Student Probability/PDE Seminar: Pathwise Otto Calculus
Seminar | April 26 | 2:10-3:30 p.m. | 891 Evans Hall
Walter Schachermayer, University of Vienna and Stanford
We revisit the [R. Jordan, D. Kinderlehrer, and F. Otto. The variational formulation of the Fokker-Planck equation (1998)] variational characterization of diffusion as entropic gradient flux, 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 minimizes the rate of relative entropy dissipation, as measured by the distance traveled 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 transparently, the backward, directions of time, using a very direct perturbation analysis; the original results follow then simply by taking expectations. As a bonus, we derive the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.