Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains that satisfy the following clause:

if $P_{ij} >= P_{ji}$, then $\pi_i <= \pi_j$ for all $i,j$

I understand the time reversible chains are a special case of the above type, but in general what other properties can be said about these chains?

Thanks in advance.

Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains that satisfy the following clause:

if $P_{ij} \ge P_{ji}$, then $\pi_i \le \pi_j$ for all $i,j$

I understand the time reversible chains are a special case of the above type, but in general what other properties can be said about these chains?

Thanks in advance.