# Dynamical large deviations for boundary driven gradient symmetric exclusion
processes in mild contact with reservoirs

Angèle Bouley  Angèle Bouley CNRS UMR 6085, Université de Rouen, Avenue de
l’Université, BP.12, Technopôle du Madrillet, F76801 Saint-Étienne-du-Rouvray,
France<EMAIL_ADDRESS>and  Claudio 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 a one-dimensional gradient symmetric exclusion process in mild
contact with boundary reservoirs. The hydrodynamic limit of the empirical
measure is given by a non-linear second-order parabolic equation with non-
linear Robin boundary conditions. We prove the dynamical large deviations
principle.

##  1\. Introduction

A major challenge addressed in recent years consists in the physical
description of non-equilibrium states (cf. [18, 3] and references therein).

The analysis of interacting particle systems stationary states, as the number
of degrees of freedom diverges, has proved to be an important step in this
program. The thermodynamical functional of the empirical measure, the
observable of interest in lattice gases, corresponds to the large deviation
rate functional for the empirical measure when the particles are distributed
according to the stationary state.

In equilibrium, when there is no current of particles, the invariant measure
is given by the Gibbs distribution specified by the Hamiltonian, and the
thermodynamical functional is called the free energy. In non-equilibrium, in
the presence of currents, the invariant measure is usually not known
explicitly, and a basic problem consists in characterizing the rate
functional, known as the non-equilibrium free energy.

In certain special cases [11, 10], combinatorial arguments based on special
feature of the exclusion processes provide a formula for the non-equilibrium
free energy. For general dynamics, a dynamical approach has been proposed [1].
One fist prove a dynamical large deviations principle for the empirical
measure in a fixed time interval $[0,T]$. The non-equilibrium free energy of a
density profile $\gamma$ is given by the minimal cost among all trajectories
which start from the stationary density profile and reaches $\gamma$ in finite
time. This procedures provides a dynamical variational formula for the non-
equilibrium free energy.

In this article we continue the investigation started in [9, 14, 8] of the
dynamical large deviations for interacting particle systems whose hydrodynamic
equation is described by a second-order parabolic equation with Robin boundary
conditions. We consider a gradient exclusion process leading to a non-linear
hydrodynamic equation. This non-linearity turns the large deviations rate
functional non-convex, a property much used in the proof of the large
deviations principle lower-bound in the linear case [14].

In possession of the dynamical large deviations, one can proceed as in [5, 13,
2, 8] to prove the large deviations for the empirical measure under the
invariant measure, deriving a dynamical variational formula for the non-
equilibrium free energy. This is left for a future work. The next step, a
challenging open problem, consists in obtaining a simple formula for the non-
equilibrium free energy, derived in [11, 1] for symmetric exclusion processis
in strong contact with reservoirs and in [10, 8] for these dynamics in mild
contact with reservoirs.

This article presents two novelties which can be useful in other contexts, as
for reaction-diffusion models without the concavity assumption on the creation
and destruction rates [17, 6, 16].

Denote by $I_{[0,T]}(\cdot)$ the large deviations rate functional. One of the
main difficulties in the proof of the lower bound consists in showing that any
trajectory $\pi(t,dx)=u(t,x)\,dx$ with finite rate function can be
approximated by a sequence of smooth paths $\pi_{n}(t,dx)=u_{n}(t,x)\,dx$ such
that $\pi_{n}\to\pi$, $I_{[0,T]}(\pi_{n})\to I_{[0,T]}(\pi)$.

The proof of this density is divided in several steps. When the rate
functional is not convex, the standard argument to settle one of them [24, 4]
relies on a Riesz representation lemma of linear functionals in negative
Sobolev spaces (cf. [4, Lemma 4.8]). We present here a much simpler and direct
proof of this step.

The second technical novelty concerns the regularity of paths with finite rate
functionals. Fix a path $\pi(t,dx)=u(t,x)\,dx$ such that
$I_{[0,T]}(\pi)<\infty$. It is known, at least since [4], that such a path is
weakly continuous in time. More precisely, that for each space-dependent
$C^{1}$ function $g$, the map $t\mapsto\langle\pi_{t},g\rangle$ is uniformly
continuous. Proposition 4.11 below asserts that this map is absolutely
continuous and its derivative belongs to $L^{1}$. This property permits to
integrate by parts $\int_{0}^{T}\langle\partial_{t}H_{t},u_{t}\rangle\,dt$,
when $H(t,x)=h(t)\,g(x)$ and plays a crucial role in the decomposition of the
rate functional presented in Section 5 below.

##  2\. Notation and results

### The model

We consider a one-dimensional boundary driven gradient symmetric exclusion
processes in weak contact with boundary reservoirs. Fix $N\geq 1$, and let
$\color[rgb]{0,0,1}\Lambda_{N}=\\{-N+1,\dots,N-1\\}$. The state-space is
represented by $\color[rgb]{0,0,1}{\mathfrak{S}}_{N}=\\{0,1\\}^{\Lambda_{N}}$
and the configurations by the Greek letters $\eta$, $\xi$ so that
$\color[rgb]{0,0,1}\eta(x)$, $x\in\Lambda_{N}$, represents the number of
particles at site $x$ for the configuration $\eta$. Here and below all
notation introduced in the text and not in displayed equations is indicated in
blue.

The theorems stated in this article apply to all gradient symmetric exclusion
dynamics in mild contact with reservoirs. To avoid heavy notation, we consider
a specific example. Fix  $a>-1/2$, $0<{\mathfrak{a}}$, ${\mathfrak{b}}<1$. The
variables ${\mathfrak{a}}$, ${\mathfrak{b}}$ represent the density of the
left, right reservoirs, respectively. Assume, without loss of generality, that
${\mathfrak{a}}\leq{\mathfrak{b}}$. Let
$r_{x,x+1}\colon{\mathfrak{S}}_{N}\to{\mathbb{R}}_{+}$, $-N+2\leq x\leq N-3$,
be the rates at which the variables $\eta(x)$, $\eta(x+1)$ are exchanged, and
given by

$\displaystyle r_{x,x+1}(\eta)=1+a\,[\,\eta(x-1)+\eta(x+2)\,]\;.$

At the boundary, replace the occupation variables which do not belong to
$\Lambda_{N}$ by the density of the reservoirs:

$\displaystyle r_{-N+1,-N+2}(\eta)=1+a\,[\,{\mathfrak{a}}+\eta(-N+3)\,]\;,$
$\displaystyle r_{N-2,N-1}(\eta)=1+a\,[\,\eta(N-3)+{{\mathfrak{b}}}\,]\;.$

The generator $L_{N}$ of the Markov process is given by

$\displaystyle L_{N}=L_{N,0}+L_{N,b}$

where $L_{N,0}$, $L_{N,b}$ correspond to the bulk, boundary dynamics,
respectively. The action of the generator $L_{N,0}$ on functions
$f\colon{\mathfrak{S}}_{N}\to\mathbb{R}$ is given by

$\displaystyle(L_{N,0}f)(\eta)=N^{2}\sum_{x}r_{x,x+1}(\eta)\,[\,f(\eta^{x,x+1})-f(\eta)\,]\;,$

where the sum is carried over all $x\in{\mathbb{Z}}$ such that
$\\{x,x+1\\}\subset\Lambda_{N}$. Moreover, $\eta^{x,x+1}$ stands for the
configuration obtained from $\eta$ by exchanging the occupation variables
$\eta(x)$ and $\eta(x+1)$:

$\displaystyle\eta^{x,x+1}(z)=\begin{cases}\eta(z)&\mbox{ if }z\neq x,x+1\\\
\eta(x+1)&\mbox{ if }z=x\\\ \eta(x)&\mbox{ if }z=x+1\;.\end{cases}$

The generator of the boundary dynamics acts on functions
$f\colon{\mathfrak{S}}_{N}\to\mathbb{R}$ as

$\displaystyle(L_{N,b}f)(\eta)=N\,\Big{\\{}\,r_{L}(\eta)\,[\,f(\eta^{-N+1})-f(\eta)\,]\,+\,r_{R}(\eta)\,[\,f(\eta^{N-1})-f(\eta)\,]\,\Big{\\}}\;.$

In this formula, $\eta^{x}$ represents the configuration obtained from $\eta$
by flipping the occupation variable at site $x$:

$\displaystyle\eta^{x}(z)=\begin{cases}\eta(z)&\mbox{ if }z\neq x\;,\\\
1-\eta(x)&\mbox{ if }z=x\;,\\\ \end{cases}$

and the rates $r_{L}$, $r_{R}$ are given by

$\displaystyle
r_{L}(\eta)=\eta(-N+1)\,(\,1-{{\mathfrak{a}}}\,)\,+\,[\,1-\eta(-N+1)\,]\,{{\mathfrak{a}}}\;,$
$\displaystyle
r_{R}(\eta)=\eta(N-1)\,(\,1-{{\mathfrak{b}}}\,)\,+\,[\,1-\eta(N-1)\,]\,{{\mathfrak{b}}}\;.$

Note that the bulk dynamics has been accelerated by $N^{2}$, while the
boundary one by $N$. Denote by $(\eta^{N}_{t})_{t\geq 0}$ the Markov process
associated to the generator $L_{N}$. For a smooth function
$\rho\colon\Omega\to]0,1[$, let $\nu_{\rho(\cdot)}^{N}$ be the Bernoulli
product measure on ${\mathfrak{S}}_{N}$ with marginals given by:

$\displaystyle\nu_{\rho(\cdot)}^{N}(\eta(x)=1)=\rho(x/N)\;,\quad
x\in\Lambda_{N}\;.$

When the function $\rho$ is constant equal to ${\mathfrak{c}}\in[0,1]$, we
denote $\nu_{\rho(\cdot)}^{N}$ by
$\color[rgb]{0,0,1}\nu_{{\mathfrak{c}}}^{N}$. An elementary computation shows
that $\nu_{{\mathfrak{a}}}^{N}$ is a reversible stationary state for the
Markov chain $\eta^{N}_{t}$ when ${\mathfrak{a}}={\mathfrak{b}}$.

For a metric space $\mathcal{E}$ and $T>0$, denote by
$\color[rgb]{0,0,1}D([0,T],\mathcal{E})$, the space of right-continuous
functions with left limits ${\bm{x}}\colon[0,T]\to{\mathcal{E}}$ endowed with
the Skorohod topology and its Borel $\sigma$-algebra. For a probability
measure $\mu$ on ${\mathfrak{S}}_{N}$, let
$\color[rgb]{0,0,1}\mathbb{P}^{N}_{\mu}$ be the measure on
$D([0,T],{\mathfrak{S}}_{N})$ induced by the continuous-time Markov process
associated to the generator $L_{N}$ starting from $\mu$. Expectation with
respect to $\mathbb{P}^{N}_{\mu}$ is denoted by
$\color[rgb]{0,0,1}\mathbb{E}^{N}_{\mu}$.

### Gradient condition

For $x\in{\mathbb{Z}}$ such that $\\{x,x+1\\}\subset\Lambda_{N}$, denote by
$j_{x,x+1}$ the instantaneous current over the bond $(x,x+1)$. This is the
rate at which a particle jumps from $x$ to $x+1$ minus the rate at which it
jumps from $x+1$ to $x$:

$j_{x,x+1}(\eta)=r_{x,x+1}(\eta)\,[\,\eta(x)-\eta(x+1)\,]\;.$

By definition of the rates $r_{x,x+1}$, the current is given by

$j_{x,x+1}(\eta)=[\,\eta(x)-\eta(x+1)\,]\,+\,a\,(\,\tau_{x-1}f_{1}-\tau_{x+1}f_{1}\,)\,+\,a\,(\,\tau_{x}f_{2}-\tau_{x-1}f_{2}\,)\;,$
(2.1)

where $\\{\tau_{x}:x\in{\mathbb{Z}}\\}$ stands for the group of translations,
$(\tau_{x}\eta)(y)=\eta(x+y)$, $y\in{\mathbb{Z}}$, and
$(\tau_{x}f)(\eta)=f(\tau_{x}\eta)$,

$f_{1}(\eta)\,=\,\eta(0)\,\eta(1)\;,\quad f_{2}(\eta)\,=\,\eta(0)\,\eta(2)\;.$

Note that the instantaneous current can be written as a sum of functions minus
their translations. This property is called the gradient condition, cf. [19,
Chapter 7]. We carefully defined the jump rates $r_{x,x+1}(\eta)$ in order to
fullfil this condition.

Denote by $\color[rgb]{0,0,1}\Omega$ the open set $]-1,1[$. Let
$\color[rgb]{0,0,1}\mathcal{M}$ be the set of non-negative measures on
$\Omega$ with total mass bounded by 2 endowed with the weak topology. For a
continuous function $F\colon[-1,1]\to\mathbb{R}$ and a measure
$\pi\in\mathcal{M}$, $\langle\pi,F\rangle$ represents the integral of $F$ with
respect to $\pi$:

$\displaystyle{\color[rgb]{0,0,1}\langle\pi,F\rangle}=\int_{\Omega}F(x)\,\pi(dx)\;.$

Denote by $\pi^{N}=\pi^{N}(\eta)$, $\eta\in{\mathfrak{S}}_{N}$, the empirical
measure obtained by assigning mass $1/N$ to each particle of a configuration
$\eta$:

${\color[rgb]{0,0,1}{\pi^{N}}}\,=\,\frac{1}{N}\sum_{x\in\Lambda_{N}}\eta(x)\,\delta_{x/N}\;\;\text{and
let}\;\;{\color[rgb]{0,0,1}\pi^{N}_{t}}=\pi^{N}(\eta^{N}_{t})\,:=\,\frac{1}{N}\sum_{x\in\Lambda_{N}}\eta^{N}_{t}(x)\,\delta_{x/N}\;,$
(2.2)

where $\delta_{y}$ represents the Dirac mass at $y$.

For an open subset ${\mathcal{A}}$ of $\mathbb{R}$,
$\color[rgb]{0,0,1}C^{m}({\mathcal{A}})$, $1\leq m\leq+\infty$, stands for the
space of real functions defined on ${\mathcal{A}}$ that are $m$ times
continuously differentiable. Let $\color[rgb]{0,0,1}C^{m}_{0}({\mathcal{A}})$,
$\color[rgb]{0,0,1}C_{K}^{m}({\mathcal{A}})$, $1\leq m\leq+\infty$, be the
subset of functions in $C^{m}({\mathcal{A}})$ which vanish at the boundary of
${\mathcal{A}}$, with compact support in ${\mathcal{A}}$, respectively.

Fix a function $G$ in $C^{3}(\Omega)$, which can be extended continuously, as
well as its derivatives to $\overline{\Omega}$. By the gradient condition
(2.1), which permits two summation by parts and, in consequence, a Taylor
expansion of $G$ up to the second order, providing in this way a factor
$1/N^{2}$,

$\displaystyle
L_{N}\langle\pi^{N},G\rangle\;=\;\frac{1}{N}\sum_{x=-N+2}^{N-2}(\Delta
G)(x/N)\,\big{\\{}\,\eta(x)+2\,a\,\tau_{x}f_{1}\,-\,a\,\tau_{x}f_{2}\,\big{\\}}\;+\;B_{N}\;+\;R_{N}\;,$

where $B_{N}=B^{L}_{N}+B^{R}_{N}$ is the boundary term

$\displaystyle B^{L}_{N}\;$
$\displaystyle=\;G(-1)\,[\,{\mathfrak{a}}-\eta(-N+1)\,]\;+\;(\nabla
G)(-1)\,\eta(-N+2)$ $\displaystyle+\,a\,(\nabla
G)(-1)\,\big{\\{}\,\tau_{-N+1}\,f_{1}\,+\,\tau_{-N+2}\,f_{1}\,-\,\tau_{-N+1}\,f_{2}\,\big{\\}}$
$\displaystyle-\,(\nabla
G)(-1)\,\big{[}\,\eta(-N+2)-\eta(-N+1)\,\big{]}\,\big{\\{}\,1+a\,{\mathfrak{a}}+a\eta(-N+3)\,\big{\\}}\;,$
$\displaystyle B^{R}_{N}\;$
$\displaystyle=\;G(1)\,[\,{\mathfrak{b}}-\eta(N-1)\,]\;-\;(\nabla
G)(1)\,\eta(N-2)$ $\displaystyle-\,a(\nabla
G)(1)\,\big{\\{}\,\tau_{N-2}\,f_{1}\,+\,\tau_{N-3}\,f_{1}\,-\,\tau_{N-3}\,f_{2}\,\big{\\}}$
$\displaystyle-\,(\nabla
G)(1)\,\big{[}\,\eta(N-1)-\eta(N-2)\,\big{]}\,\big{\\{}\,1+a\,{\mathfrak{b}}+a\eta(N-3)\,\big{\\}}\;,$

and $R_{N}$ is a remainder whose absolute value is bounded by $C_{0}(G)/N$ for
some finite constant $C_{0}$ which depends on $G$ but not on $N$. All
constants below are allowed to depend on the parameters $a$, ${\mathfrak{a}}$,
${\mathfrak{b}}$ with any mention to that in the notation. In the previous
formula, $\color[rgb]{0,0,1}\Delta G$, $\color[rgb]{0,0,1}\nabla G$ stand for
the Laplacian, gradient of $G$, respectively.

### Heuristics

Consider the martingale $M_{N}(t)$ defined by

$\displaystyle
M_{N}(t)\;=\;\langle\pi^{N}_{t},G\rangle\,-\,\langle\pi^{N}_{0},G\rangle\,-\,\int_{0}^{t}L_{N}\langle\pi^{N}_{s},G\rangle\,ds\;.$

where $G\colon[-1,1]\to{\mathbb{R}}$ is a smooth function. A computation of
its quadratic variation, similar to the one performed in [19, Chapter 4]
yields that it vanishes in $L^{2}$ as $N\to\infty$.

For a function $f\colon\\{0,1\\}^{{\mathbb{Z}}}\to{\mathbb{R}}$ which depends
only on a finite number of coordinates, let
$\widehat{f}\colon[0,1]\to{\mathbb{R}}$ be given by

$\displaystyle\widehat{f}({\mathfrak{c}})\,=\,E_{\nu^{N}_{{\mathfrak{c}}}}[\,f\,]\;.$

In this formula, $N$ provided that $N$ has been chosen sufficiently large to
contain the support of $f$.

If one is allowed to replace $(\tau_{x}f)(\eta^{N}_{s})$ by
$\widehat{f}(u(s,x/N))$ for some measurable function
$u\colon[0,T]\times[-1,1]\to[0,1]$, which is essentially the content of the
so-called two blocks estimate, it follows from the convergence of the
martingale $M_{N}(t)$ to $0$ and from the computation of the previous
subsection that

$\displaystyle\int_{-1}^{1}G(x)\,u(t,x)\,dx\;-\;\int_{-1}^{1}G(x)\,u(0,x)\,dx\,-\,\int_{0}^{t}\int_{-1}^{1}(\Delta
G)(x)\,P_{a}(u(s,x))\,dx\,ds$
$\displaystyle\quad=\;\int_{0}^{t}\Big{\\{}\,G(-1)\,[\,{\mathfrak{a}}-u(s,-1)\,]\,+\,(\nabla
G)(-1)\,P_{a}(u(s,-1))\,\Big{\\}}\,ds$
$\displaystyle\quad-\;\int_{0}^{t}\Big{\\{}\,G(1)\,[\,{\mathfrak{b}}-u(s,1)\,]\,+\,(\nabla
G)(1)\,P_{a}(u(s,1))\,\Big{\\}}\,ds\,+R_{N}\;,$

where $P_{a}(z)=z+a^{2}z$. The last term in $B^{L}_{N}$, $B^{R}_{N}$ does not
contribute because it has mean-zero with respect to all measures
$\nu^{N}_{{\mathfrak{c}}}$.

