# Steady state large deviations for one-dimensional, symmetric exclusion
processes in weak contact with reservoirs

A. Bouley, C. Erignoux, C. Landim  Laboratoire de Mathématiques Raphaël Salem
UFR des Sciences et Techniques Université de Rouen Normandie Avenue de
l’Université, BP.12 76801 Saint-Étienne-du-Rouvray, France e-mail:
<EMAIL_ADDRESS>Equipe PARADYSE, Bureau B211
Centre INRIA Lille Nord-Europe
Park Plaza, Parc scientifique de la Haute-Borne, 40 Avenue Halley
Bâtiment B, 59650 Villeneuve-d’Ascq
France. e-mail<EMAIL_ADDRESS>IMPA, Estrada Dona Castorina 110,
CEP 22460 Rio de Janeiro, Brasil and CNRS UMR 6085, Université de Rouen,
France.
e-mail<EMAIL_ADDRESS>
###### Abstract.

Consider the symmetric exclusion process evolving on an interval and weakly
interacting at the end-points with reservoirs. Denote by $I_{[0,T]}(\cdot)$
its dynamical large deviations functional and by $V(\cdot)$ the associated
quasi-potential, defined as $V(\gamma)=\inf_{T>0}\inf_{u}I_{[0,T]}(u)$, where
the infimum is carried over all trajectories $u$ such that $u(0)=\bar{\rho}$,
$u(T)=\gamma$, and $\bar{\rho}$ is the stationary density profile. We derive
the partial differential equation which describes the evolution of the optimal
trajectory, and deduce from this result the formula obtained by Derrida,
Hirschberg and Sadhu [14] for the quasi-potential through the representation
of the steady state as a product of matrices.

##  1\. Introduction

Non-equilibrium steady states have attracted a lot of interest in the last
decades, as a first step towards the understanding of far from equilibrium
behavior. We refer to the reviews [12, 13, 6], the recent works [14, 20] and
references therein. These states display many interesting phenomena, such as
non-local thermodynamic functionals, dynamical phase transitions and long
range correlations, [15, 9, 5]. Many of these properties can be derived from
the quasi-potential, the functional which plays a role analogous to the free
energy in equilibrium.

We consider the symmetric exclusion process evolving in the interval $[0,1]$
and in contact with reservoirs at the end points. In the case of a strong
interaction of the system with the reservoirs, the boundary conditions do not
appear in the thermodynamic functionals and the effect of the boundary is not
clear. To investigate the influence of the boundaries, we examine in this
article the case of weak interactions.

With strong interactions with the reservoirs, the density at the boundaries
take immediately the value of the reservoirs densities and remain fixed, while
the density in the bulk evolves according to the heat equation. That is to
say, the density profile evolves according to the heat equation with Dirichlet
boundary conditions. Even at the level of the dynamical large deviations, the
densities at the boundary are kept fixed, and only the density at the interior
may fluctuate, [21, 3, 4].

In constrast, for exclusion processes with weak interactions with the
reservoirs, the densities at the boundaries evolve in time. Actually, the
particles’ density $u$ solves the heat equation with Robin boundary conditions
[2]. Namely,

$\begin{cases}\partial_{t}u\,=\,\Delta
u\quad(t,x)\,\in\,(0,T)\,\times\,(0,1)\\\ (\nabla
u)(t,0)\,=\,A^{-1}[\,u(t,0)-\alpha\,]\quad t\,\in\,(0,T)\\\ (\nabla
u)(t,1)\,=\,B^{-1}[\,\beta-u(t,1)]\quad t\,\in\,(0,T)\\\
u(0,x)\,=\,\gamma(x)\quad x\,\in\,[0,1]\;.\end{cases}$ (1.1)

In this formula, $\alpha$, $\beta\in(0,1)$ represent the density at the left,
right reservoirs, respectively, $A$, $B>0$ the intensity of the interaction
with the left, right reservoirs, respectively, and $\gamma:[0,1]\to[0,1]$ the
initial density profile. Moreover, $\nabla u$ stands for the partial
derivative in space of $u$, $\partial_{t}u$ for its partial derivative in time
and $\Delta u$ for the Laplacian of $u$ in the space variable.

The weak interaction of the system with the boundaries also modifies the
thermodynamical variables by adding boundary terms. The Hamiltonian, denoted
by ${\mathscr{H}}(\gamma,F)$, becomes

$\displaystyle{\mathscr{H}}(\gamma,F)\;=$
$\displaystyle-\;\big{\langle}\nabla\gamma\,,\,\nabla
F\big{\rangle}\;+\;\big{\langle}\sigma(\gamma)\,,\,\big{(}\nabla
F\big{)}^{2}\big{\rangle}$ (1.2)
$\displaystyle+\;{\mathfrak{b}}_{\alpha,A}\big{(}\,\gamma(0)\,,\,F(0)\,\big{)}\;+\;{\mathfrak{b}}_{\beta,B}\big{(}\,\gamma(1)\,,\,F(1)\,\big{)}\;.$

In this formula and below, $\langle\,\cdot\,,\,\cdot\,\rangle$ represents the
scalar product in ${\mathscr{L}}^{2}([0,1])$, $\sigma:[0,1]\to{\mathbb{R}}$,
given by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\sigma(a)=a(1-a)$,
is the mobility of the exclusion process, and for $0<\varrho<1$, $D>0$,
$0<a<1$, $M\in{\mathbb{R}}$,

${\mathfrak{b}}_{\varrho,D}(a,M)\;=\;\frac{1}{D}\,\Big{\\{}[1-a]\,\varrho\,[e^{M}-1]\;+\;a\,[1-\varrho]\,[e^{-M}-1]\,\Big{\\}}\;.$
(1.3)

In the Hamiltonian formalism of classical mechanics, density profiles
$\gamma:[0,1]\to[0,1]$ play the role of position and external fields
$F:[0,1]\to{\mathbb{R}}$, the one of momentum.

The dynamical large deviations functional associated to the Hamiltonian
${\mathscr{H}}$ is given by

$I_{[0,T]}(u)\;=\;\sup_{H}\int_{0}^{T}\big{\\{}\,\big{\langle}\partial_{t}u_{t},H_{t}\big{\rangle}\,-\,{\mathscr{H}}(u_{t},H_{t})\,\big{\\}}\;dt\;,$
(1.4)

where the supremum is carried over all smooth functions
$H:[0,T]\times[0,1]\to{\mathbb{R}}$. The functional $I_{[0,T]}(u)$ specifies
the cost of observing a fluctuation $u(t)$, $0\leq t\leq T$. In particular,
$I_{[0,T]}(u)=0$ if $u$ follows the hydrodynamic equation (1.1).

Let $\bar{\rho}$ be the unique stationary solution to the equation (1.1). That
is, $\bar{\rho}$ is the solution to the elliptic equation

$\begin{cases}\Delta\rho\,=\,0\\\
(\nabla\rho)(0)\,=\,A^{-1}[\,\rho(0)-\alpha\,]\\\
(\nabla\rho)(1)\,=\,B^{-1}[\,\beta-\rho(1)]\;.\end{cases}$ (1.5)

An elementary computation yields that $\bar{\rho}$ is given by

$\bar{\rho}(x)\;=\;\frac{\alpha(1+B)+\beta
A}{1+B+A}\;+\;\frac{(\beta-\alpha)\,x}{1+B+A}\;\cdot$

Note that $\bar{\rho}$ is the linear interpolation between
$\bar{\rho}(-A)=\alpha$ and $\bar{\rho}(1+B)=\beta$.

Denote by $V(\,\cdot\,)$ the quasi-potential associated to the rate function
$I_{[0,T]}(\,\cdot\,)$. It is given by

$V(\gamma)\;:=\;\inf_{T>0}\;\inf_{u(\cdot)}\;I_{[0,T]}(u)\;,$

where the infimum is carried over all paths $u$ such that $u(0)=\bar{\rho}$,
$u(T)=\gamma$. The quasi-potential $V(\gamma)$ measures the minimal cost to
produce a profile $\gamma$ starting from $\bar{\rho}$. It is also the rate
functional of the large deviations principle for the density profile under the
steady state [3].

By using a representation of the steady state of the exclusion process as a
product of matrices, Derrida, Hirschberg and Sadhu [14] proved that the quasi-
potential can be expressed as

$\begin{gathered}V(\gamma)\;=\;\int_{0}^{1}\Big{\\{}\,\gamma(x)\,\log\frac{\gamma(x)}{F(x)}+\big{[}1-\gamma(x)\big{]}\log\frac{1-\gamma(x)}{1-F(x)}+\log\frac{\nabla
F(x)}{[\beta-\alpha]}\,\Big{\\}}\;dx\\\
\;+\;A\,\ln\frac{F(0)-\alpha}{A(\beta-\alpha)}\;+\;B\,\ln\frac{\beta-F(1)}{B(\beta-\alpha)}\;,\end{gathered}$

where $F$ solves the non linear boundary value problem

$\left\\{\begin{aligned} &\vphantom{\Big{(}}{\displaystyle\Delta
F=\big{(}\gamma-F\big{)}\frac{\big{(}\nabla
F\big{)}^{2}}{F(1-F)}\qquad\textrm{ in $(0,1)$}}\;,\\\
&\vphantom{\Big{(}}{\displaystyle\nabla F(0)=A^{-1}[F(0)-\alpha]\;,\quad\nabla
F(1)=B^{-1}[\beta-F(1)]}\;.\end{aligned}\right.$ (1.6)

This result extends to exclusion processes with weak interactions at the
boundaries a theorem of Derrida, Lebowitz and Speer [15] for the case with
strong interactions,

In this article, we provide an alternative proof of this result, based on the
strategy delineated in [3] and carried out in [4] for exclusion processes with
strong interactions at the boundaries.

Using the Hamiltonian formalism, we derive a formal equation for the path
which solves the variational problem (1.4). The optimal trajectory corresponds
to a pair $(u(t),F(t))$ which solves a system of coupled equations, see
(3.5)–(3.6) below. While it might seem, at a first glance, hopeless to prove
any property of the solutions to this pair of equations, it turns out that
$F_{t}$ evolves according to an autonomous equation, actually, according to
the hydrodynamic equation (1.1). This remarkable property is the key point and
permits to prove all properties needed to show that the candidate obtained
from the heuristic argument is indeed the optimal path.

According to [3], this optimal path, which describe how the system adjusts to
create a fluctuation of the density, corresponds to the typical trajectory for
the adjoint dynamics, reversed in time. In particular, this approach reveals
the adjoint hydrodynamic equation. It is obtained from the hydrodynamic
equation by adding a non-local drift and modifying the boundary densities and
intensities of interaction which become time-dependent (cf. Remark 3.1).

To our knowledge this is one of the few examples of an interacting particle
system whose steady state is not known explicitly and whose quasi-potential
can be computed [15, 4, 1, 14]. Moreover, the presence of boundary terms in
the Hamiltonian-Jacobi equation for the quasi-potential modifies entirely the
analysis of thermodynamic transformations of non-equilibrium states carried
out in [7] in the case of Dirichlet boundary conditions. This is left for a
future work, together with the emergence of dynamical phase transitions [5]
and the static large deviations [10, 17, 22, 18].

##  2\. Notation and Results

The model. We consider one-dimensional, symmetric exclusion processes in weak
contact with boundary reservoirs. Fix $N\geq 1$, and let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathfrak{e}}_{N}=1/N$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathfrak{r}}_{N}=1-(1/N)$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Lambda_{N}=\\{{\mathfrak{e}}_{N},\dots,(N-2)\,{\mathfrak{e}}_{N},{\mathfrak{r}}_{N}\\}$.
The sate space is represented by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Omega_{N}=\\{0,1\\}^{\Lambda_{N}}$
and the configurations by the Greek letters $\eta$, $\xi$ so that $\eta_{x}$,
$x\in\Lambda_{N}$, represents the number of particles at site $x$ for the
configuration $\eta$.

Fix throughout this article,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}0<\alpha\leq\beta<1$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}A>0$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}B>0$.
The generator of the Markov process considered here, represented by
${\mathscr{L}}_{N}={\mathscr{L}}^{\alpha,A,\beta,B}_{N}$, is given by

${\mathscr{L}}_{N}\;=\;L^{\rm lb}_{N}\;+\;L^{\rm bulk}_{N}\;+\;L^{\rm
rb}_{N}\;.$

In this formula, for every function $f:\Omega_{N}\to{\mathbb{R}}$,

$(L^{\rm
bulk}_{N}f)(\eta)\;=\;N^{2}\,\sum_{x\in\Lambda^{o}_{N}}[\,f(\sigma^{x,x+{\mathfrak{e}}_{N}}\eta)\,-\,f(\eta)\,]\;,$

where $\Lambda^{o}_{N}$ represents the interior of $\Lambda_{N}$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Lambda^{o}_{N}:=\Lambda_{N}\setminus\\{{\mathfrak{r}}_{N}\\}=\\{{\mathfrak{e}}_{N},\dots,(N-2)\,{\mathfrak{e}}_{N}\\}$,
and

$\begin{gathered}(L^{\rm
lb}_{N}f)(\eta)\;=\;\frac{N}{A}\,\big{[}\,(1-\eta_{{\mathfrak{e}}_{N}})\,\alpha\,+\,(1-\alpha)\,\eta_{{\mathfrak{e}}_{N}}\,\big{]}\big{[}\,f(\sigma^{{\mathfrak{e}}_{N}}\eta)\,-\,f(\eta)\,\big{]}\;,\\\
(L^{\rm
rb}_{N}f)(\eta)\;=\;\frac{N}{B}\,\big{[}\,(1-\eta_{{\mathfrak{r}}_{N}})\,\beta\,+\,(1-\beta)\,\eta_{{\mathfrak{r}}_{N}}\,\big{]}\big{[}\,f(\sigma^{{\mathfrak{r}}_{N}}\eta)\,-\,f(\eta)\,\big{]}\;.\end{gathered}$

From now on, we omit the subindex $N$ of ${\mathfrak{e}}_{N}$. In the formulas
above,

$(\sigma^{x,x+{\mathfrak{e}}}\eta)_{y}\;=\;\begin{cases}\eta_{y}&\mbox{ if
}y\neq x,x+{\mathfrak{e}}\\\ \eta_{x+{\mathfrak{e}}}&\mbox{ if }y=x\\\
\eta_{x}&\mbox{ if }y=x+{\mathfrak{e}}\end{cases}\quad\mbox{ and
}\quad(\sigma^{x}\eta)_{y}\;=\;\begin{cases}\eta_{y}&\mbox{ if }y\neq x\\\
1-\eta_{x}&\mbox{ if }y=x\;.\end{cases}$

For a metric space ${\mathbb{X}}$, denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}D([0,T],{\mathbb{X}})$,
$T>0$, the space of right-continuous functions
${\mathfrak{x}}\colon[0,T]\to{\mathbb{X}}$, with left-limits, endowed with the
Skorohod topology and its associated Borel $\sigma$-algebra. The elements of
$D([0,T],\Omega_{N})$ are represent by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\boldsymbol{\eta}}(\cdot)$.

For a probability measure $\mu$ on $\Omega_{N}$, let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{P}}^{N}_{\mu}$
be the measure on $D([0,T],\Omega_{N})$ induced by the continuous-time Markov
process associated to the generator ${\mathscr{L}}_{N}$ starting from $\mu$.
When the measure $\mu$ is the Dirac measure concentrated at a configuration
$\eta\in\Omega_{N}$, that is $\mu=\delta_{\eta}$, we represent
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{P}}^{N}_{\delta_{\eta}}$
simply by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{P}}^{N}_{\eta}$.
Expectation with respect to ${\mathbb{P}}^{N}_{\mu}$,
${\mathbb{P}}^{N}_{\eta}$ is denoted by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{E}}^{N}_{\mu}$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{E}}^{N}_{\eta}$,
respectively. When the context permits we remove the index $N$ from the
notation.

Hydrodynamic limit. Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\mathscr{M}$
the set of non-negative measures on $[0,1]$ with total mass bounded by $1$
endowed with the weak topology. Recall that this topology is metrisable and
that, with this topology, ${\mathscr{M}}$ is a relatively compact space. For a
continuous function $F:[0,1]\to{\mathbb{R}}$ and a measure
$\pi\in{\mathscr{M}}$, denote by $\langle\pi,F\rangle$ the integral of $F$
with respect to $\mu$:

$\langle\pi,F\rangle\;=\;\int F(x)\,\pi(dx)\;.$

Given a configuration $\eta\in\Omega_{N}$, denote by $\pi=\pi(\eta)$ the
measure in ${\mathscr{M}}$ obtained by assigning a mass $N^{-1}$ to the
position of each particle:

$\pi\;=\;\pi(\eta)\;=\;\frac{1}{N}\sum_{x\in\Lambda_{N}}\eta_{x}\,\delta_{x}\;.$

The measure $\pi$ is called the _empirical measure_.

Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\boldsymbol{\pi}}:D([0,T],\Omega_{N})\to
D([0,T],{\mathscr{M}})$ the map which associates to a trajectory
${\boldsymbol{\eta}}(\cdot)$ its empirical measure:

${\boldsymbol{\pi}}(t)\;=\;\pi({\boldsymbol{\eta}}(t))\;=\;\sum_{x\in\Lambda_{N}}\eta_{x}(t)\,\delta_{x}\;.$

For a probability measure $\mu$ in $\Omega_{N}$, let ${\mathbb{Q}}^{N}_{\mu}$
be the measure on $D([0,T],{\mathscr{M}})$ given by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathbb{Q}}^{N}_{\mu}={\mathbb{P}}^{N}_{\mu}\,\circ\,{\boldsymbol{\pi}}^{-1}$.

The first result, due to [2], establishes the hydrodynamic behavior of the
empirical measure.

######  Theorem 2.1.

Fix $T>0$, a density profile $\gamma\colon[0,1]\to[0,1]$, and sequence
$(\nu^{N}:N\geq 1)$ of probability measures on $\Omega_{N}$ associated to
$\gamma$ in the sense that

$\lim_{N\to\infty}\nu^{N}\Big{[}\,\Big{|}\,\langle\pi\,,\,G\rangle\,-\,\int_{0}^{1}\gamma(x)\,G(x)\,dx\,\Big{|}\,>\,\delta\,\Big{]}\;=\;0$

for all continuous functions $G\colon[0,1]\to{\mathbb{R}}$ and $\delta>0$.
Then, the sequence of probability measures ${\mathbb{Q}}^{N}_{\nu^{N}}$
converges to the probability measure ${\mathbb{Q}}$ concentrated on the
trajectory $\pi(t,dx)=u(t,x)\,dx$, where $u$ is the unique weak solution to
the heat equation with Robin’s boundary conditions (1.1).

We refer to Appendix B for the definition of weak solutions to equation (1.1)
and some of its properties.

Dynamical large deviations. For $T>0$ and positive integers $m,n$, denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}C^{m,n}([0,T]\times[0,1])$
the space of functions $G\colon[0,T]\times[0,1]\to{\mathbb{R}}$ with $m$
derivatives in time, $n$ derivatives in space which are continuous up to the
boundary. Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}C^{m,n}_{0}([0,T]\times[0,1])$
the set of functions in $C^{m,n}([0,T]\times[0,1])$ which vanish at the
endpoints of $[0,1]$, i.e. $G\in C^{m,n}([0,T]\times[0,1])$ belongs to
$C^{m,n}_{0}([0,T]\times[0,1])$ if and only if $G(t,0)=G(t,1)=0$ for all
$t\in[0,T]$.

Denote by ${\mathscr{M}}_{\rm ac}$ the subset of ${\mathscr{M}}$ of all
measures which are absolutely continuous with respect to the Lebesgue measure
and whose density takes values in the interval $[0,1]$:
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathscr{M}}_{\rm
ac}=\\{\pi\in{\mathscr{M}}:\pi(dx)=\gamma(x)\,dx\;\;\text{and}\;\;0\,\leq\,\gamma(x)\,\leq\,1\,\\}$.

For $T>0$, let the energy ${\mathcal{Q}}_{[0,T]}:D([0,T],{\mathscr{M}}_{\rm
ac})\to[0,\infty]$ be given by

$\displaystyle{\mathcal{Q}}_{[0,T]}(\pi)\;=\;$
$\displaystyle\quad\sup_{G}\Big{\\{}\int_{0}^{T}dt\int_{0}^{1}u(t,x)\,(\nabla
G)(t,x)\;dx\;-\;\frac{1}{2}\int_{0}^{T}dt\int_{0}^{1}\sigma(u(t,x))\,G(t,x)^{2}\;dx\,\Big{\\}}\;,$

where $\pi(t,dx)=u(t,x)\,dx$ and the supremum is carried over all smooth
functions $G:[0,T]\times(0,1)\to{\mathbb{R}}$ with compact support.

######  Remark 2.2.

Hereafter, we abuse of notation writing $\gamma\in{\mathscr{M}}_{\rm ac}$ to
mean that the measure $\gamma(x)\,dx$ belongs to ${\mathscr{M}}_{\rm ac}$.
Moreover, for functionals $\Phi\colon D([0,T],{\mathscr{M}}_{\rm
ac})\to{\mathbb{R}}$, $W:{\mathscr{M}}_{\rm ac}\to{\mathbb{R}}$, we often
write $\Phi(u)$, $W(\gamma)$ instead of $\Phi(\pi)$, $W(\mu)$ when
$\pi(t,dx)=u(t,x)\,dx$, $\mu(dx)=\gamma(x)\,dx$.

Notational convention: For a function $v:I\times[0,1]\to{\mathbb{R}}$, where
$I$ is a subset of ${\mathbb{R}}$, $v_{t}$ and $v(t)$, $t\in I$, represent the
function $w:[0,1]\to{\mathbb{R}}$ defined by $w(x)=v(t,x)$. We use the letters
$\gamma$, $\phi$, $\psi$ to represent densities [elements of
${\mathscr{M}}_{\rm ac}$], $u$, $v$, $w$ for trajectories of densities
[elements of $D({\mathbb{R}}_{+},{\mathscr{M}}_{\rm ac})$], and $F$, $G$, $H$
for external fields, usually functions in $C({\mathbb{R}}_{+}\times[0,1])$.

By [8, Lemma 4.1], the energy ${\mathcal{Q}}_{[0,T]}$ is convex and lower
semicontinuous. Moreover, if ${\mathcal{Q}}_{[0,T]}(u)$ is finite, $u$ has a
generalized space derivative, denoted by $\nabla u$, and

${\mathcal{Q}}_{[0,T]}(u)\;=\;\frac{1}{2}\int_{0}^{T}dt\,\int_{0}^{1}\frac{(\nabla
u_{t})^{2}}{\sigma(u_{t})}\;dx\;\cdot$

Fix a trajectory $\pi\in D([0,T],{\mathscr{M}}_{\rm ac})$,
$\pi(t,dx)=u(t,x)\,dx$, with finite energy, ${\mathcal{Q}}_{[0,T]}(u)<\infty$.
In particular, $\int_{0}^{T}dt\int_{0}^{1}(\nabla u_{t})^{2}\;dx$ is finite.
By [27, Assertion 48, page 1030], the trace of $u$ at the spatial boundary of
the cylinder
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Omega_{T}=[0,T]\times[0,1]$
is well defined. That is, the maps $t\mapsto u(t,0)$, $t\mapsto u(t,1)$ are
well defined and belong to ${\mathscr{L}}^{2}([0,T])$. Moreover, since for
almost all $t\in[0,T]$, $\int_{0}^{1}(\nabla u_{t})^{2}\;dx$ is finite, for
these values of $t$, $u(t,\cdot)$ is Hölder-continuous and $u(t,0)$ and
$u(t,1)$ are well defined.

Denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the usual scalar product in
${\mathscr{L}}^{2}([0,1])$:

$\langle\,f\,,\,g\,\rangle\;=\;\int_{0}^{1}f(x)\,g(x)\;dx\;,\quad
f\,,\,g\,\in\,{\mathscr{L}}^{2}([0,1])\;.$

Fix a function $\gamma\colon[0,1]\to[0,1]$, which corresponds to the initial
profile. Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}D_{{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm
ac})$ the set of trajectories in $D([0,T],{\mathscr{M}}_{\rm ac})$ with finite
energy, and by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}D_{\gamma,{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm
ac})$ the set of trajectories with finite energy and which start from
$\gamma$, $u_{0}(\cdot)=\gamma(\cdot)$ a.s.

Recall, from (1.3), the definition of ${\mathfrak{b}}_{\varrho,D}(a,M)$ and,
from Remark 2.2, the convention on notation. For each $H$ in
$C^{1,2}([0,T]\times[0,1])$, let $J_{T,H}\colon
D_{{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm ac})\longrightarrow{\mathbb{R}}$ be
the functional given by

$\displaystyle
J_{T,H}(u)\;=\;\big{\langle}u_{T},H_{T}\big{\rangle}\;-\;\langle
u_{0},H_{0}\rangle\;-\;\int_{0}^{T}\big{\langle}u_{t},\partial_{t}H_{t}\big{\rangle}\;dt$
(2.1) $\displaystyle\quad\;-\;\int_{0}^{T}\big{\langle}u_{t},\Delta
H_{t}\big{\rangle}\;dt\;+\;\int_{0}^{T}u_{t}(1)\,\nabla
H_{t}(1)\;dt\;-\;\int_{0}^{T}u_{t}(0)\,\nabla H_{t}(0)\;dt$
$\displaystyle\quad\;-\;\int_{0}^{T}\big{\langle}\sigma(u_{t}),\big{(}\nabla
H_{t}\big{)}^{2}\big{\rangle}\;dt$
$\displaystyle\quad\;-\;\int_{0}^{T}\Big{\\{}\,{\mathfrak{b}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\;+\;{\mathfrak{b}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Big{\\}}\,dt\;.$

The right-hand side is well defined because the functions $u(\cdot,0)$,
$u(\cdot,1)$ belong to ${\mathscr{L}}^{2}([0,T])$.

Since trajectories in $D_{{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm ac})$ have
generalized space-derivatives, we may integrate by parts the second line and
write the functional $J_{T,H}(\,\cdot\,)$ as

$\displaystyle J_{T,H}(u)$
$\displaystyle=\;\big{\langle}u_{T},H_{T}\big{\rangle}\;-\;\langle
u_{0},H_{0}\rangle\;-\;\int_{0}^{T}\big{\langle}u_{t},\partial_{t}H_{t}\big{\rangle}\;dt$
(2.2) $\displaystyle+\;\int_{0}^{T}\big{\langle}\nabla u_{t},\nabla
H_{t}\big{\rangle}\;dt-\;\int_{0}^{T}\big{\langle}\sigma(u_{t}),\big{(}\nabla
H_{t}\big{)}^{2}\big{\rangle}\;dt$
$\displaystyle-\;\int_{0}^{T}\Big{\\{}\,{\mathfrak{b}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\;+\;{\mathfrak{b}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Big{\\}}\,dt\;.$

Let $I_{[0,T]}\colon D_{{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm
ac})\rightarrow[0,+\infty]$ be the functional defined by

$I_{[0,T]}(\pi)\;:=\;\sup_{H\in C^{1,2}([0,T]\times[0,1])}J_{T,H}(\pi)\;.$

Fix a density profile $\gamma$ in ${\mathscr{M}}_{\rm ac}$, and let
$I_{[0,T]}(\,\cdot\,|\gamma)\colon D([0,T],{\mathscr{M}})\to{\mathbb{R}}$ be
given by

$I_{[0,T]}(\pi|\gamma)\;=\;\left\\{\begin{aligned}
&I_{[0,T]}(\pi)\quad\text{if}\;\;\pi\,\in\,D_{\gamma,{\mathcal{E}}}([0,T],{\mathscr{M}}_{\rm
ac})\;,\\\ &\infty\quad\text{otherwise}\;.\end{aligned}\right.$ (2.3)

We review in Section 4 some properties of the functional
$I_{[0,T]}(\cdot|\gamma)$ obtained in [19]. Next result is the main theorem in
[19].

######  Theorem 2.3.

Fix $T>0$ and a measure $\pi(dx)=\gamma(x)\,dx$ in ${\mathscr{M}}_{\rm ac}$.
Consider a sequence $\eta^{N}$ of configurations associated to $\gamma$. Then
the measure ${\mathbb{Q}}_{\eta^{N}}$ satisfies a large deviation principle
with speed $N$ and rate function $I_{[0,T]}(\cdot|\gamma)$. Namely, for each
closed set ${\mathcal{C}}\subset D([0,T],{\mathscr{M}})$ and each open set
${\mathcal{O}}\subset D([0,T],{\mathscr{M}})$,

$\displaystyle\limsup_{N\to\infty}\frac{1}{N}\log{\mathbb{P}}_{\eta^{N}}[{\boldsymbol{\pi}}\in{\mathcal{C}}]\;\leq\;-\inf_{\pi\in{\mathcal{C}}}I_{[0,T]}(\pi|\rho)$
$\displaystyle\quad\liminf_{N\to\infty}\frac{1}{N}\log{\mathbb{P}}_{\eta^{N}}[{\boldsymbol{\pi}}\in{\mathcal{O}}]\;\geq\;-\inf_{\pi\in{\mathcal{O}}}I_{[0,T]}(\pi|\rho)\;.$

The quasi-potential. Denote by $V:{\mathscr{M}}_{\rm ac}\to{\mathbb{R}}_{+}$
the quasi-potential associated to the rate function
$I_{[0,T]}(\,\cdot\,|\,\gamma)$. It is given by

$V(\gamma)\;:=\;\inf_{T>0}\;\inf_{u(\cdot)}\;I_{[0,T]}(u\,|\,\bar{\rho})\;,$
(2.4)

where the infimum is carried over all paths $u$ in $D([0,T],{\mathscr{M}}_{\rm
ac})$ such that $u(0)=\bar{\rho}$, $u(T)=\gamma$. The quasi-potential
$V(\gamma)$ measures the minimal cost to produce a profile $\gamma$ starting
from $\bar{\rho}$.

Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}C^{1}\big{(}[0,1]\big{)}$
the space of once continuously differentiable functions
$F:[0,1]\to{\mathbb{R}}$ endowed with the norm
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\|F\|_{C^{1}}:=\sup_{x\in[0,1]}\big{\\{}|F(x)|+|\nabla
F(x)|\big{\\}}$. Let ${\mathcal{F}}$ be the space of monotone $C^{1}$
functions:

${\mathcal{F}}:=\big{\\{}\,F\in
C^{1}\big{(}[0,1]\big{)}\,:\>\alpha\,<\,F(x)\,<\,\beta\;,\;\nabla
F(x)>0\;\;\forall\;x\in[0,1]\,\big{\\}}\;.$ (2.5)

Denote by ${\mathscr{G}}_{\rm bulk}$, ${\mathscr{G}}\colon{\mathscr{M}}_{\rm
ac}\times{\mathcal{F}}\to{\mathbb{R}}$ the functionals given by

$\begin{gathered}{\mathscr{G}}_{\rm
bulk}(\gamma,F):=\int_{0}^{1}\Big{\\{}\,\gamma(x)\,\log\frac{\gamma(x)}{F(x)}+\big{[}1-\gamma(x)\big{]}\log\frac{1-\gamma(x)}{1-F(x)}+\log\frac{\nabla
F(x)}{[\beta-\alpha]}\,\Big{\\}}\;dx\;,\\\
{\mathscr{G}}(\gamma,F)\;:=\;{\mathscr{G}}_{\rm
bulk}(\gamma,F)\;+\;A\,\ln\frac{F(0)-\alpha}{A(\beta-\alpha)}\;+\;B\,\ln\frac{\beta-F(1)}{B(\beta-\alpha)}\;\cdot\end{gathered}$

Define $S_{0}$, $S\colon{\mathscr{M}}_{\rm ac}\to{\mathbb{R}}$, by

$S_{0}(\gamma)\;:=\;\sup_{F\in{\mathcal{F}}}{\mathscr{G}}(\gamma,F)\;,\quad
S(\gamma)\;:=\;S_{0}(\gamma)\,-\,S_{0}(\bar{\rho})\;.$ (2.6)

Main results. The first main assertion of the article, Theorem 5.2, affirms
that, for each $\gamma\in{\mathscr{M}}_{\rm ac}$, the non-linear boundary-
value problem (1.6) has a unique solution in ${\mathcal{F}}$. Its precise
statement is left to Section 5 because it requires some notation.

######  Theorem 2.4.

The functional $S\colon{\mathscr{M}}_{\rm ac}\to{\mathbb{R}}$ defined in (2.6)
is bounded, convex and lower semi-continuous. Moreover,
$S_{0}(\gamma)={\mathscr{G}}(\,\gamma,F(\gamma)\,)$, where $F(\gamma)$ is the
solution to (1.6).

######  Remark 2.5.

When $\gamma=\bar{\rho}$, $F=\bar{\rho}$ is the solution to (1.6). Replacing
$F$ by $\bar{\rho}$ in the formula for ${\mathscr{G}}$ yields that
$S_{0}(\bar{\rho})\,=\,-\,(1+A+B)\,\log(1+A+B)$.

Next theorem asserts that the functionals defined through the variational
problems (2.4) and (2.6) coincide. In particular, it gives an “explicit”
formula for the dynamical variational problem (2.4) defining the quasi-
potential.

######  Theorem 2.6.

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, $V(\gamma)=S(\gamma)$. In
particular, the functional $S$ is non-negative. That is, the functional
$S_{0}$ attains its minimum at $\bar{\rho}$.

######  Remark 2.7.

Theorems 2.4 and 2.6 formalize the arguments presented in [14], where Derrida,
Hirschberg and Sadhu derived the steady state large deviations functional by
representing the steady state as a product of matrices.

######  Remark 2.8.

If $\alpha=\beta$, the exclusion dynamics is reversible and the stationary
state is the Bernoulli product measure with density $\alpha$. In particular,
in this case

$S(\gamma)\;=\;\int_{0}^{1}\Big{\\{}\,\gamma(x)\,\log\frac{\gamma(x)}{\alpha}+\big{[}1-\gamma(x)\big{]}\log\frac{1-\gamma(x)}{1-\alpha}\,\Big{\\}}\;dx\;.$

The same strategy as in the proof of Theorem 2.6 yields that $V=S$. The
arguments are much simpler because the adjoint dynamics coincides with the
original one as the process is reversible.

The method of the proof of Theorems 2.4 and 2.6 is the one proposed in [3] and
carried out in [4] for exclusion processes with strong interaction with the
reservoirs.

The fact that the densities are not fixed by the dynamics at the boundary and
the presence of exponential terms at the boundary in the dynamical large
deviations functionals (cf. the definition of ${\mathfrak{b}}_{\varrho,D}$),
introduce many new difficulties. The proof of the uniqueness of solutions to
(1.6) is one of them (cf. Proof of Theorem 5.2).

The proof of the upper bound for the quasi-potential is a second example. As
the boundary conditions are not fixed, to prove that the solutions to the
adjoint hydrodynamic equations are bounded away from $0$ and $1$, we need to
investigate the behavior of the solutions at the boundary. This is done in the
proof of Proposition 6.7.

The article is organized as follows. In Section 3 we present a heuristic
derivation of Theorem 2.6 based on the Hamiltonian formalism of rational
mechanics. This argument explains the strategy adopted in the following
sections. In Section 4, we recall some properties of the dynamical rate
function $I_{[0,T]}(\,\cdot\,|\gamma)$ obtained in [19]. Theorems 2.4, 2.6 are
proved in Sections 5, 6, respectively. In Appendices A and B we present some
results on the Robin Laplacian and on solutions to heat equations with mixed
boundary conditions needed in the proofs of the main theorems.

##  3\. Sketch of the proof of Theorem 2.6.

The arguments below are formal, but explain the idea of the proof. We follow
the strategy proposed in [4], in the context of boundary driven symmetric
simple exclusion processes with strong interaction with the boundaries, to
derive a formula for the quasi-potential based on the Hamiltonian formalism.
We also introduce the hydrodynamic equation of the adjoint process, which
describes how the dynamics acts to create an anomalous density profile. This
section also serves as a road map to prove Theorem 2.6 in other contexts.

Recall from (1.2) the definition of the Hamiltonian ${\mathscr{H}}$. With this
notation and an integration by parts in time, the functional $I_{[0,T]}$ can
be written as

$I_{[0,T]}(\pi|\gamma)\;=\;\sup_{H}\int_{0}^{T}\big{\\{}\,\big{\langle}\partial_{t}u_{t},H_{t}\big{\rangle}\,-\,{\mathscr{H}}(u_{t},H_{t})\,\big{\\}}\;dt\;.$
(3.1)

Hence, the functional $I_{[0,T]}(\,\cdot\,|\gamma)$ corresponds to the action
functional associated to the Hamiltonian ${\mathscr{H}}$.

A variational calculation yields that the quasi-potential satisfies the
Hamilton-Jacobi equation: for every $\gamma\in{\mathscr{M}}_{\rm ac}$,

${\mathscr{H}}\big{(}\,\gamma\,,\,\frac{\delta
V}{\delta\gamma}\,\big{)}\;=\;0\;.$ (3.2)

where $\delta V/\delta\gamma$ stands for the functional derivative of $V$.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$ and let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Gamma=\log[\gamma/(1-\gamma)]-\log[F/(1-F)]$
for some function $F$ taking values in the interval $(0,1)$. Lemma 6.2 asserts
that, if $\gamma$ is smooth and bounded away from $0$ and $1$, $\Gamma$ solves
the Hamilton-Jacobi equation

${\mathscr{H}}(\gamma,\Gamma)\;=\;0$ (3.3)

if $F$ is the solution to (1.6), that is, if $F=F(\gamma)$ with the notation
introduced in the statement of Theorem 2.4. By (3.2) and (3.3),

$\frac{\delta
V}{\delta\gamma}\;=\;\log\frac{\gamma}{1-\gamma}\;-\;\log\frac{F}{1-F}\;\cdot$
(3.4)

To build a functional $S$ which satisfies (3.4), we look for a functional
${\mathscr{W}}(\gamma,F)$, with two properties:

  * (a)

For every $\gamma\in{\mathscr{M}}_{\rm ac}$,

$\frac{\delta{\mathscr{W}}}{\delta\gamma}(\gamma,F)\;=\;\log\frac{\gamma}{1-\gamma}\,-\,\log\frac{F}{1-F}\;,$

  * (b)

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, the solution $F(\gamma)$ of
equation (1.6) is a critical point of ${\mathscr{W}}(\gamma,\cdot)$;

Under these assumptions, defining $S(\gamma)$ as
${\mathscr{W}}(\gamma,F(\gamma))$, we have

$\frac{\delta
S}{\delta\gamma}(\gamma)\;=\;\frac{\delta{\mathscr{W}}}{\delta\gamma}(\gamma,F(\gamma))\;+\;\frac{\delta{\mathscr{W}}}{\delta
F}(\gamma,F(\gamma))\,\frac{\delta
F}{\delta\gamma}(\gamma)\;=\;\frac{\delta{\mathscr{W}}}{\delta\gamma}(\gamma,F(\gamma))\;\cdot$

The last identity follows from property (b) of the functional ${\mathscr{W}}$
[$\delta{\mathscr{W}}/\delta F=0$ at $(\gamma,F(\gamma))$]. By property (a),
the right-hand side is equal to
$\log[\gamma/(1-\gamma)]-\log[F(\gamma)/(1-F(\gamma))]$, proving that (3.4) is
fulfilled.

This computation explains the introduction of the functional
${\mathscr{G}}(\gamma,F)$, defined below (2.6). It is obtained by integrating
(3.4) in $\gamma$ and adding terms which depend only on $F$ to match condition
(b). The functional ${\mathscr{G}}$ satisfies properties (a) and (b), as it is
easy to show that (1.6) corresponds to the Euler-Lagrange equation of the
functional ${\mathscr{G}}(\gamma,\,\cdot\,)$.

We turn to the proof that $V=S$. Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, $T>0$
and a trajectory $u_{t}$, $0\leq t\leq T$, such that $u_{0}=\bar{\rho}$,
$u_{T}=\gamma$. Let $F_{t}$ be the solution to (1.6) with $u_{t}$ replacing
$\gamma$, $F_{t}=F(u_{t})$. By (3.1), (3.3),

$I_{[0,T]}(\pi|\gamma)\;\geq\;\int_{0}^{T}\big{\langle}\partial_{t}u_{t},\Gamma_{t}\big{\rangle}\;dt\;,$

where $\Gamma_{t}=\log[u_{t}/(1-u_{t})]-\log[F_{t}/(1-F_{t})]$. In view of
(3.4), replacing $\Gamma_{t}$ by $(\delta S/\delta\gamma)(u_{t})$ yields that

$I_{[0,T]}(\pi|\gamma)\;\geq\;S(u_{T})\,-\,S(u_{0})\;=\;S(\gamma)\,-\,S(\bar{\rho})\;,$

so that

$V(\gamma)\;\geq\;S(\gamma)\,-\,S(\bar{\rho})\;.$

We proceed with the upper bound. By [3], the optimal trajectory for the
variational problem (2.4) is the hydrodynamic trajectory of the adjoint
dynamics reversed in time. Moreover, according to [3], the adjoint dynamics is
given by

$\partial_{t}v\,=\,-\,\Delta
v\,+\,2\,\nabla\big{(}\,\sigma(v)\,\nabla\frac{\delta S}{\delta v}\,\big{)}$

In view of (3.4), replacing $\delta S/\delta v$ by
$\log[v_{t}/(1-v_{t})]-\log[F_{t}/(1-F_{t})]$, where $F_{t}$ is the solution
to (1.6) with $v_{t}$ in place of $\gamma$ yields the equation

$\partial_{t}v\,=\,\Delta
v\,-\,2\,\nabla\big{(}\,\sigma(v)\,\nabla\log\frac{F}{1-F}\,\big{)}\;.$

Adding the boundary and initial conditions, as well as the equation for $F$,
the previous equation becomes the system of equations

$\left\\{\begin{aligned} &\partial_{t}v_{t}\,=\,\Delta
v_{t}\,-\,2\,\nabla\big{(}\,\sigma(v_{t})\,\nabla
R_{t}\,\big{)}\quad(t,x)\in(0,\infty)\times(0,1)\;,\\\ &\nabla
v_{t}(1)\,-\,2\,\sigma(v_{t}(1))\,\nabla
R_{t}(1)\,=\,{\mathfrak{p}}_{1-\beta,B}\big{(}\,v_{t}(1)\,,\,R_{t}(1)\,\big{)}\;,\\\
&\nabla v_{t}(0)\,-\,2\,\sigma(v_{t}(0))\,\nabla
R_{t}(0)\,=\,-\,{\mathfrak{p}}_{1-\alpha,A}\big{(}\,v_{t}(0)\,,\,R_{t}(0)\,\big{)}\;,\\\
&v_{0}(\cdot)\,=\,\gamma(\cdot)\;,\quad x\in[0,1]\;,\end{aligned}\right.$
(3.5) $\left\\{\begin{aligned} &\Delta
F_{t}=\big{(}v_{t}-F_{t}\big{)}\frac{\big{(}\nabla
F_{t}\big{)}^{2}}{F_{t}(1-F_{t})}\quad(t,x)\in(0,\infty)\times(0,1)\;,\\\
&\nabla F_{t}(0)=A^{-1}[F_{t}(0)-\alpha]\;,\quad\nabla F_{t}(1)=B^{-1}[\beta-
F_{t}(1)]\;.\end{aligned}\right.$ (3.6)

In this formula, $R_{t}=\log[F_{t}/(1-F_{t})]$ and, for $0<\varrho<1$, $D>0$,
$0<a<1$, $M\in{\mathbb{R}}$,

${\mathfrak{p}}_{\varrho,D}(a,M)\;=\;\frac{1}{D}\,\Big{\\{}[1-a]\,\varrho\,e^{M}\;-\;a\,[1-\varrho]\,e^{-M}\,\Big{\\}}\;.$
(3.7)

The first part of the proof of the upper bound consists in showing that this
trajectory is indeed the optimal one. Lemma 6.5, whose proof relies on the
explicit expression for the rate functional presented in Lemma 4.4, states
that this trajectory is optimal provided the solution $v(t)$ to this equation
relaxes to $\bar{\rho}$ as $t\to\infty$.

To prove that $v_{t}$ relaxes to $\bar{\rho}$ or any other property of the
non-local system of equations (3.5)–(3.6) looks hopeless. It turns out,
however, that these equations can be expressed in a simple form. The reason is
that $F_{t}$ in (3.5)–(3.6), which corresponds to the momentum in the
Hamiltonian formalism, evolves according to an autonomous equation, a
remarkable and unexpected property.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and denote by $F^{(\gamma)}$ the
solution to (1.6). Let $F^{(\gamma)}_{t}$ be the solution to the heat equation
(1.1) with initial condition $F^{(\gamma)}$ instead of $\gamma$. Define
$v^{(\gamma)}_{t}$ as

$v^{(\gamma)}(t)\;:=\;F^{(\gamma)}(t)\;+\;F^{(\gamma)}(t)\,[1-F^{(\gamma)}(t)]\,\frac{\Delta
F^{(\gamma)}(t)}{\big{(}\nabla F^{(\gamma)}(t)\big{)}^{2}}\;\cdot$ (3.8)

By (1.6), $v^{(\gamma)}(0)=\gamma$. Actually,
$F^{(\gamma)}_{t}=F(v^{(\gamma)}_{t})$ for all $t\geq 0$, where
$F(v^{(\gamma)}_{t})$ is the solution to (1.6) with $\gamma$ replaced by
$v^{(\gamma)}_{t}$.

Proposition 6.7 asserts that for each $\gamma\in C^{1}([0,1])$ the pair
$(v^{(\gamma)}_{t},F^{(\gamma)}_{t})$ solves the system of equations
(3.5)–(3.6). This result provides, therefore, an alternative and simple
formulation of these equations. Moreover, by Lemma 6.10,
$\lim_{t\to\infty}v^{(\gamma)}(t)=\bar{\rho}$, and, by Lemma 6.12, the optimal
path which solves the variational problem (2.4), represented by
$u^{(\gamma)}_{\rm opt}(t)$, defined on the time interval $(-\infty,0]$
instead of $[0,+\infty)$ as in (2.4), is given by $u^{(\gamma)}_{\rm
opt}(t)=v^{(\gamma)}(-t)$. Note that $u^{(\gamma)}_{\rm opt}(0)=\gamma$,
$\lim_{t\to-\infty}u^{(\gamma)}_{\rm opt}(t)=\bar{\rho}$. Hence,
$u^{(\gamma)}_{\rm opt}$ connects $\bar{\rho}$ to $\gamma$ in a infinite time
window.

