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

In this paper the stability of the Vlasov-Poisson-Fokker-Planck with respec t 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 t he scaled thermal velocity is the inverse of the scaled thermal mean free p ath and the latter tends to zero, a parabolic limit equation is obtained fo r 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 t ime or strong and local in time.