Green function of a random walk

Suppose we have a transient random walk, or one that is “killed” when it reaches the subset of state $Z$ in a finite irredicible markov chain. An interesting question is the quantity:

Def Let $G$ be a finite connected network. The Green function of the random walk on $G$ killed ad $Z$ is defined as

$${\mathcal{G}}_Z(a,x):={\mathbb{E}}_a\left[\sum _{n=0}^{\infty }1(X_n=x)\right]$$

the expected number of visits to state $x$ strictly before reaching $Z$. Since we assume the network is finite and connected, this quantity is finite. Also if $z\in Z$ then ${\mathcal{G}}_Z(a,z)=0$.

Proposition (Green function as voltage) Let $G$ be a finite connected network. When a voltage is imposed at $\{a\}\cup Z$ such that $v(Z)=0$ and at $a$ such that $||i||=1$ (unit flow), then

$$v(x)=\frac{{\mathcal{G}}_Z(a,x)}{c(x)}$$

is the voltage function of the network.

Proof It satisfies the boundary conditions since $v(z)=0$ and $v(a)$ equals the Escape probabiliy divided by $c(a)$.

Also, since $\mathcal{G}$ satisfies Detailed Balance:

$$\frac{{\mathcal{G}}_Z(a,x)}{c(x)}=\frac{{\mathcal{G}}_Z(x,a)}{c(a)}$$

we can show that it’s harmonic by first step analysis:

$${\mathcal{G}}_Z(x,a)=\sum _{y\sim x}{\mathcal{G}}_Z(y,a)p_{xy},\square$$