  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.
