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Zero-range, Random rates, invariant measures, Convergence to the  
maximal invariant measure 
---------------9911251040272
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\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
$v<c$,
\begin{equation}
  \label{cc1}
  R(p,v)\,:=\, \lim_{n\to\infty}{1\over n}\sum_{x=-n+1}^0 \int
  \nu_{p,v}(d\xi)\, \xi(x)\, = \,\lim_{n\to\infty}{1\over n}
  \sum_{x=-n+1}^0 \frac {v}{p_x -v}.
\end{equation}
We interpret $R(p,v)$ as the global expected left density per site
of the configurations distributed according to $\nu_{p,v}$. A
consequence of the existence of the limits is that for all $v<c$,
$\nu_{p,v}$ concentrates in configurations with asymptotic left
density $R(p,v)$:
\begin{equation}
  \label{cc5}
  \nu_{p,v}\Bigl(\lim_{n\to\infty}{1\over n}\sum_{x=-n+1}^0 \xi(x) =
R(p,v)\Bigr)\,=\,1\; .
\end{equation}
It is easy to prove that $R(p,v)$ is a strictly convex
increasing function of $v$, hence the limit
\begin{equation}
  \label{cc6}
  \rho^*(p):=\lim_{v\to c} R(p,v)
\end{equation}
is well defined (but may be infinite). In the sequel we assume
$\rho^*(p)<\infty$. We do not assume the existence of the limit
in \reff{cc1} for $v=c$, nor the $\nu_{p,c}$ almost sure
convergence of the density.

Our second and main result states that under the condition
$\rho^*(p)<\infty$, initial measures concentrating on configurations
with asymptotic left density strictly bigger than $\rho^*(p)$ converge
towards the maximal invariant measure $\nu_{p,c}$. We do not know in
general if this measure concentrates on configurations with density.
But if the limit $R(p,c)$ of \reff{cc1} exists, equals $\rho^*(p)$ and
is finite our result says that the process starts with global density
strictly above $\rho^*(p)$ and converges to a measure with density
$\rho^*(p)$. This behavior is remarkable as the process is
\emph{conservative}, i.e. the total number of particles is conserved,
but in the above limit ``looses mass''.  Informally speaking, what
happens is that many clients remain trapped in far away slow servers.
More precisely, denoting by $S_p(t)$ the semigroup of the process, we
first show that for any initial measure $\nu$, all weak limits of the
sequence $\{ \nu S_p(t),\, t\ge 0\}$ are dominated, in the natural
partial order, by $\nu_{p,c}$.  We then show that if $\nu$ is a
measure concentrated on configurations with asymptotic left density
strictly greater than $\rho^*(p)$, all weak limits of $\nu S_p(t)$
dominate $\nu_{p,c}$. Surprisingly enough, the proof of the second
statement is much more demanding than the proof of the first one.

It follows from the two previous results that the domain of
attraction of $\nu_{p,c}$ includes all measures with asymptotic
density strictly above $\rho^*(p)$. It remains an open question to
describe the domain of attraction of a product invariant measure
$\nu_{p,v}$ for $0< v<c$ or to show the convergence to $\nu_{p,c}$
of initial measures with asymptotic density $\rho^*(p)$.

Our results hold $m$-a.s.\ for measures $m$ concentrating on
environments satisfying \reff{cc1}.

The paper is organized as follows: in Section 2 we introduce the
notation and state the main results. In Section 3 we characterize
the set of invariant measures and show that the maximal invariant
measure dominates all the weak limits of the process. In Section 4
we obtain the asymptotic velocity of a second class particle for
the zero-range process in non homogeneous environment and use this
result to prove the main theorem.

\section{Notation and Results}

Fix $0<c\leq 1$ and consider a sequence $(p_x)_{x\in \Z}$ taking
values in $[c,1]$ such that $c=\inf_x p_x$. We consider a totally
asymmetric zero-range process in the environment $p$. This is a
Markov process that can be informally described as follows. We
initially distribute particles on the lattice $\Z$ . If there is
at least one particle at some site $x$, then at rate $ p_x$ one of
them jumps to site $x+1$. To construct a Markov process $\eta_t$
on $\X=\N^{\Z}$ corresponding to the above description, let
$N_x(t)\ ( x\in \Z)$ be a collection of independent Poisson
processes such that for all $x\in \Z,\ \E(N_x(t))=p_xt$. The
evolution of $\eta_t$ is now given by the following rule: if the
Poisson process $N_x(. )$ jumps at time t and $\eta_{t-}(x)>0$
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 $y<x$ such that
$N_y(t)=0$ .  Hence the value of $\eta_t(x)$ depends only on the
initial configuration and on a finite number of jumps.

