We consider the Vlasov-Poisson equation for a system of interacting particles, and we describe the physical model from which it can be heuristically derived, also attempting to a rigorous proof in the case of the "mollified" approximation. Then, we focus on the models supplemented with a stochastic term, which are issued from dynamical systems immersed in a thermal reservoir, to obtain the so-called Vlasov-Poisson-Fokker-Planck equations (VPFP). We address the question of solutions to the Cauchy problem, and we present the state-of-the-art existence theory according to the regularity of the initial data. We determine the structural properties of (VPFP), and in particular we prove a new estimate on the macroscopic density uniformly with respect to the diffusion parameter. The specific difficulty encountered in solving the Cauchy problem for (VPFP) relies on the singularity of the potential term, and this difficulty increases with the dimension of the physical space. The approach we use is based on a careful analysis of the characteristics associated with the underlying nonlinear diffusion process, by means of a control on the velocity moments of the distribution function in the three-dimensional space. This ensures sufficient regularity properties of the potential field to construct uniquely the solution, and moreover to infer the convergence to the Vlasov-Poisson equation in the vanishing viscosity limit.
