On Levy Processes Conditioned to Stay Positive

We construct the law of Levy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of Levy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying Levy process reflected at its minimum. Then, when the Levy process creeps upwards, we study the lower tail at 0 of the law of the height this excursion