  We solve the initial value problem for the linearized mean field Kramers
equation describing Brownian particles with long-range interactions in the
$N\rightarrow +\infty$ limit. We show that the dielectric function can be
expressed in terms of incomplete Gamma functions. The dielectric functions
associated with the linearized Vlasov equation and with the linearized mean
field Smoluchowski equation are recovered as special cases corresponding to the
no friction limit or to the strong friction limit respectively. Although the
stability of the Maxwell-Boltzmann distribution is independent on the friction
parameter, the evolution of the perturbation depends on it in a non-trivial
manner. For illustration, we apply our results to self-gravitating systems,
plasmas, and to the attractive and repulsive BMF models.