######  Remark 3.1.

The boundary conditions in (3.5) can be written as

$\left\\{\begin{aligned} &\nabla v_{t}(0)\,-\,2\,\sigma(v_{t}(0))\,\nabla
R_{t}(0)\,=\,\frac{1}{A^{*}_{t}}\,[\,v(t,0)-\alpha^{*}_{t}\,]\;.\\\ &\nabla
v_{t}(1)\,-\,2\,\sigma(v_{t}(1))\,\nabla
R_{t}(1)\,=\,\frac{1}{B^{*}_{t}}\,[\,\beta^{*}_{t}-v(t,1)\,]\;,\end{aligned}\right.$
(3.9)

where

$\displaystyle\frac{1}{B^{*}_{t}}\;=\;\frac{1}{B}\,\big{\\{}\,(1-\beta)\,e^{R_{t}(1)}\;+\;\beta\,e^{-R_{t}(1)}\,\big{\\}}\;,\quad\beta^{*}_{t}\;=\;\frac{(1-\beta)\,e^{R_{t}(1)}}{(1-\beta)\,e^{R_{t}(1)}\;+\;\beta\,e^{-R_{t}(1)}}$
$\displaystyle\frac{1}{A^{*}_{t}}\;=\;\frac{1}{A}\,\big{\\{}\,(1-\alpha)\,e^{R_{t}(0)}\;+\;\alpha\,e^{-R_{t}(0)}\,\big{\\}}\;,\quad\alpha^{*}_{t}\;=\;\frac{(1-\alpha)\,e^{R_{t}(0)}}{(1-\alpha)\,e^{R_{t}(0)}\;+\;\alpha\,e^{-R_{t}(0)}}\;\cdot$

In view of (3.9), equation (3.5) corresponds to the hydrodynamic equation of
the weakly asymmetric exclusion process with weak interactions at the boundary
(cf. [19]). Note that this equation carries a positive drift to the right
because $\nabla\log[F/(1-F)]>0$. Moreover, the boundary densities and the
intensity of the interactions $\alpha$, $\beta$, $A$, $B$ are time-dependent
and given by $\alpha^{*}_{t}$, $\beta^{*}_{t}$, $A^{*}_{t}$, $B^{*}_{t}$,
respectively.

######  Remark 3.2.

As mentioned above, equations (3.5)–(3.6) represent the adjoint hydrodynamic
equation [that is, the PDE which describes the evolution of the density under
the adjoint dynamics]. Hence, in the adjoint dynamics, the density evolves
according to a weakly asymmetric exclusion process. The drift at time $t$ is
$\log[F_{t}/(1-F_{t})]$, where $F_{t}$ is the solution to (1.6) with $\gamma$
replaced by the density profile at time $t$. The boundary densities and
intensities are given by the equations below (3.9).

######  Remark 3.3.

Notwithstanding the fact that the boundary densities have been modified and a
drift added, a straightforward computation shows that the stationary profile
of equation (3.5) is still $\bar{\rho}$, the stationary profile of the
hydrodynamic equation (1.1). More precisely, as $F_{t}=F^{(\gamma)}$ solves
equation (1.1), $F_{t}\to\bar{\rho}$ as $t\to\infty$. Replace in equation
(3.5) $R_{t}$ by $\log[\bar{\rho}/(1-\bar{\rho})]$ and consider the associated
stationary equation [that is, replace $\partial_{t}v_{t}$ by $0$, consider
this equation in the space variable only and remove the initial condition].
It’s easy to check that $\bar{\rho}$ fulfills this stationary equation.

##  4\. The dynamical rate function

For the reader’s convenience, we recall here some properties of the rate
functional $I_{[0,T]}(\,\cdot\,|\gamma)$ proved in [19].

The first estimate asserts that the cost of a trajectory in a interval $[0,T]$
is bounded by the sum of its cost in the intervals $[0,S]$ and $[S,T]$.

Let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\tau_{r}u:{\mathbb{R}}_{+}\times[0,1]\to{\mathbb{R}}$,
$r>0$, be the function defined by $\tau_{r}u(t,x)=u(t+r,x)$. For all
$\pi(t,dx)=u(t,x)\,dx$ in $D([0,T],{\mathscr{M}}_{\rm ac})$ and $0<S<T$,

$I_{[0,T]}(u\,|\,\gamma)\;\leq\;I_{[0,S]}(u\,|\,\gamma)\;+\;I_{[0,T-S]}(\tau_{S}u\,|\,u(S,\,\cdot\,))\;.$
(4.1)

######  Theorem 4.1.

Fix $T>0$ and $\gamma\in{\mathscr{M}}_{\rm ac}$. The function
$I_{[0,T]}(\cdot|\gamma):D([0,T],\mathcal{M})\to[0,\infty]$ is convex, lower
semicontinuous and has compact level sets.

######  Definition 4.2.

Given $\gamma\in{\mathscr{M}}_{\rm ac}$, let $\Pi_{\gamma}$ be the collection
of all paths $\pi(t,dx)=u(t,x)dx$ in $D([0,T],{\mathscr{M}}_{\rm ac})$ such
that

  * (a)

There exists ${\mathfrak{t}}>0$, such that $u$ follows the hydrodynamic
equation (1.1) in the time interval $[0,{\mathfrak{t}}]$. In particular,
$u(0,\cdot)=\gamma(\cdot)$.

  * (b)

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 $[\delta,T]\times[0,1]$;

  * (c)

$u$ is smooth on $(0,T]\times[0,1]$.

######  Theorem 4.3.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$. For all $\pi$ in
$D([0,T],{\mathscr{M}})$ such that $I_{[0,T]}(\pi|\gamma)<\infty$, there
exists a sequence $\\{\pi^{n}:n\geq 1\\}$ in $\Pi_{\gamma}$ such that
$\pi^{n}$ converges to $\pi$ in $D([0,T],{\mathscr{M}})$ and
$I_{[0,T]}(\pi^{n}|\gamma)$ converges to $I_{[0,T]}(\pi|\gamma)$. Moreover, if
there exists $\epsilon_{0}>0$ such that $\epsilon_{0}\leq\gamma\leq
1-\epsilon_{0}$, condition (b) in Definition 4.2 can be replaced by the
existence of $\epsilon>0$ such that $\epsilon\leq u(t,x)\leq 1-\epsilon$ for
all $(t,x)\in[0,T]\times[0,1]$.

Let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Omega_{T}$
be the cylinder $(0,T)\times(0,1)$. Fix $\pi$ in $D([0,T],{\mathscr{M}}_{\rm
ac})$, $\pi(t,dx)=u(t,x)\,dx$. Let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathcal{H}}^{1}(\Omega_{T})$
be the Hilbert spaces induced by the sets $C^{\infty}(\Omega_{T})$ endowed
with the scalar products, $\langle\\!\langle G,H\rangle\\!\rangle_{1,2}$
defined by

$\begin{gathered}\langle\\!\langle
G,H\rangle\\!\rangle_{1,2}\;=\;\int_{0}^{T}dt\int_{0}^{1}G_{t}\,H_{t}\;dx\;+\;\int_{0}^{T}dt\int_{0}^{1}\nabla
G_{t}\,\nabla H_{t}\;dx\;.\end{gathered}$

Recall from (3.7) the definition of ${\mathfrak{p}}_{\varrho,D}(a,M)$. For
$0<\varrho<1$, $D>0$, $0<a<1$, $M\in{\mathbb{R}}$, let

${\mathfrak{c}}_{\varrho,D}(a,M)\;=\;\frac{1}{D}\,\Big{\\{}[1-a]\,\varrho\,[1-e^{M}+Me^{M}]\;+\;a\,[1-\varrho]\,[\,1-e^{-M}-Me^{-M}]\,\Big{\\}}\;.$
(4.2)

Note: for a trajectory $u_{t}$ such that $\delta\leq u(t,x)\leq 1-\delta$ for
all $(t,x)\in[0,T]\times[0,1]$, the space ${\mathcal{H}}^{1}(\Omega_{T})$
introduced above coincides with the space ${\mathcal{H}}^{1}(\sigma(u))$
introduced in [19].

######  Lemma 4.4.

Fix a trajectory $\pi$ in $D([0,T],{\mathscr{M}}_{\rm ac})$,
$\pi(t,dx)=u(t,x)\,dx$. Assume that $u\in C^{1,2}([0,T]\times[0,1])$, there
exists $\delta>0$ such that $\delta\leq u(t,x)\leq 1-\delta$ for all
$(t,x)\in[0,T]\times[0,1]$ and $I_{[0,T]}(u\,|\,\gamma)$ is finite, where
$\gamma=u_{0}$. Then, there exists a function $H$ in
${\mathcal{H}}^{1}(\Omega_{T})$ such that $u$ is the unique weak solution to

$\left\\{\begin{aligned} &\partial_{t}u\;=\;\Delta
u\,-\,2\,\nabla\\{\sigma(u)\ \nabla H\\}\;,\\\ &\nabla
u_{t}(1)\,-\,2\,\sigma(u_{t}(1))\,\nabla
H_{t}(1)\,=\,{\mathfrak{p}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\;,\\\
&\nabla u_{t}(0)\,-\,2\,\sigma(u_{t}(0))\,\nabla
H_{t}(0)\,=\,-\,{\mathfrak{p}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\;,\\\
&u(0,\cdot)=\gamma(\cdot)\;.\end{aligned}\right.$ (4.3)

Moreover,

$\displaystyle I_{[0,T]}(u\,|\,\gamma)$
$\displaystyle=\;\int_{0}^{T}\langle\,\sigma(u_{t})\,,\,(\nabla
H_{t})^{2}\,\rangle\;dt+\;\int_{0}^{T}{\mathfrak{c}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\;dt$
(4.4)
$\displaystyle+\;\int_{0}^{T}{\mathfrak{c}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\;dt\;.$

Weak solutions to equation (4.3) are introduced in Definition B.7. Theorem B.8
states that for each $\gamma\in{\mathscr{M}}_{\rm ac}$ there exists one and
only one weak solution.

##  5\. The Euler-Lagrange equation for $S$

The Euler–Lagrange equation associated to the variational problem (2.6) is
given by the non-linear equation with Robin boundary conditions (1.6). In this
section, we provide a precise meaning to this equation, prove existence and
uniqueness of solutions, and prove Theorem 2.4. The approach is taken from
[4], but there is a serious technical difficulty in the proof of uniqueness.
The idea there is to extend the problem to the interval $[-A,1+B]$, see Lemma
5.5 and the proof of Theorem 5.2.

Recall the definition of ${\mathcal{F}}$ introduced in (2.5). For
$F\in{\mathcal{F}}$, let

${\mathcal{R}}_{\gamma}(x)\;=\;{\mathcal{R}}_{\gamma}(F\,;\,x)=\big{[}\,\gamma(x)-F(x)\,\big{]}\,\frac{\nabla
F(x)}{F(x)\,[1-F(x)]}\;,$ (5.1)

With this notation, equation (1.6) takes the form

$\left\\{\begin{aligned} &\Delta F\;=\;\nabla F\,{\mathcal{R}}_{\gamma}\;,\\\
&\vphantom{\Big{(}}{\displaystyle\nabla F(0)=A^{-1}[F(0)-\alpha]\;,\quad\nabla
F(1)=B^{-1}[\beta-F(1)]}\;.\end{aligned}\right.$ (5.2)

To prove the existence and uniqueness of a solution to (1.6), following [4],
we formulate (5.2) as the integro–differential equation

$F(x)\;=\;\alpha\;+\;(\beta-\alpha)\,\frac{A\;+\;\int_{0}^{x}\exp\\{\int_{0}^{y}{\mathcal{R}}_{\gamma}(F;z)\,dz\\}\,dy}{A+\int_{0}^{1}\exp\\{\int_{0}^{y}{\mathcal{R}}_{\gamma}(F;z)\,dz\\}\,dy+B\,\exp\\{\int_{0}^{1}{\mathcal{R}}_{\gamma}(F;y)\,dy\\}}\;.$
(5.3)

######  Remark 5.1.

If $\gamma=\bar{\rho}$, then $F=\bar{\rho}$ solves (1.6) and (5.3). Moreover,
if $F\in C^{2}([0,1])$ is a solution to the problem (1.6) such that $\nabla
F(x)>0$ for $x\in[0,1]$, then $F$ is also a solution to the
integro–differential equation (5.3). Conversely, if $F\in C^{1}([0,1])$ is a
solution to (5.3), then the boundary conditions in (1.6) are satisfied.
Moreover, $\nabla F(x)>0$, $\Delta F(x)$ exists for almost every $x$ and the
differential equation in (1.6) holds almost everywhere. Furthermore, if
$\gamma\in C([0,1])$, then $F\in C^{2}([0,1])$ and (1.6) holds everywhere.

######  Theorem 5.2.

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, there exists a unique
$F\in{\mathcal{F}}$ which solves (5.3).

The existence is proven by applying Schauder’s fixed point theorem. The
argument requires some notation. For each $\gamma\in{\mathscr{M}}_{\rm ac}$
consider the map
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathcal{K}}_{\gamma}:{\mathcal{F}}\to
C^{1}\big{(}[0,1]\big{)}$ defined by

${\mathcal{K}}_{\gamma}(F)(x)\;:=\;\alpha\;+\;(\beta-\alpha)\,\frac{A\;+\;\int_{0}^{x}\exp\\{\int_{0}^{y}{\mathcal{R}}_{\gamma}(F;z)\,dz\\}\,dy}{A+\int_{0}^{1}\exp\\{\int_{0}^{y}{\mathcal{R}}_{\gamma}(F;z)\,dz\\}\,dy+B\,\exp\\{\int_{0}^{1}{\mathcal{R}}_{\gamma}(F;y)\,dy\\}}\;.$
(5.4)

Let $p$ and $q$ be given by

$p\;:=\;\frac{\alpha\,(\beta-\alpha)}{A\alpha+(B+1)\beta}\,\frac{1-\beta}{1-\alpha}\;,\quad
q\;:=\;\frac{(1-\alpha)\,(\beta-\alpha)}{A(1-\alpha)+(B+1)(1-\beta)}\,\frac{\beta}{\alpha}\;.$
(5.5)

Note that $0<p<q$ because

$\frac{p}{\beta-\alpha}\;=\;\frac{1}{A+(B+1)(\beta/\alpha)}\,\frac{1-\beta}{1-\alpha}\;<\;\frac{1}{A+(B+1)[(1-\beta)/(1-\alpha)]}\,\frac{\beta}{\alpha}\;=\;\frac{q}{\beta-\alpha}\;\cdot$

The inequality above follows from the fact that
$(1-\beta)\alpha<\beta(1-\alpha)$ as $\alpha<\beta$.

Denote by ${\mathcal{B}}_{\rm bc}$ the subset of functions in $C^{1}([0,1])$
which satisfy the boundary conditions of the Euler-Lagrange equation (1.6):

${\mathcal{B}}_{\rm bc}\;:=\;\big{\\{}\,F\in C^{1}([0,1]):\nabla
F(0)=A^{-1}[F(0)-\alpha]\;,\;\;\nabla F(1)=B^{-1}[\beta-F(1)]\,\big{\\}}\;,$

and by ${\mathcal{B}}$ the subset of ${\mathcal{B}}_{\rm bc}$ given by

${\mathcal{B}}\;:=\;\big{\\{}\,F\in{\mathcal{B}}_{\rm bc}:p\leq\nabla F(x)\leq
q\;\;\forall\;x\in[0,1]\,\big{\\}}\;.$

Note that ${\mathcal{B}}_{\rm bc}$, ${\mathcal{B}}$ are closed and convex, and
that ${\mathcal{B}}$ is contained in ${\mathcal{F}}$. To establish this last
assertion, write that $F(x)\geq F(0)=\alpha+A\nabla F(0)\geq\alpha+Ap>\alpha$
because $\nabla F(x)\geq p>0$. A similar argument shows that $F(x)\leq
F(1)=\beta-B\nabla F(1)<\beta-Bp<\beta$. In particular, for every
$F\in{\mathcal{B}}$,

$\alpha\,+\,Ap\;\leq\;F(x)\;\leq\;\beta\,-\,Bp\;.$ (5.6)

######  Lemma 5.3.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$. Then,

  * (a)

The functional ${\mathcal{K}}_{\gamma}$ is a continuous map;

  * (b)

${\mathcal{K}}_{\gamma}({\mathcal{F}})\subset{\mathcal{B}}$;

  * (c)

There exists a finite constant $C_{0}$, such that

$\big{|}\,\nabla{\mathcal{K}}_{\gamma}(F)(x)\,-\,\nabla{\mathcal{K}}_{\gamma}(F)(y)\,\big{|}\;\leq\;C_{0}\,|x-y|$

for all $F\in{\mathcal{B}}$, $x$, $y\in[0,1]$.

###### Proof.

Assertion (a) follows from the definitions of ${\mathcal{R}}_{\gamma}$ and
${\mathcal{K}}_{\gamma}$. We turn to (b). Fix $F\in{\mathcal{F}}$. It is easy
to show that ${\mathcal{K}}_{\gamma}(F)$ satisfies the boundary condition of
(1.6). It remains to derive the bounds on the derivative of
${\mathcal{K}}_{\gamma}(F)$. As $0\leq\gamma\leq 1$, $-F\leq\gamma-F\leq 1-F$.
Therefore, as $\nabla F\geq 0$ and $\alpha\leq F\leq\beta$,

$\frac{-\nabla F}{1-F}\;\leq\;{\mathcal{R}}_{\gamma}\;\leq\;\frac{\nabla
F}{F}\;\cdot$

It follows from these inequalities that

$\frac{1-\beta}{1-\alpha}\;\leq\;\exp\Big{\\{}\int_{0}^{x}{\mathcal{R}}_{\gamma}(F;y)\;dy\,\Big{\\}}\;\leq\;\frac{\beta}{\alpha}$

for all $0\leq x\leq 1$. Reporting these bounds in the definition of
${\mathcal{K}}_{\gamma}(F)$ yields that $p\leq{\mathcal{K}}_{\gamma}(F)\leq
q$, as claimed.

The proof of the last assertion of the lemma relies on the previous two bounds
and the bound $p\leq\nabla F\leq q$ which holds for all functions in
${\mathcal{B}}$. ∎

######  Corollary 5.4.

The integro-differential equation (5.3) has a solution in ${\mathcal{B}}$.

###### Proof.

By Schauder’s fixed point theorem, it is enough to show that
${\mathcal{K}}_{\gamma}({\mathcal{B}})$ has a compact closure in
$C^{1}([0,1])$. By Ascoli–Arzela theorem, this property holds provided
$\nabla{\mathcal{K}}_{\gamma}(F)$ is Lipschitz continuous, uniformly for
$F\in{\mathcal{B}}$. This is the content of assertion (c) of the lemma. ∎

We turn to uniqueness. The proof relies on an argument used to prove
uniqueness of solutions to equation (1.6) with Dirichlet boundary conditions.
First, inspired by [14], we turn the pair $(\gamma,F)$ defined on the interval
$[0,1]$, of solutions to (5.3) [that is, with Robin boundary conditions] into
a pair $(\gamma_{\rm ext},F_{\rm ext})$ defined on the interval $[-A,1+B]$, of
solutions to (5.2) with Dirichlet boundary conditions.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and let $F\in{\mathcal{F}}$ be a
solution to equation (5.3). We extend $\gamma$ and $F$ to the interval
$[-A,1+B]$ as follows. $F_{\rm ext}$ coincides with $F$ on $[0,1]$, is linear
in the complement and $F_{\rm ext}(-A)=\alpha$, $F_{\rm ext}(1+B)=\beta$.
$\gamma_{\rm ext}$ coincides with $\gamma$ on $[0,1]$ and is equal to $F_{\rm
ext}$ in the complement. Hence, $F_{\rm ext}$, $\gamma_{\rm
ext}:[-A,1+B]\to{\mathbb{R}}$ are given by

$F_{\rm ext}(x)\;=\;\begin{cases}\alpha\,+\,\nabla F(0)\,[A+x\,]&\text{for
$x\in[-A,0)$}\;,\\\ F(x)&\text{for $x\in[0,1]$}\;,\\\ \beta\,+\,\nabla
F(1)\,[x-B-1\,]&\text{for $x\in(1,1+B]$}\;,\\\ \end{cases}$

and

$\gamma_{\rm ext}(x)\;=\;\begin{cases}\gamma(x)&\text{for $x\in[0,1]$}\;,\\\
F_{\rm ext}(x)&\text{otherwise}\;.\\\ \end{cases}$

Note that $F_{\rm ext}$ belongs to $C^{1}([-A,1+B])$, $F_{\rm
ext}(-A)=\alpha$, $F_{\rm ext}(1+B)=\beta$. Moreover, since
$F={\mathcal{K}}_{\gamma}(F)$ and
${\mathcal{K}}_{\gamma}({\mathcal{F}})\subset{\mathcal{B}}$, on the interval
$[0,1]$, $p\leq\nabla F_{\rm ext}(x)\leq q$. Hence, $p\wedge\alpha\leq\nabla
F_{\rm ext}(x)\leq q\vee\beta$, and $F_{\rm ext}$ belongs to the set
${\mathcal{B}}_{\rm ext}$ defined by

$\displaystyle{\mathcal{B}}_{\rm bc,ext}\;:=\;\big{\\{}\,G\in
C^{1}([-A,1+B]):G(-A)=\alpha\;,\;G(1+B)=\beta\,\big{\\}}\;,$
$\displaystyle{\mathcal{B}}_{\rm ext}\;:=\;\big{\\{}\,G\in{\mathcal{B}}_{\rm
bc,ext}:p\wedge\alpha\leq\nabla F(x)\leq
q\vee\beta\;\;\forall\;x\in[-A,1+B]\,\big{\\}}\;.$

Fix $\varphi$ in ${\mathscr{M}}_{\rm ac}([-A,1+B])$. With Dirichlet boundary
conditions on the interval $[-A,1+B]$, the problem (5.2) becomes the integro-
differential equation

$G(x)\;=\;\alpha\;+\;(\beta-\alpha)\,\frac{\int_{-A}^{x}\exp\\{\int_{-A}^{y}{\mathcal{R}}_{\varphi}(G;z)\,dz\\}\,dy}{\int_{-A}^{1+B}\exp\\{\int_{-A}^{y}{\mathcal{R}}_{\varphi}(G;z)\,dz\\}\,dy}\;,$
(5.7)

where ${\mathcal{R}}_{\varphi}(G;z)$ is given by (5.1).

######  Lemma 5.5.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and let $F\in{\mathcal{F}}$ be a
solution to equation (5.3). Then, $F_{\rm ext}$ is a solution to (5.7) for
$\varphi=\gamma_{\rm ext}$.

