  This article analyzes directed lattice paths, when a boundary reflecting or
absorbing condition is added to the classical models. The lattice paths are
characterized by two time-independent sets of rules (also called steps) which
have a privileged direction of increase and are therefore essentially
one-dimensional objects. Depending on the spatial coordinate, one of the two
sets of rules applies, namely one for altitude 0 and one for altitude bigger
than 0. The abscissa y=0 thus acts as a border which either absorbs or reflects
steps. The absorption model corresponds to the model analyzed by Banderier and
Flajolet ("Analytic combinatorics of directed lattice paths"), while the
reflecting model leads to a more complicated situation. We show how the
generating functions are then modified: the kernel method strikes again but
here it unfortunately does not give a nice product formula. This makes the
analysis more challenging, and, in the case of {\L}ukasiewicz walks, we give
the asymptotics for the number of excursions, arches and meanders. Limit laws
for the number of returns to 0 of excursions are given. We also compute the
limit laws of the final altitude of meanders. The full analytic situation is
more complicated than the Banderier-Flajolet model (partly because new
"critical compositions" appear, forcing us to introduce new key quantities,
like the drift at 0), and we quantify to what extent the global drift, and the
drift at 0 play a role in the "universal" behavior of such walks.
