Parabolic limit and stability of the Vlasov-Fokker-Planck system

In this paper the stability of the Vlasov-Poisson-Fokker-Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analyzed. For the case in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, a parabolic limit equation is obtained for the mass density. Depending on the space dimension and on the hypothesis for the initial data, the convergence result in L-1 is weak and global in time or strong and local in time.