Maxentropic Continuous-time Homogeneous Markov Chains

In this paper, we investigate the notion of entropy rate and its maximization for continuous-time time-homogeneous irreducible finite-state Markov chains. The definitions available in continuous-time suffer from an apparent paradox, as they do not properly account for the role of the average commutation frequency. In fact, we show that the entropy rate is the sum of a finite and an infinite component, the latter depending on the average commutation frequency. Thus, entropy maximization is meaningful only between chains that share the same average frequency. After settling this issue, we address entropy rate maximization under different constraints on the stationary probability: unconstrained, completely fixed, partially fixed. Closed-form solutions and provably convergent iterative algorithms are provided. The results are illustrated through several examples, including chains with string and lattice graph topology. Interesting connections with quantum mechanics topics (particle-in-a-box model, Born rule, and Anderson localization property) are highlighted.