Integrating twice by parts yields that $u(t,x)$ is the solution of the partial
differential equation

$\left\\{\begin{aligned} &\partial_{t}u\;=\;\nabla\big{(}D(u)\nabla
u\,)\;,\quad(t,x)\in(0,\infty)\times\Omega\;,\\\ &D(u)\nabla
u=u-{\mathfrak{a}}\;,\quad(t,x)\in(0,\infty)\times\\{-1\\}\;,\\\ &D(u)\nabla
u={\mathfrak{b}}-u\;,\quad(t,x)\in(0,\infty)\times\\{1\\}\;,\end{aligned}\right.$
(2.3)

where

${\color[rgb]{0,0,1}D(\rho)=P^{\prime}_{a}(\rho)=1+2a\rho}\;.$

### Hydrodynamic limit

Let $\Pi_{N}\colon D([0,T],{\mathfrak{S}}_{N})\to D([0,T],\mathcal{M})$ be the
map which associates to a trajectory $(\eta^{N}_{t}:0\leq t\leq T)$ the
empirical measure trajectory $(\pi^{N}_{t}:0\leq t\leq T)$. For a probability
measure $\mu$ in ${\mathfrak{S}}_{N}$, $\mathbb{Q}_{\mu}^{N}$ stands for the
measure on $D([0,T],\mathcal{M})$ given by
$\color[rgb]{0,0,1}\mathbb{Q}_{\mu}^{N}=\mathbb{P}_{\mu}^{N}\,(\Pi^{N})^{-1}$.

A sequence of measures $(\mu^{N})_{N\in\mathbb{N}}$ on ${\mathfrak{S}}_{N}$ is
said to be associated to a density profile $\rho_{0}\colon\Omega\to[0,1]$ if

$\underset{N\to\infty}{\lim}\mu_{N}\Big{[}\,\Big{|}\,\langle\pi_{N}\,,\,G\rangle\,-\,\int_{\Omega}G(x)\,\rho_{0}(x)\,dx\,\Big{|}>\delta\,\Big{]}\,=\,0$
(2.4)

for all continuous functions $G\colon[-1,1]\to\mathbb{R}$ and $\delta>0$. The
first main result of this article establishes the hydrodynamic limit of the
empirical measure.

Denote by $\color[rgb]{0,0,1}C^{m,n}([0,T]\times[-1,1])$, $1\leq
n,m\leq+\infty$, the space of functions
$G\colon[0,T]\times[-1,1]\to\mathbb{R}$ with $m$ continuous derivatives in
time and $n$ continuous derivatives in space.

Let $\color[rgb]{0,0,1}{\mathcal{H}}^{1}={\mathcal{H}}^{1}(\Omega)$ be the
Sobolev space of measurable functions $G\colon\Omega\to\mathbb{R}$ with
generalized derivatives $\nabla G$ in $L^{2}(\Omega)$. The space
${\mathcal{H}}^{1}$ endowed with the scalar product
$\langle\cdot,\cdot\rangle_{1}$, defined by

$\langle G\,,\,H\rangle_{1}\;:=\;\langle G\,,\,H\rangle\;+\;\langle\nabla
G\,,\,\nabla H\rangle\;,$ (2.5)

is a Hilbert space. The corresponding norm is denoted by
$\|\cdot\|_{{\mathcal{H}}^{1}}$:

$\|G\|_{{\mathcal{H}}^{1}}^{2}\;:=\;\int_{-1}^{1}|G(x)|^{2}\;dx\;+\;\int_{-1}^{1}|\nabla
G(x)|^{2}\;dx\;.$

Recall from [14] that any function $H$ in ${\mathcal{H}}^{1}$ has a continuous
version.

######  Definition 2.1.

Fix $T>0$ and a density profile $\rho_{0}\colon[-1,1]\to[0,1]$. A mesurable
function $u\colon[0,T]\times[-1,1]\to[0,1]$ is said to be a weak solution of
the differential equation (2.3) with initial condition
$u(0,\cdot)=\rho_{0}(\cdot)$ if

  1. (a)

$u$ belongs to $L^{2}([0,T],\mathcal{H}^{1})$:

$\int_{0}^{T}\left(\int_{-1}^{1}|\nabla u(s,x)|^{2}dx\right)ds<\infty$

  2. (b)

For every function $H$ in $C^{1,2}([0,T]\times[-1,1]$:

$\displaystyle\langle u_{T},H_{T}\rangle\,-\,\langle u_{0},H_{0}\rangle$
$\displaystyle=\int_{0}^{T}\langle
u_{s},\partial_{s}H_{s}\rangle\,ds\,-\,\int_{0}^{T}\langle D(u_{s})\,\nabla
u_{s},\nabla H_{s}\rangle\,ds$
$\displaystyle+\int_{0}^{T}[\,{{\mathfrak{a}}}-\rho_{s}(-1)\,]\,H_{s}(-1)\,ds+\int_{0}^{T}[\,{{\mathfrak{b}}}-\rho_{s}(1)\,]\,H_{s}(1)\,ds\;.$

Corollary 8.2 asserts that there is at most one weak solution of (2.3).
Existence for any initial condition $\rho_{0}\colon\Omega\to[0,1]$ follows
from the tightness part of the next theorem.

######  Theorem 2.2.

Fix $T>0$ and a profile $\rho_{0}\colon\Omega\to[0,1]$. Let
$(\mu^{N})_{N\in\mathbb{N}}$ be a sequence of measure on ${\mathfrak{S}}_{N}$
associated to $\rho_{0}$. Then, the sequence of probability measures
$(\mathbb{Q}_{\mu^{N}}^{N})_{N\in\mathbb{N}}$ converges weakly to the
probability measure $\mathbb{Q}$ concentrated on the trajectory
$\pi(t,dx)=u(t,x)\,dx$, where $u$ is the unique weak solution of the
hydrodynamic equation (2.3) with initial condition $\rho_{0}(\cdot)$.

The proof of this theorem is given in Section 3.

### The rate functional

Fix $T>0$. Let $\mathcal{M}^{0}$ be the subset of $\mathcal{M}$ of all
absolutely continuous measures with respect to the Lebesgue measure with
positive density bounded by 1:

$\displaystyle{\color[rgb]{0,0,1}\mathcal{M}^{0}}\,:=\,\\{\pi\in\mathcal{M}\mbox{
: }\pi(dx)=\gamma(x)dx\mbox{ and }0\leq\gamma(x)\leq 1\mbox{ a.e. }\\}\;.$

######  Definition 2.3.

Denote by $\mathcal{Q}\colon D([0,T],\mathcal{M}^{0})\to{\mathbb{R}}_{+}$ the
energy given by

$\displaystyle\mathcal{Q}(\pi)\,=\,\underset{G}{\sup}\,\Big{\\{}\,2\int_{0}^{T}\langle
u_{t},\nabla
G_{t}\rangle\,dt-\int_{0}^{T}\int_{-1}^{1}\chi(u(t,x))\,G(t,x)^{2}\,dx\,dt\,\Big{\\}}\;,$

where $\pi(t,dx)=u(t,x)\,dx$, the supremum is carried over all functions $G\in
C^{1,2}([0,T]\times[-1,1])$ and $\color[rgb]{0,0,1}\chi(a)=a(1-a)$ is the
static compressibility of the exclusion process.

Let $\color[rgb]{0,0,1}D_{{\mathcal{E}}}([0,T],{\mathcal{M}}^{0})$ be the
trajectories in $D([0,T],{\mathcal{M}}^{0})$ with finite energy. Fix $0\leq
r<s\leq T$, and a trajectory $\pi\in
D_{{\mathcal{E}}}([0,T],\mathcal{M}^{0})$, $\pi(t,du)=u(t,x)dx$. Denote by
$L_{[r,s]}\colon C^{1,1}([0,T]\times[-1,1])\to\mathbb{R}$ the linear
functional given by

${\color[rgb]{0,0,1}L_{[r,s]}(H)}\,=\,\langle\pi_{s},H_{s}\rangle\,-\,\langle\pi_{r},H_{r}\rangle\,-\,\int_{r}^{s}\langle\pi_{t},\partial_{t}H_{t}\rangle\,dt\;.$
(2.6)

When $r=0$, $s=T$, we represent $L_{[s,t]}$ by $L_{0}$:
$\color[rgb]{0,0,1}L_{0}(H)=L_{[0,T]}(H)$.

For each $H\in C^{1,1}([r,s]\times[-1,1])$, let
$\hat{J}_{H}=\hat{J}_{H,[r,s]}\colon
D_{{\mathcal{E}}}([0,T],\mathcal{M}^{0})\to{\mathbb{R}}$ be the functional
given by:

$\displaystyle\hat{J}_{H}(\pi)\,$
$\displaystyle=\,L_{[r,s]}(H)\,+\,\int_{r}^{s}\int_{-1}^{1}D(u(t,x))\,\nabla
u(t,x)\,\nabla H(t,x)\,dx\,dt$
$\displaystyle-\,\int_{r}^{s}\int_{-1}^{1}\sigma(u(t,x))\,(\nabla
H(t,x))^{2}\,dx\,dt\,-\ \int_{r}^{s}B(u_{t},H_{t})\,dt\;,$

where $\color[rgb]{0,0,1}\sigma(a)=\chi(a)\,D(a)$ is the mobility, and

$\displaystyle{\color[rgb]{0,0,1}B(u_{t},H_{t})}\,:=\,u_{t}(1)\,(1-{{\mathfrak{b}}})\,\left(e^{-H(t,1)}-1\right)\,+\,{{\mathfrak{b}}}\,[1-u_{t}(1)]\,\left(e^{H(t,1)}-1\right)$
(2.7)
$\displaystyle\;\;+u_{t}(-1)\,(1-{{\mathfrak{a}}})\,\left(e^{-H(t,-1)}-1\right)\,+\,{{\mathfrak{a}}}\,[1-u_{t}(-1)]\,\left(e^{H(t,-1)}-1\right)\;.$

We extend the definition of the functional $\hat{J}_{H}$ to the space
$D([0,T],\mathcal{M})$ by setting

${\color[rgb]{0,0,1}J_{H}(\pi)}\;=\;\left\\{\begin{array}[]{ll}\hat{J}_{H}(\pi)&\mbox{if
}\pi\in D_{{\mathcal{E}}}([0,T],\mathcal{M}^{0})\;,\\\ +\infty&\mbox{otherwise
}\;.\end{array}\right.$

Fix a profile $\gamma\colon\Omega\to[0,1]$ and $0\leq r<s\leq T$. Denote by
$I_{[r,s]}\colon D([0,T],\mathcal{M})\to[0,+\infty]$ the functional given by

$\displaystyle{\color[rgb]{0,0,1}I_{[r,s]}(\pi)}\,:=\,\underset{H\in
C^{1,1}([r,s]\times[-1,1])}{\sup}\,J_{H}(\pi)\;,$

and let

${\color[rgb]{0,0,1}I_{[r,s]}(\pi|\gamma)}\;=\;\left\\{\begin{array}[]{ll}I_{[r,s]}(\pi)&\mbox{if
}\pi_{0}(dx)\,=\,\gamma(x)\,dx\;,\\\ +\infty&\mbox{otherwise
}\;.\end{array}\right.$

######  Theorem 2.4.

Fix $T>0$ and a measurable function $\gamma\colon[-1,1]\to[0,1]$. The
functional $I_{[0,T]}(\cdot|\gamma)\colon D([0,T],\mathcal{M})\to[0,\infty]$
is lower semicontinuous and has compact level sets.

This result is proved in Section 4, where we present some properties of the
rate functional $I_{[0,T]}(\cdot)$.

### Dynamical large deviations

For $0<\beta<1$, denote by $\color[rgb]{0,0,1}C^{n+\beta}([-1,1])$, $n$ a non-
negative integer, the functions in $C^{n}([-1,1])$ whose $n$-th derivative is
Hölder continuous with parameter $\beta$. The main theorem of this article
reads as follows.

######  Theorem 2.5.

Fix $T>0$, a density profile $\gamma\in C^{2+\beta}([-1,1])$ for some
$0<\beta<1$ satisfying the boundary conditions

$D(\gamma(-1))\,\nabla\gamma(-1)\,=\,\gamma(-1)-{\mathfrak{a}}\;,\quad
D(\gamma(1))\,\nabla\gamma(1)\,=\,{\mathfrak{b}}-\gamma(1)\;,$ (2.8)

and a sequence of configurations $\\{\eta^{N}\\}_{N\in{\mathbb{N}}}$. Assume
that $\delta_{\eta^{N}}$ is associated to $\gamma$ in the sense of (2.4).
Then, the sequence of probability measures
$\\{{\mathbb{Q}}_{\eta^{N}}\\}_{N\geq 1}$ satisfies a large deviation
principle with speed $N$ and good rate function $I_{[0,T]}(\cdot|\gamma)$.
Namely, for each closed set $C\subset D([0,T],\mathcal{M})$,

$\displaystyle\limsup_{N\to\infty}\frac{1}{N}\log\mathbb{Q}_{\eta^{N}}(C)\leq-\inf_{\pi\in
C}I_{[0,T]}(\pi|\gamma)$

and for each open set $O\subset D([0,T],\mathcal{M})$,

$\displaystyle\liminf_{N\to\infty}\frac{1}{N}\log\mathbb{Q}_{\eta^{N}}(O)\geq-\,\inf_{\pi\in
O}I_{[0,T]}(\pi|\gamma)\;.$

We prove this theorem in Section 7.

######  Remark 2.6.

The hypotheses that $\gamma$ belongs to $C^{2+\beta}([-1,1])$ for some
$0<\beta<1$ and satisfies the boundary conditions (2.8) are only used in the
proof of Proposition 8.3 where we derive a maximum principle for the solutions
of the hydrodynamic equation (2.3).

In other words, let $u$ be the solution of the hydrodynamic equation (2.3)
with initial condition $\gamma$. If one is able to prove, under weaker
assumptions on $\gamma$, that for each $\delta>0$, there exists $\epsilon>0$
such that $\epsilon\leq u(t,x)\leq 1-\epsilon$ for every
$(t,x)\in[\delta,T]\times[-1,1]$, then Theorem 2.5 holds under these weaker
hypotheses.

###  $I_{[0,T]}$-density

The main step in the proof of the large deviations lower bound consists in
showing that any trajectory $\pi$ in $D([0,T],{\mathcal{M}})$ with finite rate
function can be approximated by a sequence of regular trajectories $\pi^{n}$
in such a way that $I_{[0,T]}(\pi^{n}|\gamma)$ converges to
$I_{[0,T]}(\pi|\gamma)$. The precise statement requires some notation.

######  Definition 2.7.

A subset $A$ of $D([0,T],\mathcal{M})$ is said $I_{[0,T]}(.|\gamma)$-dense if
for any $\pi\in D([0,T],\mathcal{M})$ such that
$I_{[0,T]}(\pi|\gamma)<\infty$, there exists a sequence
$(\pi^{n})_{n\in\mathbb{N}}$ in $A$ such that $\pi^{n}\to\pi$ in
$D([0,T],\mathcal{M})$, and $I_{[0,T]}(\pi^{n}|\gamma)\to
I_{[0,T]}(\pi|\gamma)$.

######  Definition 2.8.

Given $\gamma\colon[-1,1]\to[0,1]$, let $\Pi_{\gamma}$ be the collection of
all paths $\pi(t,dx)=u(t,x)dx$ in $D([0,T],{\mathcal{M}}^{0})$ such that

  * (a)

There exists ${\mathfrak{t}}>0$, such that $u$ follows the hydrodynamic
equation (2.3) with initial condition $\gamma(\cdot)$ in the time interval
$[0,{\mathfrak{t}}]$. In particular, $u(0,\cdot)=\gamma(\cdot)$.

  * (b)

There exists $\epsilon>0$ such that $\epsilon\leq u(t,x)\leq 1-\epsilon$ for
all $(t,x)$ in $[{\mathfrak{t}},T]\times[-1,1]$, and $u$ is smooth on
$({\mathfrak{t}},T]\times[-1,1]$.

Mind that in this definition we do not assume that $\gamma$ fulfils the
conditions of Theorem 2.5. Proposition 6.1 states that for density profiles
$\gamma:[-1,1]\to[0,1]$ satisfying the hypotheses of Theorem 2.5, the set
$\Pi_{\gamma}$ is $I_{[0,T]}(\cdot|\gamma)$-dense. Moreover, under the
hypotheses of Theorem 2.5, the first requirement in condition (b) can be
extended to the set $(0,T]\times[-1,1]$. Indeed, by Proposition 8.3, for every
$0<\delta\leq T$, there exists $\epsilon>0$ such that $\epsilon\leq u(t,x)\leq
1-\epsilon$ for all $(t,x)$ in $[\delta,T]\times[0,1]$.

At the end of Section 6, we provide an explicit formula for the rate function
of trajectories in $\Pi_{\gamma}$. This result does not require the density
profile $\gamma$ to satisfy the hypotheses of Theorem 2.5. For
$0\leq\varrho\leq 1$, let ${\mathfrak{p}}_{\varrho}$,
${\mathfrak{c}}_{\varrho}\colon[0,1]\times T\to{\mathbb{R}}$ be given by

$\displaystyle{\mathfrak{p}}_{\varrho}(a,M)\;=\;[1-a]\,\varrho\,e^{M}\;-\;a\,[1-\varrho]\,e^{-M}\;,$
(2.9)
$\displaystyle{\mathfrak{c}}_{\varrho}(a,M)\;=\;[1-a]\,\varrho\,[1-e^{M}+Me^{M}]\;+\;a\,[1-\varrho]\,[\,1-e^{-M}-Me^{-M}]\;.$

######  Proposition 2.9.

Fix a density profile $\gamma:[0,1]\to[-1,1]$ and a trajectory $\pi$ in
$\Pi_{\gamma}$. Then, for each $t>0$, the elliptic equation (for $H$)

$\left\\{\begin{aligned} &\partial_{t}u\;=\;\nabla(D(u)\nabla
u)\,-\,2\,\nabla\\{\sigma(u)\ \nabla H\\}\;,\\\ &[D(u)\nabla
u](t,1)\,-\,2\,\sigma(u(t,1))\,\nabla
H(t,1)\,=\,{\mathfrak{p}}_{{\mathfrak{b}}}\big{(}\,u(t,1)\,,\,H(t,1)\,\big{)}\;,\\\
&[D(u)\nabla u](t,-1)\,-\,2\,\sigma(u(t,-1))\,\nabla
H(t,-1)\,=\,-\,{\mathfrak{p}}_{{\mathfrak{a}}}\big{(}\,u(t,-1)\,,\,H(t,-1)\,\big{)}\;,\end{aligned}\right.$
(2.10)

has a unique solution, denoted by $H_{t}$. The function $H$ belongs to
$C^{1,1}(]{\mathfrak{t}},T]\times[-1,1])$, and the rate functional
$I_{[0,T]}(u)$ takes the form

$\displaystyle I_{[0,T]}(u)$
$\displaystyle=\;\int_{0}^{T}dt\int_{0}^{1}\sigma(u_{t})\,(\nabla
H_{t})^{2}\,dx\;+\;\int_{0}^{T}{\mathfrak{c}}_{{\mathfrak{b}}}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\;dt$
(2.11)
$\displaystyle+\;\int_{0}^{T}{\mathfrak{c}}_{{\mathfrak{a}}}\big{(}\,u_{t}(-1)\,,\,H_{t}(-1)\,\big{)}\;dt\;.$

Mind that $H_{t}=0$ for $0<t\leq{\mathfrak{t}}$ since $u$ follows the
hydrodynamic equation on the interval $[0,{\mathfrak{t}}]$. By the
construction presented in Section 6, the time derivative of $H$ at
$t={\mathfrak{t}}$ might be discontinuous.

