#  Concurrent Donsker-Varadhan and hydrodynamical large deviations

L. Bertini  Lorenzo Bertini Dipartimento di Matematica, Università di Roma ‘La
Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy<EMAIL_ADDRESS>,  D.
Gabrielli  Davide Gabrielli DISIM, Università dell’Aquila 67100 Coppito,
L’Aquila, Italy<EMAIL_ADDRESS>and  C. Landim  Claudio Landim
IMPA Estrada Dona Castorina 110, J. Botanico, 22460 Rio de Janeiro, Brazil and
CNRS UMR 6085, Université de Rouen, Avenue de l’Université, BP.12, Technopôle
du Madrillet, F76801 Saint-Étienne-du-Rouvray, France<EMAIL_ADDRESS>
###### Abstract.

We consider the weakly asymmetric exclusion process on the $d$-dimensional
torus. We prove a large deviations principle for the time averaged empirical
density and current in the joint limit in which both the time interval and the
number of particles diverge. This result is obtained both by analyzing the
variational convergence, as the number of particles diverges, of the Donsker-
Varadhan functional for the empirical process and by considering the large
time behavior of the hydrodynamical rate function. The large deviations
asymptotic of the time averaged current is then deduced by contraction
principle. The structure of the minimizers of this variational problem
corresponds to the possible occurrence of dynamical phase transitions.

###### Key words and phrases:

Exclusion processes, Hydrodynamical limits, Large deviations, Empirical
process, Dynamical phase transitions

###### 2010 Mathematics Subject Classification:

Primary 60K35, 60F10; Secondary 82C22, 82C70.

##  1\. Introduction

Stochastic lattice gases, that describe the evolution of interacting random
particles on a lattice of mesh $1/N$, have been an instrumental tool in the
development of non-equilibrium statistical mechanics [4, 15, 20]. Their
macroscopic behavior, usually referred to as hydrodynamic scaling limit, is
described as follows. Given a microscopic realization of the process, the
empirical density ${\boldsymbol{\pi}}_{N}$ is defined by counting locally the
average number of particles while the empirical current ${\boldsymbol{J}}_{N}$
is defined by counting the net flow of particles. By the local conservation of
the number of particles, ${\boldsymbol{\pi}}_{N}$ and ${\boldsymbol{J}}_{N}$
satisfy the continuity equation. The content of the hydrodynamical limit is
the law of large numbers for the pair
$({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})$ in the limit $N\to\infty$. For
driven-diffusive systems the limiting evolution is given by

$\begin{cases}\partial_{t}{\boldsymbol{\rho}}+\nabla\cdot{\boldsymbol{j}}=0,\\\
{\boldsymbol{j}}=-D({\boldsymbol{\rho}})\nabla{\boldsymbol{\rho}}+\sigma({\boldsymbol{\rho}})E,\end{cases}$
(1.1)

where $E=E(x)$ is the applied external field, $D$ is the diffusion matrix, and
$\sigma$ is the mobility. In particular, the density profile
${\boldsymbol{\rho}}={\boldsymbol{\rho}}(t,x)$ solves the non-linear driven
diffusive equation

$\partial_{t}{\boldsymbol{\rho}}+\nabla\cdot\big{(}\sigma({\boldsymbol{\rho}})E\big{)}=\nabla\cdot\big{(}D({\boldsymbol{\rho}})\nabla{\boldsymbol{\rho}}\big{)}.$
(1.2)

We refer to [15] for the details on the derivation of (1.2), while the scaling
limit of the empirical current leading to (1.1) is discussed in [3] in the
case of the symmetric exclusion process.

The large deviations with respect to the hydrodynamic limit in the time window
$[0,T]$ are characterized by the rate function

$A_{T}({\boldsymbol{\rho}},{\boldsymbol{j}})=\int_{0}^{T}\\!dt\\!\int\\!dx\,\frac{\big{|}{\boldsymbol{j}}+D({\boldsymbol{\rho}})\nabla{\boldsymbol{\rho}}-\sigma({\boldsymbol{\rho}})E\big{|}^{2}}{4\,\sigma({\boldsymbol{\rho}})},$
(1.3)

that is at the base of the Macroscopic Fluctuations Theory and it is widely
used in non-equilibrium statistical mechanics [4]. We refer to [15, 16] for
the derivation of this rate function when the empirical current is
disregarded.

A significant problem is the behavior of the average of empirical current over
the time interval $[0,T]$ in the limit when $N\to\infty$ and then
$T\to\infty$. By the hydrodynamical large deviations principle and contraction
principle, this amounts to analyze the behavior as $T\to\infty$ of the
minimizers to (1.3) with the constraint
$\frac{1}{T}\int_{0}^{T}\\!dt\,{\boldsymbol{j}}=J$. This problem has been
initially raised in [7] while in [1] it has been pointed out that the
minimizers can exhibit a non-trivial time dependent behavior. In [2, 8] it has
been then shown that this is actually the case for the weakly asymmetric
exclusion process and the Kipnis-Marchioro-Presutti model where, for suitable
value of the parameters, traveling waves are more convenient than constant
profiles.

Denote by $I^{(2)}(J)$ the limiting value as $T\to\infty$ of the minimum to
$T^{-1}\,A_{T}$ with the constraint
$\frac{1}{T}\int_{0}^{T}\\!dt\,{\boldsymbol{j}}=J$. Varadhan [21] proposed the
following representation for $I^{(2)}$

$I^{(2)}(J)=\inf\bigg{\\{}\\!\\!\int\\!\\!dP\\!\\!\int\\!\\!dx\,\frac{\big{|}{\boldsymbol{j}}(t)+D({\boldsymbol{\rho}}(t))\nabla{\boldsymbol{\rho}}(t)-\sigma({\boldsymbol{\rho}}(t))E\big{|}^{2}}{4\sigma({\boldsymbol{\rho}}(t))};\,\int\\!\\!dP\,{\boldsymbol{j}}(t)=J\\!\bigg{\\}}$
(1.4)

where the infimum is carried out over the probabilities $P$ invariant by time
translations on the set of paths $({\boldsymbol{\rho}},{\boldsymbol{j}})$
satisfying the continuity equation
$\partial_{t}{\boldsymbol{\rho}}+\nabla\cdot{\boldsymbol{j}}=0$. Note that
$I^{(2)}$ is convex and that, by the stationarity of $P$, the actual value of
$t$ on the right hand side of (1.4) is irrelevant.

The purpose of the present analysis is to prove the validity of the
representation (1.4) in the context of the weakly asymmetric exclusion process
for which $D=1$ and $\sigma(\rho)=\rho(1-\rho)$. This will be achieved both
when the limit $T\to\infty$ is carried out after the hydrodynamic limit
$N\to\infty$ and when the limits are carried out in the opposite order. In
fact, the representation (1.4) will be deduced by the contraction principle
from a large deviation result at the level of the empirical processes that we
next introduce.

Consider first the case in which the limit $N\to\infty$ is taken after
$T\to\infty$. By the Donsker-Varadhan result, see e.g. [11, 22], as
$T\to\infty$ the empirical process associated to the weakly asymmetric
exclusion process satisfies a large deviation principle in which the affine
rate function is the relative entropy per unit of time with respect to the
stationary process. By projecting this functional to the stationary
probabilities on the empirical density and current, and analyzing its
variational convergence as $N\to\infty$ we deduce the desired large deviation
principle with affine rate function given by

${\boldsymbol{I}}(P)=\int\\!P(d{\boldsymbol{\rho}},d{\boldsymbol{j}})\,\int\\!dx\,\,\frac{\big{|}{\boldsymbol{j}}(t)+D({\boldsymbol{\rho}}(t))\nabla{\boldsymbol{\rho}}(t)-\sigma({\boldsymbol{\rho}}(t))E\big{|}^{2}}{4\,\sigma({\boldsymbol{\rho}}(t))}\;\cdot$
(1.5)

The main ingredient in this derivation is, as for hydrodynamical large
deviations, the validity of local equilibrium with probability super-
exponentially close to one as $N\to\infty$. Observe that the rate function in
(1.4) is obtained from (1.5) by contraction. The proof for the case in which
the limit $T\to\infty$ is taken after $N\to\infty$ is achieved by lifting the
hydrodynamical rate function (1.3) to the set of stationary probabilities on
density and current and analyzing its variational convergence as $T\to\infty$.

As a corollary of the analysis here presented, we also deduce the “level two”
large deviations relative to the family of random probability measures
$\frac{1}{T}\int_{0}^{T}\\!dt\,\delta_{{\boldsymbol{\pi}}_{N}(t)}$ in the
joint limit $N,T\to\infty$. Letting
$\imath_{t}({\boldsymbol{\rho}},{\boldsymbol{j}})={\boldsymbol{\rho}}(t)$, the
corresponding rate function is

${\mathscr{I}}(\wp)=\inf\Big{\\{}{\boldsymbol{I}}(P)\;;\;P\circ\imath_{t}^{-1}\,=\wp\Big{\\}}$
(1.6)

Since ${\mathscr{I}}(\wp)=0$ if and only if $\wp$ is a stationary measure for
the flow associated to the hydrodynamic equation (1.2), this large deviations
statement implies the hydrostatic limit for the weakly asymmetric exclusion
process: in the limit $N\to\infty$ the empirical density constructed by
sampling the particles according to the stationary measure converges to the
unique stationary solution to (1.2).

##  2\. Notation and Results

### Microscopic dynamics

Denote by ${\mathbb{T}}^{d}=[0,1)^{d}$ the $d$-dimensional torus of length $1$
and let $dx$ be the corresponding Haar measure. Fix $N\geq 1$, and let
${\mathbb{T}}^{d}_{N}$ the discretization of ${\mathbb{T}}^{d}$:
${\mathbb{T}}^{d}_{N}={\mathbb{T}}^{d}\cap(N^{-1}\mathbb{Z})^{d}$. The
elements of ${\mathbb{T}}^{d}$ and ${\mathbb{T}}^{d}_{N}$ are represented by
$x$ and $y$. Let $\color[rgb]{.2,0.2,.8}{\mathbb{B}}_{N}$ be the set of
ordered, nearest-neighbor pairs $(x,y)$ in ${\mathbb{T}}^{d}_{N}$.

Denote by $\color[rgb]{.2,0.2,.8}\Sigma_{N}:=\\{0,1\\}^{{\mathbb{T}}^{d}_{N}}$
the space of configurations. Elements of $\Sigma_{N}$ are represented by
$\eta$, so that $\eta_{x}=1$, resp. $0$, if site $x$ is occupied, resp.
vacant, for the configuration $\eta$. Fix $E$ in
$\color[rgb]{.2,0.2,.8}C^{1}({\mathbb{T}}^{d};{\mathbb{R}}^{d})$, the space of
continuously differentiable vector fields defined on ${\mathbb{T}}^{d}$. In
some statements we assume that $E$ is _orthogonally decomposable_ : there are
$U\in C^{2}({\mathbb{T}}^{d})$ and $\tilde{E}\in
C^{1}({\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with vanishing divergence,
$\nabla\cdot\tilde{E}=0$, satisfying the pointwise orthogonality $\nabla
U(x)\cdot\tilde{E}(x)=0$, $x\in{\mathbb{T}}^{d}$, such that $E=-\nabla
U+\tilde{E}$.

The weakly asymmetric exclusion process (WASEP) with external field $E$ is the
Markov process on $\Sigma_{N}$ whose generator $L_{N}$ acts on functions
$f\colon\Sigma_{N}\to{\mathbb{R}}$ as

$(L_{N}f)(\eta)\;=\;N^{2}\sum_{(x,y)\in{\mathbb{B}}_{N}}\eta_{x}\,[1-\eta_{y}]\,e^{(1/2)\,E_{N}(x,y)}\big{[}f(\sigma^{x,y}\eta)-f(\eta)\big{]}\;.$
(2.1)

In this formula, the configuration $\sigma^{x,y}\eta$ is obtained from $\eta$
by exchanging the occupation variables $\eta_{x}$, $\eta_{y}$:

$(\sigma^{x,y}\eta)_{z}\;:=\;\begin{cases}\eta_{y}&\textrm{ if \ }z=x\;,\\\
\eta_{x}&\textrm{ if \ }z=y\;,\\\ \eta_{z}&\textrm{ if \ }z\neq
x,y\;,\end{cases}$

and $E_{N}(x,y)$ represents the line integral of $E$ along the oriented
segment from $x$ to $y$:

$E_{N}(x,y)\;=\;\int_{x}^{y}E\cdot\,d\ell\;=\;\int_{0}^{1}E(x+r\,[y-x])\cdot[y-x]\,dr\;,$
(2.2)

where $a\cdot b$ is the inner product in ${\mathbb{R}}^{d}$. Note that
$E_{N}(\cdot,\cdot)$ is antisymmetric and that $E_{N}$ is of order $1/N$.

Denote by
$\color[rgb]{.2,0.2,.8}\Sigma_{N,K}=\\{\eta\in\Sigma_{N}:\sum_{x\in{\mathbb{T}}^{d}_{N}}\eta_{x}=K\\}$,
$K=0,\ldots,N^{d}$, the set of configurations with $K$ particles. The Markov
process with generator $L_{N}$ is irreducible in the finite state space
$\Sigma_{N,K}$. It has therefore a unique stationary probability measure,
denoted by $\color[rgb]{.2,0.2,.8}\mu_{N,K}$.

Hereafter, $\color[rgb]{.2,0.2,.8}{\mathcal{R}}$ represents either
${\mathbb{R}}_{+}$ or ${\mathbb{R}}$. Denote by
$\color[rgb]{.2,0.2,.8}D({\mathcal{R}},\Sigma_{N})$, the set of right-
continuous functions with left-limits from ${\mathcal{R}}$ to $\Sigma_{N}$,
endowed with the Skorohod topology and the corresponding Borel
$\sigma$-algebra. Elements of $D({\mathcal{R}},\Sigma_{N})$ are represented by
${\boldsymbol{\eta}}$.

For a probability measure $\nu$ on $\Sigma_{N}$, denote by
$\color[rgb]{.2,0.2,.8}{\mathbb{P}}^{N}_{\nu}$ the probability measure on
$D({\mathbb{R}}_{+},\Sigma_{N})$ induced by the Markovian dynamics associated
to the generator $L_{N}$ starting from $\nu$. When the measure $\nu$ is
concentrated on a configuration $\eta\in\Sigma_{N}$, $\nu=\delta_{\eta}$, we
write $\color[rgb]{.2,0.2,.8}{\mathbb{P}}^{N}_{\eta}$ instead of
${\mathbb{P}}^{N}_{\delta_{\eta}}$. For $K=0,\ldots,N^{d}$, the stationary
processes associated to the WASEP dynamics with $K$ particles is denoted by
$\color[rgb]{.2,0.2,.8}{\mathbb{P}}_{\mu_{N,K}}^{N}$ that we regard as a
probability measure on $D({\mathbb{R}},\Sigma_{N,K})$ invariant with respect
to time-translations. Expectation with respect to
${\mathbb{P}}_{\mu_{N,K}}^{N}$ is represented by
$\color[rgb]{.2,0.2,.8}{\mathbb{E}}_{\mu_{N,K}}^{N}$.

### Empirical density

Let $\color[rgb]{.2,0.2,.8}{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ be the set of
positive measures on ${\mathbb{T}}^{d}$ with total mass bounded by $1$,
endowed with the weak topology and the corresponding Borel $\sigma$-algebra.
Let also $\color[rgb]{.2,0.2,.8}{\mathcal{M}}_{m}({\mathbb{T}}^{d})$,
$m\in[0,1]$, be the closed subset of ${\mathcal{M}}_{+}({\mathbb{T}}^{d})$
given by the measures whose total mass is equal to $m$.

The _empirical density_ is the map
$\pi_{N}\colon\Sigma_{N}\to{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ defined by

$\pi_{N}(\eta)\;:=\;\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}\eta_{x}\,\delta_{x}\;,$
(2.3)

where $\delta_{x}$, $x\in{\mathbb{T}}^{d}$, is the point mass at $x$. For a
continuous function $f\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$ and a measure
$\nu$ in ${\mathcal{M}}_{+}({\mathbb{T}}^{d})$, we represent by
$\langle\nu\,,\,f\rangle$ the integral of $f$ with respect to $\nu$ so that

$\langle\pi_{N}\,,\,f\rangle\;=\;\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}\eta_{x}\,f(x)\;.$
(2.4)

We also call empirical density the map ${\boldsymbol{\pi}}_{N}\colon
D({\mathcal{R}},\Sigma_{N})\to
D({\mathcal{R}},{\mathcal{M}}_{+}({\mathbb{T}}^{d}))$ defined by

$[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(t)\;:=\;\pi_{N}({\boldsymbol{\eta}}(t))\;=\;\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}{\boldsymbol{\eta}}_{x}(t)\,\delta_{x}\;,\quad
t\in{\mathcal{R}}\;.$ (2.5)

### Empirical current.

For an oriented bond $(x,y)\in{\mathbb{B}}_{N}$ and $s<t$, let
${\mathscr{N}}^{x,y}_{(s,t]}({\boldsymbol{\eta}})$ be the number of jumps from
$x$ to $y$ in the time interval $(s,t]$ of the path ${\boldsymbol{\eta}}\in
D({\mathcal{R}},\Sigma_{N})$:

${\mathscr{N}}^{x,y}_{(s,t]}({\boldsymbol{\eta}})\;=\;\sum_{s<r\leq
t}{\boldsymbol{\eta}}_{x}(r-)\,[1-{\boldsymbol{\eta}}_{y}(r-)]\,{\boldsymbol{1}}\\{{\boldsymbol{\eta}}(r)=\sigma^{x,y}{\boldsymbol{\eta}}(r-)\\}\;.$
(2.6)

Fix a trajectory ${\boldsymbol{\eta}}\in D({\mathcal{R}},\Sigma_{N})$, and
denote by $\color[rgb]{.2,0.2,.8}C({\mathbb{T}}^{d};{\mathbb{R}}^{d})$ the
space of continuous vector fields on ${\mathbb{T}}^{d}$. If
${\boldsymbol{\eta}}$ is a trajectory compatible with the WASEP dynamics, i.e.
such that for each jump time $t$ we have
${\boldsymbol{\eta}}(t)=\sigma^{x,y}{\boldsymbol{\eta}}(t-)$ for some
$(x,y)\in{\mathbb{B}}_{N}$, we define the _integrated empirical current_
${\boldsymbol{J}}_{N}({\boldsymbol{\eta}})$ as follows. Let
$[\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,]\,(0)=0$, and, for $t\,>\,0$,
let $[\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,]\,(t)$ be the linear
functional on $C({\mathbb{T}}^{d};{\mathbb{R}}^{d})$ defined by

$\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,(t)\,,\,F\rangle\;:=\;\frac{1}{N^{d}}\sum_{(x,y)\in{\mathbb{B}}_{N}}{\mathscr{N}}^{x,y}_{(0,t]}({\boldsymbol{\eta}})\int_{x}^{y}F\cdot
d\ell\;,\quad F\,\in\,C({\mathbb{T}}^{d};{\mathbb{R}}^{d})\;.$ (2.7)

For $t<0$, we replace in the previous formula
${\mathscr{N}}^{x,y}_{(0,t]}({\boldsymbol{\eta}})$ by
$-\,{\mathscr{N}}^{x,y}_{(t,0]}({\boldsymbol{\eta}})$. If
$F=(F_{1},\ldots,F_{d})$, then for $t\geq 0$

$\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,(t)\,,\,F\rangle=\frac{1}{N^{d}}\sum_{j=1}^{d}\sum_{x\in{\mathbb{T}}^{d}_{N}}\frac{1}{N}\,\Big{\\{}{\mathscr{N}}^{x,x+{\mathfrak{e}}_{j}}_{(0,t]}({\boldsymbol{\eta}})-{\mathscr{N}}^{x+{\mathfrak{e}}_{j},x}_{(0,t]}({\boldsymbol{\eta}})\Big{\\}}\int_{0}^{1}F_{j}(x+r{\mathfrak{e}}_{j})\,dr\;,$

where, $\color[rgb]{.2,0.2,.8}{\mathfrak{e}}_{j}=e_{j}/N$ and
$\\{e_{1},\dots,e_{d}\\}$ represents the canonical basis of
${\mathbb{R}}^{d}$.

Discrete vector fields. For technical issues, we need to define the empirical
current also for paths ${\boldsymbol{\eta}}$ not coming from the WASEP
dynamics. Consider an arbitrary path ${\boldsymbol{\eta}}\in
D\big{(}{\mathcal{R}};\Sigma_{N,K}\big{)}$. If for some $t\in{\mathcal{R}}$ it
happens ${\boldsymbol{\eta}}(t)\neq{\boldsymbol{\eta}}(t-)$ some of the
particles in the configuration ${\boldsymbol{\eta}}(t-)$ have rearranged
themselves to construct the configuration ${\boldsymbol{\eta}}(t)$. The
definition of the empirical current requires to decide the actual path taken
by those particles.

A _discrete vector field_ $W$ is an antisymmetric function
$W\colon{\mathbb{B}}_{N}\to{\mathbb{R}}$. The _discrete divergence_ of the
discrete vector field $W$ is the function $\nabla_{N}\cdot
W\colon{\mathbb{T}}_{N}^{d}\to{\mathbb{R}}$ defined by

$(\nabla_{N}\cdot W)\,(x):=\sum_{y\colon(x,y)\in{\mathbb{B}}_{N}}W(x,y)\;.$

Fix $0\leq K\leq N^{d}$, and consider two configurations $\eta$,
$\xi\in\Sigma_{N,K}$. Let $W_{\eta,\xi}$ be a discrete vector field which
solves

$(\nabla_{N}\cdot W_{\eta,\xi})\,(x)\;=\;\eta_{x}\,-\,\xi_{x}\;,\quad
x\,\in\,{\mathbb{T}}^{d}_{N}\;.$ (2.8)

Such a discrete vector field $W_{\eta,\xi}$ always exists. The configuration
$\zeta=\eta-\xi$ belongs to $\\{-1,0,1\\}^{{\mathbb{T}}^{d}_{N}}$ and
$\sum_{x\in{\mathbb{T}}^{d}_{N}}\zeta_{x}=0$. To define $W_{\eta,\xi}$, one
just needs to create nearest-neighbor flows from each
$x\in{\mathbb{T}}^{d}_{N}$ such that $\zeta_{x}=1$ to each
$x^{\prime}\in{\mathbb{T}}^{d}_{N}$ such that $\zeta_{x^{\prime}}=-1$, and add
all these flows.