###### Proof.

The assertion follows from a straightforward computation. The result holds for
the following reason. On the interval $[0,1]$, the identity (5.2) is in force
because $F$ is a solution to (5.3). On the other hand, On the complement,
(5.2) holds because both sides of the identity (5.2) vanish. The left-hand
side because $F_{\rm ext}$ is linear on $[0,1]^{c}$, and the right-hand side
because $\gamma_{\rm ext}=F_{\rm ext}$. ∎

###### Proof of Theorem 5.2.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$. Existence has been proven in Corollary
5.4. To prove uniqueness, consider two solutions $F^{(1)}$, $F^{(2)}$, and
recall the definition of $\gamma^{(j)}_{\rm ext}$, $F^{(j)}_{\rm ext}$, $j=1$,
$2$.

By Lemma 5.5, $F^{(1)}_{\rm ext}$, $F^{(2)}_{\rm ext}$ are solutions to (5.7),
with $\varphi=\gamma^{(1)}_{\rm ext}$, $\gamma^{(2)}_{\rm ext}$, respectively.
Therefore, since these functions solve (5.2) almost everywhere,

$(\nabla F^{(j)}_{\rm ext})(x)\;=\;(\nabla F^{(j)}_{\rm
ext})(-A)\;+\;\int_{-A}^{x}(\nabla F^{(j)}_{\rm
ext})(y)\;{\mathcal{R}}_{\gamma}(F^{(j)}_{\rm ext};y)\;dy$ (5.8)

for $j=1$, $2$ and all $x$ in $[-A,1+B]$.

Assume that $(\nabla F^{(1)}_{\rm ext})(-A)=(\nabla F^{(2)}_{\rm ext})(-A)$.
In this case, by definition of $\gamma^{(j)}_{\rm ext}$, $\gamma^{(1)}_{\rm
ext}(x)=\gamma^{(2)}_{\rm ext}(x)$ for all $x\in[-A,0)$. Since this identity
always holds for $x\in[0,1]$, $\gamma^{(1)}_{\rm ext}(x)=\gamma^{(2)}_{\rm
ext}(x)$ for all $x\in[-A,1]$. By (5.8), elementary bounds and Gronwall’s
inequality, $(\nabla F^{(1)}_{\rm ext})(x)=(\nabla F^{(2)}_{\rm ext})(x)$ for
all $x\in[-A,1]$ so that $F^{(1)}_{\rm ext}(x)=F^{(2)}_{\rm ext}(x)$ for all
$x$ in this interval due to the boundary condition satisfied by $F^{(1)}_{\rm
ext}$, $F^{(2)}_{\rm ext}$ at $-A$.

Since $(\nabla F^{(1)}_{\rm ext})(1)=(\nabla F^{(2)}_{\rm ext})(1)$,
$\gamma^{(1)}_{\rm ext}(x)=\gamma^{(2)}_{\rm ext}(x)$ also for $x\in[1,1+B]$.
The same Gronwall’s argument permits to extends the identity $F^{(1)}_{\rm
ext}(x)=F^{(2)}_{\rm ext}(x)$ to $x\in[1,1+B]$. This concludes the argument in
the case $(\nabla F^{(1)}_{\rm ext})(-A)=(\nabla F^{(2)}_{\rm ext})(-A)$.

Assume, by contradiction, that $(\nabla F^{(1)}_{\rm ext})(-A)<(\nabla
F^{(2)}_{\rm ext})(-A)$. The argument in this case relies on the following
identity. As $F^{(1)}_{\rm ext}$ is strictly increasing, (5.2) yields that

$\nabla\,\frac{F^{(j)}_{\rm ext}\,[1-F^{(j)}_{\rm ext}]}{\nabla F^{(j)}_{\rm
ext}}\;=\;1-F^{(j)}_{\rm ext}-\gamma^{(j)}_{\rm ext}$

holds almost everywhere. Hence, as $F^{(j)}_{\rm ext}(-A)=\alpha$,

$\frac{F^{(j)}_{\rm ext}(x)\,[1-F^{(j)}_{\rm ext}(x)]}{(\nabla F^{(j)}_{\rm
ext})(x)}\;=\;\frac{\alpha\,[1-\alpha]}{(\nabla F^{(j)}_{\rm
ext})(-A)}\;+\;\int_{-A}^{x}[\,1-F^{(j)}_{\rm ext}(y)-\gamma^{(j)}_{\rm
ext}(y)\,]\;dy$ (5.9)

for all $x$ in $[-A,1+B]$.

Let $x_{0}:=\inf\\{y\in(-A,1+B]:F^{(j)}_{\rm ext}(y)=F^{(2)}_{\rm ext}(y)\\}$.
The point $x_{0}$ belongs to $(-A,1+B]$ because $F^{(1)}_{\rm
ext}(-A)=F^{(2)}_{\rm ext}(-A)$, $(\nabla F^{(1)}_{\rm ext})(-A)<(\nabla
F^{(2)}_{\rm ext})(-A)$ and $F^{(1)}_{\rm ext}(1+B)=F^{(2)}_{\rm ext}(1+B)$.
Actually, it can not belong to $[-A,0]$ because the functions $F^{(j)}_{\rm
ext}$ are linear in this interval.

Suppose that $x_{0}\in(0,1]$. By definition of $x_{0}$, $F^{(1)}_{\rm
ext}(x)<F^{(2)}_{\rm ext}(x)$ for all $x\in(-A,x_{0})$. Thus,
$\gamma^{(1)}_{\rm ext}(x)\leq\gamma^{(2)}_{\rm ext}(x)$ for all $x$ in this
interval. On the other hand, $F^{(1)}_{\rm ext}(x_{0})=F^{(2)}_{\rm
ext}(x_{0})$ and $(\nabla F^{(1)}_{\rm ext})(x_{0})\geq(\nabla F^{(1)}_{\rm
ext})(x_{0})$. Therefore, by (5.9),

$\frac{F^{(1)}_{\rm ext}(x_{0})[1-F^{(1)}_{\rm ext}(x_{0})]}{(\nabla
F^{(1)}_{\rm ext})(x_{0})}\;>\;\frac{F^{(2)}_{\rm ext}(x_{0})[1-F^{(2)}_{\rm
ext}(x_{0})]}{(\nabla F^{(2)}_{\rm ext})(x_{0})}$

or, equivalently, $(\nabla F^{(1)}_{\rm ext})(x_{0})<(\nabla F^{(2)}_{\rm
ext})(x_{0})$, which is a contradiction.

We turn to the case where $x_{0}\in(1,1+B]$. By definition of $x_{0}$,
$F^{(1)}_{\rm ext}(x)<F^{(2)}_{\rm ext}(x)$ for all $x\in(-A,x_{0})$. Since
the functions $F^{(j)}_{\rm ext}$ are linear in $[1,1+B]$, this entails that
$x_{0}=1+B$ and that $\gamma^{(1)}_{\rm ext}(x)\leq\gamma^{(2)}_{\rm ext}(x)$
for all $x\in[-A,1+B]$. We may repeat the argument of the previous paragraph
to conclude that $(\nabla F^{(1)}_{\rm ext})(1+B)<(\nabla F^{(2)}_{\rm
ext})(1+B)$, which is a contradiction. This completes the proof of the
theorem. ∎

######  Proposition 5.6.

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, denote by $F=F(\gamma)$ the unique
solution in ${\mathcal{F}}$ of (5.3). Then,

  * (i)

If $\gamma\in C([0,1])$, then $F(\gamma)\in C^{2}([0,1])$ and it is the unique
solution in ${\mathcal{F}}\cap C^{2}([0,1])$ of (5.3);

  * (ii)

If $\gamma_{n}$ converges to $\gamma$ in ${\mathscr{M}}_{\rm ac}$ as
$n\to\infty$, then $F_{n}=F(\gamma_{n})$ converges to $F=F(\gamma)$ in
$C^{1}([0,1])$;

###### Proof.

Existence in assertion (i) follows from Theorem 5.2 and identity (5.3), which
holds for all points $x$ in $[0,1]$ because $\gamma$ is continuous. Uniqueness
follows from Theorem 5.2.

To prove (ii), let $\gamma_{n}$ be a sequence converging to $\gamma$ in
${\mathscr{M}}_{\rm ac}$ and denote by $F_{n}=F(\gamma_{n})$ the corresponding
solution to (5.3). By Lemma 5.3.(c) and Ascoli–Arzela theorem, the sequence
$F_{n}$ is relatively compact in $C^{1}\big{(}[0,1]\big{)}$. It remains to
show uniqueness of its limit points. Consider a subsequence $n_{j}$ and assume
that $F_{n_{j}}$ converges to $G$ in $C^{1}\big{(}[0,1]\big{)}$. Since
$\gamma_{n_{j}}$ converges to $\gamma$ in ${\mathscr{M}}_{\rm ac}$ and
$F_{n_{j}}$ converges to $G$ in $C^{1}\big{(}[0,1]\big{)}$, by (5.4)
${\mathcal{K}}_{\gamma_{n_{j}}}(F_{n_{j}})$ converges to
${\mathcal{K}}_{\gamma}(G)$. In particular,
$G=\lim_{j}F_{n_{j}}=\lim_{j}{\mathcal{K}}_{\gamma_{n_{j}}}(F_{n_{j}})={\mathcal{K}}_{\gamma}(G)$.
Hence, by uniqueness of the solutions to (5.3), $G=F(\gamma)$. This shows that
$F(\gamma)$ is the unique limit point of the sequence $F_{n}$, and concludes
the proof of (ii). ∎

Fix a trajectory $u(t,\cdot)$, and denote by $F(t,\cdot)$ the function given
by $F(t,x)=F(u(t,\cdot))(x)$. In the next lemma, we derive smoothness
properties of $F$ in terms of the ones of $u$. To prove this result, it is
convenient to introduce a new variable. Let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\varphi_{-}:=\log[\alpha/(1-\alpha)]$,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\varphi_{+}:=\log[\beta/(1-\beta)]$
and denote by $\widetilde{{\mathcal{F}}}$ be the space of monotone $C^{1}$
functions given by

$\widetilde{{\mathcal{F}}}:=\Big{\\{}\,\varphi\in
C^{1}([0,1])\,:\>\varphi_{-}\,<\,\varphi(x)\,<\,\varphi_{+}\;,\;\varphi^{\prime}(x)>0\;\;\forall\;x\in[0,1]\,\Big{\\}}\;.$
(5.10)

Denote by $\Phi:{\mathcal{F}}\to\widetilde{{\mathcal{F}}}$ the map given by

$\Phi(F)\;=\;\log\frac{F}{1-F}\;\cdot$ (5.11)

Clearly,
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Phi^{-1}(\varphi)=e^{\varphi}/[1+e^{\varphi}]$.
The advantage of working with $\varphi=\Phi(F)$ instead of $F$ lies in the
fact that, as a function of $\varphi$, the functional ${\mathscr{G}}_{\rm
bulk}$, defined above (2.6), is concave. This property plays a crucial role in
the sequel.

In terms of the variable $\varphi$ the Euler-Lagrange equation (1.6) becomes

$\left\\{\begin{aligned}
&-\,\nabla\,\Big{(}\frac{1}{\nabla\varphi}\Big{)}\;+\;\frac{1}{1+e^{\varphi}}\;=\;\gamma\qquad\text{for}\;\;x\in(0,1)\\\
&\nabla\varphi(0)\;=\;-\,\frac{1}{A}\,\big{\\{}\,(1+e^{-\varphi(0)})\,\alpha\,-\,(1-\alpha)(1+e^{\varphi(0)})\,\big{\\}}\;;\\\
&\vphantom{\Big{(}}\nabla\varphi(1)\;=\;\frac{1}{B}\,\big{\\{}\,(1+e^{-\varphi(1)})\,\beta\,-\,(1-\beta)\,(1+e^{\varphi(1)})\,\big{\\}}\;.\end{aligned}\right.$
(5.12)

By Lemma 5.3(b), there exists a constant
$C_{1}=C_{1}(\alpha,\beta,A,B)\in(0,\infty)$ such that

$\frac{1}{C_{1}}\;\leq\;(\nabla\varphi)(x)\;\leq\;C_{1}\qquad\text{for
all}\;x\in[0,1]\;,\,\gamma\in{\mathscr{M}}_{\rm ac}\;.$ (5.13)

Fix $T>0$ and a trajectory $u(t,\cdot)$, $0\leq t\leq T$, in
$C^{1,0}([0,T]\times[0,1])$ such that $0\leq u(t,x)\leq 1$ for all $(t,x)$.
Denote by $F(t,x)$ the function given by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}F(t,x)=F(u(t,\cdot))(x)$.
By Proposition 5.6, $F$ belongs to $C^{0,2}([0,T]\times[0,1])$. Next result
asserts that $F\in C^{1,2}([0,T]\times[0,1])$. Let

$\varphi(t,x)\;:=\;\Phi(F(t,\cdot))(x)\;=\;\log\frac{F(t,x)}{1-F(t,x)}\;\;,\qquad(t,x)\in[0,T]\times[0,1]\;.$

As $F$ belongs to $C^{0,2}([0,T]\times[0,1])$ and solves (5.2), an elementary
computation yields that $\varphi\in C^{0,2}\left([0,T]\times[0,1]\right)$.
Moreover, for each $t\in[0,T]$, $\varphi(t)$ is the unique strictly increasing
(w.r.t. $x$) solution to the problem (5.12) with $\gamma=u(t)$. By (5.13),
there exists a constant $C_{1}=C_{1}(\alpha,\beta,A,B)\in(0,\infty)$ such that

$\frac{1}{C_{1}}\;\leq\;(\nabla\varphi)(t,x)\;\leq\;C_{1}\qquad\forall\,(t,x)\in[0,T]\times[0,1]\;.$
(5.14)

######  Lemma 5.7.

Fix $u\in C^{1,0}([0,T]\times[0,1])$ and let $\varphi$ be the corresponding
solution to (5.12). Then $\varphi\in C^{1,2}\left([0,T]\times[0,1]\right)$ and
for each $0\leq t<T$, $\psi:=\partial_{t}\varphi$ is the unique classical
solution to the linear boundary value problem

$\left\\{\begin{aligned}
&\nabla\Big{[}\frac{\nabla\psi}{\big{(}\nabla\varphi\big{)}^{2}}\Big{]}-\frac{e^{\varphi}}{\big{(}1+e^{\varphi}\big{)}^{2}}\,\psi=\partial_{t}u\qquad
x\in(0,1)\\\
&\nabla\psi\,=\,\frac{1}{A}\Big{\\{}(1-\alpha)\,e^{\varphi}\,+\,\alpha\,e^{-\varphi}\,\Big{\\}}\,\psi\qquad
x=0\\\
&\nabla\psi\,=\,-\,\frac{1}{B}\Big{\\{}(1-\beta)\,e^{\varphi}\,+\,\beta\,e^{-\varphi}\,\Big{\\}}\,\psi\qquad
x=1\;.\end{aligned}\right.$ (5.15)

###### Proof.

Fix $t\in[0,T]$. For $h\neq 0$ such that $t+h\in[0,T]$ let
$\psi_{h}(t,x):=h^{-1}\,[\varphi(t+h,x)-\varphi(t,x)]$. By Proposition 5.6,
$\psi_{h}(t,\cdot)\in C^{2}\left([0,1]\right)$, and, by (5.12), for
$x\in(0,1)$, $\psi_{h}$ solves

$\nabla\Big{[}\,\frac{\nabla\psi_{h}(t)}{\nabla\varphi(t)\,\nabla\varphi(t+h)}\,\Big{]}\,-\,\frac{e^{\varphi(t)}}{\big{(}1+e^{\varphi(t)}\big{)}\,\big{(}1+e^{\varphi(t+h)}\big{)}}\>\frac{e^{h\,\psi_{h}(t)}-1}{h}\;=\;u_{h}(t)$
(5.16)

where $u_{h}(t)=h^{-1}\,[u(t+h)-u(t)]$. At the boundary $x=0$,

$\nabla\psi_{h}\,=\,-\,\frac{1}{A}\,\Big{\\{}\,\alpha\,e^{-\varphi(t)}\,\frac{e^{-h\,\psi_{h}(t)}-1}{h}\;-\;(1-\alpha)\,e^{\varphi(t)}\,\frac{e^{h\,\psi_{h}(t)}-1}{h}\,\Big{\\}}\;,$

and at the boundary $x=1$,

$\nabla\psi_{h}\,=\,\frac{1}{B}\,\Big{\\{}\,\beta\,e^{-\varphi(t)}\,\frac{e^{-h\,\psi_{h}(t)}-1}{h}\;-\;(1-\beta)\,e^{\varphi(t)}\,\frac{e^{h\,\psi_{h}(t)}-1}{h}\,\Big{\\}}\;.$

Claim 1: The sequence $\psi_{h}(t)$ is relatively compact in $C([0,1])$.

To prove this claim, multiply equation (5.16) by $-\,\psi_{h}(t)$ and
integrate by parts the first term. Since
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Upsilon(a):=a\,(e^{a}-1)\geq
0$ for all $a\in{\mathbb{R}}$ and in view of the expression for
$\nabla\psi_{h}$ at the boundary, the boudary terms resulting from the
integration by parts are positive. Therefore,

$\displaystyle\int_{0}^{1}\frac{\nabla\psi_{h}(t)^{2}}{\nabla\varphi(t)\,\nabla\varphi(t+h)}\,dx\;+\;\int_{0}^{1}\frac{e^{\varphi(t)}}{\big{(}1+e^{\varphi(t)}\big{)}\,\big{(}1+e^{\varphi(t+h)}\big{)}}\,\psi_{h}(t)\,\frac{e^{h\,\psi_{h}(t)}-1}{h}\;dx$
$\displaystyle\qquad\;\leq\;-\,\int_{0}^{1}\psi_{h}(t)\,u_{h}(t)\;dx\;.$

As $\Upsilon(a)\geq 0$ for all $a\in{\mathbb{R}}$, by (5.14), there exists a
finite constant $C_{0}$, which depends only on the parameters, such that

$\int_{0}^{1}\nabla\psi_{h}(t)^{2}\,dx\;+\;\frac{1}{h^{2}}\,\int_{0}^{1}\Upsilon\big{(}\,h\,\psi_{h}(t)\,\big{)}\;dx\;\leq\;C_{0}\,\Big{|}\,\int_{0}^{1}\psi_{h}(t)\,u_{h}(t)\;dx\,\Big{|}\;.$
(5.17)

On the right-hand side, adding and subtracting
$\int_{0}^{1}\psi_{h}(t)\,dx\int_{0}^{1}u_{h}(t)\,dx$ inside the absolute
value, we estimate this term by

$C_{0}\,\|u_{h}(t)\|_{\infty}\,\Big{\\{}\,\int_{0}^{1}|\,\nabla\psi_{h}(t)\,|\;dx\;+\;\int_{0}^{1}\big{|}\,\psi_{h}(t)\,\big{|}\;dx\,\Big{\\}}$
(5.18)

for some finite constant $C_{0}$ which depends only on the parameters and may
change from line to line. We estimate each term separately. By Young’s
inequality $2ab\leq Aa^{2}+A^{-1}b^{2}$, $A>0$, the first one is bounded by

$\frac{1}{2}\,\int_{0}^{1}|\,\nabla\psi_{h}(t)\,|^{2}\;dx\;+\;C_{0}\,\|u_{h}(t)\|_{\infty}^{2}\;.$

To bound the second integral in (5.18), let
$C_{1}=C_{0}\,\|u_{h}(t)\|_{\infty}$, and $\delta>0$ be such that
$\Upsilon(a)\geq\delta a^{2}$ for $|a|\leq 1$, and
$\Upsilon(a)\geq\delta\,|a|$ for $|a|\geq 1$. Rewrite the second integral as

$\frac{C_{1}}{|h|}\,\Big{\\{}\int_{0}^{1}\big{|}\,h\psi_{h}(t)\,\big{|}\,\chi_{|\,h\,\psi_{h}(t)\,|\leq
1}\;dx\;+\;\int_{0}^{1}\big{|}\,h\psi_{h}(t)\,\big{|}\,\chi_{|\,h\,\psi_{h}(t)\,|\geq
1}\;dx\,\Big{\\}}\;,$

where
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\chi_{{\mathscr{A}}}$
stands for the indicator of the set ${\mathscr{A}}$. By Young’s inequality and
the definition of $\delta$, the previous expression is bounded by

$\frac{C_{1}}{|h|}\,\Big{\\{}\frac{A}{2}\;+\;\frac{1}{2A}\int_{0}^{1}\big{|}\,h\psi_{h}(t)\,\big{|}^{2}\,\chi_{|\,h\,\psi_{h}(t)\,|\leq
1}\;dx\;+\;\frac{1}{\delta}\int_{0}^{1}\Upsilon(h\psi_{h}(t))\;dx\,\Big{\\}}$

for all $A>0$. By definition of $\delta$ and choosing $A=|h|C_{1}/\delta$, the
previous expression is less than or equal to

$\frac{C_{1}^{2}}{2\delta}\;+\;\frac{1}{h^{2}}\Big{\\{}\frac{1}{2}\,+\,\frac{C_{1}\,|h|}{\delta}\,\Big{\\}}\int_{0}^{1}\Upsilon(h\psi_{h}(t))\;dx\;.$

Therefore, (5.18) is bounded above by

$\frac{1}{2}\,\int_{0}^{1}|\,\nabla\psi_{h}(t)\,|^{2}\;dx\;+\;C_{0}\,\|u_{h}(t)\|_{\infty}^{2}\;+\;\frac{C_{1}^{2}}{2\delta}\;+\;\frac{1}{h^{2}}\Big{\\{}\frac{1}{2}\,+\,\frac{C_{1}\,|h|}{\delta}\,\Big{\\}}\int_{0}^{1}\Upsilon(h\psi_{h}(t))\,dx\,,$

where $C_{1}=C_{0}\,\|u_{h}(t)\|_{\infty}$.

Reporting this estimate in (5.17) yields that

$\frac{1}{2}\int_{0}^{1}\nabla\psi_{h}(t)^{2}\,dx\;+\;\frac{1}{4h^{2}}\,\int_{0}^{1}\Upsilon\big{(}\,h\,\psi_{h}(t)\,\big{)}\;dx\;\leq\;C_{0}\,\Big{(}\,1\,+\,\frac{1}{2\delta}\,\Big{)}\,\|u_{h}(t)\|_{\infty}^{2}$
(5.19)

for $|h|\leq\delta/4C_{1}$.

This shows that the sequence $\psi_{h}(t)$ is uniformly Lispchitz continuous,
and thus relatively compact in $C([0,1])$, proving the assertion.

Claim 2: The sequence $\psi_{h}(t)$ converges in $C([0,1])$ to the unique
classical solution to (5.15).

Recall from Appendix B the definition of the Sobolev space
${\mathcal{H}}^{1}([0,1])$ and of the associated norm. Fix a subsequence
$(\psi_{h(k)}:k\geq 1)$, still denoted by $\psi_{h}$, which converges to a
limit, represented by $\psi$. By (5.19), $\psi$ belongs to
${\mathcal{H}}^{1}([0,1])$ and $\nabla\psi_{h}(t)$ converges weakly in
${\mathscr{L}}^{2}([0,1])$ to $\nabla\psi$.

