Geodesic
A trajectorial approach to the gradient flow properties of Langevin--Smoluchowski diffusions
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 839-888

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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).
Keywords: relative entropy, Wasserstein distance, Fisher information, gradient flow, time reversal, functional inequalities.
Mots-clés : optimal transport, diffusion processes 
@article{TVP_2021_66_4_a10,
     author = {I. Karatzas and W. Schachermayer and B. Tschiderer},
     title = {A trajectorial approach to the gradient flow properties of {Langevin--Smoluchowski} diffusions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {839--888},
     publisher = {mathdoc},
     volume = {66},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a10/}
}
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I. Karatzas; W. Schachermayer; B. Tschiderer. A trajectorial approach to the gradient flow properties of Langevin--Smoluchowski diffusions. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 839-888. http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a10/