  This article is about right inverses of Levy processes as first introduced by
Evans in the symmetric case and later studied systematically by the present
authors and their co-authors. Here we add to the existing fluctuation theory an
explicit description of the excursion measure away from the (minimal) right
inverse. This description unifies known formulas in the case of a positive
Gaussian coefficient and in the bounded variation case. While these known
formulas relate to excursions away from a point starting negative continuously,
and excursions started by a jump, the present description is in terms of
excursions away from the supremum continued up to a return time. In the
unbounded variation case with zero Gaussian coefficient previously excluded,
excursions start negative continuously, but the excursion measures away from
the right inverse and away from a point are mutually singular. We also provide
a new construction and a new formula for the Laplace exponent of the minimal
right inverse.
