Asymptotics of the Green function of killed random walks in a cone of ${\mathbb{Z}}^d$

In this paper, we obtain the exact asymptotic behavior of Green functions of homogeneous random walks in ${\mathbb{Z}}^d$ killed at the first exit from and open cone of ${\mathbb{R}}^d$. Our approach combines methods of functional equations, integral representations of the Green function and Woess’ approach for the case of homogeneous random walks in ${\mathbb{Z}}^d$.