Velocity formulae between entropy and hitting time for Markov chains

TL;DR

This paper establishes a probabilistic velocity formula linking entropy and hitting time in certain Markov chains, introducing entropic analogues of classical hitting time parameters and deriving related estimates.

Contribution

It proves a novel velocity formula between entropy and hitting time for specific Markov chains and introduces entropic versions of hitting time metrics.

Findings

Velocity formula between entropy and hitting time proven for circulant Markov chains.

Defined entropic counterparts of hitting time parameters.

Derived estimates and analogous velocity formulae for these entropic quantities.

Abstract

In the absence of acceleration, the velocity formula gives "distance travelled equals speed multiplied by time". For a broad class of Markov chains such as circulant Markov chains or random walk on complete graphs, we prove a probabilistic analogue of the velocity formula between entropy and hitting time, 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 Markov trajectories. IEEE Trans. Inform. Theory 39(4): 1418-1421.], speed is the classical entropy rate of the chain, and the time variable is the expected hitting time between $i$ and $j$. This motivates us to define new entropic counterparts of various hitting time parameters such as average hitting time or commute time, and prove analogous velocity formulae and estimates between these quantities.