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The \emph{Schr\"odinger problem} is obtained by replacing the mean squaredistance with the relative entropy in the Monge-Kantorovich problem. It wasfirst addressed by Schr\"odinger as the problem of describing the most likelyevolution of a large number of Brownian particles conditioned to reach an"unexpected configuration". Its optimal value, the \textit{entropictransportation cost}, and its optimal solution, the \textit{Schr\"odingerbridge}, stand as the natural probabilistic counterparts to the transportationcost and displacement interpolation. Moreover, they provide a natural way oflifting 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 orderequation in the Riemannian structure of optimal transport. Roughly speaking,the equation says that its acceleration is the gradient of the Fisherinformation. Using this result, we obtain a fine quantitative description ofthe dynamics, and a new functional inequality for the entropic transportationcost, that generalize Talagrand's transportation inequality. Finally, we studythe convexity of the Fisher information along Schr\"odigner bridges, under thehypothesis 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.

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