We apply the shifted composition rule -- an information-theoretic principle introduced in our earlier work [AC23] -- to establish shift Harnack inequalities for the Langevin diffusion. We obtain sharp constants for these inequalities for the first time, allowing us to investigate their relationship with other properties of the diffusion. Namely, we show that they are equivalent to a sharp "local gradient-entropy" bound, and that they imply curvature upper bounds in a compelling reflection of the Bakry-Emery theory of curvature lower bounds. Finally, we show that the local gradient-entropy inequality implies optimal concentration of the score, a.k.a. the logarithmic gradient of the density.

翻译：我们应用平移组合规则——一种在我们先前工作[AC23]中引入的信息论原理——来建立朗之万扩散的平移哈纳克不等式。我们首次获得这些不等式的锐化常数，从而能够探讨它们与扩散过程其他性质的关系。具体而言，我们证明其等价于一个锐化的"局部梯度-熵"界，并表明它们以令人信服的方式蕴含曲率上界，这反映了巴克里-埃默里曲率下界理论的对偶性。最后，我们证明局部梯度-熵不等式蕴含分数（即密度的对数梯度）的最优集中性。