The random walk Green function is defined by $G(x,1) = \sum_{n\in \mathbb{N}_0} P(S_n = x)$, with $x\in\mathbb{Z}^d$.
$G(x,1)$ equals the expected number of visits to $x$.
Now I want to prove that $G(0,1)=\infty$ if and only if $G(x,1)=\infty$ for all $x\in \mathbb{Z}^d$. This looks clear for $x = 0$. But how can I prove this for general $x$?