Using a Markov duality satisfied by ASEP on the integer line, we deduce similar
dualities for half-line open ASEP and open ASEP on a segment. This leads to closed
systems of ODEs characterizing observables of the models. In the half-line case, we
solve the system of ODEs using Bethe ansatz and prove an integral formula for
-moments of the
integrated current at
distinct spatial locations. We then use this formula to confirm predictions
for the moments of the multiplicative noise stochastic heat equation on
with
Robin type boundary condition and we obtain new formulas in the case of a Dirichlet
boundary condition.