论文标题
马尔可夫链中相对熵的轨迹耗散和梯度流
Trajectorial dissipation and gradient flow for the relative entropy in Markov chains
论文作者
论文摘要
我们研究了连续时间的偏红色马尔可夫链的方差和相对熵的时间耗散，并明确计算相应的耗散速率。众所周知，就适当的希尔伯特规范而言，这些被确定为差异。在相对熵的情况下，就Dirichlet形式而言，该形式在详细的平衡条件下变成了熟悉的Fisher信息的版本。在这里，我们获得了这些结果的轨迹版本，几乎沿着随机运动的每个路径有效，并且在时间方向上最透明。 Martingale的争论和时间逆转扮演着关键的角色，就像Karatzas，Schachermayer和Tschiderer在保守扩散的最新作品中一样。扩展是开发到一般的“凸差”和可计数状态空间的。在详细的平衡条件下，研究了方差，相对熵和适当的概括的最陡峭下降和梯度流的特性，以及它们各自的几何形状，从而在当前情况下对Otto和Villani的HWI不平等进行了非常直接的证明。
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.
