  We propose and study a fully discrete finite volume scheme for the
Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite
polynomials in velocity. This approach naturally preserves the stationary
solution and the weighted L 2 relative entropy. Then, we adapt the arguments
developed in [12] based the hypocoercivity method to get quantitative estimates
on the convergence to equilibrium of the discrete solution. Finally, we prove
that in the diffusive limit, the scheme is asymptotic preserving with respect
to both the time variable and the scaling parameter at play.
