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Exclusion process, Open systems, Steady states,
Large Deviations, Hydrodynamic limit, Freidlin--Wentzell approach
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\begin{document}
\title[Large Deviations for steady states]
{From dynamic to static Large Deviations \\
in boundary driven exclusion particle systems}
\author[T. Bodineau]{Thierry Bodineau}
\address{T. Bodineau,
Universit{\'e} Paris 7 and Laboratoire de
Probabilit{\'e}s et Mod{\`e}les Al{\'e}atoires C.N.R.S. UMR 7599, U.F.R.
Math{\'e}matiques, Case 7012, 2 Place Jussieu, F-75251 Paris, France
}
\email{bodineau\@@dmi.ens.fr}
\author[G. Giacomin]{Giambattista Giacomin}
\address{G. Giacomin, Universit{\'e} Paris 7 and Laboratoire de
Probabilit{\'e}s et Mod{\`e}les Al{\'e}atoires C.N.R.S. UMR 7599, U.F.R.
Math{\'e}matiques, Case 7012, 2 Place Jussieu, F-75251 Paris, France
\hfill\break
\phantom{br.}{\it Home page:}
{\tt http://felix.proba.jussieu.fr/pageperso/giacomin/GBpage.html}}
\email{giacomin\@@math.jussieu.fr}
\date{\today}
\begin{abstract}
We consider the large deviations for the stationary measures associated
to a boundary driven symmetric simple exclusion process.
Starting from the large deviations for the hydrodynamics and
following the Freidlin and Wentzell's strategy, we prove that the
rate function is given by the quasi--potential of the Freidlin and Wentzell
theory.
This result is motivated by the recent developments on the
non-equilibrium stationary measures by Derrida, Lebowitz and Speer
\cite{cf:DLS} and the more closely related dynamical approach by
Bertini, De Sole, Gabrielli, Jona Lasinio, Landim \cite{cf:BDGJL}.
\\
\\
2000
\textit{Mathematics Subject Classification:} 60K35
\\
\\
\textit{Keywords:
Particle systems, Exclusion process, Open systems,
Steady states,
Large Deviations, Hydrodynamic limit, Freidlin--Wentzell approach
}
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
\setcounter{equation}{0}
A rigorous understanding of the steady states associated to
non equilibrium systems is far from being complete.
In particular, the transport phenomena which take place in some
non equilibrium systems induce, in general, long range correlations
in the stationary measures, see e.g. \cite{cf:Herbert}.
For the moment there is no analog to the
Gibbs equilibrium formalism
and it is typically a very challenging problem to describe
the stationary measures of systems which are defined only
by dynamical prescriptions.
A mathematical idealization of open systems is provided by
stochastic models of interacting particles systems.
Consider a system of particles performing a reversible hopping
dynamics ({\sl Kawasaki} dynamics) in a domain and some external mechanism
of creation and annihilation of particles on the boundary of the
domain which make the full process non--reversible.
The hydrodynamic behavior, namely the law of large numbers
followed by the stationary measures, has been derived for
important general classes of models (we signal in particular
\cite{cf:ELS1} and \cite{cf:KLO}).
In the case of the symmetric simple exclusion process (SSEP)
hydrodynamic behavior as well as
further results on the fluctuations can be obtained by using
the specific structure of the dynamics (see in particular
\cite{cf:Herbert} and \cite{cf:DMFIP}).
\bigskip
More recently breakthroughs were achieved by the derivation of a
large deviation principle for the stationary measures of the
one dimensional boundary driven SSEP.
Using exact computations, Derrida, Lebowitz and Speer
\cite{cf:DLS} obtained the explicit form of the
rate function for the large deviation principle.
Another approach, relying on the large deviations for the
hydrodynamics, has been pursued by Bertini, De Sole, Gabrielli,
Jona Lasinio, Landim \cite{cf:BDGJL}.
By generalizing the Freidlin and Wentzell's theory in this
context, they were able to formulate a dynamical fluctuation
theory for the stationary non-equilibrium states.
This approach relies on the hypotheses that
the rate function associated to the steady states
is given by a dynamical variational formula (the quasi-potential).
As a consequence of these hypotheses some general principles
are deduced among which an extension of the Onsager-Machlup
theory and a nonlinear fluctuation dissipation relation.
The non-local structure of the rate function is extremely hard
to interpret physically and therefore the result in \cite{cf:DLS}
raises many questions for the generalization to a broader class
of models.
On the other hand the dynamical approach
seems to be very promising
since the static rate function can be identified in a systematic way
with the quasi-potential.
Unfortunately, the quasi-potential provides a very indirect
information and, at the moment, only partial results can be extracted
from it : there is no general procedure to analyze the
quasi-potential.
Inspired by the exact formula in \cite{cf:DLS}, Bertini et al.
were able, in the case of SSEP, to integrate the
dynamical information contained in the quasi-potential and to
recover a tractable expression of the rate function by using
a purely dynamical method \cite{cf:BDGJL2}.
This important step may open the way towards further generalizations.
\bigskip
In this paper our modest goal is to address
one
of the hypotheses on which the dynamical theory in \cite{cf:BDGJL} rests.
In fact we implement
the Freidlin--Wentzell theory in the context of the SSEP, by proving
that the quasi-potential is the large deviation
functional of the steady state.
This complements the results in \cite{cf:BDGJL2}, providing
thus an alternative proof of the result in \cite{cf:DLS}.
We stress that, contrary to the original heuristic in
\cite{cf:BDGJL}, the proof requires no hypotheses on the adjoint
dynamics. We point out that this proof
uses essentially nothing of the details of the SSEP dynamics:
a {\sl good large deviation principle}
\cite{cf:DS} is the key ingredient, along
with some properties of the macroscopic
dynamics and therefore a good control
of the hydrodynamic equation and, above all,
of the large deviation functional would lead
to the generalization of the result to
a large class of interacting exclusion systems.
We will address in detail this issue
in the last section.
As a last remark, let us mention that an exact solution for the
rate function of the totally asymmetric
exclusion process has been also derived \cite{cf:DLS2} and it is an
open problem to provide a dynamical counterpart similar to the
results obtained for the SSEP.
\section{The model and the results}
\label{sec:results}
\setcounter{equation}{0}
\subsection{Boundary driven SSEP}
Let $\gL _N= \{-N,-N+1,\ldots, N\}$ and $N$ be a positive integer.
The configuration space is $\gO_N=\{0,1\}^{\gL _N}$.
