# Is a reversible distribution unique?

For a time-homogeneous discrete time Markov chain, a reversible distribution of the chain is defined as $\pi$ that satisfies: $$π_i p_{ij} = π_j p_{ji}, \forall i, j.$$

I was wondering if a reversible distribution is unique when exists?

Thanks!

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No. Check some simple cases. –  Did Dec 1 '12 at 13:22
@did: Thanks! When the states are all isolated, $p_{ij}$ are all zero, so any distribution can be reversible. –  Tim Dec 1 '12 at 13:48

Thanks to did! When the states are all isolated, $p_{ij}$ are all zero for $i \neq j$, so any distribution can be reversible.