Ioannis Karatzas - TRAJECTORIAL APPROACH TO GRADIENT FLOW PROPERTIES OF CONSERVATIVE DIFFUSIONS
| Dates: | 24 February 2021 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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Ioannis Karatzas (Columbia University) will speak in the Probability seminar.
Abstract: We provide a detailed, probabilistic interpretation for the variational characterization of conservative diffusion as entropic gradient flow. Jordan, Kinderlehrer, and Otto showed in 1998 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 in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features; these are valid along almost every trajectory of the motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by “aggregating”, i.e., taking expectations. As a bonus of the analysis, the HWI inequality of Otto and Villani, relating relative entropy, Fisher information, and Wasserstein distance, falls in our lap; and with it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis.
(Joint work with Walter Scahchermayer and Bertram Tschiderer, University of Vienna.)
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