Regarding $\eta$ and $\xi$ as positive measures on ${\mathbb{T}}^{d}_{N}$,
both of mass $K$, by the analysis of the discrete Beckmann’s problem, see e.g.
[19], we deduce that there exists a constant $C_{0}$ such that for all $N\geq
1$, $0\leq K\leq N^{d}$, $\eta$, $\xi\in\Sigma_{N,K}$ there exists a discrete
vector field $W_{\eta,\xi}$ such that

$\sum_{(x,y)\in{\mathbb{B}}_{N}}\big{|}\,W_{\eta,\xi}(x,y)\,\big{|}\;\leq\;C_{0}\,N^{d+1}\;.$
(2.9)

Of course, equation (2.8) admits more than one solution and we do not claim
that there is only one satisfying the previous bound. It turns out, however,
that the scaling limit of the empirical current does not depend on the
specific selection among those fulfilling (2.9). Hence, in the sequel and
without further mention, we assume that, for each pair
$(\eta,\xi)\in\Sigma^{2}_{N,K}$, a discrete vector field which match (2.9)
$W_{\eta,\xi}$ has been chosen.

When $\xi=\sigma^{x,y}\eta$ for some $(x,y)\in{\mathbb{B}}_{N}$ and
$\eta_{x}=1$, $\eta_{y}=0$, we define $W_{\eta,\xi}$ as

$W_{\eta,\xi}(x^{\prime},y^{\prime})\;=\;\begin{cases}1&\text{if
}(x^{\prime},y^{\prime})=(x,y)\;,\\\ -1&\text{if
}(x^{\prime},y^{\prime})=(y,x)\;,\\\ 0&\text{otherwise}\;.\end{cases}$ (2.10)

This discrete vector field clearly satisfies (2.8) and (2.9).

Integrated currents for generic paths. Fix a generic path
${\boldsymbol{\eta}}\in D({\mathcal{R}};\Sigma_{N,K})$ and denote by
$\tau_{i}$ its jump times. Let $W_{i}$ be the discrete vector field given by

$W_{i}\;=\;W_{{\boldsymbol{\eta}}(\tau_{i}-),{\boldsymbol{\eta}}(\tau_{i})}\;.$
(2.11)

For $t>0$, the integrated empirical current of the path ${\boldsymbol{\eta}}$
is then defined by

$\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})(t),F\rangle=\frac{1}{2N^{d}}\,\sum_{i\colon\tau_{i}\in(0,t]}\sum_{(x,y)\in{\mathbb{B}}_{N}}W_{i}(x,y)\,F_{N}(x,y)$
(2.12)

where $F_{N}$ is the discrete vector field constructed from $F\in
C({\mathbb{T}}^{d};{\mathbb{R}}^{d})$ by (2.2). In view of (2.10), for
trajectories ${\boldsymbol{\eta}}$ coming from the WASEP dynamics, this
definition corresponds to the original one, given in (2.7).

As before, ${\boldsymbol{J}}_{N}({\boldsymbol{\eta}})(0)=0$ and for $t<0$

$\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})(t),F\rangle=-\frac{1}{2N^{d}}\sum_{i\colon\tau_{i}\in[t,0)}\sum_{(x,y)\in{\mathbb{B}}_{N}}W_{i}(x,y)F_{N}(x,y).$

In view of the bound (2.9) and noticing that $|F_{N}|\leq\|F\|_{\infty}/N$, if
${\boldsymbol{\eta}}$ has a single jump then
$|\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})(t),F\rangle|\leq
C\|F\|_{\infty}$ for some constant $C$ independent of $N$.

### Sobolev spaces.

Let $h_{n}\in L^{2}({\mathbb{T}}^{d})$,
$n=(n_{1},\dots,n_{d})\in{\mathbb{Z}}^{d}$, be the orthonormal basis of
$L^{2}({\mathbb{T}}^{d})$ given by $\color[rgb]{.2,0.2,.8}h_{n}(x)=\exp\\{2\pi
i(n\cdot x)\\}$.

Denote by $\langle\cdot,\cdot\rangle$ the inner product in
$L^{2}({\mathbb{T}}^{d})$, and by
${\mathfrak{f}}\colon{\mathbb{Z}}^{d}\to{\mathbb{C}}$ the Fourier coefficients
of the function $f$ in $L^{2}({\mathbb{T}}^{d})$:

${\mathfrak{f}}(n):=\langle
f,h_{n}\rangle\;=\;\int_{{\mathbb{T}}^{d}}f(x)\,\overline{h_{n}(x)}\,dx\;,\quad
n\,\in\,{\mathbb{Z}}^{d}\;,$

where $\overline{z}$ represents the complex conjugate of $z\in{\mathbb{C}}$.
Hence,

$f\;=\;\sum_{n\in{\mathbb{Z}}^{d}}{\mathfrak{f}}(n)\,h_{n}\;$

Denote by $\color[rgb]{.2,0.2,.8}{\mathcal{H}}_{p}$, $p\in{\mathbb{R}}$, the
Hilbert space obtained by completing the space of smooth complex-valued
functions on ${\mathbb{T}}^{d}$ endowed with the scalar product
$\langle\,\cdot\,,\,\cdot\,\rangle_{p}$ defined by

$\langle f\,,\,g\rangle_{p}\;=\;\langle(1-\Delta)^{p}f\,,\,g\rangle\;,$ (2.13)

where $\Delta$ represents the Laplacian. An elementary computation yields that

$\langle
f\,,\,g\rangle_{p}\;:=\;\sum_{n\in{\mathbb{Z}}^{d}}\big{(}1+4\pi^{2}|n|^{2}\big{)}^{p}\,{\mathfrak{f}}(n)\,\overline{{\mathfrak{g}}(n)}\;,$
(2.14)

where $|n|^{2}=|(n_{1},\ldots,n_{d})|^{2}=\sum_{1\leq j\leq d}n^{2}_{j}$.
Denote by $\|\,\cdot\,\|_{p}$ the norm of ${\mathcal{H}}_{p}$:
$\color[rgb]{.2,0.2,.8}\|f\|^{2}_{p}=\langle f\,,\,f\rangle_{p}$. It is well
known that ${\mathcal{H}}_{-p}$, is the dual of ${\mathcal{H}}_{p}$ relatively
to the pairing $\langle\,\cdot\,,\,\cdot\rangle$ defined by

$\langle
j\,,\,g\rangle\;:=\;\sum_{n\in{\mathbb{Z}}^{d}}{\mathfrak{j}}(n)\,\overline{{\mathfrak{g}}(n)}\;,\quad
j\,\in\,{\mathcal{H}}_{-p}\;,\,g\,\in\,{\mathcal{H}}_{p}\;.$ (2.15)

Moreover, it follows from the definition that
${\mathcal{H}}_{p}\subset{\mathcal{H}}_{p^{\prime}}$ for $p>p^{\prime}$. Let
$\color[rgb]{.2,0.2,.8}{\mathcal{H}}^{d}_{p}={\mathcal{H}}_{p}\times\cdots\times{\mathcal{H}}_{p}$
that we consider endowed with the strong topology. We represent below by
$\color[rgb]{.2,0.2,.8}\langle J\,,\,H\rangle$ the value at
$H\in{\mathcal{H}}^{d}_{p}$ of a bounded linear functional $J$ defined on
${\mathcal{H}}^{d}_{p}$.

Fix $p>d/2$. By the Sobolev embedding, ${\mathcal{H}}^{d}_{p}\subset
C({\mathbb{T}}^{d};{\mathbb{R}}^{d})$. In particular, by the definition of the
empirical current, for each $t\in{\mathcal{R}}$, the functional
$[{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})]\,(t)$ is bounded on
${\mathcal{H}}^{d}_{p}$. Therefore, for each ${\boldsymbol{\eta}}\in
D({\mathcal{R}};\Sigma_{N,K})$ and $t\in{\mathcal{R}}$ it belongs to dual of
${\mathcal{H}}^{d}_{p}$, that is, to ${\mathcal{H}}^{d}_{-p}$. Furthermore, it
is easy to check that ${\boldsymbol{J}}_{N}$ is right-continuous and has left-
limits. Hence, the empirical current ${\boldsymbol{J}}_{N}$ is a map from
$D({\mathcal{R}},\Sigma_{N,K})$ to $D({\mathcal{R}},{\mathcal{H}}^{d}_{-p})$.

Fix $p\geq 0$ and let $J$ be a bounded linear functional on
${\mathcal{H}}^{d}_{p}$, i.e. $J\in{\mathcal{H}}^{d}_{-p}$. By Riesz
representation theorem, there exists $G=G(J)$ in ${\mathcal{H}}^{d}_{p}$ such
that $J(F)=\langle F\,,\,G\rangle_{p}$ for all $F$ in ${\mathcal{H}}^{d}_{p}$,
and $\|J\|^{2}_{-p}=\langle G\,,\,G\rangle_{p}$. Letting
$G=(G_{1},\ldots,G_{d})$ and defining $J_{k}(n)$ as
$\\{1+4\pi^{2}|n|^{2}\\}^{p}\,{\mathfrak{G}}_{k}(n)$, where
${\mathfrak{G}}_{k}(n)$, $n\in{\mathbb{Z}}^{d}$, are the Fourier coefficients
of $G_{k}$, $k=1,\ldots,d$, by (2.14) we deduce that

$J(F)\;=\;\sum_{k=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}{\mathfrak{F}}_{k}(n)\,\overline{J_{k}(n)}\,,\;\qquad
F=(F_{1},\ldots,F_{d})\;.$

For $n\in{\mathbb{Z}}^{d}$, $j=1,\ldots,d$, let
$H^{j,n}\colon{\mathbb{T}}^{d}\to{\mathbb{C}}^{d}$ be the vector fields given
by $H^{j,n}=(H^{j,n}_{1},\dots,H^{j,n}_{d})$ where
$H^{j,n}_{k}=\delta_{j,k}\,h_{n}$. Taking $F=H^{j,n}$ in the previous
displayed identity, one concludes that $\overline{J_{k}(n)}=J(H^{k,n})$, so
that

$J(F)\;=\;\sum_{k=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}{\mathfrak{F}}_{k}(n)\,J(H^{k,n})\,,\qquad\|J\|^{2}_{-p}\;=\;\sum_{k=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}\frac{\big{|}\,J(H^{k,n})\,\big{|}^{2}}{\big{(}1+4\pi^{2}|n|^{2}\big{)}^{p}}\;\cdot$
(2.16)

### Continuity equation.

It follows from the conservation of mass that for each
$x\in{\mathbb{T}}^{d}_{N}$, $t>0$, and each path ${\boldsymbol{\eta}}$
compatible with the WASEP dynamics

${\boldsymbol{\eta}}_{x}(t)\;-\;{\boldsymbol{\eta}}_{x}(0)\;=\;\sum_{j=1}^{d}\Big{\\{}\,{\mathscr{N}}^{x+{\mathfrak{e}}_{j},x}_{(0,t]}({\boldsymbol{\eta}})\,-\,{\mathscr{N}}^{x,x+{\mathfrak{e}}_{j}}_{(0,t]}({\boldsymbol{\eta}})\,+\,{\mathscr{N}}^{x-{\mathfrak{e}}_{j},x}_{(0,t]}({\boldsymbol{\eta}})\,-\,{\mathscr{N}}^{x,x-{\mathfrak{e}}_{j}}_{(0,t]}({\boldsymbol{\eta}})\,\Big{\\}}\;.$

Let $f$ be a function in $C^{\infty}({\mathbb{T}}^{d})$ and recall the
notation introduced in (2.4). Multiply the previous equation by $f(x)$, sum
over $x\in{\mathbb{T}}^{d}_{N}$ and divide by $N^{d}$ to get, after a
summation by parts, that

$\displaystyle\langle[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(t)\,,\,f\rangle\;-\;\langle[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(0)\,,\,f\rangle$
$\displaystyle\qquad\;=\;\frac{1}{N^{d}}\sum_{j=1}^{d}\sum_{x\in{\mathbb{T}}^{d}_{N}}\big{\\{}\,f(x+{\mathfrak{e}}_{j})-f(x)\,\big{\\}}\,\big{\\{}\,{\mathscr{N}}^{x,x+{\mathfrak{e}}_{j}}_{(0,t]}({\boldsymbol{\eta}})\,-\,{\mathscr{N}}^{x+{\mathfrak{e}}_{j},x}_{(0,t]}({\boldsymbol{\eta}})\,\big{\\}}\;.$

Since $f(x+{\mathfrak{e}}_{j})-f(x)=\int_{x}^{x+{\mathfrak{e}}_{j}}(\nabla
f)\cdot d\ell$, in view of the definition of the map ${\boldsymbol{J}}_{N}$,

$\langle[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(t)\,,\,f\rangle\;-\;\langle[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(0)\,,\,f\rangle\;=\;\langle{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,(t)\,,\,\nabla
f\rangle\;.$ (2.17)

This is the microscopic version of the continuity equation. Observe that for
paths ${\boldsymbol{\eta}}$ not coming from the WASEP dynamics, the definition
of the integrated empirical current ${\boldsymbol{J}}_{N}$ has been engineered
so that (2.17) always holds.

### Hydrodynamical limit

Let $(\eta^{N}:N\geq 1)$, $\eta^{N}\in\Sigma_{N}$, be a sequence of
configurations associated to a density profile
$\rho\colon{\mathbb{T}}^{d}\to[0,1]$ in the sense that for each continuous
function $f\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$,

$\lim_{N\to\infty}\langle\pi_{N}(\eta^{N}),f\rangle\;=\;\lim_{N\to\infty}\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}f(x)\,\eta^{N}_{x}\;=\;\int_{{\mathbb{T}}^{d}}f(x)\,\rho(x)\,dx\;.$

It is proven in [3] that $\big{(}{\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})$
converge in ${\mathbb{P}}^{N}_{\eta^{N}}$-probability, as $N\to\infty$, to
$\big{(}{\boldsymbol{\rho}}(t,\cdot)\,dx\,,\,\int_{0}^{t}\\!ds\,{\boldsymbol{j}}(s,\,\cdot\,)\big{)}_{t\geq
0}$ where $({\boldsymbol{\rho}},{\boldsymbol{j}})$ is the unique weak solution
to the Cauchy problem

$\left\\{\begin{aligned}
&\partial_{t}{\boldsymbol{\rho}}\;+\;\nabla\cdot{\boldsymbol{j}}\;=\;0\;,\\\
&{\boldsymbol{j}}\;=\;-\,\nabla{\boldsymbol{\rho}}\;+\;\sigma({\boldsymbol{\rho}})\,E\;,\\\
&{\boldsymbol{\rho}}(0,\,\cdot\,)\;=\;\rho(\,\cdot\,)\;,\end{aligned}\right.$
(2.18)

in which ${\color[rgb]{.2,0.2,.8}\sigma}\colon[0,1]\to{\mathbb{R}}_{+}$, given
by $\sigma(\rho)=\rho(1-\rho)$, is the mobility of the exclusion process.

If one considers only the empirical density and disregards the empirical
current, the above result has been proven in [10, 16], see also [15, Ch. 10].
The case in which one considers also the empirical current is discussed in [3]
for the SEP. The topology on the set of currents used in [3] is different from
the one employed in the present paper. Actually, the proof of the tightness of
the empirical current in [3] is incomplete but the arguments presented below
in Section 4 fix this issue (in the topology here introduced). In view of the
super-exponential estimates in [16] or [15, Ch. 10], the hydrodynamical limit
extends directly to the WASEP dynamics.

### Empirical process

Given $K=0,\ldots,N^{d}$, $T>0$, and ${\boldsymbol{\eta}}\in
D({\mathbb{R}}_{+},\Sigma_{N,K})$, let ${\boldsymbol{\eta}}^{T}\in
D({\mathbb{R}},\Sigma_{N,K})$ be the $T$-periodization of the trajectory
${\boldsymbol{\eta}}$, defined by

${\boldsymbol{\eta}}^{T}(t)\;=\;{\boldsymbol{\eta}}\Big{(}t-\Big{\lfloor}\frac{t}{T}\Big{\rfloor}\,T\Big{)}\;,$

where $\color[rgb]{.2,0.2,.8}\lfloor a\rfloor$ represents the largest integer
less than or equal to $a\in{\mathbb{R}}$. A probability measure on
$D({\mathbb{R}},\Sigma_{N,K})$ is stationary if it is invariant with respect
to the group of time-translations $(\vartheta_{t}:t\in{\mathbb{R}})$, defined
by

$(\vartheta_{t}{\boldsymbol{\eta}})(s)\;:=\;{\boldsymbol{\eta}}(s-t)\;,\quad
s\,\in\,{\mathbb{R}}\;.$

Denote by $\color[rgb]{.2,0.2,.8}{\mathscr{P}}^{N,K}_{\rm stat}$ the set of
stationary probability measures on $D({\mathbb{R}},\Sigma_{N,K})$ that we
consider endowed with the topology induced by the weak convergence and the
corresponding Borel $\sigma$-algebra. For a trajectory ${\boldsymbol{\eta}}\in
D({\mathbb{R}}_{+},\Sigma_{N,K})$ and $T>0$, the _empirical process_
$R_{T}({\boldsymbol{\eta}})$ is the element in ${\mathscr{P}}^{N,K}_{\rm
stat}$ given by

$R_{T}({\boldsymbol{\eta}})\;:=\;\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{t}{\boldsymbol{\eta}}^{T}}\,dt\;.$
(2.19)

Observe that the path ${\boldsymbol{\eta}}^{T}$ is not necessarily compatible
with the WASEP dynamics; indeed at the times that are integral multiples of
$T$ there is a jump that is not - in general - coming from the WASEP dynamics.
This is the reason for which we needed to define the empirical current for
generic paths. However, in view of the bound (2.9), there exists a finite
constant $C_{0}$, depending only on the space dimension $d$, such that

$\sup_{t\in[0,T]}\Big{|}\,\langle{\boldsymbol{J}}_{N}({{\boldsymbol{\eta}}}^{T})(t)-{\boldsymbol{J}}_{N}({{\boldsymbol{\eta}}})(t)\,,\,F\rangle\,\Big{|}\;\leq\;C_{0}\,\|F\|_{\infty}$
(2.20)

for all vector field $F$ in $C({\mathbb{T}}^{d},{\mathbb{R}}^{d})$. Indeed, by
the definition of ${{\boldsymbol{\eta}}}^{T}$ and by (2.12), the left-hand
side is equal to

$\Big{|}\,\langle{\boldsymbol{J}}_{N}({{\boldsymbol{\eta}}}^{T})(T)-{\boldsymbol{J}}_{N}({{\boldsymbol{\eta}}})(T)\,,\,F\rangle\,\Big{|}\;=\;\Big{|}\,\frac{1}{2N^{d}}\sum_{(x,y)\in{\mathbb{B}}_{N}}W(x,y)\,F_{N}(x,y)\,\Big{|}\;,$

where $W(x,y)=W_{{\boldsymbol{\eta}}(T-),{\boldsymbol{\eta}}(0)}$. Hence, by
(2.9), the right-hand side of the previous identity is bounded by
$C_{0}\|F\|_{\infty}$, as claimed in (2.20).

Fix $p>(d+2)/2$, and let
$\color[rgb]{.2,0.2,.8}{\mathscr{M}}={\mathcal{M}}_{+}({\mathbb{T}}^{d})\times{\mathcal{H}}^{d}_{-p}$.
Denote by $\color[rgb]{.2,0.2,.8}{\mathscr{S}}$ the closed subset of
$D({\mathbb{R}},{\mathscr{M}})\equiv
D\big{(}{\mathbb{R}};{\mathcal{M}}_{+}({\mathbb{T}}^{d}))\times
D\big{(}{\mathbb{R}};{\mathcal{H}}^{d}_{-p})$ given by the pairs
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ which satisfy the continuity equation
in the sense that for each $s<t$ and $f$ in $C^{\infty}({\mathbb{T}}^{d})$,

$\int_{s}^{t}ds_{1}\int_{s_{1}}^{t}ds_{2}\;\Big{\\{}\langle{\boldsymbol{\pi}}(s_{2})\,,\,f\rangle\;-\;\langle{\boldsymbol{\pi}}(s_{1})\,,\,f\rangle\;-\;\langle{\boldsymbol{J}}(s_{2})-{\boldsymbol{J}}(s_{1})\,,\,\nabla
f\rangle\Big{\\}}\;=\;0\;.$ (2.21)

We consider ${\mathscr{S}}$ endowed with the relative topology and the
associated Borel $\sigma$-algebra. Denote by
$\vartheta_{t}\colon{\mathscr{S}}\to{\mathscr{S}}$, $t\in{\mathbb{R}}$, the
time-translation defined by

$\vartheta_{t}({\boldsymbol{\pi}},{\boldsymbol{J}})\;=\;(\vartheta_{t}{\boldsymbol{\pi}},\vartheta_{t}{\boldsymbol{J}})\;,$
(2.22)

where $(\vartheta_{t}{\boldsymbol{\pi}})(s)={\boldsymbol{\pi}}(s-t)$, and
$(\vartheta_{t}{\boldsymbol{J}})(s)={\boldsymbol{J}}(s-t)-{\boldsymbol{J}}(-t)$,
$s\in{\mathbb{R}}$. Note that the time translations defined on
$D({\mathbb{R}};\Sigma_{N,K})$ and ${\mathscr{S}}$ are compatible in the sense
that
$\vartheta_{t}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})=({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\circ\vartheta_{t}$.

