  We consider the out-of-equilibrium dynamics of an interacting integrable
system in the presence of an external dephasing noise. In the limit of large
spatial correlation of the noise, we develop an exact description of the
dynamics of the system based on a hydrodynamic formulation. This results in an
additional term to the standard generalized hydrodynamics theory describing
diffusive dynamics in the momentum space of the quasiparticles of the system,
with a time- and momentum-dependent diffusion constant. Our analytical
predictions are then benchmarked in the classical limit by comparison with a
microscopic simulation of the non-linear Schrodinger equation, showing perfect
agreement. In the quantum case, our predictions agree with state-of-the-art
numerical simulations of the anisotropic Heisenberg spin in the accessible
regime of times and with bosonization predictions in the limit of small
dephasing times and temperatures.