Fix a function $v$ in ${\mathcal{H}}^{1}([0,1])$. Multiply both sides of
(5.16) by $v$ and integrate by parts to get that

$\displaystyle{\mathfrak{a}}_{h}(1)\,v(1)\,\nabla\psi_{h}(t,1)\;-\;{\mathfrak{a}}_{h}(0)\,v(0)\,\nabla\psi_{h}(t,0)$
$\displaystyle\quad-\,\int_{0}^{1}{\mathfrak{a}}_{h}\,\nabla
v\,\nabla\psi_{h}(t)\,dx\,+\,\int_{0}^{1}{\mathfrak{b}}_{h}\,v\,\frac{e^{h\,\psi_{h}(t)}-1}{h}\,dx\;=\;\int_{0}^{1}u_{h}(t)\,v\,dx\;,$

where ${\mathfrak{a}}_{h}=[\,\nabla\varphi(t)\,\nabla\varphi(t+h)\,]^{-1}$,
${\mathfrak{b}}_{h}\,=\,-\,e^{\varphi(t)}/(\,1+e^{\varphi(t)}\,)\,(\,1+e^{\varphi(t+h)}\,)$.
Replace in this equation $\nabla\psi_{h}(t,0)$, $\nabla\psi_{h}(t,1)$ by the
expressions appearing in the equations below (5.16). As ${\mathfrak{a}}_{h}$,
${\mathfrak{b}}_{h}$, $\psi_{h}$ converge in $C([0,1])$, and since
$\nabla\psi_{h}$ converges weakly to $\nabla\psi$ in
${\mathscr{L}}^{2}([0,1])$, passing to the limit in the previous equation
yields that

$\displaystyle{\mathfrak{a}}(1)\,v(1)\,{\mathfrak{c}}_{1}\,\psi(t,1)\;-\;{\mathfrak{a}}(0)\,v(0)\,{\mathfrak{c}}_{0}\,\psi(t,0)$
$\displaystyle\quad-\,\int_{0}^{1}{\mathfrak{a}}\,\nabla
v\,\nabla\psi(t)\,dx\,+\,\int_{0}^{1}{\mathfrak{b}}\,v\,\psi\,dx\;=\;\int_{0}^{1}(\partial_{t}u)(t)\,v\,dx\;,$

where
${\mathfrak{c}}_{1}\,=\,-\,B^{-1}\\{(1-\beta)\,e^{\varphi(t,1)}\,+\,\beta\,e^{-\varphi(t,1)}\,\\}$,
${\mathfrak{c}}_{0}=A^{-1}\\{(1-\alpha)\,e^{\varphi(t,0)}\,+\,\alpha\,e^{-\varphi(t,0)}\,\\}$.
Hence, according to [23, IV, Section 1], $\psi$ is a generalized solution to
(5.15). By [23, Theorem IV.1.2], the generalized solution is unique, which
proves that $\psi_{h}$ converges in $C([0,1])$ to the unique generalized
solution to (5.15). As $\partial_{t}u(t,\cdot)\in C([0,1])$, by [23, Theorem
IV.2.1], the generalized solution belongs to $C^{2}([0,1])$ and is a classical
solution to (5.15). This proves the claim.

It remains to prove the continuity $t\mapsto\psi(t,\cdot)$. According to [23,
Theorem IV.1.2], there exists a constant $C_{0}$, independent of
$\partial_{t}u$, such that $\|\psi(t)\|_{{\mathcal{H}}^{1}([0,1])}\leq
C_{0}\,\|(\partial_{t}u)(t)\|_{{\mathscr{L}}^{2}([0,1])}$. Since there exist a
finite constant $C_{0}$ such that $\|v\|_{\infty}\leq
C_{0}\|v\|_{{\mathcal{H}}^{1}([0,1])}$ for all $v\in{\mathcal{H}}^{1}([0,1])$,

$\|\,\psi(t+h)-\psi(t)\,\|_{\infty}\;\leq\;C_{0}\,\big{\|}\,(\partial_{t}u)(t+h)\,-\,(\partial_{t}u)(t)\,\big{\|}_{{\mathscr{L}}^{2}([0,1])}\;.$

This proves that $\psi$ belongs to $C^{0,2}([0,T]\times[0,1])$, and therefore
that $\varphi$ belongs to $C^{1,2}([0,T]\times[0,1])$, as claimed. ∎

###### Proof of Theorem 2.4.

For each $F\in{\mathcal{F}}$, ${\mathscr{G}}(\cdot,F)$ is a convex, lower
semi-continuous functional on ${\mathscr{M}}_{\rm ac}$. The functional
$S_{0}(\cdot)$ inherits these properties. By choosing $F=\bar{\rho}$ in (2.6)
we obtain that for every $\gamma\in{\mathscr{M}}_{\rm ac}$,

${\mathscr{G}}(\gamma,\bar{\rho})\;=\;S_{\rm
eq}(\gamma)\;-\;(1+A+B)\,\log(1+A+B)\;,$

where $S_{\rm eq}\colon{\mathscr{M}}_{\rm ac}\to{\mathbb{R}}$ is the convex
and nonnegative functional

$S_{\rm
eq}(\gamma)\;=\;\int_{0}^{1}\Big{\\{}\,\gamma(x)\,\log\frac{\gamma(x)}{\bar{\rho}(x)}+\big{[}1-\gamma(x)\big{]}\log\frac{1-\gamma(x)}{1-\bar{\rho}(x)}\,\Big{\\}}\;dx\;.$

As $S_{\rm eq}$ is non-negative,
$S_{0}(\gamma)\,\geq\,-\,(1+A+B)\,\log(1+A+B)$. On the other hand, as
$a\mapsto\log a$ is concave, by Jensen’s inequality and since $\alpha\leq
F(x)\leq\beta$, for every $F\in{\mathcal{F}}$, $\gamma\in{\mathscr{M}}_{\rm
ac}$,

${\mathscr{G}}_{\rm
bulk}(\gamma,F)\;\leq\;\log\frac{1}{\alpha}\;+\;\log\frac{1}{1-\beta}\;\cdot$

Hence, there exists a finite constant $C_{0}=C_{0}(\alpha,\beta,A,B)$ such
that $S_{0}(\gamma)\,\leq\,C_{0}$ for all $\gamma\in{\mathscr{M}}_{\rm ac}$.
This proves the first assertion of the theorem. We turn to the second.

Recall from (5.11) the definition of
$\varphi=\Phi(F)\in\widetilde{{\mathcal{F}}}$, and that
$F=\Phi^{-1}(\varphi)=e^{\varphi}/[1+e^{\varphi}]$. Set
$\widetilde{{\mathscr{G}}}_{\rm bulk}(\gamma,\varphi)={\mathscr{G}}_{\rm
bulk}(\gamma,\Phi^{-1}(\varphi))$ so that

$\displaystyle\widetilde{{\mathscr{G}}}_{\rm
bulk}(\gamma,\varphi)\;=\;\int_{0}^{1}\Big{\\{}{\mathfrak{h}}(\gamma)\,+\,(1-\gamma)\,\varphi\,-\,\log\big{[}\,1+e^{\varphi}\,\big{]}\,\,+\;\log\frac{\varphi^{\prime}}{\beta-\alpha}\Big{\\}}\;dx\;,$
(5.20)

where ${\mathfrak{h}}(a)=a\log a\,+\,(1-a)\log(1-a)$.

To prove the second assertion of the theorem, we have to show that for each
$\gamma\in{\mathscr{M}}_{\rm ac}$, the supremum over the set
$\widetilde{{\mathcal{F}}}$ of

$\widetilde{{\mathscr{G}}}(\gamma,\varphi)\;:=\;\widetilde{{\mathscr{G}}}_{\rm
bulk}(\gamma,\varphi)\;+\;A\,\ln\frac{F(0)-\alpha}{A(\beta-\alpha)}\;+\;B\,\ln\frac{\beta-F(1)}{B(\beta-\alpha)}$

is uniquely attained at $\varphi=\Phi(F(\gamma))$, where, recall, $F(\gamma)$
represents the unique solution to (5.3). In the previous equation, $F(x)$
stands for $\exp\\{\varphi(x)\\}/[1+\exp\\{\varphi(x)\\}]$, $x=0$, $1$.

Since the functions $a\mapsto\log a$, $a\mapsto-\log(1+e^{a})$,
$a\mapsto-\log\\{\,[e^{a}/(1+e^{a})]-\alpha\\}$,
$a\mapsto-\log\\{\,\beta-[e^{a}/(1+e^{a})]\,\\}$ are strictly concave, the
last two in the interval $(\varphi_{-},\varphi_{+})$ defined above (5.10), for
each $\gamma\in{\mathscr{M}}_{\rm ac}$, the functional
${\widetilde{{\mathscr{G}}}}(\gamma,\cdot)$ is strictly concave on
${\widetilde{{\mathcal{F}}}}$. Moreover it is easy to show that
${\widetilde{{\mathscr{G}}}}(\gamma,\cdot)$ is Gateaux differentiable on
${\widetilde{{\mathcal{F}}}}$ with derivative given by

$\displaystyle\Big{\langle}\frac{\delta{\widetilde{{\mathscr{G}}}}(\gamma,\varphi)}{\delta\varphi}\,,\,g\Big{\rangle}\;=\;\int_{0}^{1}\Big{\\{}\,\frac{g^{\prime}}{\varphi^{\prime}}\,+\,\Big{[}\,\frac{1}{1+e^{\varphi}}\,-\,\gamma\,\Big{]}\,g\,\Big{\\}}\;dx$
$\displaystyle\qquad+\;\frac{A\,g(0)}{(1-\alpha)(1+e^{\varphi(0)})-\alpha(1+e^{-\varphi(0)})}\;+\;\frac{B\,g(1)}{(1-\beta)(1+e^{\varphi(1)})-\beta(1+e^{-\varphi(1)})}$

for all $g$ in $C^{1}([0,1])$. By (5.12), the right-hand side vanishes for
$\varphi=\Phi(F(\gamma))$.

By [16, Proposition 1.5.4] and since
${\widetilde{{\mathscr{G}}}}(\gamma,\,\cdot\,)$ is strictly concave, for any
$\psi\neq\varphi$ in ${\widetilde{{\mathcal{F}}}}$,

${\widetilde{{\mathscr{G}}}}(\gamma,\psi)\;<\;{\widetilde{{\mathscr{G}}}}(\gamma,\varphi)\;+\;\Big{\langle}\,\frac{\delta{\widetilde{{\mathscr{G}}}}(\gamma,\varphi)}{\delta\varphi}\,,\,\psi-\varphi\,\Big{\rangle}\;.$

Since $\delta{\widetilde{{\mathscr{G}}}}(\gamma,\varphi)/{\delta\varphi}=0$
for $\varphi=\Phi(F(\gamma))$, the supremum on ${\widetilde{{\mathcal{F}}}}$
of ${\widetilde{{\mathscr{G}}}}(\gamma,\cdot)$ is uniquely attained when
$\varphi=\Phi(F(\gamma))$. ∎

######  Remark 5.8.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and consider a sequence
$\gamma_{n}\in{\mathscr{M}}_{\rm ac}$ such that

  * (i)

For each $n\geq 1$, there exists $\delta_{n}>0$ such that
$0<\delta_{n}\leq\gamma_{n}(x)\leq 1-\delta_{n}$ for all $x\in[0,1]$;

  * (ii)

$\gamma_{n}$ converges to $\gamma$ a.e.

Then, by the dominated convergence theorem and Proposition 5.6.(ii),

$\lim_{n\to\infty}S_{0}(\gamma_{n})\;=\;\lim_{n\to\infty}{\mathscr{G}}\big{(}\gamma_{n},F(\gamma_{n})\big{)}\;=\;{\mathscr{G}}\big{(}\gamma,F(\gamma)\big{)}\;=\;S_{0}(\gamma)\;.$

##  6\. The quasi-potential

Let $\delta_{0}>0$ be such that
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\delta_{0}\leq\alpha<\beta\leq
1-\delta_{0}$. For $\delta\in(0,\delta_{0}]$ and $T>0$, let

$\begin{gathered}{\mathscr{M}}_{\delta}\;:=\;\big{\\{}\gamma\in
C^{2}([0,1])\,:\,\delta\leq\gamma(x)\leq 1-\delta\,\big{\\}}\\\
D_{T,\delta}\;:=\;\big{\\{}\,u\in C^{1,2}([0,T]\times[0,1])\,:\>\delta\leq
u(t,x)\leq 1-\delta\,\big{\\}}\;.\end{gathered}$ (6.1)

Unless otherwise stated, throughout this section, $T>0$ and
$0<\delta\leq\delta_{0}$ are fixed.

######  Lemma 6.1.

Fix $u$ in $D_{T,\delta}$ and denote by $F(t,x)=F(u(t,\cdot))\,(x)$ the
solution to the boundary value problem (1.6) with $\gamma$ replaced by $u(t)$.
Set

$\Gamma(t,x)\;=\;\log\frac{u(t,x)}{1-u(t,x)}\;-\;\log\frac{F(t,x)}{1-F(t,x)}\;\cdot$
(6.2)

Then, for each $T\geq 0$,

$S_{0}\big{(}u(T)\big{)}-S_{0}\big{(}u(0)\big{)}\;=\;\int_{0}^{T}\langle\,\partial_{t}u(t)\,,\,\Gamma(t)\,\rangle\;dt\;.$
(6.3)

###### Proof.

Recall that $F(t,\cdot)$ is strictly increasing for any $t\in[0,T]$. By Lemma
5.7, $F$ belongs to $C^{1,2}\big{(}[0,T]\times[0,1]\big{)}$. By Theorem 2.4
and the dominated convergence theorem,

$\displaystyle\frac{d}{dt}S_{0}(\,u(t)\,)\;=\;\frac{d}{dt}\>{\mathscr{G}}(\,u(t),F(t)\,)$
$\displaystyle\quad\;=\;\big{\langle}\,\partial_{t}u(t)\,,\,\Gamma(t)\,\big{\rangle}\;+\;\big{\langle}\,\partial_{t}F(t),\frac{F(t)-u(t)}{F(t)[1-F(t)]}\,\big{\rangle}\;+\;\big{\langle}\,\frac{1}{\nabla
F(t)}\,,\,\partial_{t}\nabla F(t)\,\big{\rangle}$
$\displaystyle\quad\;+\;A\,\frac{(\partial_{t}F)(t,0)}{F(t,0)-\alpha}\;+\;B\,\frac{(\partial_{t}F)(t,1)}{F(t,1)-\beta}\;\cdot$

As $F$ belongs to ${\mathcal{B}}$ it satisfies mixed boundary conditions at
$x=0$, $x=1$. An integration by parts yields that the previous expression is
equal to

$\big{\langle}\,\partial_{t}u(t)\,,\,\Gamma(t)\,\big{\rangle}\;+\;\big{\langle}\,\partial_{t}F(t),\frac{F(t)-u(t)}{F(t)[1-F(t)]}\,+\,\frac{\Delta
F(t)}{\big{(}\nabla F(t)\big{)}^{2}}\,\big{\rangle}\;.$

To conclude the proof, it remains to recall Remark 5.1, which asserts that $F$
solves (1.6) almost everywhere. ∎

Recall the definition of the Hamiltonian ${\mathscr{H}}$, given in (1.2), and
the one of ${\mathscr{M}}_{\delta}$, introduced at the beginning of this
section.

######  Lemma 6.2.

Fix $\gamma\in{\mathscr{M}}_{\delta}$, and let $F=F(\gamma)$ be the solution
of the boundary value problem (1.6). Set

$\Gamma(x)\;=\;\log\frac{\gamma(x)}{1-\gamma(x)}\;-\;\log\frac{F(x)}{1-F(x)}\;\cdot$

Then,

${\mathscr{H}}(\,\gamma\,,\,\Gamma\,)\;=\;0\;.$

###### Proof.

By Corollary 5.4 and Proposition 5.6, $F\in{\mathscr{M}}_{\delta}$. By
definition of $\Gamma$ and since $F$ belongs to ${\mathcal{B}}$,

$\displaystyle{\mathfrak{b}}_{\alpha,A}\big{(}\,\gamma(0)\,,\,\Gamma(0)\,\big{)}\;+\;{\mathfrak{b}}_{\beta,B}\big{(}\,\gamma(1)\,,\,\Gamma(1)\,\big{)}$
(6.4) $\displaystyle\quad=\;[\,F(0)-\gamma(0)\,]\,\frac{(\nabla
F)(0)}{F(0)\,[1-F(0)]}\;-\;[\,F(1)-\gamma(1)\,]\,\frac{(\nabla
F)(1)}{F(1)\,[1-F(1)]}\;\cdot$

On the other hand, a straightforward computation yields that

$\displaystyle\big{\langle}\gamma(1-\gamma)\,,\,\big{(}\nabla\Gamma\big{)}^{2}\big{\rangle}\;-\;\big{\langle}\nabla\gamma\,,\,\nabla\Gamma\big{\rangle}$
$\displaystyle\quad=\;\Big{\langle}\gamma(1-\gamma)\,,\,\Big{(}\frac{\nabla
F}{F(1-F)}\Big{)}^{2}\Big{\rangle}\;-\;\Big{\langle}\nabla\gamma\,,\,\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;.$

Rewrite the second term as

$-\;\Big{\langle}\nabla(\gamma-F)\,,\,\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;-\;\Big{\langle}\nabla F\,,\,\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;,$

and integrate by parts the first expression. The boundary terms cancel with
the ones appearing in (6.4).

Up to this point, we proved that

${\mathscr{H}}(\,\gamma\,,\,\Gamma\,)\;=\;\Big{\langle}\,\gamma(1-\gamma)\,,\,\Big{(}\frac{\nabla
F}{F(1-F)}\Big{)}^{2}\,\Big{\rangle}\;+\;\Big{\langle}\gamma-F\,,\,\nabla\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;-\;\Big{\langle}\nabla F\,,\,\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;.$

Since $\gamma(1-\gamma)-F(1-F)=(\gamma-F)(1-\gamma-F)$, we may rewrite this
sum as

$\Big{\langle}\,\gamma-F\,,\,(1-\gamma-F)\Big{(}\frac{\nabla
F}{F(1-F)}\Big{)}^{2}\,\Big{\rangle}\;+\;\Big{\langle}\gamma-F\,,\,\nabla\frac{\nabla
F}{F(1-F)}\Big{\rangle}\;.$

On the other hand, as $\nabla\\{\nabla F/F(1-F)\\}=\Delta
F/F(1-F)-(1-2F)[\nabla F/F(1-F)]^{2}$, the previous expression is equal to

$\Big{\langle}\,\gamma-F\,,\,(F-\gamma)\Big{(}\frac{\nabla
F}{F(1-F)}\Big{)}^{2}\,\Big{\rangle}\;+\;\Big{\langle}\gamma-F\,,\,\frac{\Delta
F}{F(1-F)}\Big{\rangle}\;.$

This sum vanishes because $F$ is the solution to (1.6). ∎

######  Remark 6.3.

Lemma 6.1 identifies $\Gamma$ as the functional derivative of $S$,

$\Gamma\;=\;\frac{\delta S_{0}}{\delta\gamma}\;=\;\frac{\delta
S}{\delta\gamma}\;,$

and Lemma 6.2 states that this derivative $\Gamma=\delta S/\delta\gamma$
satisfies the Hamilton–Jacobi equation.

###  6.1. Lower bound for the quasi-potential

In this subsection, we prove that $V\geq S$.

######  Lemma 6.4.

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, $V(\gamma)\geq S(\gamma)$.

###### Proof.

In view of the variational definition of $V$, we have to show that
$S(\gamma)\leq I_{[0,T]}(u|\bar{\rho})$ for any $T>0$ and any path $u\in
D\big{(}[0,T];{\mathscr{M}}\big{)}$ which connects the stationary profile
$\bar{\rho}$ to $\gamma$ in the time interval $[0,T]$: $u(0)=\bar{\rho}$,
$u(T)=\gamma$.

Fix such a path $u$ and assume first that $u$ belongs to $D_{T,\delta}$ for
some $\delta>0$. For $0\leq t\leq T$, let $F(t)=F(u(t))$ be the solution to
the elliptic problem (1.6) with $u(t)$ in place of $\gamma$. In view of the
variational definition of $I_{[0,T]}(u|\bar{\rho})$ given in (2.3), to prove
that $S(\gamma)\leq I_{[0,T]}(u|\bar{\rho})$ it is enough to exhibit some
function $H\in C^{1,2}([0,T]\times[0,1])$ for which $S(\gamma)\leq
J_{T,H}(u)$. We claim that $\Gamma$ given in (6.2) fulfills these conditions.

Note that $\Gamma$ belongs to $C^{1,2}([0,T]\times[0,1])$ because, on the one
hand, $u$ belongs to this set as it is assumed to be in $D_{T,\delta}$. On the
other hand, by Lemma 5.7, $F\in C^{1,2}([0,T]\times[0,1])$.

Recall the definition of the Hamiltonian ${\mathscr{H}}$ introduced in (1.2).
By (2.2), integrating by parts in time yields that

$J_{T,\Gamma}\big{(}\,u\,\big{)}\;=\;\int_{0}^{T}\big{\\{}\,\big{\langle}\,\partial_{t}u(t)\,,\,\Gamma(t)\,\big{\rangle}\,-\,{\mathscr{H}}(\,u(t)\,,\,\Gamma(t)\,\big{)}\,\big{\\}}\;dt\;.$

By Lemmata 6.1 and 6.2,
$J_{T,\Gamma}(u)=S_{0}(u(T))\,-\,S_{0}(u(0))=S_{0}(\gamma)-S_{0}(\bar{\rho})=S(\gamma)$.

Up to this point we have shown that $S(\gamma)\leq I_{[0,T]}(u|\bar{\rho})$
for smooth paths $u$ bounded away from $0$ and $1$. We extend this result to
arbitrary paths

Fix a path $u$ with finite rate function: $I_{[0,T]}(u|\bar{\rho})<\infty$.
Since $\alpha\leq\bar{\rho}\leq\beta$, by Theorem 4.3, there exists a sequence
$\\{u^{n},\,n\geq 1\\}$, $u^{n}\in D_{T,\delta_{n}}$ for some $\delta_{n}>0$,
such that $u_{n}$ converges to $u$ and $I_{[0,T]}(u_{n}|\bar{\rho})$ converges
to $I_{[0,T]}(u|\bar{\rho})$. Therefore, by the result on smooth paths and the
lower semi continuity of $S$,

$I_{[0,T]}(u|\bar{\rho})\;=\;\lim_{n\to\infty}I_{[0,T]}(u_{n}|\bar{\rho})\geq\liminf_{n\to\infty}S\big{(}u_{n}(T)\big{)}\geq
S(u(T))\;,$

which concludes the proof of the lemma. ∎

###  6.2. Upper bound, the adjoint hydrodynamic equation

The following lemma explains which is the right candidate for the optimal path
for the variational problem (2.4). For $0<\varrho<1$, $D>0$, $0<a<1$,
$M\in{\mathbb{R}}$, let

${\mathfrak{q}}_{\varrho,D}(a,M)\;=\;\frac{1}{D}\,\Big{\\{}[1-a]\,\varrho\,\big{[}\,e^{M}\,-\,M\,-\,1\,\big{]}\;+\;a\,[1-\varrho]\,\big{[}\,e^{-M}\,+\,M\,-\,1\,\big{]}\,\Big{\\}}\;.$
(6.5)

Note that ${\mathfrak{q}}_{\varrho,D}(a,\cdot\,)$ is a nonnegative, convex
function which vanishes at the origin. Recall the definition of
${\mathfrak{p}}_{\varrho,D}(a,M)$, introduced in (4.2).

