Is a reversible distribution unique?

Question

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!

Answer 1

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