On the Coupling Property of L\'evy Processes
Abstract
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\'evy process admits a successful coupling, if it is a strong Feller process or if the L\'evy (jump) measure has an absolutely continuous component.
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