From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted without invoking the Markovian approximation.
However, the RPA master equation agrees with the one derived in quantum optics by the approaches of Scully and Lamb and of Haake which invoke the rotating wave and Markovian approximations.
Unlike the unperturbed momentum which is defined fully within the phonon subspace, the kinetic momentum contains contributions from the reservoir. The RPA accounts for both momenta without modifying the master equation. In this respect, it contrasts with the approaches of Scully and Lamb and of Agarwal. It also contrasts with the phenomenological phase space quantization.