The SSEP with reservoirs is defined as the Markov process
$\{\eta_t \}_{t \ge 0}$, with $\eta_t \in \gO _N$ for every $t\ge 0$,
generated by
\begin{equation}
\label{eq:L}
(L_N f)(\eta)=
\frac {N^2}2
\sum_{x,y\in \La}
\left[ f (\eta ^{x,y}) -f (\eta)\right]+
N^2\sum_{x: \vert x\vert =N}
c(x,\eta_x)\left[ f (\eta ^{x}) -f (\eta)\right],
\end{equation}
where $f$ is any function from $\Omega_N$ to $\R$ and
$\eta^x$ and $\eta ^{x,y}$ are defined in the standard way,
that is
\begin{equation}
\eta^x (z)=
\begin{cases}
1-\eta(x) &\text{ if } z=x,\\
\eta (z) &\text{ otherwise, }
\end{cases}
\phantom{leavespacele}
\eta^{x,y} (z)=
\begin{cases}
\eta(x) &\text{ if } z=y,\\
\eta(y) &\text{ if } z=x,\\
\eta (z) &\text{ otherwise. }
\end{cases}
\end{equation}
The rates $c(\pm N, \cdot)=c_\pm(\cdot)$ depend on the
activities $\gga^\pm \in (0,1)$ of the reservoirs
\begin{equation}
c_+(\eta_N) = \gga^+ + (1 - \gga^+) \eta_N \, \qquad \qquad
c_-(\eta_{-N}) = \gga^- + (1 - \gga^-) \eta_{-N} \, .
\end{equation}
Let us remark that if $\lambda_+=\lambda_-(=\lambda)$ then the model
is reversible. Therefore to
every $\lambda$ is naturally associated the value of the (uniform)
density of the equilibrium measure in the infinite volume
measure: we call $\rho_+$ (respectively $\rho_-$)
the density associated to $\lambda_+$ (respectively $\lambda_-$)
\begin{equation}
\gr^+ = \frac{\gga^+}{1 + \gga^+}
\, , \qquad
\gr^- = \frac{\gga^-}{1 + \gga^-} \, .
\end{equation}
Call $\Prob _\eta \equiv \Prob _{N, \eta}$
the path measure of the process $\{\eta_t \}_{t\ge 0}$
with $\eta _0=\eta$: it is of course a measure
on $D([0,\infty);\Omega_N)$, the (Skorohod) space
of CADLAG functions. If $\mu $ is a probability measure
on (all subsets of) $\Omega_N$, $\Prob _\mu (\cdot)=
\int_{\Omega_N} \Prob _\eta (\cdot) \mu (\dd \eta)$.
\subsection{Hydrodynamics, invariant measure and hydrostatics }
\label{subsec:Hydrodynamics}
In \cite{cf:ELS2}
it has been proven the hydrodynamic limit
scaling for this system. More precisely:
we introduce the empirical measure for $r\in [-1,1]$
\begin{equation}
\label{eq:empmeas}
\empirical{\eta} (r)=
\sum_{x=-N}^{N-1}
\eta_x \mathbf{1}_{[x,x+1)} (rN),
\end{equation}
so $\empirical{\eta} \in \Fspace \equiv \{\rho\in \bbL ^\infty ([-1,1]):
0\le \rho \le 1$ a.e. $\}$\footnote{the topology of $\Fspace$ is induced by
the weak convergence: $\rho_n \stackrel{n\to \infty}{\longrightarrow} \rho$
if
$\int_{-1}^1 \rho_n (r) f(r) \dd r$
tends to $\int _{-1}^1 \rho (r) f(r) \dd r$ for every $f\in C_b^0([-1,1];\R)$.
See later for more on this (metrizable) topology.
The $\sigma$--algebra of the measurable sets of $\Fspace$ is chosen
to be the Borel one.},
and we assume
that we are given a sequence $\{ \nu_N \}_N$
and a function $\rho_0 \in C^0([-1,1];[0,1])$
such that the law of $\nu_N \circ (\empirical{\cdot})^{-1}$, measure
on $\Fspace$, tends weakly, as $N \to \infty$,
to the measure concentrated on $\rho_0$.
Then
the law of the process $\{\phi_{\pi_{t N^2}}\}_{t\in [0,T]}$,
under $\Prob_{\nu_N}$,
converges weakly to the measure on $C^0([0,T]; [0,1])$
concentrated on $\rho$, satisfying the
energy condition $\int_{[0,T]\times [-1,1]}
\vert \nabla \rho (t,r)\vert^2 \dd r \dd t<\infty$,
{\sl unique weak solution of}
\begin{equation}
\label{eq:weak1}
\begin{cases}
\partial_t \rho (t,r) =\Delta \rho (t,r)
&\text{ for every } (t,r) \in \R^+ \times (-1,1), \\
\rho (\pm 1, t)=\rho_{\pm} &\text{ for every } t\in \R^+,\\
\rho (0)=\rho_0. & \\
\end{cases}
\end{equation}
Moreover
for every fixed $N$, the unique invariant measure
({\sl steady state}) is denoted by $\Smeasure$.
It has in fact been proven that the law
of $\empirical{\eta}$, under $\Smeasure (\dd \eta)$ converges
weakly as $N$ tends to infinity to the measure
on $\Fspace$ concentrated on the stationary solution of
\eqref{eq:weak1} which is
\begin{equation}
\Sprofile (r)= \frac{(\gr _+ - \gr _-)}{2} r +
\frac{(\gr _+ + \gr _-)}{2}.
\end{equation}
For a proof see
\cite{cf:Herbert} or \cite{cf:ELS1}.
\subsection{From dynamic to static large deviations}
\label{subsec: static large deviations}
Call $\bra \cdot, \cdot \ket$
the scalar product in $\bbL^2([-1,1])$.
For $H\in C_0^{1,2} ([0,T]\times [-1,1])$ (that is $H(\cdot, \pm 1)\equiv 0)$) let
\begin{equation}
\label{eq:Jfunct}
\begin{split}
\ld{H} (\pi)=&
\langle \pi (T) , H(T) \rangle- \langle \rho_0 , H(0)\rangle-
\int_0^T
\bra \pi (t) , \partial_t H (t) + \Delta H(t) \ket \dd t
\\
&+ \rho^+ \int_0^T \nabla H(t,1) \dd t - \rho^- \int_0^T \nabla H(t,-1) \dd t
-\frac12 \int_0^T \langle \sigma (\pi(t)) , (\nabla H (t))^2 \rangle \dd t,
\end{split}
\end{equation}
where $\sigma (x) = x(1-x)$ is the mobility.
Set also
\begin{equation}
I_T (\pi) =
\sup_{H\in C_0^{1,2} ([0,T]\times [-1,1])}
\ld{H} (\pi).
\end{equation}
For $\gr_0 \in \Fspace $ we define the {\sl LD rate
function} as
\begin{equation}
\label{eq:LDrate}
\LDfunc{\pi}{\gr_0}
=
\begin{cases}
I_T (\pi) & \text{if } \pi (0)=\rho_0 \\
+\infty & \text{otherwise.}
\end{cases}
\end{equation}
By exploiting the concavity of $\sigma$
one can show that $I_T$
is convex. Moreover one can show also
that it is lower semicontinuous
(l.s.c.) and that the level sets $\{ \pi\in D([0,T];\Fspace )
\, :\, I_{[0,T]}(\pi)\leq a\}$
are compact for every $a\geq 0$.
