Suppose $X_n$ is a Markov chain on an uncountable state space $\Omega$ (We can assume $\Omega=\mathbb{R}$), and suppose that $X_n$ is irreducible, aperiodic and Harris recurrent.
What are the extra conditions that will guarantee that $X_n$ has a stationary distribution $\pi$?
In the discrete case, every chain with these conditions has a (unique) stationary distribution, but for the uncountable case I am struggling to find a reference.