######  Lemma 6.5.

Fix a profile $\psi\in{\mathscr{M}}_{\delta}$, and a path $u\in D_{T,\delta}$
with finite rate function, $I_{[0,T]}(u|\psi)<\infty$. For $0\leq t\leq T$,
denote by $F(t)=F(u(t))$ the unique solution to the boundary value problem
(1.6) with $\phi$ replaced by $u(t)$. Then, there exists a function
$K\in{\mathcal{H}}^{1}(\Omega_{T})$ such that $u$ is the weak solution to

$\left\\{\begin{aligned} &\partial_{t}u\;=\;-\,\Delta
u\;+\;2\,\nabla\Big{(}\sigma(u)\,\nabla\big{[}\log\frac{F}{1-F}+K\big{]}\Big{)}\;,\quad(t,x)\in[0,T]\times(0,1)\\\
&\nabla u_{t}(1)\,-\,2\,\sigma(u_{t}(1))\,\nabla
G_{t}(1)\,=\,{\mathfrak{p}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,G_{t}(1)\,\big{)}\;,\\\
&\nabla u_{t}(0)\,-\,2\,\sigma(u_{t}(0))\,\nabla
G_{t}(0)\,=\,-\,{\mathfrak{p}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,G_{t}(0)\,\big{)}\;,\\\
&u(0,x)\;=\;\psi(x)\;,\quad x\in[0,1]\;.\end{aligned}\right.$ (6.6)

where $G_{t}=\Gamma_{t}-K_{t}$ and $\Gamma_{t}$ is given by (6.2). Moreover,

$\displaystyle
I_{[0,T]}(u|\psi)\;=\;S_{0}(u(T))\,-\,S_{0}(\psi)\,+\,\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,,\,[\nabla
K(t)]^{2}\,\big{\rangle}$ (6.7)
$\displaystyle\;+\;\int_{0}^{T}e^{G_{t}(1)}\,{\mathfrak{q}}_{\beta,B}(u_{t}(1),K_{t}(1))\;dt\;+\;\int_{0}^{T}e^{G_{t}(0)}\,{\mathfrak{q}}_{\alpha,A}(u_{t}(0),K_{t}(0))\;dt\;.$

Strategy of the proof of the upper bound: We present below the main steps of
the proof in light of Lemma 6.5. Fix a density profile
$\gamma\in{\mathscr{M}}_{\rm ac}$ and denote by $(v^{(\gamma)},F^{(\gamma)})$
the solution to the time-reversed equation (6.6) with $K=0$ and starting from
$\gamma$:

$\displaystyle\left\\{\begin{aligned} &\partial_{t}v\,=\,\Delta
v\,-\,2\,\nabla\big{(}\,\sigma(v)\,\nabla\log\frac{F}{1-F}\,\big{)}\quad(t,x)\in(0,\infty)\times(0,1)\;,\\\
&\nabla v_{t}(1)\,-\,2\,\sigma(v_{t}(1))\,\nabla
H_{t}(1)\,=\,{\mathfrak{p}}_{\beta,B}\big{(}\,v_{t}(1)\,,\,H_{t}(1)\,\big{)}\;,\\\
&\nabla v_{t}(0)\,-\,2\,\sigma(v_{t}(0))\,\nabla
H_{t}(0)\,=\,-\,{\mathfrak{p}}_{\alpha,A}\big{(}\,v_{t}(0)\,,\,H_{t}(0)\,\big{)}\;,\\\
&v(0,\cdot)=\gamma(\,\cdot\,)\;,\quad x\in[0,1]\;,\end{aligned}\right.$ (6.8)
$\displaystyle\left\\{\begin{aligned} &\Delta
F_{t}=\big{(}v_{t}-F_{t}\big{)}\frac{\big{(}\nabla
F_{t}\big{)}^{2}}{F_{t}(1-F_{t})}\quad(t,x)\in(0,\infty)\times(0,1)\;,\\\
&\nabla F_{t}(0)=A^{-1}[F_{t}(0)-\alpha]\;,\quad\nabla F_{t}(1)=B^{-1}[\beta-
F_{t}(1)]\;.\end{aligned}\right.$ (6.9)

In equation (6.8), $H_{t}=\log[v_{t}/(1-v_{t})]-\log[F_{t}/(1-F_{t})]$ and
${\mathfrak{p}}_{\varrho,D}$ has been introduced in (4.2). Equation (6.8) will
be shown to be equivalent to (3.5). Proposition 6.7 provides a precise meaning
to the coupled equation (6.8)–(6.9).

The second step consists in proving that the solution $v^{(\gamma)}_{t}$ of
equation (6.8) converges to $\bar{\rho}$ as $t\to\infty$. This is the content
of Lemma 6.10, where we prove that this convergence takes place in
${\mathscr{L}}^{\infty}$.

Fix $T_{1}$ large enough for $v^{(\gamma)}(T_{1})$ to be close to $\bar{\rho}$
in ${\mathscr{L}}^{\infty}$. Reverse in time the path $v^{(\gamma)}$ by
setting $w^{(1)}(t)=v^{(\gamma)}(T_{1}-t)$, $0\leq t\leq T_{1}$. The path
$w^{(1)}$ satisfies equation (6.6) with $K=0$ and $\psi=v^{(\gamma)}(T_{1})$.
Therefore, by Lemma 6.5,

$I_{[0,T_{1}]}(\,w^{(1)}\,|\,v^{(\gamma)}(T_{1})\,)\;=\;S_{0}(\gamma)\,-\,S_{0}(v^{(\gamma)}(T_{1}))\;.$

It remains to replace in the previous formula $v^{(\gamma)}(T_{1})$ by
$\bar{\rho}$, keeping in mind that $v^{(\gamma)}(T_{1})$ is close to
$\bar{\rho}$ in the ${\mathscr{L}}^{\infty}$ norm. This is done in Lemma 6.11,
where we show that if $\phi$ is close to $\bar{\rho}$ in
${\mathscr{L}}^{\infty}$, then there exists a path $w^{(2)}_{t}$, $0\leq t\leq
1$, which connects $\bar{\rho}$ to $\phi$ and such that
$I_{[0,1]}(\,w^{(2)}\,|\,\bar{\rho}\,)\leq
C_{0}\,\|\phi-\bar{\rho}\|^{2}_{2}$.

Define the path $w(t)$, $0\leq t\leq T_{1}+1$, by $w(t)=w^{(2)}(t)$ for $0\leq
t\leq 1$, $w(t)=w^{(1)}(t-1)$, $1\leq t\leq T_{1}+1$. By definition
$w(0)=\bar{\rho}$ and $w(T_{1}+1)=\gamma$. Moreover, by the previous bounds of
the rate functional, and since $w(1)=v^{(\gamma)}(T_{1})$,

$\displaystyle I_{[0,T_{1}+1]}(\,w\,|\,\bar{\rho}\,)$
$\displaystyle=\;I_{[0,1]}(\,w^{(2)}\,|\,\bar{\rho}\,)\;+\;I_{[1,T_{1}+1]}(\,w^{(1)}(1+\,\cdot\,)\,|\,v^{(\gamma)}(T_{1})\,)$
$\displaystyle\leq\;S_{0}(\gamma)\,-\,S_{0}(v^{(\gamma)}(T_{1}))\;+\;C_{0}\,\,\|v^{(\gamma)}(T_{1})-\bar{\rho}\|^{2}_{2}\;.$

The first identity says that the cost of a path in the time-interval
$[0,T_{1}+1]$ is equal to its cost in the interval $[0,1]$ plus its cost in
the interval $[1,T_{1}+1]$. Equation (4.1) states that the inequality holds,
which is enough for the argument. By lower semicontinuity of $S_{0}$, and
since $v^{(\gamma)}(T_{1})\to\bar{\rho}$ as $T_{1}\to\infty$,
$S_{0}(\bar{\rho})\leq\liminf_{T_{1}\to\infty}S_{0}(v^{(\gamma)}(T_{1}))$.
Hence, for all $\epsilon>0$, there exists $T_{1}$ large enough such that

$I_{[0,T_{1}+1]}(\,w\,|\,\bar{\rho}\,)\;\leq\;S_{0}(\gamma)\,-\,S_{0}(\bar{\rho})\;+\;\epsilon\;.$

This proves that $V(\gamma)\leq S(\gamma)$, as claimed.

###### Proof of Lemma 6.5.

Denote by $H$ the function in ${\mathcal{H}}^{1}(\Omega_{T})$ introduced in
Lemma 4.4, and recall from (6.2) the definition of $\Gamma$. Set
$K:=\Gamma-H$, so that $G=H$. The function $K$ belongs to
${\mathcal{H}}^{1}(\Omega_{T})$ because, by hypothesis, $u\in D_{T,\delta}$
and, by Lemma 5.7, $F\in C^{1,2}(\,[0,T]\times[0,1]\,)$. Then (6.6) follows
easily from (4.3).

We turn to the identity (6.7), Note that $\partial_{t}u\;=\;\Delta
u\;-\;2\,\nabla(\,\sigma(u)\,\nabla\,[\,\Gamma\,-\,K\,]\,)$. In (6.3), replace
$\partial_{t}u(t)$ by the right-hand side of this identity and integrate by
parts to get that

$\displaystyle
S_{0}(u(T))\,-\,S_{0}(\psi)\;=\;-\,\int_{0}^{T}\big{\langle}\nabla
u(t)\,,\,\nabla\Gamma(t)\big{\rangle}\;dt$
$\displaystyle\qquad+\;2\,\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,\nabla(\,\Gamma(t)-K(t)\,)\,,\,\nabla\Gamma(t)\,\big{\rangle}\;dt$
$\displaystyle\qquad+\;\int_{0}^{T}{\mathfrak{p}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Gamma_{t}(1)\,dt\;+\;\int_{0}^{T}{\mathfrak{p}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\,\Gamma_{t}(0)\;dt\;.$

By Lemma 6.2, the previous expression is equal to

$\displaystyle-\,\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,(\,\nabla\Gamma(t)\,)^{2}\,\big{\rangle}\;dt\;+\;2\,\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,\nabla(\,\Gamma(t)-K(t)\,)\,,\,\nabla\Gamma(t)\,\big{\rangle}\;dt$
$\displaystyle-\;\int_{0}^{T}{\mathfrak{b}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,\Gamma_{t}(1)\,\big{)}\;dt\;-\;\int_{0}^{T}{\mathfrak{b}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,\Gamma_{t}(0)\,\big{)}\;dt$
$\displaystyle+\;\int_{0}^{T}{\mathfrak{p}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Gamma_{t}(1)\;dt\;+\;\int_{0}^{T}{\mathfrak{p}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\,\Gamma_{t}(0)\;dt\;.$

Up to this point, we expressed the difference $S_{0}(u(T))\,-\,S_{0}(\psi)$ as
a sum of many terms. Add on both sides of this identity
$\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,(\,\nabla
K(t)\,)^{2}\,\big{\rangle}\;dt$. Add and subtract on the right-hand side

$\int_{0}^{T}{\mathfrak{c}}_{\beta,B}\big{(}\,u_{t}(1)\,,\,H_{t}(1)\,\big{)}\;dt\;+\;\int_{0}^{T}{\mathfrak{c}}_{\alpha,A}\big{(}\,u_{t}(0)\,,\,H_{t}(0)\,\big{)}\;dt\;.$

Recall that $H=\Gamma-K=G$ and, from (4.4), the identity satisfied by
$I_{[0,T]}(u|\psi)$, to get after these summations that

$\displaystyle
S_{0}(u(T))\;-\;S_{0}(\psi)\;+\;\int_{0}^{T}\big{\langle}\,\sigma(u(t))\,(\,\nabla
K(t)\,)^{2}\,\big{\rangle}\;dt$
$\displaystyle\;=\;I_{[0,T]}(u|\psi)\;-\;\int_{0}^{T}e^{H_{t}(1)}\,{\mathfrak{q}}_{\beta,B}(u_{t}(1),K_{t}(1))\;dt\;-\;\int_{0}^{T}e^{H_{t}(0)}\,{\mathfrak{q}}_{\alpha,A}(u_{t}(0),K_{t}(0))\;dt\;,$

as claimed [because $H=G$]. ∎

We turn to the proof of the upper bound for the quasi-potential, as described
in the Strategy of the proof. We first simplify the boundary conditions in
equations (6.8)–(6.9).

As $H_{t}=\log[v_{t}/(1-v_{t})]-\log[F_{t}/(1-F_{t})]$, the boundary terms are
given by

$\begin{gathered}-\,\nabla
v_{t}\,+\,2\,\frac{\sigma(v_{t})}{\sigma(F_{t})}\,\nabla
F_{t}\,=\,\frac{1}{B\,\sigma(F_{t})}\,\big{\\{}\,\beta\,v_{t}\,[\,1-F_{t}\,]^{2}\,-\,(1-\beta)\,(1-v_{t})\,F_{t}^{2}\,\big{\\}}\;,\\\
-\,\nabla v_{t}\,+\,2\,\frac{\sigma(v_{t})}{\sigma(F_{t})}\,\nabla
F_{t}\,=\,\frac{-\,1}{A\,\sigma(F_{t})}\,\big{\\{}\,\alpha\,v_{t}\,[\,1-F_{t}\,]^{2}\,-\,(1-\alpha)\,(1-v_{t})\,F_{t}^{2}\,\big{\\}}\;,\end{gathered}$
(6.10)

for $x=1$, $0$, respectively. Let
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}R=\log[F/(1-F)]$.
With this notation, equation (6.8) can be written as

$\left\\{\begin{aligned} &\partial_{t}v\,=\,\Delta
v\,-\,2\,\nabla\big{(}\,\sigma(v)\,\nabla
R\,\big{)}\quad(t,x)\in(0,\infty)\times(0,1)\;,\\\ &\nabla
v_{t}(1)\,-\,2\,\sigma(v_{t}(1))\,\nabla
R_{t}(1)\,=\,{\mathfrak{p}}_{1-\beta,B}\big{(}\,v_{t}(1)\,,\,R_{t}(1)\,\big{)}\;,\\\
&\nabla v_{t}(0)\,-\,2\,\sigma(v_{t}(0))\,\nabla
R_{t}(0)\,=\,-\,{\mathfrak{p}}_{1-\alpha,A}\big{(}\,v_{t}(0)\,,\,R_{t}(0)\,\big{)}\;,\\\
&v(0,\cdot)=\gamma\;,\quad x\in[0,1]\;,\end{aligned}\right.$ (6.11)

where $F_{t}$ is the solution to (6.9). Note that this equation corresponds to
equation (3.5).

Proposition 6.7 provides a precise meaning for the system of equations
(6.8)–(6.9), or, equivalently, (6.11)–(6.9). This proposition requires an
estimate on the solutions to equation (1.1). Denote by $u^{(\gamma)}$,
$\gamma\in{\mathscr{M}}_{\rm ac}$, the solution to (1.1) with initial
conditions $\gamma$, and by $F=F(\gamma)$ the solution to (1.6). Recall the
definition of the constant $p$ and $q$ introduced in (5.5).

######  Lemma 6.6.

For every $\gamma\in{\mathscr{M}}_{\rm ac}$,
$(t,x)\in{\mathbb{R}}_{+}\times[0,1]$,
$\alpha\,+\,Ap\,\leq\,u^{(F(\gamma))}(t,x)\,\leq\,\beta\,-\,Bp$. Moreover, for
every $T>0$, there exists a constant $c_{1}=c_{1}(A,B,\alpha,\beta,T)>0$ such
that $c_{1}\leq\nabla u^{(F(\gamma))}(t,x)\leq c^{-1}_{1}$ for all
$\gamma\in{\mathscr{M}}_{\rm ac}$, $(t,x)\in[0,T]\times[0,1]$.

###### Proof.

By (5.6) and Corollary 5.4,
$\alpha\,+\,Ap\,\leq\,F(\gamma)(x)\,\leq\,\beta\,-\,Bp$ for all
$\gamma\in{\mathscr{M}}_{\rm ac}$, $x\in[0,1]$. The first assertion of the
lemma follows from Theorem B.4.

By Corollary 5.4, $F=F(\gamma)$ belongs to ${\mathcal{B}}$. Therefore,
$p\leq\nabla F(x)\leq q$ for all $0\leq x\leq 1$, $\gamma\in{\mathscr{M}}_{\rm
ac}$. Let $v=\nabla u^{(F(\gamma))}$. Then, $v$ solves the equation

$\begin{cases}\partial_{t}v\,=\,\Delta v\\\
v(t,0)\,=\,A^{-1}\,[\,u^{(F(\gamma))}(t,0)-\alpha\,]\\\
v(t,1)\,=\,B^{-1}\,[\beta\,-\,u^{(F(\gamma))}(t,1)\,]\\\ v(0,\cdot)\,=\,\nabla
F(\cdot)\;.\end{cases}$ (6.12)

The maximum principle, Theorem 2 of [24, Chapter 3], states that the maximum
and the minimum of $v$ are attained at the boundary. The assertion of the
lemma follows from the bounds on $u^{(F(\gamma))}$ and $\nabla F$ obtained
above. These estimates are uniform over $\gamma\in{\mathscr{M}}_{\rm ac}$. ∎

######  Proposition 6.7.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and denote by
$F^{(\gamma)}(t)=u^{(F(\gamma))}(t)$ the solution to the heat equation (1.1)
with initial condition $F^{(\gamma)}(0,\,\cdot\,)=F(\gamma)(\,\cdot\,)$.
Define $v^{(\gamma)}=v^{(\gamma)}(t,x)$ by (3.8). Then,
$v^{(\gamma)}(0,\,\cdot\,)=\gamma(\,\cdot\,)$, $v^{(\gamma)}$ is smooth in
$(0,\infty)\times[0,1]$ and $(v^{(\gamma)},F^{(\gamma)})$ satisfies (6.9),
(6.11) in $(0,\infty)\times[0,1]$.

###### Proof.

Fix $\gamma\in{\mathscr{M}}_{\rm ac}$, and let $F(\gamma)$ be the solution of
(1.6). By Corollary 5.4, $F(\gamma)$ belongs to $C^{1}([0,1])$ and there is a
constant $c_{0}\in(0,\infty)$, depending only on the parameters, such that
$c_{0}\leq[\nabla F(\gamma)]\,(x)\leq c^{-1}_{0}$ for all $x\in[0,1]$.

Let $F^{(\gamma)}(t)$ be the solution to (1.1) with initial condition
$F^{(\gamma)}(0)=F(\gamma)$. By Theorem B.4, $F^{(\gamma)}$ is smooth in
$(0,\infty)\times[0,1]$. By Lemma 6.6, for every $T>0$, there exists
$c_{1}=c_{1}(A,B,\alpha,\beta,T)\in(0,\infty)$ such that $c_{1}\leq(\nabla
F^{(\gamma)})(t,x)\leq c^{-1}_{1}$ for all $(t,x)\in[0,T]\times[0,1]$.

Define $v^{(\gamma)}_{t}$ by equation (3.8) which we reproduce here:

$v^{(\gamma)}_{t}\;=\;F^{(\gamma)}_{t}\;+\;\sigma(\,F^{(\gamma)}_{t}\,)\;\frac{\Delta
F^{(\gamma)}_{t}}{(\nabla F^{(\gamma)}_{t})^{2}}\;,\quad t\,\geq\,0\;.$ (6.13)

For $t=0$, we may replace on the right-hand side $F^{(\gamma)}_{0}$ by
$F(\gamma)$ to get that $v^{(\gamma)}_{0}=\gamma$, as claimed.

In view of the regularity and the bounds obtained for $F^{(\gamma)}$ in the
previous paragraph, $v^{(\gamma)}$ is smooth in $(0,\infty)\times[0,1]$.
Moreover, by (6.13), the pair $(v^{(\gamma)},F^{(\gamma)})$ satisfies (6.9).

It remains to show that $v^{(\gamma)}$ fullfils (6.11).

Claim 1: The function $v^{(\gamma)}_{t}$ complies with the boundary conditions
of (6.11).

We prove this assertion for $x=1$, the other one being similar. By (6.13), at
the boundary, as $\nabla F^{(\gamma)}_{t}(1)=[\beta-F^{(\gamma)}_{t}(1)]/B$,
taking a time-derivative on both sides of the identity yields that
$(\nabla^{3}F^{(\gamma)}_{t})(1)=-\,(1/B)\,(\Delta F^{(\gamma)}_{t})(1)$
because $\partial_{t}F^{(\gamma)}=\Delta F^{(\gamma)}$.

Compute, separately, the right and left-hand sides of the right boundary
condition in (6.11). We start with the left-hand side of the identity. Since
$(\nabla^{3}F^{(\gamma)}_{t})(1)=-(1/B)(\Delta F^{(\gamma)}_{t})(1)$, taking a
space derivative in (6.13) yields that, at $x=1$,

$-\,\nabla v^{(\gamma)}\;=\;-\,\nabla
F^{(\gamma)}\,-\,(1-2F^{(\gamma)})\,\frac{\Delta F^{(\gamma)}}{\nabla
F^{(\gamma)}}\;+\;\frac{\sigma(F^{(\gamma)})}{B}\,\frac{\Delta
F^{(\gamma)}}{(\nabla
F^{(\gamma)})^{2}}\;+\;2\,\sigma(F^{(\gamma)})\,\frac{(\Delta
F^{(\gamma)})^{2}}{(\nabla F^{(\gamma)})^{3}}\;\cdot$

On the other hand, by (6.13) and a straightforward computation,

$2\,\frac{\sigma(v^{(\gamma)})}{\sigma(F^{(\gamma)})}\,\nabla
F^{(\gamma)}\;=\;2\,\nabla
F^{(\gamma)}\,\Big{\\{}\,1\,+\,(1-2F^{(\gamma)})\,\frac{\Delta
F^{(\gamma)}}{(\nabla
F^{(\gamma)})^{2}}\,-\,\sigma(F^{(\gamma)})\,\frac{(\Delta
F^{(\gamma)})^{2}}{(\nabla F^{(\gamma)})^{4}}\,\Big{\\}}\;.$

Summing the previous two identities yields that, at $x=1$,

$-\,\nabla
v^{(\gamma)}\;+\;2\,\frac{\sigma(v^{(\gamma)})}{\sigma(F^{(\gamma)})}\,\nabla
F^{(\gamma)}\;=\;\nabla F^{(\gamma)}\;+\;\\{\,\beta\,-\,2\beta
F^{(\gamma)}\,+\,(F^{(\gamma)})^{2}\,\\}\,\frac{\Delta
F^{(\gamma)}}{B\,(\nabla F^{(\gamma)})^{2}}\;\cdot$

A simple calculation gives that the right-hand side of the right boundary
condition in (6.11) is also equal to this quantity. This proves Claim 1.