Given a path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in ${\mathscr{S}}$ and
$T>0$, denote by $({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})$ its
$T$-periodization:

$\begin{split}&{\boldsymbol{\pi}}^{T}(t)\;=\;{\boldsymbol{\pi}}\Big{(}t-\Big{\lfloor}\frac{t}{T}\Big{\rfloor}\Big{)}\;,\\\
&{\boldsymbol{J}}^{T}(t)\;=\;{\boldsymbol{J}}\big{(}t-\big{\lfloor}\tfrac{t}{T}\big{\rfloor}\big{)}\;+\;\big{\lfloor}\tfrac{t}{T}\big{\rfloor}\big{(}{\boldsymbol{J}}(T)+A\big{)}\end{split}$
(2.23)

where $A$ is an element of ${\mathcal{H}}^{d}_{-p}$ satisfying $\nabla\cdot
A\;+\;{\boldsymbol{\pi}}(T)\,-\,{\boldsymbol{\pi}}(0)=0$. A straightforward
computation shows that $({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})$
satisfies the continuity equation (2.21). We hereafter assume that the choice
of $A$ in the previous definition and of the discrete vector field $W$ in
(2.8) are compatible in the sense that whenever
${\boldsymbol{\pi}}(0)=N^{-d}\sum_{x\in{\mathbb{T}}^{d}_{N}}\eta_{x}$ and
${\boldsymbol{\pi}}(T)=N^{-d}\sum_{x\in{\mathbb{T}}^{d}_{N}}\xi_{x}$ for some
$\eta,\xi\in\Sigma_{N}$ then

$A=\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}\sum_{k=1}^{d}W_{\eta,\xi}(x,x+{\mathfrak{e}}_{k})\,e_{k}\,{\mathfrak{H}}^{1}_{\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits[x,x+{\mathfrak{e}}_{k}]}$

where ${\mathfrak{H}}^{1}_{\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits[x,x+{\mathfrak{e}}_{k}]}$ is
the one-dimensional Hausdorff measure of the interval
$[x,x+{\mathfrak{e}}_{k}]$. This compatibility implies that
$(\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}}^{T})\,,\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}}^{T})\,)=({\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})^{T},{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})^{T})$,
for each $\eta\in D({\mathbb{R}}_{+};\Sigma_{N,K})$.

Given ${\boldsymbol{\eta}}\in D({\mathbb{R}}_{+};\Sigma_{N,K})$, let finally
${\mathfrak{R}}_{T,N}({\boldsymbol{\eta}})$ be the stationary probability
(with respect to the map $\vartheta_{t}$ defined above) on ${\mathscr{S}}$
given by

${\mathfrak{R}}_{T,N}({\boldsymbol{\eta}}):=\frac{1}{T}\int_{0}^{T}\delta_{\big{(}{\boldsymbol{\pi}}_{N}(\vartheta_{t}{\boldsymbol{\eta}}^{T})\,,\,{\boldsymbol{J}}_{N}(\vartheta_{t}{\boldsymbol{\eta}}^{T})\big{)}}\,dt=\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{t}\big{(}{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})^{T}\,,\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})^{T}\big{)}}\,dt$
(2.24)

and observe that, according to definitions (2.5), (2.7), (2.19), we have

${\mathfrak{R}}_{T,N}=R_{T}\circ\big{(}{\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N}\big{)}^{-1}\;.$

### Large deviations asymptotic

Our main result establishes the large deviations principle for
${\mathfrak{R}}_{T,N}({\boldsymbol{\eta}})$ in the joint limit $T\to\infty$
and $N\to\infty$ when ${\boldsymbol{\eta}}$ is sampled according to the WASEP
dynamics. We prove this result both when $T\to\infty$ before $N\to\infty$ and
when $N\to\infty$ before $T\to\infty$. The corresponding rate function is
independent of the limiting procedure.

The statement of this result requires further notation. Denote by
$\color[rgb]{.2,0.2,.8}{\mathscr{S}}_{m}$, $m\in(0,1)$, the closed set of
trajectories $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in ${\mathscr{S}}$ such
that ${\boldsymbol{\pi}}(t,{\mathbb{T}}^{d})=m$ for all $t\in{\mathbb{R}}$.
Recalling that $\sigma(\rho)=\rho(1-\rho)$ is the mobility of the exclusion
process, let finally $\color[rgb]{.2,0.2,.8}{\mathscr{S}}_{m,{\rm ac}}$ be the
subset of elements $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in
${\mathscr{S}}_{m}$ such that

  * (a)

${\boldsymbol{\pi}}\in C({\mathbb{R}},{\mathcal{M}}_{m}({\mathbb{T}}^{d}))$,
${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$ for some
${\boldsymbol{\rho}}$ such that $0\leq{\boldsymbol{\rho}}(t,x)\leq 1$, and,
for any $T>0$,

$\int_{-T}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|\,\nabla{\boldsymbol{\rho}}\,|^{2}}{\sigma({\boldsymbol{\rho}})}\><\;\infty\;;$

  * (b)

${\boldsymbol{J}}\in C({\mathbb{R}},{\mathcal{H}}^{d}_{-p})$, and
${\boldsymbol{J}}(t)\,=\,\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$,
$t\in{\mathbb{R}}$, for some ${\boldsymbol{j}}$ in $L^{2}_{\rm
loc}({\mathbb{R}}\times{\mathbb{T}}^{d},\sigma({\boldsymbol{\rho}}(t,x))^{-1}\,dt\,dx;{\mathbb{R}}^{d})$.
Thus, for any $T>0$

$\int_{-T}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|\,{\boldsymbol{j}}\,|^{2}}{\sigma({\boldsymbol{\rho}})}\><\;\infty\;.$

Let the _action_ $A_{m,T}\colon{\mathscr{S}}\to[0,+\infty]$, $m\in(0,1)$,
$T>0$, be defined by

$A_{m,T}({\boldsymbol{\pi}},{\boldsymbol{J}})=\begin{cases}\displaystyle\int_{0}^{T}dt\,\int_{{\mathbb{T}}^{d}}dx\,\frac{|{\boldsymbol{j}}+\nabla{\boldsymbol{\rho}}-\sigma({\boldsymbol{\rho}})\,E|^{2}}{4\,\sigma({\boldsymbol{\rho}})}&\text{
if $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m,{\rm ac}}$\;,
}\\\ \\\ +\infty&\text{ otherwise.}\end{cases}$ (2.25)

By the arguments presented in [6, § 4], the functional $A_{m,T}$ is lower
semicontinuous. Note that if ${\boldsymbol{J}}$ has a density
${\boldsymbol{j}}$ as in item (b) above, then the density of
$\vartheta_{t}{\boldsymbol{J}}$ is given by
$(\vartheta_{t}{\boldsymbol{j}})(s):={\boldsymbol{j}}(s-t)$.

For $m\in(0,1)$, let $\color[rgb]{.2,0.2,.8}{\mathscr{P}}_{{\rm stat}}$,
$\color[rgb]{.2,0.2,.8}{\mathscr{P}}_{{\rm stat},m}$, be the set of
translation invariant probability measures on ${\mathscr{S}}$,
${\mathscr{S}}_{m}$, respectively. We consider ${\mathscr{P}}_{{\rm stat}}$
and ${\mathscr{P}}_{{\rm stat},m}$ endowed with the topology induced by the
weak convergence and the corresponding Borel $\sigma$-algebra. Let
${\boldsymbol{I}}_{m}\colon{\mathscr{P}}_{{\rm stat}}\to[0,+\infty]$ be the
functional defined by

${\boldsymbol{I}}_{m}(P)\;:=\;\frac{1}{T}\,E_{P}\big{[}\,A_{m,T}\,\big{]}\;,$
(2.26)

observing that the right-hand side does not depend on $T>0$ by stationarity.
Moreover, by using the continuity equation (2.21), the identity
$\nabla{\boldsymbol{\rho}}/\sigma({\boldsymbol{\rho}})=\nabla
h^{\prime}({\boldsymbol{\rho}})$, where
$h(\rho):=\rho\log\rho+(1-\rho)\log(1-\rho)$ is the Bernoulli entropy, and the
stationarity of $P$, integrating by parts we deduce that if
${\boldsymbol{I}}_{m}(P)<+\infty$ then for any $T>0$ (equivalently for some
$T>0$)

$\frac{1}{T}\,E_{P}\bigg{[}\int_{-T}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{(}\frac{|\,\nabla{\boldsymbol{\rho}}\,|^{2}}{4\,\sigma({\boldsymbol{\rho}})}+\frac{|\,{\boldsymbol{j}}\,|^{2}}{4\,\sigma({\boldsymbol{\rho}})}\Big{)}\bigg{]}<+\infty\;.$
(2.27)

In the next statements, by $\limsup_{T,N}$, we mean either
$\limsup_{N}\limsup_{T}$ or $\limsup_{T}\limsup_{N}$. Analogously,
$\liminf_{T,N}$ stands for either $\liminf_{N}\liminf_{T}$ or
$\liminf_{T}\liminf_{N}$.

######  Theorem 2.1.

Fix $m\in(0,1)$, $p>(d+2)/2$, and a sequence $K_{N}$ such that $K_{N}/N^{d}\to
m$. For each closed subset ${\mathscr{F}}$ of ${\mathscr{P}}_{\rm stat}$

$\limsup_{N,T\to\infty}\;\sup_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\,\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}{\mathfrak{R}}_{T,N}\in{\mathscr{F}}\,]\;\leq\;-\,\inf_{P\in{\mathscr{F}}}{\boldsymbol{I}}_{m}(P)\;.$

Moreover, if $E$ is orthogonally decomposable, then, for each open subset
${\mathscr{G}}$ of ${\mathscr{P}}_{\rm stat}$,

$\liminf_{N,T\to\infty}\;\inf_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\,\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}{\mathfrak{R}}_{T.N}\in{\mathscr{G}}\,]\;\geq\;-\,\inf_{P\in{\mathscr{G}}}{\boldsymbol{I}}_{m}(P)\;.$

Finally, the functional ${\boldsymbol{I}}_{m}\colon{\mathscr{P}}_{\rm
stat}\to[0,+\infty]$ is good and affine.

In the lower bound, the technical condition that the external field $E$ is
orthogonally decomposable is only used to guarantee that the quasi-potential
of the WASEP is bounded, an ingredient that enters in the proof of Lemma 4.12.
In the common terminology of large deviations, the previous statement is
analogous to a level-three large deviation principle, for which the rate
function has an explicit expression. The contraction principle permits to
derive from this result large deviations principles for relevant observables.

### Level two large deviations

Let $\color[rgb]{.2,0.2,.8}{\mathscr{P}}({\mathcal{M}}_{+}({\mathbb{T}}^{d}))$
be the space of probability measures on ${\mathcal{M}}_{+}({\mathbb{T}}^{d})$
endowed with the weak topology. Recalling (2.5), define $\wp_{T,N}$ as the map
from $D\big{(}{\mathbb{R}}_{+};\Sigma_{N,K}\big{)}$ to
${\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}$ by

$\wp_{T,N}({\boldsymbol{\eta}}):=\frac{1}{T}\int_{0}^{T}\\!dt\,\delta_{\pi_{N}({\boldsymbol{\eta}}(t))}\,,$

i.e. $\wp_{T,N}$ is the time empirical measure associated to the path
$\pi_{N}({\boldsymbol{\eta}})$. Letting
$\imath_{t}\colon{\mathscr{S}}\to{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ be the
map $({\boldsymbol{\pi}},{\boldsymbol{J}})\mapsto{\boldsymbol{\pi}}(t)$, then
${\mathfrak{R}}_{T,N}\circ\imath_{t}^{-1}=\wp_{T,N}$. Denote by
${\mathscr{I}}_{m}\colon{\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}\to[0,+\infty]$,
$m\in(0,1)$, the functional given by

${\mathscr{I}}_{m}(\wp)=\inf\big{\\{}{\boldsymbol{I}}_{m}(P):P\in{\mathscr{P}}_{{\rm
stat}}\,,\,P\circ\imath_{t}^{-1}\,=\wp\big{\\}}\,.$ (2.28)

Given $m\in(0,1)$, let $(\Phi^{m}_{t}:t\geq 0)$ be the flow induced by the
hydrodynamic equation (2.18) on the set of densities with total mass equal to
$m$. Namely, when $\int\rho\,dx=m$ we set
$\Phi^{m}_{t}(\rho)={\boldsymbol{\rho}}(t)$ where
$({\boldsymbol{\rho}},{\boldsymbol{j}})$ is the unique weak solution to
(2.18). By identifying measures absolutely continuous with respect to $dx$
with their densities, we regard $(\Phi^{m}_{t}:t\geq 0)$ as a flow on
${\mathcal{M}}_{m}({\mathbb{T}}^{d})$. The following result is obtained from
Theorem 2.1 by the contraction principle and implies the _hydrostatic limit_ :
in the limit $N\to\infty$ the empirical density constructed by sampling the
particles according to the stationary measure $\mu_{N,K}$ converges to the
unique stationary solution to the hydrodynamic equation.

######  Corollary 2.2.

Fix $m\in(0,1)$ and a sequence $K_{N}$ such that $K_{N}/N^{d}\to m$. For each
closed subset ${\mathscr{F}}$ of
${\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}$

$\limsup_{N,T\to\infty}\;\sup_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\,\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}\wp_{T,N}\in{\mathscr{F}}\,]\;\leq\;-\,\inf_{\wp\in{\mathscr{F}}}{\mathscr{I}}_{m}(\wp)\;.$

If $E$ is orthogonally decomposable, then for each open subset ${\mathscr{G}}$
of ${\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}$

$\liminf_{N,T\to\infty}\;\inf_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\,\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}\wp_{T,N}\in{\mathscr{G}}\,]\;\geq\;-\,\inf_{\wp\in{\mathscr{G}}}{\mathscr{I}}_{m}(\wp)\;.$

Finally, the functional
${\mathscr{I}}_{m}\colon{\mathscr{P}}\big{(}{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}\to[0,+\infty]$
is good, convex, and vanishes only on the invariant probabilities for the flow
$\Phi^{m}$. In particular, if $E$ is orthogonally decomposable, then
${\mathscr{I}}_{m}(\wp)=0$ if and only if $\wp=\delta_{\bar{\rho}\,dx}$ where
$\bar{\rho}$ is the unique stationary solution to the hydrodynamic equation
with mass $m$.

### Level one large deviations

For $T>0$ the _time-averaged empirical density_ is the map
$\pi_{T,N}:D({\mathbb{R}}_{+},\Sigma_{N})\to{\mathcal{M}}_{+}({\mathbb{T}}^{d})$
defined by

$\pi_{T,N}({\boldsymbol{\eta}})\;:=\;\frac{1}{T}\int_{0}^{T}[\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})\,]\,(t)\,dt\;.$
(2.29)

Likewise, for $p>d/2$, the _time-averaged empirical current_ is the map
$J_{T,N}$ from $D({\mathbb{R}}_{+},\Sigma_{N,K})$ to ${\mathcal{H}}^{d}_{-p}$
defined by

$J_{T,N}({\boldsymbol{\eta}})\;=\;\frac{1}{T}\,[\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\,]\,(T)\;,$
(2.30)

which can also be written as

$\langle
J_{T,N}({\boldsymbol{\eta}})\,,\,F\rangle\;=\;\frac{1}{T}\frac{1}{N^{d}}\sum_{(x,y)\in{\mathbb{B}}_{N}}{\mathscr{N}}^{x,y}_{(0,T]}({\boldsymbol{\eta}})\int_{x}^{y}F\cdot
d\ell\;,\quad F\,\in\,C^{\infty}({\mathbb{T}}^{d};{\mathbb{R}}^{d})\;.$

Note that

$(\pi_{T,N},J_{T,N})=\int\\!d{\mathfrak{R}}_{T,N}\,\Big{(}{\boldsymbol{\pi}}(t),\frac{1}{t}{\boldsymbol{J}}(t)\Big{)}+\frac{1}{T}\big{(}0,{\mathcal{E}}_{T,N}\big{)}$
(2.31)

where the first term on right hand side does not depend on $t\neq 0$ by the
stationarity of ${\mathfrak{R}}_{T,N}$ and

${\mathcal{E}}_{T,N}({\boldsymbol{\eta}})={\boldsymbol{J}}_{N}({\boldsymbol{\eta}}^{T}(T))-{\boldsymbol{J}}_{N}({\boldsymbol{\eta}}(T))\;.$

By (2.20) and the Sobolev embedding, for $p>d/2$ we deduce that
$\|{\mathcal{E}}_{T,N}({\boldsymbol{\eta}})\|_{-p}$ is bounded uniformly in
${\boldsymbol{\eta}}$, $T$, and $N$. Hence the second term on the right hand
side of (2.31) is irrelevant for large deviations in the asymptotics
$T\to\infty$.

Let $I_{m}\colon{\mathscr{M}}\to[0,+\infty]$, $m\in(0,1)$, be the functional
defined by

$I_{m}(\pi,J)\;:=\;\inf\big{\\{}\,{\boldsymbol{I}}_{m}(P):P\in{\mathscr{P}}_{{\rm
stat}}\,,\,E_{P}[\,{\boldsymbol{\pi}}(t)\,]\,=\,\pi\,,\,E_{P}[\,{\boldsymbol{J}}(\,t\,)\,]\,=\,tJ\,\\}\;,$

which does not depend on $t\neq 0$. If the vector field $J$ is not divergence
free, the set on the right-hand side is empty. Indeed, by stationarity and the
continuity equation (2.21), if the above constraints are satisfied, we deduce
that for each smooth function $f$ on ${\mathbb{T}}^{d}$ and $t>0$

$0\;=\;E_{P}\Big{[}\int_{0}^{t}\langle{\boldsymbol{\pi}}(s)-{\boldsymbol{\pi}}(0)\,,\,f\rangle\,ds\,\big{]}\;=\;E_{P}\big{[}\int_{0}^{t}\langle{\boldsymbol{J}}(s)\,,\,\nabla
f\rangle\,ds\,\Big{]}\;=\;\frac{t^{2}}{2}\,\langle J\,,\,\nabla f\rangle\;.$

By the contraction principle, Theorem 2.1 implies the following statement.

######  Corollary 2.3.

Fix $m\in(0,1)$, $p>(d+2)/2$, and a sequence $K_{N}$ so that $K_{N}/N^{d}\to
m$. For each closed subset ${\mathscr{F}}$ of ${\mathscr{M}}$

$\limsup_{T,N\to\infty}\;\sup_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}(\pi_{T,N},J_{T,N})\in{\mathscr{F}}\,\big{]}\;\leq\;-\,\inf_{(\pi,J)\in{\mathscr{F}}}I_{m}(\pi,J)\;.$

Moreover, if $E$ is orthogonally decomposable, then for each open subset
${\mathscr{G}}$ of ${\mathscr{M}}$,

$\liminf_{T,N\to\infty}\;\inf_{\eta\in\Sigma_{N,K_{N}}}\frac{1}{N^{d}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}(\pi_{T,N},J_{T,N})\in{\mathscr{G}}\,\big{]}\;\geq\;-\,\inf_{(\pi,J)\in{\mathscr{G}}}I_{m}(\pi,J)\;.$

Finally, the functional $I_{m}\colon{\mathscr{M}}\to[0,+\infty]$ is good and
convex.

The projections of $I_{m}$ on the two components can be further analyzed and
computed explicitly under additional conditions which are satisfied, for
instance, in the SEP case. Denote by
$I_{m}^{(1)}\colon{\mathcal{M}}_{+}({\mathbb{T}}^{d})\to[0,+\infty]$ the
projection of the functional $I_{m}$ on the density, i.e.,

$I_{m}^{(1)}(\pi)\;=\;\inf\big{\\{}\,{\boldsymbol{I}}_{m}(P):P\in{\mathscr{P}}_{{\rm
stat},m}\,,\,E_{P}[\,{\boldsymbol{\pi}}(\,t\,)\,]\,=\,\pi\,\\}\;.$ (2.32)

It turns that when the external field $E$ is a gradient, so that the WASEP
dynamics is reversible, then $I_{m}^{(1)}$ can be computed explicitly. Assume
$E=-\nabla U$ for some $U\in C^{2}({\mathbb{T}}^{d})$ and let
${\mathscr{V}}_{m}\colon{\mathcal{M}}_{m}({\mathbb{T}}^{d})\to[0,+\infty]$ be
the functional defined by

${\mathscr{V}}_{m}(\pi):=\begin{cases}\displaystyle{\int_{{\mathbb{T}}^{d}}\\!dx\,\frac{\big{|}\nabla\rho+\sigma(\rho)\nabla
U\big{|}^{2}}{4\,\sigma(\rho)}}&\textrm{ if $\pi({\mathbb{T}}^{d})=m$ and
$\pi(dx)=\rho\,dx$,}\\\ \vphantom{\Big{\\{}}+\infty&\textrm{
otherwise.}\end{cases}$

Let also $\textrm{co}({\mathscr{V}}_{m})$ be the convex hull of
${\mathscr{V}}_{m}$ and observe that, in view of the concavity of $\sigma$, if
$\nabla U=0$ then $\mathrm{co}({\mathscr{V}}_{m})={\mathscr{V}}_{m}$. The
functional ${\mathscr{V}}_{m}$ can be seen as a non-linear version of the
level two Donsker-Varadhan functional for reversible diffusions (sometimes
called Fisher information). Indeed, in the case of independent particles
$\sigma(\rho)=\rho$ and the functional ${\mathscr{V}}_{m}$ reduces to the
Dirichlet form of the square root for the diffusion on ${\mathbb{T}}^{d}$ with
generator $\Delta-\nabla U\cdot\nabla$.

