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Zero-range, Random rates, invariant measures, Convergence to the
maximal invariant measure
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\def\vep{\varepsilon}
\def\reff#1{(\ref{#1})}
\def\gbt{{t\gamma\beta }}
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\def\proof{\noindent{\bf Proof. }}
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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
$v0$
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
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\end{document}
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