A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost
Abstract
The \emph{Schrödinger problem} is obtained by replacing the mean square distance with the relative entropy in the MongeKantorovich problem. It was first addressed by Schrödinger as the problem of describing the most likely evolution of a large number of Brownian particles conditioned to reach an "unexpected configuration". Its optimal value, the \textit{entropic transportation cost}, and its optimal solution, the \textit{Schrödinger bridge}, stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation. Moreover, they provide a natural way of lifting from the point to the measure setting the concept of Brownian bridge. In this article, we prove that the Schrödinger bridge solves a second order equation in the Riemannian structure of optimal transport. Roughly speaking, the equation says that its acceleration is the gradient of the Fisher information. Using this result, we obtain a fine quantitative description of the dynamics, and a new functional inequality for the entropic transportation cost, that generalize Talagrand's transportation inequality. Finally, we study the convexity of the Fisher information along Schrödigner bridges, under the hypothesis that the associated \textit{reciprocal characteristic} is convex. The techniques developed in this article are also well suited to study the \emph{FeynmanKac penalisations} of Brownian motion.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.04821
 arXiv:
 arXiv:1704.04821
 Bibcode:
 2017arXiv170404821C
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis;
 58J65;
 47D07;
 35Q70
 EPrint:
 to appear in Probability Theory and Related Fields