The recurrence dichotomy via Green's functions

Let $P$ be an irreducible Markov chain on a countable state space $V$: nonnegative entries with each row summing to one as a convergent series, and every state reaching every other at some time. For a state $x$, the first return time is $\tau^+_x=\min\{t\ge1:X_t=x\}$; the state is recurrent when return is certain, $\mathbb P_x\{\tau^+_x>t\}\to0$ as $t\to\infty$, and transient otherwise. The Green's function at $x$ is the expected number of visits to $x$,

The theorem (Proposition 21.3 of Levin–Peres–Wilmer) asserts:

1. a state $x$ is recurrent if and only if its Green's series diverges — equivalently, transience is exactly summability of the return probabilities $P^t(x,x)$;
2. recurrence is a class property: if one state of the irreducible chain is recurrent, every state is.

The equivalence comes from the renewal structure: the number of returns to $x$ is geometric with success probability $\mathbb P_x\{\tau^+_x<\infty\}$, so its expectation $G(x,x)$ is finite exactly when return is uncertain. This dichotomy is the standard tool for deciding recurrence — Pólya's theorem, the goal of this mission, is proved by estimating the series $\sum_tP^t(0,0)\asymp\sum_tt^{-d/2}$ for the walk on $\mathbb Z^d$.