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\rightline {\today}
%\rightline {March 23, 1994}
\centerline {\bf Second class particles in the rarefaction fan}
\vskip 2truemm
\centerline { P.\ A.\ Ferrari, C. Kipnis}
\vskip 1truemm
\centerline {\it Universidade de S\~ao Paulo and Universit\'e de Paris Dauphine}
\vskip 1truemm
\noindent {\bf Summary.} We consider the one dimensional totally asymmetric
nearest neighbors simple exclusion process with drift to the right starting
with the configuration ``all one'' to the left and ``all zero'' to the right
of the origin. We prove that a second class particle initially added at the
origin chooses randomly one of the characteristics with the uniform law on the
directions and then moves at constant speed along the chosen one. The result
extends to the case of a product initial distribution with densities
$\rho>\la$ to the left and right of the origin respectively. Furthermore we
show that, with a positive probability, two second class particles in the
rarefaction fan never meet.
\vskip 3truemm
\noindent {\bf R\'esum\'e.} On consid\`ere le processus d'exclusion simple
totalement asym\'etrique unidimensionel avec d\'erive \`a droite. Le processus
commence avec la configuration ``tout un'' \`a gauche de l'origine et ``tout
z\'ero'' \`a droite. On prouve qu'une particule de deuxi\`eme classe plac\'ee
\`a l'origine \`a l'instant z\'ero choisit une des charact\'eristiques avec une
loi uniforme et ensuite suit cette charact\'eristique \`a vitesse constante. Le
r\'esultat s'etend au cas d'une mesure initiale produit avec densit\'es
$\rho$ \`a gauche et $\la$ \`a droite de l'origine, satisfaisant $\rho>\la$.
Finalement, on prouve que deux particules de deuxi\`eme classe dans le front
de rar\'efaction ont une probabilit\'e positive de ne pas se rencontrer.
\vskip 5truemm
\noindent {\it Keywords and phrases.}
Asymmetric simple exclusion. Second class particle. Law of large numbers.
Rarefaction fan.
\vskip 2truemm
\noindent {\it AMS 1991 Classification.} 60K35, 82C22, 82C24,
82C41.
\vskip 2truemm
\noindent {\it Short title:} Second class particles in the rarefaction fan.
\vskip 3truemm
\noindent {\bf 1. Introduction.}
\numsec=1\numfor=1
The one dimensional asymmetric
exclusion process is known to have the inviscid Burgers equation as
hydrodynamic limit.
Usually one takes advantage of the deep knowledge
accumulated through the years on this non linear pde to prove results for the
exclusion process. In this short note we intend to do exactly the contrary.
We prove a result for the exclusion process and use it to guess a result for
the pde.
For this purpose we study the trajectory of a second class particle which is
known to give information on the characteristics of the pde. More precisely,
it has been proved (Ferrari (1992), Rezakhanlou (1993)) except for the case of
the rarefaction fan, that a second class particle added at a macroscopic site
$a$ has a position at macroscopic time determined by the characteristic
emanating from $a$. Of course in these cases there is only one characteristic
issued from $a$. When dealing with a rarefaction fan one has an infinite
number of characteristics issued from $a$. We prove that the second class
particle chooses instantaneously {\it at random} (uniformly) among the possible
characteristics and then of course follows it.
This suggests the following result for the pde: if a small perturbation is
added at a point of discontinuity that would give rise to a rarefaction fan,
then the perturbation is smeared uniformly in the fan.
Informally the one dimensional asymmetric simple exclusion process we study
here is described as follows. Only a particle is allowed per site and at rate
one each particle independently of the others attemps to jump to its right
nearest neighbor; the jump is realized only if the destination site is
empty. A second class particle is a particle that jumps over empty sites to
the right of it at rate $1$ and interchanges positions with the other
particles to the left of it at rate $1$. Let $S(t)$ be the semigroup
corresponding to the process without the second class particle.
