  We introduce a family of multi-dimensional Askey-Wilson signed measures. We
offer an explicit description of the stationary measure of the open asymmetric
simple exclusion process (ASEP) in the full phase diagram, in terms of
integrations with respect to these Askey-Wilson signed measures. Using our
description, we provide a rigorous derivation of the density profile and limit
fluctuations of open ASEP in the entire shock region, including the high and
low density phases as well as the coexistence line. This in particular confirms
the existing physics postulations of the density profile.
