Content-Type: multipart/mixed; boundary="-------------9911251040272" This is a multi-part message in MIME format. ---------------9911251040272 Content-Type: text/plain; name="99-444.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-444.keywords" Zero-range, Random rates, invariant measures, Convergence to the maximal invariant measure ---------------9911251040272 Content-Type: application/x-tex; name="AFGL.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="AFGL.tex" \documentclass[12pt]{article} %\documentstyle{article} \usepackage{amssymb} %\usepackage{showkeys} %\renewcommand{\baselinestretch}{1.5} %\oddsidemargin -7mm % Remember this is 1 inch less than actual %\evensidemargin 7mm %\textwidth 18cm %\topmargin -9mm % Remember this is 1 inch less than actual %\headsep 0.9in % Between head and body of text %\headsep 20pt % Between head and body of text %\textheight 24cm %\textheight 19cm %to see the full page in the laptop \def\vep{\varepsilon} \def\reff#1{(\ref{#1})} \def\gbt{{t\gamma\beta }} \def\E{{\bf E}} \def\P{{\bf P}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\N{{\mathbb N}} \def\X{{\bf X}} \def\1{{\mathbb 1}} \def\ze{{\zeta}} \def\be{{\beta}} \def\de{{\delta}} \def\la{{\lambda}} \def\ga{{\gamma}} \def\th{{\theta}} \def\proof{\noindent{\bf Proof. }} \def\rate{{e^{- \beta|\ga|}}} \def\A{{\bf A}} \def\B{{\bf B}} \def\C{{\bf C}} \def\D{{\bf D}} \def\one{{\bf 1}} \begin{document} \title{Convergence to the maximal invariant measure\\ for a zero-range process with random rates.} \author{E.D. Andjel\kern -0.5pt \renewcommand{\thefootnote}{\alph{footnote}}\footnotemark% \ \ P.A. Ferrari\kern -2pt \addtocounter{footnote}{4} \renewcommand{\thefootnote}{\alph{footnote}}\footnotemark% \ \ H. Guiol\kern -0.5pt \renewcommand{\thefootnote}{\alph{footnote}}\footnotemark% \ \ and\ C. Landim\kern -2pt \addtocounter{footnote}{4}\renewcommand{\thefootnote}{\alph{footnote}} \footnotemark} \maketitle \renewcommand{\thefootnote}{\alph{footnote}} \addtocounter{footnote}{1} \footnotetext{LATP-CMI, 36 Rue Joliot-Curie, 13013 Marseille, France.} \renewcommand{\thefootnote}{\alph{footnote}} \addtocounter{footnote}{5} \footnotetext{IME-USP, P.B. 66281, 05315-970 S\~ao Paulo, SP, Brasil.} \renewcommand{\thefootnote}{\alph{footnote}} \addtocounter{footnote}{1} \footnotetext{IMECC-UNICAMP, P.B. 6065, 13053-970, Campinas, SP, Brasil.} \renewcommand{\thefootnote}{\alph{footnote}} \addtocounter{footnote}{5} \footnotetext{IMPA, Estrada Dona Castorina 110, Jardim Bot\^anico, Rio de Janeiro, Brasil and CNRS UPRES-A 6085, Universit\'e de Rouen, BP 118, 76821 Monts Saint Aignan Cedex, France.} \newcommand{\carn}{\hfill\rule{0.25cm}{0.25cm}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \abstract {We consider a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates ---an \emph{environment}. For each environment $p$ we prove that the set of all invariant measures is the convex hull of a set of product measures with geometric marginals. As a consequence we show that for environments $p$ satisfying certain asymptotic property, there are no invariant measures concentrating on configurations with critical density bigger than $\rho^*(p)$, a critical value. If $\rho^*(p)$ is finite we say that there is phase-transition on the density. In this case we prove that if the initial configuration has asymptotic density strictly above $\rho^*(p)$, then the process converges to the maximal invariant measure.\\ {\em AMS 1991 subject classifications.} Primary 60K35; Secondary 82C22.\\ {\em Key words and Phrases.