### Comments

Since the first papers on the subject [12, 20], robust methods and tools have
been developed to derive dynamical large deviations principles for the
empirical measures of diffusive interacting particle systems [19, 24, 4, 15,
14]. With these tools, the proof of the large deviations principle boils down,
essentially, to the $I_{[0,T]}(\cdot|\gamma)$-density of a set of smooth
trajectories.

The proof of the $I_{[0,T]}(\cdot|\gamma)$-density of the set $\Pi_{\gamma}$,
introduced in Definition 2.8, presents two novelties which might be helpful
for other dynamics, as reaction-diffusion models without the concavity
assumption on the creation and destruction rates [17, 6, 16].

Here, as in [24, 4, 15], the rate functional $I_{[0,T]}(\cdot|\gamma)$ is not
convex. The lack of convexity complicates the second step in the proof of the
$I_{[0,T]}(\cdot|\gamma)$-density, handled in Lemma 6.3. There, we approximate
a trajectory $\pi(t,dx)=u(t,x)dx$ by
$\pi^{\epsilon}(t,dx)=u^{\epsilon}(t,x)dx$, where
$u^{\epsilon}=(1-\epsilon)u+\epsilon\rho$ and $\rho$ is the solution of the
hydrodynamic equation. If $I_{[0,T]}(\cdot|\gamma)$ were convex, to proof that
$I_{[0,T]}(\pi^{\epsilon}|\gamma)$ converges to $I_{[0,T]}(\pi|\gamma)$ would
be straightforward. One direction would be obtained by the lower semi-
continuity of the functional and the other from the convexity.

The proof of this step presented in [24, 4] (cf. [4, Lemma 5.5]) for non-
convex rate functionals is based on the representation of a linear functionals
given in [4, Lemma 4.8], and relies on Lemmata 5.2 and 5.3 in [4]. We present
here a much simpler and direct proof which might be useful in other contexts.

The second difficulty in the proof of the $I_{[0,T]}(\cdot|\gamma)$-density
arises, as in [14], from the presence of a boundary term in the rate
functional. Propositions 5.4–5.7 state that the cost $I_{[0,T]}(\pi)$ of a
trajectory $\pi\in D([0,T],{\mathcal{M}})$ can be expressed as the sum
$I^{(1)}_{[0,T]}(\pi)+I^{(2)}_{[0,T]}(\pi)$, where the first term accounts for
the cost due to evolution in the interior of the interval $[-1,1]$, while the
second one for the evolution at the boundary.

This decomposition is needed in Lemma 6.5 for trajectories bounded away from
$0$ and $1$ and presented in Propositions 5.4, 5.5 under this assumption. In
[14] (cf. equation (4.2)) the trajectory is assumed to be continuous and
smooth in time. The lack of regularity of the trajectory is responsible for
the presence of cross terms and a convoluted expression for the rate
functionals. Assuming regularity, Propositions 5.7 presents a similar formula
to the one obtained [14].

The decomposition of the rate functional requires the following property of
trajectories. Fix a path $\pi(t,dx)=u(t,x)\,dx$ in $D([0,T],{\mathcal{M}})$
with finite rate function, $I_{[0,T]}(u)<\infty$. It is known, at least since
[4], that such a path is weakly continuous in time. More precisely, that for
each $g\in C^{1}([-1,1])$, the map $t\mapsto\langle\pi_{t},g\rangle$ is
uniformly continuous (cf. Lemma 4.3). Proposition 4.11 asserts that this map
is, actually, absolutely continuous and its derivative belongs to $L^{1}$.
This property permits an integration by parts on the right-hand side of (2.6)
when $H(t,x)=h(t)\,g(x)$ for some $h\in C^{1}([0,T])$, $g\in C^{1}([-1,1])$
and plays a crucial role in the decomposition of the rate functional.

######  Remark 2.10.

As in [15, 14] and in contrast to [20, 4], the large deviations principle is
formulated here for the empirical measure and not for the empirical density.
More precisely, the empirical measure $\pi(t)$, defined in (2.2), is a sum of
Dirac measures, while in [20, 4] $\pi(t)$ is defined as a function taking
values in the set $\\{0,1\\}$. Defining $\pi(t)$ as a singular measure
requires to prove that $I_{[0,T]}(\pi)=+\infty$ if $\pi_{t}$ is not absolutely
continuous with respect to the Lebesgue measure. The proof presented in
Subsection 6.3 of [15] applies to the present context.

The paper is organized as follows. In Section 3, we prove the hydrodynamic
limit. In Section 4, we present the main properties of the rate functional
$I_{[0,T]}$ and, in Section 5, its decomposition. In Section 6, we prove the
$I_{[0,T]}$-density of the set $\Pi_{\gamma}$, and, in Section 7 the large
deviations principle. In Section 8, we discuss uniqueness of weak solutions of
the equation (2.3).

##  3\. Hydrodynamic limit

In Section 8, we prove uniqueness of weak solutions of the hydrodynamic
equation. The proof of Theorem 2.2 is similar to the classical one presented
in Section 5 in [19]. The slight difference lies in the energy estimate.

######  Lemma 3.1.

Fix a profile $\rho_{0}\colon[-1,1]\to[0,1]$, and a sequence of probability
measure $(\mu^{N})_{N\in\mathbb{N}}$ associated to $\rho_{0}$. Consider a
dense family of functions $\\{H_{l}\\}_{l\geq 1}$ in
$C^{0,1}([0,1]\times[-1,1])$. There exist a constant $K_{0}$ such that

$\displaystyle\underset{N\to+\infty}{\mbox{ lim sup
}}\mathbb{E}_{\mu^{N}}\Big{[}\,\underset{1\leq j\leq
k}{\max}\Big{\\{}\int_{0}^{T}W_{N}(H_{j}(s),\eta_{s})\,ds\Big{\\}}\,\Big{]}\,\leq\,K_{0}$

for every $k\geq 1$, where

$\displaystyle
W_{N}(H,\eta)=\sum_{x}H\left(\frac{x}{N}\right)(\eta(x)-\eta(x+1))-\frac{1}{2AN}\sum_{x}H\left(\frac{x}{N}\right)^{2}(\eta(x)-\eta(x+1))^{2}\;,$

the sum is performed over all $-N+1\leq x\leq N-2$, and $A=\min\\{1,1+2a\\}$.

Since $r_{x,x+1}\leq A$ and the Bernoulli product measure is reversible for
the process with generator $L_{N,0}$ ([15]), the proof is similar to that of
Lemma 5.7.3 in [19]. As stated in [19, Theorem 5.7.1], Lemma 3.1 ensures that
all limit points of the sequence ${\mathbb{Q}}^{N}_{\mu^{N}}$ are concentrated
on paths with finite energy.

##  4\. The rate function

In this section, we present some properties of the rate function
$I_{[0,T]}(.)$ and the energy ${\mathcal{Q}}(\cdot)$. These properties are
crucial for proving Theorem 2.5.

### The energy $\mathcal{Q}$

The proof of the next result can be found in Section 4 of [4].

######  Proposition 4.1.

The functional $\mathcal{Q}$ is convex and lower semicontinuous. Let $\pi\in
D([0,T],\mathcal{M})$ such that $\pi(t,dx)=u(t,x)dx$. Then $u$ has finite
energy if, and only if, $u\in L^{2}([0,T],\mathcal{H}^{1}(\Omega))$ and
$\displaystyle\int_{0}^{T}\int_{-1}^{1}\frac{|\nabla
u(t,x)|^{2}}{\chi(u(t,x))}dxdt<+\infty$. In this case
$\displaystyle\mathcal{Q}(\pi)=\int_{0}^{T}\int_{-1}^{1}\frac{|\nabla
u(t,x)|^{2}}{\chi(u(t,x))}dxdt$.

######  Corollary 4.2.

The weak solution of the hydrodynamic equation has finite energy.

###### Proof.

In view of Lemma B.2, the continuity of $D(\cdot)$ and the boundedness of the
weak solution of the hydrodynamic equation, the proof of Corollary 4.2 is
similar to the one of Lemma B.5 in [14]. ∎

### The rate functional $I_{[0,T]}$

In this section, we present certain properties of the rate function
$I_{[0,T]}$: we establish that it is lower semicontinuity and that its level
sets are compact. The proof of the next result is similar to the one of Lemma
4.1 in [15], Lemma 3.1 in [14].

######  Lemma 4.3.

Fix $T>0$. For each $M>0$, $g$ in $C^{1}([0,1])$ and $\epsilon>0$, there
exists $\delta>0$ such that

$\sup_{|t-s|\leq\delta}\;\sup_{\pi:I_{[0,T]}(\pi)\leq
M}\,\big{|}\,\langle\pi_{t},g\rangle-\langle\pi_{s},g\rangle\,\big{|}\;\leq\;\epsilon\;.$

In particular, $\pi$ belongs to $C([0,T],{\mathcal{M}}^{0})$.

As $D$ is continuous and $u$ is bounded, the proofs of the next two results
are similar to the ones of Proposition 3.5 and Corollary 3.6 of [14].

######  Lemma 4.4.

There exists a constant $C_{0}>0$ such that:

$\displaystyle\int_{0}^{T}\int_{-1}^{1}\frac{|\nabla
u(t,x)|^{2}}{\chi(u(t,x))}\leq C_{0}(I_{[0,T]}(u)+1)$

for all $u\in D([0,T],\mathcal{M}^{0})$.

######  Corollary 4.5.

Let $\pi\in D([0,T],\mathcal{M}^{0})$ such that $\pi(t,dx)=u(t,x)dx$. Then,
$u$ is the weak solution of the hydrodynamic equation (2.3) if, and only if,
$I_{[0,T]}(\pi|\rho_{0})=0$.

Let $(E_{q})_{q\geq 0}$ be the level set of the rate function $I_{[0,T]}(.)$:

$\displaystyle E_{q}=\\{\pi\in D([0,T],\mathcal{M})|I_{[0,T]}(\pi)\leq q\\}$

######  Theorem 4.6.

Fix $T>0$. Then, $I_{[0,T]}\colon D([0,T],\mathcal{M}^{0})\to[0,\infty]$ is
lower semicontinuous and has compact level sets.

###### Proof.

The proof of the lower semicontinuity is similar to the one of Theorem 2.4 of
[14]. We turn to the compactness of $E_{q}$ in $D([0,T],\mathcal{M})$. Fix
$q\geq 0$. By lower semicontinuity, $E_{q}$ is a closed set. It remains to
prove the relative compactness. By Lemma 4.3, $E_{q}\subset
C([0,T],\mathcal{M})$. This follows from Ascoli theorem.

Since for each $\pi(t,dx)=u(t,x)dx\in E_{q}$, $0\leq u\leq 1$, for every
sequence $(\pi^{n})_{n\in\mathbb{N}}$ in $E_{q}$, $t\in[0,T]$, there exists a
subsequence $(\varphi(n))_{n\in\mathbb{N}}$ such that
$(\pi_{t}^{\varphi(n)})_{n\in\mathbb{N}}$ converges. On the other hand, by
definition of the distance $d(.,.)$ in $\mathcal{M}$, by (1.1) in Section 4.1
of [19], and by Lemma 4.3,

$\displaystyle\underset{\delta\to 0}{\lim}\sup_{\pi\in
E_{q}}\underset{|r-s|\leq\delta}{\mbox{ sup }}d(\pi_{s},\pi_{r})=0\;.$

This prove that $E_{q}$ is relatively compact. ∎

######  Remark 4.7.

Fix a profile $\gamma\colon[-1,1]\to[0,1]$. The same proof yields that the
rate function $I_{[0,T]}(.|\gamma)$ is lower semicontinuous and has compact
level sets. Details are given in Theorem 3.2.24 in [7].

Next two results show that the cost of a trajectory in an interval
$[0,S]\cup[S,T]$ is equal to the sum of its cost on each interval. Denote by
$\color[rgb]{0,0,1}C_{K}^{1,2}(]0,T[\times[-1,1])$ the space of functions in
$C^{1,2}([0,T]\times[-1,1])$ with compact support in $]0,T[\times[-1,1]$.

######  Lemma 4.8.

Fix $T>0$. Let $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Then,

$\displaystyle I_{[0,T]}(\pi)=\underset{H\in
C_{K}^{1,2}(]0,T[\times[-1,1])}{\sup}\hat{J}_{H}(\pi)$

###### Proof.

Clearly, the right-hand side is bounded by the left-hand side. We turn to the
reversed inequality. The proof is presented in details in Lemma 3.6.1 in [7].

Fix $\epsilon>0$. Let $\psi_{\epsilon}$ be a function in $C^{1}([0,T])$ such
that $\psi_{\epsilon}(t)=1$ for every $t\in[\epsilon,T-\epsilon]$ and
$\psi_{\epsilon}(t)=0$ for all
$t\in[0,\frac{\epsilon}{2}]\cup[T-\frac{\epsilon}{2},T]$. Fix $G\in
C^{1,2}([0,T]\times[-1,1])$. Consider $G^{\epsilon}=G\psi_{\epsilon}$. Then,
$G^{\epsilon}\in C_{K}^{1,2}(]0,T[\times[-1,1])$ and by the dominated
convergence theorem and Lemma 4.3:

$\displaystyle\hat{J}_{G}(\pi)=\underset{\epsilon\to
0}{\lim}\hat{J}_{G^{\epsilon}}(\pi)\leq\underset{C_{K}^{1,2}(]0,T[\times[-1,1])}{\mbox{
sup }}\hat{J}_{H}(\pi)\;,$

which completes the proof of the lemma. ∎

######  Corollary 4.9.

Fix $T>0$. Let $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Then,
for each $0<s<T$,

$\displaystyle I_{[0,T]}(\pi)=I_{[0,s]}(\pi)+I_{[s,T]}(\pi)\;.$

In the remaining part of this section, we examine the regularity of the map
$t\mapsto\langle\pi_{t},g\rangle$ for a trajectory $\pi$ with finite rate
functional and a smooth function $g$.

######  Lemma 4.10.

Fix $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Then, for each
$g\in C^{2}([-1,1])$, $t\mapsto\langle\pi_{t},g\rangle$ is a bounded variation
function.

###### Proof.

Fix $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$, $g\in
C^{2}([-1,1])$, $n\in\mathbb{N}^{*}$, and a partition $0=t_{0}<\cdots<t_{n}=T$
of the interval $[0,T]$. Consider $0\leq r\leq s\leq T$. By definition of the
rate functional $I_{[r,s]}(\cdot)$, and since $\sigma$, $D$, $b$ and $u$ are
bounded,

$\displaystyle\langle\pi_{s},g\rangle-\langle\pi_{r},g\rangle$
$\displaystyle\leq C_{0}\int_{r}^{s}\int_{-1}^{1}|\nabla u(t,x)|\,|\nabla
g(x)|\,dx\,dt$ $\displaystyle+C_{0}(s-r)\int_{-1}^{1}\Big{\\{}(\nabla
g(x))^{2}+e^{|g|}+1\Big{\\}}\,dx\,+\,I_{[r,s]}(\pi)$

for some finite constant $C_{0}$ which does not depend on $g$ or $\pi$.
Replacing $g$ by $-g$ shows that the absolute value of
$\langle\pi_{s},g\rangle-\langle\pi_{r},g\rangle$ is bounded by the right-hand
side. Hence, by Corollary 4.9, the linearity of the integral and Young’s
inequality $2xy\leq x^{2}+y^{2}$,

$\displaystyle\sum_{k=0}^{n-1}|\langle\pi_{t_{k+1}},g\rangle-\langle\pi_{t_{k}},g\rangle|\leq
C_{0}\,T\,\int_{-1}^{1}\Big{\\{}(\nabla
g(x))^{2}+e^{|g|}+1\Big{\\}}\,dx\,+\,{\mathcal{Q}}(\pi)\,+\,I_{[0,T]}(\pi)$

for some finite constant $C_{0}$ which does not depend on $g$ or $\pi$. This
completes the proof of the lemma. ∎

######  Proposition 4.11.

Fix $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Then, for each
$g\in C^{1}([-1,1])$, there exists $P\in L^{1}([0,T])$ such that,

$\displaystyle H_{T}\langle\pi_{T},g\rangle-
H_{0}\langle\pi_{0},g\rangle-\int_{0}^{T}H^{\prime}_{t}\,\langle\pi_{t},g\rangle\,dt=\int_{0}^{T}H_{t}\,P_{t}\,dt$

for all $H\in C^{1}([0,T])$.

###### Proof.

Fix $g\in C^{2}([-1,1])$. By Lemma 4.10, $t\mapsto\langle\pi_{t},g\rangle$ is
a bounded variation function. Denote by $\nu$ the signed measure on $[0,T]$
induced by this function. Integrating by parts yields that

$\displaystyle H_{T}\langle\pi_{T},g\rangle-
H_{0}\langle\pi_{0},g\rangle-\int_{0}^{T}H^{\prime}_{t}\,\langle\pi_{t},g\rangle\,dt=\int_{0}^{T}H_{t}\,\nu(dt)\;.$

Our goal is to show that $\nu$ is absolutely continuous with respect to the
Lebesgue measure.

For every $t\in[0,T]$, denoted by $V_{t}$ the total variation of
$\langle\pi_{.},g\rangle$ on $[0,t]$. Since $t\mapsto\langle\pi_{t},g\rangle$
is a bounded variation function, $V$ is a finite increasing function, and
$\mu\colon\displaystyle A\mapsto\int_{A}dV_{t}$ is a positive measure.

Claim 1: The measure $\mu$ has no atoms.

Fix $s\in[0,T]$. Since $V$ is a bounded variation function, the set
$D=\\{s:\mu(\\{s\\})>0\\}$ is countable. Define sequences
$(a_{k})_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$ such that $0\leq
a_{k}\leq a_{k+1}\leq b_{k+1}\leq b_{k}\leq T$,
$\underset{k\to\infty}{\lim}a_{k}=\underset{k\to\infty}{\lim}b_{k}=s$ and
$\mu(\\{b_{k}\\})=\mu(\\{a_{k}\\})=0$ for all $k\in\mathbb{N}$. Then,
$0\leq\mu(\\{s\\})\leq\mu([a_{k},b_{k}])=V_{b_{k}}-V_{a_{k}}$.