######  Theorem 2.4.

If $E=-\nabla U$, then $I^{(1)}_{m}=\mathrm{co}({\mathscr{V}}_{m})$.

As discussed in the introduction, the projection of $I_{m}$ on the second
component is related to the possible occurrence of dynamical phase transitions
for the current. For $p>(d+2)/2$, $m\in(0,1)$, denote by
$I_{m}^{(2)}\colon{\mathcal{H}}^{d}_{-p}\to[0,+\infty]$ the projection of the
functional $I_{m}$ on the current, i.e.

$I_{m}^{(2)}(J)\;=\;\inf\big{\\{}\,{\boldsymbol{I}}_{m}(P):P\in{\mathscr{P}}_{{\rm
stat},m}\,,\,E_{P}[\,{\boldsymbol{J}}(\,t\,)\,]\,=\,t\,J\,\\}\;,$ (2.33)

that corresponds to the Varadhan’s proposal informally presented in (1.4). By
the contraction principle, Corollary 2.3 implies that the time-averaged
empirical current $J_{T,N}$ satisfies a large deviation principle with rate
function $I_{m}^{(2)}$.

It has been pointed out in [2, 3, 8] that the variational problem (2.33) has a
non-trivial solution when $E$ is constant and large enough. Such behavior is
interpreted as a dynamical phase transition. Strictly speaking, the problem
(2.33) is not really considered in [2, 3, 8], but the analysis performed there
implies the results summarized in the next statement. We restrict to the one-
dimensional case with constant external field. Since $I_{m}^{(2)}(J)<+\infty$
implies $\nabla\cdot J=0$, in the one-dimensional case, $I^{(2)}_{m}$ is
finite only if $J(x)=j$ for some constant $j\in{\mathbb{R}}$.

######  Theorem 2.5.

Let $d=1$, $m\in(0,1)$, and $E\geq 0$ be constant.

  * (i)

There exists $E_{0}>0$ such that if $E\leq E_{0}$ then for $J\,=\,j$,
$j\in{\mathbb{R}}$,

$I^{(2)}_{m}(J)\;=\;\frac{(j-\sigma(m)E)^{2}}{4\,\sigma(m)}\;\cdot$

The optimal $P$ for the variational problem (2.33) is $\delta_{(m,j)}$.

  * (ii)

There exists $E_{1}>E_{0}$ such that if $E\geq E_{1}$ then for $J=j$,
$j\in{\mathbb{R}}$, with $j$ large enough

$I^{(2)}_{m}(J)\;<\;\frac{(j-\sigma(m)E)^{2}}{4\,\sigma(m)}\;\cdot$

Furthermore, taking the time average of a probability concentrated on a
traveling wave provides a measure $P$ in ${\mathscr{P}}_{{\rm stat},m}$ such
that $E_{P}[\,{\boldsymbol{J}}(\,t\,)\,]\,=\,t\,J$,
${\boldsymbol{I}}_{m}(P)<{\boldsymbol{I}}_{m}(\delta_{(m,j)})$.

Regarding the higher dimensional case, we mention that the argument in [3,
Prop. 5.1] implies that in the SEP case ($E=0$) for $J$ with vanishing
divergence we have

$I^{(1)}_{m}(J)=\inf_{\rho}\int_{{\mathbb{T}}^{d}}\\!\frac{|J+\nabla\rho|^{2}}{4\sigma(\rho)}\,dx\;,$

where the infimum is carried out over the density profiles $\rho$ of mass $m$.
In other words, the infimum in (2.33) is achieved for a probability measure of
the form $P=\delta_{(\rho\,dx\,,\,J)}$ and no dynamical phase transition
occurs.

##  3\. Donsker-Varadhan large deviations principle

In this section, we recall the Donsker-Varadhan large deviations principle in
the context of the WASEP dynamics with fixed number of particles.

Recall from (2.19) the definition of the empirical process
$R_{T}({\boldsymbol{\eta}})$. Referring to [22] for equivalent
characterizations, we introduce the rate functional for the family
$(R_{T}:T>0)$ by a variational representation that will be most useful for our
purposes. For $t>0$, let $H_{N,K}(t,\,\cdot\,):{\mathscr{P}}^{N,K}_{\rm
stat}\to[0,+\infty]$ be the functional given by

$H_{N,K}(t,{\mathbb{Q}})\;=\;\sup_{\Phi}\int\\!d{\mathbb{Q}}({\boldsymbol{\eta}})\Big{[}\Phi({\boldsymbol{\eta}})\,-\,\log{\mathbb{E}}_{{\boldsymbol{\eta}}(0)}^{N}\big{(}e^{\Phi}\big{)}\Big{]}\;,$

where the supremum is carried over the bounded and continuous functions $\Phi$
on $D({\mathbb{R}},\Sigma_{N,K})$ that are measurable with respect to
$\sigma\\{{\boldsymbol{\eta}}(s),\>s\in[0,t]\big{\\}}$. Let
$H_{N,K}\colon{\mathscr{P}}^{N,K}_{\rm stat}\to[0,+\infty]$ be the functional
defined by

$H_{N,K}({\mathbb{Q}})\;:=\;\sup_{t>0}\frac{1}{t}\,H_{N,K}(t,{\mathbb{Q}})\;=\;\lim_{t\to\infty}\frac{1}{t}\,H_{N,K}(t,{\mathbb{Q}})\;,$
(3.1)

where the second identity follows from the inequality before [22, Theorem
10.9]. By [22, Theorems 10.6 and 10.8], the functional $H_{N,K}$ is good and
affine.

The classical Donsker-Varadhan theorem, see [11] or Theorems 11.6 and 12.5 in
[22], states that, uniformly on the initial configuration
$\eta\in\Sigma_{N,K}$, the family of probability measures
$({\mathbb{P}}^{N}_{\eta}\circ R_{T}^{-1}:T>0)$ satisfies a large deviations
principle with rate function $H_{N,K}$.

######  Theorem 3.1.

Fix $N$ and $K$. For each closed set ${\mathscr{F}}$ and each open set
${\mathscr{G}}$ in ${\mathscr{P}}^{N,K}_{\rm stat}$,

$\displaystyle\limsup_{T\to\infty}\sup_{\eta\in\Sigma_{N,K}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}R_{T}\in{\mathscr{F}}\,\big{]}\;\leq\;-\,\inf_{{\mathbb{Q}}\in{\mathscr{F}}}H_{N,K}({\mathbb{Q}})\;,$
$\displaystyle\liminf_{T\to\infty}\inf_{\eta\in\Sigma_{N,K}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}R_{T}\in{\mathscr{G}}\,\big{]}\;\geq\;-\,\inf_{{\mathbb{Q}}\in{\mathscr{G}}}H_{N,K}({\mathbb{Q}})\;.$

The rate function $H_{N,K}({\mathbb{Q}})$ can also be understood as the
relative entropy _per unit of time_ of the stationary probability
${\mathbb{Q}}$ with respect to the stationary process
${\mathbb{P}}^{N}_{\mu_{N,K}}$. Given $T_{0}<T_{1}$, denote by
$i_{T_{0},T_{1}}\colon D({\mathbb{R}},\Sigma_{N,K})\to
D([T_{0},T_{1}],\Sigma_{N,K})$ the canonical projection. Given two probability
measures ${\mathbb{Q}}^{1},{\mathbb{Q}}^{2}$ on
$D({\mathbb{R}},\Sigma_{N,K})$, let ${\mathbb{H}}_{[T_{0},T_{1}]}$ be the
relative entropy between the marginal of ${\mathbb{Q}}^{1}$ relative to the
time interval $[T_{0},T_{1}]$ and the marginal of ${\mathbb{Q}}^{2}$ on the
same interval:

${\mathbb{H}}_{[T_{0},T_{1}]}({\mathbb{Q}}^{1}|{\mathbb{Q}}^{2})\;=\;\textrm{Ent}\big{(}{\mathbb{Q}}^{1}_{[T_{0},T_{1}]}\big{|}{\mathbb{Q}}^{2}_{[T_{0},T_{1}]}\big{)}:=\int\log\frac{d\,{\mathbb{Q}}^{1}_{[T_{0},T_{1}]}}{d\,{\mathbb{Q}}^{2}_{[T_{0},T_{1}]}}\;d\,{\mathbb{Q}}^{1}_{[T_{0},T_{1}]}\;,$
(3.2)

where ${\mathbb{Q}}^{j}_{[T_{0},T_{1}]}={\mathbb{Q}}^{j}\circ
i_{T_{0},T_{1}}^{-1}$, $j=1$, $2$. We also shorthand ${\mathbb{H}}_{[0,T]}$ by
${\mathbb{H}}^{(T)}$. By [13, Theorem 5.4.27], for each ${\mathbb{Q}}$ in
${\mathscr{P}}^{N,K}_{\rm stat}$

$H_{N,K}({\mathbb{Q}})\;=\;\lim_{T\to\infty}\frac{1}{T}\,{\mathbb{H}}^{(T)}\big{(}\,{\mathbb{Q}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K}}\,\big{)}\;=\;\sup_{T>0}\frac{1}{T}\,{\mathbb{H}}^{(T)}\big{(}\,{\mathbb{Q}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K}}\,\big{)}\;,$
(3.3)

where the second identity follows by a super-additivity argument which stems
from [22, Lemma 10.3]. Actually, Theorem 5.4.27 in [13] states that the
empirical process $(R_{T}:T>0)$ satisfies a large deviations principle with
good rate function given by
$H^{\star}_{N,K}({\mathbb{Q}}):=\lim_{T\to\infty}T^{-1}\,{\mathbb{H}}^{(T)}(\,{\mathbb{Q}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K}}\,)$.
Since, by Theorem 3.1, $(R_{T}:T>0)$ also satisfies a large deviations
principle with good rate function given by $H_{N,K}({\mathbb{Q}})$, a simple
argument using the lower semi-continuity of the functionals yields that
$H_{N,K}({\mathbb{Q}})=H^{\star}_{N,K}({\mathbb{Q}})$ for all
${\mathbb{Q}}\in{\mathscr{P}}^{N,K}_{\rm stat}$.

Recall that we denote by ${\mathscr{S}}$ the set of trajectories
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ which satisfy the continuity equation
(2.21). By (2.17), the map
$({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\colon
D({\mathbb{R}},\Sigma_{N,K})\to D({\mathbb{R}},{\mathscr{M}})$ takes values in
${\mathscr{S}}$. As already observed,
$({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})(\vartheta_{t}{\boldsymbol{\eta}})=[\vartheta_{t}({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})]({\boldsymbol{\eta}})$
so that $({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})$ induces a map from the
stationary probabilities on $D({\mathbb{R}},\Sigma_{N,K})$ to the stationary
probabilities on ${\mathscr{S}}$. More precisely, if
${\mathbb{P}}\in{\mathscr{P}}_{\rm stat}^{N,K}$, then
${\mathbb{P}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}$ belongs
to ${\mathscr{P}}_{\rm stat}$. Let ${\boldsymbol{I}}_{N,K}:{\mathscr{P}}_{\rm
stat}\to[0,\infty]$ be defined by

${\boldsymbol{I}}_{N,K}(P)\;=\;\inf\Big{\\{}H_{N,K}({\mathbb{P}})\colon\;{\mathbb{P}}\in{\mathscr{P}}^{N,K}_{\rm
stat},\;{\mathbb{P}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}=P\Big{\\}}\;.$
(3.4)

Note that the set on the right-hand side is either empty (for example, if the
$P$-measure of the set of piece-wise constant paths is not equal to $1$) or it
is a singleton because the map
$({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\colon
D({\mathbb{R}},\Sigma_{N,K})\to D({\mathbb{R}},{\mathscr{M}})$ is injective.

######  Corollary 3.2.

Fix $N$ and $K=0,\ldots,N^{d}$. The functional
${\boldsymbol{I}}_{N,K}\colon{\mathscr{P}}_{\rm stat}\to[0,+\infty]$ is affine
and good. Moreover, for each closed ${\mathscr{F}}$ and each open
${\mathscr{G}}$ in ${\mathscr{P}}_{\rm stat}$,

$\displaystyle\limsup_{T\to\infty}\sup_{\eta\in\Sigma_{N,K}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}\,{\mathfrak{R}}_{T,N}\in{\mathscr{F}}\,\big{]}\;\leq\;-\,\inf_{P\in{\mathscr{F}}}{\boldsymbol{I}}_{N,K}(P)\;,$
$\displaystyle\liminf_{T\to\infty}\inf_{\eta\in\Sigma_{N,K}}\frac{1}{T}\,\log{\mathbb{P}}^{N}_{\eta}\,\big{[}\,{\mathfrak{R}}_{T,N}\in{\mathscr{G}}\,\big{]}\;\geq\;-\,\inf_{P\in{\mathscr{G}}}{\boldsymbol{I}}_{N,K}(P)\;.$

###### Proof.

It is enough to show that the map ${\mathscr{P}}^{N,K}_{\rm
stat}\ni{\mathbb{Q}}\mapsto{\mathbb{Q}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\in{\mathscr{P}}_{\rm
stat}$ is continuous. The statement then follows from Theorem 3.1 by the
contraction principle.

Since the map ${\boldsymbol{\pi}}_{N}\colon D({\mathbb{R}};\Sigma_{N,K})\to
D({\mathbb{R}};{\mathcal{M}}_{+}({\mathbb{T}}^{d}))$ introduced in (2.5) is
continuous, we directly deduce the continuity of the map
${\mathscr{P}}^{N,K}_{\rm
stat}\ni{\mathbb{P}}\mapsto{\mathbb{P}}\circ{\boldsymbol{\pi}}_{N}^{-1}\in{\mathscr{P}}_{\rm
stat}\big{(}D({\mathbb{R}};{\mathcal{M}}_{+}({\mathbb{T}}^{d}))\big{)}$. In
contrast, the map
${\boldsymbol{\eta}}\mapsto{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})$ is not
continuous. Indeed, consider the sequence ${\boldsymbol{\eta}}^{(k)}$ in which
${\boldsymbol{\eta}}^{(k)}$ has a unique jump at time $1/k$ from
${\boldsymbol{\eta}}(0)$ to $\sigma^{x,y}{\boldsymbol{\eta}}(0)$ for some
$(x,y)\in{\mathbb{B}}_{N}$. Then, ${\boldsymbol{\eta}}^{(k)}$ converges to the
path ${\boldsymbol{\eta}}$ with a single jump at time $t=0$ but
${\boldsymbol{J}}_{N}({\boldsymbol{\eta}}^{(k)})$ does not converge to
${\boldsymbol{J}}_{N}({\boldsymbol{\eta}})$. In contrast, the map
${\boldsymbol{\eta}}\mapsto{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})$ is
continuous if ${\boldsymbol{\eta}}$ does not have a jump at time $t=0$.
Moreover, if ${\mathbb{Q}}$ is a stationary probability on
$D\big{(}{\mathbb{R}};\Sigma_{N,K}\big{)}$, then the
${\mathbb{Q}}$-probability of the paths ${\boldsymbol{\eta}}$ which have a
jump a time $t=0$ is necessarily zero. This implies that the map
${\mathscr{P}}^{N,K}_{\rm
stat}\ni{\mathbb{Q}}\mapsto{\mathbb{Q}}\circ{\boldsymbol{J}}_{N}^{-1}\in{\mathscr{P}}_{\rm
stat}\big{(}D({\mathbb{R}};{\mathcal{H}}^{d}_{-p})\big{)}$ is continuous. ∎

##  4\. Variational convergence of the Donsker-Varadhan functional.

Referring to [9] for an overview, we recall the definition of
$\Gamma$-convergence. Fix a Polish space ${\mathcal{X}}$ and a sequence
$(U_{n}:n\in{\mathbb{N}})$ of functionals on ${\mathcal{X}}$,
$U_{n}\colon{\mathcal{X}}\to[0,+\infty]$. The sequence $U_{n}$ is _equi-
coercive_ if for each $\ell\geq 0$ there exists a compact subset
${\mathcal{K}}_{\ell}$ of ${\mathcal{X}}$ such that
$\\{x\in{\mathcal{X}}\colon U_{n}(x)\leq\ell\\}\subset{\mathcal{K}}_{\ell}$
for any $n\in{\mathbb{N}}$. The sequence $U_{n}$ _$\Gamma$ -converges_ to the
functional $U\colon{\mathcal{X}}\to[0,+\infty]$, i.e.
$U_{n}\stackrel{{\scriptstyle\Gamma}}{{\longrightarrow}}U$, if and only if the
two following conditions are met:

  * (i)

_$\Gamma$ -liminf._ The functional $U$ is a $\Gamma$-liminf for the sequence
$U_{n}$: For each $x\in{\mathcal{X}}$ and each sequence $x_{n}\to x$, we have
that $\liminf_{n}U_{n}(x_{n})\geq U(x)$.

  * (ii)

_$\Gamma$ -limsup._ The functional $U$ is a $\Gamma$-limsup for the sequence
$U_{n}$: For each $x\in{\mathcal{X}}$ there exists a sequence $x_{n}\to x$
such that $\limsup_{n}U_{n}(x_{n})\leq U(x)$.

Recall the definition of the functionals ${\boldsymbol{I}}_{m}$,
${\boldsymbol{I}}_{N,K}$ introduced in (2.26) and (3.4), respectively. The
main result of this section reads as follows.

######  Theorem 4.1.

Fix $0<m<1$, $p>(d+2)/2$, and a sequence $K_{N}$ so that $K_{N}/N^{d}\to m$.
The sequence $(N^{-d}{\boldsymbol{I}}_{N,K_{N}}:N\geq 1)$ is equi-coercive.
The functional ${\boldsymbol{I}}_{m}$ is a $\Gamma$-liminf for
$N^{-d}{\boldsymbol{I}}_{N,K_{N}}$. If $E$ is orthogonally decomposable, then
the functional ${\boldsymbol{I}}_{m}$ is also a $\Gamma$-limsup for
$N^{-d}{\boldsymbol{I}}_{N,K_{N}}$. Therefore, under this hypothesis on $E$,
$N^{-d}{\boldsymbol{I}}_{N,K_{N}}\stackrel{{\scriptstyle\Gamma}}{{\longrightarrow}}{\boldsymbol{I}}_{m}$.

The proof of this theorem is divided in three parts. In Subsection 4.1, we
prove that the sequence $N^{-d}{\boldsymbol{I}}_{N,K_{N}}$ is equi-coercive.
In Subsection 4.2, that ${\boldsymbol{I}}_{m}$ is a $\Gamma$-liminf, and, in
Subsection 4.3, that ${\boldsymbol{I}}_{m}$ is a $\Gamma$-limsup provided $E$
is decomposable. For the rest of this section, fix $m\in(0,1)$, $p>(d+2)/2$,
and a sequence $K_{N}$ such that $K_{N}/N^{d}\to m$.

4.1 Equi-coercivity. Set
$\color[rgb]{.2,0.2,.8}P_{N}:={\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\circ\big{(}{\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N}\big{)}^{-1}\in{\mathscr{P}}_{{\rm
stat}}$. We first establish the exponential tightness of the sequence
$(P_{N}:N\geq 1)\subset{\mathscr{P}}_{{\rm stat}}$.

######  Proposition 4.2.

There exists a sequence $({\mathscr{K}}_{\ell}:\ell\geq 1)$ of compact subsets
of ${\mathscr{S}}$ such that

$\lim_{\ell\to+\infty}\limsup_{N\to+\infty}\frac{1}{N^{d}}\log
P_{N}\big{(}\,{\mathscr{K}}_{\ell}^{\complement}\,\big{)}\;=\;-\,\infty\;.$

###### Proof.

In view of Ascoli-Arzelà theorem, the compactness of
${\mathcal{M}}_{+}({\mathbb{T}}^{d})$, and the compact embedding
${\mathcal{H}}_{-p}\hookrightarrow{\mathcal{H}}_{-p^{\prime}}$ for
$p^{\prime}>p$, the assertion of this proposition follows from the next three
lemmata. ∎

Let $\color[rgb]{.2,0.2,.8}D_{T,\delta}:=\\{(s,t)\in{\mathbb{R}}^{2}:0\leq
s\leq t\leq T\,,\,|t-s|\leq\delta\\}$.

######  Lemma 4.3.

For each $T>0$, $\varepsilon>0$, and smooth
$g\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$,

$\displaystyle\lim_{\delta\to
0}\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\sup_{(s,t)\in
D_{T,\delta}}\big{|}\,\langle{\boldsymbol{\pi}}_{N}(t)\,-\,{\boldsymbol{\pi}}_{N}(s)\,,\,g\rangle\big{|}\;>\;\varepsilon\Big{]}\;=\;-\infty\;.$

######  Lemma 4.4.

For each $T>0$

$\displaystyle\lim_{A\to\infty}\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq
T}\big{\|}\,{\boldsymbol{J}}_{N}(t)\big{\|}^{2}_{-p}\;>\;A\,\Big{]}\;=\;-\infty\;.$

######  Lemma 4.5.

For each $\varepsilon>0$, $T>0$, and smooth
$H\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$

$\lim_{\delta\to
0}\,\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{(s,t)\in
D_{T,\delta}}\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,-\,{\boldsymbol{J}}_{N}(s)\,,\,H\rangle\,\big{|}\;\;>\;\varepsilon\,\Big{]}\;=\;-\infty\;.$

Lemma 4.3 is a standard result in the large deviations theory of
hydrodynamical limits, see e.g. [15, § 10.4]. Note that this result can be
deduced from Lemma 4.5 by taking $H=\nabla g$ and using the continuity
equation (2.17). On the other hand, the exponential tightness of the empirical
current is stated in [3] but the proof presented there is incomplete. For this
reason, we present below a detailed proof of Lemmata 4.4 and 4.5.