These properties extend to $\LDfunc{\cdot}{\gr_0}$:
in particular $\LDfunc{\cdot}{\gr_0}$ is a {\sl good rate function}
(i.e. it is l.s.c. and it has compact level sets, \cite{cf:DS}).
A proof of these properties can be found in \cite[Ch. 10]{cf:Landim}
and in \cite{cf:BDGJL2}.
The large deviation principle for boundary driven SSEP
is derived in \cite{cf:BDGJL2}:
\bigskip
\begin{teo}
\label{th:LDdyn}
For every choice of $\{ \eta^N\}_N$, $\eta^N \in \Omega _N$
such that $\empirical{\eta^N}\in \Fspace$ converges to $\gr_0$,
the sequence of random functions $\{\empirical{\eta _{N^2 \cdot} }\}_{N=1,2,
\ldots}$ in $D([0,T], \Fspace )$, $\eta _0 =\eta^N$,
obeys a full Large Deviations principle with speed $N$ and rate function
$\LDfunc{\cdot}{\gr_0}$, that is
for every $A\subset D([0,T],\Fspace)$, we have that
\begin{equation}
\label{eq: dynamical LD}
-\inf_{\pi \in A^{\circ}} \LDfunc{\pi}{\gr_0} \leq
\liminf_{N \to \infty}
\frac 1N \log \Dmeasure{\eta^N} \left( \empirical{\cdot} \in
A^{\circ} \right)
\le
\limsup_{N \to \infty}
\frac 1N \log \Dmeasure{\eta^N} \left( \empirical{\cdot} \in \overline{A} \right)
\leq
-\inf_{\pi \in \overline{A}} \LDfunc{\pi}{\gr_0},
\end{equation}
where $A^{\circ}$ denotes the interior of $A$ and $\overline{A}$
its closure.
\end{teo}
\bigskip
% More precisely, for every closed set $\cC \subset D([0,T],\Fspace)$
% and every open set $\cO \subset D([0,T],\Fspace)$, we have
% \begin{eqnarray}
% \label{eq: dynamical LD}
% &&-\inf_{\rho \in \cO} \LDfunc{\rho}{\gr_0} \leq
% \liminf_{N \to \infty}
% \frac 1N \log \Dmeasure{\eta^N} \left( \empirical{\cdot} \in \cO \right)\\
% &&
% \limsup_{N \to \infty}
% \frac 1N \log \Dmeasure{\eta^N} \left( \empirical{\cdot} \in \cC \right)
% \leq
% -\inf_{\rho \in \cC} \LDfunc{\rho}{\gr_0} \, , \nonumber
% \end{eqnarray}
Let us introduce the {\sl quasipotential}, \cite{cf:FW},
\cite{cf:BDGJL} :
for every $\gr \in \Fspace$
\begin{equation}
\label{eq:quasipotential}
\quasipot (\gr) =\inf
\left\{\LDfunc{\pi}{\Sprofile}
\; : \; \pi (T)=\gr \text{ and } T>0
\right\}.
\end{equation}
Of course the infimum can be restricted to
$\pi \in D([0,T]; \Fspace)$ such that $\pi (0)=\Sprofile$.
Moreover it is not too difficult to show
(cf. \cite{cf:BDGJL2} and \cite{cf:Landim}) that
$I_T(\pi)=+\infty$ unless $\pi \in C^0([0,T]; \Fspace)$, therefore
we may
restrict further this extremum
to trajectories which are continuous in time.
Starting with the next statement,
we will commit abuse of notation calling $\Smeasure$
also the measure $\Smeasure \circ ({\empirical{\cdot}})^{-1}$ on $\Fspace$.
\bigskip
We can now state the main result of this paper
\begin{teo}
\label{th:LDstationary}
The stationary measure $\Smeasure$ obeys a full Large Deviations
principle with rate function
$\quasipot$ and speed $N$.
%, namely: for every $A \subset \Fspace$
%\begin{equation}
%-\inf_{\rho \in A^{\circ}}
%\quasipot (\rho) \le
%\liminf_{N \to \infty}
%\frac 1N \log \Smeasure \left( A\right)
%\le \limsup_{N \to \infty}
%\frac 1N \log \Smeasure \left( A\right)\le
%-\inf_{\rho \in \overline{A}} \quasipot (\rho) \, ,
%\end{equation}
%where $A^{\circ}$ denotes the interior of $A$ and $\overline{A}$
%its closure.
\end{teo}
\section{The proof}
\label{sec:proof}
\setcounter{equation}{0}
The scheme of the proof follows closely the one introduced by
Freidlin and Wentzell \cite[\S 4]{cf:FW}.
This requires further notations on the topology of the functional
spaces.
\bigskip
Recall that the space
$$
\Fspace \equiv \big\{ \rho\in \bbL ^\infty ([-1,1]): 0\leq \rho \leq 1
\big\}
$$
was introduced in subsection
\ref{subsec:Hydrodynamics}.
The space $\Fspace$ is metrizable:
if we set $f_{2n+1}(r)=\sin (\pi n r)$
and $f_{2n}(r)=\cos (\pi n r)$,
$n =0,1,\ldots$, we may define
the distance as
\begin{eqnarray}
\label{eq:weak metric}
\text{dist} (\gr_1,\gr_2) = \sum_{k =1}^\infty
\frac{1}{2^k}
| \bra \gr_1 , f_k \ket - \bra \gr_2 , f_k \ket | \, ,
\end{eqnarray}
for $\gr_1$, $\gr_2 \in \Fspace$. Of course
$\bra \cdot , \cdot \ket$ is the scalar product
in $\bbL^2$.
Moreover for $\gep >0$ and $\gr\in \Fspace$, then the
closed
$\gep$-ball around $\gr$ in the weak topology is denoted by
\begin{eqnarray}
\label{eq:Boule}
\bbB_\gep (\gr) = \{ \gp \in \Fspace \ | \quad
\text{dist} (\gr,\gp) \leq \gep \} \, .
\end{eqnarray}
On the dynamical level we will work with several spaces, but
the basic one is the Skorohod space $D([0,T],\Fspace)$:
observe that $\empirical{\cdot} \in D([0,T],\Fspace)$.
Let us be more precise about this space and let us recall
that it is a metric space:
if we let $\gL$ be the set of increasing continuous
functions $\gl$ of $[0,T]$ into itself,
then a distance associated to the Skorohod topology is given by
\begin{equation}
\label{eq:Skorohod}
\dd (\pi, \pi^\prime) = \inf_{\gl \in \gL} \sup_{t\in[0,T]}
\left\{ \text{dist} \left(\pi_t,
\pi^\prime_{\gl(t)} \right) +\vert \lambda (t)-t \vert
\right\},
\ \ \ \pi, \pi^\prime \in D([0,T],\Fspace).
\end{equation}
For any $\pi$ in $D([0,T],\Fspace)$, the $\gep$-neighborhood of $\pi$
in the Skorohod topology is denoted by $\cV_{[0,T]}^\gep (\pi)$.