Let $\nurl$ be the product distribution with
marginals $\nurl(\eta(x)=1) = \rho 1\{x\le 0\} + \la 1\{x>0\}$. Let $\nu_\rho
= \nu_{\rho,\rho}$.
The process with initial distribution $\nurl$ has hydrodynamic limit
$$
\lime \nurl S(\vep^{-1}t) \tau_{[\vep^{-1}r]}f = \nu_{u(r,t)}f \Eq(12)
$$
where $f$ is a cylinder function, $\tau_x$ is the translation by $x$ operator,
$[.]$ is the integer part and for $t\ge 0$, $r\in\R$,
$u(r,t)$ is the entropy solution of
$$
\cases \displaystyle{{\partial u \over \partial t} + {\partial (u(1-u))\over
\partial r}=0} & \cr
u(r,0) = u_0(r)& \cr
\endcases \Eq(bur)
$$
where $u_0(r) = \rho 1\{r\le 0\} + \la 1\{r>0\}$.
We consider $\rho>\la$. In this case the explicit form of $u(r,t)$ is the
following:
$$
u(r,t) = \cases \rho & \text {if } r\le (1-2\rho)t \cr
(t-r)/2t & \text {if } (1-2\rho)t < r \le (1-2\la)t \cr
\la & \text {if } r> (1-2\la)t \cr \endcases \Eq(urt)
$$
(See Rost (1982), Andjel and Vares (1987),
Rezakhanlou (1990), Landim (1993) and references therein.)
The characteristics emanating from $a$
related to this equation are the solutions $r(t)$ of the ode
$$
\cases
\displaystyle{\frac{dr}{dt}} = 1-2 u(r,t) \\
r(0) = a \ .
\endcases \Eq(1.6)
$$
Here $1-2u$ is the derivative with respect to $u$ of the current $u(1-u)$
appearing in the non linear part of the Burgers equation \equ(bur).
``Since $u$ is not continuous, \equ(1.6) is understood in the Filippov
sense: an absolutely continuous function $r$ is a solution if
for almost all $t$, $\frac{dr}{dt}$ is between the essential
infimum and the essential supremum of $1-2 u(r,t)$ evaluated at the
point $(r(t),t)$.'' (Rezakhanlou (1993); see Filippov (1960)).
Under our initial condition there is only one characteristic for every $a\ne
0$ but there are infinitely many characteristics departing from the origin
producing the fan in the region $[(1-2\rho)t,(1-2\la)t]$. Our first result
says that the second class particle chooses one uniformly among those and follows
it.
\proclaim {Theorem 1} Consider the simple exclusion process starting
with the product measure $\nurl$ with $1\ge\rho>\la\ge 0$. At time zero put a
second class particle in the origin regardless the configuration value at
this point.
Let $X_t$ be the position of the second class particle at time $t$
and let $X^\vep_t = \vep X_{\vep^{-1}t}$. Then
$$
\lime X^\vep_t = U_t\ \ \ \text{in distribution,}\ \Eq(lln1)
$$
where $U_t$ is a random variable uniformly distributed in the interval
$[(1-2\rho)t,(1-2\la)t]$.
