University of Vienna
Title: Trajectorial Otto calculus.
Date: Friday, April 16, 2021
Place and Time: Zoom, 1:25-2:15 pm
We revisit the variational characterization of diffusion as entropic
gradient flux, established by Jordan, Kinderlehrer, and Otto in , and
provide for it a probabilistic interpretation based on stochastic calculus. It was shown in  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.
 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