Some applications of duality for Lévy processes in a half-line

The central result of this paper is an analytic duality relation for real-valued Lévy processes killed upon exiting a half-line. By Nagasawa’s theorem, this yields a remarkable time-reversal identity involving the Lévy process conditioned to stay positive. As examples of applications, we construct a version of the Lévy process indexed by the entire real line and started from −∞ which enjoys a natural spatial-stationarity property, and point out that the latter leads to a natural Lamperti-type representation for self-similar Markov processes in (0,∞) started from the entrance point 0+.