} Zero-range; Random rates; invariant measures; Convergence to the maximal invariant measure } \section{Introduction} The interest on the behavior of interacting particle systems in random environment has grown recently: Benjamini, Ferrari and Landim (1996), Evans (1996) and Ferrari and Krug (1996) observed the existence of phase transition in these models; Benjamini, Ferrari and Landim (1996), Krug and Sepp\"al\"ainen (1999) and Koukkous (1999) investigated the hydrodynamic behavior of conservative processes in random environments; Landim (1996) and Bahadoran (1998) considered the same problem for non-homogeneous asymmetric attractive processes; Gielis, Koukkous and Landim (1998) deduced the equilibrium fluctuations of a symmetric zero range process in random environment. In this article we consider a one-dimensional, totally asymmetric, nearest-neighbor zero-range process in a non-homogeneous environment. The evolution can be informally described as follows. Fix $c\in(0,1)$ and provide each site $x$ of $\Z$ with a rate function $p_x\in[c,1]$. If there is at least one particle at some site $x$, one of these particles jumps to $x+1$ at rate $p_x$. A rate configuration $p=(p_x:x\in\Z)$ is called an \emph {environment} and a measure $m$ on the set of possible environments a \emph{random environment.} Benjamini, Ferrari and Landim (1996) and Evans (1996) for an asymmetric exclusion process with rates associated to the particles ---which is isomorphic to a zero range process with rates associated to the sites--- and Ferrari and Krug (1996) for the model considered here, proved the existence of a phase transition in the density. More precisely, they proved that, under certain conditions on the distribution $m$, specified in Theorem \ref{2.4}, there exists a finite critical value $\rho^*$ such that for $m$-almost-all $p$ there are no product invariant measures for the process with rates $p$ concentrating on configurations with asymptotic density bigger than $\rho^*$ and that there are product invariant measures concentrating on configurations with asymptotic density smaller than or equal to $\rho^*$. (The density of a configuration is essentially the average number of particles per site and is defined in \reff{dens} below). Our first result is that the set of extremal invariant measures for the process with fixed environment $p=(p_x:x\in\Z)$ is the set $\{\nu_{p,v}: v< p_x, \forall x\}$, where $\nu_{p,v}$ is the product measure on $\N^\Z$ with marginals \begin{equation} \label{889} \nu_{p,v}\{\xi : \, \xi(x)=k\} =\Bigl(\frac v{ p_x}\Bigr)^k \Bigl(1-\frac v{ p_x}\Bigr)\; . \end{equation} The above result does not surprise specialists in queuing theory. In fact we are dealing with an infinite series of M/M/1 queues with service rate $p_x$ at queue $x$. The value $v$ can be interpreted as the arrival rate at ``queue'' $-\infty$. Since Burke's theorem guarantees that in equilibrium the departure process of a M/M/1 queue is the same as the arrival process (both Poisson of rate $v$), there is an invariant measure for each arrival rate $v$ strictly smaller than all service rates. Assume $c=\inf_x p_x$ and that the following limits exist. For $v0$ then one particle is moved from $x$ to $x+1$ at that time. That is, for cylinder functions $f:\X\to\R$, \[ df(\eta_t) := \sum_{x\in\Z} dN_x(t)\,\one\{\eta(x)>0\} \,[f(\eta^x)-f(\eta)]\; , \] where $dN_x(t)$ is one if there is an event of the Poisson process $N_x$ at time $t$, otherwise it is zero. In the above formula $\eta^x=\eta-\mathfrak d_{x}+\mathfrak d_{x+1}$, where $\mathfrak d_y$ stands for a configuration with just one particle at $y$ and addition of configurations is performed componentwise. To see that the process is well defined by this rule, just note that in any time interval $[0,t]$ for any $ x$ there exists with probability $1$ a $y0\} \left[f(\eta^x)-f(\eta)\right]\;. \end{equation} We denote by $\{S_p(t),\, t\ge 0\}$ the semigroup associated to the generator $L_p$, i.e.