Trajectorial dissipation and gradient flow for the relative entropy in Markov chains

TL;DR

This paper investigates the dissipation of variance and relative entropy in ergodic Markov chains, providing trajectorial insights, explicit dissipation rates, and connections to gradient flows and geometric inequalities.

Contribution

It introduces trajectorial versions of dissipation results, extends to convex divergences and countable spaces, and offers a direct proof of the HWI inequality in this context.

Findings

Explicit dissipation rates for variance and relative entropy.

Trajectorial formulations valid along almost every path.

Gradient flow structures and geometric inequalities derived.

Abstract

We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extension are developed to general "convex divergences" and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.