  This paper investigates the asymptotic behavior of Green functions associated
to partially homogeneous random walks in the quadrant $Z_+^2$. There are four
possible distributions for the jumps of these processes, depending on the
location of the starting point: in the interior, on the two positive axes of
the boundary, and at the origin $(0,0)$.
  With mild conditions on the positive jumps of the random walk, which can be
unbounded, a complete analysis of the asymptotic behavior of the Green function
of the random walk killed at $(0,0)$ is achieved. The main result is that {\em
eight} regions of the set of parameters determine completely the possible
limiting behaviors of Green functions of these Markov chains. These regions are
defined by a set of relations for several characteristics of the distributions
of the jumps.
  In the transient case, a description of the Martin boundary is obtained and
in the positive recurrent case, our results give the exact limiting behavior of
the invariant distribution of a state whose norm goes to infinity along some
asymptotic direction in the quadrant. These limit theorems extend results of
the literature obtained, up to now, essentially for random walks whose jump
sizes are either $0$ or $1$ on each coordinate.
  Our approach relies on a combination of several methods: probabilistic
representations of solutions of analytical equations, Lyapounov functions,
convex analysis, methods of homogeneous random walks, and complex analysis
arguments.
