After an introduction to the problem, the aim of the talk will be to present a recent approach to the stability of homogeneous equilibria for  some Vlasov-Poisson systems in the whole space  which is alternative to the one of Bedrossian,  Masmoudi and Mouhot. This approach is based on the derivation and the  use of  pointwise dispersive estimates in the physical space.

(Joint work with D. Han-Kwan and T. Nguyen)