In this talk, I will present a Hamiltonian approach to show decay for massless Vlasov fields on the exterior of black hole backgrounds. Vlasov fields are transported along the geodesic flow, and so, the existence of trapped geodesics is an important difficulty of the problem. We address this issue by working with a well-chosen defining function of the set of future-trapped null geodesics. This function captures concentration properties of the geodesic flow. By using a projection of the associated symplectic gradient, we show decay in time of a suitable energy flux. We also obtain decay in time for the energy-momentum tensor of matter. This is based on joint work with Léo Bigorgne (Université de Rennes).