2020/05/28 by Ioannis Karatzas, Jan Maas, Karatzas, Ioannis +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2005.14177
openalex publication_date 2020/05/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the temporal dissipation of variance and relative entropy for\nergodic Markov Chains in continuous time, and compute explicitly the\ncorresponding dissipation rates. These are identified, as is well known, in the\ncase of the variance in terms of an appropriate Hilbertian norm; and in the\ncase of the relative entropy, in terms of a Dirichlet form which morphs into a\nversion of the familiar Fisher information under conditions of detailed\nbalance. Here we obtain trajectorial versions of these results, valid along\nalmost every path of the random motion and most transparent in the backwards\ndirection of time. Martingale arguments and time reversal play crucial roles,\nas in the recent work of Karatzas, Schachermayer and Tschiderer for\nconservative diffusions. Extension are developed to general "convex\ndivergences" and to countable state-spaces. The steepest descent and gradient\nflow properties for the variance, the relative entropy, and appropriate\ngeneralizations, are studied along with their respective geometries under\nconditions of detailed balance, leading to a very direct proof for the HWI\ninequality of Otto and Villani in the present context.\n