The Vlasov-Poisson-Focker-Planck system with infinite kinetic energy F. Castella Vlasov-Poisson-Fokker-Planckpropagation of momentsinfinite kinetic energyhypoellipticity We build solutions to the three dimensional Vlasov-Poisson-Fokker-Planck System having infinite kinetic energy. For that purpose, we first derive a new conservation law, which states the propagation of the second space moment for solutions to this system. The existence of such infinite kinetic energy solutions relies essentialy on the dispersive effects of the kinetic transport. We also show strong regularizing effects, such as an $L_x^{\infty}$ bound on the force field, combining dispersive properties and smoothing effects of the diffusion. Indiana University Mathematics Journal 1998 text pdf 10.1512/iumj.1998.47.1472 10.1512/iumj.1998.47.1472 en Indiana Univ. Math. J. 47 (1998) 939 - 964 state-of-the-art mathematics http://iumj.org/access/