Green function of a transient weighted graph
= Green function of a transient weighted graph
{title2=$g(x,y)$}
For the walk on a <weighted graph>, the Green function normalized by vertex weight is
$$
g(x,y)=\frac1{\mu_y}\mathbb E_x\sum_{n\geq0}\mathbf1_{\{X_n=y\}}.
$$
Symmetry of the conductances makes $g(x,y)=g(y,x)$, and $\mathcal E(g(\mathord\cdot,y),f)=f(y)$ for finitely supported $f$.