In the same way if $A\subset D([0,T],\Fspace)$,
$\cV_{[0,T]}^\gep (A)=\cup_{\pi \in A} \cV_{[0,T]}^\gep (\pi)$.
\subsection{Lower Bound}
It is sufficient to check that for any $\gep >0$ and any
$\gr$ in $\Fspace$
\begin{eqnarray}
\label{eq:lbound}
\liminf_{N \to \infty} \
\frac{1}{N} \; \log \Smeasure \left( \bbB_\gep (\gr) \right)
\geq - \quasipot (\gr) \, .
\end{eqnarray}
By definition of $\quasipot$ (recall that the infimum
may be restricted to continuous functions), for every $\gd >0$, there exists
$T$ and $\pi \in C^0([0,T];\Fspace)$ such that
\begin{equation}
\LDfunc{\pi}{\Sprofile} \leq \quasipot (\gr) + \gd
\qquad \text{and} \qquad
\pi (T) = \gr \, .
\end{equation}
By using the definition \eqref{eq:Skorohod},
for any trajectory $\nu$
in $\cV_{[0,T]}^\gep(\pi)$ we see
that $\nu_T \in \bbB_\gep (\gr)$
(because $\gl(T) = T$).
Since $\Smeasure$ is the stationary measure of the dynamics
\begin{equation}
\Smeasure \left( \bbB_\gep (\gr) \right) \geq
\Prob_{\Smeasure}
\left( \empirical{\eta_T} \in \bbB_\gep (\gr) \right) \geq
\E_{\Smeasure} \left[ \
\Prob_{\eta_0}
\left(
\empirical{\eta_{\cdot}}
\in \cV_{[0,T]}^\gep (\pi)
\right) \, ; \, \empirical{\eta_0} \in \bbB_{\gep_N}
(\Sprofile) \right] \, ,
\end{equation}
for every $\gep _N>0$.
The hydrostatics results recalled in subsection \ref{subsec:Hydrodynamics}
can be rephrased as
\begin{equation*}
\text {for every } \gd >0, \qquad
\lim_{N \to \infty}
\Smeasure \left( \bbB_\gd (\Sprofile) \right) =1 \, .
\end{equation*}
This is equivalent to the existence of
a sequence $\{ \gep _N\}_{N=1,2,\ldots}$,
$ \gep _N \searrow 0$ as $N \to \infty$,
such that $\Smeasure ( \bbB_{\gep _N} (\Sprofile))$ converges to 1. Therefore
for $N$ sufficiently large
\begin{equation}
\label{eq:lbound2}
\Smeasure \left( \bbB_\gep (\gr) \right)
\geq
\frac 12
\inf \left \{
\Prob_{\eta^N}
\left(
\empirical{\eta_\cdot}
\in \cV_{[0,T]}^\gep (\pi)
\right)
\, : \, \eta^N \; s.t. \; \empirical{\eta^N} \in \bbB_{\gep _N}
(\Sprofile)\right\} .
\end{equation}
Since $\gep_N \searrow 0$ as $N\nearrow \infty$,
we may apply the lower bound in Theorem \ref{th:LDdyn}
to obtain that
\begin{equation}
\label{eq:lbound3}
\liminf_{N \to \infty}
\frac 1N \log
\Smeasure \left( \bbB_\gep (\gr) \right)
\geq
- \inf_{\pi^\prime \in \cV_{[0,T]}^{\gep /2} (\pi) }
\LDfunc{\pi^{\prime}}{\Sprofile}
\ge - \LDfunc{\pi}{\Sprofile}
\ge - \quasipot (\rho) -\delta.
\end{equation}
Since $\delta$ can be chosen arbitrarily small,
\eqref{eq:lbound} is proven and the lower bound
in Theorem \ref{th:LDstationary} is established.
\qed (Lower bound)
\bigskip
\subsection{Upper Bound}
We are now going to check that for any closed subset $\cC$ of $\Fspace$
\begin{eqnarray}
\label{eq:ubound}
\limsup_{N \to \infty} \
\frac{1}{N} \; \log \Smeasure \left( \cC \right)
\leq - \quasipot (\cC) \, ,
\end{eqnarray}
where $\quasipot (\cC) = \inf_{\pi \in \cC} \quasipot (\pi)$.
Since if $\Sprofile\in \cC$ the result is trivial, let us assume
$\Sprofile\not\in \cC$.
Therefore there exists $\gd>0$ such that $\bbB_{4 \gd} (\Sprofile)\cap \cC =
\emptyset$.
We fix $\gd$ throughout the proof and set
\begin{eqnarray*}
\gt = \bbB_\gd (\Sprofile)
\qquad \text{and} \qquad
\gG = \left\{ \gr \in \Fspace \ | \ 3 \gd \leq \text{dist}
(\Sprofile, \gr)\leq 4 \gd \right\} \, .\\
\end{eqnarray*}
For any subset $A$ of $\Fspace$, let $\tau_A$ be the first return
time in $A$ of the process $\{ \empirical{\eta_t} \}_{t \geq 0}$.
We introduce also $\tau_1$ defined as follows
\begin{equation}
\tau_1 = \inf \left\{ t > 0 : \text{ there exists }
s \in [0,t) \text{ such that } \empirical{\eta_s} \in \gG
\text{ and }
\empirical{\eta_t} \in \gt \right\} \,.
\end{equation}
In order to state a classical representation of the invariant measure
$\Smeasure$ for the Markov chain $\{ \eta_t \}_{t \geq 0}$, we need to
introduce some more notation.
The first step is to define a notion of {\sl discrete external boundary} for $\gt$.
Let $\partial \gt^N$ be the set of configurations $\eta^N$ such that
there exists $k\in \N$ and a sequence of configurations $\eta^{N,0}, \dots,
\eta^{N,k} = \eta^N $ which satisfy the
constraints:
\medskip
\begin{enumerate}
\item for every $i$, the configuration $\eta^{N, i+1}$ can be deduced from
the configuration $\eta^{N,i}$ by spin exchange or spin creation according
to the rule prescribed by the dynamics.
\item $\eta^{N,0} \in \gG$ and for every
$ i 0: \text{ there exists }
s \in (0,t) \text{ such that }
\empirical{\eta_s} \in \gG \text{ and } \eta_t \in \partial \gt^N \right\} \,.
\end{equation}
The sequence of stopping times obtained by iterating this procedure is
denoted by $\{ \tau^N_k \}$.
In this way an irreducible Markov chain $\{ X_k\}_{k=1,2,\ldots}$ is
defined on $\partial \gt^N$ by setting $X_k = X_{\tau^N_k}^\eta$
(see Remark~\ref{th:irred} at the end of the proof).
Since the irreducible chain $\{ X_k \}_{k=1,2,\ldots}$
evolves on a finite state space, it has a unique stationary measure
$\nu_N$ on $\partial \gt^N$.
