  We describe some analogy between optimal transport and the Schr\"odinger
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.
