Kemeny's Constant for Markov Processes

Abstract

The mean time taken by an irreducible Markov chain on a finite state space to hit a target chosen at random according to the stationary distribution does not depend on the initial state of the chain. This mean time is known as Kemeny's constant. I present a new approach, based on time reversal and a mean occupation time formula. The method is used to prove an analogous result for continuous-time Markov processes. We also present a second approach, based on work of N.~Eisenbaum and H.~Kaspi, when all states are regular. Examples are provided.