###### Proof of Lemma 4.4.

We use the notation and statements introduced in (2.13)-(2.16), and denote
${\boldsymbol{J}}_{N}(t)({H}^{j,n})$ by
$\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}\rangle$. By (2.16),

$\big{\|}\,{\boldsymbol{J}}_{N}(t)\big{\|}^{2}_{-p}\;=\;\sum_{j=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}\frac{1}{\gamma^{p}_{n}}\;\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}\rangle\,\big{|}^{2}\;,\qquad\gamma_{n}\,:=\,1+4\pi^{2}|n|^{2}\;.$

Let $\beta_{n}=\gamma^{p}_{n}/[c_{p}\,(1+|n|)^{2(p-1)}]$. Here, $c_{p}$ is a
constant such that $d\,\sum_{n}(\beta_{n}/\gamma^{p}_{n})=1$, that is,
$c_{p}=d\sum_{n}(1+|n|)^{-2(p-1)}$. Note that this sum is finite because we
assumed $p>1+(d/2)$. Introducing the supremum inside the sum yields that

$\displaystyle{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq t\leq
T}\sum_{j=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}\frac{1}{\gamma^{p}_{n}}\,\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}\rangle\,\big{|}^{2}\;>\;A\,\Big{]}$
$\displaystyle\quad\leq\;{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\bigcup_{j=1}^{d}\bigcup_{n\in{\mathbb{Z}}^{d}}\Big{\\{}\frac{1}{\beta_{n}}\,\sup_{0\leq
t\leq
T}\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}\rangle\,\big{|}^{2}\;>\;A\,\Big{\\}}\,\Big{]}\;.$

Fix $1\leq j\leq d$, $n\in{\mathbb{Z}}^{d}$ and denote by ${H}^{j,n}_{1}$,
${H}^{j,n}_{-1}$ the real and the imaginary part of ${H}^{j,n}$, respectively.
The previous expression is then bounded by

$\sum_{b=\pm
1}\sum_{j=1}^{d}\sum_{n\in{\mathbb{Z}}^{d}}{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq
T}\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}_{b}\rangle\,\big{|}\;>\;\sqrt{A\,\beta_{n}/2}\,\Big{]}\;.$

We may remove the absolute value from the previous expression at the cost of
an extra factor $2$ in front of the sum and an estimation of ${H}^{j,n}_{b}$
and $-\,{H}^{j,n}_{b}$. We next bound the probability of the event
$\\{\,\sup_{0\leq t\leq
T}\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}_{b}\rangle\,\,>\,\sqrt{A\,\beta_{n}/2}\,\\}$,
the other one being similar.

Recall the notation introduced in (2.2) and let
$B_{x,x+{\mathfrak{e}}_{k}}(\eta)=B_{x,x+{\mathfrak{e}}_{k}}(\eta,{H}^{j,n}_{b})$,
$1\leq k\leq d$, $x\in{\mathbb{T}}^{d}_{N}$, be given by

$\displaystyle B_{x,x+{\mathfrak{e}}_{k}}(\eta)\;$
$\displaystyle=\;N^{2}\,\eta_{x}\,[\,1-\eta_{x+{\mathfrak{e}}_{k}}\,]\,e^{(1/2)\,E_{N}(x,x+{\mathfrak{e}}_{k})}\,\Big{[}e^{{H}^{j,n}_{b,N}(x,x+{\mathfrak{e}}_{k})}\,-\,1\Big{]}$
$\displaystyle+\;N^{2}\,\eta_{x+{\mathfrak{e}}_{k}}\,[\,1-\eta_{x}\,]\,e^{(1/2)\,E_{N}(x+{\mathfrak{e}}_{k},x)}\,\Big{[}e^{{H}^{j,n}_{b,N}(x+{\mathfrak{e}}_{k},x)}\,-\,1\Big{]}\;,$

By [15, Proposition A1.2.6], for each $\eta\in\Sigma_{N}$ the process

$\displaystyle{\mathbb{M}}_{N}(t)\;:=\;\exp\Big{\\{}N^{d}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,{H}^{j,n}_{b}\rangle\;-\;\int_{0}^{t}\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}_{N}^{d}}B_{x,x+e_{k}}({\boldsymbol{\eta}}(s))\;ds\,\Big{\\}}$

is a mean-one ${\mathbb{P}}^{N}_{\eta}$-martingale.

Since $N\big{|}{H}^{j,n}_{b,N}(x,x+{\mathfrak{e}}_{k})\big{|}$ is bounded
uniformly in $b,j,k,x,n,N$, a Taylor expansion yields

$\displaystyle\bigg{|}\,\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}_{N}^{d}}B_{x,x+e_{k}}(\eta)\;-\;N^{2}\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}^{d}_{N}}{H}^{j,n}_{b,N}(x,x+{\mathfrak{e}}_{k})\,(\eta_{x}\,-\,\eta_{x+{\mathfrak{e}}_{k}}\,)\bigg{|}\;\leq\;C_{1}\,N^{d}\;,$

for some constant $C_{1}$ independent of $b,j,n,N$. Summing by parts and using
that $\big{|}\partial_{x_{k}}{H}^{j,n}_{b}(x)\big{|}\leq C_{2}|n|$ for some
constant $C_{2}$ independent of $b,j,k,x,n$, we conclude that

$\Big{|}\,\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}^{d}_{N}}B_{x,x+e_{k}}(\eta,{H}^{j,n}_{b})\,\Big{|}\;\leq\;C_{0}N^{d}(1+|n|)$

for some constant $C_{0}$ independent of $b,j,n,N$.

In view of the previous estimate, adding and subtracting the sum of the time-
integrals of $B_{x,x+e_{k}}$ and taking exponentials, we get that

$\displaystyle{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq t\leq
T}\langle{\boldsymbol{J}}_{N}(t)\,,\,H^{j,n}_{b}\rangle\;>\;\sqrt{A\,\beta_{n}/2}\;\Big{]}$
$\displaystyle\qquad\;\leq\;{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq
T}{\mathbb{M}}_{N}(t)\;>\;e^{\sqrt{A\,\beta_{n}/2}N^{d}-C_{0}TN^{d}(1+|n|)}\,\Big{]}$
$\displaystyle\qquad\;\leq\;e^{-\sqrt{A\,\beta_{n}/2}N^{d}+C_{0}TN^{d}(1+|n|)}\;,$

where we used Doob’s inequality in the last step and the fact that
${\mathbb{M}}_{N}(t)$ is a mean-one martingale.

We have thus shown that

$\displaystyle{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq t\leq
T}\big{\|}{\boldsymbol{J}}_{N}(t)\|^{2}_{{\mathcal{H}}^{d}_{-p}}\;>\;A\,\Big{]}\;\leq\;4\,d\,\sum_{n\in{\mathbb{Z}}^{d}}e^{-N^{d}\big{[}\sqrt{A\,\beta_{n}/2}-C_{0}T(1+|n|)\big{]}}\;.$

By definition of $\beta_{n}$, there exists a positive constant $c_{0}$ such
that $\beta_{n}\geq c_{0}(1+|n|)^{2}$. The statement follows. ∎

###### Proof of Lemma 4.5.

By a standard inclusion of events and the stationarity of
$P^{N}_{\mu_{N,K_{N}}}$, it is enough to prove that for each $\varepsilon>0$
and smooth $H\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$

$\lim_{\delta\to
0}\,\limsup_{N\to\infty}\frac{1}{N^{d}}\log{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq\delta}\big{|}\,\langle{\boldsymbol{J}}_{N}(t)\,,\,H\rangle\big{|}\;\;>\;\varepsilon\,\Big{]}\;=\;-\,\infty\;,$
(4.1)

where we used that ${\boldsymbol{J}}_{N}(0)=0$. As in the proof of the
previous lemma, we can furthermore remove the modulus in the above bound.

Given a smooth vector-valued function
$H\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ and $\ell>0$, let
$B_{x,x+{\mathfrak{e}}_{k}}^{\ell}(\eta)=B_{x,x+{\mathfrak{e}}_{k}}(\eta,\ell
H)$, $x\in{\mathbb{T}}^{d}_{N}$, be given by

$\displaystyle B_{x,x+{\mathfrak{e}}_{k}}^{\ell}(\eta)\;$
$\displaystyle=\;N^{2}\,\eta_{x}\,(\,1-\eta_{x+{\mathfrak{e}}_{k}}\,)\,e^{(1/2)\,E_{N}(x,x+{\mathfrak{e}}_{k})}\,\Big{[}e^{{\ell
H}_{N}(x,x+{\mathfrak{e}}_{k})}\,-\,1\Big{]}$
$\displaystyle+\;N^{2}\,\eta_{x+{\mathfrak{e}}_{k}}\,[\,1-\eta_{x}\,]\,e^{(1/2)\,E_{N}(x+{\mathfrak{e}}_{k},x)}\,\Big{[}e^{{\ell
H}_{N}(x+{\mathfrak{e}}_{k},x)}\,-\,1\Big{]}\;.$

By [15, App. 1, Prop. 2.6], for each $\eta\in\Sigma_{N}$, the process

$\displaystyle{\mathbb{M}}_{N}^{\ell}(t)\;:=\;\exp\Big{\\{}N^{d}\,\ell\,\langle{\boldsymbol{J}}_{N}(t)\,,\,H\rangle\;-\;\int_{0}^{t}\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}_{N}^{d}}B_{x,x+e_{k}}^{\ell}({\boldsymbol{\eta}}(s))\;ds\,\Big{\\}}$

is a mean-one ${\mathbb{P}}^{N}_{\eta}$-martingale.

The same computation of the previous lemma yields that there exists a constant
$C_{1}=C_{1}(H)$ such that

$\Big{|}\,\sum_{k=1}^{d}\sum_{x\in{\mathbb{T}}_{N}^{d}}B_{x,x+{\mathfrak{e}}_{k}}^{\ell}(\eta)\,\Big{|}\;\leq\;C_{1}\,N^{d}e^{C_{1}\ell/N}(1+\ell^{2})$

for all $N$, $\ell$ and $\eta$. Therefore, by Doob’s inequality,

$\begin{split}&{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq\delta}\langle{\boldsymbol{J}}_{N}(t)\,,\,H\rangle\;>\;\varepsilon\,\Big{]}={\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq\delta}N^{d}\,\ell\,\langle{\boldsymbol{J}}_{N}(t)\,,\,H\rangle\;>\;N^{d}\,\ell\,\varepsilon\,\Big{]}\\\
&\quad\leq P^{N}_{\mu_{N,K_{N}}}\Big{[}\,\sup_{0\leq
t\leq\delta}{\mathbb{M}}_{N}^{\ell}(t)\;>\;\exp\big{\\{}N^{d}\big{[}\ell\,\varepsilon-\delta\,C_{1}\,e^{C_{1}\ell/N}(1+\ell^{2})\big{]}\big{\\}}\Big{]}\\\
&\quad\leq\exp\Big{\\{}-N^{d}\big{[}\ell\,\varepsilon-\delta\,C_{1}\,e^{C_{1}\ell/N}(1+\ell^{2})\big{]}\Big{\\}}\;,\end{split}$

which yields (4.1) by taking the limit $\ell\to\infty$ after the limits in $N$
and $\delta$. ∎

###### Proof of Theorem 4.1. Equi-coercivity..

For $\ell\geq 1$, let

${\mathscr{E}}_{\ell}\;:=\;\bigcup_{N\geq 1}\Big{\\{}P\in{\mathscr{P}}_{\rm
stat}\colon\frac{1}{N^{d}}{\boldsymbol{I}}_{N,K_{N}}(P)\;\leq\;\ell\Big{\\}}\;.$

In view of Ascoli-Arzelà theorem, the compactness of
${\mathcal{M}}_{+}({\mathbb{T}}^{d})$, and the compact embedding
${\mathcal{H}}_{-p}\hookrightarrow{\mathcal{H}}_{-p^{\prime}}$ for
$p^{\prime}>p$, to show that the set ${\mathscr{E}}_{\ell}$ is pre-compact, it
is enough to prove that for each $p>(d+2)/2$, $\varepsilon>0$, $T>0$, and
smooth functions $g\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$,
$H\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$,

$\begin{gathered}\lim_{\delta\to
0}\,\sup_{P\in{\mathscr{E}}_{\ell}}P\Big{[}\,\sup_{(s,t)\in
D_{T,\delta}}\,\big{|}\,\langle{\boldsymbol{\pi}}(t)\,-\,{\boldsymbol{\pi}}(s)\,,\,g\rangle\big{|}\;>\;\varepsilon\,\Big{]}\;=\;0\;,\\\
\lim_{A\to\infty}\,\sup_{P\in{\mathscr{E}}_{\ell}}P\Big{[}\sup_{0\leq t\leq
T}\big{\|}\,{\boldsymbol{J}}(t)\,\big{\|}_{-p}\;>\;A\,\Big{]}\;=\;0\;,\\\
\lim_{\delta\to 0}\,\sup_{P\in{\mathscr{E}}_{\ell}}P\Big{[}\sup_{(s,t)\in
D_{T,\delta}}\big{|}\,\langle{\boldsymbol{J}}(t)\,-\,{\boldsymbol{J}}(s)\,,\,H\rangle\big{|}\;>\;\varepsilon\,\Big{]}\;=\;0\;,\end{gathered}$
(4.2)

where $D_{T,\delta}$ has been introduced before the statement of Lemma 4.3.

To prove the first assertion in (4.2), fix $\varepsilon>0$, $T>0$ and a smooth
function $g\colon{\mathbb{T}}^{d}\to{\mathbb{R}}$. For $\delta>0$, let

${\mathscr{B}}\;=\;{\mathscr{B}}^{\delta,\varepsilon}_{g,T}\;:=\;\Big{\\{}({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}\colon\sup_{(s,t)\in
D_{T,\delta}}\,\big{|}\,\langle{\boldsymbol{\pi}}(t)\,-\,{\boldsymbol{\pi}}(s)\,,\,g\rangle\big{|}\;>\;\varepsilon\,\Big{\\}}\;.$

Fix $P\in{\mathscr{E}}_{\ell}$. By definition of the set
${\mathscr{E}}_{\ell}$, there exists $N\geq 1$ such that
${\boldsymbol{I}}_{N,K_{N}}(P)\leq\ell\,N^{d}$. Furthermore, by definition
(3.4) of the rate function ${\boldsymbol{I}}_{N,K_{N}}$,
$P={\mathbb{P}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}$ for
some ${\mathbb{P}}\in{\mathscr{P}}_{\rm stat}^{N,K}$, and
${\boldsymbol{I}}_{N,K_{N}}(P)=H_{N,K_{N}}({\mathbb{P}})$.

Since the set ${\mathscr{B}}$ is measurable with respect to
$\sigma\big{\\{}({\boldsymbol{\pi}}(t),{\boldsymbol{J}}(t)),\,t\in[0,T]\big{\\}}$,
by the definition (3.2) of the relative entropy ${\mathbb{H}}^{(T)}$ and by
the entropy inequality (see e.g. [15, Proposition A1.8.2]),

$P[{\mathscr{B}}]\;=\;{\mathbb{P}}\big{[}\,({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{B}}\,\big{]}\;\leq\;\frac{\log
2\,+\,{\mathbb{H}}^{(T)}({\mathbb{P}}\,|\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}})}{\log\Big{(}1+\big{(}{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\big{[}\,({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{B}}\,\big{]}\big{)}^{-1}\Big{)}}\;\cdot$

By (3.3), and since
$H_{N,K_{N}}({\mathbb{P}})={\boldsymbol{I}}_{N,K_{N}}(P)\leq\ell\,N^{d}$,

$P[{\mathscr{B}}]\;\leq\;\frac{\log
2\,+\,T\,\ell\,N^{d}}{\log\Big{(}1+\big{(}{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\big{[}\,({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{B}}\,\big{]}\big{)}^{-1}\Big{)}}\;\cdot$

This bound is uniform over $P\in{\mathscr{E}}_{\ell}$ provided we take the
supremum over $N\geq 1$ on the right-hand side.

Fix $a>0$, and let $\gamma=(\log 2+T\,\ell)/a$. By Lemma 4.3, there exists
$\delta_{0}=\delta_{0}(T,g,\varepsilon,\gamma)$ and
$N_{0}=N_{0}(T,g,\varepsilon,\gamma)$ such that

${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\big{[}\,({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{B}}^{\delta_{0},\varepsilon}_{g,T}\,\big{]}\;\leq\;e^{-\,\gamma\,N^{d}}$

for all $N\geq N_{0}$. By changing the value of $\delta_{0}$ we may extend
this inequality to all $N\geq 1$. In particular, by definition of $\gamma$,

$\sup_{P\in{\mathscr{E}}_{\ell}}P\big{[}\,{\mathscr{B}}^{\delta_{0},\varepsilon}_{g,T}\,\big{]}\;\leq\;\sup_{N\geq
1}\frac{\log 2\,+\,T\,\ell\,N^{d}}{\gamma\,N^{d}}\;\leq\;a\;.$

As
${\mathscr{B}}^{\delta,\varepsilon}_{g,T}\subset{\mathscr{B}}^{\delta_{0},\varepsilon}_{g,T}$
for $0<\delta\leq\delta_{0}$, the previous inequality holds for all
$0<\delta\leq\delta_{0}$. Since $a>0$ is arbitrary, this proves the first
assertion of (4.2).

The second and third assertions in (4.2) are proven similarly, based on
Lemmata 4.4 and 4.5. ∎

4.2 The $\Gamma$-liminf. Let $(P_{N}\colon N\geq 1)$ be a sequence of
probability measures in ${\mathscr{P}}_{\rm stat}$ such that
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})<\infty$. The following
lemma lists properties of the cluster points of these sequences.

######  Lemma 4.6.

Let $(P_{N}\colon N\geq 1)$ be a sequence of probability measures in
${\mathscr{P}}_{\rm stat}$ such that
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})<+\infty$. Assume that
$P_{N}\to P$ for some $P\in{\mathscr{P}}_{\rm stat}$. Then, $P$-almost surely
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ belongs to
${\mathscr{S}}_{m,\mathrm{ac}}$ and there exists a constant $C_{0}$ such that
for all $T>0$,

$\begin{split}&E_{P}\Big{[}\,\int_{0}^{T}dt\,\int_{{\mathbb{T}}^{d}}dx\,\frac{|{\boldsymbol{j}}(t,x)|^{2}}{\sigma({\boldsymbol{\rho}}(t,x))}\;+\;\int_{0}^{T}dt\,\int_{{\mathbb{T}}^{d}}dx\,\frac{|\nabla{\boldsymbol{\rho}}(t,x)|^{2}}{\sigma({\boldsymbol{\rho}}(t,x))}\,\Big{]}\\\
&\qquad\;\leq\;C_{0}\,(1\,+\,T)\,+\,2\,T\,\liminf_{N}\frac{1}{N^{d}}{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;,\end{split}$

where ${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$ and
${\boldsymbol{J}}(t)\;=\;\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$.

###### Proof.

By passing to a subsequence, we may assume
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})=\lim_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})$.
By (3.4) and (3.3), for every $T>0$,

${\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\geq\;\frac{1}{T}\,{\mathbb{H}}^{(T)}\big{(}\,{\mathbb{Q}}_{N}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\big{)}\;,$

where ${\mathbb{Q}}_{N}$ is the unique stationary probability on
$D({\mathbb{R}},\Sigma_{N,K})$ such that
${\mathbb{Q}}_{N}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}=P_{N}$.
In particular, for every $T>0$,

$\limsup_{N}\frac{1}{N^{d}}\,{\mathbb{H}}^{(T)}\big{(}\,{\mathbb{Q}}_{N}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\big{)}\;\leq\;T\,\lim_{N}\frac{1}{N^{d}}{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;.$
(4.3)

By this bound, the marginal of ${\mathbb{Q}}_{N}$ in the time interval $[0,T]$
is absolutely continuous with respect to the marginal of
${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$ in the same interval. Moreover, for each
continuous function $g\colon{\mathbb{R}}\times{\mathbb{T}}^{d}\to{\mathbb{R}}$
with support on $(0,T)\times{\mathbb{T}}^{d}$, $P_{N}$-almost surely,

$\Big{|}\,\int_{{\mathbb{R}}}dt\int_{{\mathbb{T}}^{d}}{\boldsymbol{\pi}}(t,dx)\,g(t,x)\,\Big{|}\;\leq\;\frac{1}{N^{d}}\sum_{x\in{\mathbb{T}}^{d}_{N}}\int_{{\mathbb{R}}}dt\,\big{|}\,g(t,x)\,\big{|}$

because there is at most one particle per site. Since the left-hand side is a
continuous function of ${\boldsymbol{\pi}}$ in the Skorohod topology, taking
the limit $N\to\infty$ we deduce that $P$-almost surely

$\Big{|}\,\int_{{\mathbb{R}}}dt\int_{{\mathbb{T}}^{d}}{\boldsymbol{\pi}}(t,dx)\,g(t,x)\,\Big{|}\;\leq\;\int_{{\mathbb{R}}}dt\,\int_{{\mathbb{T}}^{d}}\big{|}\,g(t,x)\,\big{|}\,dx\;.$

This implies that $P$-almost surely, for Lebesgue almost all $t$, the measure
${\boldsymbol{\pi}}(t,dx)$ is absolutely continuous with respect to the
Lebesgue measure: ${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$ for
some density ${\boldsymbol{\rho}}$ satisfying $0\leq{\boldsymbol{\rho}}\leq
1$.

