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In the absence of acceleration, the velocity formula gives "distancetravelled equals speed multiplied by time". For a broad class of Markov chainssuch as circulant Markov chains or random walk on complete graphs, we prove aprobabilistic analogue of the velocity formula between entropy and hittingtime, where distance is the entropy of the Markov trajectories from state $i$to state $j$ in the sense of [L. Ekroot and T. M. Cover. The entropy of Markovtrajectories. IEEE Trans. Inform. Theory 39(4): 1418-1421.], speed is theclassical entropy rate of the chain, and the time variable is the expectedhitting time between $i$ and $j$. This motivates us to define new entropiccounterparts of various hitting time parameters such as average hitting time orcommute time, and prove analogous velocity formulae and estimates between thesequantities.

Velocity formulae between entropy and hitting time for Markov chains