Content-Type: multipart/mixed; boundary="-------------0009111001867" This is a multi-part message in MIME format. ---------------0009111001867 Content-Type: text/plain; name="00-355.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-355.comments" 24 pages. ---------------0009111001867 Content-Type: text/plain; name="00-355.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-355.keywords" Asymmetric zero range, Hydrodynamical limit, Large deviations, Random environment. ---------------0009111001867 Content-Type: application/x-tex; name="ldpJSP.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ldpJSP.tex" \documentstyle[a4wide,12pt]{article} \pagestyle{myheadings} \def\sym{\fam\comfam\com} \font\tensym=msbm10 \font\sevensym=msbm7 \font\fivesym=msbm5 \newfam\symfam \textfont\symfam=\tensym \scriptfont\symfam=\sevensym \scriptscriptfont\symfam=\fivesym \def\sym{\fam\symfam\relax} \def\N{{\sym N}} \def\Z{{\sym Z}} \def\Q{{\sym Q}} \def\D{{\sym D}} \def\R{{\sym R}} \def\C{{\sym C}} \def\T{{\sym T}} \def\X{{\sym X}} \def\E{{\sym E}} \def\P{{\sym P}} \def\1{{\bf 1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\nit}{\noindent} \newcommand{\ti}{\partial} \newcommand{\bh}{\bar H} \newcommand{\ab}{\bg<} \newcommand{\ba}{\bg>} \newcommand{\hh}{{\cal H}} \newcommand{\cl}{{\cal C}} \newcommand{\dl}{{\cal D}} \newcommand{\mm}{{\cal M}_+} \newcommand{\jj}{{\cal J}_H} \newcommand{\ii}{{\cal I}} \newcommand{\tf }{{\cal T}} \newcommand{\Cf}{{\cal B}} \newcommand{\fkn}{\f{k}{N}} \newcommand{\bl}{\label} \newcommand{\fxn}{\bg(\f{x}{N}\bg)} \newcommand{\Bq}{\begin{equation}} \newcommand{\Eq}{\end{equation}} \newcommand{\bg}{\bigg} \newcommand{\fl}{\frac{1}{2l+1}} \newcommand{\fN}{\f{1}{N}} \newcommand{\Bg}{\Bigg} \newcommand{\pro}{\P_{\rho,p}^N} \newcommand{\hpro}{\P_{\ma,N}^{p,H}} \newcommand{\esp}{\E^N_{\rho,p}} \newcommand{\f}{\frac} \newcommand{\lspA}{\overline{\lim_{A\ra\ty}}\ } \newcommand{\lspN}{\overline{\lim_{N\ra\ty}}\ } \newcommand{\lifN}{\liminf_{N\ra\ty}\ } \newcommand{\lspl}{\overline{\lim_{l\ra\ty}}\ } \newcommand{\lspk}{\overline{\lim_{k\ra\ty}}\ } \newcommand{\lspbt}{\overline{\lim_{\bt\ra 0}}\ } \newcommand{\lspdt}{\overline{\lim_{\dt\ra 0}}\ } \newcommand{\nbr}{{\bar\nu}_\rho} \newcommand{\nbrl}{{\bar\nu_{\rho,0,l}^p}} \newcommand{\nrl}{{\bar\nu_{\rho,0,l}}} \newcommand{\nbrxl}{{\bar\nu_{\rho,x,l}}} \newcommand{\lspe}{\overline{\lim_{\e\ra 0}}\ } \newcommand{\lspfa}{\overline{\lim_{\fa\ra 0}}\ } \newcommand{\Ww}{W_{N,\e}^{H,\Psi}} \newcommand{\sx }{\sum_x} %%%%%%%%%%%%%%%%%%%%%%% \newcommand{\wg}{\Lambda} \newcommand{\ld}{\lambda} \newcommand{\ta}{\theta} \newcommand{\fa}{\alpha} \newcommand{\ma}{\gamma} \newcommand{\lm}{\lambda} \newcommand{\e}{\varepsilon} \newcommand{\nd}{\frac{1}{N}} \newcommand{\bt}{\beta} \newcommand{\vp}{\varphi} \newcommand{\dt}{\delta} \newcommand{\al}{(2l+1)^d} \newcommand{\eN}{(2\e N+1)} \newcommand{\om}{\omega} \newcommand{\ty}{\infty} \newcommand{\ra}{\rightarrow} \newcommand{\dw}{\downarrow} \newcommand{\adl}{A_{\delta,\fa}^l} \newcommand{\di} {{\ \rm d}} \newcommand{\n} {\nu_\vp^p} \newcommand{\mn}{\mu^N} \newcommand{\vn}{\nu_{\vp,0,l}} \newcommand{\fx}{\vp p_x^{-1}} \newcommand{\fkl}{\nu_{\vp,x,l}} \newcommand{\fk}{f_{x,y,l}^p} \newcommand{\lglim}{\longrightarrow} \newcommand{\si}{\sigma} \newcommand{\pn}{\pi_t^N} \newcommand{\feN}{\frac{1}{\eN}} \newcommand{\A}{A^{l,k,\dt}} \newcommand{\mx}{\max_{0\leq K\leq A\al}} \newcommand{\Pm}{\P^{N,p}_{\mn}} \newcommand{\vk}{\vskip0.3truecm} \newcommand{\Em}{\E^{N,p}_{\mn}} \renewcommand{\baselinestretch}{1.3} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%Title%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{\large{Large deviations for a zero mean asymmetric zero range process in random media.}}\vskip0.2cm \author{A. Koukkous\kern -0.5pt \addtocounter{footnote}{10} \renewcommand{\thefootnote}{\alph{footnote}}\footnotemark% \ \ and\ H. Guiol\kern -2pt \addtocounter{footnote}{-5 }\renewcommand{\thefootnote}{\alph{footnote}} \footnotemark} \maketitle \renewcommand{\thefootnote}{\alph{footnote}} \addtocounter{footnote}{7}\footnotetext{$^{,k}$IMECC-UNICAMP, P.B. 6065, 13083-970, Campinas, SP, Brasil.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%Abstract%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We consider an asymmetric zero range process in infinite volume with zero mean and random jump rates starting from equilibrium. We investigate the large deviations from the hydrodynamical limit of the empirical distribution of particles and prove an upper and a lower bound for the large deviation principle. Our main argument is based on a super-exponential estimate in infinite volume. For this we extend to our case a method developed by Kipnis \& al. (1989) and Benois \& al. (1995). \vk\nit {\it Keywords: } Asymmetric zero range, Hydrodynamical limit, Large deviations, Random environment. \vk\nit {\it AMS 2000 classification } 60K35, 60K37, 82C22. \end{abstract} %} \noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%INTRODUCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} The so called zero range process is one of the simplest particle systems that has been systematically and successfully investigated in random or inhomogeneous media in the last few years (see for instance Benjamini \& al. (1996), Evans (1996), Krug-Ferrari (1996), Landim (1996), Gielis \& al. (1998), Bahadoran (1998), Sepp\"{a}l\"{a}inen-Krug (1999), Koukkous (1999), Andjel \& al. (2000)) . The zero process can be described informally as