Given disjoint nonempty subsets $X_1, X_2$ of the state space of a recurrent Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1$-random point $x$, if one walks until one first hits $X_2$ at some point $y$, the point $y$ is distributed according to $\mu_2$, and (b) vice versa (with the roles of $X_1$ and $X_2$ reversed). E.g., if the Markov chain is (unbiased) random walk on $Z/5Z$ and $X_1$ is {0,1} and $X_2$ is {3,4}, then $\mu_1(0)=\mu_2(4)=2/3$ and $\mu_1(1)=\mu_2(3)=1/3$. Is there a way to think about these measures in terms of standard constructions in discrete probability theory?

Given disjoint nonempty subsets $X_1, X_2$ of the state space of a finite irreducible Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1$-random point $x$, if one walks until one first hits $X_2$ at some point $y$, the point $y$ is distributed according to $\mu_2$, and (b) vice versa (with the roles of $X_1$ and $X_2$ reversed). E.g., if the Markov chain is (unbiased) random walk on $Z/5Z$ and $X_1$ is {0,1} and $X_2$ is {3,4}, then $\mu_1(0)=\mu_2(4)=2/3$ and $\mu_1(1)=\mu_2(3)=1/3$. Is there a way to think about these measures in terms of standard constructions in discrete probability theory?