A new fluctuation identity for Levy processes and some applications

Let tau and H be respectively the ladder time and ladder height processes a ssociated with a given Levy process X. We give an identity in law between ( tau, H) and (X, H*), H* being the right-continuous inverse of the process H . This allows us to obtain a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected pro cess (X-t - inf(s less than or equal tot)X(s), t greater than or equal to 0 ). In the stable case, some explicit calculations are provided. These resul ts also lead to an explicit form of the entrance law of the Levy process co nditioned to stay positive.