varrho\in\mathbb{R}_{>0}$.
Provided any initial particle configuration measures
$\\{\mu_{0}^{N}\\}_{N=1}^{\infty}$ that is $\varrho$-stable, for any smooth
test function $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ the sequence
$\\{\mathscr{Y}_{T,J}^{N}\\}_{N=1}^{\infty}$ is tight in the Skorokhod space
$\mathscr{D}([0,T],\mathbb{R})$, and moreover every limit point is equal to
the solution of the _SBE_ acting on
$J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$:

$\displaystyle\partial_{T}\mathscr{Y}_{T}\ $ $\displaystyle=\
\frac{1}{2}c_{\varrho}^{\prime}\partial_{x}^{2}\mathscr{Y}_{T}\ +\
\frac{1}{2}\gamma\frac{\mathrm{d}^{2}}{\mathrm{d}\varrho^{2}}c_{\varrho}\partial_{x}\mathscr{Y}_{T}^{2}\
+\ \sqrt{\frac{1}{2}c_{\varrho}}\partial_{x}\dot{\mathscr{W}}_{T},$ (2.19)
$\displaystyle\mathscr{Y}_{0,\bullet}^{\infty}\ $ $\displaystyle=\
\lim_{N\to\infty}\mathscr{Y}_{0,\bullet}^{N}.$ (2.20)

###  2.5. Strategy

Our proof of Theorem 2.6 and Theorem 2.7 is rather natural, quite simple to
implement, and quite brief. It consists of three ingredients which we outline
as follows.

  * •

Construct approximations of the density fluctuation field at the initial time,
so sampled from $\mu_{0}^{N}$, that are "almost" in statistical equilibrium;
namely, these approximations will be close to $\mu_{0}^{N}$ at the level of
the density fluctuation field, but as probability measures on
$\Omega_{\mathbb{T}_{N}}$ they will be of uniformly stable relative entropy
with respect to statistical equilibrium. Via standard arguments concerning
relative entropy, we may just inherit the results of [11] to obtain Theorem
2.6 and Theorem 2.7 for these approximations – this is the primary utility of
the energy solution theory adopted therein in the context of this article.

  * •

Show that the approximations above that hold at the initial time propagates
globally in time. This takes advantage of the assumed attractiveness of the
dynamics, and is rather standard.

  * •

Identify the relevant SPDE limits for the approximations – for Theorem 2.6
this amounts to standard linear theory of SPDEs, and for Theorem 2.7 it
amounts to a slightly more involved application of linear theory for SPDEs
upon appealing to the multiplicative-noise stochastic heat equation.

###  2.6. Remarks

The current subsection is dedicated towards potential reasonable concerns
about the above results.

######  Remark 2.8.

As promised, we address the role and importance of compactness, namely
analyzing the SPDEs on the torus $\mathbb{T}^{1}$ and the particle system on a
discretization of it. As we will point out, the relevant analysis of the SPDEs
is sufficiently robust to extend to the infinite-volume setting of the real
line $\mathbb{R}$. The primary technical difficulty of extending the analysis
of the particle systems to the infinite-volume lattice
$\mathbb{Z}\subseteq\mathbb{R}$ is roughly summarized as follows:

  * •

We recall that the strategy taken within this article is a transfer of the
problem at statistical non-equilibrium to the problem at statistical
equilibrium. We perform this through finding approximations of statistical
non-equilibrium by particle systems sufficiently close to statistical
equilibrium for which the results of Theorem 2.6 and Theorem 2.7 apply for
straightforward reasons.

The construction of the "almost-equilibrium" approximations is done through
re-interpreting particle configurations as random walks and afterwards
analyzing random walks via classical tools. As is often the case, analysis of
random walks is significantly more robust on compact time-intervals as opposed
to infinite-time random walks; in our re-interpretation, the former
corresponds to a compact geometry and the latter corresponds to an infinite-
volume geometry. We certainly believe that the extension to infinite-volume is
a rather straightforward adaptation of the construction within this article,
however.

######  Remark 2.9.

As alluded to in the introduction for this article, the work herein provides
the first example outside statistical equilibrium and particle systems whose
spin spaces are larger than the Ising spin space; at the level of the particle
system, this feature corresponds to allowing more than one particle at any
particular site as is the case for the zero-range process. We reemphasize that
a microscopic Cole-Hopf approach certainly fails for the zero-range process
for a number of reasons; to name an example, one of the resulting quantities
in the Cole-Hopf transformation at the level of the particle system is the
possibly bounded particle number at a given site with a diverging constant in
the large-$N$ limit – for exclusion processes, this quantity becomes a
deterministic constant.

In view of the above paragraph, however, a natural question to ask is _if_ the
height function of the particle system actually converges to the solution of
the KPZ equation after suitable renormalization for the zero-range process,
for example. This is exceptionally unclear, both in view of the above
paragraph and also provided that convergence at the level of density
fluctuation fields to the stochastic Burgers equation does not necessarily
imply a convergence of the height functions after deterministic
renormalization. However, provided that this question can be answered for
particle systems at stationarity, we emphasize the methods within this article
would immediately answer this question for non-stationary particle systems.

### Acknowledgements

The author would like to thank both Amir Dembo and Li-Cheng Tsai and for
helpful conversations. Lastly, the author would like to thank Alexander Dunlap
and Jimmy He for interesting discussions.

##  3\. Invariant Measures and Bounded Entropy Perturbations

The primary goal of the current section is to prove the following statement.

