论文标题
一种轨迹方法，用于Langevin-Smoluchowski扩散的梯度流量特性
A trajectorial approach to the gradient flow properties of Langevin-Smoluchowski diffusions
论文作者
论文摘要
我们重新审视保守性扩散作为熵梯度流的变异表征，并为其提供基于随机演算的概率解释。 Jordan，Kinderlehrer和Otto展示了，对于Langevin-Smoluchowski类型的扩散，Fokker-Planck概率密度流量最大化相对熵耗散的速率，可通过在有限的第二矩的环境空间中行进的距离来衡量，以有限的第二矩量的距离进行，以实现有限的第二矩量的距离。我们使用非常直接的扰动分析获得了这些特征的新颖的，随机的版本，几乎沿着时间向后的扩散运动的每个轨迹有效。通过对路径空间上的基础度量的平均结果进行平均，我们建立了沿Fokker-Planck流的最大熵耗散速率，并精确地测量与此对应于任何给定扰动相对应的最大值的偏差。作为我们轨迹方法的奖励，我们得出了相对熵（H），Wasserstein距离（W）和相对Fisher Information（i）的HWI不等式。
We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backward direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker-Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. As a bonus of our trajectorial approach we derive the HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).
