The convex minorant of a Lévy process

Abstract

We offer a unified approach to the theory of convex minorants of Lévy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Lévy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson–Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a Lévy process with continuous distributions above its convex minorant on the interval $[0,1]$.