for a stationary distribution, I am familiar with the interpretation that the stationary distribution: $\mu$ is the distribution satisfying:
$$\mu=\mu*P$$
However, there is a different characterization I just came across which has a bit of ambiguous notation. In this characterization, $\mu$ solves the following:
$$\mu_j=\sum_{i}\mu_iP_{ij}\quad \forall j$$
My confusion is with the subscripts of $\mu$. Are these meant to denote the $P({X_t}=j)$? If so, then what does the summation over 'i' imply: is that only relative to the position in the transition matrix ie summation across every row of P? In that case $\mu_i$ seems like it would be constant which would make for an odd choice of variable. So I'm guessing now that $m_i$ must be referencing something else or the book chose an oddly convoluted subscript.
Any help interpreting this would be greatly appreciated