follows. Particles are distributed on the $d$-dimensional lattice $\Z^d$. Each particle at site $x$ of $\Z^d$ jumps, with a rate depending only on the total number of particles standing at this site, to the left or to the right. In what follows, we consider a sequence of random variables $p=(p_x)_{x\in\Z^d}$ ({\it called an environment}) in $[a_0,a_1]$ (where $00$, $m\{p: p_0\in [a_0,a_0+\e)\}m\{p: p_0\in (a_1-\e,a_1]\}>0$. \vk We denote by $\X_d:=\N^{\Z^d}$ the configuration space and by Greek letters $\eta$ and $\xi$ its elements. As usual $\eta(x)$ stands for the total number of particles at site $x$ for the configuration $\eta$. For each environment $p$, we are interested in the Markov process ${(\eta_t)}_{t\geq 0}$ on $\X_d$ whose generator is defined by \Bq \label{generateur} ({\cal L}_pf)(\eta)=\sum_{\scriptstyle{x,y\in\Z^d}} p_xg(\eta(x))T(x,y)[f(\eta^{x,y})-f(\eta)], \Eq where $f:\X_d\rightarrow\R$ is a bounded cylinder function, that is $f$ only depends on $\eta$ through a finite number of coordinates. $T(\cdot,\cdot)$ is a transition probability on $\Z^d$. The function $g$ is positive and vanishes at $0$: $g(0)=0 < g(k)\ \mbox{for all}\ k\geq 1$. In the previous formula, $\eta^{x,y}(z)$ is the configuration obtained from $\eta$ when a particle jumps from $x$ to $y$: \[ \eta^{x,y}(z) = \left\{\begin{array}{ll} \eta(z) & \ \mbox{if $z\neq x,y$} \\ \eta(x)-1& \ \mbox{if $z=x$} \\ \eta(y)+1& \ \mbox{if $z=y$\ \ \ .} \end{array}\right. \] For every non-negative real $\vp$ we denote by $\nu_\vp^p$ the product measure on $\X_d$ whose marginals are defined by \[ \nu_\vp^p\{\eta : \eta(x) =k\} = \frac{1}{Z(\fx)} \frac{(\fx)^k}{g(k)!}, \ \ \ \mbox{for all}\ \ k\geq 0, \] where $g(k)!=g(1)g(2)...g(k)$ if $k>0$ and $g(0)!=1$. Under some hypotheses (for instance [H1] and [H2] in what follows), those measures are invariant for the process. In this formula, $Z : \R_+\ra\R_+$ is the partition function \[ Z(\vp)=\sum_{k \geq 0} \frac{\vp^k}{g(k)!}. \] Let $\vp^*$ be the radius of convergence of $Z(\cdot)$; we assume that \Bq \bl{eq1} \lim_{\vp\uparrow\vp^*}Z(\vp) =+\infty. \Eq Denote by $\nu_{\vp}(\cdot):=\nu_\vp^1(\cdot)$ the invariant measure of the process $(\eta_t)_{t\geq 0}$ when $m$ is the Dirac measure concentrated on the set $\{p : p_x=1,x\in \Z^d\}$ (see Andjel (1982)). We define $M : [0,\vp^*)\ra\R_+$ by $M(\vp)=\nu_\vp[\eta(0)]$, the expected number of particles at $0$ with respect to $\nu_{\vp}$. \vk A simple computation shows that $M(\vp)=\vp\partial_\vp \log Z(\vp)$ and from assumption (\ref{eq1}) we check that $M$ is an increasing, continuous, one-to-one function from $[0,\vp^*)$ to $\R_+$. We define the ``density" of particles ({\it i.e.} the expected number of particles at $0$) with respect to the random media by the continuous and increasing function $R : [0,a_0\vp^*)\ra\R^+$ such that \[ R(\vp)=m[M(\vp p_0^{-1})] \] and in order to ensure the existence of an invariant measure for any given value of the density, we assume that \Bq \bl{eq2} \lim_{\vp\uparrow a_0\vp^*} R(\vp) =\infty. \Eq Under this assumption the function $R$ is one to one from $[0,a_0\vp^*)$ to $\R_+$. We denote by $\Phi$ its inverse (which is also a continuous increasing bijection). \vk For a density $\rho>0$ we write $$\nbr^p=\nu_{\Phi(\rho)}^p.$$ In the following we state all the hypotheses assumed throughout this paper. \vk {\bf [H1]}\ \ The transition probability $T(\cdot,\cdot)$ on $\Z^d$ is a zero-mean irreducible translation invariant probability with finite range. That is \[ \begin{array}{c} \hspace{2.1cm}T(x,y)=T(0,y-x)=:T(y-x), \ \hspace{2.5cm}\\ \mbox{there exists a constant }A>0\mbox{ such that } T(x)=0\ \ \mbox{if}\ \ |x|\geq A\\ \ \mbox{and }\ \ \displaystyle{ \sum_{x\in\Z^d}x~T(x)=0}. \end{array} \] {\bf [H2]}\ \ The rate function $g$ has bounded variation: \[ \hspace{3.1cm}g^*=\sup_k|g(k+1)-g(k)|<\infty.\hspace{4.5cm} \] Under the hypotheses [H1] and [H2] there exists a unique Markov process with corresponding generator defined by (\ref{generateur}) for the deterministic case {\it i.e.} $p\equiv 1$ (see Andjel (1982)). Andjel's proof applies also in the case we consider. Let $(\si_{ij})_{\{1\leq i,j\leq d\}}$ be a symmetric nonnegative definite matrix defined by the covariance matrix of the transition probability $T(\cdot)$: \[ \si_{ij}=\sum_{y\in\Z^d}y_iy_jT(y)\ \ \ \mbox{where}\ \ y=(y_1,\cdots,y_d). \] {\bf [H3]}\ In order to avoid the degenerate case of the hydrodynamic equation, we assume $(\si_{ij})_{\{1\leq i,j\leq d\}}$ to be a positive definite matrix. That is there exists $\kappa>0$ such that \[\hspace{3.1cm}\sum_{i,j}\si_{ij}x_ix_j\geq\kappa\sum_i x_i^2,\ \ \mbox{for all}\ \ x=(x_1,\cdots,x_d)\in \R^d.\hspace{3.5cm} \] {\bf [H4]} To ensure some finite exponential moments of $\eta(x)$ under the measures $\nu_\vp^p$ we shall assume that there exists a convex and increasing function $\om :\R_+\longrightarrow\R_+$ such that\\ (i) $ \om(0)=0,$ \\ (ii) $\lim_{x\ra\infty}(\f{\om(x)}{x})=\infty$\ and\\ (iii) for all density $\vp$ there exists a positive constant $\ta:=\ta(\vp)$ such that \[ \nu_\vp\bigg[\exp\big\{\ta\om(\eta(0))\big\}\bigg]<\infty. \] This last assumption ensures also that $Z(\cdot)$ has infinite radius of convergence. It holds for exemple if $g(k+1)-g(k)\geq g^*_0$ for some constant $g^*_0$ and $k$ sufficiently large.