We study ‘nondegenerate’ SDE's with jumps. These include SDE satisfying ‘point-wise positive’ condition and that satisfying (nonstationary) Hörmander's condition. We show that solutions of these SDE's have hypoelliptic properties. Our result is based on the Malliavin calculus on the Wiener-Poisson space. In case of continuous SDE, it extends and refines works based on the Malliavin calculus on the Wiener space.
Hiroshi KUNITA. "Nondegenerate SDE's with jumps and their hypoelliptic properties." J. Math. Soc. Japan 65 (3) 993 - 1035, July, 2013. https://doi.org/10.2969/jmsj/06530993