Convex minorants and the fluctuation theory of Lévy processes

We establish a novel characterisation of the law convex minorant any L\'evy process. Our self-contained elementary proof is based on analysis piecewise linear functions and requires only very basic properties processes. main result provides new simple approach to fluctuation theory processes, circumventing local time excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities Wiener-Hopf factorisation, well factorisation identity.