######  Proposition 3.1.

Consider $\varrho\in\mathbb{R}_{\geqslant 0}$ and a sequence of
$\varrho$-stable initial particle configuration probability measures
$\\{\mu_{0}^{N}\\}_{N=1}^{\infty}$. Provided any
$\varepsilon\in\mathbb{R}_{>0}$, we may construct a family of initial particle
configuration probability measures
$\\{\mu_{\varepsilon}^{N}\\}_{\varepsilon,N}$ satisfying the following
constraints:

  * •

Define $\mathscr{H}_{0,x}^{N}$ to be the height function sampled from
$\mu_{0}^{N}$, and define $\mathscr{H}_{0,x}^{N,\varepsilon}$ to be the height
function sampled from $\mu_{\varepsilon}^{N}$. We have the estimate

$\displaystyle\lim_{\varepsilon\to 0}\
\sup_{N\in\mathbb{Z}_{>0}}\sup_{x\in\mathbb{T}_{N}}\left|\mathscr{H}_{0,x}^{N}-\mathscr{H}_{0,x}^{N,\varepsilon}\right|\
=\ 0$ (3.1)

where the convergence is in probability.

  * •

The results within _Theorem 2.6_ and _Theorem 2.7_ are valid for the sequence
of probability measure $\\{\mu_{\varepsilon}^{N}\\}_{N=1}^{\infty}$ provided
any $\varepsilon\in\mathbb{R}_{>0}$.

Although the content of later sections within this article, we comment briefly
that if one could propagate the estimate on height functions to arbitrary
compact temporal intervals, then from Proposition 3.1 and propagation of such
an estimate one could deduce Theorem 2.6 and Theorem 2.7 upon identifying the
limit of the resulting SPDEs in the limit $\varepsilon\to 0$. This will be the
procedure adopted in this article; we emphasize that the initial data estimate
within Proposition 3.1 is the key ingredient to this strategy.

###  3.1. Product Invariant Measures

A classical exercise provides the following identification of invariant
measures for the zero-range process.

######  Lemma 3.2.

Provided any $\alpha\in\mathbb{R}_{\geqslant 0}$ and function
$g\in\mathbb{Z}_{\geqslant 0}\to\mathbb{R}$ satisfying the constraints of the
generator coefficients, consider the probability measure

$\displaystyle\nu_{\alpha}^{N}\ $ $\displaystyle\overset{\bullet}{=}\
\bigotimes_{x\in\mathbb{T}_{N}}\nu_{\alpha}(x),\quad\nu_{\alpha}(x)\ \sim\
\sum_{k=0}^{\infty}\left[\frac{\alpha^{k}}{\mathfrak{Z}(\alpha)}\prod_{j=1}^{k}\frac{1}{g(j)}\right]\delta_{k};$
(3.2)

above, $\delta_{k}$ denotes a Dirac point mass at $k\in\mathbb{Z}_{\geqslant
0}$ and $\mathfrak{Z}(\alpha)$, if finite, is the normalization constant.

The probability measure $\nu_{\alpha}^{N}$, assuming
$\mathfrak{Z}(\alpha)\neq\infty$, is invariant under the dynamics with
generator $\mathrm{L}^{N}$; moreover, $\mathrm{L}^{N}_{\mathrm{Sym}}$ is the
symmetric component of $\mathrm{L}^{N}$ with respect to $\nu_{\alpha}^{N}$,
and $\mathrm{L}^{N}_{\wedge}$ is the corresponding asymmetric component.

The second result we cite without proof in this discussion is the primary
results in [11].

######  Theorem 3.3 (Theorem 2.2 and Theorem 2.3 in [11]).

The results of _Theorem 2.6_ and _Theorem 2.7_ are valid if
$\mu_{0}^{N}=\nu_{\alpha_{\varrho}}^{N}$ for
$\alpha_{\varrho}\in\mathbb{R}_{\geqslant 0}$ the unique parameter so that

$\displaystyle\sum_{k=0}^{\infty}\frac{\alpha_{\varrho}^{k}}{\mathfrak{Z}(\alpha_{\varrho})}\prod_{j=1}^{k}\frac{1}{g(j)}\cdot
k\ =\ \varrho$ (3.3)

if one exists.

Again, for those $\varrho\in\mathbb{R}_{\geqslant 0}$ for which there exists
no such $\alpha_{\varrho}\in\mathbb{R}_{\geqslant 0}$, we simply discard these
values of $\varrho\in\mathbb{R}_{\geqslant 0}$.

The preceding discussion provides the required preliminary discussion on the
product invariant measures. We will now focus on establishing a dual
interpretation of the invariant measures $\nu_{\alpha}^{N}$ in terms of non-
simple random walks under the diffusive space-time scaling yielding Brownian
motions. The first step is constructing the auxiliary space of random walks in
bijection with the state space $\Omega_{\mathbb{T}_{N}}$.

######  Lemma 3.4.

Consider the following space of $N$-dependent random walks:

$\displaystyle\mathrm{RW}_{N}\ $ $\displaystyle\overset{\bullet}{=}\
\left\\{(S_{n})_{n=0}^{N}:\ S_{0}=0,\ S_{n+1}-S_{n}\in
N^{-\frac{1}{2}}\mathbb{Z}_{\geqslant 0}\right\\}.$ (3.4)

Further, consider the map

$\displaystyle\mathrm{RW}_{N}\ \longrightarrow\
\Omega_{\mathbb{T}_{N}},\quad(S_{n})_{n=0}^{N}\ \mapsto\
\left[N^{-\frac{1}{2}}\eta_{x}=S_{Nx+N}-S_{Nx}\right].$ (3.5)

The above map is a bijection; this allows us to define the pullback of
$\nu_{\alpha}^{N}$ as a probability measure on $\mathrm{RW}_{N}$. In fact,
this pullback is given by a random walk whose increments are independent and
stationary with law given by $\nu_{\alpha}(x)$ scaled by $N^{-\frac{1}{2}}$.

