Stochastic processes and semigroups associated with degenerate lévy generating operators

We consider three types of Levy generating operators with smooth data (diffusion matrix, drift vector, and Levy measure). These operators are characterized by the types of their Levy measures: the first type is given in general form, but a strong integrability condition is assumed; the second and third types are given as certain generalizations of the Levy measures of stable processes. For such Levy generating operators with degenerate diffusion matrix, we prove a uniqueness theorem for the martingale problem and study a smoothness preserving property of the time-inhomogeneous semigroups associated with the operators.