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Department of Mathematics pres.

Financial/Actuarial Mathematics Seminar

Van Eenam Lecture #2: Conservative Diffusion as Entropic Gradient Flow

We provide a detailed probabilistic interpretation, based on stochastic calculus, for the variational characterization of conservative diffusion as entropic gradient flux. 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 diffusive motion, in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by taking expectations. As a bonus of the approach we obtain the HWI inequality of Otto and Villani relating relative entropy, Fisher information, and Wasserstein distance; and from it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis. (Joint work with W. Schachermayer and B. Tschiderer.) Speaker(s): Ioannis Karatzas (Columbia University)

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