Following \cite{cf:FW}, we represent the stationary measure
of the process $\{\eta_t\}_{t\ge 0}$ as
\begin{equation}
\Smeasure ( A ) =
\frac{1}{C_N}
\int_{\partial \gt^N} \,
\bbE_\eta \left(
\int_0^{\tau_1^N} \mathbf{1}_A ( \eta_s ) \, ds
\right)
\dd \nu_N( \eta)
\, ,
\end{equation}
for every $A\subset \Fspace$, with
\begin{equation}
C_N = \int_{\partial \gt^N} \,
\bbE_\eta ( \tau_1^N ) \dd\nu_N( \eta)\, .
\end{equation}
In order to estimate the probability of the set $\cC$, we
observe that the strong Markov property implies
\begin{equation}
\Smeasure ( \cC ) \leq
\frac{1}{C_N}
\sup_{\eta \in \partial \gt^N}
\bbP_{\eta} \left( \tau_\cC < \tau_1^N \right)
\;
\sup_{\eta \in \cC}
\bbE_{\eta} \left( \tau_1^N \right)\,.
\end{equation}
Moreover we notice that
there is $c>0$ such that $C_N \geq 1/{c N^2}$:
this comes from the fact that the process, which jumps with
jump rates of the order of $N^2$,
has to leave $\partial \gt^N$ before returning to it.
By construction $\tau_1^N \leq \tau_1$, thus
\begin{equation}
\label{eq:FW1}
\Smeasure ( \cC ) \leq
c N^2 \sup_{\eta \in \partial \gt^N}
\bbP^\eta_N \left( \tau_\cC < \tau_1 \right)
\;
\sup_{\eta \in \cC}
\bbE^\eta_N \left( \tau_1 \right)\,.
\end{equation}
The upper bound of the large deviations \eqref{eq:ubound} will
therefore follow from the following lemma. Recall that
most of the definitions we gave depend
on a positive (and sufficiently small) parameter $\delta$.
\medskip
\begin{lem}
\label{th:expest}
We have that
\begin{enumerate}
\item for every $\delta$
\begin{eqnarray}
\label{eq:step1}
\limsup_{N \to \infty} \
\frac{1}{N} \; \log \; \sup_{\eta \in \cC}
\bbE_\eta \big( \tau_1 \big)
\leq 0 \, ;
\end{eqnarray}
\item
for every $\epsilon>0$ there is $\delta_0$ such that
for $\delta \in (0, \delta_0)$
\begin{eqnarray}
\label{eq:step2}
\limsup_{N \to \infty} \
\frac{1}{N} \; \log \; \sup_{\eta \in \partial \gt^N}
\bbP_\eta \left( \tau_\cC < \tau_1 \right)
\leq - \quasipot (\cC) + \epsilon \, .
\end{eqnarray}
\end{enumerate}
\end{lem}
\bigskip
In the proof of Lemma
\ref{th:expest} we will make use of the following technical result:
\bigskip
\begin{lem}
\label{th:techresub}
There exists $T_0 > 0$, $c>0$ and $N_0>0$ such that
\begin{eqnarray}
\sup_{\eta \not \in \gt} \bbP_\eta \left( \tau_\gt > T \right)
\leq \exp \left( - c (T-T_0) N\right) \, ,
\end{eqnarray}
for every $T\geq T_0$ and $N \geq N_0$.
\end{lem}
\bigskip
\noindent
{\it Proof of Lemma~\ref{th:techresub}.}
The first step is to check that there is $T_0>0$ and $a >0$
such that if $\pi \in D([0,T_0], \Fspace )$
is such that $\pi (t)\in \Fspace\setminus \gt$ for every $t$ then
\begin{equation}
\label{eq:relax}
I_{[0,T_0]} (\pi) > a \, .
\end{equation}
To establish this start by considering the following Cauchy problem:
for given $\gr_0\in \Fspace$,
we look for $\gr(\cdot)\in C^0([0,T], \Fspace )$ such that
\begin{multline}
\label{eq:evlaw}
\bra J (T),\rho(T)\ket
-\bra J (0), \rho_0\ket
-\int_0^T \bra (\partial _t + \Delta) J(t), \rho (t) \ket \, dt\\
+\rho_+ \int_{0}^T \nabla J(t,1) \dd t
-\rho_- \int_{0}^T \nabla J(t,-1) \dd t=0,
\ \ \ \text{ for every } J \in C^{1,2}_0.
\end{multline}
This Cauchy problem is well posed and the solution
is classical for positive times. We can see this by first observing, for example
via Fourier analysis,
that there exists a solution $\rho \in C^{1,2}((0,T]\times [-1,1]$
satisfying
$\partial _t \rho (t)= \Delta \rho (t)$ for $t \in (0,T]$,
$\rho (t,\pm 1)= \rho^\pm$ and $ \lim_{t \searrow 0}
\rho (t)=\rho_0$ (in $\Fspace$, but also in $\bbL ^1$). Uniqueness follows from the following argument:
for any function $f_0 \in C^2_0([-1,1])$, we consider the
classical solution $f\in C^{1,2}_0([0,T] \times [-1,1])$ of the heat equation with $f(0)=f_0$.
We are now going to insert into
equation \eqref{eq:evlaw} the test function
$J(r,t) = f(r,T-t)$. Since
$$
\partial_t \, J(t) + \Delta J(t) = 0 \qquad \text{for every }t \in [0,T],
$$
the differential term in \eqref{eq:evlaw} disappears.
Let us then assume that $\rho$, $\bar \rho \in C^{0}([0,T^\prime];
\Fspace)$, $T\le T^\prime$, are two solutions of \eqref{eq:evlaw}
with the same initial data.
Let us set $\tilde \rho =\rho -\bar \rho$.
By using the test function $J$ and by linearity, we obtain
\begin{equation}
\label{eq:foruniq}
\int_{-1}^{+1} f_0 (r) \tilde \rho (T, r) \, \dd r
=0 \, .
\end{equation}
Finally, by approximation we can extend the validity
of \eqref{eq:foruniq} to every $f_0\in C^0_b([-1,1])$.
This implies that $\tilde \rho (T)=0$ and, since
$T$ is arbitrary, $\tilde \rho \equiv 0$.
\medskip
We claim now that the solution to \eqref{eq:evlaw}
relaxes in $\bbL^2 ([-1,1])$ exponentially fast to the equilibrium
profile.
In fact since by uniqueness $\rho (\cdot)$ is smooth,
for $t>0$ we have
\begin{equation}
\frac 12
\partial_t \| \gr (t) - \Sprofile \|_2^2 =
- \int_{-1}^{+1} \,
\left[
\nabla (\gr (t,r) - \Sprofile (r) ) \right]^2 \dd r \, .
\end{equation}
But the spectral gap of the Laplacian with Dirichlet boundary conditions
is strictly positive,
so we get that for some $c_1>0$
\begin{equation}
\label{eq:exprelax}
\partial_t \| \gr(t) - \Sprofile \|_2^2 \leq - c_1
\| \gr(t) - \Sprofile \|_2^2 \, ,
\end{equation}
for every $t>0$, and therefore the exponentially
fast convergence to equilibrium.
