The general solution at infinity is either an ergodic process (mixing across all cells), mixing across sets of subcells or absorbtion (the process halts at a finite time). Each of these is due to the characteristics of the Chapman-Kolmogorov transition matrix in an obvious way
The Markov chain Like DTMC is the sequence of Random variables with sequence of probability functions. Properties of MC states Like Transit and Recurrent are computing with sum of probabilities.