About the analogy between optimal transport and minimal entropy

2015/10/28 by Ivan Gentil, Christian Léonard, Gentil, Ivan +3 · 8 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1510.08230

arxiv created 2016/05/20 · arxiv updated 2016/05/23

Abstract

We describe some analogy between optimal transport and the Schrödinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role.

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