Since $0 \le \rho_0 \le 1$, \eqref{eq:exprelax}
implies that
for every $\gep>0$ there exists $T>0$ such that
\begin{equation}
\label{eq:relaxation}
\forall t \geq T, \qquad
\sup_{\rho_0 \in \Fspace}
\| \gr(t) - \Sprofile \|_2 \leq \gep \, .
\end{equation}
This ensures the existence of $T>0$ such
that $\gr(t) \in \gt$ for every $t >T$.
\medskip
We set $T_0 = 2T$ and we want to show
that \eqref{eq:relax} holds with this choice.
Let us assume that
this is not the case: then there exists a sequence $\pi_k$ of
trajectories in $D([0,T_0], \Fspace \setminus \gt)$ such that
$I_{[0,T_0]} (\pi_k) \leq 1/k$.
This, together with the fact that $I_{T_0}$ is l.s.c.
and has compact level sets,
implies the existence of $\pi$ in $D([0,T_0], \Fspace)$ taking
values in the closure of $\Fspace \setminus \gt$ and such that
$I_{T_0} (\pi) = 0$.
Then $\pi$ solves \eqref{eq:evlaw},
and, as we saw in \eqref{eq:relaxation}, $\pi (t) \in \gt$
for $t \ge T$, which contradicts the assumption and we are done
with proving \eqref{eq:relax}.
\medskip
From \eqref{eq:relax}, we know that for some $a >0$ there
exists $T_a$ such that any trajectory in
\begin{eqnarray}
\gP (a) = \{ \pi \in C^0([0,T_a); \Fspace)\, :\,
I_{T_a} (\pi ) \leq a \},
\end{eqnarray}
enters in the neighborhood $\bbB_{\gd/2} (\gr_S)$.
Notice that the interior of the set $\gP (a)$ is empty
(recall that we are working with the Skorohod
topology) and
we therefore choose to work with an open neighborhood
of $\gP(a)$:
\begin{equation}
\gP^\prime (a) = \cV^{\gd/2}_{[0,T_a]} \left( \gP (a) \right),
\end{equation}
and if $\pi \in \gP^\prime (a)$, then
$\pi (t) \in \gt$ for some $t\in [0,T_a]$.
This implies that
\begin{eqnarray*}
\left\{ \pi \in D([0,\infty); \Fspace) \, : \,
\tau_\gt >T_a \right\} \subset \left( \gP^\prime (a) \right)^\complement \, .
\end{eqnarray*}
Furthermore, by construction,
for any $\pi$ in $\big( \gP^\prime (a) \big)^\complement$
we have $I_{T_a} (\pi) > a$.
We are now in the position of applying
the dynamical large deviation principle: observe that
we can select a sequence $\{ \widetilde \eta ^N\}_{N=1,2,\ldots}$
such that
\begin{equation}
\max_{\eta^N: \empirical{\eta^N} \in \Fspace \setminus \gt}
\bbP_{\eta^N} \left( \tau_\gt > T_a \right)=
\bbP_{\widetilde \eta^N} \left( \tau_\gt > T_a \right)
\end{equation}
and by compactness of $\Fspace$ we can apply the large
deviations upper bound \eqref{eq: dynamical LD}
to every subsequence of $\{\widetilde \eta ^N\}_{N=1,2,\ldots}$
such that $\empirical{\widetilde \eta ^N}$ converges in $\Fspace$
to obtain that there exists $N_0$ such that for $N >N_0$
\begin{equation}
\sup_{\eta^N: \empirical{\eta^N} \in \Fspace \setminus \gt}
\bbP_{\eta^N} \left( \tau_\gt > T_a \right)
\leq
\sup_{\eta^N: \empirical{\eta^N} \in \Fspace \setminus \gt}
\bbP_{\eta^N} \left( \left( \gP^\prime
(a) \right)^\complement \right)
\leq
\exp \left( - \frac{a}{2}N \right) \, .
\end{equation}
By using the Markov property we can iterate this procedure
to get that for $N>N_0$
\begin{equation}
\begin{split}
\sup_{\eta^N: \empirical{\eta^N} \in \Fspace \setminus \gt}
\bbP_{\eta^N} \left( \tau_\gt > k T_a \right)
&\leq \sup_{\eta^N: \empirical{\eta^N} \in \Fspace \setminus \gt}
\bbE_{\eta^N} \left( 1_{ \{ \tau_\gt > (k-1) T_a \}} \;
\bbP_{\eta_{(k-1) T_a}} \left( \tau_\gt > T_a \right)
\right) \\
&\leq \exp \left( - \frac{a k}{2}N \right) \, ,
\end{split}
\end{equation}
where $k$ is an arbitrary positive integer number.
The proof is therefore complete.
$\stackrel{\text{Lemma } \ref{th:techresub}}{\qed}$
\bigskip
\noindent{\sl Proof of Lemma \ref{th:expest}.}
By construction $\cC \cap \bbB_{4 \gd} = \emptyset$.
Therefore, for $N$ large enough, any trajectory $\empirical{\eta_\cdot}$ starting from
$\cC$ will cross $\gG$ before touching $\gt$ (the jumps of
$\text{dist} (\empirical{\eta _\cdot}, \Sprofile)$ are in fact
of order $1/N$).
This implies that $\tau_1$ can be replaced by $\tau_\gt$ in
\eqref{eq:step1}.
By applying Lemma \ref{th:techresub}, we see that uniformly
in $\eta^N$ such that $\empirical{\eta^N} \in \cC$ for $N >N_0$
\begin{equation}
\bbE_{\eta^N} \big( \tau_\gt \big)
\leq T_a\left( 1+ \sum_{k =1}^\infty \, \bbP_{\eta^N} \left(
\tau_\gt \geq k T_a
\right)\right)
\leq T_a \sum_{k =0}^\infty \, \exp \left( - \frac{a k}{2} N \right) \, .
\end{equation}
Therefore \eqref{eq:step1} holds and the first
part of Lemma \ref{th:expest} is established.
\medskip
In order to prove \eqref{eq:step2}, it is enough to check that for every
$\gep>0$ we can find $\delta >0$ such that
\begin{equation}
\limsup_{N \to \infty} \
\frac{1}{N} \; \log \; \sup_{\eta \in \widetilde \gt}
\bbP_\eta \left( \tau_\cC < \tau_1 \right)
\leq - \quasipot (\cC) + \gep \, ,
\end{equation}
where $\widetilde \gt = \bbB_{2 \gd} (\Sprofile)$.
Lemma \ref{th:techresub} ensures that there is $T>0$ and $N_0>0$
large enough such that for $N >N_0$
\begin{equation}
\forall N \geq N_0, \qquad
\sup_{\eta \in \tilde \gt} \bbP_\eta \left( \tau_1 > T \right)
\leq \exp \big( - N ( \quasipot(\cC) + 1) \big) \, .
\end{equation}
Thus it remains to check that for $N$ large
\begin{equation}
\sup_{\eta \in \tilde \gt} \bbP_\eta
\left( \tau_\cC \leq \tau_1 \leq T \right)
\leq \exp \left( - ( \quasipot(\cC) - \gep ) N\right) \, .
