Generalized inverses of Markovian kernels in terms of properties of the Markov chain

All one-condition generalized inverses of the Markovian kernel I−PI−P, where P   is the transition matrix of a finite irreducible Markov chain, can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include the group inverse of I−PI−P, Kemeny and Snell's fundamental matrix of the Markov chain and the Moore–Penrose g-inverse. The elements of some sub-families of the generalized inverses can also be re-expressed involving the second moments of the recurrence time variables. Some applications to Kemeny's constant and perturbations of Markov chains are also considered.