\/ $S_p(t)f(\eta) = \E[f(\eta_t)\,|\,\eta_0=\eta]$ and by ${\cal I}_p$ the set of invariant measures of $\eta_t$ (the Markov process with generator $L_p$). Let $v$ be a real number such that $ 0c$ for all $x$, respectively. In the first case to prove the theorem we only need to follow the proof of Theorem 1.11 in Andjel (1982), but in the second case a complementary argument is needed. In both cases the proof relies on the standard partial order for probability measures on $\X$. To define it, first say that $\eta \leq \xi$ if $\eta (x) \leq \xi (x)$ for all $x\in \Z$. Then say that a real valued function $f$ defined on $\X$ is increasing if $ \eta \leq \xi$ implies that $f(\eta) \leq f(\xi)$. Finally if $\mu$ and $\nu $ are two probability measures on $\X$, say that $\mu \leq \nu$ if $\int fd\mu \leq \int fd\nu$ for all bounded increasing cylinder functions $f$. The complementary argument alluded above depends on the following proposition: \begin{proposition} \label{2.2} Assume that $p$ is an environment such that \begin{equation} \label{ccc} p_x>c \mbox{ for all }x\in\Z\mbox{ and }\liminf_{x\to -\infty}\ p_x=c\,, \end{equation} and let $\nu$ be an arbitrary probability measure on $\X$. Then the set of measures $\{\nu S_p(t)\; :\; t>0\}$ is tight and its weak limits as $t$ goes to infinity are bounded above by $\nu_{p,c}$. \end{proposition} An immediate corollary of Proposition \ref{2.2} is that under \reff{ccc} all invariant measures are dominated by~$\nu_{p,c}$. To state our main result let $\eta$ be an element of $\X$ and consider \begin{eqnarray} \label{dens} \underline D(\eta )&=& \liminf_{n\to\infty} \frac {1}{n}\sum_{x=-n+1}^0 \eta (x)\;,\nonumber\\ \overline D(\eta )&=& \limsup_{n\to\infty} \frac {1}{n}\sum_{x=-n+1}^0 \eta (x)\;,\nonumber \end{eqnarray} the \emph{lower}, respectively \emph{upper asymptotic left density} of $\eta$. If both limits are equal to $\alpha$ we say that $\eta$ has \emph{left density} $\alpha$ and write $D(\eta)= \alpha$. Assume that $p$ is an environment for which the limits defined in \reff{cc1} exist. Then, by Kolmogorov's law of large numbers (see {\sl e.g.} Shiryayev (1984), Theorem 2 p. 364) $\nu_{p,v}$ concentrates on configurations with left density $R(p,v)$: \begin{equation} \label{cc2} \nu_{p,v}\{\eta\in\X: D(\eta) = R(p,v)\}= 1 \end{equation} for all $vc$ for all $x$. In this case $\nu_{p,c}$ is well defined and concentrates on configurations with infinite asymptotic left density, and for any $\rho\in[0,\infty]$ there exists $v=v(p,\rho)$ such that $\nu_{p,v}\{\eta\in\X: D(\eta) = \rho\}= 1$. If $\lim_{v\to c}R(p,v)= \rho^*(p)<\infty$ and $p_x>c$ for all $x$, the measure $\nu_{p,c}$ is well defined and Theorem \ref{2.1} tells us that there are no invariant measures bigger than $\nu_{p,c}$. Our next theorem describes what happens in this case when one starts with a density strictly bigger than $\rho^*(p)$. This is our main result. \begin{theorem}\label{115} Let $p$ be an environment satisfying \reff{ccc} such that $\rho^*(p)<\infty$ and $\eta$ be a configuration such that $\underline D(\eta) > \rho^*(p)$. Then \[ \lim_{t\to\infty} \delta_\eta S_p(t)=\nu_{p,c}\,, \] where $\delta_\eta$ is the measure giving weight one to the configuration $\eta$. \end{theorem} \vskip 3mm As a corollary to Theorem \ref{115} we obtain the asymptotic behavior of the system when the environment is randomly chosen. Let $m$ be the distribution of a stationary ergodic sequence $p$ on $[c,1]$ such that $m(\{p:p_0=c\})=0$, $m(\{p: c0$ for all $\varepsilon>0$. The measure $m\nu_{\cdot,v}$ defined by $m\nu_{\cdot,v}f = \int m(dp)\int \nu_{p,v}(d\eta) f(\eta)$ is an ergodic distribution on $\X$ and, by the Ergodic Theorem, for all $v< c$ and for $m$-almost all $p$, the asymptotic density exists $\nu_{p,v}$ a.s. and is equal to: \[ R(v)=\int \frac {v}{p_0 -v}m(dp). \] Let $\rho^* := \lim_{v\to c} R(v)$ and assume $\rho^*<\infty$. In this case for $m$-almost all environment $p$ any invariant measure for $L_p$ is dominated by $\nu_{p,c}$. The following theorem concerns the behavior of the process when the initial measure concentrates on configurations with density strictly higher than $\rho^*$. \begin{theorem}\label{2.4} Let $m$ be the distribution of a stationary ergodic sequence $p=(p_x)_{x\in\Z}$ on $(c,1]$ such that $m(\{p: c< p_0 < c +\varepsilon \})>0$ for all $\varepsilon>0$ and for which $\rho^*<\infty $. Let $\nu$ be a measure for which $\nu \ a.s.$ $\underline D(\eta)$ is strictly bigger than $\rho^*$. Then, for $m$-almost all $p$ \[ \lim_{t\to\infty} \nu S_p(t)=\nu_{p,c}. \] \end{theorem} \section{Domination and Invariant measures} In this section we prove Proposition \ref{2.2} and Theorem \ref{2.1}. \noindent{\bf Proof of Proposition 2.2.} Fix an arbitrary site $y$ and let $x_n$ be a decreasing sequence such that $x_10\} p_z[f(\eta^z)-f(\eta)] \nonumber \\ &&\quad +\; p_{x_n}[f(\eta+\mathfrak d_{x_n +1}) - f(\eta)] \\ &&\quad +\; {\bf 1 }\{\eta (y)>0\} p_y [f(\eta -\mathfrak d_y)-f(\eta)] \; .\nonumber \end{eqnarray} Let $S_{p,n}$ be the semigroup associated to this process and for an arbitrary probability measure $\nu$ let $\nu _n$ be its projection on ${\N}^{\{x_n +1,...,y\}}$. Standard coupling arguments show that \[ (\nu S_p(t))_n \leq \nu_n S_{p,n}(t) \; . \] The coupling of the two processes is done using the same Poisson processes $N_x(t)$ defined in Section 2. The reason why the domination holds is that for the process $S_{p,n}(t)$, each time the Poisson process $N_{x_n}(t)$ jumps, a new particle appears in $x_n+1$, while the same happens for the process $S_p(t)$ only when there is at least a particle in the site $x_n$. The process with generator $L_{p,n} $ is irreducible and has a countable state space, moreover a simple computation shows that the product measure $\mu_{n,p}$ with marginals given by \[ \mu _{n,p} \{\eta :\eta (z)=k\} = \Bigl(1-\frac {p_{x_n}}{p_z}\Bigr)\Bigl(\frac {p_{x_n}}{p_z}\Bigr)^k, \] where $x_nk \}=1$ for all $k$. Therefore $\nu_p \geq \nu_{c,p}$ and either $\liminf_{x\to -\infty}\ p_x=\inf p_y$ or $\liminf_{x\to \infty}\ p_x=\inf p_y$. In the first of these cases, Proposition \ref{2.2} allows us to conclude immediately. In the second case we argue by contradiction: let $\widetilde\nu$ be a probability measure on ${\N}^{\Z}\times{\N}^{\Z}$ admitting as first marginal and second marginal $\nu _p$ and $\nu _{c,p}$ respectively and such that $\widetilde\nu \{(\eta ,\xi): \eta \geq \xi \}=1$. Consider the standard coupled process with initial measure $\widetilde\nu$. Denote by $\overline{S}(t)$ the semigroup associated to this process and assume that for some $x$, $\widetilde\nu \{(\eta ,\xi): \eta (x) > \xi (x) \}>0$. It then follows that for all $t>0$ \[ \widetilde\nu \overline{S}(t) \left\{(\eta ,\xi): \eta (x) > \xi (x)=0 \right\}>0. \] Hence \[ \nu _p\left\{ \eta :\eta (x)>0\right\}> \nu _{c,p}\left\{ \eta :\eta (x)>0\right\}=\frac {c}{p_x}. \] Pick $y>x$ and such that $p_y < p_x\nu _p\{ \eta :\eta (x)>0\}$. Then let $f(\eta )=\sum_{z=x+1}^y \eta (z)$. Now a simple calculation shows that $\int L_p f(\eta) d\nu _p>0$ contradicting the invariance of $\nu_p$.