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 , ∞ ) × R d − 1 , d ≥ 2 , is given in terms of three measures μ , μ ̂ and μ ̇ on [ 0 , ∞ ) × R d − 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 μ ̂ 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.