Regularity of density for SDEs driven by degenerate L\'evy noises

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.Comment: 25 page

Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop

It is provedin this paper that the maximum number of limit cycles of system [formula] is equal to two in the finite plane, where [formula]. This is partial answer to the seventh question in [2], posed by Arnold