The generator $L_p$ of this process, defined by $L_pf(\eta) =
d\E[f(\eta_t) \,|\,\eta_0=\eta]/dt\Big|_{t=0}$, acts on cylinder
functions $f$ as follows:
\begin{equation}
 \label{a3}
(L_p f)(\eta)=\sum_{x\in{\Z}}\,p_x\,\one\{\eta (x)>0\}
\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 $ 0<v<p_x$ for all
$x$. Then a standard calculation shows that the product measure
$\nu_{p,v}$ with marginals given by (\ref{889})
%\[
%\nu_{p,v}\{\xi : \, \xi(x)=k\} =\Bigl(\frac v{ p_x}\Bigr)^k
%\Bigl(1-\frac v{ p_x}\Bigr)\;,  \quad k\ge 0\; ,
%\]
is an invariant measure for the process. Benjamini, Ferrari and
Landim (1998) raised the question of whether or not there exist
invariant measures which are not convex combinations of the
$\nu_{p,v}$'s.  In Section 2 we prove the following theorem which,
combined with Theorem 12.2 in Dynkin (1978), gives a negative
answer to that question.  In its statement we denote by $({\cal
I}_p)_e$ the set of extremal invariant measures for the process.
\begin{theorem}
\label{2.1} Let $p$ be an arbitrary environment then
 \[
 ({\cal I}_p)_e=\{\nu_{p,v}\ :\ v<p_x,\ \forall x\in \Z \}.
 \]
\end{theorem}

In this theorem the range of the parameter $v$ may be either
$[0,c)$ or $[0,c]$ - when $p_x=c$ for some $x$ or $p_x>c$ 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 $v<c$.

The values assumed by $R(p,v)$ for $v<c$
are crucial for the characterization of the set of invariant measures
for the process with rates $p$. If $\lim_{v\to c}R(p,v)=\infty$, then
the range of allowed densities is $[0,\infty)$ or $[0,\infty]$. The
first case occurs when $p_x= c $ for some~$x$. In this case
$\nu_{p,v}$ is defined for any $v<c$, but not for $v=c$.  Moreover,
since $R(p, \cdot)$ is continuous and increases to $\infty$ as $v\to
c$, then for all $\rho\in[0,\infty)$ there exists $v=v(p,\rho)$ such
that $\nu_{p,v}\{\eta\in\X: D(\eta) = \rho\}= 1$. The second case
occurs when $p_x>c$ 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 $m(\{p:p_0=c\})=0$, $m(\{p:
c<p_0 < c +\varepsilon \})>0$ 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 $x_1<y$, $p_{x_n}
<p_z$ for $x_n<z\le y$ and $p_{x_n}$ decreases to $c$. The
existence of such a sequence is guaranteed by \reff{ccc}. Consider
a process on ${\N}^{\{x_n +1,...,y\}}$ with generator given by:
\begin{eqnarray}
  \label{ln}
  L_{p,n}f(\eta )&=& \sum_{z=x_{n}+1}^{y-1}
{\bf 1 }\{\eta (z)>0\} 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_n<z\leq y $, is invariant for the process. Therefore
$\nu_n S_{n,p}(t)$ converges to $\mu _{n,p}$ and any weak limit
point of $(\nu S_p(t))_n$ is bounded above by $\mu_{n,p}$. Since
as $n$ goes to infinity the marginals of $\widetilde\nu_{n,p}$
converge to the marginals of $\nu_{c,p}$ the proposition is
proved. $\carn$

