I want to ask why the Green's function of simple random walk is defined on $\Bbb Z^d$, $d=3,4,\dots$.
Let $S_n$ be the position of the walker at time $n$.
The Green's function is defined to be $$G(x,y)=\sum_{n=0}^\infty P(S_n=y|S_o=x)$$
$x,y \in \Bbb Z^d$.
My attempt:
Let $\tau_y = \min \{n : S_n=y\}$.
For $n>\tau_y$,
$$P(S_n=y|S_o=x)=P(S_n=y,S_{\tau_y}=y|S_o=x)=P(S_n=y|S_{\tau_y}=y)P(S_{\tau_y}=y|S_o=x)=P(S_n=y|S_{\tau_y}=y)P(\tau_y<\infty|S_0=x).$$
We have $$G(x,y)=\sum_{n=0}^\infty P(S_n=y|S_o=x)=P(\tau_y<\infty|S_0=x)\sum_{n=0}^\infty P(S_n=y|S_{\tau_y}=y)=P(\tau_y<\infty|S_0=x)G(y,y)<\infty$$ since when $d=3,4,\dots$, $G(y,y)<\infty$.
I'm not quite sure whether I have missed someting, appreciate any suggestion.