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

TL;DR

This paper establishes necessary and sufficient conditions for successful coupling of Lévy processes with non-degenerate jumps, linking coupling success to the process being strong Feller or having an absolutely continuous Lévy measure.

Contribution

It provides explicit criteria for successful coupling of Lévy processes, utilizing transition semigroup formulas and prior coupling results, advancing understanding of process coupling conditions.

Findings

Successful coupling occurs if the process is strong Feller.

Lévy processes with an absolutely continuous Lévy measure admit successful coupling.

Explicit formulas for compound Poisson process transition semigroups are derived.

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\'{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.