On the Coupling Property of L\'{e}vy Processes

We give necessary and sufficient conditions guaranteeing that the coupling for L'evy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a L'{e}vy process admits a successful coupling, if it is a strong Feller process or if the L'evy (jump) measure has an absolutely continuous component.