Claim 2: The function $v^{(\gamma)}_{t}$ fullfils (6.11) in the interior.

The proof of this claim is identical to the one presented in [3, Appendix B]
and in [4, Lemma 5.5]. We reproduce it here in sake of completeness.

From (6.13),

$\frac{v^{(\gamma)}(1-v^{(\gamma)})}{F^{(\gamma)}(1-F^{(\gamma)})}=1+(1-2F^{(\gamma)})\frac{\Delta
F^{(\gamma)}}{\big{(}\nabla
F^{(\gamma)}\big{)}^{2}}-F^{(\gamma)}(1-F^{(\gamma)})\frac{\big{(}\Delta
F^{(\gamma)}\big{)}^{2}}{\big{(}\nabla F^{(\gamma)}\big{)}^{4}}\;\cdot$

As $F^{(\gamma)}$ solves the heat equation (1.1), a long computation yields
that

$\big{(}\,\partial_{t}\,-\,\Delta\,\big{)}\,\Big{(}\,\sigma(F^{(\gamma)})\,\frac{\Delta
F^{(\gamma)}}{\big{(}\nabla
F^{(\gamma)}\big{)}^{2}}\,\Big{)}\;=\;-\,2\,\nabla\Big{(}\frac{\sigma(v^{(\gamma)})}{\sigma(F^{(\gamma)})}\,\nabla
F^{(\gamma)}\Big{)}\;.$

By (6.13), $v^{(\gamma)}$ satisfies the differential equation in (6.11), as
claimed. ∎

######  Lemma 6.8.

Under the hypotheses of Proposition 6.7, assume that $\gamma$ belongs to
$C^{2}([0,1])$. Then, $v^{(\gamma)}$ belongs to
$C^{1,2}([0,\infty)\times[0,1])\cap C([0,\infty);{\mathscr{M}}_{\rm ac})$.

###### Proof.

Assume that $\gamma$ belongs to $C^{2}([0,1])$. By Remark 5.1 and (1.6),
$\Delta F(\gamma)$ belongs to $C^{2}([0,1])$. Taking time derivatives in (1.1)
yields that $\Delta F^{(\gamma)}(t)$ is the solution to (B.2) with initial
condition $\Delta F(\gamma)$. By Theorem B.2 (a), $(t,x)\mapsto\Delta
F^{(\gamma)}(t,x)$ belongs to $C^{1,2}([0,\infty)\times[0,1])$. Therefore, by
(6.13), $v^{(\gamma)}$ belongs to $C^{1,2}([0,\infty)\times[0,1])$

Claim 1: The function $v^{(\gamma)}_{t}$ belongs to
$C({\mathbb{R}}_{+},{\mathscr{M}}_{\rm ac})$.

We have to show that $0\leq v^{(\gamma)}(t,x)\leq 1$ for all $(t,x)$. In view
of (3.9), equation (6.11) describes the macroscopic evolution of the density
for weakly asymmetric boundary driven exclusion processes with weak boundary
interaction. The drift is given by $\nabla R_{t}$ where
$R_{t}=\log[F^{(\gamma)}_{t}/(1-F^{(\gamma)}_{t})]$. By the first part of the
proof and Lemma 6.6, $R_{t}$ belongs to $C^{1,2}([0,\infty)\times[0,1])$.

As $v^{(\gamma)}_{0}=\gamma$ and $0\leq\gamma\leq 1$, by the hydrodynamic
limit of theses systems, derived in [19], this equation has a weak solution
taking values in the interval $[0,1]$, cf. Theorem B.8. Since $v^{(\gamma)}$
belongs to $C^{1,2}([0,\infty)\times[0,1])$ and solves (6.11) pointwisely, it
is a weak solution to equation (6.11) in the sense of Definition B.7.
Therefore, by the uniqueness of weak solutions of (6.11), Theorem B.8,
$v^{(\gamma)}$ coincides with solution obtained in the proof of the
hydrodynamic limit which takes values in $[0,1]$. This completes the proof of
the lemma. ∎

######  Lemma 6.9.

Under the hypotheses of Proposition 6.7, assume that $\delta\leq\gamma(x)\leq
1-\delta$ a.e. for some $\delta>0$. Then, there exists
$\delta^{\prime}=\delta^{\prime}(A,B,\alpha,\beta,\delta)\in(0,1)$, such that
$\delta^{\prime}\leq v^{(\gamma)}(t,x)\leq 1-\delta^{\prime}$ for all
$(t,x)\in(0,\infty)\times[0,1]$.

###### Proof.

The proof is divided in several assertions. Fix $t>0$.

Claim 1: If $v^{(\gamma)}(t,\cdot)$ has a local maximum at $x_{0}\in(0,1)$ and
$v^{(\gamma)}(t,x_{0})>1-\alpha$, then $\partial_{t}v^{(\gamma)}(t,x_{0})<0$.

Assume that $v^{(\gamma)}(t,\cdot)$ has a local maximum at $x_{0}\in(0,1)$.
Since $v^{(\gamma)}$ is a smooth solution to (6.8), $(\nabla
v^{(\gamma)})(t,x_{0})=0$. By (6.13) and a straightforward computation,
$\Delta\log\\{F^{(\gamma)}/(1-F^{(\gamma)})\\}=(v^{(\gamma)}+F^{(\gamma)}-1)\,(\nabla
F^{(\gamma)})^{2}/\sigma(F^{(\gamma)})^{2}$. Therefore, by (6.11), at the
point $(t,x_{0})$,

$\partial_{t}v^{(\gamma)}\;=\;\Delta
v^{(\gamma)}\;-\;2\,\sigma(v^{(\gamma)})\,(v^{(\gamma)}+F^{(\gamma)}-1)\,\frac{\,(\nabla
F^{(\gamma)})^{2}}{\sigma(F^{(\gamma)})^{2}}\;\cdot$

As $x_{0}$ is a local maximum, $\Delta v^{(\gamma)}\leq 0$. On the other hand,
since $v^{(\gamma)}(t,x_{0})>1-\alpha$, and, by Lemma 6.6, $\alpha\leq
F^{(\gamma)}$, $v^{(\gamma)}+F^{(\gamma)}-1>0$ so that
$\partial_{t}v^{(\gamma)}<0$, which proves the claim.

The same argument shows that $(\partial_{t}v^{(\gamma)})(t,x_{1})>0$ if
$x_{1}\in(0,1)$ is a minimum of $v^{(\gamma)}(t,\cdot)$ and
$v^{(\gamma)}(t,x_{1})<1-\beta$.

We turn to the possibility that the maximum is attained at the boundary. By
Lemma 6.6, $F^{(\gamma)}$ takes value in the interval $[\alpha+pA,\beta-pB]$.
Let

$\nu\;:=\;\max_{\alpha+pA\leq\varphi\leq\beta-
pB}\frac{(1-\beta)\varphi^{2}}{\beta-2\beta\varphi+\varphi^{2}}\;<\;1\;.$

As $\nu<1$, there exists $a>0$ such that $\nu/(1-2a)<1$ and $aBp<1$.

Claim 2: If
$v^{(\gamma)}_{t}(1)\,>\,{\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\nu_{r}\,:=\,\max\\{\nu/(1-2a)\,,\,1-aBp\\}}$,
then $\nabla v^{(\gamma)}_{t}(1)<0$.

Since $v^{(\gamma)}$ solves equation (6.11), in view of the definition of $R$,
the first equation in (6.10) holds with $v^{(\gamma)}$, $F^{(\gamma)}$ in
place of $v$, $F$, respectively. At the boundary, $\nabla
F^{(\gamma)}_{t}(1)=(1/B)[\beta-F^{(\gamma)}_{t}(1)]$. Therefore,

$\displaystyle\nabla v^{(\gamma)}_{t}$
$\displaystyle=\,\frac{1}{B\,\sigma(F^{(\gamma)}_{t})}\,\Big{\\{}2\,\sigma(v^{(\gamma)}_{t})\,[\beta-F^{(\gamma)}_{t}]\,+\,(1-\beta)\,(1-v^{(\gamma)}_{t})\,[F^{(\gamma)}_{t}]^{2}\,-\,\beta\,v^{(\gamma)}_{t}\,[\,1-F^{(\gamma)}_{t}\,]^{2}\,\Big{\\}}$
$\displaystyle=\,\frac{-1}{B\,\sigma(F^{(\gamma)}_{t})}\,\Big{\\{}\,v^{(\gamma)}_{t}\,\Big{[}\,\\{\,\beta-2\beta
F^{(\gamma)}_{t}+[F^{(\gamma)}_{t}]^{2}\,\\}\,-\,2\,(1-v^{(\gamma)}_{t})\,(\beta-F^{(\gamma)}_{t})\,\Big{]}\,-\,(1-\beta)\,[F^{(\gamma)}_{t}]^{2}\,\Big{\\}}\;.$

In these formulas, we wrote $v^{(\gamma)}_{t}$, $F^{(\gamma)}_{t}$ for
$v^{(\gamma)}_{t}(1)$, $F^{(\gamma)}_{t}(1)$, respectively. As
$v^{(\gamma)}_{t}(1)>1-a\,B\,p$ and $Bp\leq\beta-F^{(\gamma)}_{t}$ , the last
term is bounded by

$\displaystyle\frac{-1}{B\,\sigma(F^{(\gamma)}_{t})}\,\Big{\\{}\,v^{(\gamma)}_{t}\,\Big{[}\,\\{\,\beta-2\beta
F^{(\gamma)}_{t}+[F^{(\gamma)}_{t}]^{2}\,\\}\,-\,2\,a\,(\beta-F^{(\gamma)}_{t})^{2}\,\Big{]}\,-\,(1-\beta)\,[F^{(\gamma)}_{t}]^{2}\,\Big{\\}}$
$\displaystyle\quad\leq\;\frac{-1}{B\,\sigma(F^{(\gamma)}_{t})}\,\Big{\\{}\,(1-2a)\,v^{(\gamma)}_{t}\,\\{\,\beta-2\beta
F^{(\gamma)}_{t}+[F^{(\gamma)}_{t}]^{2}\,\\}\,-\,(1-\beta)\,[F^{(\gamma)}_{t}]^{2}\,\Big{\\}}\;.$

This expression is negative by definition of $\nu$ and because we assumed that
$v^{(\gamma)}_{t}(1)>\nu/(1-2a)$. This proves Claim 2.

Let

$\mu_{r}\;:=\;\min_{\alpha+pA\leq\varphi\leq\beta-
pB}\frac{(1-\beta)\varphi^{2}}{\beta-2\beta\varphi+\varphi^{2}}\;>\;0\;.$

Claim 3: If $v^{(\gamma)}_{t}(1)\,<\,\mu_{r}$, then $\nabla
v^{(\gamma)}_{t}(1)>0$.

Recall the formula for $\nabla v^{(\gamma)}_{t}(1)$ presented at the beginning
of the proof of Claim 2. Since
$2(1-v^{(\gamma)}_{t})\,(\beta-F^{(\gamma)}_{t})\,>\,0$ and
$v^{(\gamma)}_{t}(1)\,<\,\mu_{r}$,

$\nabla
v^{(\gamma)}_{t}\,>\,\frac{1}{B\,\sigma(F^{(\gamma)}_{t})}\,\Big{\\{}\,(1-\beta)\,[F^{(\gamma)}_{t}]^{2}\,-\,\mu_{r}\,\\{\,\beta-2\beta
F^{(\gamma)}_{t}+[F^{(\gamma)}_{t}]^{2}\,\\}\,\Big{\\}}\;.$

As $F^{(\gamma)}_{t}$ takes values in the interval $[\alpha+pA,\beta-pB]$, the
right-hand side is non-negative by definition of $\mu_{r}$. This proves Claim
3.

Similarly, one can be prove the existence of $\mu_{l}>0$ and $\nu_{l}<1$ such
that $\nabla v^{(\gamma)}_{t}(0)>0$ if $v^{(\gamma)}_{t}(0)\,>\,\nu_{l}$;
$\nabla v^{(\gamma)}_{t}(0)<0$ if $v^{(\gamma)}_{t}(0)\,<\,\mu_{l}$.

This result together with Claims 1, 2, 3, yield that
$\min\\{\mu_{l}\,,\,\mu_{r}\,,\,\delta\,,\,1-\beta\,\\}\,\leq\,v^{(\gamma)}(t,x)\,\leq\,\max\\{\nu_{l}\,,\,\nu_{r}\,,\,1-\delta\,,\,1-\alpha\,\\}$
for all $(t,x)\in{\mathbb{R}}_{+}\times[0,1]$, which concludes the proof of
the lemma. ∎

Next result states that the solution to (6.8), as constructed in Proposition
6.7, converges to $\bar{\rho}$, as $t\to\infty$, uniformly with respect to the
initial condition $\gamma$.

######  Lemma 6.10.

Let $v^{(\gamma)}$, $\gamma\in{\mathscr{M}}_{\rm ac}$, be given by (6.13).
Then,

$\lim_{t\rightarrow\infty}\sup_{\gamma\in{\mathscr{M}}_{\rm
ac}}\;\big{\|}\,v^{(\gamma)}(t)-\bar{\rho}\,\big{\|}_{\infty}\;=\;0\;.$

###### Proof.

Write the solution $F^{(\gamma)}(t)$ of (1.1) as
$F^{(\gamma)}(t,x)=\bar{\rho}(x)+\Psi^{(\gamma)}(t,x)$. Then,
$\Psi^{(\gamma)}$ solves the equation (B.2) with
$\phi=F(\gamma)\,-\,\bar{\rho}$. In particular, by Theorem B.2,
$\Psi^{(\gamma)}(t)$ can be represented as
$\Psi^{(\gamma)}(t)=P^{(R)}_{t}\Psi^{(\gamma)}(0)$. Since
$\Psi^{(\gamma)}(0)=F(\gamma)-\bar{\rho}$ and the solution $F(\gamma)$ of
(1.6) as well as $\bar{\rho}$ are contained in the interval $[\alpha,\beta]$,
we have that $\|\Psi^{(\gamma)}(0)\|_{\infty}\leq\beta-\alpha<1$, uniformly
over $\gamma\in{\mathscr{M}}_{\rm ac}$. Therefore, expressing
$\Psi^{(\gamma)}(t)=P^{(R)}_{t}\Psi^{(\gamma)}(0)$ in the basis $(f_{j}:j\geq
1)$ by standard arguments (see [26, Corollary 2 of Section 4.4]),

$\lim_{t\to\infty}\sup_{\gamma\in{\mathscr{M}}_{\rm
ac}}\;\|\Psi^{(\gamma)}(t)\|_{\infty}\;=\;0\;.$

Taking a time derivative at the boundary yields that
$\Delta\Psi^{(\gamma)}(t)$ also solves equation (B.2). By (1.6), as
$F(\gamma)$ belongs to ${\mathcal{B}}$,
$\|\Delta\Psi^{(\gamma)}(0)\|_{\infty}\leq C_{0}(A,B,\alpha,\beta)$. Thus, as
$\Delta\bar{\rho}=0$, by the same argument,

$\lim_{t\to\infty}\sup_{\gamma\in{\mathscr{M}}_{\rm
ac}}\;\|\Delta\Psi^{(\gamma)}(t)\|_{\infty}\;=\;0\;.$

Expressing the derivative $(\nabla\Psi^{(\gamma)})(t,x)$ as
$(\nabla\Psi^{(\gamma)})(t,0)+\int_{0}^{x}(\Delta\Psi^{(\gamma)})(t,y)\,dy$,
and since $(\nabla\Psi^{(\gamma)})(t,0)=A^{-1}\,\Psi^{(\gamma)}(t,0)$, we
deduce from the two previous results that

$\lim_{t\to\infty}\sup_{\gamma\in{\mathscr{M}}_{\rm
ac}}\;\|(\nabla\Psi^{(\gamma)})(t)\|_{\infty}\;=\;0\;.$

By (6.13), the assertion of the lemma follows from the previous estimates. ∎

Lemma 6.10 shows that we may join a profile $\gamma$ in ${\mathscr{M}}_{\rm
ac}$ to a neighborhood of the stationary profile by using the equation (6.8)
for a time interval $[0,T_{1}]$ which at the same time regularizes the
profile. As described in the Strategy of the proof, it remains to connect
$v^{(\gamma)}(T_{1})$ to $\bar{\rho}$. In the next lemma we show that this can
be done by paying only a small price. Denote by $\|\,\cdot\,\|_{2}$ the norm
in ${\mathscr{L}}^{2}([0,1])$, and recall the definition of $\delta_{0}$,
given at the beginning of this section, and of the set $D_{T,\delta}$,
introduced in (6.1). In the lemma below, $\lambda_{1}$ represents the smallest
eigenvalue of the Robin Laplacian (cf. Appendix A).

######  Lemma 6.11.

Let $\gamma\in{\mathscr{M}}_{\rm ac}$ be a smooth profile such that
$\|\gamma-\bar{\rho}\|_{\infty}\leq\delta_{0}\,\min\\{(1/4)$,
$(1/\Lambda)\\}$, where $\Lambda=16\sqrt{A/\lambda_{1}}$. Then, there exist a
smooth path $w(t)$, $t\in[0,1]$, with $\delta_{0}/2\leq w(t)\leq
1-\delta_{0}/2$, $w(0)=\bar{\rho}$, $w(1)=\gamma$ and a finite constant
$C_{0}=C_{0}(\delta_{0})$ such that

$I_{[0,1]}(w|\bar{\rho})\;\leq\;C_{0}\,\|\gamma-\bar{\rho}\|_{2}^{2}\;.$

In particular, for profiles $\gamma$ satisfying the hypotheses of this lemma,
$V(\gamma)\,\leq\,C_{0}\,\|\gamma-\bar{\rho}\|_{2}^{2}$.

The “straight path” $w(t)=\bar{\rho}\,(1-t)+\gamma\,t$ yields a bound in terms
of the ${\mathcal{H}}_{1}([0,1])$ norm of $\gamma-\bar{\rho}$. In contrast,
the path below, similar to the one proposed in [4], provides a bound in terms
of the ${\mathscr{L}}^{2}$ norm. We assume in the proof below that the reader
is familiar with the notation and results presented in Appendix A.

###### Proof.

Recall that we denote by $\\{f_{j}:j\geq 1\\}$ the orthonormal eigenfunctions
of the Robin Laplacian and by $\lambda_{j}$ the associated eigenvalues.

We claim that the path $w(t)=w(t,x)$, $(t,x)\in[0,1]\times[0,1]$ given by

$w(t)\;=\;\bar{\rho}\;+\;\sum_{k\geq
1}\frac{e^{\lambda_{k}t}-1}{e^{\lambda_{k}}-1}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle\,f_{k}$
(6.14)

fulfills the conditions stated in the lemma. By [26, Corollary 2 of Section
4.4], this sum is absolutely convergent, uniformly in $x\in[0,1]$.

Clearly, $w(0)=\bar{\rho}$, $w(1)=\gamma$ and $w$ satisfies the boundary
conditions

$(\nabla w)(t,0)\,=\,A^{-1}\,[w(t,0)-\alpha]\;,\quad(\nabla
w)(t,1)\,=\,B^{-1}[\beta-w(t,1)]\;.$ (6.15)

By the smoothness assumption on $\gamma$, $w\in C^{1,2}([0,1]\times[0,1])$.

In order to show that $\delta_{0}/2\leq w\leq 1-\delta_{0}/2$, write $w(t)$ as
$w(t)=\bar{\rho}+q(-t)$. Clearly, $q(t)=q(t,x)$, $(t,x)\in[-1,0]\times[0,1]$
solves the equation

$\left\\{\begin{aligned} &\partial_{t}q(t)\;=\;\Delta q(t)\,-\,g\\\ &(\nabla
q)(t,0)\,=\,A^{-1}\,q(t,0)\;,\\\ &(\nabla q)(t,1)\,=\,-\,B^{-1}q(t,1)\;,\\\
&q(-1)=\gamma-\bar{\rho}\;,\end{aligned}\right.$ (6.16)

where $g=g(x)$ is given by

$g\;=\;\sum_{k\geq
1}\frac{\lambda_{k}}{e^{\lambda_{k}}-1}\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle\,f_{k}\;.$

Recall the definition of the ${\mathcal{H}}_{R}$-norm
$\|g\|_{{\mathcal{H}}_{R}}$ induced by the Robin Laplacian. By definition of
$g$ and since $a^{2}\leq 2(e^{a}-1)$, $a>0$,
$\|\gamma-\bar{\rho}\|_{\infty}\leq\delta_{0}/\Lambda$,

$\displaystyle\|g\|_{{\mathcal{H}}_{R}}^{2}$ $\displaystyle=\;\sum_{k\geq
1}\lambda_{k}\,\Big{(}\frac{\lambda_{k}}{e^{\lambda_{k}}-1}\Big{)}^{2}\,\langle\gamma-\bar{\rho},f_{k}\rangle^{2}\;\leq\;\frac{4}{\lambda_{1}}\sum_{k\geq
1}\langle\gamma-\bar{\rho},f_{k}\rangle^{2}$
$\displaystyle=\;\frac{4}{\lambda_{1}}\,\|\gamma-\bar{\rho}\|_{2}^{2}\;\leq\;\frac{4}{\lambda_{1}}\frac{\delta_{0}^{2}}{\Lambda^{2}}\;\cdot$

The solution $q(t)$ of (6.16) can be expressed as
$P_{t+1}^{(R)}(\gamma-\bar{\rho})+\int_{-1}^{t}P^{(R)}_{t-s}g\,ds$. Therefore,
since, by hypothesis, $\|\,\gamma-\bar{\rho}\,\|_{\infty}\leq\delta_{0}/4$, by
(B.4), (A.11) and the previous bound,

$\sup_{t\in[-1,0]}\|q(t)\|_{\infty}\;\leq\;\|\,\gamma-\bar{\rho}\,\|_{\infty}\;+\;\|\,g\,\|_{\infty}\;\leq\;\frac{\delta_{0}}{4}\;+\;\sqrt{2(A\vee
1)}\,\frac{2}{\sqrt{\lambda_{1}}}\,\frac{\delta_{0}}{\Lambda}\;\cdot$

In particular, by definition of $\Lambda$, $w$ belongs to
$D_{1,\delta_{0}/2}$.

