  In this paper we study the Hamiltonian dynamics of charged particles subject
to a non-self-consistent stochastic electric field, when the plasma is in the
so-called weak turbulent regime. We show that the asymptotic limit of the
Vlasov equation is a diffusion equation in the velocity space, but homogeneous
in the physical space. We obtain a diffusion matrix, quadratic with respect to
the electric field, which can be related to the diffusion matrix of the
resonance broadening theory and of the quasilinear theory, depending on whether
the typical autocorrelation time of particles is finite or not. In the
self-consistent deterministic case, we show that the asymptotic distribution
function is homogenized in the space variables, while the electric field
converges weakly to zero. We also show that the lack of compactness in time for
the electric field is necessary to obtain a genuine diffusion limit. By
contrast, the time compactness property leads to a "cheap" version of the
Landau damping: the electric field converges strongly to zero, implying the
vanishing of the diffusion matrix, while the distribution function relaxes, in
a weak topology, towards a spatially homogeneous stationary solution of the
Vlasov-Poisson system.