Fix $\lambda>0$. By definition of $V$, for each $\epsilon>0$, there exists a
subdivision $(t_{i})_{1\leq i\leq n}$ of $[a_{k},b_{k}]$ such that:

$\displaystyle\lambda\mu(\\{s\\})\;=\;\lambda(V_{b_{k}}-V_{a_{k}})\leq\lambda\epsilon+\sum_{i=0}^{n-1}|\langle\pi_{t_{i+1}},\lambda
g\rangle-\langle\pi_{t_{i}},\lambda g\rangle|\;.$

By the proof of the previous lemma, the right-hand side is bounded by

$\displaystyle\lambda\epsilon+C_{0}\,\lambda\,\int_{a_{k}}^{b_{k}}\int_{-1}^{1}|\nabla
u(t,x)|\,|\nabla g(x)|\,dx\,dt$
$\displaystyle\quad\,+\,C_{0}\,(b_{k}-a_{k})\,\int_{-1}^{1}\Big{\\{}(\lambda\nabla
g(x))^{2}+e^{\lambda|g|}+1\Big{\\}}\,dx\,+\,I_{[a_{k},b_{k}]}(\pi)$

for some finite constant $C_{0}$ which does not depend on $\lambda$
$\epsilon$, $g$ or $\pi$. Let $\epsilon\to 0$, and then $k\to+\infty$ to
conclude that

$\displaystyle\mu(\\{s\\})\,\leq\,\frac{1}{\lambda}\,I_{[0,T]}(\pi)$

for all $\lambda>0$. It remains to let $\lambda\to\infty$ to complete the
proof of Claim 1.

Claim 2: The measure $\mu$ is absolutely continuous.

Fix a set $K\subset[0,T]$ of zero Lebesgue measure, $\epsilon>0$ and
$\lambda>0$. Let $G$ be an open set such that $K\subset G$ and
$\Lambda(G)\leq\epsilon_{1}$ where $\epsilon_{1}$ will be fixed afterwards and
$\Lambda$ represents the Lebesgue measure. Let $G=\cup_{k}]a_{k},b_{k}[$. By
the same computation as before,

$\displaystyle 0\leq\lambda\,\mu(K)\,\leq\,\lambda\,\mu(G)$ $\displaystyle\leq
C_{0}\,\lambda\,\int_{G}\int_{-1}^{1}|\nabla u(t,x)|\,|\nabla g(x)|\,dx\,dt$
$\displaystyle\quad\,+\,C_{0}\,\Lambda(G)\,\int_{-1}^{1}\Big{\\{}(\lambda\nabla
g(x))^{2}+e^{\lambda|g|}+1\Big{\\}}\,dx\,dt\,+\,I_{[0,T]}(\pi)$

for some finite constant $C_{0}$ which does not depend on $\lambda$
$\epsilon$, $g$ or $\pi$. Choose $\lambda>0$ and then
$\epsilon_{1}=\epsilon_{1}(\lambda,\epsilon)>0$ so that
$(1/\lambda)\,I_{[0,T]}(\pi)\leq\epsilon/2$ and

$C_{0}\,\int_{G}\int_{-1}^{1}|\nabla u(t,x)|\,|\nabla
g(x)|\,dx\,dt\,+\,\frac{C_{0}}{\lambda}\,\Lambda(G)\,\int_{-1}^{1}\Big{\\{}(\lambda\nabla
g(x))^{2}+e^{\lambda|g|}+1\Big{\\}}\,dx\,dt\,\leq\,\frac{\epsilon}{2}\;\cdot$

Therefore, $\mu(K)\leq\epsilon$ for every $\epsilon>0$ so that $\mu(K)=0$, and
Claim 2 is proved.

It follows from Claim 2 that $\nu$ is absolutely continuous, and there exists
$P\in L^{1}([0,T])$ such that $\nu(dt)=P_{t}dt$. This completes the proof of
the proposition. ∎

######  Remark 4.12.

One can show that $P\in L^{1}\log L^{1}$.

##  5\. Deconstructing the rate functional

The main results of this section, stated in Propositions 5.4 and 5.5 below,
show that the rate function $I_{[0,T]}(\,\cdot\,)$ can be decomposed as the
sum of two rate functions. The first one measures the cost of the trajectory
due to its evolution in the bulk, while the second one measures the costs due
to the boundary evolution. This decomposition of the rate function is an
important tool in the proof that any trajectory $u$ with finite rate function
can be approximated by a sequence of regular trajectories $(u^{n}:n\geq 1)$ in
such a way that $I_{[0,T]}(u^{n}\,|\gamma)\to I_{[0,T]}(u\,|\gamma)$, the
content of the next section.

### Weighted Sobolev spaces

Recall that $\Omega=[-1,1]$, and let
$\color[rgb]{0,0,1}\Omega_{T}=[-1,1]\times[0,T]$. For a _bounded_ positive
function $f:\Omega\to{\mathbb{R}}_{+}$ (resp.
$f:\Omega_{T}\to{\mathbb{R}}_{+}$), denote by $L^{2}(f)$ (resp.
${\mathbb{L}}^{2}(f)$) the Hilbert space of (equivalence classes of)
measurable functions
$\\{H:\Omega\to{\mathbb{R}}:\int_{\Omega}H(x)^{2}f(x)dx<\infty\\}$ (resp.
$\\{H:\Omega_{T}\to{\mathbb{R}}:\int_{\Omega_{T}}H(t,x)^{2}f(t,x)dtdx<\infty\\}$)
endowed with the scalar product $\langle\cdot,\cdot\rangle_{f}$ (resp.
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{f}$) induced by

$\langle
H\,,\,G\rangle_{f}\;=\;\int_{\Omega}H(x)\,G(x)\,f(x)\,dx\;,\quad\langle\\!\langle
H\,,\,G\rangle\\!\rangle_{f}\;=\;\int_{0}^{T}\langle
H_{t},G_{t}\rangle_{f(t,\cdot)}\;dt\;.$

The norm associated to the above scalar products is denoted by
$\|\cdot\|_{f}$. When $f=1$ we omit the index $f$ and denote the spaces
$L^{2}(1)$, ${\mathbb{L}}^{2}(1)$ by $L^{2}(\Omega)$, $L^{2}(\Omega_{T})$,
respectively.

Since $f$ is bounded, $C^{\infty}_{K}(\Omega_{T})$ is dense in
${\mathbb{L}}^{2}(f)$. Moreover, the space of bounded linear functionals on
${\mathbb{L}}^{2}(f)$ can be identified with ${\mathbb{L}}^{2}(1/f)$: any
bounded linear functional $\ell$ on ${\mathbb{L}}^{2}(f)$ can be represented
as

$\ell(G)\;=\;\int_{0}^{T}\langle H_{t}\,,\,G_{t}\rangle\;dt$ (5.1)

for some $H$ in ${\mathbb{L}}^{2}(1/f)$. Indeed, by Riesz’s representation
theorem, for each bounded linear functional $\ell$ on ${\mathbb{L}}^{2}(f)$,
there exists a unique element $\hat{H}=\hat{H}_{\ell}\in{\mathbb{L}}^{2}(f)$
such that $\ell(G)=\langle\\!\langle\hat{H}\,,\,G\rangle\\!\rangle_{f}$. Let
$H=f\,\hat{H}$. Clearly, $H$ belongs to $L^{2}(1/f)$ and we obtain the
representation claimed above.

Denote by $\color[rgb]{0,0,1}{\mathcal{H}}^{1}_{0}(f)$ the Hilbert spaces
induced by the set $C^{\infty}_{K}(\Omega_{T})$ endowed with the scalar
products, $\langle\\!\langle G,H\rangle\\!\rangle_{1,f}$ defined by

$\langle\\!\langle G,H\rangle\\!\rangle_{1,f}\;=\;\langle\\!\langle\nabla
G,\nabla H\rangle\\!\rangle_{f}\;.$

Let $\color[rgb]{0,0,1}\|\cdot\|_{1,f}$ be the norm associated to the scalar
product $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{1,f}$. Denote
$\mathcal{H}^{-1}(f)$ the dual of $\mathcal{H}_{0}^{1}(f)$, it is a Hilbert
space equipped with the norm $\|\cdot\|_{-1,f}$ defined by

$\displaystyle\|L\|_{-1,f}^{2}=\underset{G\in
C_{K}^{\infty}(]0,T[\times]-1,1[)}{\mbox{ sup
}}\\{\,2L(G)\,-\,\|G\|^{2}_{1,f}\,\\}\;.$

In this formula, $L(G)$ stands for the value of the linear form $L$ at $G$.
The proof of the next proposition is similar to the one of Lemma 4.8 in [4]
and is therefore omitted.

######  Proposition 5.1.

A linear functional $L\colon\mathcal{H}_{0}^{1}(\sigma(u))\to\mathbb{R}$
belongs to $\mathcal{H}^{-1}(\sigma(u))$ if and only if there exists $P\in
L^{2}(\sigma(u)^{-1})$ such that $L(H)=\langle\langle P,\nabla
H\rangle\rangle$ for every $H\in C^{\infty}_{K}(]0,T[\times]-1,1[)$. In this
case,

$\displaystyle\|L\|_{-1,\sigma(u)}^{2}=\int_{0}^{T}\left\\{\langle
P_{t},P_{t}\rangle_{\sigma(u_{t})^{-1}}-c_{t}\right\\}dt\mbox{ , }$

where $c_{t}=\langle
P_{t}\sigma(u_{t})^{-1}\rangle^{2}\langle\sigma(u_{t})^{-1}\rangle^{-1}\,\mathbb{1}_{\\{\langle\sigma(u_{t})^{-1}\rangle<\infty\\}}$
.

######  Lemma 5.2.

Fix $T>0$. Let $\pi(t,x)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Then,
$H\mapsto\int_{0}^{T}\langle\pi_{t},\partial_{t}H_{t}\rangle dt$ is continuous
on $\mathcal{H}_{0}^{1}(\sigma(u))$ and there exists $P\in
L^{2}(\sigma(u)^{-1})$ such that

$-\int_{0}^{T}\langle\pi_{t},\partial_{t}H_{t}\rangle
dt\,=\,\langle\langle\nabla H,P\rangle\rangle\;\;\text{for all}\;\;H\in
C^{\infty}_{K}(]0,T[\times]-1,1[)\;.$

###### Proof.

Define the function $A\colon
C^{\infty}_{K}(]0,T[\times[-1,1])\mapsto\mathbb{R}$ by

$\displaystyle A(H)=-\int_{0}^{T}\langle\pi_{t},\partial_{t}H_{t}\rangle
dt\mbox{ . }$

Let $H\in C^{\infty}_{K}(]0,T[\times[-1,1])$. By definition of
$\hat{J}_{H}(\pi)$:

$\displaystyle A(H)$
$\displaystyle=\hat{J}_{H}(\pi)+\int_{0}^{T}\int_{-1}^{1}\sigma(u(t,x))(\nabla
H(t,x))^{2}dxdt$ $\displaystyle-\int_{0}^{T}\int_{-1}^{1}D(u(t,x))\nabla
u(t,x)\nabla H(t,x)dxdt\,+\,\int_{0}^{T}B(u_{t},H_{t})dt\mbox{ . }$

Since $H(t,.)$ has compact support in $]-1,1[$, the last term on the right-
hand side vanishes. Replace $H$ by $RH$ for some $R>0$. We may estimate
$\hat{J}_{RH}(\pi)$ by $I_{[0,T]}(\pi)$. By linearity of $A$, since $D$ is
bounded, and $2xy\leq ax^{2}+\frac{1}{a}y^{2}$ for all $a>0$,

$\displaystyle
A(H)\,\leq\,\frac{1}{R}\Big{\\{}I_{[0,T]}(\pi)+2R^{2}\|H\|^{2}_{1,\sigma(u)}\,+\,C_{0}\int_{0}^{T}\int_{-1}^{1}\frac{(\nabla
u(t,x))^{2}}{\sigma(u(t,x))}dxdt\,\Big{\\}}$

for some finite constant $C_{0}$ which may change from line to line.

By Lemma 4.4, the previous expression is less than or equal to

$\displaystyle\frac{C_{0}}{R}\left\\{I_{[0,T]}(\pi)+1\right\\}+2R\|H\|_{1,\sigma(u)}^{2}\;.$

Replacing $H$ by $-H$, and optimizing over $R$ yields that

$\displaystyle
A(H)^{2}\,\leq\,C_{0}\,(I_{[0,T]}(\pi)+1)\,\|H\|_{1,\sigma(u)}^{2}\;.$

This proves that the linear functional $A$ is bounded in
$\mathcal{H}_{0}^{1}(\sigma(u))$. To complete the proof, it remains to recall
the assertion of Proposition 5.1. ∎

### First decomposition

Until the end of this section, fix $T>0$ and $\pi(t,dx)=u(t,x)dx$ such that
$I_{[0,T]}(\pi)<+\infty$. Recall the definition of the linear operator $L_{0}$
introduced in (2.6). Denote $L\colon C^{1,2}([0,T]\times[-1,1])\to\mathbb{R}$
the functional given by

$\displaystyle L(H)$
$\displaystyle=L_{0}(H)+\int_{0}^{T}\int_{-1}^{1}D(u(t,x))\nabla u(t,x)\nabla
H(t,x)dxdt$
$\displaystyle+\int_{0}^{T}\Bigl{\\{}(u_{t}(1)-{{\mathfrak{b}}})H_{t}(1)+(u_{t}(-1)-{{\mathfrak{a}}})H_{t}(-1)\Bigl{\\}}dt\mbox{
, }$ (5.2)

Denote by $\color[rgb]{0,0,1}C_{0}^{1,2}([0,T]\times[-1,1])$ the space of
functions $H\in C^{1,2}([0,T]\times[-1,1])$ such that $H_{T}=H_{0}=0$. We use
the same notation $L_{0}$, $L$ when these functionals are defined on
$C_{0}^{1,2}([0,T]\times[-1,1])$.

For functions $g\in C^{2}([-1,1])$, $H\in C_{0}^{1}([0,T])$, denote by $Hg$
the function defined by $Hg(t,x)=H(t)g(x)$. Next result is an elementary
consequence of Proposition 4.11.

######  Lemma 5.3.

Fix $g\in C^{2}([-1,1])$. Then, there exists $P^{g}\in L^{1}([0,T])$ such that

$\displaystyle L_{0}(Hg)\,=\,\int_{0}^{T}H(t)P^{g}_{t}dt$

for every $H\in C_{0}^{1}([0,T])$.

Next result requires some notation. Denote by $P\in L^{2}(\sigma(u)^{-1})$ the
function introduced in Lemma 5.2 and such that:

$\displaystyle-\int_{0}^{T}\langle\pi_{t},\partial_{t}H_{t}\rangle
dt=\langle\langle\nabla H,P\rangle\rangle$ (5.3)

for every $H\in C_{K}^{\infty}(]0,T[\times]-1,1[)$. If we replace $P_{t}$, by
$P_{t}-\langle P_{t}/\sigma(u_{t})\rangle/\langle 1/\sigma(u_{t})\rangle$, we
may assume that $\langle P_{t}/\sigma(u_{t})\rangle=0$ for all $0\leq t\leq
T$.

Let $X$, $\mathbb{1}\colon[-1,1]\to\mathbb{R}$ be the functions given by
$X(x)=x$, $\mathbb{1}(x)=1$. By Proposition 4.11, there exist $P^{g}$ in
$L^{1}([0,T])$ such that

$\displaystyle-\int_{0}^{T}H^{\prime}(t)\langle\pi_{t},g\rangle
dt\,=\,\int_{0}^{T}H(t)\,P_{t}^{g}\,dt$

for $g=X$, $\mathbb{1}$ and every $H\in C_{0}^{1}([0,T])$.

Denote by $M\in L^{2}(\sigma(u)^{-1})$, $g,h\in L^{1}([0,T])$ the functions
defined by:

$\displaystyle M_{t}=P_{t}+D(u_{t})\nabla u_{t}\mbox{ , }$ (5.4)
$\displaystyle g(t)\,=\,2\,\Big{\\{}\,P^{\mathbb{1}}(t)-P^{X}(t)-\langle
D(u_{t})\nabla u_{t}\rangle+2(u_{t}(-1)-{{\mathfrak{a}}})+\langle
M_{t}\rangle-2\langle
M_{t}\sigma(u_{t})^{-1}\rangle\mathcal{S}_{t}\,\Big{\\}}$ $\displaystyle
h(t)\,=\,2\,\Big{\\{}\,P^{\mathbb{1}}(t)+P^{X}(t)+\langle D(u_{t})\nabla
u_{t}\rangle+2(u_{t}(1)-{{\mathfrak{b}}})-\langle M_{t}\rangle+2\langle
M_{t}\sigma(u_{t})^{-1}\rangle\mathcal{S}_{t}\,\Big{\\}}\mbox{ , }$

where $\color[rgb]{0,0,1}\mathcal{S}_{t}=1/\langle\sigma(u_{t})^{-1}\rangle$.
Lastly, denote by
$\Psi\colon[0,1]^{2}\times{\mathbb{R}}^{2}\to{\mathbb{R}}_{+}$ the function
given by:

$\displaystyle\Psi(r,s,\alpha,\beta)$
$\displaystyle=4{{\mathfrak{a}}}(1-r)(e^{\alpha}-\alpha-1)+4r(1-{{\mathfrak{a}}})(e^{-\alpha}+\alpha-1)$
$\displaystyle+4{{\mathfrak{b}}}(1-s)(e^{\beta}-\beta-1)+4s(1-{{\mathfrak{b}}})(e^{-\beta}+\beta-1)\mbox{
. }$

Note that $\Psi$ is non-negative, linear in $(r,s)$ and convex in
$(\alpha,\beta)$.

######  Proposition 5.4.

$I_{[0,T]}(\pi)=I_{[0,T]}^{(1)}(\pi)+I_{[0,T]}^{(2)}(\pi)$ where:

$\displaystyle
I_{[0,T]}^{(1)}(\pi)\,=\,\frac{1}{4}\,\int_{0}^{T}\Big{\\{}\,\langle
M_{t},M_{t}\rangle_{\sigma(u_{t})^{-1}}\,-\,\langle
M_{t}\sigma(u_{t})^{-1}\rangle^{2}\mathcal{S}_{t}\,\Big{\\}}\,dt\;,$
$\displaystyle
I_{[0,T]}^{(2)}(\pi)\;=\;\frac{1}{4}\,\underset{\alpha,\beta}{\sup}\int_{0}^{T}\Big{\\{}\,\alpha(t)\,g(t)+\beta(t)\,h(t)\,\,-\,\Upsilon_{t}(u_{t}(-1),u_{t}(1),\alpha(t),\beta(t))\Big{\\}}\,dt\;.$

where the supremum is carried out over all $\alpha$, $\beta\in
C_{0}^{1}([0,T])$, and

${\color[rgb]{0,0,1}\Upsilon_{t}(r,s,\alpha,\beta)}\,:=\,4\,(\alpha-\beta)^{2}\,\mathcal{S}_{t}\,+\,\Psi(r,s,\alpha,\beta)\;.$
(5.5)

###### Proof.

By Lemma 4.8, and replacing in the supremum the function $H$ by $H/2$, yields
that

$\displaystyle I_{[0,T]}(\pi)\,=\,\frac{1}{4}\underset{H}{\mbox{ sup
}}\Big{\\{}2L(H)-\int_{0}^{T}\|\nabla
H_{t}\|_{\sigma(u_{t})}^{2}dt-\int_{0}^{T}\Psi(u_{t},H_{t})\,dt\Big{\\}}\;,$

where $L$ has been introduced in (5) and the supremum is carried over all
functions $H\in C_{K}^{1,2}(]0,T[\times[-1,1])$. In this formula and below,
$\Psi(u_{t},H_{t})=\Psi(u_{t}(-1),u_{t}(1),$ $H_{t}(-1)/2,H_{t}(1)/2)$.

