With Nicolas Besse we propose some justifications of the quasilinear approximation

∂tF(t,v) − ∂v(D(t,v)∂vF(t,v)) = 0. (1)

for the space average of the solutions of the Vlasov equation.

∂tF +v⋅∇xF +E⋅∇vF =0, .

E=∇Φ,−∆Φ(t,x)= ∫ F(v,x,t)dv−1, (2)

This is a subtle problem because (2) is hamiltonian hence reversible while (1) is a parabolic equation and for a non trivial limit D(t,v) =/ 0 strong convergence is excluded. Hence I will consider the following issues:

1. Replacing ∇Φ solution of the Poisson equation by a E given potential, elaborating on previous contributions of Vasseur and coworkers, we show that it is only when E is a stochastic potential that the quasilinear approximation is valid on finite, but large, time.

2. Using the recent proof of Grenier, Nguyen and Rodnianski we compare the issue D(t, v) = 0 with the Landau damping.

3. From spectral analysis deducing the validity of a non stochastic short time approximation.

4. In conclusion bridging the gap between short time deterministic and large time stochastic in term of interaction between ”wave and particles”.