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This paper investigates the asymptotic behavior of Green functions associatedto partially homogeneous random walks in the quadrant $Z_+^2$. There are fourpossible distributions for the jumps of these processes, depending on thelocation of the starting point: in the interior, on the two positive axes ofthe boundary, and at the origin $(0,0)$.  With mild conditions on the positive jumps of the random walk, which can beunbounded, a complete analysis of the asymptotic behavior of the Green functionof the random walk killed at $(0,0)$ is achieved. The main result is that {\emeight} regions of the set of parameters determine completely the possiblelimiting behaviors of Green functions of these Markov chains. These regions aredefined by a set of relations for several characteristics of the distributionsof the jumps.  In the transient case, a description of the Martin boundary is obtained andin the positive recurrent case, our results give the exact limiting behavior ofthe invariant distribution of a state whose norm goes to infinity along someasymptotic direction in the quadrant. These limit theorems extend results ofthe literature obtained, up to now, essentially for random walks whose jumpsizes are either $0$ or $1$ on each coordinate.  Our approach relies on a combination of several methods: probabilisticrepresentations of solutions of analytical equations, Lyapounov functions,convex analysis, methods of homogeneous random walks, and complex analysisarguments.

A Classification of Asymptotic Behaviors of Green Functions of Random Walks in the Quadrant