We turn to the cost of the path $w$. Since $w$ is a smooth path such that
$w(0)=\bar{\rho}$, in formula (2.1), integrate by parts twice in space and
once in time to get that

$\displaystyle J_{1,H}(w)$
$\displaystyle=\;\int_{0}^{1}\big{\langle}\partial_{t}w_{t}\,-\,\Delta
w_{t}\,,\,H_{t}\big{\rangle}\;dt\;-\;\int_{0}^{1}\big{\langle}\sigma(w_{t}),\big{(}\nabla
H_{t}\big{)}^{2}\big{\rangle}\;dt$ $\displaystyle-\;\int_{0}^{1}\nabla
w_{t}(1)\,H_{t}(1)\;dt\;+\;\int_{0}^{1}\nabla w_{t}(0)\,H_{t}(0)\;dt$
$\displaystyle-\;\int_{0}^{1}\Big{\\{}\,{\mathfrak{b}}_{\alpha,A}\big{(}\,w_{t}(0)\,,\,H_{t}(0)\,\big{)}\;+\;{\mathfrak{b}}_{\beta,B}\big{(}\,w_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Big{\\}}\,dt\;.$

for all $H$ in $C^{1,2}([0,1]\times[0,1])$. Recall the definition of
${\mathfrak{q}}_{\varrho,D}(a,M)$, introduced in (6.5). As $w$ satisfies the
boundary conditions (6.15), we may rewrite the previous identity as

$\displaystyle J_{1,H}(w)$
$\displaystyle=\;\int_{0}^{1}\big{\langle}\partial_{t}w_{t}\,-\,\Delta
w_{t}\,,\,H_{t}\big{\rangle}\;dt\;-\;\int_{0}^{1}\big{\langle}\sigma(w_{t}),\big{(}\nabla
H_{t}\big{)}^{2}\big{\rangle}\;dt$
$\displaystyle-\;\int_{0}^{1}\Big{\\{}\,{\mathfrak{q}}_{\alpha,A}\big{(}\,w_{t}(0)\,,\,H_{t}(0)\,\big{)}\;+\;{\mathfrak{q}}_{\beta,B}\big{(}\,w_{t}(1)\,,\,H_{t}(1)\,\big{)}\,\Big{\\}}\,dt\;.$

As $\delta_{0}/2\leq w\leq 1-\delta_{0}/2$,
$\sigma(w_{t})\geq(\delta_{0}/2)^{2}$. On the other hand, for
$\delta_{0}/2\leq a,\varrho\leq 1-\delta_{0}/2$,
${\mathfrak{q}}_{\varrho,D}(a,M)\geq(2/D)\,(\delta_{0}/2)^{2}\,[\cosh
M-1]\geq(1/D)\,(\delta_{0}/2)^{2}\,M^{2}$. Therefore,

$\displaystyle J_{1,H}(w)$
$\displaystyle\leq\;\int_{0}^{1}\big{\langle}\partial_{t}w_{t}\,-\,\Delta
w_{t}\,,\,H_{t}\big{\rangle}\;dt\;-\;\Big{(}\frac{\delta_{0}}{2}\Big{)}^{2}\,\int_{0}^{1}\big{\langle}\big{(}\nabla
H_{t}\big{)}^{2}\big{\rangle}\;dt$
$\displaystyle-\;\Big{(}\frac{\delta_{0}}{2}\Big{)}^{2}\int_{0}^{1}\Big{\\{}\,\frac{1}{A}H_{t}(0)^{2}\;+\;\frac{1}{B}H_{t}(1)^{2}\Big{\\}}\,dt\;.$

By (A.7), we may rewrite this inequality as

$J_{1,H}(w)\;\leq\;\int_{0}^{1}\big{\langle}\partial_{t}w_{t}\,-\,\Delta
w_{t}\,,\,H_{t}\big{\rangle}\;dt\;-\;\Big{(}\frac{\delta_{0}}{2}\Big{)}^{2}\,\int_{0}^{1}\big{\|}\,H_{t}\,\big{\|}_{{\mathcal{H}}_{R}}^{2}\;dt\;.$

Since $Q_{[0,1]}(w)<\infty$, the rate functional $I_{[0,1]}(w|\bar{\rho})$ is
given by the variational formula (2.3). Therefore, maximizing over $H$ on both
sides of the previous displayed equation yields that

$I_{[0,1]}(w|\bar{\rho})\;\leq\;\int_{0}^{1}\Big{(}\frac{2}{\delta_{0}}\Big{)}^{2}\sup_{G\in
C^{2}([0,1])}\Big{\\{}\,\big{\langle}h(t)\,,\,G\big{\rangle}\;-\;\big{\|}\,G\,\big{\|}_{{\mathcal{H}}_{R}}^{2}\,\Big{\\}}\;dt\;,$
(6.17)

where $h(t)=\partial_{t}w_{t}\,-\,\Delta w_{t}$.

By (6.14),

$h(t)\;=\;\sum_{k\geq
1}\lambda_{k}\,\frac{2e^{\lambda_{k}t}-1}{e^{\lambda_{k}}-1}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle\,f_{k}$

Hence, by Young’s inequality and (A.10), for all $G\in C^{2}([0,1])$,

$\langle h(t)\,,\,G\rangle\;=\;\sum_{k\geq 1}\langle
h(t)\,,\,f_{k}\big{\rangle}\,\langle
G\,,\,f_{k}\big{\rangle}\;\leq\;\sum_{k\geq 1}\frac{1}{4\,\lambda_{k}}\langle
h(t)\,,\,f_{k}\big{\rangle}^{2}\;+\;\big{\|}\,G\,\big{\|}_{{\mathcal{H}}_{R}}^{2}\;.$

By the formula for $h(t)$, the last sum is equal to

$\frac{1}{4}\sum_{k\geq
1}\lambda_{k}\,\Big{(}\frac{2e^{\lambda_{k}t}-1}{e^{\lambda_{k}}-1}\Big{)}^{2}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle^{2}\;\leq\;J^{2}\sum_{k\geq
1}\lambda_{k}\,e^{2\lambda_{k}(t-1)}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle^{2}\;,$

where $J=(1-e^{-\lambda_{1}})^{-1}$.

Reporting the previous estimate to (6.17) yields that

$\displaystyle I_{[0,1]}(w|\bar{\rho})$
$\displaystyle\leq\;\Big{(}\frac{2J}{\delta_{0}}\Big{)}^{2}\int_{0}^{1}\sum_{k\geq
1}\lambda_{k}\,e^{2\lambda_{k}(t-1)}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle^{2}\;dt$
$\displaystyle\leq\;\Big{(}\frac{2J}{\delta_{0}}\Big{)}^{2}\sum_{k\geq
1}\,\langle\gamma-\bar{\rho}\,,\,f_{k}\rangle^{2}\;=\;\Big{(}\frac{2J}{\delta_{0}}\Big{)}^{2}\,\|\,\gamma-\bar{\rho}\,\|^{2}_{2}\;,$

which concludes the proof of the lemma. ∎

We can now prove the upper bound for the quasi-potential and conclude the
proof of Theorem 2.6.

######  Lemma 6.12.

For each $\gamma\in{\mathscr{M}}_{\rm ac}$, we have $V(\gamma)\leq S(\gamma)$.

###### Proof.

Fix $0<\varepsilon<(\delta_{0}/2)\,\min\\{(1/4),(1/\Lambda)\\}$, where
$\Lambda$ has been introduced in the statement of Lemma 6.11. Let
$\gamma\in{\mathscr{M}}_{\rm ac}$, and recall that we denote by
$v^{(\gamma)}(t,x)$ the solution to (6.8) with initial condition $\gamma$. By
Lemma 6.10, there exists $T_{1}=T_{1}(\varepsilon)$ such that
$\|v^{(\gamma)}(t)-\bar{\rho}\|_{\infty}<\varepsilon$ for any $t\geq T_{1}$.
Since $v^{(\gamma)}(T_{1})$ fullfils the hypotheses of Lemma 6.11, let $w$ be
the path which connects $\bar{\rho}$ to $v^{(\gamma)}(T_{1})$ in the interval
$[0,1]$ constructed in that lemma.

Let $T:=T_{1}+1$ and $w^{*}(t)$, $t\in[0,T]$, be the path

$w^{*}(t)\;=\;\left\\{\begin{array}[]{ll}w(t)&\text{for $0\leq t\leq 1$}\\\
v^{(\gamma)}(T-t)&\text{for $1\leq t\leq T$}\end{array}\right.$ (6.18)

Recall the definition of ${\mathscr{M}}_{\delta}$ given in (6.1). Let
$(\gamma_{n},\,n\geq 1)$ be a sequence such that
$\gamma_{n}\in{\mathscr{M}}_{\delta_{n}}$ for some $\delta_{n}>0$ and which
converges to $\gamma$ a. s. Denote by $v^{(\gamma_{n})}$ the solution to (6.8)
with initial condition $\gamma_{n}$.

Claim 1: $v^{(\gamma_{n})}(T_{1})$ converges to $v^{(\gamma)}(T_{1})$ in
$C([0,1])$.

To prove this claim, let $F=F(\gamma)$, $F_{n}=F(\gamma_{n})$, and denote by
$F^{(\gamma)}$, $F^{(\gamma_{n})}$ the solutions to (1.1) with initial
conditions $F$, $F_{n}$, respectively.

By Proposition 5.6, $F_{n}$ converges to $F$ in $C^{1}([0,1])$. Hence, by
Lemma B.5, $F^{(\gamma_{n})}(T_{1})$ converges to $F^{(\gamma)}(T_{1})$ in
$C^{2}([0,1])$. On the other hand, by Lemma 6.6, there exists a constant
$c_{1}>0$ such that $c_{1}\leq\nabla F^{(\gamma_{n})}(T_{1})\leq c^{-1}_{1}$,
$c_{1}\leq\nabla F^{(\gamma)}(T_{1})\leq c^{-1}_{1}$ for all $n\geq 1$. Hence,
by (6.13), $v^{(\gamma_{n})}(T_{1})$ converges to $v^{(\gamma)}(T_{1})$ in
$C([0,1])$, as claimed.

Since $\|v^{(\gamma)}(T_{1})-\bar{\rho}\|_{\infty}<\varepsilon$, by Claim 1,
there exists $n_{0}$ such that
$\|v^{(\gamma_{n})}(T_{1})-\bar{\rho}\|_{\infty}<2\,\varepsilon$ for all
$n\geq n_{0}$. Fix such $n$, and let $w^{n}(t)$ be the path joining
$\bar{\rho}$ to $v^{(\gamma_{n})}(T_{1})$ in the time interval $[0,1]$
constructed in Lemma 6.11. Define the path $w^{n,*}(t)$, $0\leq t\leq T_{1}+1$
as

$w^{n,*}(t)\;=\;\left\\{\begin{array}[]{ll}w^{n}(t)&\text{for $0\leq t\leq
1$}\\\ v^{(\gamma_{n})}(T-t)&\text{for $1\leq t\leq T$}\end{array}\right.$
(6.19)

Claim 2: The path $w^{n,*}$ converges in $D([0,T],{\mathscr{M}})$ to $w^{*}$.

Before proving this claim, we conclude the proof of the lemma. By the lower
semi-continuity of the functional $I_{[0,T]}(\,\cdot\,|\,\bar{\rho})$,

$I_{[0,T]}(w^{*}|\bar{\rho})\;\leq\;\liminf_{n}I_{[0,T]}(w^{n,*}|\bar{\rho})\;.$
(6.20)

On the other hand, by definition of the rate function and (4.1),

$I_{[0,T]}\big{(}w^{n,*}\,|\,\bar{\rho}\big{)}\;\leq\;I_{[0,1]}\big{(}w^{n}\,|\,\bar{\rho}\big{)}\;+\;I_{[0,T_{1}]}\big{(}v^{(\gamma_{n})}(T_{1}-\cdot)\,|\,v^{(\gamma_{n})}(T_{1})\big{)}\;.$
(6.21)

By Lemma 6.11, for $n\geq n_{0}$

$I_{[0,1]}\big{(}w^{n}\big{|}\bar{\rho}\big{)}\,\leq\;C_{0}\,\|v^{(\gamma_{n})}(T_{1})-\bar{\rho}\|_{2}^{2}$
(6.22)

for some constant $C_{0}=C_{0}(\delta_{0})$.

By Proposition 6.7 and Lemmata 6.8, 6.9, $(x,t)\mapsto
v^{(\gamma_{n})}(T_{1}-t,x)$ belongs to $C^{1,2}([0,T_{1}]\times[0,1])$ and is
bounded away from $0$ and $1$, namely it belongs to $D_{T_{1},\delta_{n}}$ for
some $\delta_{n}>0$. Hence, by Lemma 6.5, as
$v^{(\gamma_{n})}(T_{1}-\,\cdot\,)$ solves (6.6) with $K=0$,

$I_{[0,T_{1}]}\big{(}v^{(\gamma_{n})}(T_{1}-\cdot)\,|\,v^{(\gamma_{n})}(T_{1})\big{)}\;=\;S_{0}(\gamma_{n})\;-\;S_{0}(v^{(\gamma_{n})}(T_{1}))\;.$
(6.23)

By equations (6.20)–(6.23),

$I_{[0,T]}(w^{*}\,|\,\bar{\rho})\;\leq\;\liminf_{n}\big{\\{}\,S_{0}(\gamma_{n})\;-\;S_{0}(v^{(\gamma_{n})}(T_{1}))\;+\;C_{0}\,\|v^{(\gamma_{n})}(T_{1})-\bar{\rho}\|_{2}^{2}\,\big{\\}}\;.$

By Remark 5.8, $S_{0}(\gamma_{n})$ converges to $S_{0}(\gamma)$. Thus, by the
convergence of $v^{(\gamma_{n})}(T_{1})$ to $v^{(\gamma)}(T_{1})$ in
$C([0,1])$, the lower semicontinuity of $S_{0}$, and the bound
$\|v^{(\gamma)}(T_{1})-\bar{\rho}\|_{\infty}<\varepsilon$, the right-hand side
is less than or equal to

$S_{0}(\gamma)\;-\;S_{0}(\bar{\rho})\;+\;C_{0}\,\|v^{(\gamma)}(T_{1})-\bar{\rho}\|_{2}^{2}\;\leq\;S(\gamma)\;+\;C_{0}\,\varepsilon^{2}\;.$

To completes the proof of the lemma, it remains to show that Claim 2 is in
force.

Proof of Claim 2. It is enough to show that $w^{n,*}$ converges to $w^{*}$ in
$C([0,T],{\mathscr{M}})$. Equivalently, to show that $v^{(\gamma_{n})}$
converges to $v^{(\gamma)}$ in $C([0,T_{1}],{\mathscr{M}})$ and $w^{n}$
converges to $w$ in $C([0,1],{\mathscr{M}})$.

Fix $0<t_{0}<T_{1}$ small. By the arguments presented in the proof of Claim 1
and since the convergence in Lemma B.5 is uniform for $t\geq t_{0}$,
$v^{(\gamma_{n})}$ converges to $v^{(\gamma)}$ in
$C([t_{0},T_{1}]\times[0,1])$.

On the other hand, by Lemma 5.3, $\nabla F^{(\gamma_{n})}(t)$ and $\nabla
F^{(\gamma)}(t)$ are uniformly bounded, and, by Lemma 6.6, $\nabla
v^{(\gamma_{n})}(t)$ and $\nabla v^{(\gamma)}(t)$ are uniformly bounded in
$[0,T_{1}]$. As $v^{(\gamma_{n})}$, $v^{(\gamma)}$ are weak solutions to
(6.11), for each $G\in C([0,1])$,

$\lim_{t_{0}\downarrow 0}\;\limsup_{n}\;\sup_{t\in[0,t_{0}]}\big{|}\,\langle
v^{(\gamma_{n})}(t),G\rangle\,-\,\langle
v^{(\gamma)}(t),G\rangle\,\big{|}\;=\;0\;.$

This concludes the proof that $v^{(\gamma_{n})}$ converges to $v^{(\gamma)}$
in $C([0,T],{\mathscr{M}})$.

By (6.13) and Remark B.6, we can extend the result obtained in Claim 1 and
show that $v^{(\gamma_{n})}(T_{1})$ converges to $v^{(\gamma)}(T_{1})$ in
$C^{2}([0,1])$. Hence, by the explicit formula (6.14), $w^{n}$ converges to
$w$ in $C([0,1]\times[0,1])$. This completes the proof of Claim 2 and of the
lemma. ∎

##  Appendix A The Robin Laplacian

We present in this section some results on the Robin Laplacian needed in the
previous section. Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\Delta_{R}$
the Laplacian on $[0,1]$ with Robin boundary conditions, sometimes called the
Robin Laplacian [25, Section 4.3].

Consider the eigenvalue problem

$\left\\{\begin{aligned} &-\,\Delta f=\lambda\,f\;,\\\ &(\nabla
f)(0)\,=\,A^{-1}\,f(0)\;,\\\ &(\nabla
f)(1)\,=\,-\,B^{-1}f(1)\;.\end{aligned}\right.$ (A.1)

The equation $-\,\Delta f=\lambda\,f$ can be turned into a two-dimensional ODE
which yields that the solutions to (A.1) are given by
$f(x)=a\,[\,\cos(\sqrt{\lambda}x)+b\,\sin(\sqrt{\lambda}x)\,]$ for some $a$,
$b\in{\mathbb{R}}$. The boundary conditions are satisfied if and only if

$\tan\sqrt{\lambda}\;=\;(A+B)\,\frac{\sqrt{\lambda}}{\lambda AB-1}\;,$ (A.2)

in which case $b=(A\sqrt{\lambda})^{-1}$. This identity excludes $\lambda=0$
from the set of eigenvalues of the Robin Laplacian.

An analysis of (A.2) shows that it has a countable set of solutions
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\\{\lambda_{j}:j\geq
1\\}$, where $0<\lambda_{1}$, $\lambda_{j}<\lambda_{j+1}$ and $\lambda_{j}\sim
j^{2}$ in the sense that there exists $0<c_{0}<c_{1}<\infty$ such that

$c_{0}\,j^{2}\;\leq\;\lambda_{j}\;\leq\;c_{1}\,j^{2}\;.$ (A.3)

Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\\{f_{j}:j\geq
1\\}$ the associated orthonormal eigenvectors, which form a basis of
${\mathscr{L}}^{2}([0,1])$. By the previous analysis,

$f_{j}(x)\;=\;a_{j}\,\big{\\{}\,\cos(\sqrt{\lambda_{j}}x)\;+\;\frac{1}{A\sqrt{\lambda_{j}}}\,\sin(\sqrt{\lambda_{j}}x)\,\big{\\}}\;,$
(A.4)

where $a_{j}$ is chosen for $f_{j}$ to have ${\mathscr{L}}^{2}$-norm equal to
$1$. It can be shown that $|a_{j}|\leq C_{0}$ for all $j\geq 1$, where $C_{0}$
is a finite constant depending only on $A$ and $B$. Therefore, by (A.3),

$\|\,f_{j}\,\|_{\infty}\;\leq\;C_{0}\;,\quad\|\,\nabla^{n}f_{j}\,\|_{\infty}\;\leq\;C_{0}\,(\lambda_{j})^{n/2}\;\leq\;C_{0}\,j^{n}$
(A.5)

for all $j\geq 1$, $n\geq 1$.

A straightforward computation provides a formula for the Green function of the
Robin Laplacian: Let $K_{R}:[0,1]\times[0,1]\to{\mathbb{R}}_{+}$ be given by

$K_{R}(x,y)\;=\;\frac{1}{1+A+B}\,\left\\{\begin{aligned}
&(B+1-x)\,(A+y)\;,\quad 0\leq y\leq x\leq 1\;,\\\ &(B+1-y)\,(A+x)\;,\quad
0\leq x\leq y\leq 1\;.\end{aligned}\right.$ (A.6)

Denote by $K_{R}$ the integral operator defined by

$(K_{R}f)(x)\;=\;\int_{0}^{1}K_{R}(x,y)\,f(y)\;dy\;.$

Then, $K_{R}=(-\Delta_{R})^{-1}$.

Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}{\mathcal{H}}_{R}$
the Hilbert space obtained by completing the space
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}C^{2}_{A,B}([0,1])=\\{f\in
C^{2}([0,1]):(\nabla f)(0)\,=\,A^{-1}\,f(0)\;,\,(\nabla
f)(1)\,=\,-\,B^{-1}f(1)\\}$ endowed with the scalar product
$\langle\,\cdot\,,\,\cdot\,\rangle_{{\mathcal{H}}_{R}}$ defined by

$\displaystyle\langle\,f\,,\,g\,\rangle_{{\mathcal{H}}_{R}}$
$\displaystyle=\;\langle\,f\,,\,(-\,\Delta_{R})g\,\rangle$ (A.7)
$\displaystyle=\;\frac{1}{A}\,f(0)\,g(0)\;+\;\int_{0}^{1}(\nabla
f)(x)\,(\nabla g)(x)\;dx\;+\;\frac{1}{B}\,f(1)\,g(1)\;.$

Denote by
$\color[rgb]{.2,0.2,.8}\definecolor[named]{pgfstrokecolor}{rgb}{.2,0.2,.8}\|f\|_{{\mathcal{H}}_{R}}$
the norm induced by the scalar product
$\langle\,\cdot\,,\,\cdot\,\rangle_{{\mathcal{H}}_{R}}$. We have that

$\|f\|^{2}_{{\mathcal{H}}_{R}}\;=\;\sum_{k\geq 1}\lambda_{k}\,\langle
f\,,\,f_{k}\rangle^{2}\;.$ (A.8)

for all $f\in{\mathcal{H}}_{R}$.

The norms $\|\,\cdot\,\|_{{\mathcal{H}}_{R}}$ and
$\|\,\cdot\,\|_{{\mathcal{H}}_{1}}$ are equivalent. There exist finite
constants $0<C_{1}<C_{2}<\infty$ such that

$C_{1}\,\|f\|_{{\mathcal{H}}_{1}}\;\leq\;\|f\|_{{\mathcal{H}}_{R}}\;\leq\;C_{2}\,\|f\|_{{\mathcal{H}}_{1}}$
(A.9)

for all $f\in C^{2}([0,1])$. In particular, the spaces ${\mathcal{H}}_{R}$ and
${\mathcal{H}}_{1}$ coincide.

In terms of the eigenfunctions $f_{k}$,

$\|\,f\,\|^{2}_{{\mathcal{H}}_{R}}\;=\;\sum_{k\geq
1}\lambda_{k}\,|\,\langle\,f\,,\,f_{k}\,\rangle\,|^{2}\;.$ (A.10)

Moreover, a straightforward computation yields that for all $f\in
C^{2}_{A,B}([0,1])$,

$\|\,f\,\|^{2}_{\infty}\;\leq\;2\,(A\vee 1)\,\|f\|^{2}_{{\mathcal{H}}_{R}}\;.$
(A.11)

Fix a function $f$ in ${\mathcal{H}}_{1}$. It is well known that there exists
a continuous function $f^{(c)}:[0,1]\to{\mathbb{R}}$ (actually Hölder
continuous, $|f^{(c)}(y)-f^{(c)}(x)|\leq\|f\|_{2}|y-x|^{1/2}$) such that
$f=f^{(c)}$ almost surely. Moreover, for all $h\in C^{1}([0,1])$,

$\int_{0}^{1}f\,\nabla
h\;dx\;=\;f^{(c)}(1)\,h(1)\;-\;f^{(c)}(0)\,h(0)\;-\;\int_{0}^{1}\nabla
f\,h\;dx\;.$ (A.12)

The next result provides an explicit formula for $f^{(c)}$ in terms of the
eigenvectors $f_{k}$.

######  Lemma A.1.

There exists a finite constant $C_{0}$ such that

$\sum_{k\geq 1}\big{|}\,\langle
f\,,\,f_{k}\rangle\,\big{|}\;\leq\;C_{0}\,\|\,f\,\|_{{\mathcal{H}}_{R}}$