Splitting at the infimum and excursions in half-lines for random walks and Lévy processes

The central result of this paper is that, for a process X with independent and stationary increments, splitting at the infimum on a compact time interval amounts (in law) to the juxtaposition of the excursions of X in half-lines according to their signs. This identity yields a pathwise construction of X conditioned (in the sense of harmonic transform) to stay positive or negative, from which we recover the extension of Pitman's theorem for downwards-skip-free processes. We also extend for Lévy processes an identity that Karatzas and Shreve obtained for the Brownian motion. In the special case of stable processes, the sample path is studied near a local infimum.