Moreover for any $0~~0$ and
$u_{0,\de}(r)$ be a density that differs from $u_0(r)$ only in the interval
$[0,\de]$ and in this interval the density is equal to $\rho$ (instead of
$\la$). Then $u_\de(r,t)$, the entropic solution of the Burgers equation
\equ(bur) but with initial condition $u_{0,\de}(r)$ is given by
$$
u_\de(r,t) = \cases \rho & \text {if } r\le (1-2\rho)t +\de \cr
(t-r+\de)/2t & \text {if } (1-2\rho)t+\de < r \le (1-2\la)t +\de \cr
\la & \text {if } r> (1-2\la)t+\de \cr \endcases \Eq(udrt)
$$
and the difference between this solution and the solution of the unperturbed
system is given by $m_\de(r,t) = u_\de(r,t)-u(r,t)$. For $\de < 2(\rho-\la)t$,
$$
m_\de(r,t)=
\cases
0 & \text {if } r\le (1-2\rho)t \cr
(r-(1-2\rho)t)/2t & \text {if } (1-2\rho)t \le r \le (1-2\rho)t + \de \cr
\de/2t &\text {if } (1-2\rho)t + \de \le r \le (1-2\la)t \cr
((1-2\la)t-r+\de)/2t & \text {if } (1-2\la)t \le r \le (1-2\la)t +\de \cr
0 & \text {if } r> (1-2\la)t+\de \cr
\endcases \Eq(drt)
$$
In other words, the perturbation is smeared in the rarefaction front. The same
result is presumably true for more general initial conditions and more general
type of perturbations. To show \equ(drt) it suffices to
compute $u(r,t)$ for the two different initial conditions and subtract. To do
this computation observe that $u_\de$ is just a translation of $u$ by $\de$.
Theorem 1 is shown in the next section. Its proof is based in computing the
same quantity using two different couplings. In Section 3 we consider the
process starting with $\nu_{1,0}$, that is with the configuration that has
$1$'s to the left of the origin (including it) and $0$'s to its right. We show
that if two second class particles are added in sites $0$ and $1$ at time
zero, then there is a positive probability that they never meet.
\vskip 3truemm
%\vfill\eject
\noindent {\bf 2. Couplings.}
\numsec=2\numfor=1
A coupling is a joint realization of two versions of the process with
different initial configurations. To realize the ``basic coupling'' of Liggett
(1985) one attaches a Poisson clock of parameter one to each site of $\Z$.
When the clock rings for site $x$, if there is a particle in $x$ and there is no
particle in $x+1$, then the particle jumps one unit to the right. Under this
coupling the two configurations use the same realization of the clocks.
Under the ``particle to particle'' coupling we have to label the particles of
the two configurations. We can also use the same realizations of the clocks
attached to the sites, but only one of the configurations (say the
first one) looks at the clocks. When a clock rings for the
$i$-th particle of the first configuration, then the $i$-th particles of both
configurations try to jump. On each marginal the jump is actually performed
if the exclusion rules of the configuration of that marginal allow it.
\noindent {\bf Proof of \equ(lln1).} We
want to show that for $r\in [(1-2\rho)t,(1-2\la)t]$,
$$
\lime \P(X^\vep_t > r) = {(1- 2\la)t - r
\over 2(\rho-\la)t}. \Eq(lln2)
$$
For a given initial configuration $\eta$, let $J_{r,t}(\eta)$ be the number of
particles of $\eta$ to the left of the origin (including it) that end up at
time $t$ strictly to the right of $r$ minus the number of particles of $\eta$
strictly to the right of the origin that end up at time $t$ to the left of $r$
(including it). We call $J_{r,t}$ the current through $r$ up to time $t$.
Let $J^\vep_{r,t}
= J_{r\vep^{-1},t\vep^{-1}}$. Now we compute in two different ways
$$
\int d\nurl(\eta)\E J^\vep_{r,t}(\eta)-
\int d(\tau_{-1}\nurl)(\eta)\E J^\vep_{r,t}(\eta).
$$
where $\tau_x$ is the translation by $x$ operator: $(\tau_x\eta)(z) =
\eta(z-x)$.
For {\it any} coupling $\bar\mu$ of $\nurl$ and $\tau_{-1}\nurl$ and {\it any}
coupling $\bar \P$ of the two processes the previous quantity is also equal to
$$
\int d\bar\mu(\eta^0,\eta^1)
\bar\E(J^\vep_{r,t}(\eta^0)- J^\vep_{r,t}(\eta^1)). \Eq(diff)
$$
In the sequel we write $\E$ for the expectation with respect to the coupled
process.