\par We will denote by $\om^*$ the Legendre transform of $\om$ given by: \Bq \bl{ome} \om^*(x)=\sup_{\fa>0}\{\fa x-\om(\fa)\}. \Eq In the next paragraphs, we define the state space of the process and its topology. Denote by ${\cal C}(\R^d)$ (resp. ${\cal C}_K(\R^d)$) the space of continuous (resp. with compact support) functions on $\R^d$ with classic uniform norm. Let $\mm$ denote the space of positive Radon measures on $\R^d$ with the weak topology induced by ${\cal C}_K(\R^d)$ via $\big<\pi,H\big>:={\large\int} H\di\pi$ for $H\in{\cal C}_K(\R^d)$ and $\pi\in\mm$. We fix a positive time parameter $\tf >0$. For each realization of the environment $p$ and all fixed positive density $\rho$, $\pro$ will denote the probability measure on the path space ${\cal D}([0,\tf ],\X_d)$ corresponding to the Markov process $(\eta_t)_{t\in[0,\tf ]}$ with generator $N^2{\cal L}_p$ starting from the measure $\nbr^p$. By $\esp$ we denote the expectation under $\pro$. \vk Let $\pi^N_.$ be the empirical measure defined on ${\cal D}([0,\tf ],\mm)$ by \[ \pn(du)=\f{1}{N^d}\sum_{x\in\Z^d}\eta_t(x)\dt_{x/N}(du), \] for $0\leq t\leq \tf $. Let $Q^N_{\rho,p}$ denote the measure on the path space $D([0,{\cal T}],\mm)$ associated to the process $\pi_.^N$ with generator $N^2{\cal L}_p$ starting from $\nbr^p$ . \vk To investigate the large deviations of the empirical measure, we shall consider some small perturbations of the zero range process as mentionned earlier. For this, we will need the following notation.\par Let ${\cal C}^{l,k}_K([0,\tf ]\times\R^d)$ denote the space of compact support functions with $l\in\N$ continuous derivatives in time and $k\in\N$ continuous derivatives in space. Let ${\cal C}_\rho(\R^d)$ be the set defined by \[ {\cal C}_\rho(\R^d)={\cal C}(\R^d)\cap\{u:\R^d\ra\R^+;\ \ u(x)=\rho \ \mbox{for}\ \ |x|\ \mbox{sufficiently large}\}. \] For a fixed $\ma$ in ${\cal C}_\rho(\R^d)$ and for some smooth function $H$ in $\cl^{1,2}_K([0,\tf ]\times\R^d)$ we consider the Markov process generated by \[ N^2({\cal L}_{N,t}^{p,H}f)(\eta)=N^2\sum_{\scriptstyle{x,y\in\Z^d}} p_xg(\eta(x))T(y)e^{\{H(t,\f{x+y}{N})-H(t,\f{x}{N})\}}[f(\eta^{x,x+y})-f(\eta)], \] where $f$ is a cylinder function. Let $\bar\nu_{\ma,N}^p$ be the initial product measure of this process with marginals \[ \bar\nu_{\ma,N}^p\{\eta, \eta(x)=k\}=\bar\nu_{\ma(x/N)}^p\{\eta, \eta(x)=k\} \] for all $x\in\Z^d$ and $k\in \N$. We therefore denote by $ \P_{\ma,N}^{p,H}$ and $Q^{p,H}_{\ma,N}$ the small perturbations of $\pro$ and $Q^N_{\rho,p}$ respectively. For any path $\pi_.\in\dl([0,\tf ],\mm)$, denote by $u_t$ the Radon-Nikodym derivative of $\pi_t$ with respect to the Lebesgue measure $\lm$: $u_t:=\f{\di\pi_t}{\di\lm}$. Let $\cal A=\cal A(\rho)$ be the space path of $\pi\in\dl([0,\tf ],\mm)$ such that $u_t$ is the solution of the PDE \[ \mbox{\bf(E)} \hspace{3cm}\left\{\begin {array}{ll} \partial_t u &=(\si/2)\triangle(\Phi(u))-\sum_{i=1}^{d}\partial_{x_i}(\Phi(u)\partial_{x_i}H)\\ u(0,\cdot)&=\gamma(\cdot)\ \ .\hspace{4.5cm} \end{array} \right.\hspace{2cm} \] for some $\gamma\in{\cal C}_\rho(\R^d)$ and some $H\in{\cal C}^{1,3}_K([0,\tf ]\times\R^d).$ $\triangle$ stands the Laplacian operator. \vk The following notation is devoted to the definition of the rate functional of the large deviation principle for $(\pi_.^N)_{0\leq t\leq\tf }$. \vk For $H\in\cl^{1,2}_K([0,\tf ]\times\R^d)$, we define $\jj\ :\dl([0,\tf ],\mm)\ra\R\cup\{\ty\}$ by \[ \jj(\pi)=\jj^1(\pi)-\jj^2(\pi) \] where \[ \jj^1(\pi)=\ab u_\tf ,H_\tf \ba-\ab u_0,H_0\ba-\int_0^\tf \ab u_t,\partial_tH_t\ba\di t, \] \[ \jj^2(\pi)=\f{\si}{2}\int_0^\tf \ab\Phi(u_t),\sum_{i=1}^{d}\bg(\partial_{x_i}^2H_t+(\partial_{x_i}H_t)^2\bg)\ba\di t, \] such that $\jj(\cdot)=\ty$ outside $\dl([0,\tf ],\mm)$ or if $\pi_t$ is not absolutely continuous with respect to the Lebesgue measure $\lm$ for some $0\leq t\leq\tf $. We are now ready to define the part of the large deviations rate function, $\ii_0(\cdot): \dl([0,\tf ],\mm)\ra[0,\ty]$ coming from the stochastic evolution: \[ \ii_0(\pi)=\sup_{H\in\cl^{1,2}_K([0,\tf ]\times\R^d)}\jj(\pi). \] The other part of the large deviations rate function coincides with the behaviour of deviations coming from the initial state. Let $h(\cdot|\rho)$ be the entropy defined for a positive function $\ma:\R^d\ra\R^+$ by \[ h(\ma|\rho)=\int_{\R^d}\Bg\{ \ma(x)\log\bg(\f{\Phi(\ma(x))}{\Phi(\rho)}\bg)- \E_m\bg[\log\bg(\f{Z(\Phi(\ma(x))p_0^{-1})}{Z(\Phi(\rho)p_0^{-1})}\bg)\bg]\Bg\}\ \di x. \] Thus, the rate function of the large deviation principle is defined for a density $\rho>0$ by \[ \ii_\rho(\pi)=\ii_0(\pi)+h(u_0|\rho). \] \vk From now on, for each $x\in\Z^d$, we denote by $\eta^l(x)$ the mean density of particles in a box of length $(2l+1)$ centered at $x$ : $$\eta^l(x)=\frac{1}{(2l+1)^d}\sum_{|y-x|\leq l}\eta(y).$$ For each cylinder function $\Psi : \X_d\ra\R$, we define \Bq \label{tilde} \tilde{\Psi}(\rho):=m\bg[\nu_{\Phi(\rho)}^p(\Psi)\bg]\ , \Eq and we say that $\Psi$ is a Lipschitz function if \[ \exists k_0\in\N\mbox{ and } c_0>0\mbox{ such that }\ \ \ \bg|\Psi(\eta)-\Psi(\xi)\bg|\leq c_0\sum_{|x|\leq k_0}\bg|\eta(x)-\xi(x)\bg|, \] for all $\eta$ and $\xi$ in $\X_d$.\vk\nit Denote by $\tau_x$ the shift operator defined by $\tau_x\Psi(\eta(\cdot)):= \Psi(\tau_x\eta(\cdot))$ where $\tau_x\eta(y)=\eta(x+y)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%SUPER-EXPONENTIaL ESTIMATE%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We can now state our results: \newtheorem{theorem}{Theorem}[section] \begin{theorem} Let $\Psi$ be a cylinder Lipschitz function and $H\in\cl^{0,2}_K([0,\tf ]\times\R^d)$. Under hypotheses [H1] to [H4], for all $\dt>0$ we have \bl{th2} \Bq \bl{superexp} \lspe\lspN\frac{1}{N^d}\log \pro\Bigg[\Bigg|\int_0^\tf W_{N,\e}^{H,\Psi}(t,\eta_t)\di t\Bigg|>\dt\Bigg]=-\infty \Eq $m$-almost surely, where $$W_{N,\e}^{H,\Psi}(t,\eta)=\frac{1}{N^d}\sx H(t,x/N)\bigg[\tau_x\Psi(\eta)-\tilde\Psi(\eta^{\e N}(x))\bigg].