Regularity of semigroups generated by Lévy type operators via coupling

By adopting the coupling method, we obtain new verifiable sufficient conditions about the -Feller continuity, the Lipschitz continuity and the strong Feller continuity of the semigroups associated with Lévy type operators. These results easily apply to jump-diffusion processes and stochastic differential equations driven by Lévy processes. Our results also yield the criterion for the e-property (namely the characterization about the equi-continuity of semigroups acting on bounded Lipschitz functions) of Lévy type operators, and show that both genuine Lévy processes and the Ornstein-Uhlenbeck type processes are e-processes.