\bigskip
\noindent{\bf Proof of Theorem \ref{2.1}.} Since only the final
step of the proof is different from the proof of Theorem 1.11 in
Andjel (1982) we refer the reader to that paper. Exactly as there
one proves that if $\nu _p$ is an extremal invariant measure then
for each $v<\inf_x p_x$ either $\nu _p\leq \nu _{v,p}$ or $\nu
_p\geq \nu _{v,p}$.  This implies that either $\nu _p =\nu _{v,p}$
for some $v$ or $\nu _p \geq \nu _{v,p}$ for all $v$.  The latter
case cannot occur if there exists $x$ such that $p_x=\inf_y p_y$
because this would imply that $\nu _p \{\eta\ :\eta (x)>k \}=1$
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 $v<c$,
\[
\lim_{t\to\infty} (\delta_\zeta\times\nu_{p,v})\overline{S}_p(t)
\Big \{(\eta,\xi):\, \eta (x)<\xi(x) \Big \} \; =\; 0
\]
for all $x$ in ${\Z}$.
\end{lemma}

The proof of this lemma requires the following result. It states that
for each $v<c$ the asymptotic velocity of a second class particle in
the zero-range process in the environment $p$ under the invariant
measure $\nu_{p,v}$ is strictly positive.

Fix a starting site $z$ and consider a coupled zero range process
with initial condition $(\eta,\eta+\mathfrak d_{z})$ and semigroup
$\bar S_p(t)$. Under the coupled dynamics the number of sites
where the marginals differ does not increase in time. Let $X^z_t$
be the site where the marginals differ at time $t$. We can think
that $X^z_t$ stands for the position of a ``second class
particle''. Indeed, if the second class particle is at $x$ at time
$t$ it jumps to $x+1$ at rate $p_x\,\one\{\eta_t(x)=0\}$. In other
words, the second class particle jumps only if there is no other
particle at the site where it is.


For an environment $p$ and a probability measure $\nu$ on $\X$,
denote by $\P_\nu$ the measure on $D(\R_+, \X)$ induced by $\nu$
and the Markov process with generator $L_p$ defined in \reff{a3}.
In the next lemma we write $\P_{(\nu,z)}$ for a coupled process
whose initial configuration is $(\eta,\eta+\mathfrak d_z)$, with
$\eta$ distributed according to $\nu$. Since $R(p,v)$
is convex and strictly increasing
\begin{equation}
  \label{602}
  \ga(p,v) := [R'(p,v)]^{-1}
\end{equation}
exists in a dense subset of $(0,c)$. In the sequel we abuse notation
by not writing integer parts where necessary.


\begin{lemma}
\label{secondclass} Let $p$ be an environment for which the limits in
\reff{cc1} exist for $v<c$. Pick $v\in (0,c)$ such that $ \ga(p,v)$
exists. Then, 
\begin{equation}
\label{600} \lim_{t\to\infty}
\P_{(\nu_{p,v},-at)}\Bigl(\Bigl|\frac{X^{-at}_t}{t}-
(\ga(p,v)-a)\Bigr|\,>\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 $u<w<c$ let
$\bar \nu_{p,u,w}$ be the product measure on $\X\times {\X}$ whose
first marginal is equal to $\nu_{p,u}$, whose second marginal is equal
to $\nu_{p,w}$ and which is concentrated above the diagonal: $\bar
\nu_{p,u,w} \{(\eta, \xi):\, \eta \le \xi\}=1$.  Denote by $(\eta_t,
\xi_t)$ the coupled Markov process starting form $\bar \nu_{p,u,w}$.

Denote by $\zeta_t$ the difference $\xi_t - \eta_t$ and observe
that the $\zeta$-particles evolve as second class particles in the
sense that a $\zeta$-particle jumps from $x$ to $x+1$ at rate
\[
p_x\, [\one\{\eta (x) + \zeta (x)\ge 1\} - \one\{\eta (x)\ge 1\}];
\]
that is, when there are no $\eta$ particles present.  In this
case we say that the $\eta$ particles have \emph{priority} over
the $\zeta$ particles. We label the $\zeta$-particles at time $0$
in the following way. Without losing much we can assume that
there is a $\zeta$ particle at site (integer part of) $-at$. The
measure conditioned on this event is absolutely continuous with
respect to $\bar \nu_{p,u,w}$, an this will be enough for our
purposes, as we shall only use laws of large numbers. Call particle
$0$ this particle, and complete the labeling in such a way that a
particle  with label $j$ is at the same site or at the left of a
particle with label $k$ if $j<k$. Denote by $Y^j_t$ the position
at time $t$ of the particle labeled $j$. By construction, we have
$\cdots \le Y^{-1}_0 < Y^{0}_0 = -at \le Y^1_0 \le \cdots$. We let
the second class particles evolve in a way to preserve this order.
To keep track of the densities involved in the definition we call
$Y^{u,w}_t = Y^0_t$.

