article published yes Ioannis Karatzas author Jan Maas author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338 Walter Schachermayer author JaMa department Optimal Transport and Stochastic Dynamics project Taming Complexity in Partial Differential Systems project 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. Extensions 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. International Press2021 eng Markov Chainrelative entropytime reversalsteepest descentgradient flow 1526-7555 2005.1417710.4310/CIS.2021.v21.n4.a1 214481-536 Karatzas, I., Maas, J., &#38; Schachermayer, W. (2021). Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. <i>Communications in Information and Systems</i>. International Press. <a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a> Karatzas, Ioannis, et al. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” <i>Communications in Information and Systems</i>, vol. 21, no. 4, International Press, 2021, pp. 481–536, doi:<a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">10.4310/CIS.2021.v21.n4.a1</a>. Karatzas, Ioannis, Jan Maas, and Walter Schachermayer. “Trajectorial Dissipation and Gradient Flow for the Relative Entropy in Markov Chains.” <i>Communications in Information and Systems</i>. International Press, 2021. <a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">https://doi.org/10.4310/CIS.2021.v21.n4.a1</a>. I. Karatzas, J. Maas, and W. Schachermayer, “Trajectorial dissipation and gradient flow for the relative entropy in Markov chains,” <i>Communications in Information and Systems</i>, vol. 21, no. 4. International Press, pp. 481–536, 2021. Karatzas I, Maas J, Schachermayer W. Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. <i>Communications in Information and Systems</i>. 2021;21(4):481-536. doi:<a href="https://doi.org/10.4310/CIS.2021.v21.n4.a1">10.4310/CIS.2021.v21.n4.a1</a> I. Karatzas, J. Maas, W. Schachermayer, Communications in Information and Systems 21 (2021) 481–536. Karatzas I, Maas J, Schachermayer W. 2021. Trajectorial dissipation and gradient flow for the relative entropy in Markov chains. Communications in Information and Systems. 21(4), 481–536. 100232021-09-19T08:53:19Z2025-04-14T07:27:45Z