Conditionings and path decompositions for Lévy processes

We first give an interpretation for the conditioning to stay positive (respectively, to die at 0) for a large class of Lévy processes starting at x > 0. Next, we specify the laws of the pre-minimum and post-minimum parts of a Lévy process conditioned to stay positive. We show that, these parts are independent and have the same law as the process conditioned to die at 0 and the process conditioned to stay positive starting at 0, respectively. Finally, in some special cases, we prove the Skorohod convergence of this family of laws when x goes to 0.