Lastly, provided any density $\varrho\in\mathbb{R}_{\geqslant 0}$, under the
pullback of $\nu_{\alpha_{\varrho}}$, the height function
$\mathscr{H}_{0,X}^{N}$ of density parameter $\varrho\in\mathbb{R}_{\geqslant
0}$ is a martingale indexed by $X\in\mathbb{T}_{N}$.

###### Proof.

The inverse of the proposed map is well-defined due to the constraint
$S_{0}=0$. Concerning the law of the pullback of $\nu_{\alpha}^{N}$, observe
the increments $N^{\frac{1}{2}}S_{n+1}-N^{\frac{1}{2}}S_{n}$ are distributed
via a product measure $\nu_{\alpha}^{N}$, from which the remainder of the
lemma follows immediately. ∎

As an immediate consequence of Lemma 3.4 and standard martingale theory, we
obtain the following functional CLT.

######  Corollary 3.5.

Retain the setting in _Lemma 3.4_, and further assume the following moment
condition holds for some universal constant $\delta\in\mathbb{R}_{>0}$:

$\displaystyle\sum_{k=0}^{\infty}\frac{\alpha_{\varrho}^{k}}{\mathfrak{Z}(\alpha_{\varrho})}\prod_{j=1}^{k}\frac{1}{g(j)}\cdot
k^{2+\delta}\ <\ \infty.$ (3.6)

Provided any $\varrho\in\mathbb{R}_{\geqslant 0}$, with respect to the
pullback of $\nu_{\alpha_{\varrho}}$ the sequence of height functions
$\mathscr{H}_{0,NX}^{N}$ of density $\varrho\in\mathbb{R}_{\geqslant 0}$
converges uniformly in $X\in\mathbb{T}_{N}$ to a Brownian motion of diffusion
coefficient $D(\varrho)\in\mathbb{R}_{>0}$ in probability.

We now explain the upshot of Lemma 3.4 and Corollary 3.5. The strategy adopted
within this article towards proving both Theorem 2.6 and Theorem 2.7 amounts
to finding perturbations of $\varrho$-stable probability measures that admit
relative entropies with respect to $\nu_{\alpha_{\varrho}}^{N}$ that are
stable in the large-$N$ limit. Via the straightforward arguments within [11]
to address particle configurations with stable relative entropy, the results
of Theorem 2.6 and Theorem 2.7 therefore hold with respect to initial particle
configuration measures with stable relative entropy in the large-$N$ limit. A
suitable approximation of the density fluctuation field for generic
$\varrho$-stable initial particle configuration measures by these
perturbations of stable relative entropy in the latter sections in this
article complete the proof of both Theorem 2.6 and Theorem 2.7.

To find such perturbations of stable relative entropy, however, we appeal to
Lemma 3.4 and Corollary 3.5. In particular, the former of these statements
transforms the problem of constructing probability measures on the particle
configuration space $\Omega_{\mathbb{T}_{N}}$ into a problem of constructing
probability measures on the space of random walks; this latter problem admits
a _dynamical_ approach, especially with the additional ingredient provided by
Corollary 3.5 as our perturbations will be on the scale at which the
diffusive-Brownian scaling in Corollary 3.5 is applied. We comment further in
detail when necessary.

###  3.2. Brownian Envelopes

Because the aforementioned perturbations of stable relative entropy will be
constructed using random-walk-dynamical methods, we call these perturbations
"Brownian envelopes".

The first ingredient we require for the construction of these perturbations is
a continuum result, namely at the level of the limiting Brownian motions in
Corollary 3.5.

######  Lemma 3.6.

Consider any compact and non-degenerate interval
$\mathscr{I}\subseteq\mathbb{R}$ and a deterministic continuous function
$f:\mathscr{I}\to\mathbb{R}$ satisfying the constraint $f(\inf\mathscr{I})=0$.
Furthermore, denote by $B_{\bullet}$ a Brownian motion on $\mathscr{I}$ with
diffusion coefficient $D\in\mathbb{R}_{>0}$ satisfying
$B_{\inf\mathscr{I}}=0$. Then for any $\varepsilon\in\mathbb{R}_{>0}$, we have

$\displaystyle\mathbf{P}\left[\left\|f-B_{\bullet}\right\|_{\mathrm{L}^{\infty}(\mathscr{I})}\
<\ \varepsilon\right]\ \gtrsim_{\varepsilon}\ 1;$ (3.7)

in particular, the implied constant depends only on
$\varepsilon\in\mathbb{R}_{>0}$.

###### Proof.

The proof is provided through the following steps.

  * •

Because $f\in\mathscr{C}(\mathscr{I})$ is continuous, there exists a Lipschitz
function $\psi_{\varepsilon}\in\mathscr{C}^{0,1}(\mathscr{I})$ satisfying both
$\psi_{\varepsilon}(\inf\mathscr{I})=0$ and
$\|f-\psi_{\varepsilon}\|_{\mathrm{L}^{\infty}(\mathscr{I})}\leqslant\frac{1}{2}\varepsilon$.
Thus it remains to prove the estimate for $\psi_{\varepsilon}$ replacing $f$,
since this estimate is deterministic.