$$ \end{theorem} This theorem, called the super-exponential estimate, will be a crucial argument in the proof of the following large deviations principle: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%PRINCIPE DES GRANDES DEVIATIONS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%555 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \bl{th1} Under hypotheses [H1] to [H4], for every closed subset ${\cal C}$ and every open subset ${\cal O}$ of $\dl([0,\tf ],\mm)$, we have $$\limsup_{N\ra\ty}\frac{1}{N^d}\log Q^N_{\rho,p}({\cal C})\leq-\inf_{\pi\in{\cal C}}\ii_\rho(\pi)$$ and $$\lifN\frac{1}{N^d}\log Q^N_{\rho,p}({\cal O})\geq-\inf_{\pi\in{\cal O}\cap{\cal A}}\ii_\rho(\pi)$$ $m$-almost surely. \end{theorem} \nit{\bf\Large Remarks } Before starting to prove our results, we would like to mention some facts and claims that will be used and whose proofs are omitted. For more details the reader is refered to Kipnis-Landim's book (1999) and Benois \& al. (1995). {\bf [R1]\ \ \ } From Lemma I.3.5 of Kipnis-Landim's book (1999), the function defined by $\vp\longrightarrow\nu_\vp$ for $\vp>0$, is an increasing function (see also the proof of lemma 4.3 in Benois \& al. (1995)). Therefore, assumption [H4] implies that for a fixed environment $p$ defined in the beginning of the last section, for all $x\in\Z^d$ and $\vp>0$, there exists $\ta:=\ta(x,\vp)>0$ such that \[ \nu_\vp^p\bigg[\exp\big\{\ta\om(\eta(x))\big\}\bigg]<\infty\quad m \mbox{-almost surely.} \] {\bf [R2]\ \ \ } Assumption [H4] ensures that the function $\om^*$ defined by (\ref{ome}) is also a continuous convex function such that $\om^*(0)=0$. \par {\bf [R3]\ \ \ } A simple computation shows that from the second condition in [H4], for every $\e>0$ the function $\om^{-1}(r)-\e r$ is negative for each $r\geq C_2(\e)$, for some constant $C_2(\e)$ dependent only on $\e$.\par {\bf [R4]\ \ \ } By definition of $\om$ in [H4], the function defined on $\R^*_+$ by $\Omega(r)=\f{\om(r)}{r}$ is an increasing function.\par {\bf [R5]\ \ \ } For each cylinder Lipschitz function $\Psi(\cdot)$, the function $\tilde\Psi(\cdot)$ given by (\ref{tilde}) is also a Lipschitz function (see Lemma I.3.6 of Kipnis-Landim (1999)). Moreover one can check that $\tilde\Psi(k)\leq Ck$ for all $k\in\Z$ for some constant $C$.\par The strategy we adopted to prove the results is similar to the one presented in Benois \& al. (1995). However, we need some arguments developed in Koukkous (1999) in order to overcome the lack of translation invariance of the invariant measures for the zero range process in random media. We will thus focus only on the main differences. \vk From now on, to keep the notation simple, we will restrict our study to the one-dimensional case. The reader can extend the proofs to any dimension without any difficulty.\par %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%PROOF OF SUPER-EXPONENTIAL ESTIMATE %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{\bf Proof of Theorem \ref{th2}} \bl{pvsupexpo} Let $G$ be a positive continuous function on $\R$ defined by \Bq \bl{GG} G(x)=\sup_{y\in[x-1,x+1]}\max\bg\{|H(y)|,|\partial_yH(y)|,|\partial^2_yH(y)|\bg\}. \Eq We have \begin{eqnarray} \bl{pr1} \pro\bigg[\int_0^\tf W_{N,\e}^{H,\Psi}(t,\eta_t)\di t>\dt\Bigg]\nonumber\\ &\ &\hskip-3cm\leq\pro\bigg[\int_0^\tf \bigg\{W_{N,\e}^{H,\Psi}(t,\eta_t)\di t-\f{\bt}{N}\sx G\bigg(\f{x}{N}\bigg)\om(\eta_t(x))\bigg\}\di t>\dt/2\bigg]\nonumber\\ &\ &\hskip0.9cm+\pro\bigg[\int_0^\tf \f{\bt}{N}\sx G\bigg(\f{x}{N}\bigg)\om(\eta_t(x)) \di t>\dt/2\bigg] \end{eqnarray} for every $\bt>0$.\\ By Tchebycheff exponential inequality the first term in the left hand side in (\ref{pr1}) is bounded above by $$\exp\{-N\ta\dt/2\}\esp\Bigg[\exp\ta\int_0^\tf \bigg\{N\Ww(t,\eta_t)-\bt\sx G\bigg(\f{x}{N}\bigg)\om(\eta_t(x))\bigg\}\di t\Bigg]$$ for every $\ta>0$.\\ Therefore, we have to prove two Lemmas: \newtheorem{lemme}{Lemma}[section] \begin{lemme} \bl{L1} For every $G\in{\cal C}_K(\R)$, \Bq \bl{l1} \lspA\lspN\fN\log\pro\bigg[\int_0^\tf \fN\sx G(x/N)\om(\eta_t(x)) \di t>A\bigg]=-\ty \Eq $m$-almost surely. \end{lemme} \begin{lemme} \bl{L2} For any $\ta>0$ and $\bt>0$ \Bq \bl{l2} \lspe\lspN\fN\log\esp\Bigg[\exp\ta\int_0^\tf \bigg\{\Ww(t,\eta_t)-\bt\sx G\bigg(\f{x}{N}\bigg)\om(\eta_t(x))\bigg\}\di t\Bigg]=0. \Eq $m$-almost surely. \end{lemme} {\bf Proof of Lemma \ref{L1}}.\\ Using respectively Tchebycheff exponential inequality and Jensen inequality, we show that for every positive constant $\ta$, the logarithmic term in (\ref{l1}) is bounded above by $$-\ta AN+\log\esp\Bg[\f{1}\tf \int_0^\tf \exp\bigg\{\sx \ta \tf G(x/N)\om(\eta_t(x))\bigg\}\di t\Bg].$$ From the begining of [R1] and since the product measure $\bar\nu_\rho^p$ is invariant for the process and $p_x\in[a_0,a_1]$, a simple computation shows that the right hand side term in (\ref{l1}) is bounded above by \Bq \bl{ep2} \lspA\lspN\inf_{\ta>0}\Bigg\{-\ta A+\fN\sx \log\nu_{\Phi(\rho)a_0^{-1}}\bigg[\exp\bigg\{\ta\tf G(x/N)\om(\eta(0))\bigg\}\bigg]\Bg\}. \Eq Let $B>0$ be such that $$\mbox{supp}G\subset[-B,B].$$ From {\bf [H4]}, there exists $\ta_0>0$ such that $$\nu_{\Phi(\rho)a_0^{-1}}\Bg[\exp\bigg\{\ta_0 \tf\|G\|_\infty\om(\eta(0))\bigg\}\Bg]<\ty.