  This paper reviews the formulation of the Feynman-Vernon model of linear
dissipative systems for a standard Brownian particle moving in an external
potential $V(x,t)$ and introduces the formulation of a generalized oscillator
model of a Brownian particle coupled to a thermal environment moving with a
given velocity $v_{env}$. Diffusion processes in a moving environment are of
interest e.g. in the study of the motion of vortices in superfluids. The
starting point of the paper is the formulation of the oscillator model that
takes into account space and time invariance of a thermal environment [M.
Patriarca, Statistical correlations in the oscillator model of quantum Brownian
motion, Il Nuovo Cimento B, 111(1), 61-72 (1996), doi: 10.1007/BF02726201,
arXiv:1801.02429], which has the property of being finite and consistent with
the classical limit. The Langevin equation and the influence functional for a
Brownian particle in a moving environment are derived.
