Ομιλία του κου. Iωάννη Καρατζά (Columbia University, http://www.math.columbia.edu/~ik/), στο Σεμινάριο του Τομέα Μαθηματικών της ΣΕΜΦΕ, την Τρίτη 27 Αυγούστου στις 13:05, στην Αίθουσα Σεμιναρίων του τομέα Μαθηματικών (2ος όροφος, κτίριο Ε – Χάρτης )

Τίτλος : “CONSERVATIVE DIFFUSION AS ENTROPIC GRADIENT FLOW OF STEEPEST DESCENT”

Περίληψη : “We provide a probabilistic and trajectorial interpretation, based on stochastic calculus, for the variational characterization of diffusion as entropic gradient flux. This was first established, through a discretiza- tion scheme, in the seminal paper by Jordan-Kinderlehrer-Otto (1998). It was shown by those authors that, for diffusions of the Langevin- Smoluchowski type dX (t ) = −∇Ψ􏰁X (t )􏰂 dt + dW (t ) the associated Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation – measured by the quadratic Wasserstein distance traveled in the ambient space (Riemannian manifold), of probability measures with finite second moments.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 (as in Fontbona-Jourdain 2016), 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 maximality of the entropy dissipation rate along the Fokker-Planck flow – and measure precisely the deviation from this maximum that corresponds to any given perturbation. As a bonus of this perturbation analysis, the famous so-called HWI inequality of Otto and Villani (2000), relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I), literally falls in our lap. And with it, so do the Talagrand, log-Sobolev and Poincaré inequalities of Functional Analysis Joint work with Walter Schachermayer and Bertram Tschiderer (Vienna)”