We first couple $\nurl$ and $\tau_{-1}\nurl$ in such a way that if $\eta^0$
and $\eta^1$ are two configurations with those distributions respectively,
then $\eta^0(x) = \eta^1(x)$ for all $x\ne 0$ and with probability $\rho-\la$
there is a particle in the origin for the first marginal and no particle for
the second marginal: $\bar\mu(\eta^0(0) = 1-\eta^1(0) = 1)=\rho-\la$. Now, in
the event that there is a discrepancy in the origin, we use the basic coupling
and observe that this discrepancy behaves like a second class particle. If we
label the particles at the other sites and call them first class particles,
then under this coupling the positions of these particles are exactly the same
for both marginals. This implies that the current produced by the first class
particles are identical for both marginals and that the only difference can
arise from the second class particle. It is then easy to see that the currents
through $\vep^{-1}r$ at time $\vep^{-1}t$ for the two marginals differ if and
only if at time $\vep^{-1}t$ the second class particle is beyond
$\vep^{-1}r$. Hence, taking expectations, from this coupling we see that
\equ(diff) is equal to
$$
(\rho-\la)\P(X^\vep_t > r). \Eq(coup1)
$$
We now couple $\nurl$ and $\tau_{-1}\nurl$ in such a way that
$\eta^1=\tau_{-1}\eta^0$. Then we use the particle to particle coupling to
obtain that the currents through $\vep^{-1}r$ for the two marginals differ by
one if and only if for the first marginal there is a particle at
$[\vep^{-1}r]+1$ at time $\vep^{-1}t$ and no particle in site $1$ at time $0$.
Those currents differ by $-1$ if and only if for the first marginal there is a
particle in site $1$ at time $0$ and there is no particle in site
$[\vep^{-1}r]+1$ at time $\vep^{-1}t$. Taking expectations and noting that the
above events depend only on the first marginal,
\equ(diff) is also equal to
$$
\eqalign{
&\P (\eta_{\vep^{-1}t}([\vep^{-1}r]+1)=1,\eta_0(1)=0) -
\P (\eta_{\vep^{-1}t}([\vep^{-1}r]+1)= 0,\eta_0(1)=1) \cr
&\quad= \P (\eta_{\vep^{-1}t}([\vep^{-1}r]+1)=1) - \P( \eta_0(1)=1).\cr
}\Eq(coup3)
$$
Since for the first marginal the initial distribution is $\nurl$, $\P( \eta_0(1)=1)=\la$.
Letting $\vep$ tending to zero we obtain by standard convergence to local
equilibrium \equ(12) that
$$
\lime \P(\eta_{\vep^{-1}t}([\vep^{-1}r]+1) = 1) = u(r,t), \Eq(coup2)
$$
where $u(r,t)$ is defined by \equ(urt). Putting \equ(coup1), \equ(coup3)
and \equ(coup2)
together we get \equ(lln2). \square
To show \equ (lln) we need the following lemma.
\ppclaim \Lemma(rrr). Consider the simple exclusion process starting
with the product measure $\nurl$ with $1\ge\rho>\la\ge 0$.
Let $r\in (1-2\rho,1-2\la)$. Assume that at time
$\vep^{-1}s$ we put a second class particle in position
$[\vep^{-1}rs]$ disregarding the occupation number in this position. For $t\ge s$ let
$R^\vep_t$ be $\vep$ times the position of this particle at time
$\vep^{-1}t$. Then
$$
\lime R^\vep_t = rt \ \ \ \text{in probability.}
$$
\proof We first take $\de>0$ and use the basic
coupling for the process starting with densities
$u_0$ and $u_\de$, respectively, where $u_\de$ is defined in \equ(udrt).
