Coupling property and gradient estimates of Lévy processes via the symbol

We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Levy processes, including stable Levy processes, layered stable processes, tempered stable processes and relativistic stable processes.