Characterization of stationary measures for one-dimensional exclusion processes

Abstract

The product Bernoulli measures $\nu_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process on $\mathbb{Z}$ with irreducible random walk kernel $p(\cdot)$. Stationary measures that are not translation invariant are known to exist for finite range $p(\cdot)$ with positive mean. These measures have particle densities that tend to 1 as $x\to\infty$ and tend to 0 as $x\to -\infty$; the corresponding extremal measures form a one-parameter family and are translates of one another. Here, we show that for an exclusion process where $p(\cdot)$ is irreducible and has positive mean, there are no other extremal stationary measures. When $\sum_{x<0} x^2 p(x) =\infty$, we show that any nontranslation invariant stationary measure is not a blocking measure; that is, there are always either an infinite number of particles to the left of any site or an infinite number of empty sites to the right of the site. This contrasts with the case where $p(\cdot)$ has finite range and the above stationary measures are all blocking measures. We also present two results on the existence of blocking measures when $p(\cdot)$ has positive mean, and $p(y)\leq p(x)$ and $p(-y)\leq p(-x)$ for $1\leq x\leq y$. When the left tail of $p(\cdot)$ has slightly more than a third moment, stationary blocking measures exist. When $p(-x)\leq p(x)$ for $x>0$ and $\sum_{x<0}x^2p(x)>\infty$, stationary blocking measures also exist.