For each $H\in C^{1,2}_{0}([0,T]\times[-1,1])$, decompose $H$ as
$H=H^{(0)}+H^{(1)}$ where

$\displaystyle
H^{(1)}(t,x)=\frac{H(t,1)-H(t,-1)}{2}x+\frac{H(t,-1)+H(t,1)}{2}\mbox{ . }$

Note that $H^{(0)}(t,\pm 1)=0$, $H^{(1)}(t,\pm 1)=H(t,\pm 1)$ for all $0\leq
t\leq T$. By linearity of $L$, the previous expression is equal to

$\displaystyle\frac{1}{4}\underset{H^{(1)}}{\mbox{ sup
}}\Big{\\{}2L(H^{(1)})-\int_{0}^{T}\|\nabla
H_{t}^{(1)}\|_{\sigma(u_{t})}^{2}dt-\int_{0}^{T}\Psi(u_{t},H_{t}^{(1)})\,dt$
$\displaystyle\quad+\underset{H^{(0)}}{\mbox{ sup
}}\Big{[}\,2L(H^{(0)})-2\int_{0}^{T}\langle\nabla H_{t}^{(0)},\nabla
H_{t}^{(1)}\rangle_{\sigma(u_{t})}dt-\int_{0}^{T}\|\nabla
H_{t}^{(0)}\|_{\sigma(u_{t})}^{2}dt\,\Big{]}\,\Big{\\}}\;.$

We first compute the supremum over the function $H^{(0)}$. Since $H^{(0)}$
belongs to $C^{1,2}_{0}([0,T]\times[-1,1])$ and vanishes at $x=\pm 1$, and
since the set of smooth functions $C_{K}^{\infty}(]0,T[\times]-1,1[)$ is dense
in $C_{0}^{1,2}([0,T]\times]-1,1[)$, the supremum over $H^{(0)}$ is equal to:

$\displaystyle\underset{H\in C_{K}^{\infty}(]0,T[\times]-1,1[)}{\mbox{ sup
}}\Big{\\{}\,2L(H)-2\int_{0}^{T}\langle\nabla H_{t},\nabla
H_{t}^{(1)}\rangle_{\sigma(u_{t})}dt-\int_{0}^{T}\|\nabla
H_{t}\|_{\sigma(u_{t})}^{2}\,dt\,\Big{\\}}\mbox{ . }$

By the definition of $P$, given in (5.3), and the one of $M$, introduced in
(5.4), the previous supremum is equal to

$\displaystyle\underset{H\in C_{K}^{\infty}(]0,T[\times]-1,1[)}{\mbox{ sup
}}\Big{\\{}2\int_{0}^{T}\big{\langle}\,M_{t}-\sigma(u_{t})\,\nabla
H_{t}^{(1)}\,,\,\nabla H_{t}\,\big{\rangle}\,dt-\int_{0}^{T}\|\nabla
H_{t}\|_{\sigma(u_{t})}^{2}\,dt\Big{\\}}\;.$

Since $M_{t}-\sigma(u_{t})\nabla H_{t}^{(1)}$ belongs to
$L^{2}(\sigma(u)^{-1})$, by Proposition 5.1, the previous supremum is equal to
:

$\displaystyle\int_{0}^{T}\Big{\\{}\,\|\,M_{t}-\sigma(u_{t})\nabla
H_{t}^{(1)}\|^{2}_{\sigma(u_{t})^{-1}}\,-\,\langle\,[M_{t}-\sigma(u_{t})\nabla
H_{t}^{(1)}]\,\sigma(u_{t})^{-1}\rangle^{2}\,\mathcal{S}_{t}\,\Big{\\}}\,dt\;.$

Expanding the squares and by definition of $I_{[0,T]}^{(1)}(\pi)$, this
expression is equal to

$\displaystyle 4\,I_{[0,T]}^{(1)}(\pi)+\int_{0}^{T}\|\nabla
H_{t}^{(1)}\|_{\sigma(u_{t})}^{2}\,dt\,-\,\int_{0}^{T}\langle\nabla
H_{t}^{(1)}\rangle^{2}\,\mathcal{S}_{t}\,dt$
$\displaystyle\quad-\,2\,\int_{0}^{T}\langle M_{t},\nabla H_{t}^{(1)}\rangle
dt\,+\,2\,\int_{0}^{T}\langle M_{t}\sigma(u_{t})^{-1}\rangle\langle\nabla
H_{t}^{(1)}\rangle\,\mathcal{S}_{t}\,dt\;.$

Note that the term $\int_{0}^{T}\|\nabla
H_{t}^{(1)}\|_{\sigma(u_{t})}^{2}\,dt$ cancels with one which appears in the
formula for $I_{[0,T]}(\pi)$

Up to this point, we proved that $I_{[0,T]}(\pi)$ is equal to

$\displaystyle I_{[0,T]}^{(1)}(\pi)+\frac{1}{4}\underset{H^{(1)}}{\mbox{ sup
}}\Big{\\{}\,$ $\displaystyle
2L(H^{(1)})-\int_{0}^{T}\Psi\bigl{(}u_{t},H^{(1)}_{t})dt-\int_{0}^{T}\langle\nabla
H_{t}^{(1)}\rangle^{2}\mathcal{S}_{t}dt$ $\displaystyle-2\int_{0}^{T}\langle
M_{t},\nabla H_{t}^{(1)}\rangle dt+2\int_{0}^{T}\langle
M_{t}\sigma(u_{t})^{-1}\rangle\langle\nabla
H_{t}^{(1)}\rangle\mathcal{S}_{t}dt\Big{\\}}\mbox{ . }$

Let $\alpha(t)=H^{(1)}(t,-1)/2$, $\beta(t)=H^{(1)}(t,1)/2$ so that $\alpha$,
$\beta\in C_{0}^{1}([0,T])$, and
$H^{(1)}=(\beta-\alpha)X+(\beta+\alpha)\,{\mathbb{1}}$, where the functions
$X$, ${\mathbb{1}}$ have been introduced just before the statement of the
proposition. Replace $H^{(1)}$ by
$(\beta-\alpha)X+(\beta+\alpha)\,{\mathbb{1}}$ in the previous formula. A
straightforward computation yields that the expression inside braces coincides
with the one appearing in the definition of $I_{[0,T]}^{(2)}(\pi)$. This
completes the proof of the lemma since the supremum over $H^{(1)}$ corresponds
to one over $\alpha$, $\beta\in C_{0}^{1}([0,T])$. ∎

Let $\Phi_{t}\colon[0,1]^{2}\times\mathbb{R}^{2}\to\mathbb{R}$, $0\leq t\leq
T$, be given by:

${\color[rgb]{0,0,1}\Phi_{t}(r,s,x,y)}\,:=\,\underset{\alpha,\beta\in\mathbb{R}^{2}}{\mbox{
sup }}\Bigl{\\{}\alpha x+\beta y-\Upsilon_{t}(r,s,\alpha,\beta)\Bigl{\\}}$
(5.6)

where $\Upsilon_{t}$ has been introduced in (5.5). Taking $\alpha=\beta=0$
yields that $\Phi_{t}(r,s,x,y)\geq 0$. Moreover, since $\Psi$ is strictly
convex in $(\alpha,\beta)$, by Theorem 26.6 in [25], for each
$(x,y)\in\mathbb{R}^{2}$ there exists $(\alpha,\beta)\in\mathbb{R}^{2}$ such
that

$\displaystyle\Phi_{t}(r,s,x,y)=\alpha x+\beta
y-\Upsilon_{t}(r,s,\alpha,\beta)\mbox{ . }$

######  Proposition 5.5.

Fix $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Assume that there
exists $\epsilon_{0}>0$ such that $\epsilon_{0}\leq u\leq 1-\epsilon_{0}$.
Then,

$\displaystyle I_{[0,T]}^{(2)}(\pi)$
$\displaystyle=\frac{1}{4}\int_{0}^{T}\Phi_{t}\bigl{(}u_{t}(-1),u_{t}(1),g(t),h(t)\bigl{)}\,dt\mbox{
. }$ (5.7)

Moreover, there exists a finite constant $C(\epsilon_{0})$ such that

$\displaystyle e^{|\alpha(t)|}+e^{|\beta(t)|}\leq
C(\epsilon_{0})\,\big{\\{}\,1+|g(t)|+|h(t)|\,\big{\\}}$ (5.8)

for all $0\leq t\leq T$. In this formula, $g$, $h$ are the functions
introduced in (5.4), and $(\alpha(t),\beta(t))$ the pair which optimizes the
variational problem (5.6) with $r=u_{t}(-1)$, $s=u_{t}(1)$, $x=g(t)$,
$y=h(t)$.

###### Proof.

We first prove (5.8). Fix $t\in[0,T]$ such that $|g(t)|$, $|h(t)|<+\infty$.
Recall the definition of $\alpha(t),\beta(t)$. Clearly,

$U(\alpha)=g(t)-8(\alpha-\beta)\mathcal{S}_{t}\;\;\text{and}\;\;V(\beta)=h(t)+8(\alpha-\beta)\mathcal{S}_{t}\;,$

where

$\displaystyle
U(\alpha)=4{{\mathfrak{a}}}(1-u_{t}(-1))(e^{\alpha}-1)\,-\,4u_{t}(-1)(1-{{\mathfrak{a}}})(e^{-\alpha}-1)\;,$
$\displaystyle
V(\beta)=4{{\mathfrak{b}}}(1-u_{t}(1))(e^{\beta}-1)-4u_{t}(1)(1-{{\mathfrak{b}}})(e^{-\beta}-1)\;.$

Suppose, to fix ideas, that $\alpha\leq 0\leq\beta$. The other cases are
handled similarly. By definition of $U$,

$\displaystyle
4u_{t}(-1)\,(1-{{\mathfrak{a}}})\,(e^{-\alpha}-1)\,=\,4{{\mathfrak{a}}}\,(1-u_{t}(-1))\,(e^{\alpha}-1)-g(t)+8\,(\alpha-\beta)\,\mathcal{S}_{t}\mbox{
. }$

Since $\epsilon_{0}\leq u\leq 1-\epsilon_{0}$ and $\alpha\leq 0$, there exists
$C(\epsilon)<+\infty$:

$\displaystyle
e^{-\alpha}\,\leq\,C(\epsilon_{0})\,\big{\\{}\,1+|g(t)|+|\alpha|+|\beta|\,\big{\\}}\mbox{
. }$

A similar argument yields that:

$\displaystyle
e^{\beta}\,\leq\,C(\epsilon_{0})\,\big{\\{}\,1+|h(t)|+|\alpha|+|\beta|\,\big{\\}}\mbox{
. }$

Since there exists a finite constants $C_{0}$ such that $a\leq
C_{0}+(1/4)e^{a}$ for all $a\geq 0$, summing the previous bounds yields that

$\displaystyle e^{|\alpha|}+e^{|\beta|}=e^{-\alpha}+e^{\beta}\leq
C(\epsilon_{0})\,\big{\\{}\,1+|g(t)|+|h(t)|\,\big{\\}}$

as claimed.

We turn to the proof of identity (5.7). It is clear that

$\displaystyle I_{[0,T]}^{(2)}(\pi)$
$\displaystyle\leq\frac{1}{4}\int_{0}^{T}\Phi_{t}\bigl{(}u_{t}(-1),u_{t}(1),g(t),h(t)\bigl{)}\,dt\mbox{
. }$

To prove the reversed inequality, fix $t\in[0,T]$, and recall the definition
of the pair $(\alpha(t),\beta(t))$ introduced in the statement of the
proposition, and the one of $\Upsilon_{t}$ given in (5.5).

Claim A: Denote by $B([0,T])$ the subset of bounded functions. Then,

$\displaystyle\int_{0}^{T}\Phi_{t}\bigl{(}u_{t}(-1),u_{t}(1),g(t),h(t)\bigl{)}\,dt$
$\displaystyle\quad\leq\underset{\alpha,\beta\in B([0,T])}{\mbox{ sup
}}\int_{0}^{T}\Bigl{\\{}\alpha(t)g(t)+\beta(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha(t),\beta(t)\bigl{)}\Bigl{\\}}\,dt\mbox{
. }$

Let $(\alpha_{n}(t),\beta_{n}(t))$ be given by

$\displaystyle\alpha_{n}(t)=\alpha(t)\,\mathbb{1}_{\\{|\alpha(t)|\leq
n\\}}\mathbb{1}_{\\{|\beta(t)|\leq
n\\}}\;,\quad\beta_{n}(t)=\beta(t)\,\mathbb{1}_{\\{|\alpha(t)|\leq
n\\}}\mathbb{1}_{\\{|\beta(t)|\leq n\\}}\mbox{ . }$

By definition, $(\alpha_{n},\beta_{n})$ belong to $B([0,T])$. As
$\alpha(\cdot)$, $\beta(\cdot)$ are finite almost surely,
$\lim_{n}\alpha_{n}(t)=\alpha(t)$ and $\lim_{n}\beta_{n}(t)=\beta(t)$ almost
surely. Moreover:

$\displaystyle\alpha_{n}(t)g(t)+\beta_{n}(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha_{n}(t),\beta_{n}(t)\bigl{)}\geq
0$

because this expression is either equal to
$-\Psi\bigl{(}u_{t}(-1),u_{t}(1),0,0\bigl{)}\geq 0$ or it is equal to
$\Phi_{t}\bigl{(}u_{t}(-1),u_{t}(1),g(t),h(t)\bigl{)}\geq 0$. Therefore, by
Fatou’s Lemma,

$\displaystyle\int_{0}^{T}\Phi_{t}\bigl{(}u_{t}(-1),u_{t}(1),g(t),h(t)\bigl{)}dt$
$\displaystyle=$ $\displaystyle\int_{0}^{T}\underset{n\to+\infty}{\mbox{ lim
inf
}}\Bigl{\\{}\alpha_{n}(t)g(t)+\beta_{n}(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha_{n}(t),\beta_{n}(t)\bigl{)}\Bigl{\\}}\,dt$
$\displaystyle\leq$ $\displaystyle\underset{n\to+\infty}{\mbox{ lim inf
}}\int_{0}^{T}\Bigl{\\{}\alpha_{n}(t)g(t)+\beta_{n}(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha_{n}(t),\beta_{n}(t)\bigl{)}\Bigl{\\}}\,dt\;.$

Since $(\alpha_{n},\beta_{n})\in B([0,T])^{2}$, the previous expression is
bounded by:

$\displaystyle\underset{\alpha,\beta\in B([0,T])}{\mbox{ sup
}}\int_{0}^{T}\Bigl{\\{}\alpha(t)g(t)+\beta(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha(t),\beta(t)\bigl{)}\Bigl{\\}}\,dt\mbox{
, }$ (5.9)

as claimed.

Fix $\epsilon>0$ and $(\alpha,\beta)\in B([0,T])^{2}$. Let
$\varphi\colon\mathbb{R}\to[0,+\infty[$ be a smooth function such that ${\rm
supp}\,\varphi\subset[-1,1]$, $\int_{-1}^{1}\varphi(x)dx=1$ and
$\varphi(-x)=\varphi(x)$ for every $x\in\mathbb{R}$. Let
$\varphi_{\epsilon}(x)=\frac{1}{\epsilon}\varphi(\frac{x}{\epsilon})$,
$\epsilon>0$. Extend the definition of $\alpha$, $\beta$, setting
$\alpha(t)=\beta(t)=0$ for $t<0$, $t>T$. Let
$\alpha_{\epsilon}=\alpha*\psi_{\epsilon}$ and
$\beta_{\epsilon}=\beta*\psi_{\epsilon}$. Since $g,h\in L^{1}([0,T])$ and
$\alpha$, $\beta$, $u$ are bounded:

$\displaystyle\underset{\epsilon\to
0}{\lim}\int_{0}^{T}\Big{\\{}\,\alpha_{\epsilon}(t)g(t)+\beta_{\epsilon}(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha_{\epsilon}(t),\beta_{\epsilon}(t)\bigl{)}\,\Big{\\}}\,dt$
$\displaystyle\quad=\,\int_{0}^{T}\Bigl{\\{}\alpha(t)g(t)+\beta(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha(t),\beta(t)\bigl{)}\Bigl{\\}}\,dt\mbox{
. }$

Since $\alpha_{\epsilon}$ and $\beta_{\epsilon}$ are smooth functions, (5.9)
is equal to

$\displaystyle\underset{\alpha,\beta\in C^{\infty}([0,T])}{\mbox{ sup
}}\int_{0}^{T}\Bigl{\\{}\alpha(t)g(t)+\beta(t)h(t)-\Upsilon_{t}\bigl{(}u_{t}(-1),u_{t}(1),\alpha(t),\beta(t)\bigl{)}\Bigl{\\}}\,dt\mbox{
. }$

It remains to approximate smooth functions $\alpha$ and $\beta$ in
$C^{\infty}$ by functions which vanish at the boundary. This is standard and
left to the reader. Details can be found in Theorem 3.7.8 in [7]. ∎

######  Remark 5.6.

Proposition 5.5 can be proved without the condition
$\epsilon_{0}<u<1-\epsilon_{0}$. We refer to Section 3.7.2 in [7].

The proof of the next result is similar to the one presented in Section 4 of
[14].

