On L\'{e}vy processes conditioned to stay positive
  We construct the law of L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy process reflected at its minimum. Then, when the L\'{e}vy process creeps upwards, we study the lower tail at 0 of the law of the height this excursion. 
Lo\"ic Chaumont (LPMA), Ron A. Doney