  * •

Define $X_{\bullet}=B_{\bullet}-\psi_{\varepsilon}$. Because
$\psi_{\varepsilon}\in\mathscr{C}^{0,1}(\mathscr{I})$ is Lipschitz, the
process $X_{\bullet}$ is an Ito diffusion. This provides that the Radon-
Nikodym derivative of its path-space law with respect to the Wiener measure is
an $\mathrm{L}^{2}$-martingale indexed by $\mathscr{I}$ and vice versa, per
Novikov’s condition. Thus, we have

$\displaystyle\mathbf{P}\left[\|\widetilde{B}_{\bullet}\|_{\mathrm{L}^{\infty}(\mathscr{I})}\
<\ \frac{1}{2}\varepsilon\right]\ $ $\displaystyle\lesssim\
\sqrt{\mathbf{P}\left[\|X_{\bullet}\|_{\mathrm{L}^{\infty}(\mathscr{I})}\right]},$
(3.8)

where $\widetilde{B}_{\bullet}$ is another Brownian motion of the same
diffusion coefficient $D\in\mathbb{R}_{>0}$ such that
$\widetilde{B}_{\inf\mathscr{I}}=0$. The desired estimate now estimate for
standard Brownian motion computations, for example via the reflection
principle.

This completes the proof. ∎

The next ingredient is a technical argument which obtains a lattice-variation
of Lemma 3.6 above, namely a statement for the pullback of
$\nu_{\alpha_{\varrho}}$ through the map in Lemma 3.4. The following estimate
certainly holds in wider generality, but for concreteness we specialize to the
height functions $\mathscr{H}_{0,X}^{N}$ sampled from the $\varrho$-stable
initial particle configuration measures.

######  Lemma 3.7.

Provided any $\varrho\in\mathbb{R}_{\geqslant 0}$, we let
$\mathscr{H}_{0,X}^{N,\mathrm{stat}}$ denote the height function associated to
the initial particle configuration measure $\nu_{\alpha_{\varrho}}$. Then
provided any $\varepsilon\in\mathbb{R}_{>0}$ and sufficiently large
$N\in\mathbb{Z}_{>0}$, we have

$\displaystyle\mathbf{P}\left[\left\|\mathscr{H}_{0,X}^{N}-\mathscr{H}_{0,X}^{N,\mathrm{stat}}\right\|_{\mathrm{L}^{\infty}(\mathbb{T}^{1})}\
\leqslant\ \varepsilon\right]\ \gtrsim_{\varepsilon}\ 1.$ (3.9)

###### Proof.

Denote by $\mathscr{H}_{0,X}^{\infty}$ the limit
$\lim_{N\to\infty}\mathscr{H}_{0,X}^{N}$ sampled from the generic
$\varrho$-stable initial particle configuration measures. In particular, for
sufficiently large $N\in\mathbb{Z}_{>0}$, we have the estimate

$\displaystyle\mathbf{P}\left[\left\|\mathscr{H}_{0,X}^{N}-\mathscr{H}_{0,X}^{\infty}\right\|_{\mathrm{L}^{\infty}(\mathbb{T}^{1})}\
\geqslant\ \frac{1}{3}\varepsilon\right]\ \leqslant\ \delta_{N},$ (3.10)

where $\delta_{N}\to_{N\to\infty}0$. Second, a direct consequence of Lemma
3.6, we obtain the estimate

$\displaystyle\mathbf{P}\left[\left\|\mathscr{H}_{0,X}^{\infty}-B_{X}\right\|_{\mathrm{L}^{\infty}(\mathbb{T}^{1})}\
\leqslant\ \frac{1}{3}\varepsilon\right]\ $
$\displaystyle\gtrsim_{\varepsilon,D}\ 1,$ (3.11)

where $B_{\bullet}$ is a Brownian motion with any diffusion coefficient
$D\in\mathbb{R}_{>0}$ satisfying $B_{0}=0$. Lastly, by a similar token as for
the first estimate upon additionally employing Corollary 3.5, we have

$\displaystyle\mathbf{P}\left[\left\|\mathscr{H}_{0,X}^{N,\mathrm{stat}}-B_{X}\right\|_{\mathrm{L}^{\infty}(\mathbb{T}^{1})}\
\geqslant\ \frac{1}{3}\varepsilon\right]\ $ $\displaystyle\leqslant\
\delta_{N}$ (3.12)

for an appropriate Brownian motion. ∎

We are now in position to construct the Brownian envelope measures of primary
interest in Proposition 3.1.

######  Definition 3.8.

Consider any density $\varrho\in\mathbb{R}_{\geqslant 0}$ and any
$\varepsilon\in\mathbb{R}_{>0}$; the former is the density parameter for the
particle system and the latter is the Brownian envelope parameter.
Furthermore, in spirit of Lemma 3.7 the following construction will always
rely on taking $N\in\mathbb{Z}_{>0}$ sufficiently large depending only on
$\varepsilon\in\mathbb{R}_{>0}$.

