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Mathematics Colloquium

Walter Schachermayer
University of Vienna

Title: Trajectorial Otto calculus.
Date: Friday, April 16, 2021
Place and Time: Zoom, 1:25-2:15 pm

Abstract. We revisit the variational characterization of diffusion as entropic gradient flux, established by Jordan, Kinderlehrer, and Otto in [1], and provide for it a probabilistic interpretation based on stochastic calculus. It was shown in [1] that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second mo- ments, 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 both the forward and, most trans- parently, the backward, directions 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 minimum rate of entropy dissipation along the Fokker-Planck flow and measure exactly the deviation from this minimum that corresponds to any given perturbation. As a bonus of the perturbation analysis, we derive the so-called HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I). Joint work with I. Karatzas and B. Tschiderer.
[1] R. Jordan, D. Kinderlehrer, and F. Otto (1998) The variational formula of the Fokker-Planck equation. SIAM journal on mathematical analysis 29,1,1-17