Consider now a single second class particle for the $\eta$ process
initially at the position of $-at$. This is obtained by
considering the coupled initial condition $(\eta,\eta+\mathfrak
d_{-at})$. Denote the position of the single second class particle
at time $t$ by $X^u_t$ (for $u=v$, this has the same law as the
particle denoted by $X^{-at}_t$ in the statement of the
proposition). Since $Y^{u,w}_0= X^u_0$, in the coupled evolution
obtained by using the same Poisson processes $(N_x(t))$ we have
$Y^{u,w}_t\le X^u_t$ for all $t$. Indeed, in this coupling $Y^k_t$
for $k>0$ 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 $0\le u<w\le c$,
\begin{equation}
  \label{0vwc}
  X^w_t\le Y^{u,w}_t\le X^u_t\,,
\end{equation}
$\P_{\bar \nu_{p,u,w}}$ almost surely.

Denote by $J_t^1$, $J_t^{1+2}$ and $J_t^2$ the the total number of
$\eta$, $\xi$ and $\zeta$ particles that jumped from $-at$ to
$-at+1$ before time $t$. In particular, $J_t^2 = J_t^{1+2} -
J_t^1$. By Burke's theorem, the number of $\eta$-particles (resp.
$\xi$-particles) that jump from $-at$ to $-at+1$ is a Poisson
process of parameter $u$ (resp. $w$). Hence the number of
$\zeta$-particles that jump from $-at$ to $-at+1$ in the interval
$[0,t]$ is the difference of two Poisson process and satisfies the
law of large numbers:
\[
\lim_{t\to \infty} \frac{J_t^2}t \; =\; \lim_{t\to \infty}
\frac{J_t^{1+2} - J^1_t}t \; =\; w  - u
\]
in $\P_{\bar \nu_{p,u,w}}$ probability. On the other hand, for
every $t\ge 0$,
\[
J_t^2 \; := \; \sum_{x=-at}^{Y^{u,w}_t}\zeta_t(x)-A_t \; = \;
\sum_{x=-at}^{Y^{u,w}_t} \xi_t(x) \; - \;
\sum_{x=-at}^{Y^{u,w}_t} \eta_t(x)-A_t \; .
\]
where  $\vert A_t\vert \leq \zeta_t(Y^{u,w}_t)$. Note that $\zeta_t(
Y^{u,w}_t)$ is stochastically bounded above by a geometric random
variable of parameter $w/c$. Therefore $\vert {A_t}/{t}\vert$
converges to $0$ in in $\P_{\bar \nu_{p,u,w}}$ probability as $t$
goes to infinity. As in Ferrari (1992), it follows from the
previous equation and the law of large numbers for $\eta_t$ and
$\xi_t$ ---that are distributed according to product (invariant)
measures with densities $R(p,u)$ and $R(p,w)$ respectively--- that
for $u$ and $w$ strictly smaller than $c$,
\begin{equation}
  \label{ytp}
  \lim_{t\uparrow\infty} {Y^{u,w}_t\over t} \;=\;{w-u \over R(p,w) -
R(p,u)}
\end{equation}
in $\P_{\bar \nu_{p,u,w}}$ probability. Hence, from \reff{0vwc} we
have
\begin{equation}
  \label{xcw}
  \lim_{t\uparrow\infty} {X^{w}_t\over t} \;\le\;{w-u \over R(w) -
  R(u)} \;\le\;\lim_{t\uparrow\infty} {X^{u}_t\over t}
\end{equation}
in $\P_{\bar \nu_{p,u,w}}$ probability. Fixing $w=v$ and taking the
limit $u\to v$ and then fixing $u=v$ and taking the limit $u\to v$ and
taking account of the differentiability of $R$ in $v$, we get \reff{600} and \reff{601}. $\carn$

\bigskip
We are now in a position to prove Lemma \ref{eq:lim}.

\noindent {\bf Proof of lemma \ref{eq:lim}} We shall prove that
for all $v<c$, any weak limit $\tilde\nu$ of $\delta_\eta S_p(t)$
dominates stochastically $\nu_{p,v}$: $\tilde\nu\ge\nu_{p,v}$. The
result follows from this because $\{\nu_{p,v}:0\le v<c\}$ is an
increasing sequence of measures converging to $\nu_{p,c}$.