  * •

Define the event

$\displaystyle\mathscr{A}_{\varepsilon,N}\ $
$\displaystyle\overset{\bullet}{=}\ \left\\{\eta\in\Omega_{\mathbb{T}_{N}}:\
\left\|\mathscr{H}_{0,X}^{N}-\mathscr{H}_{0,X}^{N,\mathrm{stat}}\right\|_{\mathrm{L}^{\infty}(\mathbb{T}^{1})}\
\leqslant\ \varepsilon\right\\};$ (3.13)

above, $\mathscr{H}_{0,X}^{N}$ is _fixed_ and independent of the
$\eta\in\Omega_{\mathbb{T}_{N}}$ explicitly written, which in turn defines
$\mathscr{H}_{0,X}^{N,\mathrm{stat}}$. Moreover, in view of the bijection in
Lemma 3.4, this event $\mathscr{A}_{\varepsilon,N}$ is well-defined and may be
viewed as a subset of $\Omega_{\mathbb{T}_{N}}$ or a random-walk-path-space
similar to $\mathrm{RW}_{N}$ upon shift by deterministic drift.

We define the $\varepsilon$-_Brownian envelope_ to be the measure
$\mu_{\varepsilon}^{N}$ on $\Omega_{\mathbb{T}_{N}}$ defined via the following
conditional expectation for any bounded function
$\varphi:\Omega_{\mathbb{T}_{N}}\to\mathbb{R}$:

$\displaystyle\operatorname{\mathbf{E}}^{\mu_{\varepsilon}^{N}}\varphi(\eta)\
$ $\displaystyle\overset{\bullet}{=}\
\operatorname{\mathbf{E}}^{\nu_{\alpha_{\varrho}}}\left[\varphi(\eta)\middle|\mathscr{A}_{\varepsilon,N}\right].$
(3.14)

In words, $\mu_{\varepsilon}^{N}$ is the probability measure
$\nu_{\alpha_{\varrho}}^{N}$ _conditioning_ on $\mathscr{A}_{\varepsilon,N}$.

###### Proof of _Proposition 3.1_.

It suffices to check that $\\{\mu_{\varepsilon}^{N}\\}_{N,\varepsilon}$
satisfies the two constraints within the statement of the proposition.
Obviously the height function estimate is satisfied by definition. To verify
the second constraint, per the remark in [11] concerning initial particle
configuration measures with stable relative entropy with respect to
$\nu_{\alpha_{\varrho}}$, it suffices to show

$\displaystyle\mathrm{H}\left(\mu_{\varepsilon}^{N}\middle|\nu_{\alpha_{\varrho}}\right)\
$ $\displaystyle\overset{\bullet}{=}\
\int_{\Omega_{\mathbb{T}_{N}}}\frac{\mathrm{d}\mu_{\varepsilon}^{N}}{\mathrm{d}\nu_{\alpha_{\varrho}}}\log\frac{\mathrm{d}\mu_{\varepsilon}^{N}}{\mathrm{d}\nu_{\alpha_{\varrho}}}\
\mathrm{d}\nu_{\alpha_{\varrho}}\ \lesssim_{\varepsilon}\ 1;$ (3.15)

above, the implied constant cannot depend on $N\in\mathbb{Z}_{>0}$. By
definition of conditional probability and Lemma 3.7, however, the result
follows immediately; this completes the proof. ∎

##  4\. Analysis of Relevant SPDEs

The content within the current subsection consists only of analysis of linear
SPDEs, but we include it both for completeness and to indicate the robustness
of the overall methods within this article.

We emphasize that throughout the entirety of this section, we fix a Gaussian
white noise $\dot{\mathscr{W}}_{T}$, so that all solutions to a given SPDE are
coupled directly through the underlying noise.

###  4.1. Additive SHE

We first show the $\partial$-ASHE admits the following Feller continuity
property which is a straightforward though key ingredient in our analysis.

######  Lemma 4.1.

Suppose $\mathscr{Y}_{T}^{\varepsilon}$ is a sequence of solutions to _( 2.3)_
with initial conditions $\mathscr{Y}_{0}^{\varepsilon}\to\mathscr{Y}_{0}$
converging in the following sense:

  * •

Provided any test function $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$, we have

$\displaystyle\langle\mathscr{Y}_{0}^{\varepsilon},J\rangle\
\to_{\varepsilon\to 0}\ \langle\mathscr{Y}_{0},J\rangle.$ (4.1)

Then provided any test function $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$, we
have
$\langle\mathscr{Y}_{T}^{\varepsilon},J\rangle\to\langle\mathscr{Y}_{T},J\rangle$
for all $T\in\mathbb{R}_{>0}$; here, $\mathscr{Y}_{T}$ denotes the solution to
the ASHE with initial condition $\mathscr{Y}_{0}$.

###### Proof.

Denoting
$\mathscr{X}_{T}^{\varepsilon}=\mathscr{Y}_{T}^{\varepsilon}-\mathscr{Y}_{T}$,
we see that $\mathscr{X}_{T}^{\varepsilon}$ solves the classical heat equation
on the torus $\mathbb{T}^{1}$ _without_ noise. We then have

$\displaystyle\langle\mathscr{X}_{T}^{\varepsilon},J\rangle\ $
$\displaystyle=\
\langle\mathrm{e}^{\alpha_{1}T\partial_{x}^{2}}\mathscr{X}_{0}^{\varepsilon},J\rangle$
(4.2) $\displaystyle=\
\langle\mathscr{X}_{0}^{\varepsilon},\mathrm{e}^{\alpha_{1}T\partial_{x}^{2}}J\rangle$
(4.3)

which vanishes as $\varepsilon\to 0$ by assumption. This completes the proof.
∎

###  4.2. Stochastic Burgers Equation

The analysis of the stochastic Burgers equation is noticeably more involved
than the preceding ASHE; to circumvent the nonlinearity of the equation, we
will employ the Cole-Hopf linearization for the KPZ equation. Alternatively,
upon taking spatial derivatives the discussion in this subsection follows from
the continuity with respect to initial data from Hairer’s regularity
structures approach to the KPZ equation in [13]. However, linearization via
the Cole-Hopf map is robust in non-compact geometries and geometries with
boundary, so we present it here.

