  We complete the kinetic theory of inhomogeneous systems with long-range
interactions initiated in previous works. We use a simpler and more physical
formalism. We consider a system of particles submitted to a small external
stochastic perturbation and determine the response of the system to the
perturbation. We derive the diffusion tensor and the friction by polarization
of a test particle. We introduce a general Fokker-Planck equation involving a
diffusion term and a friction term. When the friction by polarization can be
neglected, we obtain a secular dressed diffusion (SDD) equation sourced by the
external noise. When the external perturbation is created by a discrete
collection of $N$ field particles, we obtain the inhomogeneous Lenard-Balescu
kinetic equation reducing to the inhomogeneous Landau kinetic equation when
collective effects are neglected. We consider a multi-species system of
particles. When the field particles are at statistical equilibrium (thermal
bath), we establish the proper expression of the fluctuation-dissipation
theorem for systems with long-range interactions relating the power spectrum of
the fluctuations to the response function of the system. In that case, the
friction and diffusion coefficients satisfy the Einstein relation and the
Fokker-Planck equation reduces to the inhomogeneous Kramers equation. We also
consider a gas of Brownian particles with long-range interactions described by
$N$ coupled stochastic Langevin equations and determine its mean and mesoscopic
evolution. We discuss the notion of stochastic kinetic equations and the role
of fluctuations possibly triggering random transitions from one equilibrium
state to the other. Our presentation parallels the one given for
two-dimensional point vortices in a previous paper [P.H. Chavanis, Eur. Phys.
J. Plus 138, 136 (2023)].

Statistical Mechanics;Astrophysics of Galaxies