Decay properties of Vlasov fields on non-trapping asymptotically hyperbolic manifolds

In this paper, we study pointwise decay estimates in time for Vlasov fields on non-trapping asymptotically hyperbolic manifolds. We prove optimal decay estimates in time for the spatial density induced by Vlasov fields on these geometric backgrounds in dimension two. First, we show exponential decay for Vlasov fields on hyperbolic space supported away from the zero velocity set. In contrast, we obtain inverse polynomial decay for general Vlasov fields on hyperbolic space. In the second part of the article, we prove exponential decay for Vlasov fields on non-trapping asymptotically hyperbolic manifolds supported away from the zero velocity set. The proofs are obtained through a commuting vector field approach. We exploit the hyperbolicity of the geodesic flow in these geometric backgrounds, by making use of a commuting vector field in the unstable invariant distribution of phase space.