######  Proposition 5.7.

Fix $T>0$ and $\pi(t,dx)=u(t,x)dx$ such that $I_{[0,T]}(\pi)<+\infty$. Assume
that there exists $\epsilon_{0}>0$ such that $\epsilon_{0}\leq u(t,x)\leq
1-\epsilon_{0}$ for all $(t,x)$ in $[0,T]\times[-1,1]$ and that $u(\cdot,x)\in
C^{\infty}([0,T])$ for all $x\in[-1,1]$. Then,
$I_{[0,T]}(\pi)=I_{[0,T]}^{(a)}(\pi)+I_{[0,T]}^{(b)}(\pi)$, where

$\displaystyle
I_{[0,T]}^{(a)}(\pi)=\frac{1}{4}\int_{0}^{T}\left\\{\|P_{t}+D(u_{t})\nabla
u_{t}\|_{\sigma(u_{t})^{-1}}^{2}-R_{t}\right\\}\,dt$ $\displaystyle
I_{[0,T]}^{(b)}(\pi)=\frac{1}{4}\,\int_{0}^{T}\Phi\bigl{(}u_{t}(-1),u_{t}(1),a(t),b(t)\bigl{)}\,dt$

$P$ is defined in (5.3), and

$\displaystyle\Xi(t,x)=\frac{1}{\langle\sigma(u_{t})^{-1}\rangle}\,\int_{-1}^{x}\frac{1}{\sigma(u_{t}(y))}\,dy\;,\quad
R_{t}=\left\langle\frac{D(u_{t})\nabla
u_{t}}{\sigma(u_{t})}\right\rangle^{2}\mathcal{S}_{t}\mbox{ , }$
$\displaystyle\vphantom{\Big{\\{}}a(t)\,=\,4\,\Big{\\{}\,\langle\partial_{t}u_{t},(1-\Xi(t))\rangle-\langle
D(u_{t})\nabla
u_{t},\nabla\Xi_{t}\rangle+u_{t}(-1)-{\mathfrak{a}}\,\Big{\\}}\;,$
$\displaystyle\vphantom{\Big{\\{}}b(t)\,=\,4\,\Big{\\{}\,\langle\partial_{t}u_{t},\Xi(t)\rangle+\langle
D(u_{t})\nabla
u_{t},\nabla\Xi_{t}\rangle+u_{t}(1)-{\mathfrak{b}}\,\Big{\\}}\;.$

##  6\. $I_{[0,T]}(.|\gamma)$-density

Fix a profile $\gamma\colon[-1,1]\to[0,1]$ satisfying the hypotheses of
Theorem 2.5. The main result of this section states that any trajectory
$\pi\in D([0,T],\mathcal{M})$ with finite rate function can be approximated by
a sequence of smooth trajectories $(\pi^{n})_{n\in\mathbb{N}}$ such that
$I_{[0,T]}(\pi^{n}|\gamma)\to I_{[0,T]}(\pi|\gamma)$.

Denote by $\rho$ the weak solution of the hydrodynamic equation (2.3) with
initial condition $\gamma$. Recall from Definition 2.8 that $\Pi_{\gamma}$
represents the set of paths $\pi(t,dx)=u(t,x)dx$ in $D([0,T],\mathcal{M})$
such that:

  * (a)

There exist $\delta>0$ such that $u_{t}=\rho_{t}$ on $[0,\delta]$;

  * (b)

$u\in C^{\infty}([\delta,T]\times[-1,1])$ and $u\in C([0,T]\times[-1,1])$

  * (c)

For each $\delta^{\prime}>0$, there exists $\epsilon>0$ such that
$\epsilon\leq u(t,x)\leq 1-\epsilon$ for all
$(t,x)\in[\delta^{\prime},T]\times[-1,1]$.

The main result of this section reads as follows.

######  Proposition 6.1.

Let $\gamma$ be a density profile satisfying the hypotheses of Theorem 2.5.
Then, the set $\Pi_{\gamma}$ is $I_{[0,T]}(.|\gamma)$-dense.

The proof is divided in several steps.

###  $\mathcal{F}_{1}$ density

Let $\mathcal{F}_{0}$ be the set of all paths $\pi(t,dx)=u(t,x)dx$ in
$D([0,T],\mathcal{M}^{0})$ with finite energy and for which there exists
$\delta>0$ for which $u_{t}=\rho_{t}$ for $0\leq t\leq\delta$. The proof of
the next result is similar to one of Lemma 5.3 in [14].

######  Lemma 6.2.

The set $\mathcal{F}_{0}$ is $I_{[0,T]}(.|\gamma)$-dense.

Let $\mathcal{F}_{1}$ be the set of all paths $\pi(t,dx)=u(t,x)dx$ in
$\mathcal{F}_{0}$ with the property that for every $\delta>0$ there exists
$\epsilon>0$ such that $\epsilon\leq u(t,x)\leq 1-\epsilon$ for all
$(t,x)\in[\delta,T]\times[-1,1]$.

######  Lemma 6.3.

The set $\mathcal{F}_{1}$ is $I_{[0,T]}(.|\gamma)$-dense.

###### Proof.

Fix $\pi(t,dx)=u(t,x)dx$ in $\mathcal{F}_{0}$. By Lemma 6.2, it is enough to
show that there exists a sequence $(\pi^{n})_{n\in\mathbb{N}}$ in
$\mathcal{F}_{1}$ such that $I_{[0,T]}(\pi^{n}|\gamma)\to
I_{[0,T]}(\pi|\gamma)$ and $\pi^{n}\to\pi$.

Let $\pi^{\epsilon}(t,dx)=u^{\epsilon}(t,x)dx$, where
$u^{\epsilon}=(1-\epsilon)u+\epsilon\rho$, $0<\epsilon<1$. We claim that
$\pi^{\epsilon}$ belongs to $\mathcal{F}_{1}$. Clearly $\pi^{\epsilon}\in
D([0,T],\mathcal{M}^{0})$. By the definition of $\mathcal{F}_{0}$, by
Corollary 4.2 and by convexity of $\mathcal{Q}_{[0,T]}$ (Proposition 4.1),
$\pi^{\epsilon}$ has finite energy because

$\displaystyle\mathcal{Q}_{[0,T]}(u^{\epsilon})\,\leq\,\epsilon\,\mathcal{Q}_{[0,T]}(\rho)\,+\,(1-\epsilon)\,\mathcal{Q}_{[0,T]}(u)<+\infty\;,$

Since $\pi\in\mathcal{F}_{0}$, there exists $\delta>0$ such that
$u_{t}=\rho_{t}$ for every $0\leq t\leq\delta$. Hence, by construction of
$\pi^{\epsilon}$, $u^{\epsilon}_{t}=\rho_{t}$ for every $0\leq t\leq\delta$.
Moreover, by Proposition 8.3, for each $\delta_{1}>0$, there exist
$\epsilon_{1}>0$ such that $\epsilon_{1}\leq\rho_{t}\leq 1-\epsilon_{1}$, for
every $\delta_{1}\leq t\leq T$. Therefore, for $t$ in this interval,

$\displaystyle\epsilon\epsilon_{1}\leq(1-\epsilon)u_{t}+\epsilon\rho_{t}\leq
1-\epsilon+\epsilon(1-\epsilon_{1})=1-\epsilon\epsilon_{1}\;.$

This shows that $\pi^{\epsilon}$ belongs to $\mathcal{F}_{1}$.

It’s clear that $\pi^{\epsilon}$ converges to $\pi$ in $D([0,T],\mathcal{M})$
as $\epsilon\to 0$. To conclude the proof, it remains to show that
$I_{[0,T]}(\pi^{\epsilon}|\gamma)$ converges to $I_{[0,T]}(\pi|\gamma)$. By
Theorem 4.6, $I_{[0,T]}$ is lower semicontinuous, so that
$\liminf_{\epsilon\to 0}I_{[0,T]}(\pi^{\epsilon}|\gamma)\geq
I_{[0,T]}(\pi|\gamma)$. We turn to the reverse inequality,
$\limsup_{\epsilon\to 0}I_{[0,T]}(\pi^{\epsilon})\leq I_{[0,T]}(\pi)$.

Recall from (2.6) the definition of the functionals $L_{0}$, $L$. To stress
its dependence on $\pi(t,dx)=u(t,x)\,dx$, we denote them below by
$L_{0}(u,H)$, $L(u,H)$, respectively. Fix $H\in C^{1,2}([0,T]\times[-1,1])$.
Since $\chi$ is concave and $L_{0}$, $B$ are linear in $u$,
$\hat{J}_{H}(\pi^{\epsilon})$ is less than or equal to

$\displaystyle\epsilon\,L_{0}(\rho,H)\,+\,(1-\epsilon)\,L_{0}(u,H)-\epsilon\,\int_{0}^{T}B(\rho_{t},H_{t})dt-(1-\epsilon)\,\int_{0}^{T}B(u_{t},H_{t})dt$
(6.1)
$\displaystyle+\epsilon\int_{0}^{T}\int_{-1}^{1}\Bigl{\\{}D(u^{\epsilon}_{t})\nabla\rho_{t}\nabla
H_{t}-D(u^{\epsilon}_{t})\chi(\rho_{t})(\nabla H_{t})^{2}\Bigl{\\}}dx\,dt$
$\displaystyle+(1-\epsilon)\int_{0}^{T}\int_{-1}^{1}\Bigl{\\{}D(u^{\epsilon}_{t})\nabla
u_{t}\nabla H_{t}-D(u^{\epsilon}_{t})\chi(u_{t})(\nabla
H_{t})^{2}\Bigl{\\}}dxdt\;.$

We first estimate the terms involving the solution of the hydrodynamic
equation. We claim that

$\displaystyle\underset{\epsilon\to 0}{\lim}$ $\displaystyle\underset{H\in
C^{1,2}([0,T]\times[-1,1])}{\mbox{ sup
}}\Big{\\{}\epsilon\,L_{0}(\rho,H)\,-\,\epsilon\,\int_{0}^{T}B(\rho_{t},H_{t})dt$
(6.2)
$\displaystyle\qquad\quad+\,\epsilon\int_{0}^{T}\int_{-1}^{1}\Big{[}\,D(u^{\epsilon}_{t})\nabla\rho_{t}\nabla
H_{t}-D(u^{\epsilon}_{t})\chi(\rho_{t})(\nabla
H_{t})^{2}\Big{]}\,dx\,dt\,\Big{\\}}\leq 0\;.$

Since $\rho$ is a weak solution of hydrodynamic equation (2.3), and
${\mathfrak{c}}-u={\mathfrak{c}}(1-u)-u(1-{\mathfrak{c}})$, the expression
inside braces is equal to

$\displaystyle-\epsilon\int_{0}^{T}B_{0}(\rho_{t},H_{t})dt\,+\,\epsilon\int_{0}^{T}\int_{-1}^{1}\Big{\\{}\,\big{[}\,(D(u^{\epsilon}_{t})-D(\rho_{t})\,\big{]}\nabla\rho_{t}\nabla
H_{t}-D(u^{\epsilon}_{t})\chi(\rho_{t})(\nabla H_{t})^{2}\Bigl{\\}}dx\,dt\;,$

where

$\displaystyle B_{0}(u_{t},H_{t})\,$
$\displaystyle=\,u_{t}(1)\,[1-{{\mathfrak{b}}}]\,\Lambda(-H_{t}(1))\,+\,[1-u_{t}(1)]\,{{\mathfrak{b}}}\,\Lambda(H_{t}(1))$
$\displaystyle=\,u_{t}(-1)\,[1-{{\mathfrak{a}}}]\,\Lambda(-H_{t}(-1))\,+\,[1-u_{t}(-1)]\,{{\mathfrak{a}}}\,\Lambda(H_{t}(-1))\;,$

and $\Lambda(x)=e^{x}-x-1$. Clearly, $B_{0}(u_{t},H_{t})\geq 0$, and the first
term of the penultimate displayed equation is negative. On the other hand, by
Young’s inequality the second term is bounded above by

$\displaystyle\epsilon\,\int_{0}^{T}\int_{-1}^{1}\frac{[\,D(u^{\epsilon}_{t})-D(\rho_{t})\,]^{2}(\nabla\rho_{t})^{2}}{4D(u^{\epsilon}_{t})\chi(\rho_{t})}\,dx\,dt\;.$

Since the diffusion coefficient $D$ is bounded below by a strictly positive
constant and above by a finite constant, the previous expression is less than
or equal to $C_{0}\,\epsilon\,\mathcal{Q}_{[0,T]}(\rho)$. By Corollary 4.2.
the solution of the hydrodynamic equation has finite energy, which proves
claim (6.2).

We turn to the terms in (6.1) which are multiplied by $1-\epsilon$. Our goal
is to replace $u^{\epsilon}$ by $u$ to get an expression bounded by
$(1-\epsilon)I_{[0,T]}(u)$. Rewrite $D(u^{\epsilon}_{t})\chi(u_{t})$ as
$D(u_{t})\chi(u_{t})\times[D(u^{\epsilon}_{t})/D(u_{t})]$. Since
$u^{\epsilon}=u+\epsilon(\rho-u)$, and $|\rho-u|\leq 1$, adding and
subtracting in the numerator $D(u)$ yields that
$D(u^{\epsilon}_{t})/D(u_{t})\geq 1-C_{0}\epsilon$ for some finite constant
$C_{0}$ because $D$ is bounded below by a strictly positive constant. In
conclusion, since $\chi D=\sigma$, the terms in (6.1) which are multiplied by
$1-\epsilon$ are bounded by

$L_{0}(u,H)-\int_{0}^{T}B(u_{t},H_{t})dt+\int_{0}^{T}\int_{-1}^{1}\Bigl{\\{}D(u^{\epsilon}_{t})\nabla
u_{t}\nabla H_{t}-(1-C_{0}\epsilon)\sigma(u_{t})(\nabla
H_{t})^{2}\Bigl{\\}}dxdt\;.$

Fix $\eta>0$ which converges to $0$ after $\epsilon$, and add and subtract
$D(u_{t})\nabla u_{t}\nabla H_{t}$. By Young’s inequality, and since
$|D(u^{\epsilon}_{t})-D(u_{t})|\leq C_{0}\epsilon$,

$\int_{0}^{T}\int_{-1}^{1}\\{\,D(u^{\epsilon}_{t})-D(u_{t})\\}\nabla
u_{t}\nabla
H_{t}\,dx\,dt\;\leq\;\eta\int_{0}^{T}\int_{-1}^{1}\sigma(u_{t})(\nabla
H_{t})^{2}\,dx\,dt\,+\,\frac{C_{0}\epsilon^{2}}{\eta}{\mathcal{Q}}_{[0,T]}(u)\;.$

As $u$ belongs to ${\mathcal{F}}_{0}$, it has finite energy, and the second
term on the right-hand side vanishes as $\epsilon\to 0$. Therefore, the next
to the last displayed equation is bounded by

$L(u,H)-\int_{0}^{T}B_{0}(u_{t},H_{t})dt\,-\,(1-\eta-
C_{0}\epsilon)\,\int_{0}^{T}\int_{-1}^{1}\sigma(u_{t})(\nabla
H_{t})^{2}dx\,dt\,+\,C_{0}(u)\epsilon^{2}\;.$

Mind that we changed $L_{0}$, $B$ to $L$, $B_{0}$, respectively.

Let $a=(1-\eta-C_{0}\epsilon)^{-1}>1$, $H=aG$. Since $e^{ax}-ax-1\geq
a(e^{x}-x-1)$ for $x\in{\mathbb{R}}$, the previous expression is bounded by

$a\,\Big{\\{}\,L(u,G)-\int_{0}^{T}B_{0}(u_{t},G_{t})dt\,-\,\,\int_{0}^{T}\int_{-1}^{1}\sigma(u_{t})(\nabla
G_{t})^{2}dx\,dt\,\Big{\\}}\,+\,C_{0}(u)\epsilon^{2}\;.$

This expression is bounded by $aI_{[0,T]}(u)+C_{0}(u)\epsilon^{2}$. It remains
to let $\epsilon\to 0$ and then $\eta\to 0$ to complete the proof of the
lemma. ∎

###  $\mathcal{F}_{2}$ density

Let $\mathcal{F}_{2}$ be the set of all paths $\pi(t,dx)=u(t,x)dx$ in
$\mathcal{F}_{1}$ with the property that there exist $\delta_{1},\delta_{2}>0$
such that $u_{t}=\rho_{t}$ in $[0,\delta_{1}]$ and $u$ is constant in
$[\delta_{1},\delta_{1}+\delta_{2}]$.

######  Lemma 6.4.

The set $\mathcal{F}_{2}$ is $I_{[0,T]}(.|\gamma)$-dense.

###### Proof.

Fix $\pi(t,dx)=u(t,x)dx$ in $\mathcal{F}_{1}$. By Lemma 6.2, it is enough to
show that there exists a sequence $(\pi^{n})_{n\in\mathbb{N}}$ in
$\mathcal{F}_{2}$ such that $I_{[0,T]}(\pi^{n}|\gamma)\to
I_{[0,T]}(\pi|\gamma)$ and $\pi^{n}\to\pi$.

By definition of $\mathcal{F}_{1}$, there exists $\delta>0$ such that
$u_{t}=\rho_{t}$ for $t\in[0,\delta]$. Fix a sequence $r_{n}\uparrow\delta$.
Since $\rho$ has a finite energy, there exists $r_{n}<t^{n}_{1}<\delta$ such
that

$\int_{-1}^{1}\frac{D(\rho_{t^{n}_{1}})\,(\nabla\rho_{t^{n}_{1}})^{2}}{\sigma(\rho_{t^{n}_{1}})}\,dx\,\leq\,\frac{2}{\delta-
r_{n}}\int_{r_{n}}^{\delta}\int_{-1}^{1}\frac{D(\rho_{t})\,(\nabla\rho_{t})^{2}}{\sigma(\rho_{t})}\,dx\,dt<+\infty\;.$
(6.3)

Consider a sequence $t^{n}_{2}$ such that $t^{n}_{2}\to 0$,
$t^{n}_{1}+t^{n}_{2}<\delta$. Define the path
$\pi^{(n)}(t,dx)=u^{(n)}(t,x)dx$, $n\in\mathbb{N}^{*}$, by $u_{t}^{(n)}=u_{t}$
on $[0,t^{n}_{1}]$, $u_{t}^{(n)}=u_{t^{n}_{1}}$ on
$[t^{n}_{1},t^{n}_{1}+t^{n}_{2}]$ and $u_{t}^{(n)}=u_{t-t^{n}_{2}}$ on
$[t^{n}_{1}+t^{n}_{2},T]$. By construction, $\pi^{(n)}$ belongs to
$\mathcal{F}_{1}$ and $\pi^{(n)}\to\pi$. By lower semicontinuity of the rate
function, it remains to show that $\limsup_{n}I_{[0,T]}(\pi^{(n)}|\gamma)\leq
I_{[0,T]}(\pi|\gamma)$.

By Corollary 4.9,

$\displaystyle
I_{[0,T]}(\pi^{(n)}|\gamma)\,=\,I_{[0,t^{n}_{1}]}(\pi^{(n)}|\gamma)\,+\,I_{[t^{n}_{1},s^{n}_{2}]}(\pi^{(n)})\,+\,I_{[s^{n}_{2},T]}(\pi^{(n)})\;.$

where $s^{n}_{2}=t^{n}_{1}+t^{n}_{2}$. By definition of $\pi^{(n)}$,
$I_{[0,t^{n}_{1}]}(\pi^{(n)}|\gamma)=I_{[0,t^{n}_{1}]}(\rho|\gamma)=0$ for all
$n$.

