= Solution
For an <irreducible Markov chain> on a countable <state space>, \b[the set has either zero or one element]. By <uniqueness of stationary distributions for an irreducible countable chain>, a stationary probability distribution exists exactly in the <positive recurrent> case, and then
$$
\pi_i=\frac1{\mathbb E_iT_i^+}
$$
for the <first return time> $T_i^+$. The right-hand side fixes every entry, proving uniqueness.
For a finite state space, existence is guaranteed, so there is \b[exactly one]. A countably infinite state space can have none: the <simple symmetric random walk> on the integers is irreducible and null recurrent. Periodicity does not affect this existence-and-uniqueness statement.
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