The analog for the stochastic Burgers equation of Lemma 4.1 is the following
Feller continuity result.

######  Lemma 4.2.

Suppose $\mathscr{Y}_{T}^{\varepsilon}$ is a sequence of energy solutions to
the stochastic Burgers equation with initial conditions
$\mathscr{Y}_{0}^{\varepsilon}\to\mathscr{Y}_{0}$ converging in the following
sense:

  * •

There exists constants $\kappa_{1},\kappa_{2}\in\mathbb{R}_{>0}$ such that

$\displaystyle\kappa_{1}\ \leqslant\
\inf_{\varepsilon\in(0,1]}\inf_{x\in\mathbb{T}^{1}}\mathscr{Z}_{0,x}^{\varepsilon}\
\leqslant\
\sup_{\varepsilon\in(0,1]}\sup_{x\in\mathbb{T}^{1}}\mathscr{Z}_{0,x}^{\varepsilon}\
\leqslant\ \kappa_{2};$ (4.4)

above, $\mathscr{Z}_{T,x}^{\varepsilon}$ is the corresponding solution to the
multiplicative stochastic heat equation to the energy solution
$\mathscr{Y}_{T}^{\varepsilon}$ of the stochastic Burgers equation.

  * •

We have

$\displaystyle\sup_{x\in\mathbb{T}^{1}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{0,x}^{\varepsilon}-\mathscr{Z}_{0,x}\right|^{2}\right]\
\to_{\varepsilon\to 0}\ 0.$ (4.5)

Then provided any test function $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ and
any $T\in\mathbb{R}_{\geqslant 0}$, we have the following convergence in
probability:

$\displaystyle\langle J,\mathscr{Y}_{T}^{\varepsilon}\rangle-\langle
J,\mathscr{Y}_{T}\rangle\ \to_{\varepsilon\to 0}\ 0,$ (4.6)

where $\mathscr{Y}_{T}$ is the energy solution of the stochastic Burgers
equation with initial condition
$\mathscr{Y}_{0}=\partial_{x}\log\mathscr{Z}_{0,x}$.

###### Proof.

Consider any $\delta\in\mathbb{R}_{>0}$; by the comparison theorem for the
multiplicative-noise SHE in [17], with probability at least
$1-\kappa_{\delta}$ with $\kappa_{\delta}=\mathrm{o}_{\delta\to 0}(1)$, we
have the following uniform estimate in space-time for constants
$\kappa_{1,\delta},\kappa_{2,\delta}\in\mathbb{R}_{>0}$ depending only on
$\delta,\kappa_{1},\kappa_{2}\in\mathbb{R}_{>0}$:

$\displaystyle\kappa_{1,\delta}\ \leqslant\
\inf_{\varepsilon\in(0,1]}\inf_{t\in[0,T]}\inf_{x\in\mathbb{T}^{1}}\mathscr{Z}_{t,x}^{\varepsilon}\
\leqslant\
\sup_{\varepsilon\in(0,1]}\sup_{t\in[0,T]}\sup_{x\in\mathbb{T}^{1}}\mathscr{Z}_{t,x}^{\varepsilon}\
\leqslant\ \kappa_{2,\delta};$ (4.7)

this estimate may be obtained by considering the solutions to the SHE with the
two initial conditions
$\inf_{\varepsilon}\inf_{x\in\mathbb{T}^{1}}\mathscr{Z}_{0,x}^{\varepsilon}$
and
$\sup_{\varepsilon}\sup_{x\in\mathbb{T}^{1}}\mathscr{Z}_{0,x}^{\varepsilon}$,
respectively, and applying the aforementioned comparison theorem in [17].

On this event, we see

$\displaystyle\left|\log\mathscr{Z}_{T,x}^{\varepsilon}-\log\mathscr{Z}_{T,x}\right|\
$ $\displaystyle\lesssim\
\left|\mathscr{Z}_{T,x}^{\varepsilon}-\mathscr{Z}_{T,x}\right|,$ (4.8)

where the implied constant depends only on
$\kappa_{1,\delta},\kappa_{2,\delta}\in\mathbb{R}_{>0}$, so we will assume the
above estimate at the cost of an event of probability $\mathrm{o}_{\delta\to
0}(1)$.

Recall

$\displaystyle\left|\langle J,\mathscr{Y}_{T}^{\varepsilon}\rangle-\langle
J,\mathscr{Y}_{T}\rangle\right|\ $ $\displaystyle\leqslant\
\left\langle|\partial_{x}J|,\left|\log\mathscr{Z}_{T,\bullet}^{\varepsilon}-\log\mathscr{Z}_{T,\bullet}\right|\right\rangle$
(4.9) $\displaystyle\leqslant\
\left\langle|\partial_{x}J|,\left|\mathscr{Z}_{T,x}^{\varepsilon}-\mathscr{Z}_{T,x}\right|\right\rangle.$
(4.10)

Thus, it remains to show that

$\displaystyle\sup_{x\in\mathbb{T}^{1}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{T,x}^{\varepsilon}-\mathscr{Z}_{T,x}\right|^{2}\right]\
\to_{\varepsilon\to 0}\ 0.$ (4.11)