$$ The lemma is proved in fact that (\ref{ep2}) is bounded above by $$\lspA\bigg\{-\ta_0A+2B\log\nu_{\Phi(\rho)a_0^{-1}}\bigg[e^{\big\{\ta_0\tf\|G\|_\ty\om(\eta(0))\big\}}\bigg]\bigg\}.\hspace{3cm}$$ \nit {\bf Proof of Lemma \ref{L2}}.\\ Let $$V(\eta)=\ta\bigg\{N\Ww(0,\eta)-\bt\sx G\bigg(\f{x}{N}\bigg)\om(\eta(x))\bigg\}.$$ Let ${\cal L}^p_V$ be the generator $N^2{\cal L}_p+V$ and ${\cal L}^{p,*}_V$ its adjoint operator, which is equal to $N^2{\cal L}^*_p+V$. If we denote by $S_{t}^{V,p}$ the semigroup associated to the generator ${\cal L}^p_V$, by the Feyman-Kac formula the expectation in the lemma is equal to $$\big\leq\big^\f{1}{2}.$$ Now, if we denote by $\ld_V$ the largest eigenvalue of the self-adjoint operator ${\cal L}^p_V+{\cal L}^{p,*}_V$, $$\partial_t\big = \big<({\cal L}^p_V+{\cal L}^{p,*}_V)S_t ^{V,p}1,S_t ^{V,p}1\big>\leq\ld_V\big.$$ By Gronwall's lemma we show that \Bq \bl{vpr} \big\leq\exp\bigg\{\tf \ld_V\bigg\}. \Eq Recall that we did not assume $T(\cdot)$ to be symmetric and therefore $\nu_{\Phi(\rho)}^p$ can be non-reversible for the process. However, at this level, our study is dealing with the reversible generator $N^2({\cal L}_p+{\cal L}_p^*)$. Thus we can assume the generator ${\cal L}_p$ to be reversible and $T(\cdot)$ given by $T(x)=(1/2)\1_{\{|x|=1\}}$.\par\nit Let $$I_{x,x+1}^p(f)=\f{1}{2}\int p_xg(\eta(x))\left[\sqrt{f(\eta^{x,x+1})}-\sqrt{f(\eta)}\right]^2\nbr^p(\di\eta),$$ and $D_p(\cdot)$ the Dirichlet form given by $$D_p(f)=\sx I_{x,x+1}^p(f).$$ Using the variational formula for the largest eigenvalue of a self-adjoint operator (see appendix A3.1 of Kipnis-Landim (1999)), from (\ref{vpr}) we reduce the proof of the lemma to show that for every positive $\ta$ $$\lspe\lspN\sup_f\Bigg\{\int\ta\bigg[\Ww(\eta)- \f{\bt}{N}\sx G\bigg(\f{x}{N}\bigg)\om(\eta(x))\bigg]f(\eta)\nbr^p(\di\eta)-ND_p(f)\Bigg\}\leq0.$$ The supremum is taken over all positive densities functions with respect to $\nbr^p$. \par\nit We use now some computations from Benois \& al. (1995) and Kipnis \& al. (1989). Let $$W_l^\Psi(\eta)=\fl\sum_{|y|\leq l}\tau_y\Psi(\eta)-\tilde\Psi(\eta^l(0))$$ In this way, we can rewrite the term $$\Ww(\eta)- \f{\bt}{N}\sx G\bigg(\f{x}{N}\bigg)\om(\eta(x))$$ as $$\hskip-3cm\fN{\sum_x}\Bigg\{H\bg(\f{x}{N}\bg)\Bg[\tau_x\Psi(\eta)-\fl\sum_{|y-x|\leq l}\tau_y\Psi(\eta)\Bg]- \f{\bt}{3}G\bg(\f{x}{N}\bg)\om(\eta(x))\Bg\}$$ $$\hskip-5cm +\fN{\sum_x}\Bigg\{H\bg(\f{x}{N}\bg)\tau_xW_l^{\Psi}(\eta)- \f{\bt}{3}G\bg(\f{x}{N}\bg)\om(\eta(x))\Bg\}$$ $$ +\fN{\sum_x}\Bigg\{H\bg(\f{x}{N}\bg)\Bg[\tilde\Psi(\eta^l(x))-\tilde\Psi(\eta^{\e N}(x))\Bg]- \f{\bt}{3}G\bg(\f{x}{N}\bg)\om(\eta(x))\Bg\}.$$ From the assumption on $\Psi$, we chek easily that there exist $C(\Psi,p)$ such that for all $x\in\Z$ $\Psi(\eta(x))\leq C(\Psi,p)\eta(x)$. Then from the definitions of $\om^*(\cdot)$ and $G(\cdot)$ (cf. (\ref{ome}) and (\ref{GG})), the first term in the last expression is bounded above by $$\hskip-2cm\fN{\sum_x}\Bigg\{\bg|\fl\sum_{|y-x|\leq l}H\bg(\f{y}{N}\bg)-H\bg(\f{x}{N}\bg)\bg|\Psi(\eta(x))- \f{\bt}{3}G\bg(\f{x}{N}\bg)\om(\eta(x))\Bg\}$$ $$\hskip-5cm\leq\f{\bt}{3N}\sx G\fxn\Bg\{\f{3{\cal C}(\Psi,p)l}{\bt N}\eta(x)-\om(\eta(x))\Bg\}$$ $$\hskip-6cm\leq\om^*\Bg\{\f{3{\cal C}(\Psi,p)l}{\bt N}\Bg\}\f{\bt\|G\|_\ty}{3}.$$ This last term vanishes as $N\uparrow\ty$ since $\om^*(\cdot)$ is continuous and $\om^*(0)=0$. Now, to achieve the proof of the lemma \ref{L2}, we shall prove: \begin{lemme} \label{L3} For any $b>0$ $$\hskip-13cm\lspl\lspN\sup_f$$ \Bq \label{l3} \Bigg\{\fN{\sum_x}\int\bigg[H\bg(\f{x}{N}\bg)\tau_xW_l^{\Psi}(\eta)- \bt G\bg(\f{x}{N}\bg)\om(\eta(x))\bg]f(\eta)\di\nbr^p(\di\eta)-bND_p(f)\Bigg\}\leq0\Eq m-almost surely. The supremum is taken over all positive densities functions with respect to $\nbr^p$. \end{lemme} And, thanks to remarks [R5], we have to prove that: \begin{lemme} \label{L4} For any $b>0$ $$\hskip-11cm\lspl\lspe\lspN\sup_f$$ \Bq \label{l4} \Bigg\{\fN{\sum_x}\int\bigg[H\bg(\f{x}{N}\bg)\bg|\eta^{\e N}(x)-\eta^{l}(x)\bg|-\bt G\bg(\f{x}{N}\bg)\om(\eta(x))\bg]f(\eta)\di\nbr^p(\di\eta)-bND_p(f)\Bigg\}\leq0\Eq m-almost surely. The supremum is taken over all positive densities functions with respect to $\nbr^p$. \end{lemme} {\bf Proof of Lemma \ref{L3}}.\\ Using the convexity of $\om$ and definition of $G$, we check that \begin{eqnarray} \label{eq33.1} \fN\sx\bg|H\fxn\bg|\om(\eta^l(x))&\leq&\fN\sx\bg|H\fxn\bg|\fl\sum_{|y-x|\leq l}\om(\eta(y))\nonumber\\ &=&\fN\sx\om(\eta(x))\fl\sum_{|y-x|\leq l}\bg|H(y/N)\bg|\nonumber\\ &\leq& \fN\sx\om(\eta(x))G\fxn \end{eqnarray} At the beginning, we introduce some notations in order to deal in our study of (\ref{l3}) with the boxes of length $(2l+1)$. Indeed, the term $$H\bg(\f{x}{N}\bg)\tau_xW_l^{\Psi}(\eta)-\bt\bg|H\fxn\bg|\om(\eta^l(x))$$ depends on $\eta$ only through $\eta(x-l)\cdots\eta(x+l)$. Thus we may restrict the integral to microscopic blocks. Denote by $\wg_l=\{-l\cdots\l\}$ the box of length $(2l+1)$ centered at the origin. For a fixed $z\in\Z$, we denote by $\wg_{z,l}$ the box $z+\wg_l$, by $\X^l$ the configuration space $\N^{\wg_l}$, by $\bar\nu_{\rho,z,l}^p$ the product measure $\nbr^{\ta_zp}$ restricted to $\X^l$, by $f_{z,l}$ the density, with respect to $\bar\nu_{\rho,z,l}^{p}$, of the marginal of the measure $f(\eta)\nbr^{\ta_zp}(d\eta)$ on $\X^l$ and by $D_{\rho,z,l}^p(h)$ the Dirichlet form on $\X^l$ given by $$D_{\rho,z,l}^p(h)=\sum_{|x-y|=1\atop x,y \in \wg_{z,l}}\int p_xg(\eta(x))\left[\sqrt{h(\eta^{x,y})}-\sqrt{h(\eta)}\right]^2\bar\nu_{\rho,z,l}^p(\di\eta).