an:04113711
Zbl 0679.60094
de Masi, A.; Kipnis, C.; Presutti, E.; Saada, E.
Microscopic structure at the shock in the asymmetric simple exclusion
EN
Stochastics Stochastics Rep. 27, No. 3, 151-165 (1989).
00172268
1989
j
60K35
simple exclusion process; Bernoulli measure; central limit theorem
Let \(P_ t\) be the semigroup of the asymmetric simple exclusion process on \({\mathbb{Z}}\) evolving in the following way:
\[
\eta \to \eta^{x,x+1},\quad at\quad rate\quad p\eta (x)(1-\eta (z));
\]
\[
\eta \to \eta^{x,x+1},\quad at\quad rate\quad q\eta (x)(1-\eta (x)),
\]
where \(q<p<1\) and \(p+q=1\). Let \(\nu_{\rho}\) be the Bernoulli measure of parameter \(\rho\) and take as initial distribution the product measure \({\bar \nu}{}_{\rho}\) whose marginals are \({\bar \nu}{}_{\rho}\{\eta (x)=1\}=0\) for \(x<0\), \(=1\) for \(x=0\), \(=\nu_{\rho}\{\eta (x)=1\}\) for \(x>0\). Set \(u_{\rho}=(p-q)(1-\rho)\) and denote by \(r_ x\) the space shift by x to the left as an operator acting on the probability measure on \(\{0,1\}^ Z\). Finally, let \(\lambda (r,t)=P[B_ t>r],\) where \(B_ t\) is the Brownian motion with diffusion coefficient \(u_{\rho}\). The authors prove that for all \(r\in R\),
\[
\lim_{\epsilon \to 0}r_{u_{\rho}\epsilon^{-1}t+r\epsilon^{-1/2}}{\bar \nu}_{\rho}P_{\epsilon^{-1}t}=\lambda (r,t)\nu_ 0+(1-\lambda (r,t))\nu_{\rho}.
\]
Moreover, let \(X_ t\) be the position of the leftmost particle at time t, then the following central limit theorem holds: \((X_ t-u_{\rho}t)/\sqrt{t}\) converges to the distribution of \(B_ 1\).
Mufa Chen