To this end, we simply employ the mild formulation of the stochastic heat
equation with multiplicative noise; via the Ito formula, we have the following
estimate for some universal constant $\kappa\in\mathbb{R}_{>0}$:

$\displaystyle\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{T,x}^{\varepsilon}-\mathscr{Z}_{T,x}\right|^{2}\right]\
$ $\displaystyle\leqslant\
\mathrm{e}^{\alpha_{1}^{\prime}T\partial_{x}^{2}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{0,\bullet}^{\varepsilon}-\mathscr{Z}_{0,\bullet}\right|^{2}\right]\
+\
\kappa\int_{0}^{T}\frac{1}{\sqrt{T-s}}\mathrm{e}^{\alpha_{1}^{\prime}(T-s)\partial_{x}^{2}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{s,\bullet}^{\varepsilon}-\mathscr{Z}_{s,\bullet}\right|^{2}\right]\
\mathrm{d}s.$ (4.12)

Iterating and applying the semigroup property for the classical heat equation
yields

$\displaystyle\sup_{x\in\mathbb{T}^{1}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{T,x}^{\varepsilon}-\mathscr{Z}_{T,x}\right|^{2}\right]\
$ $\displaystyle\lesssim_{T}\
\sup_{x\in\mathbb{T}^{1}}\operatorname{\mathbf{E}}\left[\left|\mathscr{Z}_{0,x}^{\varepsilon}-\mathscr{Z}_{0,x}\right|^{2}\right];$
(4.13)

since the RHS vanishes as $\varepsilon\to 0$, we obtain the desired
convergence outside a set of probability $\mathrm{o}_{\delta\to 0}(1)$. As
$\delta\in\mathbb{R}_{>0}$ was arbitrary, this completes the proof. ∎

##  5\. Proof of Theorem 2.6 and Theorem 2.7

###  5.1. Coupling Strategy

The primary aim of the current subsection is to establish the propagation of
the height function estimate in Proposition 3.1 globally in time.

######  Lemma 5.1.

Retain the setting of _Proposition 3.1_. There exists a coupling of the
particle dynamics uniform over the initial particle configuration measures
$\mu_{\varepsilon}^{N}$ such that provided any test function
$J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ and any
$\varepsilon\in\mathbb{R}_{>0}$, we have

$\displaystyle\left|\mathscr{Y}_{T,J}^{N,\varepsilon}-\mathscr{Y}_{T,J}^{N}\right|\
\lesssim\ \varepsilon,$ (5.1)

where the implied constant is universal, and
$\mathscr{Y}_{T,J}^{N,\varepsilon}$ is the density fluctuation field
integrated against $J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$ when the initial
particle configuration is sampled from $\mu_{\varepsilon}^{N}$.

To clarify, the coupling claimed in Lemma 5.1 is a coupling among _all_ the
particle systems with initial configurations sampled from
$\\{\mu_{\varepsilon}^{N}\\}_{\varepsilon}$ respectively; this couplng will be
the assumed basic coupling.

###### Proof.

Consider any $\varepsilon\in\mathbb{R}_{>0}$. The first step consists of
employing the summation-by-parts identity, which provides

$\displaystyle\left|\mathscr{Y}_{T,J}^{N,\varepsilon}-\mathscr{Y}_{T,J}^{N}\right|\
$ $\displaystyle\lesssim_{J}\ \ \mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-11.66676pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-9.73344pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-7.4334pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-6.16672pt}}\\!\sum_{x\in\mathbb{T}_{N}}\left|\mathscr{H}_{T,x}^{N,\varepsilon}-\mathscr{H}_{T,x}^{N}\right|,$
(5.2)

where $\mathscr{H}_{T,x}^{N,\varepsilon}$ denotes the evolved height function
with initial particle configuration sampled from the probability measure
$\mu_{\varepsilon}^{N}$. We now define the following perturbations of these
height functions:

$\displaystyle\mathscr{H}_{T,x}^{N,\varepsilon}(\pm)\ $ $\displaystyle=\
\mathscr{H}_{T,x}^{N,\varepsilon}\pm\kappa\varepsilon,$ (5.3)

where $\kappa\in\mathbb{R}_{>0}$ will be a universal constant to be
determined; in particular, it will be chosen independent of
$N\in\mathbb{Z}_{>0}$ and $\varepsilon\in\mathbb{R}_{>0}$. Certainly it
suffices to establish the following bounds for such a value of
$\kappa\in\mathbb{R}_{>0}$:

$\displaystyle\mathscr{H}_{T,x}^{N,\varepsilon}(-)\ \leqslant\
\mathscr{H}_{T,x}^{N}\ \leqslant\ \mathscr{H}_{T,x}^{N,\varepsilon}(+),\quad
T\in\mathbb{R}_{\geqslant 0}.$ (5.4)

To this end, we first observe that the above pair of inequalities are
certainly true for $T=0$ by definition of the Brownian envelopes
$\mu_{\varepsilon}^{N}$. Employing the basic coupling for the particle system,
we make the following observations which establish a microscopic analog of the
comparison principle in [17] for the multiplicative-noise stochastic heat
equation. To illustrate the argument, we fix $\varepsilon\in\mathbb{R}_{>0}$
and couple the dynamics with initial data $\mu_{\varepsilon}^{N}$ to those
with initial data $\mu_{0}^{N}$, though this can be carried through for all
the Brownian envelope initial configuration measure $\mu_{\varepsilon}^{N}$
simultaneously on the same probability space.

  * •

Via the basic coupling, for the bounds (5.4) to fail, we necessarily have a
jump in the particle system within one configuration that does not occur in
the other.