$\carn$ \medskip \noindent {\bf Remark:} Proofs of Theorem \ref{2.1} and Proposition \ref{2.2} can easily be extended to a larger class of one-dimensional nearest-neighbors asymmetric zero range processes in non-homogeneous environment. In these systems a particle at site $x$ on configuration $\eta$ jumps at rate $p_xg(\eta(x))$ to site $x+1$, where $g:\N\to[0,\infty)$ is a non-decreasing bounded function such that $g(0)=0$. \section{Convergence} We prove in this section Theorem \ref{115}. Fix a measure $\nu$ on $\X$ concentrated on configurations with lower asymptotic left density strictly greater than $\rho^*(p)$. Theorem \ref{115} follows from Proposition \ref{2.2} and from the next lemma, which implies that any invariant measure $\nu_{p,v}$ is dominated by any weak limit of $\nu S_p(t)$. Denote $\{\overline{S}_p(t):\, t\ge 0\}$ the semigroup corresponding to the coupling between two versions of the process with (possibly) different initial configurations, by using the same Poisson processes $(N_x(t):x\in\Z)$ in its construction. \begin{lemma} \label{eq:lim} Let $p$ be an environment satisfying \reff{ccc} and such that $\rho^*(p)<\infty$ and $\zeta$ a configuration with lower asymptotic left density $\underline D(\zeta)>\rho^*(p)$. Then for any $v\vep\Bigr)\; = \;1\,, \end{equation} for all $\vep>0$ if $a> \ga(p,v)$ and \begin{equation} \label{601} \lim_{t\to\infty} \P_{(\nu_{p,v},-at)}\Bigl(\frac{X^{-at}_t}{t}> 0\Bigr) \,=\, 1 \end{equation} if $a<\ga(p,v)$. \end{lemma} \vskip 3mm \noindent{\bf Remark.} The more complete result when the starting point $a$ is greater than $\ga(p,v)$ comes from the fact that in our hypothesis we have only the asymptotic \emph{left} limits \reff{cc1}. If the limits \reff{cc1} hold for both sides, then \reff{600} is valid for all $a$. \noindent{\bf Proof:} Note that it suffices to prove (\ref{600}), since (\ref{601}) follows from (\ref{600}) and the fact that it does not depend on the environment to the right of the origin. For $u0$ have priority over $Y^{u,w}_t$ while those particles have no priority over $X^u_t$. Similarly, consider a second class particle for the $\xi$ process and denote it $X^w_t$. Since $Y^k_t$ for $k<0$ have priority over $X^w_t$ but not over $Y^{u,w}_t$, $X^w_t\le Y^{u,w}_t$. Hence, for $0\le u0$ and (possibly an infinite number of) sites $y$ such that $(\eta(y)-\xi(y))^->0$. We say that we have $\eta\xi$ discrepancies in the first case and $\xi\eta$ discrepancies in the second. The number of coupled particles at site $x$ at time $t$ is given by \begin{equation} \label{cp} \bar\xi_t(x):= \min \{\eta_t(x),\xi_t(x)\} \end{equation} The $\bar\xi$ particles move as regular (first class) zero range particles. There is at most one type of discrepancies at each site at time zero. Discrepancies of both types move as second class particles with respect to the already coupled particles. When a $\eta\xi$ discrepancy jumps to a site $z$ occupied by at least one $\xi\eta$ discrepancy, the $\eta\xi$ discrepancy and one of the $\xi\eta$ discrepancies at $z$ coalesce into a coupled $\bar\xi$ particle in $z$. The coupled particle behaves from this moment on as a regular (first class) particle. The same is true when the roles of $\xi$ and $\eta$ are reversed. The above description of the evolution implies in particular that a tagged discrepancy can not go through a region occupied by the other type of discrepancies. We will choose a negative site $y$ such that the jump rate from $y-1$ to $y$ is close to $c$. Then we follow the $\xi\eta$ discrepancies belonging to two disjoint regions of ${\Z}$ at time $0$ and give upper bounds on the probability of finding them at $y$ at time $t$. Roughly speaking, a $\xi\eta$ discrepancy at $y$ cannot come from a region ``close'' to $y$ because we prove that there is a minimum positive velocity for the $\xi\eta$ discrepancies to go. This velocity is given by the velocity of a second class particle under $\nu_{p,v}$. On the other hand, the $\xi\eta$ discrepancy cannot come from a region ``far'' from the origin because due to the difference of densities, a lot of $\eta\xi$ discrepancies will be between it and $y$ and hence they must pass site $y-1$ before it. But since we have chosen a small rate for this site, a traffic rush will prevent them to pass. With this idea in mind, we have to choose the ``close'' and ``far'' regions and the value of the rate at~$y-1$. Fix $v\; 0\; . \end{eqnarray} This allows us to fix $\varepsilon = \varepsilon(v)$ satisfying \[ 0 \,<\, \varepsilon(v)\, <\, \beta \gamma b - c + v\,. \] Finally, choose a negative site $y=y(v)$ such that $p_{y-1}< c+ \varepsilon$. We shall prove that \begin{equation} \label{a1} \lim_{t\to\infty}\; (\nu\times\nu_{p,v})\overline{S}_p (t) \Big \{(\eta,\xi):\, \eta (y)<\xi(y) \Big \} \; =\; 0 \; . \end{equation} We can order the $\xi\eta$ discrepancies and assume without loss of generality that the order is preserved in future times as we did in Lemma \ref{secondclass}. Of course some of the discrepancies will disappear. Let $Z^k=Z^k_t(\xi,\eta)$ the positions of the ordered $\xi\eta$ discrepancies at time $t$ with the convention that $Z^k_t=\infty$ if the corresponding discrepancy coalesced with a $\eta\xi$ one giving place to a $\bar\xi$ coupled particle. Let \begin{eqnarray} \label{agt} \lefteqn{ A_{\ga,t}(\eta,\xi)}\nonumber \\ &:=& \left\{\hbox{a $\xi\eta$ discrepancy in the box $[y-(t\gamma \bar b),y]$ at time $0$ }\right.\\ &&\ \left.\hbox{has moved to site $y$ at time $t$}\right\} \nonumber\\ &:=& \cup_k\left\{Z^k_0\in\left[y-(t\gamma \bar b),y\right], \;Z^k_t=y\right\} \nonumber \end{eqnarray} where $\bar b\,:=\,(1+b)/2\,\in\,(b,1)$. Hence \begin{equation} \label{lzp} \P(A_{\ga,t}(\eta,\xi)) \,\le\, \P\left(\min\left\{Z^k_t: Z^k_0\in \left[y-(t\gamma \bar b),y\right]\right\} \le \,y\right) \end{equation} We wish to give and upper bound to the event in the right hand side above. To do so we consider the coupled $(\eta , \xi) $ process and the $\xi $ process to which we add a unique second class particle at $y-(t\gamma \bar b)$, evolving together with jumps occurring at times given by the the same Poisson processes. We denote by $X_t^v$ the position of the second class particle at time $t$. If the second class particle has reached $ y+1$ no latter than time $t$, then there exists an increasing sequence of random times $0 \gbt b\right\} \right)}}\\ &&\qquad\qquad\qquad \subset \left\{ J^2_t - J^1_t >\gbt b\right\} \label{weh} \end{eqnarray} where $J^2_t$ and $J^1_t$ are the number of $\eta$, respectively $\xi$, particles jumping from $y-1$ to $y$ in the interval $[0,t]$. Since \begin{eqnarray} \label{100} \lefteqn{ \left\{ \bigcap_{z\le y-\gamma t\bar b} \left\{\sum_{x=z}^{y-1} (\eta_0(x)-\xi_0(x)) > \gbt b\right\} \right\}^c}\nonumber\\ &=& \bigcup_{z\le y-\gamma t\bar b} \left\{\sum_{x=z}^{y-1} \left(\eta_0(x)-\xi_0(x)\right) \leq \gbt b\right\}, \end{eqnarray} \noindent to bound $\P(B_{\gamma,t}(\eta,\xi))$ it suffices to bound the probabilities of the sets on the right hand side of \reff{weh} and \reff{100}. For \reff{weh} we have \begin{equation} \label{101} \P(J^2_t - J^1_t >\gbt b)\,\le\,\P(N^{c+\varepsilon}_t \,- \,N^v_t>\gbt b), \end{equation} where $N^a_t$ is a Poisson process of parameter $a$. The above inequality holds because the $\eta$-particles jump from $y-1$ to $y$ at rate not greater than $p_{y-1}$, which is by construction less than or equal to $c+\varepsilon$. On the other hand, by Burke's theorem, the number of jumps from $y-1$ to $y$ for the $\xi$-particles is a Poisson process of rate $v$. By the law of large numbers for the Poisson processes, we have \begin{eqnarray} \label{lnp} \lim_{t\to\infty} {1\over t} (N^{c+\varepsilon}_t \,- \,N^v_t) &=& c-v+\varepsilon \;<\;\beta\gamma b\,, \end{eqnarray} because we chose $\varepsilon < \gamma\beta b -c +v$. Hence \reff{101} goes to zero as $t\to\infty$. On the other hand, the probability of the set in the right hand side of \reff{100} is \begin{equation} \label{102} \P \left(\sup_{z < y-\gamma t\bar b} \sum_{x=z}^{y-1} (\eta_0(x)-\xi_0(x)) \leq \gbt b\right)\end{equation} By the ergodicity of $\xi$ and the fact that $\eta$ has left density, with probability one: \begin{eqnarray} \lim_{t\to\infty}\;{1\over t} \sum_{x=y-t\gamma \bar b}^{y-1} (\eta_0(x)-\xi_0(x))&=& {\gamma \beta \bar b} \;>\; \gamma\beta b\,, \end{eqnarray} by the way we chose $\bar b$. This implies that \reff{102} goes to zero as $t\to\infty$. This proves (\ref{a1}). Standard arguments (cf. Andjel (1982)) permit to deduce the statement of the lemma from (\ref{a1}). \quad $\carn$ \bigskip \begin{thebibliography}{9} \bibitem{andjel} Andjel, E.D. Invariant measures for the zero range process. {\em Ann. Probab.}, {\bf 10}, 525-547 (1982). \bibitem{bahadoran} Bahadoran, C. Hydrodynamical limit for spatially heterogeneous simple exclusion processes. {\em Probab. Theory Related Fields}, {\bf 110}, no. 3, 287-331 (1998). \bibitem{befela} Benjamini, I., Ferrari, P.A. and Landim, C. Asymmetric conservative processes with random rates. {\em Stoch. Proc. Appl.}, {\bf 61}, (1996), 181-204. \bibitem{dynkin} Dynkin, E.B. Sufficient statistics and extreme points. {\em Ann. Probab.}, {\bf 6}, 705-730 (1978). \bibitem{ev} Evans, M.R. Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. {\em Europhys. Lett.}, {\bf 36}, 13-18 (1996). \bibitem{ferrari} Ferrari, P.A. Shocks in the Burgers equation and the asymmetric simple exclusion process, In {\em Statistical Physics, Automata Networks and Dynamical Systems}, E. Goles and S.Martinez (eds), Kluwer Academic Publisher, 25-64 (1992). \bibitem{gkl} Gielis, G., Koukkous, A., Landim, C. {\em Equilibrium fluctuations for zero range process in random environment}, {\em Stoch. Proc. Appl.} {\bf 77}, 187-205 (1998). \bibitem{kiplan} Kipnis, C. and Landim, C. {\em Scaling Limit of Interacting Particle Systems}, Grundlheren {\bf 320}, Springer Verlag, Berlin, (1999). \bibitem{koukkous} Koukkous, A. {\sl Hydrodynamic behavior of symmetric zero range process with random rates}, to appear in {\em Stoch. Proc. Appl.} (1999). \bibitem{fk} Krug J. and Ferrari, P.A. Phase transitions in driven diffusive systems with random rates. {\em J. Phys. A:Math. Gen.}, {\bf 29}, 1465-1471 (1996). \bibitem{Claudio} Landim, C. Hydrodynamical limit for space inhomogeneous one dimensional totally asymmetric zero range processes. {\em Ann. Probab.}, {\bf 24}, 599-638 (1996). \bibitem{Timo} Sepp\"al\"ainen, T. and Krug, J. Hydrodynamics and platoon formation for a totally asymmetric exclusion model with particlewise disorder. {\em J. Statist. Phys.} {\bf 95}, no. 3-4, 525-567 (1999). \bibitem{Shir} Shiryayev A.N, {\em Probability}. Graduate Texts in Mathematics {\bf 95}, Springer Verlag, New York, (1984). \end{thebibliography} \end{document} ---------------9911251040272--