On a formula on the potential operators of absorbing Lévy processes in the half space

A representation of the potential operator of an absorbing Lévy process in the half space (0,∞)×Rd−1,d≥2, is given in terms of three measures μ,μ̂ and μ̇ on [0,∞)×Rd−1[0,∞)×Rd−1 arising in the fluctuation theory of Lévy processes. In the case of a rotation invariant stable Lévy process, the potential kernel in the half space is computed explicitly. It will also be proved that the measure μ̂ is an excessive measure (an invariant measure under some conditions) of a Markov process, which is derived from the given Lévy process in a certain way.