Regularity of density for SDEs driven by degenerate Lévy noises

Abstract: By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive L\'evy noises. Under full H\"ormander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.