\end{equation}
Since $\cC$ and $\gt$ are closed sets, the set of trajectories
such that $\{\tau_\cC \leq \tau_1 \leq T \}$ is also a closed
subset (of $D([0,T], \Fspace)$).
Therefore it is enough to check that for any $\pi$ such that
$\pi(0) \in \tilde \gt$ and $\pi(t) \in \cC$ for some $t \in [0,T]$
\begin{equation}
I_T ( \pi ) \geq \quasipot (\cC) - \gep \, ,
\end{equation}
for $\delta$ sufficiently small.
If this is not true then one can choose $\gd = 1/k$ and
a sequence $\pi_k$ in $D([0,T], \Fspace)$ such that for
some $\ga > 0$
\begin{equation}
\limsup_{k\to \infty} I_T ( \pi_k ) < \quasipot (\cC) - \ga \, .
\end{equation}
But $\{ \pi_k (0) \}_{k=1,2,\ldots}$ converges to $\Sprofile$ and,
since $I_T$ has compact level sets,
one can extract a subsequence of $\{ \pi_k\}_{k=1,2,\ldots}$ which
converges in $D([0,T];\Fspace)$ to $\pi$ such that
\begin{equation}
\pi(0) = \Sprofile, \quad \tau_\cC \leq T, \quad \text{ and } \quad
I_T (\pi) \leq \quasipot (\cC) - \ga \, ,
\end{equation}
by lower semicontinuity of the functional $I_T$.
By the definition \eqref{eq:quasipotential}
of $\quasipot$, this is a contradiction
and this completes the proof of Lemma \ref{th:expest} and, with it,
the proof of the upper bound of Theorem \ref{th:LDstationary}.
\qed (Upper bound)
\bigskip
\begin{rem}
\label{th:irred}
\rm
While the irreducibility of the Markov process $\{ \eta_t \}_{t \geq 0}$
is clear, we would like to
comment on the irreducibility of the chain $\{ X_k \}_{k }$
introduced right after \eqref{eq:tau1st}.
Let $\eta^{(1)},\eta^{(2)}$ be in $\partial \gt^N$. By definition of $\eta^{(2)}$,
there is a sequence of (particle) configurations $\{ \gz_1, \dots, \gz_k = \eta_2 \}$
leading from $\gG$ to $\eta^{(2)}$, keeping out of $\partial \gamma ^N$ except
for the last point (that is $\eta^{(2)}$).
Therefore it is enough to check that one can find a sequence of
configurations $\{ \gs_1, \dots, \gs_{k^\prime} \}$ which does not touch
$\gG$ and which leads from $\eta^{(1)}$ to $\eta^{(2)}$: notice that we are allowed
to go from one configuration to another only via the elementary
steps of the dynamics.
In fact, if we can find it by considering the sequence of configurations
$\{ \gs_1 = \eta^{(1)}, \dots, \gs_{k^\prime}, \gz_{k-1}, \dots, \gz_2,
\gz_1, \gz_2,
\dots, \gz_k = \eta^{(2)} \}$ that starts from $\eta_1$ and intersects
$\partial \gt^N$ for the first time (after having touched $\gG$) at
the point $\eta^{(2)}$, we are done.
As $\gt$ is convex, the functions
$\big\{ u_k \equiv\frac{k}{K}\empirical {\eta^{(1)}} + (1- \frac{k}{K})
\empirical{\eta^{(2)}}
\big\}_{0\le k \leq K}$ belong
to $\bbB_{2\delta} (\Sprofile)$ for any $K \in \bbN$.
Choose $K$ much bigger than $1/\delta$ and consider only integer $k$'s:
it should be clear that we are done if we show how
to go, for $N$ sufficiently large, from $\eta$ to
$\sigma$, $\mathrm{dist}(\empirical\eta , \empirical \sigma )\le 2/K$
passing through configurations $\gz$ such that $\mathrm{dist}
(\empirical\gz , \empirical\sigma)\le 4/K$.
This is achieved by taking into account
that:
\begin{enumerate}
\item
By choosing $N$ sufficiently large we may assume that
birth or death are allowed at any point of the
system: for example for a birth at a site $x$
choose the first particle on the right and displace it
by elementary hops till $x$ and restart, till there is no particle
on the right and just have one be born and displace it
till the right position. Analogous
reasoning for the death of a particle.
\item Partition $[-1,1]$ in (say) at least $K^2$ (but no more
than $2K^2$) intervals
of equal length. Two functions in $\Fspace$ which differ
only on one of these subintervals are closer than $1/K^2$.
\item
Finally, by taking $N$ sufficiently large we may assume
that we can approximate two functions $u$ and $v $ in $\Fspace$ which differ
only on one of the subintervals via two particle configurations
$\sigma$ and $\eta$ such that $\text{dist}(\empirical{\sigma}, u)\le 1/K^3$
and $\text{dist}(\empirical{\eta}, v)\le 1/K^3$.
\end{enumerate}
By using the three steps above, one performs
the requested path.
\end{rem}
\section{About more general exclusion processes}
\label{sec:extensions}
\setcounter{equation}{0}
The aim of this short section is to stress that
the proof of
Theorem
\ref{th:LDstationary} is {\sl very little model dependent},
once a result like
Theorem~\ref{th:LDdyn} is known.
Therefore we expect it to
be susceptible of generalization
to a broad class of model.
This however passes through clarifying a
number of issues, that are of analytical rather
than probabilistic nature.
%and that have been
%a bit overlooked in the hydrodynamic limits
%literature up to now.
We will not attempt to
solve these points here: we merely list them
and connect them with the argument presented in
this note.
\subsection{Boundary driven exclusion processes: hydrodynamics
and hydrostatics}
A natural generalization of the boundary driven SSEP are
boundary driven {\sl Kawasaki} dynamics. By this we mean processes
generated by operators of the form
\begin{equation}
\label{eq:generator}
\generator f(\eta)=
\frac{N^2}2
\sum_{x,y \in \gL _N}
c(x,y,\eta)\left[ f(\eta^{x,y})- f(\eta)\right]
+\sum_{x\in \gL _N :\vert x \vert =N}
c(x,\eta) \left[ f(\eta^{x}) - f(\eta)\right],
\end{equation}
which clearly generalizes \eqref{eq:L}.
The arising process is clearly the superposition of
a dynamics with a conservation law
({\sl Kawasaki dynamics}: the rates are $c(x,y,\eta)$), acting on the whole
of $\gL_N$, and a dynamics
without conservation laws ({\sl Glauber dynamics} or {\sl birth and
death dynamics}: the rates are $c(x,\eta)$),
acting only at the boundary.
Some hypotheses on
the rates should be imposed
and we present them
in a rather informal way, we refer
to \cite{cf:ELS1} for precise definitions:
consider first the class of
finite range non-degenerate models
of particles hopping
on $\gL_N$, with birth and death at the boundary,
which are reversible (cf. \cite[pp. 161--164]{cf:Spohn})
with respect to a finite volume
Gibbs measure associated to a translation invariant
family of specifications.
