Logo

Mathematical Sciences Research Institute

Home » Workshop » Schedules » The Schrödinger problem: a probabilistic analogue of optimal transport. Application to discrete metric graphs

The Schrödinger problem: a probabilistic analogue of optimal transport. Application to discrete metric graphs

Introductory Workshop on Optimal Transport: Geometry and Dynamics August 26, 2013 - August 30, 2013

August 30, 2013 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Christian Leonard (Université de Paris X (Paris-Nanterre))
Location: MSRI: Simons Auditorium
Video

Abstract To fit in with the spirit of an introductory workshop, the first part of the talk intends to be expository. The final part will be devoted to some advanced results about discrete metric graphs. In 1931, Schrödinger addressed a problem of statistical physics nature which amounts to minimize the relative entropy of Markov random processes on a state space X subject to prescribed initial and final marginal distributions. The time marginal flow of the minimizing Markov process interpolates between the prescribed marginals on the space Proba(X) of probability measures on X. Large deviations arguments show that slowing the processes down towards a no-motion process leads to a dynamical Monge-Kantorovich problem whose transport cost function is related to the random dynamics. The corresponding entropic interpolations are smooth paths on Proba(X) which converge to some displacement interpolation. For instance, if one takes the Brownian motion as reference process, the limiting interpolation is McCann's one. Considering random walks on a discrete set X with a graph structure and passing to the slowing down limit allows to define natural displacement interpolations on Proba(X) that are geodesics with respect to an intrinsic graph distance. This opens the way for investigating curvature of graphs by tracking Lott-Sturm-Villani theory
Supplements
18149?type=thumb v1149 2.21 MB application/pdf Download
Video/Audio Files

v1149

H.264 Video v1149.m4v 323 MB video/mp4 rtsp://videos.msri.org/data/000/017/657/original/v1149.m4v Download
Quicktime v1149.mov 452 MB video/quicktime rtsp://videos.msri.org/data/000/017/658/original/v1149.mov Download
Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture. The videos are sold at cost for $20USD (shipping included). Please Click Here to send an email to MSRI to purchase the DVD.

See more of our Streaming videos on our main VMath - Streaming Video page.