On the other hand, as ${\mathbb{Q}}_{N}$ is absolutely continuous with respect
to ${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$,
$P_{N}[{\boldsymbol{\pi}}(t,{\mathbb{T}}^{d})=K_{N}/N^{d}]=1$ for all
$t\in{\mathbb{R}}$. As $K_{N}/N^{d}\to m$ and $P_{N}\to P$,
$P[{\boldsymbol{\pi}}(t,{\mathbb{T}}^{d})=m]=1$ for Lebesgue almost all
$t\in{\mathbb{R}}$.

For a vector field $F$ in
$C^{1}({\mathbb{R}}\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with compact
support in $(0,T)\times{\mathbb{T}}^{d}$, $\varepsilon>0$ and $a>0$, let

$\begin{split}{\mathcal{E}}_{a,\varepsilon}(F,{\boldsymbol{\pi}})&\;=\;\int_{{\mathbb{R}}}\\!dt\,\big{\langle}{\boldsymbol{\pi}}(t)\,,\,\nabla\cdot
F(t)\big{\rangle}\;-\;a\,\int_{{\mathbb{R}}}dt\int_{{\mathbb{T}}^{d}}dx\;\sigma({\boldsymbol{\pi}}^{\varepsilon})\;|F|^{2}\;,\\\
{\mathcal{V}}_{a,\varepsilon}(F,{\boldsymbol{\pi}},{\boldsymbol{J}})&\;=\;{\boldsymbol{J}}(F)\;-\;a\,\int_{{\mathbb{R}}}\\!dt\,\int_{{\mathbb{T}}^{d}}dx\,\sigma({\boldsymbol{\pi}}^{\varepsilon})\,|F|^{2}\;.\end{split}$
(4.4)

where
$\color[rgb]{.2,0.2,.8}({\boldsymbol{\pi}}^{\varepsilon})(t,x)=(2\varepsilon)^{-d}{\boldsymbol{\pi}}(t,[x-\varepsilon,x+\varepsilon]^{d})$
and
${\boldsymbol{J}}(F)=-\int_{{\mathbb{R}}}\\!dt\,\langle{\boldsymbol{J}}(t),\partial_{t}F\rangle$.

Let $(F_{j}:j\geq 1)$ be a family of vector fields in
$C^{1}((0,T)\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$ with compact support and
dense in $L^{2}([0,T]\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$. Assume that
$F_{1}=0$. In view of Lemma A.5, the entropy bound (4.3), and a classical
argument which allows to bound a maximum over a finite set in exponential
estimates, there exist finite constants $a$ and $C_{0}$ such that for all
$k\geq 1$,

$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}E_{P_{N}}\Big{[}\max_{1\leq j\leq
k}\,{\mathcal{E}}_{a,\varepsilon}(F_{j},{\boldsymbol{\pi}})\,\Big{]}\;\leq\;A\;,$
$\displaystyle\limsup_{\varepsilon\to
0}\limsup_{N\to\infty}E_{P_{N}}\Big{[}\max_{1\leq j\leq
k}{\mathcal{V}}_{a,\varepsilon}(F_{j},{\boldsymbol{\pi}},{\boldsymbol{J}})\,\Big{]}\;\leq\;A\;,$

where

$A\;:=\;C_{0}\,(1+T)\,+\,T\,\lim_{N}\frac{1}{N^{d}}{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;.$

Since $P_{N}$ converges to $P$ that is concentrated on measures which are
absolutely continuous with respect to the Lebesgue measure,
${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$, and whose density
${\boldsymbol{\rho}}$ is bounded below by $0$ and above by $1$, taking the
limit in $N$ and $\varepsilon$ yields

$\displaystyle E_{P}\Big{[}\max_{1\leq j\leq
k}\Big{\\{}\,{\boldsymbol{J}}(F_{j})\;-\;a\int_{{\mathbb{R}}}dt\int_{{\mathbb{T}}^{d}}dx\,\sigma({\boldsymbol{\rho}})\,|F_{j}|^{2}\Big{\\}}\,\Big{]}\;\leq\;A\;,$
$\displaystyle E_{P}\Big{[}\max_{1\leq j\leq
k}\Big{\\{}\,\int_{{\mathbb{R}}}\\!dt\int_{{\mathbb{T}}^{d}}\\!dx\,\Big{[}{\boldsymbol{\rho}}\,\,\nabla\cdot
F_{j}\;-\;a\,\sigma({\boldsymbol{\rho}})\,|F_{j}|^{2}\Big{]}\,\Big{\\}}\,\Big{]}\;\leq\;A\;.$

Each maximum is positive because $F_{1}=0$. By monotone convergence, taking
the limit in $k$, we obtain a similar bound, where the maximum over $1\leq
j\leq k$ is replaced by the maximum over $j\geq 1$. Since the sequence $F_{j}$
is dense in $L^{2}([0,T]\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})$, by Riesz
representation theorem, $P$-almost surely,
${\boldsymbol{J}}(t)=\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$ for some
${\boldsymbol{j}}$ in
$L^{2}([0,T]\times{\mathbb{T}}^{d},\sigma({\boldsymbol{\rho}})^{-1}dt\,dx;{\mathbb{R}}^{d})$.
These arguments also yield the bounds stated in the lemma.

We turn to the proof that $P$-almost surely ${\boldsymbol{\pi}}\in
C\big{(}{\mathbb{R}};{\mathcal{M}}_{m}({\mathbb{T}}^{d})\big{)}$. By the
continuity equation, $P$-almost surely for all functions $g$ in
$C^{1}({\mathbb{T}}^{d})$ and $0\leq s<t\leq T$,

$\int_{{\mathbb{T}}^{d}}{\boldsymbol{\pi}}(t,dx)\,g(x)\;-\;\int_{{\mathbb{T}}^{d}}{\boldsymbol{\pi}}(s,dx)\,g(x)\;=\;\int_{s}^{t}dr\int_{{\mathbb{T}}^{d}}dx\,{\boldsymbol{j}}(r,x)\cdot(\nabla
g)(x)\;.$

Since ${\boldsymbol{j}}$ belongs to
$L^{2}([0,T]\times{\mathbb{T}}^{d},\sigma({\boldsymbol{\rho}})^{-1}dt\,dx;{\mathbb{R}}^{d})$,
$P$-almost surely ${\boldsymbol{\pi}}$ belongs to
$C([0,T],{\mathcal{M}}_{m}({\mathbb{T}}^{d}))$, as claimed. ∎

Fix $T>0$ and a continuous vector field
$w\colon[0,T]\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D({\mathbb{R}};{\mathcal{H}}_{-p}^{d})\to{\mathbb{R}}^{d}$ that is
continuously differentiable in $x$ and such that for each
$(x,\pi)\in{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})$ and
$t\in[0,T]$ the map $[0,t]\times
D({\mathbb{R}};{\mathcal{H}}_{-p}^{d})\ni(s,{\boldsymbol{J}})\to
w(s,x,\pi,{\boldsymbol{J}})$ is measurable with respect to the Borel
$\sigma$-algebra on $[0,t]\times D([0,t];{\mathcal{H}}^{d}_{-p}\big{)}$. Let
$G_{w}\colon{\mathbb{R}}\times{\mathbb{T}}^{d}\times{\mathscr{S}}_{m,{\rm
ac}}\to{\mathbb{R}}^{d}$ be the progressively measurable map defined by

$G_{w}(t,x,{\boldsymbol{\pi}},{\boldsymbol{J}})\;=\;w(t,x,{\boldsymbol{\pi}}(t),{\boldsymbol{J}})\;.$
(4.5)

Finally, for $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m,{\rm
ac}}$, let

$V_{T,w}({\boldsymbol{\pi}},{\boldsymbol{J}})\;=\;\frac{1}{T}\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{\\{}\,G_{w}\cdot\big{[}\,{\boldsymbol{j}}\,+\,\nabla{\boldsymbol{\rho}}\,-\,\sigma({\boldsymbol{\rho}})\,E\,\big{]}\,-\,\sigma({\boldsymbol{\rho}})\,|G_{w}|^{2}\,\Big{\\}}\;,$
(4.6)

where ${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$,
${\boldsymbol{J}}(t)\;=\;\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$,
$t\in{\mathbb{R}}$.

######  Lemma 4.7.

Let $(P_{N}\colon N\geq 1)$ be a sequence of probability measures in
${\mathscr{P}}_{\rm stat}$ such that
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})<+\infty$. Assume that
$P_{N}\to P$ for some $P\in{\mathscr{P}}_{\rm stat}$. Then, for each $T>0$ and
each function $w$ as above,

$\displaystyle\liminf_{N\to\infty}\frac{1}{N^{d}}\,{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\geq\;E_{P}\big{[}\,V_{T,w}\,\big{]}\;.$
(4.7)

By Lemma 4.6, $P$–almost surely $({\boldsymbol{\pi}},{\boldsymbol{J}})$ belong
to ${\mathscr{S}}_{m,\mathrm{ac}}$ so that the right hand side of (4.7) is
well defined.

###### Proof of Lemma 4.7.

By passing to a subsequence, if needed, we may assume that
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})=\lim_{N}N^{-d}\,{\boldsymbol{I}}_{N,K_{N}}(P_{N})$.
By definition of ${\mathbb{H}}_{N,K}(t,{\mathbb{Q}})$, (3.4) and (3.1), for
each $T>0$ and each bounded, continuous functions $\Phi$ on
$D({\mathbb{R}},\Sigma_{N,K_{N}})$, measurable with respect to
$\sigma\big{\\{}{\boldsymbol{\eta}}(t),\,t\in[0,T]\big{\\}}$,

${\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\geq\;\frac{1}{T}\,\int
d{\mathbb{Q}}_{N}({\boldsymbol{\eta}})\Big{\\{}\Phi({\boldsymbol{\eta}})\,-\,\log{\mathbb{E}}^{N}_{{\boldsymbol{\eta}}(0)}\big{[}e^{\Phi}\big{]}\,\Big{\\}}\;,$

where ${\mathbb{Q}}_{N}$ is the unique stationary probability on
$D({\mathbb{R}},\Sigma_{N,K_{N}})$ such that
${\mathbb{Q}}_{N}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}=P_{N}$.

Recalling (2.6), let $\Phi$ be given by

$\Phi({\boldsymbol{\eta}})=\sum_{(x,y)\in{\mathbb{B}}_{N}}\Big{\\{}\int_{0}^{T}\phi^{x,y}(t)\,{\mathscr{N}}^{x,y}_{(t,t+dt]}({\boldsymbol{\eta}})\,-\,N^{2}\int_{0}^{T}{\boldsymbol{\eta}}_{x}(t)[1-{\boldsymbol{\eta}}_{y}(t)]\big{[}e^{\phi^{x,y}(t)}-1\big{]}\,dt\Big{\\}}\;,$

where

$\phi^{x,y}(t)\;=\;\int_{x}^{y}w(t,\cdot\,,{\boldsymbol{\pi}}_{N}(t),{\boldsymbol{J}}_{N})\,\cdot
d\ell\;.$

By Lemma A.1, ${\mathbb{E}}_{\eta}^{N}\big{[}e^{\Phi}\big{]}=1$ for each
$\eta\in\Sigma_{N,K}$ and by Lemma A.6,

$\frac{1}{T}\,\lim_{N\to\infty}\frac{1}{N^{d}}\,E_{{\mathbb{Q}}_{N}}[\Phi]\;=\;E_{P}\big{[}\,V_{T,w}\,\big{]}\;,$

which completes the proof. ∎

###### Proof of Theorem 4.1. $\Gamma$-liminf.

Fix $P$ in ${\mathscr{P}}_{\rm stat}$ and a sequence $(P_{N}:N\geq 1)$,
$P_{N}\in{\mathscr{P}}_{\rm stat}$, such that $P_{N}\to P$,
$\liminf_{N}N^{-d}{\boldsymbol{I}}_{N,K_{N}}(P_{N})<+\infty$. By Lemma 4.6, we
may assume that $P$-almost surely $({\boldsymbol{\pi}},{\boldsymbol{J}})$
belongs to ${\mathscr{S}}_{m,{\rm ac}}$ and that there exists a constant
$C_{1}$ such that for all $T>0$,

$E_{P}\Big{[}\,\int_{0}^{T}dt\,\int_{{\mathbb{T}}^{d}}dx\,\frac{|{\boldsymbol{j}}(t,x)|^{2}}{\sigma({\boldsymbol{\rho}}(t,x))}\;+\;\int_{0}^{T}dt\,\int_{{\mathbb{T}}^{d}}dx\,\frac{|\nabla{\boldsymbol{\rho}}(t,x)|^{2}}{\sigma({\boldsymbol{\rho}}(t,x))}\,\Big{]}\;\leq\;C_{1}\,(1+T)\;,$
(4.8)

where ${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$,
${\boldsymbol{J}}(t)\;=\;\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$.

By Lemma 4.7, it is enough to show that for some $T>0$,

$\displaystyle{\boldsymbol{I}}_{m}(P)\;\leq\;\sup_{w}E_{P}\big{[}\,V_{T,w}\big{]}\;,$
(4.9)

where the supremum is carried over all continuous vector fields
$w\colon[0,T]\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D({\mathbb{R}};{\mathcal{H}}_{-p}^{d})\to{\mathbb{R}}^{d}$ satisfying the
assumption enunciated above (4.5).

Fix $T>0$ and a bounded and continuous function
$f\colon{\mathbb{T}}^{d}\times{\mathcal{H}}_{1}\times
L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})\to{\mathbb{R}}^{d}$ that is
continuously differentiable in $x$. For $\delta>0$, let
$f_{\delta}\colon(0,T)\times{\mathbb{T}}^{d}\times{\mathcal{M}}_{+}({\mathbb{T}}^{d})\times
D\big{(}{\mathbb{R}};{\mathcal{H}}^{d}_{-p}\big{)}\to{\mathbb{R}}^{d}$ be
given by

$f_{\delta}(t,x,\pi,{\boldsymbol{J}})\;=\;\chi_{\delta}(t)\,f\big{(}x,\pi_{\delta},{\boldsymbol{j}}_{\delta}(t)\big{)}\,.$

Here $\chi_{\delta}$, $0<\delta<1$, stands for a sequence of continuous
functions, bounded below by $0$ and above by $1$, whose support is contained
in $[\delta,T]$, and which converges in $L^{1}$ to the indicator functions of
the set $[0,T]$. Moreover,
$\pi_{\delta}(x)=\langle\pi,\kappa_{\delta}(x-\cdot)\rangle$ where
$\kappa_{\delta}\colon{\mathbb{T}}^{d}\to{\mathbb{R}}_{+}$ is a smooth
approximation of the identity, and

${\boldsymbol{j}}_{\delta}(t,x)\;=\;\int_{-\infty}^{t}\\!ds\;a^{\prime}_{\delta}(t-s)\,\langle{\boldsymbol{J}}(s),\imath_{\delta}(x-\cdot)\rangle\,,$
(4.10)

where $a_{\delta}\colon{\mathbb{R}}\to{\mathbb{R}}_{+}$ is a smooth
approximation of identity with compact support in $(0,\delta)$,
$a^{\prime}_{\delta}$ the derivative of $a_{\delta}$, and
$\imath_{\delta}\colon{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ another smooth
approximation of the identity. Observe that ${\boldsymbol{j}}_{\delta}(t)$
depends on ${\boldsymbol{J}}(s)$ only for $s\in(t-\delta,t)$. Hence, since
$\chi_{\delta}(t)=0$ for $t\leq\delta$, the function
$f_{\delta}(t,x,\pi,\cdot)$ depends on ${\boldsymbol{J}}(s)$ only for
$s\in[0,t]$. This is a requirement of the test functions $w$ introduced above
(4.5). Since $f_{\delta}$ satisfies the conditions presented above (4.5) for
each $\delta>0$, we deduce

$\sup_{w}E_{P}\big{[}\,V_{T,w}\big{]}\;\geq\;\limsup_{\delta\to
0}E_{P}\big{[}\,V_{T,f_{\delta}}\big{]}\;.$

Let $H_{f}\colon[0,T]\times{\mathbb{T}}^{d}\times
L^{2}\big{(}[0,T];{\mathcal{H}}_{1}\big{)}\times
L^{2}([0,T]\times{\mathbb{T}}^{d};{\mathbb{R}}^{d})\to{\mathbb{R}}^{d}$ be
defined by

$H_{f}(t,x,{\boldsymbol{\rho}},{\boldsymbol{j}})\;=\;f(x,{\boldsymbol{\rho}}(t),{\boldsymbol{j}}(t))\;.$

By (4.8), $P$–almost surely,
${\boldsymbol{\pi}}_{\delta}(t)\to{\boldsymbol{\rho}}(t)$ and
${\boldsymbol{j}}_{\delta}(t)\to{\boldsymbol{j}}(t)$ for Lebesgue almost all
$t\in(0,T)$. Since $f$ is bounded, by dominated convergence,

$\lim_{\delta\to
0}E_{P}\Big{[}\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{\\{}\,G_{f_{\delta}}(t,x,{\boldsymbol{\pi}},{\boldsymbol{J}})\,-\,H_{f}(t,x,{\boldsymbol{\rho}},{\boldsymbol{j}})\Big{\\}}^{2}\,\Big{]}\;=\;0\;,$
(4.11)

where $G_{f_{\delta}}$ has been introduced in (4.5).

Denote by $W_{T,f}$ the right-hand side of (4.6) when $G_{w}$ is replaced by
$H_{f}$. By (4.8), (4.11) and the dominated convergence theorem,

$\lim_{\delta\to
0}E_{P}\big{[}\,V_{T,f_{\delta}}\,\big{]}\;=\;E_{P}\big{[}\,W_{T,f}\,\big{]}\;.$

In view of the previous estimates, it remains to show that

$\sup_{f}E_{P}\big{[}\,W_{T,f}\,\big{]}\;\geq\;{\boldsymbol{I}}_{m}(P)\;.$
(4.12)

Let $\widehat{f}\colon{\mathbb{T}}^{d}\times{\mathcal{H}}_{1}\times
L^{2}({\mathbb{T}}^{d};{\mathbb{R}}^{d})\to{\mathbb{R}}^{d}$ be given by

$\widehat{f}(x,\rho,j)\;=\;\frac{j(x)+(\nabla\rho)(x)-\sigma(\rho(x))\,E(x)}{2\,\sigma(\rho(x))}$

and note that ${\boldsymbol{I}}_{m}(P)=E_{P}[\,W_{T,\widehat{f}}\,]$, which is
finite in view of (4.8). Using this bound and approximating the function
$\widehat{f}$ by a sequence of bounded and continuous functions that are
continuously differentiable in $x$, we obtain (4.12) by the dominated
convergence theorem. ∎

4.3 The $\Gamma$-limsup. Given $P\in{\mathscr{P}}_{\rm stat}$, we shall
construct a sequence $(P_{N}:N\geq 1)$, such that $P_{N}\to P$ and

$\limsup_{N\to\infty}{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\leq\;{\boldsymbol{I}}_{m}(P)\;.$

We carry this out first for $P$ satisfying certain regularity assumptions, and
then use density arguments to extend the result to any $P$ with finite rate
function, ${\boldsymbol{I}}_{m}(P)<+\infty$.

Fix $T>0$. Recalling (2.22), a path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in
${\mathscr{S}}_{m}$ is $T$-periodic if
$\vartheta_{T}({\boldsymbol{\pi}},{\boldsymbol{J}})=({\boldsymbol{\pi}},{\boldsymbol{J}})$.
An element $P$ in ${\mathscr{P}}_{\rm stat}$ is said to be _$T$ -holonomic_ if
there exists a $T$-periodic path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ such
that

$P\;=\;\frac{1}{T}\,\int_{0}^{T}\delta_{\vartheta_{s}({\boldsymbol{\pi}},{\boldsymbol{J}})}\,ds\;.$
(4.13)

An element of ${\mathscr{P}}_{\rm stat}$ is _holonomic_ if it is $T$-holonomic
for some $T>0$.

Fix $T>0$ and a $T$-periodic path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in
${\mathscr{S}}_{m,{\rm ac}}$. Denote by
$({\boldsymbol{\rho}},{\boldsymbol{j}})$ the densities so that
${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$,
${\boldsymbol{J}}(t)=\int_{0}^{t}\\!ds\,{\boldsymbol{j}}(s)$. Assume that
${\boldsymbol{\rho}}$, ${\boldsymbol{j}}$ are smooth and that there exists
$\delta>0$ such that $\delta\leq{\boldsymbol{\rho}}(t,x)\leq 1-\delta$ for all
$(t,x)$. Denote by
$F\colon{\mathbb{R}}\times{\mathbb{T}}^{d}\to{\mathbb{R}}^{d}$ the smooth,
$T$-periodic vector field defined by

$F\;=\;\frac{{\boldsymbol{j}}\,+\,\nabla{\boldsymbol{\rho}}\,-\,\sigma({\boldsymbol{\rho}})E}{\sigma({\boldsymbol{\rho}})}\;\cdot$

As the path $({\boldsymbol{\rho}},{\boldsymbol{J}})$ satisfies the continuity
equation (2.21),
$\partial_{t}{\boldsymbol{\rho}}+\nabla\cdot{\boldsymbol{j}}=0$, by definition
of $F$,

$\partial_{t}{\boldsymbol{\rho}}\;=\;\Delta{\boldsymbol{\rho}}\;-\;\nabla\cdot\big{(}\,\sigma({\boldsymbol{\rho}})\,\big{[}\,E+F(t)\,\big{]}\,\big{)}\;.$
(4.14)

Let finally $P\in{\mathscr{P}}_{\rm stat}$ be the $T$-holomic probability
corresponding, as in (4.13), to the $T$-periodic path
$({\boldsymbol{\pi}},{\boldsymbol{J}})$.