Of course the chemical potential of the Gibbs
measure will be related to the (equal at $\pm N$!) {\sl activity}
of the birth and death process at the boundary.
Moreover the value of the mean density (or expected
value of the occupation number, under the Gibbs
measure), which will be independent
of the space coordinate, is determined by the
chemical potential. Under
these prescriptions, the Kawasaki
rates are (unlike the Glauber rates) independent
of the chemical potential.
The general class of dynamics of interest
corresponds to choosing the Kawasaki rates exactly
like in the previous example, but this time we allow
the possibility of choosing
Glauber rates $c(\pm N, \eta)$
with different activities at $\pm N$.
Thus, while the dynamics is locally reversible,
in general
it is not globally reversible and
one has no expression for the invariant measure.
\subsection{Hydrodynamics, invariant measure and hydrostatics}
\label{subsec : Hydrodynamics EXT}
In \cite{cf:ELS2}
it has been proven that the hydrodynamic limit of such systems are
described by parabolic non degenerate equations
\begin{equation}
\label{eq:weak1 EXT}
\partial_t \rho (t,r) =\nabla \left[ D(\rho (t,r))\nabla \rho (t,r)\right]
\text{ for every } (t,r) \in \R^+ \times (-1,1),
\end{equation}
with $\rho (t, \pm 1)=\rho_{\pm}$ for every $ t\in \R^+$.
We remark here that in \cite{cf:ELS2}
such a result is proven only for {\sl gradient} models \cite{cf:Landim}:
in this case it is easy to see that $D(\cdot)$ is a smooth function,
see \cite[formula (3.5)]{cf:ELS1}.
The result may be extended to
{\sl non--gradient} models \cite{cf:Landim}: then
$D(\cdot)$ can be expressed in terms of a
{\sl Green--Kubo} formula, see e.g. \cite[p.180]{cf:Spohn},
and it is not as easy to obtain its regularity properties.
We would like to stress that, at least in one--dimensional cases, the hydrodynamic
limit problem (the law of large numbers) with boundaries is rather well understood as long as
the corresponding problem without boundaries (say: on a torus)
is understood.
Moreover these results rely
on the absence of phase transitions, which of course
is ensured for
local models in $d=1$.
\medskip
Once again for every fixed $N$
the assumptions we make on the rates are (largely) sufficient
to ensure the existence of a unique invariant measure
({\sl steady state}) that we will call $\Smeasure$.
In \cite{cf:ELS1}, for the gradient case, and
in \cite{cf:KLO} and \cite{cf:LMS}
for some non-gradient ones, a law of large numbers
for $\{\Smeasure \}_N$ has been established.
It has in fact been proven that the law
of the empirical field
on the steady state
converges
as $N$ tends to infinity to the measure
on $\Fspace$ concentrated on the unique
solution $\Sprofile$ of the non-degenerate elliptic
equation
\begin{equation}
\label{eq:elliptic EXT}
\nabla\left[ D(\Sprofile(r)) \nabla \Sprofile(r) \right]=0,
\ \ \ \ \ \text{ for every } r \in (-1,1),
\end{equation}
and $\Sprofile (\pm 1)=\rho_\pm$.
\subsection{From dynamic to static large deviations}
It is not difficult to guess what the dynamical large deviation function
should be in this general case: going back to
Section~\ref{subsec: static large deviations}, it suffices
to replace formula \eqref{eq:Jfunct} with
\begin{equation}
\label{eq:Jfunct EXT}
\begin{split}
\ld{H} (\pi)=&
\langle \pi (T) , H(T) \rangle- \langle \rho_0 , H(0)\rangle-
\int_0^T
\left[
\bra \pi (t) , \partial_t H (t) \ket + \bra \theta (\pi(t)),
\Delta H(t) \ket
\right]
\dd t
\\
&+ \theta (\rho^+)\int_0^T \nabla H(t,1) \dd t-
\theta(\rho^-)\int_0^T \nabla H(t,-1) \dd t
-\frac12
\int_0^T \langle \sigma (\pi(t)) , (\nabla H (t))^2 \rangle \dd t,
\end{split}
\end{equation}
where $\theta (0)=0$, $\theta^\prime =D$ and $\sigma$ is a function
from $[0,1]$ to $[0,\infty)$.
The function $\sigma$ ({\sl mobility}, {\sl conductivity})
is related to the diffusion coefficient $D$ via the so called
Einstein relation
\cite{cf:Spohn}: $D$ and $\sigma$
coincide up to a multiplicative
density dependent factor ({\sl compressibility}),
which is a thermodynamical coefficient
which depends only on the equilibrium measure,
and therefore it is regular.
Of course the expected Large Deviations functional for
$\{ \mu_N\}_N$ is still given by the quasipotential
\eqref{eq:quasipotential}.
\bigskip
The argument of this note goes through word by word if
\medskip
\begin{enumerate}
\item
One has the generalization of
Theorem~\ref{th:LDdyn}. It should be noted
that the {\sl hydrodynamic limit} technology
(\cite{cf:Landim}, \cite{cf:VY})
naturally provides the
{\sl super--exponential}
probabilistic
estimates that allow to analyze
large deviation events and leads
to the proof of a full upper bound
and a lower bound for neighbors
of smooth trajectories.
The full lower bound is recovered if one
can show that
$I_{[0,T]} (\rho_n)\to I_{[0,T]} (\rho) $
for a sequence of smooth functions
$\rho_n$ which tends to $\rho$ in $C^0([0,T]; \Fspace)$
(in the SSEP case this is shown by using some
convexity properties that are absent in the general context).
Moreover we require $I_{[0,T]}$ to be
a good rate functional: while the compactness
of the level sets follows by the standard arguments,
one has to provide a proof of lower semicontinuity.
\item One has uniqueness to the weak formulation
of the limit PDE \eqref{eq:weak1 EXT}, that is
there exists a unique $\rho \in C^0 ([0,T];\Fspace)$ such
that
$I_{[0,T]} (\rho\vert \rho_0)=0$, $\rho_0 \in \Fspace$.
This result is already known, see \cite[th. 4.1 page 365]{cf:Landim},
with periodic boundary conditions. We remark that
we used also the regularity of $\rho$ for positive
times, and therefore $\rho$
is a classical solution to \eqref{eq:weak1 EXT} for
$t>0$: however this requirement may be
weakened and the argument goes through, once uniqueness
is established, if
there exists
a standard weak solution (in the $\bbH _1$ sense)
to \eqref{eq:weak1 EXT} for positive times.
Standard parabolic regularity results
may be applied if $D$ is differentiable and in this case
there exists a classical solution to \eqref{eq:weak1 EXT}.
\end{enumerate}
\section*{Acknowledgements}
We warmly thank L.~Bertini, B.~Derrida, C.~Landim and
J.~Lebowitz
for the continuous exchanges we had with them in the last two
years on and around the topic of this work.
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\end{document}
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