  A recently proposed master equation in the Lindblad form is studied with
respect to covariance properties and existence of a stationary solution. The
master equation describes the interaction of a test particle with a quantum
fluid, the so-called Rayleigh gas, and is characterized by the appearance of a
two-point correlation function known as dynamic structure factor, which
reflects symmetry and statistical mechanics properties of the fluid. In the
case of a free gas all relevant physical parameters, such as fugacity, ratio
between the masses, momentum transfer and energy transfer are put into
evidence, giving an exact expansion of the dynamic structure factor. The limit
in which these quantities are small is then considered. In particular in the
Brownian limit a Fokker-Planck equation is obtained in which the corrections
due to quantum statistics can be explicitly evaluated and are given in terms of
the Bose function $g_0 (z)$ and the Fermi function $f_0 (z)$.
