Has a Markov chain in compact metric space a stationary distribution (possibly non-unique)?

Question

Let $Y_n$ be a Markov chain in a compact metric space. Is it true that it has a stationary distribution?

Answer 1

One criteria for deciding if a Markov chain (or process), defined on a Polish space has a stationary distribution, is the Krylov-Bogoljubov theorem.

For a better treatment of the subject than the one given by the wikipedia link, you can look at this document, especially at section 1.4 for the theorem, and previous sections for the corresponding definitions.

The tl;dr version is, if the Markov chain verifies a type of "continuity" (the Feller property) and is "concentrated on a compact set" (ie, the distributions $P^k(\cdot, x_0)$ are tight), then you can guarantee existence of a stationary distribution.

So, the answer to your question is, not necesarily.

First, to apply these results, you also need your space to be complete and to be separable. In which case the space is Polish and compact, so you verify the tightness condition. I have not seen any reference in which the convergence of Markov chains in spaces that are not Polish is studied.

But even in this case, you need the Markov chain to verify the Feller property to conclude existence.