$$ Thus, from (\ref{eq33.1}) and since the Dirichlet form is convex (by Schwarz inequality), the supremum in the lemma is bounded above by the supremum over all positive densities $f$ (with respect to $\nbr^p$) of the term \Bq \label{eq33.2} \fN{\sum_x}\Bg\{\int\bigg[H\bg(\f{x}{N}\bg)W_l^{\Psi}(\eta)- \bt\bg|H\fxn\bg|\om(\eta^l(0))\bg]f_{x,l}\bar\nu_{\rho,x,l}^p(\di\eta)-\f{bN^2}{C(l)}D_{\rho,x,l}^p(f_{x,l})\Bg\} \Eq As in the proof of lemma 3.1 of Koukkous (1999) we may now characterize the sites $x$ where the environment degenerates (behaves badly).\\ Fix $\dt>0$ , $\fa>0$ and $n\in\N$ sufficiently large such that $\frac{a_1-a_0}{n}<\dt$. For $0\leq j\leq n-2$, let $I_j^\dt=[\bt_j,\bt_{j+1}[$ where $\bt_j\in[a_0,a_1]$ is such that $$\bt_j = a_0+(a_1-a_0)\Bigg(\frac{j}{n}\Bigg) .$$ Let $I_{n-1}^\dt=[\bt_{n-1},a_1]$ and notice that, for $0\leq j\leq n-1$, we have $|\bt_{j+1}-\bt_j|<\dt$. \vskip0.2truecm Fix $k0$, we let $$\A_{x,i,\fa}=\bigg\{p,\hspace{4mm} \bigg|N_{x,j,i}^{l,k,\dt}(p)-m(I_j^\dt)\bigg|\leq\fa\ \ \mbox{for all}\ j,\ 0\leq j\leq n-1\bigg\}.$$ To keep notation simple, we denote $\A_{0,1,\fa}$ by $\A_\fa$. Let $$\A_{x,\fa}=\bigg\{p , \hspace{3mm} \frac{1}{L}\sum_{i=1}^L\1_{\{p\in \A_{x,i,\fa}\}}\geq 1-\fa\bigg\}.$$ From the definition of $\om^*$ and the property of $\Psi(\cdot)$ and $\tilde\Psi(\cdot)$ given in the remarks [R5], a simple computation shows that the integral term in (\ref{eq33.2}) is bounded by \Bq \label{eq33.3} C_1=\bt\|H\|_\ty\om^*\bg(\f{2C(\Psi,p)}{\bt}\bg). \Eq Therefore, the supremum over all positive densities $f$ (with respect to $\nbr^p$) of the term (\ref{eq33.2}) is bounded above by $$\f{1}{N}\sx\sup_{p\in\A_{0,\fa}}\sup_{h\in\Cf_p^l}\Bg\{\int\bigg[H\bg(\f{x}{N}\bg)W_l^{\Psi}(\eta)-\bt\bg|H\fxn\bg|\om(\eta^l(0))\bg]h(\eta)\nbrl (\di\eta)-\f{bN^2}{C(l)}D_{\rho,0,l}^p(h)\Bg\}$$ \Bq \label{eq33.4} \hskip-9cm+\ C_1\f{1}{N}\sx\1_{\{p\notin\A_{x,\fa}\}} \Eq where $\Cf_p^l$ is the set of positive density functions with respect to $\nbrl$.\\ By ergodicity and stationary of the environment law, the second term converges $m$-almost surely, as $N\uparrow\ty$, to $$C_1m\bg\{p\notin\A_{0,\fa}\bg\}.$$ Again the ergodicity of $m$ ensures that this expression vanishes as $l\uparrow\ty$ and $k\uparrow\ty$ afterwards. Now, let us turn to the first term in (\ref{eq33.4}). If we denote $$\E_h^p[f]=\int h(\eta)f(\eta)\di \nbrl (\eta),$$ the integral term in (\ref{eq33.4}) is bounded above by $$2C(\Psi)\bg|H\fxn\bg|\Bg\{\E_h^p\bg[\eta^l(0)\bg]-\f{\bt}{2C(\Psi)}\E_h^p\bg[\om(\eta^l(0))\bg]\Bg\}.$$ Recall that $\om$ is a convex and increasing function. Thus, by Jensen's inequality, the last expression is bounded above by $$2C(\Psi)\bg|H\fxn\bg|\Bg\{\om^{-1}\Bg[\E_h^p\bg[\om(\eta^l(0))\bg]\Bg]-\f{\bt}{2C(\Psi)}\E_h^p\bg[\om(\eta^l(0))\bg]\Bg\}.$$ From the remarks [R3], we claim that there exists a finite constant $C_2=C_2(\bt,C(\Psi))$ such that the integral term in (\ref{eq33.4}) is negative if $\E_h^p\bg[\eta^l(0)\bg]\geq C_2$. \nit Let $B>0$ be such that $suppH\subset[-B,B]$, then from (\ref{eq33.3}) and the last claim, we check that the first term in (\ref{eq33.4}) is bounded above by $$(2B+1)\|H\|_\ty\sup_{p\in \A_{0,\fa}}\sup_{f\in{\Cf_l^p(\f{2C_1C(l)}{bN^2},C_2)}}\Bg|\int W_l^{\Psi}(\eta)f(\eta)\nbrl (\di\eta)\Bg|$$ where $\Cf_l^p(a,b)$ is defined for positive constant $a$ and $b$ by $$\Cf_l^p(a,b)=\Bg\{f\in\Cf_l^p : D_{\rho,0,l}^p(f)\leq a \ \mbox{and}\ \ \E_f^p\bg[\om(\eta^l(0))\bg]\leq b\Bg\}.$$ The weak topology of the set of probability measures on $\X^l$ ensures that, by definition, $\Cf_l^p(\f{2C_1C(l)}{bN^2},C_2)$ is one of its compact subsets. Therefore, by the lower semi-continuity of the Dirichlet form, we know that $$\hskip-2cm\lspN\sup_{p\in \A_{0,\fa}}\sup_{f\in{\Cf_l^p(\f{2C_1C(l)}{bN^2},C_2)}}\Bg|\int W_l^{\Psi}(\eta)f(\eta)\nbrl (\di\eta)\Bg|$$ \Bq \label{eq33.5} \leq\sup_{p\in \A_{0,\fa}}\sup_{f\in{\Cf_l^p(0,C_2)}}\Bg|\int W_l^{\Psi}(\eta)f(\eta)\nbrl (\di\eta)\Bg|. \Eq From the assumption on $\Psi$ (and $\tilde\Psi$), for every positive constant $C_3$, the term in absolute value is bounded above by $$2C(\Psi)\int\1_{\{\eta^l(0)\geq C_3\}}\eta^l(0)f(\eta)\nbrl (\di\eta)+\Bg|\int W_l^{\Psi}(\eta)\1_{\{\eta^l(0)\leq C_3\}}f(\eta)\nbrl (\di\eta)\Bg|.$$ By remarks [R4], the first term in the last expression is bounded above by \begin{eqnarray} 2C(\Psi)\bg(\f{C_3}{\om(C_3)}\bg)\int\om(\eta^l(0))f(\eta)\nbrl(\di\eta)&=&2C(\Psi)\bg(\f{C_3}{\om(C_3)}\bg)\E_f^p\bg[\om(\eta^l(0))\bg]\nonumber\\ &\leq&2C_2C(\Psi)\bg(\f{C_3}{\om(C_3)}\bg)\nonumber \end{eqnarray} for all $f\in\Cf_l^p(0,C_2)$. From (H4), this last term vanishes as $C_3\uparrow\ty$. At this point, we achieve by proving that \Bq \label{eq33.6} \lspk\lspl\sup_{p\in \A_{0,\fa}}\sup_{f\in{\Cf_l^p(0,C_2)}}\Bg|\int W_l^{\Psi}(\eta)\1_{\{\eta^l(0)\leq C_3\}}f(\eta)\nbrl (\di\eta)\Bg|\leq {\cal C}(\dt,\fa) \Eq where ${\cal C}(\dt,\fa)$ vanishes as $\fa\downarrow0$ and $\dt\downarrow 0$ afterwards. We omit this proof since it is developed in the proof of lemma 3.1 in Koukkous (1999).