It is a well-known fact of finite-state Markov chains that every harmonic function on a stationary Markov chain is almost-surely constant. Is this true also in continuous space?
More precisely, let $X$ be a measurable space, and let $\Phi$ be a stationary, discrete-time Markov process with state space $X$ and stationary measure $p$ on $X$. Let $f:X\to R$ be a bounded harmonic function for $\Phi$. Then is there a subset $S$ of $X$ with $p(S)=1$, and such that $f$ is constant on all of $S$?
A reference would also be appreciated.
How about this one from this book: