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A Trajectorial Approach to the Gradient Flow Properties of Langevin--Smoluchowski Diffusions
Theory of Probability and Its Applications ( IF 0.6 ) Pub Date : 2022-02-03 , DOI: 10.1137/s0040585x97t990678
I. Karatzas , W. Schachermayer , B. Tschiderer

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 668-707, February 2022.
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 backwards 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. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I).


中文翻译:

Langevin--Smoluchowski 扩散梯度流动特性的轨迹方法

概率论及其应用,第 66 卷,第 4 期,第 668-707 页,2022 年 2 月。
我们重新审视作为熵梯度流的保守扩散的变分特征,并为其提供基于随机演算的概率解释。Jordan、Kinderlehrer 和 Otto 表明,对于 Langevin-Smoluchowski 类型的扩散,Fokker-Planck 概率密度流最大化了相对熵耗散的速率,通过在概率测量的环境空间中行进的距离来测量根据二次 Wasserstein 度量,具有有限的二阶矩。我们使用非常直接的微扰分析,获得了这些特征的新颖的随机过程版本,几乎沿时间向后方向的扩散运动的每条轨迹都有效。通过将我们的轨迹结果相对于路径空间的基础度量进行平均,我们确定了沿 Fokker-Planck 流的最大熵耗散率,并准确测量了与任何给定扰动相对应的该最大值的偏差。我们的轨迹方法的一个好处是它导出了与相对熵 (H)、Wasserstein 距离 (W) 和相对 Fisher 信息 (I) 相关的 HWI 不等式。
更新日期:2022-02-03
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