  * •

Suppose a jump occurs in the configuration with initial data sampled from
$\mu_{\varepsilon}^{N}$ but not in the configuration initially sampled from
$\mu_{0}^{N}$, and suppose the jump was left-ward. For this case, the value of
the height function $\mathscr{H}_{T,x}^{N,\varepsilon}$ does not decrease
anywhere by definition. Similarly, suppose a jump occurs in the configuration
with initial data from $\mu_{0}^{N}$ but not in the configuration initially
sampled from $\mu_{\varepsilon}^{N}$, and suppose the jump was right-ward.
Then, the value of the height function $\mathscr{H}_{T,x}^{N}$ does not
increase anywhere by definition. This implies the inequality

$\displaystyle\mathscr{H}_{T,x}^{N}\ \leqslant\
\mathscr{H}_{T,x}^{N,\varepsilon}(+)$ (5.5)

is preserved after this jump.

  * •

Following the argument in the previous bullet point, given a left-ward jump
occurs in the configuration with initial data sampled from $\mu_{0}^{N}$ but
not in the configuration initially sampled from $\mu_{\varepsilon}^{N}$, or if
a right-ward jump occurs in the configuration with initial data sampled from
$\mu_{\varepsilon}^{N}$ but not in the configuration initially sampled from
$\mu_{0}^{N}$, then the following inequality is preserved:

$\displaystyle\mathscr{H}_{T,x}^{N,\varepsilon}(-)\ \leqslant\
\mathscr{H}_{T,x}^{N}.$ (5.6)

  * •

Suppose now that a right-ward jump occurs in the particle configuration
initially sampled from $\mu_{\varepsilon}^{N}$ but not in the particle
configuration initially sampled from $\mu_{0}^{N}$. This situation is slightly
more detailed because the value of the height function
$\mathscr{H}_{T,x}^{N,\varepsilon}$ could decrease at the site from which the
jump was taken.

However, this implies that there are more particles at the site from which was
jump was made in the configuration initially sampled from
$\mu_{\varepsilon}^{N}$ than in the configuration initially sampled from
$\mu_{0}^{N}$. Moreover, recall the pointwise bound
$\mathscr{H}_{T,\bullet}^{N,\varepsilon}(+)\geqslant\mathscr{H}_{T,\bullet}^{N}$
everywhere before this jump, including at the site to the immediate left of
the site where the jump was taken. Because the height functions are unaffected
at this immediately-left site, it is preserved after the jump. Because at the
site from which the jump was made, the number of particles in the
configuration initially sampled from $\mu_{\varepsilon}^{N}$ is at least the
number of particles in the configuration initially sampled from $\mu_{0}^{N}$
after this jump, we deduce that the pointwise bound

$\displaystyle\mathscr{H}_{T,x}^{N}\ \leqslant\ \mathscr{H}_{T,x}^{N}(+)$
(5.7)

is still preserved pointwise.

Similar considerations show the above inequality is preserved in the situation
that a left-ward jump occurs in the configuration initially sampled from
$\mu_{0}^{N}$ but not in the configuration initially sampled from
$\mu_{\varepsilon}^{N}$.

  * •

Similar considerations as in the previous bullet point cover the remaining
cases and show the pointwise bound

$\displaystyle\mathscr{H}_{T,x}^{N,\varepsilon}(-)\ \leqslant\
\mathscr{H}_{T,x}^{N}$ (5.8)

is also preserved after any coupled jump, or attempted coupled jump.

Thus, we have propagated (5.4) globally from its validity for the initial
data. This completes the proof. ∎

###  5.2. Proof of Theorem 2.7

Courtesy of Proposition 3.1 and Lemma 5.1, we deduce that
$\mathscr{Y}_{T,J}^{N}$ is tight in the large-$N$ limit in the Skorokhod space
$\mathscr{D}([0,T],\mathbb{R})$ given any smooth test function
$J\in\mathscr{C}^{\infty}(\mathbb{T}^{1})$. Indeed, provided any
$\varepsilon\in\mathbb{R}_{>0}$, we may choose
$\varepsilon^{\prime}\in\mathbb{R}_{>0}$ sufficiently small so that

$\displaystyle\sup_{t\in[0,T]}\left|\mathscr{Y}_{t,J}^{N,\varepsilon^{\prime}}-\mathscr{Y}_{t,J}^{N}\right|\
\leqslant\ \varepsilon.$ (5.9)

In particular, as the above estimate is uniform in time and
$N\in\mathbb{Z}_{>0}$, and since
$\mathscr{Y}_{\bullet,J}^{N,\varepsilon^{\prime}}$ is tight in the large-$N$
limit in the Skorokhod space $\mathscr{D}([0,T],\mathbb{R})$, we obtain the
claimed tightness. To identify the limit points of the large-$N$ limit, it
suffices to directly cite Lemma 4.2, because under the coupling in Lemma 5.1
employed in this proof, the resulting SPDE solutions
$\mathscr{Y}_{T,J}^{\infty,\varepsilon^{\prime}}$ are coupled through the
limiting space-time white noise, as this noise is a functional of the Poisson
clocks only; also, the required a priori two-sided bounds for the initial data
$\mathscr{Z}_{0,x}^{\varepsilon}$ are certainly satisfied with high
probability by tightness of the height functions. This completes the proof.

###  5.3. Proof of Theorem 2.6

This follows from an argument identical to the argument used in the proof of
Theorem 2.7, the only difference in applying Lemma 4.1 instead of Lemma 4.2.

## References

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