Theorem 5.5.10 — the support of a stationary distribution is recurrent

If $p$ has a stationary distribution $\pi$, then every state $y$ with $\pi(y)>0$ is recurrent.

The contrast Durrett draws just before this result is the point of it. Stationary measures exist for transient chains — counting measure is stationary for simple random walk on $\mathbb{Z}^d$ in every dimension, including the transient ones $d\ge3$. What rules out transience is not stationarity but normalizability: a stationary measure with finite total mass cannot be supported on transient states.

The proof is a computation with the renewal identity. Stationarity gives $\pi(y)=\sum_x\pi(x)p^n(x,y)$ for every $n$, so summing over $n$ and using that $\sum_n p^n(x,y)$ is finite at a transient $y$ forces $\pi(y)$ to be zero.

Formalization Note The hypothesis is a stationary distribution, a probability vector, not a stationary measure; that is where the finiteness that drives the argument comes from. Only $\pi(y)>0$ is assumed at the state in question — nothing about the rest of the support.