Let $P$ be the transition matrix of an irreducible Markov chain on a finite state space $\Omega$. Let $\pi_1$ and $\pi_2$ be two stationary distributions for $P$. Is the function $h(x)={\pi_1(x) \over \pi_2(x)}, x \in \Omega $ harmonic for $P$.
I tried brute force to show $h(x)=\sum_{y \in \Omega} P(x,y)h(y), \forall x \in \Omega$, but I can't reduce the RHS to the LHS.
I know that for an irreducible Markov chain the stationary distribution is unique. I would like to use the precedent result as an alternative proof of the uniqueness of the stationary distribution.