We claim that
$\underset{n\to+\infty}{\lim}I_{[t^{n}_{1},s^{n}_{2}]}(\pi^{(n)})=0$. Fix
$H\in C^{1,2}([t^{n}_{1},s^{n}_{2}]\times[-1,1])$. Since $\pi^{(n)}$ is
constant on $[t^{n}_{1},s^{n}_{2}]$, $L_{0}(H)=0$, and by Young’s inequality,

$\displaystyle\hat{J}_{[t^{n}_{1},s^{n}_{2}],H}(\pi^{(n)})\,\leq\,(s^{n}_{2}-t^{n}_{1})\,\int_{-1}^{1}\frac{D(\rho(t^{n}_{1},x))[\nabla\rho(t^{n}_{1},x)]^{2}}{4\,\sigma(\rho(t^{n}_{1},x)}\,dx\,-\,\int_{t^{n}_{1}}^{s^{n}_{2}}B(\rho_{t^{n}_{1}},H_{t})dt\;.$

By (6.3), the first term is bounded by

$\int_{r_{n}}^{\delta}\int_{-1}^{1}\frac{D(\rho(t,x))[\nabla\rho(t,x)]^{2}}{2\,\sigma(\rho(t,x)}\,dx\,dt\;,$

which vanishes as $n\to\infty$ because $\rho$ has finite energy. On the other
hand, as $-\,B(a,H)\leq 4$, the second term is bounded by
$4(s^{n}_{2}-t^{n}_{1})\to 0$. This proves that
$I_{[t^{n}_{1},s^{n}_{2}]}(\pi^{(n)})\to 0$.

Finally, by construction,
$I_{[s^{n}_{2},T]}(\pi^{(n)})=I_{[t^{n}_{1},T-t^{n}_{2}]}(\pi)\leq
I_{[t^{n}_{1},T]}(\pi)$, which completes the proof of the lemma. ∎

###  $\mathcal{F}_{3}$ density

Let $\mathcal{F}_{3}$ be the set of all paths $\pi(t,dx)=u(t,x)dx$ in
$\mathcal{F}_{2}$ for which there exist $\delta,\delta_{1}>0$ such that
$u_{t}=\rho_{t}$ in $[0,\delta]$, $u_{t}$ is constant in
$[\delta,\delta+\delta_{1}]$, and $u(.,x)$ belongs to $C^{\infty}([\delta,T])$
for each $x\in[-1,1]$.

The proof of the $\mathcal{F}_{3}$-density relies one the representation of
the rate functional derived in Proposition 5.4. Fix $\pi\in\mathcal{F}_{2}$.
By definition, $\pi(t,dx)=u(t,x)dx$ belongs to $D([0,T],\mathcal{M})$, has
finite energy, and there exist $(\delta,\delta_{1})\in[0,T]^{2}$ such that:

$\displaystyle u_{t}=\left\\{\begin{array}[]{ccc}\rho_{t}&0\leq
t\leq\delta\;,\\\ u_{\delta}&\delta\leq
t\leq\delta+\delta_{1}\;,\end{array}\right.$

where $\rho$ is the weak solution of the hydrodynamic equation. Moreover, for
every $\delta_{2}$, there exists $\epsilon>0$ such that $\epsilon\leq
u(t,x)\leq 1-\epsilon$ for all $(t,x)\in[\delta_{2},T]\times[-1,1]$.

Fix a smooth function $\varphi$ such that supp$(\varphi)\subset\,]0,1[$ and
$\int_{0}^{1}\varphi(t)dt=1$. Let
$\varphi_{\epsilon}(t)=(1/\epsilon)\,\varphi(t/\epsilon)$, $\epsilon>0$. We
extend the definition of $u$ to $[0,2T]$ setting $u(T+t,x)=u(T-t,x)$ for
$0\leq t\leq T$. For $n$ such that $1/n<\delta_{1}$, let

$\displaystyle u^{n}(t)=\left\\{\begin{array}[]{ccc}u(t)&0\leq t\leq\delta\\\
(u*\varphi_{\frac{1}{n}})(t)&\delta\leq t\leq T\;.\end{array}\right.$ (6.6)

Note that $u(t)=u(\delta)$ for $\delta\leq t\leq\delta+\delta_{1}-(1/n)$. Set
$\pi^{n}(t,dx)=u^{n}(t,x)dx$.

######  Lemma 6.5.

Fix $\pi\in\mathcal{F}_{2}$. Each sequence in ${\mathbb{N}}$ has a subsequence
$(\psi(n))_{n\in\mathbb{N}}$ such that

$\displaystyle\underset{n\to\infty}{\lim}I_{[0,T]}^{(1)}(\pi^{\psi(n)}|\gamma)=I_{[0,T]}^{(1)}(\pi|\gamma)$

###### Proof.

By Proposition 5.4:

$\displaystyle I_{[0,T]}^{(1)}(\pi^{n}|\gamma)$
$\displaystyle=\frac{1}{4}\int_{0}^{T}\Bigl{\\{}\langle
M_{n}(t),M_{n}(t)\rangle_{\sigma(u_{t}^{n})^{-1}}-\langle
M_{n}(t)\sigma(u_{t}^{n})^{-1}\rangle^{2}\langle\sigma(u_{t}^{n})^{-1}\rangle^{-1}\Bigl{\\}}dt$

where $M_{n}=P_{n}+D(u^{n})\nabla u^{n}$ and $P_{n}\in
L^{2}(\sigma(u^{n})^{-1})$, introduced in Lemma 5.2, is such that

$\displaystyle-\int_{0}^{T}\langle\pi^{n},\partial_{t}H_{t}\rangle
dt=\langle\langle\nabla H,P_{n}\rangle\rangle$

for every $H\in C_{K}^{\infty}(]0,T[\times]-1,1[)$. An elementary computation
shows that

$\displaystyle P_{n}(t)=\left\\{\begin{array}[]{ccc}P(t)&0\leq t\leq\delta\\\
(P*\varphi_{\frac{1}{n}})(t)&\delta\leq t\leq T\;,\end{array}\right.$

provided we extend the definition of $P$ to $[0,2T]$ setting $P(T+t)=-P(T-t)$
for $t\in[0,T]$.

Since $u^{n}=u$ and $P_{n}=P$ on the time-interval $[0,\delta]$, in view of
the formula for $I_{[0,T]}^{(1)}(\pi^{n}|\gamma)$, we only need to examine the
interval $[\delta,T]$. In this interval,

$\displaystyle u^{n}=u*\varphi_{\frac{1}{n}}\;,\quad\nabla u^{n}=\nabla
u*\varphi_{\frac{1}{n}}\;,\;\;\text{ and
}\;\;P_{n}=P*\varphi_{\frac{1}{n}}\;.$

As $\pi$ belongs to ${\mathcal{F}}_{2}$, there exists $\epsilon>0$ such that
$\epsilon\leq u\leq 1-\epsilon$ on the interval $[\delta,T]$. In particular,
on the interval $[\delta,T]$, the space $L^{2}$ and
$L^{2}(\sigma(u^{n})^{-1})$ coincide.

Fix the set $[\delta,T]$. As $u^{n}$ converge to $u$ in $L^{1}$, there exists
a subsequence, still denoted by $u^{n}$, which converges a.s. to $u$. Hence,
the uniformly bounded sequences $D(u_{n})$, $\sigma(u_{n})^{-1}$ converge a.s.
to $D(u)$, $\sigma(u)^{-1}$, respectively.

Since $P$, $\nabla u$ belong to $L^{2}$, $D$ is bounded and $u_{n}$ converge
a.s. to $u$, $P_{n}$, $D(u_{n})\,\nabla u_{n}$ converge in $L^{2}$ to $P$,
$D(u)\,\nabla u$, respectively.

Putting together the previous estimates yields that there exists a subsequence
$\psi(n)$ such that

$\displaystyle\underset{n\to\infty}{\lim}\frac{1}{4}\int_{\delta}^{T}\Bigl{\\{}\,\|\,M_{\psi(n)}(t)\,\|^{2}_{\sigma(u_{t}^{\psi(n)})^{-1}}-\langle
M_{\psi(n)}(t)\,\sigma(u_{t}^{\psi(n)})^{-1}\rangle^{2}\langle\sigma(u_{t}^{\psi(n)})^{-1}\rangle^{-1}\Bigl{\\}}\,dt$
$\displaystyle\quad=\,\frac{1}{4}\int_{\delta}^{T}\Bigl{\\{}\|\,M_{\psi(n)}(t)\,\|^{2}_{\sigma(u_{t})^{-1}}-\langle
M(t)\sigma(u_{t})^{-1}\rangle^{2}\langle\sigma(u_{t})^{-1}\rangle^{-1}\Bigl{\\}}\,dt\;,$

which completes the proof of the lemma. ∎

######  Lemma 6.6.

Fix $\pi\in\mathcal{F}_{2}$. Each subsequence of ${\mathbb{N}}$ has a
subsequence $(\psi(n))_{n\in\mathbb{N}}$ such that

$\displaystyle\underset{n\to\infty}{\limsup}I_{[0,T]}^{(2)}(\pi^{\psi(n)}|\gamma)\leq
I_{[0,T]}^{(2)}(\pi|\gamma)\;.$

###### Proof.

Recall from Proposition 5.5 the formula for the functional
$I_{[0,T]}^{(2)}(\,\cdot\,|\gamma)$. As in the proof of the previous lemma, we
may concentrate on the interval $[\delta,T]$.

Since $u$ belongs to ${\mathcal{F}}_{2}$, there exists $\epsilon>0$ such that
$\epsilon\leq u\leq 1-\epsilon$ on the time-interval $[\delta,T]$.

$\displaystyle\int_{\delta}^{T}\Phi_{t}\bigl{(}u^{n}_{t}(-1),u^{n}_{t}(1),g^{n}(t),h^{n}(t)\bigl{)}\,dt$
$\displaystyle\quad=\;\int_{\delta}^{T}\sup_{\alpha,\beta}\Bigl{\\{}\alpha
g^{n}(t)+\beta
h^{n}(t)-\Upsilon_{t}(u^{n}_{t}(-1),u^{n}_{t}(1),\alpha,\beta)\Bigl{\\}}\,dt\;,$

where $g_{n}(t)$, $h_{n}(t)$ are given by (5.4) with $u^{n}$ replacing $u$.
Mind that this substitution affects the definitions of $P^{X}_{n}$,
$P^{{\mathbb{1}}}_{n}$, $M^{n}$ and ${\mathcal{S}}^{n}$. By the bound on
$\alpha$, $\beta$ presented in Proposition 5.5, we may restrict the supremum
to pairs $(\alpha,\beta)$ such that $e^{|\alpha|}+e^{|\beta|}\leq
C_{0}(1+|g_{n}(t)|+|h_{n}(t)|)$.

Most of the terms appearing in the formula for $g^{n}$ and $h^{n}$ are
convolutions with $\varphi_{\frac{1}{n}}$. For example,
$P_{n}^{X}(t)=\bigl{(}P^{X}*\varphi_{\frac{1}{n}}\bigl{)}(t)$. We wish to
replace all terms which appear in the formula for $g^{n}$, $h^{n}$,
${\mathcal{S}}^{n}$ and are not expressed as convolutions by convolutions. We
present the details for one of them. The others are handled similarly.

Consider the term $(\alpha-\beta)\langle D(u_{n})\nabla u_{n}\rangle$. Add and
subtract $(\alpha-\beta)\langle D(u)\nabla u\rangle*\varphi_{\frac{1}{n}}$.
Since the supremum of a sum is bounded by the sum of the supremums, to carry
out this replacement, it is enough to show that

$\displaystyle\underset{n\to+\infty}{\lim}\int_{\delta}^{T}\underset{\alpha,\beta}{\mbox{
sup }}\Bigl{\\{}(\alpha-\beta)\bigl{[}\langle D(u_{n})\nabla
u_{n}\rangle(t)-\langle D(u)\nabla
u\rangle*\varphi_{\frac{1}{n}}(t)\bigl{]}\Bigl{\\}}\,dt=0\;,$

where, recall, the supremum is restricted to pairs whose exponentials are
bounded. By Young’s inequality $2xy\leq A^{-1}x^{2}+Ay^{2}$, and since
$a^{2}\leq C_{0}+e^{|a|}$ for some finite constant $C_{0}$ whose value may
change from line to line, integral is bounded by:

$\displaystyle\frac{C_{0}}{A}\int_{\delta}^{T}\bigl{(}1+|g_{n}(t)|+|h_{n}(t)|\bigl{)}\,dt\,+\,A\int_{\delta}^{T}\bigl{[}\langle
D(u_{n})\nabla u_{n}\rangle(t)-\langle D(u)\nabla
u\rangle*\varphi_{\frac{1}{n}}(t)\bigl{]}^{2}\,dt$

for all $A>0$.

Since $D(.)$ is continuous and $u_{n}$, $\nabla u_{n}$ converge in
$L^{2}([\delta,T]\times[-1,1])$ to $u$, $\nabla u$, respectively, the second
term vanisheds as $n\to+\infty$. For a similar reasons, the first term
vanishes, as $n\to+\infty$, and then $A\to\infty$.

After all replacements, we obtain that

$\displaystyle\underset{n\to+\infty}{\limsup}\,I_{[\delta,T]}^{(2)}(\pi^{n})\leq\underset{n\to+\infty}{\limsup}\,\frac{1}{4}\int_{\delta}^{T}$
$\displaystyle\sup_{\alpha,\beta}\Big{\\{}\int_{0}^{1/n}\varphi_{n}(s)\,\Big{\\{}\,\alpha\,g(t+s)+\beta\,h(t+s)\,-\,\Upsilon_{t+s}(u_{t+s}(-1),u_{t+s}(1),\alpha,\beta)\Big{\\}}\,ds\,\Big{\\}}\,dt\;.$

By moving the supremum inside the integral, the right-hand side is seen to be
bounded by

$\displaystyle\underset{n\to+\infty}{\limsup}\,\frac{1}{4}\int_{\delta}^{T}[\Phi_{\cdot}(u_{\cdot}(-1),u_{\cdot}(1),g(\cdot),h(\cdot))*\varphi_{n}](t)\,dt\;.$

Since $\Phi$ is integrable, this expression is equal to

$\displaystyle\frac{1}{4}\int_{\delta}^{T}\Phi_{t}(u_{t}(1),u_{t}(-1),g(t),h(t))\,dt\;,$

which completes the proof of the lemma. ∎

######  Lemma 6.7.

The set $\mathcal{F}_{3}$ is $I_{[0,T]}(.|\gamma)$-dense.

###### Proof.

Fix $\pi(t,dx)=u(t,x)dx$ in $\mathcal{F}_{2}$ such that
$I_{[0,T]}(\pi|\gamma)<+\infty$. Consider $(\pi^{n})_{n\in\mathbb{N}^{*}}$ as
in (6.6). It’s clear that $\pi^{n}$ belongs to $\mathcal{F}_{3}$ and converges
to $\pi$. Since, by Theorem 4.6, the rate function is lower semicontinuous, to
complete the proof it remains to show that
$\limsup_{n}I_{[0,T]}(\pi^{n}|\gamma)\leq I_{[0,T]}(\pi|\gamma)$. This follows
from the two previous lemmata. ∎

###  $\Pi_{\gamma}$ density

Recall the definition of the set $\Pi_{\gamma}$ introduced at the beginning of
this section. Denote by $(P_{t}^{(D)})_{t\geq 0}$ and $(P_{t}^{(N)})_{t\geq
0}$ the semigroup associated to the Laplacian on $[-1,1]$ with Dirichlet,
Neumann boundary conditions, respectively. The proof of the next result is
similar to the one of Lemmata 5.6 and 5.7 in [14]. It relies on the
decomposition of the rate function presented in Proposition 5.7 and uses the
fact that $D$ is bounded.

######  Lemma 6.8.

Fix $\pi(t,dx)=u(t,x)dx$ in $\mathcal{F}_{3}$ such that
$I_{[0,T]}(\pi|\gamma)<\infty$. Let $\delta$, $\delta_{1}>0$ be the positive
constants such that $u_{t}=\rho_{t}$ on $[0,\delta]$ and $u_{t}=\rho_{\delta}$
on $[\delta,\delta+\delta_{1}]$. Let $\kappa\colon[0,T]\to[0,1]$ be a
nondecreasing smooth function such that

$\displaystyle\kappa(t)=\left\\{\begin{array}[]{ccc}0&0\leq t\leq\delta\;,\\\
0<\kappa(t)<1&\delta<t<\delta+\delta_{1}\;,\\\ 1&\delta+\delta_{1}\leq t\leq
T\;.\end{array}\right.\;.$

Let $\kappa_{n}(t)=(1/n)\,\kappa(t)$, $n\geq 1$, and set
$\pi^{n}(t,dx)=u^{n}(t,x)\,dx$ where
$u_{t}^{n}=w_{t}+P^{(D)}_{\kappa_{n}(t)}[u_{t}-w_{t}]$ and

$w_{t}(x)\;=\;\frac{u_{t}(1)+u_{t}(-1)}{2}\,+\,\frac{u_{t}(1)-u_{t}(-1)}{2}\,x\;.$

Then, for each $n\geq 1$, $\pi^{n}(t,dx)=u^{n}(t,x)dx$ belongs to
$\Pi_{\gamma}$ and $\limsup_{n}I_{[0,T]}(\pi^{n}|\gamma)\leq
I_{[0,T]}(\pi|\gamma)$.

###### Proof of Proposition 6.1.

Since $\pi^{n}$ converges to $\pi$, the assertion of the proposition follows
from the previous lemma. ∎

We conclude this section providing an explicit formula for the rate functions
of trajectories in $\Pi_{\gamma}$. The proof does not require the density
profile $\gamma$ to satisfy the hypotheses of Theorem 2.5.

###### Proof of Proposition 2.9.

By Corollary 4.9 and Corollary 4.5,
$I_{[0,T]}(\pi|\gamma)=I_{[\delta,T]}(\pi)$. The rest of the argument is
straightforward from the variational formula for $I_{[\delta,T]}(\pi)$. We
refer to the proof of Proposition 2.6 on [14]. ∎

##  7\. The large deviations principle

The proof of the large deviations principle follows the one presented in [4,
15]. It differes slightly from the original one [19] as the functional
$J_{H}(\cdot)$ is set to be $+\infty$ on paths with infinite energy.

The proof of the lower and upper bounds do not require $\gamma$ to satisfy the
regularity conditions stated in Theorem 2.5. It is only at the end of the
proof of the lower bound, where we use the $I_{[0,T]}$-density of the set
$\Pi_{\gamma}$, that these hypotheses are needed. See Remark 2.6.

The proof of the upper bound is identical to the one for one-dimensional
symmetric exclusion processes in weak contact with reservoirs [14] which
relies on results presented in [15]. Full details are given in Section 3.4 in
[7].

### Lower bound

The proof of the lower bound relies on the hydrodynamic limit of weakly
asymmetric dynamics and on the $I_{[0,T]}(\,\cdot\,|\gamma)$-density of the
set $\Pi_{\gamma}$ introduced in the previous section.

