Initial value problem for the linearized mean field Kramers equation with long-range interactions

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.