\par\nit {\bf Proof of Lemma \ref{L4}}. First of all, we approximate (replace) the average over a small macroscopic box by an average over large microscopic boxes. More precisely, for $N$ sufficiently large we check that \begin{eqnarray} \ &\ &\hskip-1.5cm\fN\sx\bg|H\fxn\bg|\bg|\eta^{\e N}(x)-\eta^{l}(x)\bg|\nonumber\\ &\leq&\fN\sx\bg|H\fxn\bg||\feN\sum_{2l+1<|y|\leq\e N}|\eta^l(x)-\eta^{l}(x+y)|+{\cal O}\bg(\f{l}{\e N}\bg)\sx G\fxn\eta(x)\nonumber\\ &\leq&\fN\sx\bg|H\fxn\bg|\feN\sum_{2l+1<|y|\leq\e N}|\eta^l(x)-\eta^{l}(x+y)|+\f{\bt}{N}\sx G\fxn\om(\eta(x))\nonumber \end{eqnarray} Define $$\om_l(\eta,\xi,x,z)=\Bg(\om(\eta^l(x))+\om(\xi^l(z))\Bg)$$ $$W_A^l(\eta,\xi,x,z)=|\eta^l(x)-\xi^{l}(y)|\1_{\{\eta^l(x)\vee\xi^l(z)\leq A\}}$$ and to keep notation simple, we denote $W_A^l(\eta,\xi,0,0)$ by $W_A^l(\eta,\xi)$ and $\om_l(\eta,\xi,0,0)$ by $\om_l(\eta,\xi)$. As in the previous proof, we introduce an indicator function and in the same way as in (\ref{eq33.1}), we reduce our proof to show that, for every positive constant $A$ \Bq \label{eq34.1} \hskip-4cm\lspl\lspe\lspN\sup_f\Bg\{\fN\sx\bg|H\fxn\bg|\feN\sum_{2l+1<|y|\leq\e N} \Eq $$\hskip3cm\int\Bg[W_A^l(\eta,\eta,x,x+y)-\bt\om_l(\eta,\eta,x,x+y)\Bg]f(\eta)\nbr(\di\eta)-bND_p(f)\Bg\}\leq 0$$ From the definition of $$ \feN\sum_{2l+1<|y|\leq\e N}W_A^l(\eta,\eta,x,x+y)$$ and since $\eta^l(x)$ and $\eta^l(x+y)$ depend on the configuration $\eta$ only through its values on the set $$\wg_{x,y,l}:=\wg_{x,l}\cup \Big(y+\wg_{x,l}\Big),$$ we shall replace $f$ by its conditional expectation with respect to the $\sigma\mbox{-algebra}$ generated by $\{\eta(z); z\in\wg_{x,y,l}\}$. Some notation are necessary. For all $y\in\Z$, we define the shift operator $\ta_y(\cdot)$ on environments by $(\ta_yp)(x)=p(x+y)$. For fixed integer $l$ and environments $p$ and $q$, we denote by $\tilde\X^l$ the configuration space $\N^{\wg_l}\times\N^{\wg_l}$, by $\bar\nu_{\rho,x,l}^{p,q}$ the product measure $\bar\nu_\rho^{\ta_xp}\ \otimes\bar\nu_\rho^{\ta_xq}$ restricted to $\tilde\X^l$, and by $\fk$ the conditional expectation of $f$ with respect to the $\sigma\mbox{-algebra}$ generated by $\{\eta(z); z\in\wg_{x,y,l}\}$. Thus the supremum in (\ref{eq34.1}) is bounded above by $$\hskip-8cm\sup_f\Bg\{\fN\sx\bg|H\fxn\bg|\feN\sum_{2l+1<|y|\leq\e N}$$ $$\hskip3cm\int\Bg[\tau_xW_A^l(\xi_1,\xi_2)-\bt\om_l(\xi_1,\xi_2)\Bg]f_{x,y,l}^p(\xi_1,\xi_2)\nbrxl^{p,\ta_yp}(\di\xi)-bND^p(f)\Bg\}.$$ Let us turn now to the Dirichlet form of $f_{x,y,l}^p$ into microscopic boxes $\wg_{x,y,l}$. Let $D_l^{p,q}(h)$ be $$D_l^{p,q}(h)=I_{l,1}^{p,q}(h)+I_{l,2}^{p,q}(h)+\sum_{z,z'\in \wg_l\atop |z-z'|=1}I_{z,z',1}^{p,q}(h)+\sum_{z,z'\in\wg_l\atop |z-z'|=1}I_{z,z',2}^{p,q}(h)$$ where, for each $ z,z' \in\wg_l$, such that $|z-z'|=1$, $$I_{z,z',1}^{p,q}(h)=1/2\int p_zg(\xi_1(z))\Big[\sqrt{h(\xi_1^{z,z'},\xi_2)} -\sqrt{h(\xi_1,\xi_2)}\Big]^2\nrl^{p,q}(d\xi),$$ $$I_{z,z',2}^{p,q}(h)=1/2\int q_zg(\xi_2(z))\Big[\sqrt{h(\xi_1,\xi_2^{z,z'})} -\sqrt{h(\xi_1,\xi_2)}\Big]^2\nrl^{p,q}(d\xi),$$ $$I_{l,1}^{p,q}(h)=1/2\int p_0g(\xi_1(0))\Big[\sqrt{h(\xi_1^{0,-},\xi_2^{0,+})} -\sqrt{h(\xi_1,\xi_2)}\Big]^2\nrl^{p,q}(d\xi),$$ $$I_{l,2}^{p,q}(h)=1/2\int q_0g(\xi_2(0))\Big[\sqrt{h(\xi_1^{0,+},\xi_2^{0,-})} -\sqrt{h(\xi_1,\xi_2)}\Big]^2\nrl^{p,q}(d\xi).$$ The configurations $\xi^{0,\pm}(\cdot)$ are defined by $$\xi^{0,\pm}(z) = \left\{\begin{array}{ll} \xi(z) & \ \mbox{if $z\neq 0$ } \\ \xi(0)\pm 1& \ \mbox{if $z=0$.} \end{array} \right.$$ We claim that \Bq \label{eq34.2} \fN\sx\feN\sum_{2l+1<|y|\leq\e N}D_l^{p,\ta_y p}\bg(f_{x,y,l}^p\bg)\leq C(l)\e^2ND^p(f). \Eq The proof of the claim is omitted. See Lemma 4.3 of Koukkous (1999) for more details.\\ From the same notation in the proof of Lemma \ref{L3}, we separate the sites where the environment behaves badly and repeat the computation in the begining of (\ref{eq33.3}). Using (\ref{eq34.2}) and introducing the indicator function of the environements afterwards, our lemma is a consequence of the following results \begin{lemme} \label{LL3} $$\lspl\lspe\lspN\feN\sum_{2l+1<|y|\leq\e N}\fN\sx\Bg[\1_{\{\ta_xp\notin\A_{0,\fa}\}}+\1_{\{\ta_{x+y}p\notin\A_{0,\fa}\}}\Bg]=0$$ $m$ almost surely. \end{lemme} \begin{lemme} \label{LL4} For positive constants $a$ and $b$, let $$\Cf^{p,q}_l(a,b)=\Bg\{h\geq0,\E_{{\nrl^{p,q}}}[h]=1, D_l^{p,q}(h)\leq a, \E_h^{p,q}\bg[\om_l(\xi_1,\xi_2)\bg]\leq b\Bg\}$$ \Bq \label{eq34.3} \lspk\lspl\lspe\lspN\sup_{p,q\in\A_{0,\fa}}\sup_{h\in\Cf^{p,q}_l\bg(\f{\eN}{bN^2}C_1,C_2\bg)}\E_h^{p,q}\Bg(W_A^l(\xi_1,\xi_2)\Bg)\leq{\cal C}(\dt,\fa) \Eq where ${\cal C}(\dt,\fa)$ vanishes as $\fa\downarrow0$ and $\dt\downarrow 0$ afterwards. \end{lemme} The lemma \ref{LL3} is trivially proved using the ergodicity and stationarity of $m$. (see Koukkous (1999)).\par\nit Since $\Cf^{p,q}_l\bg(\f{\eN}{bN^2}C_1,C_2\bg)$ is a compact subset of the probability measures set on $\X^l\times\X^l$ endowed with the weak topology, by the lower semi-continuity of the Dirichlet form, to prove (\ref{eq34.3}) it is enough to prove that $$\lspdt\lspfa\lspk\lspl\sup_{p,q\in\A_{0,\fa}}\ \ \sup_{h\in\Cf^{p,q}_l(0,C_2)}\E_h^{p,q}\Bg(W_A^l(\xi_1,\xi_2)\Bg)=0.$$ which is proved in Koukkous (1999) ( see the proof of lemma 4.2 at formula (23)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%Proof of LDP Principle%%%%%%%%%%%%%%%5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{\bf Proof of Theorem \ref{th1}} The proof of lower bound presented in Benois \& al. (1995) is easily adapted in this case using some computations already developed in the previous proof of super-exponential estimate and some arguments presented in the below upper bound's proof. We therefore omit details for the reader.\par \bl{LDP} Let $H\in\cl^{1,2}_K([0,\tf ]\times\R)$ and $\ma\in{\cal C}_\rho(\R)$. From Girsanov formula, the Radon-Nikodym derivative of $\hpro$ with respect to $\pro$ is given by \Bq \label{girsanov} \exp N\Bg\{\jj^1(\pi_t^N)+h_\ma^{p,N}(\pi_0^N|\rho)-N\int_0^t\sum_{x,y}p_xg(\eta_s(x))T(y)\bg[e^{\{H(t,\f{x+y}{N})-H(t,\f{x}{N})\}}\ -\ 1\bg]\di s\Bg\} \Eq where $h_\ma^{p,N}(\cdot|\rho) :{\cal M}_+\ra\R$ is defined by $$h_\ma^{p,N}(\mu|\rho)=\ab\mu,\log\Bg(\f{\Phi(\ma(\cdot))}{\Phi(\rho)}\Bg)\ba-\fN\sum_x\log\Bg[\f{Z(\Phi(\ma(x/N))p_x^{-1})}{Z(\Phi(\rho)p_x^{-1})}\bg].