Fix $0<t_{0}<T$, a profile $\gamma\colon[-1,1]\to[0,1]$, and a function $H$ in
$C([0,T]\times[-1,1])$ such that $H=0$ in $[0,t_{0}]\times[-1,1]$ and $H\in
C^{1,2}([t_{0},T]\times[-1,1])$. The concept of weak solutions, introduced in
Definition 2.1 for the hydrodynamic equation, can be extended to the equation

$\left\\{\begin{aligned} &\partial_{t}u_{t}-\nabla[D(u_{t})\,\nabla
u_{t}]=2\nabla[\sigma(u_{t})\nabla
H_{t}]\quad\text{in}\;\;[0,T]\times(-1,1)\;,\\\ &D(u_{t})\nabla
u_{t}-2\sigma(u_{t})\nabla
H_{t}=-u_{t}\,(1-{{\mathfrak{b}}})\,e^{-H_{t}}+{{\mathfrak{b}}}\,[1-u_{t}]\,e^{H_{t}}\quad\text{in}\;\;[0,T]\times\\{1\\}\;,\\\
&D(u_{t})\nabla u_{t}-2\sigma(u_{t})\nabla
H_{t}=u_{t}\,(1-{{\mathfrak{a}}})\,e^{-H_{t}}-{{\mathfrak{a}}}\,[1-u_{t}]\,e^{H_{t}}\quad\text{in}\;\;[0,T]\times\\{-1\\}\;,\end{aligned}\right.$
(7.1)

with initial condition $\gamma$. Corollary 8.2 states that there exists a
unique weak solution.

Fix a function $G$ in $C([0,T]\times[-1,1])$, and consider the generator
$L_{N}^{G}$ given by $L^{G}_{N}=L_{N,0}^{G}+L_{N,b}^{G}$, where

$(L_{N,0}^{G}f)(\eta)=N^{2}\sum_{x\in\Lambda_{N}}r_{x,x+1}(\eta)\,e^{-[\eta(x+1)-\eta(x)]\,[G_{t}(\frac{x+1}{N})-G_{t}(\frac{x}{N})]}[\,f(\eta^{x,x+1})-f(\eta)\,]\;,$

and the sum is carried over all $x\in{\mathbb{Z}}$ such that
$\\{x,x+1\\}\subset\Lambda_{N}$. On the other hand,

$\displaystyle(L_{N,b}^{G}f)(\eta)$
$\displaystyle=\,N\,r_{L}(\eta)\,e^{-\,[1\,-2\eta(-N+1)\,]\,G_{t}(-1+\frac{1}{N})}\,[\,f(\eta^{-N+1})-f(\eta)\,]$
$\displaystyle+\,N\,r_{R}(\eta)\,e^{-\,[1\,-2\eta(N-1)\,]\,G_{t}(1-\frac{1}{N})}\,[\,f(\eta^{N-1})-f(\eta)\,]\;,$

These generators are time-dependent, though this dependence does not appear in
the notation.

For a probability measure $\mu$ on ${\mathfrak{S}}_{N}$, denote by
$\mathbb{P}_{\mu}^{G,N}$ the probability measure on
$D([0,T],{\mathfrak{S}}_{N})$ induced by the Markov chain whose generator is
$L^{G}_{N}$ starting from $\mu$. Let
$\mathbb{Q}_{\mu}^{G,N}=\mathbb{P}_{\mu}^{G,N}(\Pi^{N})^{-1}$. The next result
provides a law of large numbers for the empirical measure for weakly
asymmetric gradient exclusion processes in mild contact with reservoirs.

######  Proposition 7.1.

Fix $0<t_{0}<T$, a profile $\gamma\colon[-1,1]\to[0,1]$, and a function $H$ in
$C([0,T]\times[-1,1])$ such that $H=0$ in $[0,t_{0}]\times[-1,1]$ and $H\in
C^{1,2}([t_{0},T]\times[-1,1])$. Let $(\mu_{N}:N\geq 1)$ be a sequence of
probability measures on ${\mathfrak{S}}_{N}$ associated to $\gamma$. Then, the
sequence of probability measures $(\mathbb{Q}_{\mu_{N}}^{H,N}:N\in\mathbb{N})$
converges weakly to the probability measure $\mathbb{Q}$ concentred on the
trajectory $\pi(t,dx)=u(t,x)dx$ where $u$ is the unique weak solution of
equation (7.1) with initial condition $\gamma$.

The proof of this result is similar to the one presented in Section 3, and
left to the reader. For two probability measures $\mathbb{P}$, $\mathbb{Q}$ on
$D([0,T],{\mathfrak{S}}_{N})$, denote by
$\mathcal{H}_{N}(\mathbb{P}|\mathbb{Q})$ the relative entropy of
${\mathbb{P}}$ with respect to ${\mathbb{Q}}$:

$\mathcal{H}_{N}(\mathbb{P}|\mathbb{Q})=\int\log\frac{d{\mathbb{P}}}{d{\mathbb{Q}}}\,d{\mathbb{P}}\;.$

Next result follows from the previous proposition and Proposition 2.9.

######  Lemma 7.2.

Fix a density profile $\gamma\colon[-1,1]\to[0,1]$, and let $(\eta^{N}:N\geq
1)$ be a sequence of configurations on ${\mathfrak{S}}_{N}$ associated to
$\gamma$. Under the hypotheses of the previous proposition on $H$,
$\underset{N\to\infty}{\lim}\frac{1}{N}\mathcal{H}(\mathbb{P}_{\eta^{N}}^{H,N}|\mathbb{P}_{\eta^{N}})=I_{[0,T]}(u|\gamma)$
where $u(t,x)$ is the unique weak solution of equation (7.1).

Fix a density profile $\gamma\colon[-1,1]\to[0,1]$. Let $\pi(t,dx)=u(t,x)dx$
be a trajectory in $\Pi_{\gamma}$. Recall from the end of the previous section
that for each $0<t\leq T$, there exists a unique solution, denoted by $H_{t}$,
of the elliptic equation (2.10). Moreover the solution satisfies the
hypotheses of Proposition 7.1 and Lemma 7.2.

Let $(\eta^{N}:N\geq 1)$ be a sequence of configurations on
${\mathfrak{S}}_{N}$ associated to $\gamma$. The proof of Lemma 10.5.4 in [19]
yields that for every open set ${\mathcal{O}}\subset D([0,T],\mathcal{M})$,

$\displaystyle\underset{N\to+\infty}{\mbox{ lim inf
}}\frac{1}{N}\log\mathbb{Q}_{\eta^{N}}({\mathcal{O}})\geq-\underset{\pi}{\mbox{
inf }}I_{[0,T]}(\pi|\gamma)\;,$

where the infimum is carried over all trajectories $\pi(t,dx)$ in
$\Pi_{\gamma}\cap{\mathcal{O}}$. To complete the proof of the lower bound it
remains to recall that the set $\Pi_{\gamma}$ is
$I_{[0,T]}(\,\cdot\,|\gamma)$-dense. At this point, and only here, we need
$\gamma$ to fulfil the hypotheses of Theorem 2.5.

##  8\. Uniqueness of weak solutions

In this section, we prove uniqueness of weak solutions of the hydrodynamic
equation for weakly asymmetric dynamics. As many results are standard, they
are just stated, and we refer to Section 2.4 in [7] for detailed proofs.

For $\delta>0$ small, denote by $\varphi_{\delta}$ the function defined by

$\displaystyle\varphi_{\delta}(\cdot)\,:=\,\frac{x^{2}}{2\delta}\,\chi_{[-\delta,\delta]}(\cdot)\,+\,[\,|x|-(\delta/2)\,]\chi_{[-\delta,\delta]^{c}}(\cdot)\;.$

Fix $H\in C([0,T]\times[-1,1])$, for which there exists $0<t_{H}<T$ such that
$H=0$ on $[0,t_{H}]\times[-1,1]$ and $H\in C^{1,2}([t_{H},T]\times[-1,1])$.
The proofs of next results are similar to the ones of [15, Section 7]. Details
can be found in Section 3.5 in [7].

######  Lemma 8.1.

Fix two initial profiles $\rho_{i}\colon\Omega\to[0,1]$, $i=1$, $2$. Let
$u^{(i)}$ be a weak solution of the hydrodynamic equation (7.1) with initial
condition $\rho_{i}$. Then, for all $0<t\leq T$,

$\displaystyle\int_{-1}^{1}\varphi_{\delta}(w_{t})\,dx-\int_{-1}^{1}\varphi_{\delta}(w_{0})\,dx\,=\,$
$\displaystyle-\,\int_{0}^{t}[\,{{\mathfrak{b}}}\,e^{H_{s}(1)}+(1-{{\mathfrak{b}}})\,e^{-H_{s}(1)}\,]\,w_{s}(1)\,\varphi_{\delta}^{\prime}(w_{s}(1))\,ds$
$\displaystyle-\,\int_{0}^{t}[\,{{\mathfrak{a}}}\,e^{H_{s}(-1)}+(1-{{\mathfrak{a}}})\,e^{-H_{s}(-1)}]\,w_{s}(-1)\,\varphi_{\delta}^{\prime}(w_{s}(-1))\,ds$
$\displaystyle-\,\int_{0}^{t}\int_{-1}^{1}\big{\\{}\,D(u^{(2)}_{s})\,\nabla
u^{(2)}_{s}-D(u^{(1)}_{s})\,\nabla u^{(1)}_{s}\,\big{\\}}\nabla
w_{s}\,\varphi_{\delta}^{\prime\prime}(w_{s})\,dx\,ds$
$\displaystyle+\,2\,\int_{0}^{t}\int_{-1}^{1}\big{\\{}\,\sigma(u^{(2)}_{s})-\sigma(u^{(1)}_{s})\,\big{\\}}\,\nabla
H_{s}\,\nabla w_{s}\,\varphi_{\delta}^{\prime\prime}(w_{s})\,dx\,ds\;,$

where $w=u^{(2)}-u^{(1)}$.

######  Corollary 8.2.

Fix two initial profiles $\rho_{i}\colon\Omega\to[0,1]$, $i=1$, $2$. Let
$u^{(i)}$ be a weak solution of the hydrodynamic equation (7.1) with initial
condition $\rho_{i}$. Then, $t\mapsto\|u^{(2)}_{t}-u^{(1)}_{t}\|_{1}$ is non-
increasing. In particular, there is at most one weak solution of the parabolic
equations (2.3) and (7.1). Moreover, if $H=0$, then

$\int_{0}^{\infty}\|u^{(2)}_{t}-u^{(1)}_{t}\|_{1}^{2}\,<\,\infty\;.$

Denote by $\color[rgb]{0,0,1}C^{1+\beta/2,2+\beta}([0,T]\times[-1,1])$ the
functions $H$ in $C^{1,2}([0,T]\times[-1,1])$ whose derivatives
$\partial^{2}_{x}H$, $\partial_{t}H$ are Hölder continuous in the $x$ variable
with parameter $\beta$ and Hölder continuous in the $t$ variable with
parameter $\beta/2$.

######  Proposition 8.3.

Fix an initial profile $\gamma\colon\Omega\to[0,1]$ in $C^{2+\beta}([-1,1])$
for some $0<\beta<1$ and satisfying the boundary conditions (2.8). Let $u$ be
the weak solution of the hydrodynamic equation (7.1) with initial condition
$\gamma$. Then,

  1. (1)

$u\in C^{1+\beta/2,2+\beta}([0,T]\times[-1,1])$

  2. (2)

$0<u(t,x)<1$ for every $(t,x)\in]0,T]\times[-1,1]$

In particular, for each $\delta>0$, there exists $\epsilon>0$ such that
$\epsilon\leq u(t,x)\leq 1-\epsilon$ for every
$(t,x)\in[\delta,T]\times[-1,1]$.

###### Proof.

The proof is based on [21, Theorem V.7.4]. Fix an initial profile
$\gamma\colon\Omega\to[0,1]$ satisfying the hypothesis of the proposition, and
let $u$ be the weak solution of the hydrodynamic equation (7.1) with initial
condition $\gamma$.

Let $v=u-\gamma$. Then, $v$ satisfies equations (7.1), (7.2) of Chapter V in
[21] for some coefficients $a(x,v)$, $b(x,v,\nabla v)$ and $\psi(x,v)$, and
initial condition $v(0,\cdot)=0$. It can be checked that all the hypotheses of
[21, Theorem V.7.4] are fulfilled, but the second condition in equation
(7.34), which is violated for large negative values of $v$.

Modify $b$ for this condition to hold when $v$ takes large negative values,
say when $v<-2$. With this modifications all hypotheses of [21, Theorem V.7.4]
are fulfilled. According to [21, Theorem V.7.4], there exists a solution in
$C^{1,2}({\mathbb{R}}\times[-1,1])$ to (7.1), (7.2) with the modified $b$ and
initial condition $v(0,\cdot)=0$.

Let $t_{0}$ be the first time that $v$ attains $-2$. We claim that
$t_{0}=\infty$. Assume, by contradiction that this is not the case. As the
solution starts from $v=0$ and $v$ is continuous, $t_{0}>0$. In the time
interval $[0,t_{0}]$, $v>-2$. Hence, in this interval the modified $b$
coincides with the original $b$. In particular, in the time interval
$[0,t_{0}]$, $u=v+\gamma$ is the solution of the hydrodynamic equation with
initial condition $\gamma$. Since $u$ is positive, $v=u-\gamma>-\gamma>-3/2$
on $[0,t_{0}]$, in contradiction to the fact that $v(t_{0},x)=-2$ for some
$x\in[-1,1]$.

Hence $v$ is a solution in ${\mathbb{R}}\times[-1,1]$ to the equation (7.1),
(7.2) with initial condition $v(0,\cdot)=0$ for the unmodified coefficient
$b$, and $u=v+\gamma$ is the solution of the hydrodynamic equation with
initial condition $\gamma$ on the same region. As $v\in
C^{1,2}({\mathbb{R}}\times[-1,1])$ and $\gamma\in C^{2}([-1,1])$, $u\in
C^{1,2}({\mathbb{R}}\times[-1,1])$, as claimed.

We claim that $u(t,\cdot)\in]0,1[$ at the boundary for all $t>0$. Indeed, if
$u(t,-1)=1$, say, then, by the boundary condition $\nabla u(t,-1)>0$ so that
$u(t,x)>1$ for some $x\in[-1,1[$, in contradiction with the fact that $u$
takes value in the interval $[0,1]$. To complete the proof of (2), it remains
to recall the maximum principle Theorem 3.3.5 [23]. ∎

######  Lemma 8.4.

Fix a density profile $\gamma\colon[-1,1]\to[0,1]$ and $\pi$ in
$\Pi_{\gamma}$. For all $0<t\leq T$, there exists a unique solution $H_{t}$ in
$C^{2}([-1,1])$ of the elliptic equation (2.10). Moreover, $H_{t}=0$ for
$0\leq t\leq\delta$, where $\delta>0$ is the constant appearing in item (a) of
the definition of the set $\Pi_{\gamma}$, and $H\in
C^{\infty}([\delta,T]\times[-1,1])$.

###### Proof.

The proof is straightforward. As $u$ follows the hydrodynamic equation in the
interval $]0,\delta]$, $H_{t}=0$ is a solution of (2.10) for $t$ in this
interval. On the other hand, for $\delta\leq t\leq T$, as $u$ is smooth and
bounded away from $0$ and $1$, one can derive an explicit expression for the
solution $H_{t}$, and check that it is smooth in space and time.

For uniqueness, consider two solutions $H^{(i)}_{t}$, $i=1$, $2$. Subtract the
bulk equations, multiply by the difference $H^{(2)}_{t}-H^{(1)}_{t}$,
integrate by parts, and replace at the boundary the gradients of $H^{(i)}_{t}$
using the boundary conditions. These computations yield that the sum of three
positive terms is equal to $0$, proving uniqueness. A detailed proof can be
found in Lemma 3.6.1 in [7]. ∎

## References

  * [1] L. Bertini, A. De Sole, D. Gabrielli, G. Jona–Lasinio, C. Landim: Macroscopic fluctuation theory for stationary non equilibrium state. J. Statist. Phys. 107, 635–675 (2002).
  * [2] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim: Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process. Mathematical Physics, Analysis and Geometry 6: 231-267 (2003).
  * [3] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim; Macroscopic Fluctuation Theory. Rev. Modern Phys. 87, 593–636 (2015).
  * [4] L. Bertini and C. Landim and M. Mourragui: Dynamical large deviations for the boundary driven weakly asymmetric exclusion process. Ann. Probab. 37, 2357–2403 (2009)
  * [5] T. Bodineau and G. Giacomin: From Dynamic to Static Large Deviations in boundary Driven Exclusion Particle Systems. Stochastic Procss. Appl. 110, 67–81 (2004).
  * [6] Th. Bodineau, M. Lagouge: Large deviations of the empirical currents for a boundary-driven reaction diffusion model. Ann. Appl. Probab. 2012, 22, 2282–2319 (2012).
  * [7] A. Bouley: Thèse
  * [8] A. Bouley, C. Erignoux, C. Landim: Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs. To appear in Annales de l’Institut Henri Poincaré, Prob. et Stat. arXiv: 2107.06606 (2021)
  * [9] A. Bouley, C. Landim: Thermodynamics of nonequilibrium driven diffusive systems in weak contact with boundary reservoirs. J. Stat. Phys. 188, 19 (2022)
  * [10] B. Derrida, O. Hirschberg, T Sadhu: Large deviations in the symmetric simple exclusion process with slow boundaries. J. Stat. Phys. (2021) doi.org/10.1007/s10955-020-02680-3
  * [11] B. Derrida, J. L. Lebowitz, E. R. Speer: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process, J. Statist. Phys. 107, 599–634 (2002).
  * [12] M. D. Donsker, S. R. S. Varadhan: Large deviations from a hydrodynamic scaling limit. Comm. Pure Appl. Math. 42, 243–270 (1989).
  * [13] J. Farfán: Static large deviations of boundary driven exclusion processes. arXiv:0908.1798.
  * [14] T. Franco and P. Gonçalves and C. Landim and A. Neumann: Dynamical large deviations for the boundary driven symmetric exclusion process with Robin boundary conditions. ALEA Lat. Am. J. Probab. Math. Stat. 19, 1497–1546 (2022)
  * [15] J. Farfan, C. Landim, M. Mourragui: Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes. Stochastic Process. Appl. 121, 725–758 (2011).
  * [16] J. Farfán, C. Landim, K. Tsunoda; Static large deviations for a reaction-diffusion model. Probab. Th. Rel. Fields. 174, 49–101 (2019)
  * [17] G. Jona-Lasinio, C. Landim, M. E. Vares; Large deviations for a reaction-diffusion model, Probability Theory and Related Fields 97, 339–361, (1993).
  * [18] S. R. De Groot, P. Mazur. Non-equilibrium thermodynamics. Courier Corporation, 2013.
  * [19] C. Kipnis, C. Landim _Scaling limits of interacting particle systems._ Springer-Verlag, Berlin, 1999.
  * [20] C. Kipnis, S. Olla, S. R. S. Varadhan Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989).
  * [21] O. A. Ladyzenskaja, U. A. Solonnikov, N. N. Ural’ceva: Linear and Quasi-Linear Equations of Parabolic Type. Volume 23 of Translations of mathematical monographs, A. M. S., Providence, Rhode lsland (1968):
  * [22] C. Landim: Occupation time large deviations for the symmetric simple exclusion process. Ann. Probab. 20, 206–231 (1992).
  * [23] M. H. Protter, H. F. Weinberger; _Maximum principles in differential equations_. Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984.
  * [24] J. Quastel, F. Rezakhanlou, S. R. S. Varadhan: Large deviations for the symmetric simple exclusion process in dimensions $d\geq 3$, Probab. Th. Rel. Fields 113, 1–84, (1999).
  * [25] R. T. Rockafellar: Convex analysis. Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J. 1970