Green function of a transient weighted graph ($g(x,y)$)

For the walk on a weighted graph, the Green function normalized by vertex weight isSymmetry of the conductances makes $g(x,y)=g(y,x)$, and $\mathcal{E}(g(\cdot ,y),f)=f(y)$ for finitely supported $f$.

For simple random walk on ${\mathbb{Z}}^d$ with $d\ge 3$,Consequently the probability of ever hitting $x$ from the origin has the same order.