Suppose $X$ is a complete separable metric space, and there is a continuous map $x \mapsto \mu_x$ associating to each point in $X$ a probability measure on $X$ (where we use the weak topology on the space of probabilities).

For each probability $\mu$ on $X$, let $T\mu = \int \mu_x \mathrm{d}\mu(x)$. A probability $\mu$ on $X$ is stationary if $T\mu = \mu$.

If I've done things correctly $T^n\mu$ is the distribution of the $n$-th step of a Markov chain with initial distribution $\mu$ and transition probabilities given by the family $\mu_x$.

Suppose one knows that for any $\mu$ and any open set $A$ one has $T^n\mu(A) > 0$ for all $n$ large enough.

**Question: In the conditions above is it true that only one of the following statements hold?**

- There is a unique stationary probability.
- For each probability $\mu$ and each compact sets $K$ one has $\lim\limits_{n \to +\infty}T^n\mu(K) = 0$.

If $X$ is countable then the statement above can be deduced from the law of large numbers and the Markov property. One obtains that almost surely, starting with any initial distribution, the asymptotic fraction of time spent in a state is the inverse of the expected return time to that state.

I know of some results (e.g. Derriennic's version of the zero-two law[1]) implying the dichotomy in the case where $X$ is a locally compact group, and $\mu_x$ is the translate of some fixed probability $x$ under multiplication by $x$.

I was wondering if there is a standard reference for general results of this type (or maybe some instructive counter-example). From what I can tell Harris chains are not general enough for the type of examples I have in mind. I am willing to assume $X$ is locally compact if necessary.

[1]Derriennic, Yves. "Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires." Ann. Inst. H. Poincaré Sect. B (NS) 12.2 (1976): 111-129.