Let $L_{N}^{F}$ be the time-dependent generator of a perturbed WASEP defined
by

$(L_{N}^{F}\,f)(\eta)\;=\;N^{2}\sum_{(x,y)\in{\mathbb{B}}_{N}}\eta_{x}\,[1-\eta_{y}]\,e^{(1/2)[\,E_{N}(x,y)+F_{N}(t,x,y)\,]}\big{[}f(\sigma^{x,y}\eta)-f(\eta)\big{]}\;,$
(4.15)

where $F_{N}(t,x,y)$ represents the line integral of $F(t,\cdot)$ along the
oriented segment from $x$ to $y$ introduced in (2.2). Denote by
$({\boldsymbol{\eta}}^{F}(t):t\geq 0)$ the continuous-time, time-inhomogeneous
Markov chain whose generator is $L_{N}^{F}$. By the hydrodynamic limit of the
time dependent WASEP dynamics, see e.g. [15, Section 10.5], the empirical
density ${\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}}^{F})$ associated to the
process ${\boldsymbol{\eta}}^{F}$ evolves, in the limit $N\to\infty$,
according to the solution of the PDE (4.14). This explains the introduction of
the process ${\boldsymbol{\eta}}^{F}$.

Let $(\xi_{k}:k\geq 0)$ be the discrete-time, $\Sigma_{N}$-valued Markov chain
given by $\xi_{k}={\boldsymbol{\eta}}^{F}(kT)$. Since $F$ is $T$-periodic,
$\xi_{k}$ is time-homogeneous. As it is irreducible on each set
$\Sigma_{N,K}$, $\xi_{k}$ has a unique stationary state, denoted by
$\mu^{F}_{N,K}$. Let finally, ${\mathbb{P}}^{F}_{N,K}$ be the law of
${\boldsymbol{\eta}}^{F}$ when the initial condition is sampled according to
$\mu^{F}_{N,K}$. Note that ${\mathbb{P}}^{F}_{N,K}$ is invariant by
$T$-translations:
${\mathbb{P}}^{F}_{N,K}\circ\vartheta^{-1}_{T}={\mathbb{P}}^{F}_{N,K}$. Since
this measure, defined on $D({\mathbb{R}}_{+},\Sigma_{N})$, is invariant by
$T$-translations, we may extend it to $D({\mathbb{R}},\Sigma_{N})$.

Let $P_{N}$ be the measure on ${\mathscr{S}}$ given by

$P_{N}\;=\;\Big{(}\frac{1}{T}\int_{0}^{T}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\vartheta^{-1}_{t}\,dt\Big{)}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\;,$
(4.16)

that, by construction, belongs to ${\mathscr{P}}_{\rm stat}$.

######  Proposition 4.8.

For each $T$-periodic path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in
${\mathscr{S}}_{m,\mathrm{ac}}$ with smooth densities
$({\boldsymbol{\rho}},{\boldsymbol{j}})$ with ${\boldsymbol{\rho}}$ bounded
away from zero and one, the sequence $(P_{N}:N\geq 1)$ introduced in (4.16)
converges to the $T$-holonomic probability $P$ given by (4.13) and

$\limsup_{N\to\infty}\frac{1}{N^{d}}{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\leq\;{\boldsymbol{I}}_{m}(P)\;.$

The proof of this proposition relies on the following lemma.

######  Lemma 4.9.

In the setting of Proposition 4.8, the sequence of probability measures on
$D({\mathbb{R}};{\mathcal{M}}_{+}({\mathbb{T}}^{d})\big{)}$ given by
${\mathbb{P}}_{N,K_{N}}^{F}\circ{\boldsymbol{\pi}}_{N}^{-1}$ converges to
$\delta_{{\boldsymbol{\pi}}}$.

###### Proof.

By the smoothness of the external field $F$, Lemma 4.3 holds also when
${\mathbb{P}}_{\mu_{N,K_{N}}}^{N}$ is replaced by
${\mathbb{P}}^{F}_{N,K_{N}}$. By the compactness of
${\mathcal{M}}_{+}({\mathbb{T}}^{d})$, this implies the pre-compactness of the
family
$\big{(}{\mathbb{P}}_{N,K_{N}}^{F}\circ{\boldsymbol{\pi}}_{N}^{-1}\,:\,N\geq
1\big{)}$. Let ${\mathcal{P}}^{F}$ be a cluster point of this sequence. By the
$T$-periodicity of ${\mathbb{P}}_{N,K_{N}}^{F}$ and the hydrodynamic limit for
the time dependent WASEP with generator (4.15), ${\mathcal{P}}^{F}$ is $T$
periodic and ${\mathcal{P}}^{F}$ almost surely
${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}^{F}(t,x)dx$ for some density
${\boldsymbol{\rho}}^{F}$ of mass $m$ that solves (4.14). By the uniqueness of
$T$-periodic solutions to (4.14) and the $L^{1}({\mathbb{T}}^{d})$ convergence
to this unique solution, as stated in Theorem 7.1,
${\boldsymbol{\rho}}^{F}={\boldsymbol{\rho}}$. Hence
${\mathcal{P}}^{F}=\delta_{{\boldsymbol{\pi}}}$ as claimed. ∎

###### Proof of Proposition 4.8.

By the smoothness of the external field $F$, Lemmata 4.3, 4.4, and 4.5 holds
also when ${\mathbb{P}}_{\mu_{N,K_{N}}}^{N}$ is replaced by
${\mathbb{P}}^{F}_{N,K_{N}}$. This implies the pre-compactness of the sequence
of probabilities on ${\mathscr{S}}$ given by
$\big{\\{}{\mathbb{P}}^{F}_{N,K_{N}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\big{\\}}$.
Let now $P_{0}$ be a cluster point of this sequence. By the hydrodynamic limit
for the perturbed WASEP, $P_{0}$-almost surely,
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ belongs to ${\mathscr{S}}_{m,{\rm
ac}}$ and the corresponding densities $({\boldsymbol{\rho}},{\boldsymbol{j}})$
is a $T$-periodic weak solution to the hydrodynamic equation

$\left\\{\begin{aligned}
&\partial_{t}{\boldsymbol{\rho}}\;+\;\nabla\cdot{\boldsymbol{j}}\;=\;0\;,\\\
&{\boldsymbol{j}}\;=\;-\,\nabla{\boldsymbol{\rho}}\;+\;\sigma({\boldsymbol{\rho}})\,[\,E+F\,]\;.\end{aligned}\right.$

By Lemma 4.9, $P_{0}=\delta_{({\boldsymbol{\pi}},{\boldsymbol{J}})}$. Taking
time-averages, we deduce $P_{N}\to P$.

We turn to the second claim of the proposition. By (3.4),

${\boldsymbol{I}}_{N,K_{N}}(P_{N})\;=\;H_{N,K_{N}}\Big{(}\,\frac{1}{T}\,\int_{0}^{T}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\,\vartheta^{-1}_{t}\,dt\,\Big{)}\;.$
(4.17)

Fix $\ell\in{\mathbb{N}}$. By (3.3) and the convexity of the relative entropy,
the right-hand side of the previous equation is equal to

$\begin{split}&\lim_{\ell\to\infty}\frac{1}{\ell T}\,{\mathbb{H}}^{(\ell
T)}\Big{(}\,\frac{1}{T}\,\int_{0}^{T}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\,\vartheta^{-1}_{t}\,dt\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\Big{)}\\\
&\qquad\;\leq\;\lim_{\ell\to\infty}\frac{1}{\ell\,T^{2}}\,\int_{0}^{T}{\mathbb{H}}^{(\ell
T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\,\vartheta^{-1}_{t}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\,dt\;.\end{split}$

Since ${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$ is translation invariant, for fixed
$\ell$ and $0\leq t\leq T$, by definition of the translations
$(\vartheta_{s}:s\in{\mathbb{R}})$, introduced in (2.22), recalling (3.2),

$\displaystyle{\mathbb{H}}^{(\ell
T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\,\vartheta^{-1}_{t}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;$
$\displaystyle=\;{\mathbb{H}}_{[0\,,\,\ell
T]}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\circ\,\vartheta^{-1}_{t}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\circ\,\vartheta^{-1}_{t}\,\Big{)}$
$\displaystyle=\;{\mathbb{H}}_{[-t\,,\,\ell
T-t]}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;.$

As ${\mathbb{P}}^{F}_{N,K_{N}}$ is $T$-translation invariant and
${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$ is translation invariant, the dynamical
contribution to the relative entropy of the time interval $[-t,0]$ corresponds
to the one of the time interval $[-t+\ell T,\ell T]$. Hence, the previous
expression is equal to

${\mathbb{H}}^{(\ell
T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;-\;\textrm{Ent}\big{(}\,\mu_{N,K_{N}}^{F}\,\big{|}\,\mu_{N,K_{N}}\,\big{)}\;+\;\textrm{Ent}\Big{(}\,{\mathbb{P}}_{N,K_{N}}^{F}\circ\imath_{-t}^{-1}\,\big{|}\,\mu_{N,K_{N}}\,\Big{)}\;,$

where ${\mathbb{P}}_{N,K_{N}}^{F}\circ\imath_{s}^{-1}$ is the marginal at time
$s$ of ${\mathbb{P}}_{N,K_{N}}^{F}$.

Using again that ${\mathbb{P}}^{F}_{N,K_{N}}$ is $T$-translation invariant and
${\mathbb{P}}^{N}_{\mu_{N,K_{N}}}$ is translation invariant,

$\displaystyle{\mathbb{H}}^{(\ell
T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;-\;\textrm{Ent}\big{(}\,\mu_{N,K_{N}}^{F}\,\big{|}\,\mu_{N,K_{N}}\,\big{)}$
$\displaystyle\qquad\;=\;\ell\,{\mathbb{H}}^{(T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;-\;\ell\,\textrm{Ent}\big{(}\,\mu_{N,K_{N}}^{F}\,\big{|}\,\mu_{N,K_{N}}\,\big{)}\;.$

Putting together the previous estimates and letting $\ell\to\infty$ yields
that

${\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\leq\;\frac{1}{T}\,\Big{\\{}{\mathbb{H}}^{(T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}\;-\;\textrm{Ent}\big{(}\,\mu_{N,K_{N}}^{F}\,\big{|}\,\mu_{N,K_{N}}\,\big{)}\,\Big{\\}}\;.$

By Lemma 4.9 and the large deviations lower bound in hydrodynamical limits,
see e.g. [15, Lemma 10.5.4],

$\limsup_{N\to\infty}\frac{1}{N^{d}}\,\Big{\\{}\,{\mathbb{H}}^{(T)}\Big{(}\,{\mathbb{P}}^{F}_{N,K_{N}}\,\big{|}\,{\mathbb{P}}^{N}_{\mu_{N,K_{N}}}\,\Big{)}-\textrm{Ent
}\big{(}\mu_{N,K_{N}}^{F}\big{|}\mu_{N,K_{N}}\big{)}\,\Big{\\}}\;\leq\;A_{T,m}({\boldsymbol{\pi}},{\boldsymbol{J}})\;.$

Therefore,

$\limsup_{N\to\infty}\frac{1}{N^{d}}\,{\boldsymbol{I}}_{N,K_{N}}(P_{N})\;\leq\;\frac{1}{T}\,A_{T,m}({\boldsymbol{\pi}},{\boldsymbol{J}})\;.$

The right-hand side is equal to ${\boldsymbol{I}}_{m}(P)$ in view of the
equations (2.26), (4.13), and the $T$-periodicity of the path
$({\boldsymbol{\pi}},{\boldsymbol{J}})$. ∎

To complete the proof of the $\Gamma$-limsup, we show that any $P$ in
${\mathscr{P}}_{\rm stat}$ can be approximated by convex combinations of
holonomic probability measures supported by smooth paths bounded away from
zero and one and that the corresponding rate function converges to
${\boldsymbol{I}}_{m}(P)$. Denote by
$\color[rgb]{.2,0.2,.8}{\mathscr{P}}^{\varepsilon}_{{\rm stat},m}$,
$\varepsilon>0$, the subset of ${\mathscr{P}}_{{\rm stat},m}$ formed by the
stationary measures $P$ such that $P$-almost surely
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ belongs to
${\mathscr{S}}_{m,\mathrm{ac}}$ with smooth densities
$({\boldsymbol{\rho}},{\boldsymbol{j}})$ such that
$\varepsilon\leq{\boldsymbol{\rho}}(t,x)\leq 1-\varepsilon$ for all $(t,x)$.

######  Theorem 4.10.

Assume that $E$ is orthogonally decomposable and fix $P\in{\mathscr{P}}_{\rm
stat}$ such that ${\boldsymbol{I}}_{m}(P)<+\infty$. There exist a sequence
$(\varepsilon_{n}:n\geq 1)$, a triangular array $(\alpha_{n,i}\,,\,1\leq i\leq
n\,,\,n\geq 1)$ with $\alpha_{n,i}\geq 0$, $\sum_{i}\alpha_{n,i}=1$, and a
triangular array $(P_{n,i}\,,\,1\leq i\leq n\,,\,n\geq 1)$ of holonomic
measures belonging to ${\mathscr{P}}^{\varepsilon_{n}}_{{\rm stat},m}$ such
that by setting $P_{n}:=\sum_{i}\alpha_{n,i}P_{n,i}$ we have $P_{n}\to P$ and
${\boldsymbol{I}}_{m}(P_{n})\to{\boldsymbol{I}}_{m}(P)$.

Postponing the proof of this statement, we first conclude the
$\Gamma$-convergence of the Donsker-Varadhan functional.

###### Proof of Theorem 4.1. $\Gamma$-limsup.

The statement follows, by a diagonal argument, from Proposition 4.8 and
Theorem 4.10. ∎

We turn to the proof of Theorem 4.10. It relies on two lemmata.

######  Lemma 4.11.

Fix $P$ satisfying (2.27). There exists a sequence $(P_{n}\colon n\geq 1)$
converging to $P$ and such that $P_{n}$ belongs to
${\mathscr{P}}^{\varepsilon_{n}}_{{\rm stat},m}$ for some $\varepsilon_{n}>0$,
and

$\limsup_{n\to\infty}{\boldsymbol{I}}_{m}(P_{n})\;=\;{\boldsymbol{I}}_{m}(P)\;.$

###### Proof.

Fix a smooth probability density
$\phi:{\mathbb{R}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ whose support is
contained in $[-1,1]\times[-1,1]^{d}$, so that

$\int_{{\mathbb{R}}\times{\mathbb{R}}^{d}}\phi(t,x)\,dt\,dx\;=\;1\;.$

Let
$\phi_{\varepsilon}(t,x)=\varepsilon^{-(d+1)}\phi(t/\varepsilon,x/\varepsilon)$,
$\varepsilon>0$. For a trajectory $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in
${\mathscr{S}}_{m,\mathrm{ac}}$, whose density is represented by
$({\boldsymbol{\rho}},{\boldsymbol{j}})$, let

${\boldsymbol{\rho}}_{\varepsilon}\;:=\;(1-\varepsilon)\,({\boldsymbol{\rho}}*\phi_{\varepsilon})\;+\;\varepsilon\,m\;,\quad{\boldsymbol{j}}_{\varepsilon}\;:=\;(1-\varepsilon)\,({\boldsymbol{j}}*\phi_{\varepsilon})\;,$

where $*$ denotes space-time convolution and $0<\varepsilon<1$. Observe that
$({\boldsymbol{\rho}}_{\varepsilon},{\boldsymbol{j}}_{\varepsilon})$ satisfy
the continuity equation.

Denote by $\Psi_{\varepsilon}$ the map
$({\boldsymbol{\rho}},{\boldsymbol{j}})\mapsto({\boldsymbol{\rho}}_{\varepsilon},{\boldsymbol{j}}_{\varepsilon})=:\Psi_{\varepsilon}({\boldsymbol{\rho}},{\boldsymbol{j}})$
and set $P^{\varepsilon}=P\circ\Psi^{-1}_{\varepsilon}$. Then, for each
$\varepsilon>0$, the probability $P^{\varepsilon}$ belongs to
${\mathscr{P}}^{\delta}_{{\rm stat},m}$ for some
$\delta=\delta(\varepsilon)>0$ and $P^{\varepsilon}\to P$ as $\varepsilon\to
0$.

It remains to show that
$\lim_{\varepsilon}{\boldsymbol{I}}_{m}(P^{\varepsilon})={\boldsymbol{I}}_{m}(P)$.
As

${\boldsymbol{I}}_{m}(P^{\varepsilon})\;=\;E_{P}\Big{[}\,\frac{1}{T}\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|{\boldsymbol{j}}_{\varepsilon}+\nabla{\boldsymbol{\rho}}_{\varepsilon}-\sigma({\boldsymbol{\rho}}_{\varepsilon})\,E|^{2}}{4\,\sigma({\boldsymbol{\rho}}_{\varepsilon})}\,\Big{]}\;,$

and since the sequence

$\frac{|{\boldsymbol{j}}_{\varepsilon}+\nabla{\boldsymbol{\rho}}_{\varepsilon}-\sigma({\boldsymbol{\rho}}_{\varepsilon})\,E|^{2}}{4\,\sigma({\boldsymbol{\rho}}_{\varepsilon})}$
(4.18)

converges $(dP\,dt\,dx)$-almost surely to the same expression without the
subscript $\varepsilon$, it is enough to prove that the sequence (4.18) is
uniformly integrable.

Since $P$ satisfies (2.27), by [6, Lemma 5.3] there exist increasing convex
functions $\Upsilon_{1}$, $\Upsilon_{2}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$
such that $\lim_{r\to\infty}\Upsilon_{a}(r)/r=\infty$, $a=1$, $2$, and

$E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{\\{}\,\Upsilon_{1}\Big{(}\frac{|{\boldsymbol{j}}|^{2}}{\sigma({\boldsymbol{\rho}})}\Big{)}+\Upsilon_{2}\Big{(}\frac{|\nabla{\boldsymbol{\rho}}|^{2}}{\sigma({\boldsymbol{\rho}})}\Big{)}\,\Big{\\}}\,\Big{]}\;<\;+\infty\;.$
(4.19)

Moreover, the uniform integrability of the sequence (4.18) follows from the
bound

$\limsup_{\varepsilon\to
0}E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Big{\\{}\,\Upsilon_{1}\Big{(}\frac{|{\boldsymbol{j}}_{\varepsilon}|^{2}}{\sigma({\boldsymbol{\rho}}_{\varepsilon})}\Big{)}+\Upsilon_{2}\Big{(}\frac{|\nabla{\boldsymbol{\rho}}_{\varepsilon}|^{2}}{\sigma({\boldsymbol{\rho}}_{\varepsilon})}\Big{)}\,\Big{\\}}\,\Big{]}\;<\;+\infty\;.$
(4.20)

Note that

$\displaystyle|{\boldsymbol{j}}_{\varepsilon}|^{2}\;=\;[1-\varepsilon]^{2}\,\big{|}\,{\boldsymbol{j}}*\phi_{\varepsilon}\,\big{|}^{2}\;\leq\;\big{[}\,(1-\varepsilon)\,\sigma({\boldsymbol{\rho}}*\phi_{\varepsilon})\,+\,\varepsilon\,\sigma(m)\,\big{]}\,\Big{\\{}\,[1-\varepsilon]\,\frac{|\,{\boldsymbol{j}}*\phi_{\varepsilon}\,|^{2}}{\sigma({\boldsymbol{\rho}}*\phi_{\varepsilon})}\,\Big{\\}}\;.$

By the concavity of $\sigma$, the first term on the right-hand side is bounded
by $\sigma({\boldsymbol{\rho}}_{\varepsilon})$. In conclusion,

$\frac{|{\boldsymbol{j}}_{\varepsilon}|^{2}}{\sigma({\boldsymbol{\rho}}_{\varepsilon})}\;\leq\;\,[1-\varepsilon]\,\frac{|\,{\boldsymbol{j}}*\phi_{\varepsilon}\,|^{2}}{\sigma({\boldsymbol{\rho}}*\phi_{\varepsilon})}\;\leq\;\,\frac{|\,{\boldsymbol{j}}*\phi_{\varepsilon}\,|^{2}}{\sigma({\boldsymbol{\rho}}*\phi_{\varepsilon})}\;\cdot$

On the other hand, by concavity of $\sigma$ and Cauchy-Schwarz inequality,

$\displaystyle\frac{|{\boldsymbol{j}}*\phi_{\varepsilon}|^{2}}{\sigma({\boldsymbol{\rho}}*\phi_{\varepsilon})}(t,x)$
$\displaystyle\;\leq\;\frac{1}{[\,\sigma({\boldsymbol{\rho}})*\phi_{\varepsilon}\,](t,x)}\Big{(}\int_{{\mathbb{R}}}ds\int_{{\mathbb{R}}^{d}}dy\,\phi_{\varepsilon}(t-s,x-y)\,\frac{j(s,y)}{\sqrt{\sigma({\boldsymbol{\rho}}(s,y))}}\sqrt{\sigma({\boldsymbol{\rho}}(s,y))}\,\Big{)}^{2}$
$\displaystyle\qquad\leq\;\Big{(}\,\phi_{\varepsilon}\,*\,\frac{|{\boldsymbol{j}}|^{2}}{\sigma({\boldsymbol{\rho}})}\,\Big{)}(t,x)\;.$

Whence, as $\Upsilon_{1}$ is increasing,

$E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Upsilon_{1}\Big{(}\frac{|{\boldsymbol{j}}_{\varepsilon}|^{2}}{\sigma({\boldsymbol{\rho}}_{\varepsilon})}\Big{)}\,\Big{]}\leq\;E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Upsilon_{1}\Big{(}\phi_{\varepsilon}\,*\,\frac{|{\boldsymbol{j}}|^{2}}{\sigma({\boldsymbol{\rho}})}\Big{)}\,\Big{]}\;.$

Since $\Upsilon_{1}$ is convex, integrating the convolution, we deduce that
the previous expression is bounded by

$E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\phi_{\varepsilon}\,*\,\Upsilon_{1}\Big{(}\frac{|{\boldsymbol{j}}|^{2}}{\sigma({\boldsymbol{\rho}})}\Big{)}\,\Big{]}\;=\;E_{P}\Big{[}\,\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\Upsilon_{1}\Big{(}\frac{|{\boldsymbol{j}}|^{2}}{\sigma({\boldsymbol{\rho}})}\Big{)}\,\Big{]}\;.$

Since the previous argument for ${\boldsymbol{j}}$ applies to
$\nabla{\boldsymbol{\rho}}$, the bound (4.20) follows from (4.19). ∎

######  Lemma 4.12.

Assume that $E$ is orthogonally decomposable. There exists $T_{0}>0$ and
$C_{0}>0$ such that the following holds. For any
$\pi_{0},\pi_{1}\in{\mathcal{M}}_{m}({\mathbb{T}}^{d})$ there exists
$\bar{T}\leq T_{0}$ and a path
$\big{(}{\boldsymbol{\pi}}(t),{\boldsymbol{J}}(t)\big{)}$, $t\in[0,\bar{T}]$,
with ${\boldsymbol{\pi}}(0)=\pi_{0}$, ${\boldsymbol{\pi}}(\bar{T})=\pi_{1}$,
satisfying the continuity equation (2.21) for each $0\leq s<t\leq\bar{T}$ and

$A_{m,\bar{T}}(\bar{{\boldsymbol{\pi}}},\bar{{\boldsymbol{J}}})\leq C_{0}\;.$
(4.21)

###### Proof.

In the case of the symmetric exclusion process, this statement is proven in
[3, Lemma 4.7] with the following strategy. Start from $\pi_{0}$ and follow
the hydrodynamic equation for a long but fixed time interval $[0,T_{1}]$ so
that ${\boldsymbol{\pi}}(T_{1})$ lies in small neighborhood of the stationary
solution with mass $m$. Then interpolate, in the time interval
$[T_{1},T_{1}+1]$ from ${\boldsymbol{\pi}}(T_{1})$ to a suitable
$\widehat{\pi}$ that is still close to the stationary solution. Finally, from
$\widehat{\pi}$ use the optimal path for the escape problem to reach
$\pi_{1}$. Provided that the _quasi-potential_ is bounded, this argument
applies also to the WASEP case. As discussed in [4, § V.C] and [5], if the
external field is orthogonally decomposable then the quasi-potential can be
computed explicitly and it is indeed bounded. ∎

###### Proof of Theorem 4.10.

Fix $P\in{\mathscr{P}}_{{\rm stat},m}$. By Lemma 4.11 we can assume that
$P\in{\mathscr{P}}^{\varepsilon}_{{\rm stat},m}$ for some $\varepsilon>0$.
Since $P$ can be written as a convex combination of ergodic probabilities and
${\boldsymbol{I}}_{m}$ is affine, it suffices to show that for each ergodic
$P\in{\mathscr{P}}^{\varepsilon}_{{\rm stat},m}$ with
${\boldsymbol{I}}_{m}(P)<+\infty$ there exists a sequence of holonomic
measures $P_{T}$ in ${\mathscr{P}}^{\varepsilon}_{{\rm stat},m}$ converging to
$P$ and such that
$\lim_{T}{\boldsymbol{I}}_{m}(P_{T})={\boldsymbol{I}}_{m}(P)$.

