  We present the application of a fluctuating hydrodynamic theory to study
current fluctuations in diffusive systems on a semi-infinite line in contact
with a reservoir with slow coupling. We show that the distribution of the
time-integrated current across the boundary at large times follows a large
deviation principle with a rate function that depends on the coupling strength
with the reservoir. The system exhibits a long-term memory of its initial
state, which was earlier reported on an infinite line and can be described
using quenched and annealed averages of the initial state. We present an
explicit expression of the rate function for independent particles, which we
verify using an exact solution of the microscopic dynamics. For the symmetric
simple exclusion process, we present expressions for the first three cumulants
of both quenched and annealed averages.