Define a family of initial distributions
$\{\nu^\vep\}_{\vep>0}$, where $\nu^\vep$ is a product distribution with marginals
$\nu^\vep(\eta(x) = 1) = \rho 1\{x\le \de\vep^{-1}\} + \la 1\{x>
\de\vep^{-1}\}$. We
couple the initial distributions $\nurl$ and $\nu^\vep$ in such a way that if
$(\eta,\si)$ is a pair of configurations chosen from this coupling, then $\eta(x)\le
\si(x)$ for all $x$. We use the basic coupling to construct the process
with initial configurations $(\eta,\si)$. Calling $\eta_t$ and $\si_t$
the corresponding configurations at time $t$, we have $\eta_t(x)\le\si_t(x)$
and calling $\xi_t(x) = \si_t(x)-\eta_t(x)$, we have that the $\xi$ particles
behave as second class particles interacting by exclusion among them. Define
$J^{2,\vep}_{r,t}$,
the current of second class particles through the space--time
line $(0,0)$--$(\vep^{-1}r,\vep^{-1}t)$ by
$$
J^{2,\vep}_{r,t}
= \sum_{x\ge r\vep^{-1}} \xi_{\vep^{-1}t}(x)- \sum_{x\ge
0} \xi_0(x).
$$
This is well defined because there is a finite number of
second class particles at all times for all $\vep>0$.
Rezakhanlou (1990) proved the following law of large numbers for the density
fields. Let $\Phi$ be a bounded compact support
continuous function, then (in our context),
$$
\lime \E\left\vert \vep\sum_{x} \Phi(\vep x) \si_{\vep^{-1}t}(x)
- \int \Phi(r) u_\de(r,t) dr
\right\vert = 0, \Eq(hydro)
$$
where $u_\de$ is defined in \equ(udrt) and $\E$ denotes the expectation for
the process with initial distribution $\nu^\vep$.
The limit also holds if $\Phi$ is the indicator of a finite interval. The
limit \equ(hydro) implies a law of large numbers for the
density fields of the second class particles:
$$
\lime \E\left\vert \vep\sum_{x} \Phi(\vep x) \xi_{\vep^{-1}t}(x)
- \int \Phi(r) m_\de(r,t) dr
\right\vert = 0, \Eq(hydr2)
$$
where $m_\de$ is the function given by
\equ(drt) and $\E$ denotes the expectation for the coupled
process with coupled initial distribution with marginals $\nurl$ and $\nu^\vep$ as
described above. Call
$$
M_\de(r,t) = \int_r^\infty m_\de(w,t) dw.
$$
The limit \equ(hydr2) implies that
$$
\lime \E\left\vert \vep J^{2,\vep}_{r,t}
- (M_\de(r,t) - \de(\rho-\la)) \right\vert = 0
\Eq(hydj)
$$
We claim that for each $r\in ((1-2\rho),(1-2\la))$, $0~~~~0$,
there exists $r'= r'(r,s,t,\de)$ with the
following property:
$$
\lime \E\left\vert \vep J^{2,\vep}_{r't,t}- \vep J^{2,\vep}_{rs,s}\right\vert = 0.
\Eq(difj)
$$
In other words, the limit of the rescaled current of second class particles
through the line determined by the space-time points $(rs,s)$ and $(r't,t)$ is
zero. To see that \equ(difj) is true one has to see how the extra particles
evolve. Their (macroscopic) evolution is given by \equ(drt) hence $r'$ is the
point such that the area of the perturbation at time $t$ to the right of the
point $r't$ is the same as the area of the perturbation at time $s$ to the
right of $rs$. In other words, $r'$ is the unique solution of
$$
M_\de(r't,t) = M_\de(rs,s). \Eq(mdel)
$$
To see that there is a unique solution one checks that for each $t$
and $\de$, $M_\de(.,t)$ is strictly decreasing in the interval $[(1-2\rho)t,
(1-2\la)t+\de]$. This and \equ(hydj) imply \equ(difj). From \equ(mdel)
it is easy to check
that $r' = r - O(\de)$, where
$O(\de)$ is some positive function that goes to zero as $\de$ goes to zero.
