Null recurrent chains: $P^t(x,y)\to 0$

Let $P$ be an irreducible Markov chain on a countable state space $V$ (nonnegative entries, rows summing to one as convergent series, every state reaching every other), and suppose the chain is null recurrent: some state $x_0$ is recurrent — return is certain, $\mathbb P_{x_0}\{\tau^+_{x_0}>t\}\to0$ — but not positive recurrent — the expected return time $\sum_{t\ge0}\mathbb P_{x_0}\{\tau^+_{x_0}>t\}$ diverges. (Simple random walk on $\mathbb Z$ or $\mathbb Z^2$ is the standard example.)

The theorem (Theorem 21.17 of Levin–Peres–Wilmer) asserts: for all states $x,y$,

A null recurrent chain visits every state infinitely often, yet at any late fixed time it is nowhere in particular: the mass spreads out and no stationary profile is approached — consistent with this mission's equivalence, since a stationary distribution would force positive recurrence. Together with the positive-recurrent convergence theorem, this completes the trichotomy: transient chains escape, null recurrent chains return but diffuse away, positive recurrent chains converge to their stationary distribution.