A trajectorial approach to the gradient flow properties of Langevin-Smoluchowski diffusions

TL;DR

This paper provides a probabilistic, trajectorial perspective on the gradient flow properties of Langevin-Smoluchowski diffusions, linking stochastic processes with entropy dissipation and Wasserstein geometry.

Contribution

It introduces a novel trajectorial approach to analyze the gradient flow of Langevin-Smoluchowski diffusions, including stochastic-process versions and derivation of the HWI inequality.

Findings

Trajectorial analysis valid along almost every diffusive path.

Maximal entropy dissipation rate established for Fokker-Planck flow.

Derived the HWI inequality connecting entropy, Wasserstein distance, and Fisher information.

Abstract

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).