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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.

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So do you allow here the use of irreducibility? –  Ilya Sep 28 '11 at 14:38
    
@Gortaur: Irreducibility can be used. Except this result from irreducibility: For an irreducible Markov chain, the stationary distribution is unique. –  Nicolas Essis-Breton Oct 3 '11 at 12:05
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