I would like to know whether an irreducible Markov chain on a countable state space must necessarily have at least one ($\sigma$-finite) invariant measure. (By an invariant measure I mean a possibly infinite measure which is preserved by the dynamics.)
I suspect this is not true, so I'm looking for an example of such a Markov chain without an invariant measure, or a reference to a theorem confirming the above.
It is shown in the below-cited paper of Derman (Example 1) that a transient renewal chain has no invariant measure. I paraphrase the proof here.
Let $0 < p_i < 1$ be a sequence of numbers satisfying $\prod_{i=1}^\infty p_i > 0$. Consider a chain on $\{0,1,2,\dots\}$ with $p(i,i+1) = p_i$ for $i \ge 1$, $p(i,0) = 1-p_i$, and $p(0,1) = 1$. Suppose $\pi$ is a $\sigma$-finite invariant measure and suppose without loss of generality that $\pi(0) = 1$. Then we must have $\pi(i+1) = p_i \pi(i)$, so by induction $\pi(i) = \prod_{j=1}^{i-1} p_j$ (where $\pi(1)=1$). On the other hand, $$\pi(0) = \sum_{i=1}^\infty \pi(i) (1-p_i) = \sum_{i=1}^\infty (\pi(i) - \pi(i+1)).$$ This sum telescopes giving $$\pi(0) = \pi(1) - \lim_{i \to \infty} \ \pi(i) = 1 - \prod_{i=1}^\infty p_i < 1$$ which contradicts $\pi(0)=1$.
Derman, Cyrus, Some contributions to the theory of denumerable Markov chains, Trans. Am. Math. Soc. 79, 541-555 (1955). ZBL0065.11405. http://www.ams.org/journals/tran/1955-079-02/S0002-9947-1955-0070883-3/