On Levy processes conditioned to avoid zero

The purpose of this paper is to construct the law of a Levy process conditioned to avoid zero, under mild technicals conditions, two of them being that the point zero is regular for itself and the Levy process is not a compound Poisson process. Two constructions are proposed, the first lies on the method of h-transformation, which requires a deep study of the associated excessive function; while in the second it is obtained by conditioning the underlying Levy process to avoid zero up to an independent exponential time whose parameter tends to 0. The former approach generalizes some of the results obtained by Yano (2010) in the symmetric case and recovers some of main results in Yano (2013), while the latter is reminiscent of Chaumont and Doney (2005). We give some properties of the resulting process and we describe in some detail two examples: alpha stable and spectrally negative Levy processes.