A second order equation for Schr\"odinger bridges with applications to the hot gas experiment and entropic transportation cost

The \emph{Schr\"odinger problem} is obtained by replacing the mean square distance with the relative entropy in the Monge-Kantorovich problem. It was first addressed by Schr\"odinger 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{Schr\"odinger 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 Schr\"odinger 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 Schr\"odigner 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{Feynman-Kac penalisations} of Brownian motion.