### 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