The proof is performed via coupling. We start with two different
initial configurations $\eta$ and $\xi$ with marginal
distributions $\nu$ and $\nu_{p,v}$, respectively. Hence $\eta$
has lower asymptotic density bigger than $\rho^*(p)$ and $\xi$ has
asymptotic density $R(p,v)$. We use the same Poisson processes for
both processes and call $(\eta_t,\xi_t)$ the coupled process. The
configurations $\eta$ and $\xi$ are in principle not ordered:
there are (possibly an infinite number of) sites $z$ such that
$(\eta(z)-\xi(z))^+>0$ 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 $v<c$ such that $R(p,\cdot)$ is differenciable in $v$. Let
$\gamma= \gamma(p,v)$, the (strictly positive) asymptotic speed of a
second class particle under $\nu_{p,v}$ in the sense of~(\ref{602}).
Denote by $\beta$ the difference between the asymptotic lower density
of $\eta$ and $R(p,v)$~:
\[
\beta=\beta(p,v) \; =\; \lim_{n\to \infty} \frac 1n \sum_{x=-n}^0
[\eta (x) - R(p,v)]\; .
\]
For reasons that will become clear later (cf.\/ display
\reff{lnp}), we let
\[
b\,=\,b(p,v)\,=\, {R'(p,v) (c-v) \over \rho^*(p)-R(p,v)} \,<\,1\,,
\]
by the convexity of $R$; recall that $\rho^*(p) = \lim_{v\to c}
R(p,v)$. With this choice,
\begin{eqnarray}
  \label{gb4}
   \beta\gamma b - c + v \;=\; { c-v \over \rho^*(p)-R(p,v) } \Big\{
  \beta - [\rho^*(p) - R(p,v)]\Big\} \; >\; 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 $0<T_{y-(t\gamma \bar b
  )}<T_{y-(t\gamma \bar b)+1}<...<T_y$ such that at each of these
times the corresponding site has been emptied of its $\xi $
particles. But this implies that all the $\xi \eta $ discrepancies
which at time $0$ were in the interval $[y-(t\gamma \bar b),y]$
have disappeared or are strictly to the right of $y$. Therefore:
\begin{equation}
  \label{aia}
  \P\left(A_{\ga,t}(\eta,\xi)\right)\,\le\,
  \P(X^{y-t\gamma \bar b}_t \le y )\,.
\end{equation}
By \reff{601} this tends to $0$ as $t$ tends to infinity because
$\gamma \bar b<\ga$. The above argument is independent of the
value of $p_{y-1}$.

It now suffices to check that the probability that a $\xi\eta$
discrepancy, to the left of $y-(t\gamma\bar b)$ at time 0 reaches
$y$ no later than time $t$, tends to $0$ as $t$ tends to infinity.
Let
\begin{eqnarray}
  \label{bgt}
 \lefteqn{ B_{\ga,t}(\eta,\xi)}\nonumber \\
&:=& \left\{\hbox{a $\xi\eta$ discrepancy in $(-\infty,y-(t\gamma
\bar b)]$ at time $0$}\right.\\ &&\ \left.\hbox{ has moved to site
$y$ at time $t$}\right\} \nonumber\\ &:=&
\cup_k\left\{Z^k_0\in(-\infty,y-(t\gamma \bar b)],
\;Z^k_t=y\right\}\nonumber
\end{eqnarray}
Call $W_t^k(\eta,\zeta)$ the positions of the $\eta\xi$
discrepancies at time $t$, being $W_0^0$ the first $\eta\xi$
discrepancy to the left of the origin. As before set
$W^k_t=\infty$ if the $k$th discrepancy coalesced with a $\xi\eta$
one before $t$.

Since a  $\xi \eta $ discrepancy cannot cross over an $\eta \xi $
discrepancy,
\begin{eqnarray}
\nonumber
\lefteqn{\hskip-2cm {B_{\gamma,t}(\eta,\xi)\cap
  \left( \bigcap_{z\le y-\gamma t\bar b} \left\{\sum_{x=z}^{y-1}
    \left(\eta_0(x)-\xi_0(x)\right) > \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



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