  We demonstrate a Markov duality between the dynamic ASEP and the standard
ASEP. We then apply this to step initial data, as well as a half-stationary
initial data (which we introduce). While investigating the duality for
half-stationary initial data, we uncover and utilize connections to continuous
$q^{-1}$-Hermite polynomials. Finally, we introduce a family of stationary
initial data which are related to the indeterminate moment problem associated
with these $q^{-1}$-Hermite polynomials.