Let
$Z^\vep_t$ be $\vep$ times the position at time $\vep^{-1}t$ of
the extra particle that at time $\vep^{-1}s$ is located in site
$\vep^{-1}rs$ (if there is not we add one in this site disregarding the
previous occupation number). Since by the exclusion interaction the
current of second class particles through the space-time line
$(\vep^{-1}rs,\vep^{-1}s)$-$(\vep^{-1}Z^\vep_t,\vep^{-1}t)$ is zero,
it is not hard to conclude that
$$
\lime Z^\vep_t = r't \ \ \ \text{in probability.}
$$
It is simple to check that for $t\ge s$, $Z^\vep_t\le
R^\vep_t$ if the inequality holds for a precedent time. This holds indeed
because $R^\vep_s = Z^\vep_s$. It is here where we
use that the particles jump only to the right.
Hence, for all $\de>0$ and $\ga>0$, since $r'=r-O(\de)$,
$$
\lime P(t^{-1} R^\vep_t - r < -O(\de)- \ga) = 0.
$$
If $\de<0$ we perform again the basic coupling. In this case $r'=r+O(\de)$ and
the process with initial configuration $\eta$ has extra particles. A similar
argument shows that $Z^\vep_t\ge R^\vep_t$ and as before,
$$
\lime P(t^{-1} R^\vep_t - r > O(\de)+ \ga) = 0.
$$
Putting the two limits together and taking $\de$ to zero we get the result.
\ \square
\vskip 3truemm
\noindent {\bf Proof of \equ(lln).} By the first part of the Theorem we know
that the rescaled position of the second class particle at macroscopic time
$s$ belongs to the interval $((1-2\rho)s,(1-2\la)s)$ with large
probability. We fix $\ga>0$ and
partition this interval in $N$ sub-intervals of length $\ga s$
(without loss of generality we can take $N = 2(\rho-\la)/\ga$). Let
$$
\ell^{k}_s = s (k\ga + (1-2\rho)) , \ \ k=0,\dots,N.
$$
Now
$$
\eqalign{
\P\left(\left\vert s^{-1}X^{\vep}_{s}-t^{-1}X^{\vep}_{t}\right\vert
> 2\ga\right)
&\le \sum_{k=0}^{N-1} \P\left(X^{\vep}_{s}\in [\ell^{k}_s,\ell^{k+1}_s),
\left\vert s^{-1}X^{\vep}_{s}-t^{-1}X^{\vep}_{t}\right\vert> 2\ga\right) \cr
&\quad\quad+ \P\left(s^{-1}X^{\vep}_{s}\notin [1-2\rho,1-2\la)\right). \cr
}
$$
The last term goes to zero as a consequence of the first part of the
Theorem. Since the sum above has a finite number of terms, it suffices to show
that each term goes to zero. We bound the $k$-th term by
$$
\eqalign{
&\P \left( X^{\vep}_{s}\in [\ell^{k}_s,\ell^{k+1}_s),
t^{-1}X^{\vep}_t < s^{-1}\ell^{k}_s - \ga\right) \cr
&\quad\quad
+\P \left( X^{\vep}_{s}\in [\ell^{k}_s,\ell^{k+1}_s),
t^{-1}X^{\vep}_t > s^{-1}\ell^{k+1}_s + \ga\right). \cr
} \Eq(au)
$$
As in the proof of \equ(lln1) the position of the
second class particle is given by the position of a discrepancy initially at
the origin. We consider two initial configurations
$\eta^0$ picked from $\nurl$ and $\eta^1$ that differs from $\eta^0$ only
in the origin, that is $\eta^1(0) = 1-\eta^0(0)$. Performing the basic
coupling for these configurations we get
$$
X^\vep_t = \vep \sum_x x \one\{\eta_{\vep^{-1}t}^0(x)=1- \eta_{\vep^{-1}t}^1(x) \}.
$$
Assume $\eta^0(0) = 1$ (the other case is treated similarly) and suppose
$\ell^{k}_ss$,
$$
L^{\vep,k}_t = \vep \sum_x x
\one\{\eta^{2,\vep}_{\vep^{-1}t}(x) = 1-\eta_{\vep^{-1}t}^0(x) \}.