Recalling that the $T$-periodization of paths in ${\mathscr{S}}$ as been
defined in (2.23), set

$\begin{split}{\mathscr{A}}_{P}:=\Big{\\{}&({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m}\colon\;\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{t}({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})}\,dt\to
P\\\ &\quad\textrm{ and
}\lim_{T\to+\infty}\frac{1}{T}A_{T,m}({\boldsymbol{\pi}},{\boldsymbol{J}})\to
E_{P}\big{[}A_{1,m}\big{]}\Big{\\}}\;.\end{split}$

Since $P$ is ergodic, by the Birkhoff ergodic theorem
$P({\mathscr{A}}_{P})=1$. Pick an element
$({{\boldsymbol{\pi}}}^{*},{{\boldsymbol{J}}}^{*})\in{\mathscr{A}}_{P}$. By
definition, the $T$-holonomic probability associated to the $T$-periodization
of $({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})$ converges to $P$ but, in
general, its rate function does not since when $T$-periodizing paths we may
insert jumps. By using Lemma 4.12, we now show how the path
$({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})$ can be modified to accomplish
our needs.

Given $T^{*}>0$, let $\pi_{0}={{\boldsymbol{\pi}}}^{*}(T^{*})$ and
$\pi_{1}={{\boldsymbol{\pi}}}^{*}(0)$. Let also
$(\bar{{\boldsymbol{\pi}}}(u),\bar{{\boldsymbol{J}}}(u))$, $u\in[0,\bar{T}]$
the path provided by Lemma 4.12 satisfying
$\bar{{\boldsymbol{\pi}}}(0)=\pi_{0}$,
$\bar{{\boldsymbol{\pi}}}(\bar{T})=\pi_{1}$.

Set $T:=T^{*}+\bar{T}$ and let
$({{\boldsymbol{\pi}}}(u),{{\boldsymbol{J}}}(u))$, $u\in[0,T]$ be the path
defined by

$\big{(}{{\boldsymbol{\pi}}}(u),{{\boldsymbol{J}}}(u)\big{)}=\begin{cases}\big{(}{{\boldsymbol{\pi}}}^{*}(u),{{\boldsymbol{J}}}^{*}(u)\big{)}&\textrm{
if $u\in[0,T^{*}]$,}\\\
\big{(}\bar{{\boldsymbol{\pi}}}(u-T^{*}),{\boldsymbol{J}}(T^{*})+\bar{{\boldsymbol{J}}}(u-T^{*})\big{)})&\textrm{
if $u\in(T^{*},T]$.}\end{cases}$

Observe that ${\boldsymbol{\pi}}(0)={\boldsymbol{\pi}}(T)$ and extend
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ to the path
$({\boldsymbol{\pi}}^{T},{{\boldsymbol{J}}}^{T})$ defined on ${\mathbb{R}}$ by
periodicity. By construction, $t\mapsto{\boldsymbol{\pi}}^{T}(t)$ is
continuous and denote by $P_{T}$ the $T$-holonomic measure associated to
$({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})$ as in (4.13). Since
$\bar{T}\leq T_{0}$ for some fixed $T_{0}$, $P_{T}\to P$ as $T\to\infty$.
Moreover, by construction and by Lemma 4.12,

$\begin{split}{\boldsymbol{I}}_{m}(P_{T})&=\frac{1}{T}A_{m,T}({\boldsymbol{\pi}},{\boldsymbol{J}})=\frac{1}{T}A_{m,T^{*}}({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})+\frac{1}{T}A_{m,\bar{T}}(\bar{{\boldsymbol{\pi}}},\bar{{\boldsymbol{J}}})\\\
&\leq\frac{1}{T}A_{m,T^{*}}({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})+\frac{1}{T}C_{0}\end{split}$

so that, since $({\boldsymbol{\pi}}^{*},{\boldsymbol{J}}^{*})$ belongs to
${\mathscr{A}}_{P}$,
$\limsup_{T\to\infty}{\boldsymbol{I}}_{m}(P_{T})\leq{\boldsymbol{I}}_{m}(P)$.
As ${\boldsymbol{I}}_{m}$ is lower semi-continuous, actually,
$\lim_{T\to\infty}{\boldsymbol{I}}_{m}(P_{T})={\boldsymbol{I}}_{m}(P)$, as
claimed. ∎

##  5\.  Long time behavior of the hydrodynamical rate function

In this section, we consider the asymptotic in which we take first the limit
as $N\to\infty$ and then $T\to\infty$. The former limit is essentially the
content of the large deviations from the hydrodynamical scaling limit in which
we emphasize that the corresponding $T$-dependent rate function still depends
on the initial condition. To analyze the limit as $T\to\infty$ we first lift
this rate function to the set of translation invariant probabilities on
${\mathscr{S}}$ and then analyze its variational convergence, showing in
particular that the limit is independent of the initial condition.

Hereafter, fix $m\in(0,1)$ and a sequence $K_{N}$ such that $N^{-d}K_{N}\to
m$.

###  5.1. Hydrodynamical large deviations

Recall that the sequence $\\{\eta^{N},\,N\geq 1\\}$,
$\eta^{N}\in\Sigma_{N,K_{N}}$ is _associated_ to a measurable density
$\rho\colon{\mathbb{T}}^{d}\to[0,1]$ satisfying $\int\rho\,dx=m$ (hereafter of
total mass $m$) if and only if $\pi_{N}(\eta^{N})\to\rho(x)\,dx$ in the
topology of ${\mathcal{M}}_{+}({\mathbb{T}}^{d})$.

Recalling that
${\mathscr{M}}={\mathcal{M}}_{+}({\mathbb{T}}^{d})\times{\mathcal{H}}_{-p}^{d}$,
let $\color[rgb]{.2,0.2,.8}{\mathscr{S}}^{T}$, $T>0$, be the subset of the
paths in $D([0,T];{\mathscr{M}})$ which satisfy the continuity equation (2.21)
for any $0<s<t<T$. Let also $\color[rgb]{.2,0.2,.8}{\mathscr{S}}^{T}_{m}$ be
the subset of ${\mathscr{S}}^{T}$ given by the elements
$({\boldsymbol{\pi}},{\boldsymbol{J}})$ which satisfy
${\boldsymbol{\pi}}(t,{\mathbb{T}}^{d})=m$ for all $0\leq t\leq T$.

Finally, given a measurable function $\rho\colon{\mathbb{T}}^{d}\to[0,1]$ of
total mass $m$, that plays the role of the initial datum, let
$\color[rgb]{.2,0.2,.8}{\mathscr{S}}^{T}_{m,\mathrm{ac},\rho}$ be the subset
of elements $({\boldsymbol{\pi}},{\boldsymbol{J}})$ in ${\mathscr{S}}^{T}_{m}$
such that

  * (a’)

${\boldsymbol{\pi}}\in C([0,T],{\mathcal{M}}_{m}({\mathbb{T}}^{d}))$, and
${\boldsymbol{\pi}}(t,dx)={\boldsymbol{\rho}}(t,x)\,dx$ for some
${\boldsymbol{\rho}}$ such that $0\leq{\boldsymbol{\rho}}(t,x)\leq 1$.
Moreover,

$\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|\,\nabla{\boldsymbol{\rho}}\,|^{2}}{\sigma({\boldsymbol{\rho}})}\><\;\infty\;;$

  * (b’)

${\boldsymbol{J}}\in C([0,T],{\mathcal{H}}^{d}_{-p})$, and
${\boldsymbol{J}}(t)\,=\,\int_{0}^{t}{\boldsymbol{j}}(s)\,ds$, $t\in[0,T]$,
for some ${\boldsymbol{j}}$ in
$L^{2}\big{(}[0,T]\times{\mathbb{T}}^{d},\sigma({\boldsymbol{\rho}}(t,x))^{-1}\,dt\,dx;{\mathbb{R}}^{d}\big{)}$.
Thus,

$\int_{0}^{T}dt\int_{{\mathbb{T}}^{d}}dx\,\frac{|\,{\boldsymbol{j}}\,|^{2}}{\sigma({\boldsymbol{\rho}})}\><\;\infty\;;$

  * (c’)

${\boldsymbol{\pi}}(0,dx)=\rho(x)\,dx$.

Note that conditions (a’) and (b’) are the same of conditions (a) and (b)
below equation (2.21) apart from the fact that the here the path
$\big{(}{\boldsymbol{\rho}}(t),{\boldsymbol{j}}(t)\big{)}$ is defined only for
$t\in[0,T]$.

Let the _action_ $A_{m,T,\rho}:{\mathscr{S}}^{T}\to[0,+\infty]$, be defined by

$A_{m,T,\rho}({\boldsymbol{\pi}},{\boldsymbol{J}})\,=\,\begin{cases}\displaystyle\int_{0}^{T}\\!dt\\!\int_{{\mathbb{T}}^{d}}\\!dx\,\frac{|{\boldsymbol{j}}+\nabla{\boldsymbol{\rho}}-\sigma({\boldsymbol{\rho}})\,E|^{2}}{4\,\sigma({\boldsymbol{\rho}})}&\text{
if $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{T,m,{\rm
ac},\rho}$, }\\\ \vphantom{\bigg{\\{}}+\infty&\text{ otherwise.}\end{cases}$
(5.1)

The large deviation principle with respect to the hydrodynamical limit for the
WASEP dynamics can be stated as follows. Here, we understand that the
empirical density and current $({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})$
is defined as a map from $D\big{(}[0,T];\Sigma_{N,K_{N}}\big{)}$ to
${\mathscr{S}}^{T}$.

######  Theorem 5.1.

Fix $T>0$, $m>0$ and a density profile $\rho:{\mathbb{T}}^{d}\to[0,1]$ of
total mass $m$. For each sequence $(\eta^{N}:N\geq 1)$ associated to $\rho$,
each closed set ${\mathscr{F}}\subset{\mathscr{S}}^{T}$, and each open set
${\mathscr{G}}\subset{\mathscr{S}}^{T}$,

$\displaystyle\limsup_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{P}}^{N}_{\eta^{N}}\,\big{[}({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{F}}\,]\;\leq\;-\,\inf_{({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{F}}}A_{m,T,\rho}({\boldsymbol{\pi}},{\boldsymbol{J}})\;,$
$\displaystyle\liminf_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{P}}^{N}_{\eta^{N}}\,\big{[}({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})\in{\mathscr{G}}\,]\;\geq\;-\,\inf_{({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{G}}}A_{m,T,\rho}({\boldsymbol{\pi}},{\boldsymbol{J}})\;.$

Moreover, $A_{m,T,\rho}\colon{\mathscr{S}}^{T}\to[0,+\infty]$ is a good rate
function.

If one considers only the empirical density and disregards the empirical
current, the above result has been proven in [16] in case of SEP, see also
[15, Ch. 10]. This result has been extended to WASEP in [6]. Relying on the
super-exponential estimates proven in [16], the case in which one considers
the empirical current is discussed in [3] for the SEP. However, the topology
on the set of currents there introduced is different from the one used in the
present paper and the proof of the exponential tightness is incomplete. The
issue of the exponential tightness of the empirical current is fixed in the
present paper (in the topology here introduced). Indeed, Lemmata 4.3, 4.4, and
4.5 which hold also in the present setting, yield the exponential tightness of
the sequence
$\big{(}{\mathbb{P}}^{N}_{\eta^{N}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\colon
N\geq 1\big{)}$ thus completing, together with [3], the proof of the above
result for the SEP. The extension to WASEP requires, for the lower bound, a
density argument that has been carried out in detail in [6, Thm. 5.1] and can
be adapted to include the current.

Recall, from (2.23), the definition of the $T$-periodization of a path
$({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}_{m}$, which depends
only on the restriction of the path $({\boldsymbol{\pi}},{\boldsymbol{J}})$ to
the time interval $[0,T]$. Let
$\chi_{T}\colon{\mathscr{S}}^{T}\to{\mathscr{P}}_{\mathrm{stat}}$ be the
continuous map defined by

$\chi_{T}({\boldsymbol{\pi}},{\boldsymbol{J}}):=\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{s}({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})}\,ds\;,$

where the translation $\vartheta_{s}$ acting on ${\mathscr{S}}$ has been
introduced above (2.26). Namely,
$\chi_{T}({\boldsymbol{\pi}},{\boldsymbol{J}})$ is the $T$-holonomic measure
associated to the $T$-periodic path
$({\boldsymbol{\pi}}^{T},{\boldsymbol{J}}^{T})$ obtained by $T$-periodizing
the path $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}^{T}$.

Recall the definition of the empirical process $R_{T}({\boldsymbol{\eta}})$
introduced in (2.19). Since
$(\,{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}}^{T})\,,\,{\boldsymbol{J}}_{N}({\boldsymbol{\eta}}^{T})\,)=({\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})^{T},{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})^{T})$,
for each ${\boldsymbol{\eta}}\in D\big{(}[0,T];\Sigma_{N,K_{N}}\big{)}$,

$\displaystyle{\mathfrak{R}}_{T,N}({\boldsymbol{\eta}})=\;R_{T}({\boldsymbol{\eta}})\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}=\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{s}{\boldsymbol{\eta}}^{T}}\circ({\boldsymbol{\pi}}_{N},{\boldsymbol{J}}_{N})^{-1}\,ds$
(5.2)
$\displaystyle\quad=\;\frac{1}{T}\int_{0}^{T}\delta_{\vartheta_{s}\big{(}{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}})^{T},{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})^{T}\big{)}}\,ds\;=\;\chi_{T}\big{(}{\boldsymbol{\pi}}_{N}({\boldsymbol{\eta}}),{\boldsymbol{J}}_{N}({\boldsymbol{\eta}})\big{)}\;.$

Observe that the image of ${\mathscr{S}}^{T}$ by $\chi_{T}$ corresponds to the
set of $T$-holonomic measures. For $\rho$ of total mass $m$, let
${\boldsymbol{I}}_{m,T,\rho}\colon{\mathscr{P}}_{\mathrm{stat}}\to[0,+\infty]$
be the functional defined by

${\boldsymbol{I}}_{m,T,\rho}(P)\;:=\;\inf\big{\\{}A_{m,T,\rho}({\boldsymbol{\pi}},{\boldsymbol{J}})\,,\;({\boldsymbol{\pi}},{\boldsymbol{J}})\in\chi_{T}^{-1}\big{(}P\big{)}\big{\\}}\;,$
(5.3)

where we adopted the convention that $\inf\varnothing=+\infty$. In particular,
${\boldsymbol{I}}_{m,T,\rho}(P)<+\infty$ only for $T$-holonomic measures $P$.
Moreover, ${\boldsymbol{I}}_{m,T,\rho}(P)<+\infty$ only if the $T$-periodic
path $({\boldsymbol{\pi}},{\boldsymbol{J}})\in{\mathscr{S}}$ associated to $P$
satisfies the following condition. There there exists $s\in[0,T]$ such that
the restriction of $\vartheta_{s}({\boldsymbol{\pi}},{\boldsymbol{J}})$ to
$[0,T]$ belongs to ${\mathscr{S}}^{T}_{m,\mathrm{ac},\rho}$. In particular
${\boldsymbol{\pi}}(t)=\rho$ for some $t\in{\mathbb{R}}$.

In view of the identity (5.2), Theorem 5.1, by the contraction principle,
yields the following statement.

######  Corollary 5.2.

Fix $T>0$, $m>0$ and a density profile $\rho:{\mathbb{T}}^{d}\to[0,1]$ of
total mass $m$. For each sequence $(\eta^{N}:N\geq 1)$ associated to $\rho$,
each closed set ${\mathscr{F}}\subset{\mathscr{P}}_{\mathrm{stat}}$, and each
open set ${\mathscr{G}}\subset{\mathscr{P}}_{\mathrm{stat}}$,

$\displaystyle\limsup_{N\to\infty}\frac{1}{N^{d}}\,\log{\mathbb{P}}^{N}_{\eta^{N}}\,\big{[}{\mathfrak{R}}_{T,N}\in{\mathscr{F}}\,\big{]}\;\leq\;-\,\inf_{P\in{\mathscr{F}}}{\boldsymbol{I}}_{m,T,\rho}(P)\;,$
$\displaystyle\liminf_{N\to+\infty}\frac{1}{N^{d}}\,\log{\mathbb{P}}^{N}_{\eta^{N}}\,\big{[}{\mathfrak{R}}_{T,N}\in{\mathscr{G}}\,\big{]}\;\geq\;-\,\inf_{P\in{\mathscr{G}}}{\boldsymbol{I}}_{m,T,\rho}(P)\;.$

Moreover,
${\boldsymbol{I}}_{m,T,\rho}\colon{\mathscr{P}}_{\mathrm{stat}}\to[0,+\infty]$
is a good rate function.

###  5.2. Variational convergence of the hydrodynamical rate function.

The main result of this section reads as follows.

######  Theorem 5.3.

Fix $m\in(0,1)$ and a density profile $\rho:{\mathbb{T}}^{d}\to[0,1]$ of total
mass $m$. As $T\to+\infty$, the sequence $T^{-1}{\boldsymbol{I}}_{m,T,\rho}$
is equi-coercive uniformly in $\rho$, and $\Gamma$-converges uniformly in
$\rho$ to the functional ${\boldsymbol{I}}_{m}$ introduced in (2.26). That is:

  * (i)

For each $\ell>0$ there exists a compact
${\mathcal{K}}_{\ell}\subset{\mathscr{P}}_{\mathrm{stat}}$ such that for any
$T>1$ and any $\rho$,
$\big{\\{}P:T^{-1}{\boldsymbol{I}}_{m,T,\rho}(P)\leq\ell\big{\\}}\subset{\mathcal{K}}_{\ell}$;

  * (ii)

For any $P\in{\mathscr{P}}_{\mathrm{stat}}$, any sequence of density profiles
$\rho_{T}:{\mathbb{T}}^{d}\to[0,1]$ of total mass $m$ and any sequence
$P_{T}\to P$,

$\liminf_{T\to\infty}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho_{T}}(P_{T})\geq{\boldsymbol{I}}_{m}(P)\;;$

  * (iii)

If the external field $E$ is orthogonally decomposable, then for any
$P\in{\mathscr{P}}_{\mathrm{stat}}$ and any sequence of density profiles
$\rho_{T}:{\mathbb{T}}^{d}\to[0,1]$ of total mass $m$, there exists a sequence
$P_{T}\to P$ such that

$\limsup_{T\to\infty}\frac{1}{T}{\boldsymbol{I}}_{m,T,\rho_{T}}(P_{T})\leq{\boldsymbol{I}}_{m}(P)\;.$

###### Proof.

The proof is divided in three parts.

_Equicoercivity._ In view of the compactness of
${\mathcal{M}}_{m}({\mathbb{T}}^{d})$, the compact embedding of