Recurrent extensions of self-similar Markov processes and Cramér's condition

Abstract

Let ξ be a real-valued Lévy process that satisfies Cramér's condition, and X a self-similar Markov process associated with ξ via Lamperti's transformation. In this case, X has 0 as a trap and satisfies the assumptions set out by Vuolle-Apiala. We deduce from the latter that there exists a unique excursion measure \exc, compatible with the semigroup of X and such that \exc(X₀₊>0)=0. Here, we give a precise description of \exc via its associated entrance law. To this end, we construct a self-similar process X^\natural, which can be viewed as X conditioned never to hit 0, and then we construct \exc similarly to the way in which the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of \exc is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of \exc.