$$
There are two possibilities: either (a)
$\eta^0_{\vep^{-1}s}([\vep^{-1}\ell^{k}_s])=0$, in this case it is easy to see that
$L^{\vep,k}_t$ and
$X^\vep_t$ interact by exclusion and $L^{\vep,k}_t 0 \} \ \ \ \
\text{ a.s.} \Eq(bc)
$$
where $Y^0_t$ and $ Y^1_t$ are the positions of the discrepancies that at time
$0$ were in the origin and in site $1$ respectively.
These discrepancies behave like annihilating second
class particles. Hence, up to the meeting time, $X^i_t \equiv Y^i_t$.
This implies that
$$
\{X^0_s \ne X^1_s \text{ for all $0\le s \le t$}\} \supset
\{Y^0_t \le 0,Y^1_t > 0 \}
$$
and
$$
\P(X^0_t \ne X^1_t \text{ for all $t\ge 0$})
\ge \lim_{t\to\infty} \P(Y^0_t \le 0,Y^1_t >0). \Eq(2)
$$
On the other hand,
$$
{d \over dt} \E J_t(\bar\eta) = \E[\eta_t(0)(1-\eta_t(1))].
$$
Hence, by standard convergence to local equilibrium \equ(12),
$$
\limt {d \over dt} \E J_t(\bar\eta) = u(0,1)(1-u(0,1))={1\over 4}. \Eq(3)
$$
Putting \equ(kbe), \equ(bc), \equ(2) and \equ(3) together we get \equ(1/4).
The same argument can be applied to show that for $r\in [-1,1]$,
$$
\limt \P(Y^0_t \le rt,Y^1_t > rt) = u(r,1)(1-u(r,1))-ru(r,1). \Eq(jo)
$$
Write
$$
\E (X^1_t-X^0_t)
= \sum_y P(X^0_t\le y < X^1_t)
= \sum_y P(Y^0_t\le y < Y^1_t)
$$
because up to the meeting time, $X^1_t-X^0_t = Y^1_t-Y^0_t$ almost surely.
Using \equ(jo) and dominated convergence,
$$
\eqalign {
\limt t^{-1}\E(X^1_t-X^0_t) &= \int_{-1}^1
\left[u(r,1)(1-u(r,1))-ru(r,1)\right] dr\cr
& = \int_{-1}^1 \left[ {1-r \over 2}{1+r \over 2} - r {1-r \over 2}\right] dr
= {2\over 3}.\ \ \ \square\cr}
$$
%\vskip 3truemm
\vfill\eject
\bigskip
\noindent {\bf Acknowledgments.} We thank
Fraydoun Rezakhanlou and Enrique Andjel for valuable discussions.
This paper was written while the authors
participate of the program {\it Random Spatial Processes} at Isaac
Newton Institute for Mathematical Sciences of University of Cambridge, to whom
very nice hospitality is acknowledged.
This paper is supported by FAPESP, CNPq and SERC Grant GR G59981.
\bigskip
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\bigskip
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Ann. Probab.\bf 21 }4 1782-1808
\item {-} T.\ M.\ Liggett (1985). {\sl Interacting Particle Systems.}
Springer, Ber\-lin.
\item {-} H.\ Rezakhanlou (1990) Hydrodynamic limit for attractive particle
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\item {-} H.\ Rezakhanlou (1993) Microscopic structure of shocks in one
conservation laws. Preprint submitted to {\sl Ann. Inst. H. Poincar\'e,
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\vskip 3truemm
\+Instituto de Matem\'atica e Estat\'\i stica --- %
Universidade de S\~ao Paulo \cr
\+Cx.\ Postal 20570 --- 01452-990 S\~ao Paulo SP --- Brasil \cr
\+{\tt} \cr
\bye
~~