Current fluctuations in a semi-infinite line

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.