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

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