$$ {\bf Upper bound :}\\ The proof is dealing only with a fixed compact subset ${\cal C}$ of $\dl([0,\tf ],\mm)$. To extend this result to a closed subset, we need exponential tightness for $Q^N_{\rho,p}$. It is easily obtained thanks to the proof presented in Benois (1996) (see also Lemma V.1.5 in Kipnis-Landim (1999)).\par For every $q>1$, $$Q^N_{\rho,p}(\cl)=\E^N_{\rho,p}\Bg[\Bg(\f{\di\pro}{\di\hpro}\Bg)^{1/q}\Bg(\f{\di\hpro}{\di\pro}\Bg)^{1/q}\1_{\{\pi^N\in\cl\}}\Bg].$$ Let $\vartheta_\e$ be the approximation of identity defined by $(2\e)^{-1}\1_{[-\e,\e]}(x)$ and $\ast$ the classic convolution product.\par For $0\leq s\leq\tf $, let $$u_{\e,N}^{p,H}(\eta_s)=\f{\si}{2N}\sum_k\{\ti_x^2H(s,k/N)+[\ti_xH(s,k/N)]^2\}\{p_kg(\eta_s(k))-\Phi(\eta^{\e N}_s(k))\}$$ and $$u_{N,H}^p(\eta_s)=\fN\sum_kp_kg(\eta_s(k))\Bg\{\sum_jT(j)N^2\bg[e^{\{H(t,\f{k+j}{N})-H(t,\f{k}{N})\}}\ -\1\bg]\hspace{3cm}$$ $$\hspace{9cm}-\f{\si}{2}\bg\{\ti_x^2H(s,k/N)+(\ti_xH(s,k/N))^2\bg\}\Bg\}$$ From (\ref{girsanov}), a simple computation shows that $\bg(\di\pro/\di\hpro\bg)$ is bounded above by $$\exp N\Bg\{-\jj^1(\pi_\tf^N)+\jj^2(\pi^N\ast\vartheta_\e)-h_\ma^{p,N}(\pi_0^N|\rho)+\int_0^\tf\bg\{u_{\e,N}^{p,H}(\eta_s)+u_{N,H}^p(\eta_s)\bg\}\di s\Bg\}$$ Thus, $\fN\log Q^N_{\rho,p}(\cl)$ is bounded above by \Bq \label{ldp2} \f{1}{q}\sup_{\pi\in\cl}\Bg\{-\jj^1(\pi_\tf ^N)+\jj^2(\pi^N\ast\vartheta_\e)-h_\ma^{p,N}(\pi_0^N|\rho)\Bg\}\hspace{6cm} \Eq $$\hspace{4cm}+\fN\log\E^N_{\rho,p}\Bg[\Bg(\f{\di\hpro}{\di\pro}\Bg)^{1/q}\exp\Bg\{\f{N}{q}\int_0^\tf\bg(u_{\e,N}^{p,H}(\eta_s)+u_{N,H}^p(\eta_s)\bg)\di s\Bg\}\Bg]$$ Let $\bar H$ be a real continuous function with the same support as $\sup_t|H_t|$, such that it bounds above $\sup_{0\leq t\leq\tf}[|\ti_x^2H_t|+(\ti_xH_t)^2+|H_t|]$.\par Let $C_0\in\N$ such that $supp H\subset [0,\tf ]\times[-(C_0-1),(C_0+1)]$. Using H\"older's inequality, we show that, for $q'\in\R$ such that $(1/q)+(1/q')=1$, the second term in (\ref{ldp2}) is bounded above by $$ \f{1}{3Nq'}\log\E^N_{\rho,p}\Bg[\exp\bg\{\f{3Nq'}{q}\bg(\int_0^\tf u_{\e,N}^{p,H}(\eta_s)\di s-\int_0^\tf \f{\fa}{N}\sum_k\bh(\fkn)\om(\eta_s(k))\di s\bg)\bg\}\Bg]\hspace{2cm}$$ \Bq \label{ldp3} + \f{1}{3Nq'}\log\E^N_{\rho,p}\Bg[\exp\bg\{\f{3Nq'}{q}\int_0^\tf u_{N,H}^p(\eta_s)\di s\bg\}\Bg]\hspace{4cm} \Eq $$\hspace{3cm}+ \f{1}{3Nq'}\log\E^N_{\rho,p}\Bg[\exp\bg\{ \f{3Nq'}{q}\bg(\f{\fa}{N}\int_0^\tf \sum_k\bh(\fkn)\om(\eta_s(k))\di s\bg)\bg\}\Bg]$$ Using similar arguments as in the proof of lemma \ref{L2} ( see (\ref{ep2})), we check that the last term in (\ref{ldp3}) is bounded above by $$R_1(\fa,q,H)=\f{2C_0}{3q}\log\nu_{\Phi(\rho)a_0^{-1}}\bigg[e^{\big\{\f{3\fa q'\tf }{q}\|{\bar H}\|_\ty\om(\eta(0))\big\}}\bigg]$$ which vanishes as $\fa\downarrow0$ for each fixed $q$ and $H$ thanks to assumption [H4].\par From assumption [H2], we check that $g(k)\leq g^*k$ for all $k\in\Z$ and therefore $\Phi(\rho)\leq g^*\rho$. Thus, we repeat the same argument as above, a simple computation shows that the second term in (\ref{ldp3}) is bounded above by $$R_2(q,H,N)=\f{2C_0}{3q'}\log\nu_{\Phi(\rho)a_0^{-1}}\bigg[e^{\big\{\f{\bt}{N}\eta(0)\big\}}\bigg]$$ where $\bt=\bt(\tf,g^*,H,a_1,q,\si)$. \nit For each fixed $q$ and $H$, it is easy to see that $R_2(q,H,N)$ vanishes as $N\uparrow\ty$.\par Let us turn to the first term in (\ref{ldp3}) and denote $R_3(\fa,q,H,\e)$ its limit when $N\uparrow\ty$. A similar computation as in the proof of the super-exponential estimate ( see lemma \ref{L2} and its proof), gives that $$\lim_{\e\rightarrow 0}R_3(\fa,q,H,\e)=0$$ for all $\fa>0$, $q>1$ and smooth function $H$. \par In the other hand notice that by a simple computation and from the ergodicity and stationarity of $m$, we prove that $h_\ma^{p,N}(\pi_0^N|\rho)$ converges (uniformly in $\pi\in\cal C$) to $h(\ma|\rho)$ when $N\uparrow\ty$.\par We therefore proved that $\overline{\lim}_{N\ra\ty}(1/N)\log Q^N_{\rho,p}(\cl)$ is bounded above by $$\inf_{H,\ma,q,\fa,\e}\Bg\{\f{1}{q}\sup_{\pi\in\cl}\bg\{-\jj^1(\pi)+\jj^2(\pi\ast\vartheta_\e)-h(\ma|\rho)\bg\}+R_3(\fa,q,H,\e)+R_1(\fa,q,H)\Bg\}$$ \nit where the infimum is taken over all $H\in\cl^{1,2}_K([0,\tf ]\times\R)$, $\ma\in{\cal C}_\rho(\R)$, $q>1$, $\fa>0$ and $\e>0$.\par At this level, using the continuity of $\jj^2(\cdot\ast\vartheta_\e)$ for every fixed $H$ and $\e>0$, the compacity of $\cal C$ and the arguments developed in (Kipnis \& al. (1989)) to permute the supremum and infimum, we check that this last expression is bounded above by $$-\inf_{\pi\in\cl}\sup_{H,\ma,q,\fa,\e}\Bg\{\f{1}{q}\bg\{-\jj^1(\pi)+\jj^2(\pi\ast\vartheta_\e)-h(\ma|\rho)\bg\}+R_3(\fa,q,H,\e)+R_1(\fa,q,H)\Bg\}$$ We conclude therefore our proof by letting $\e\downarrow0$. $\fa\downarrow0$ and $q\downarrow1$. \vk\noindent {\bf Acknowledgements.} % We thank C. Landim for % suggesting the problem, O. Benois, P. Ferrari, M. Mourragui and % E. Saada, for fruitful discussions. A.K. thanks Fapesp support (FAPESP n. 99/06918-0). This work is part of FAPESP Tematico n. 95/0790-1, and FINEP Pronex n. 41.96.0923.00. \begin{thebibliography}{1} \bibitem{anj} E.~D. Andjel. \newblock Invariant measures for the zero range process. \newblock {\em Ann. Probab.} Vol. {\bf 10}:\ 525--547, 1982. \bibitem{afgl} E.~D. Andjel., P.~A. 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