Minimizing relative entropy of path measures under marginal constraints

We study generalizations of the Schr"odinger problem in statistical mechanics in two directions: when the density is constrained at more than two times, and when the joint law of the initial and final positions for the particles is prescribed. This is done in agreement with the so-called Br"odinger problem recently introduced to regularize Brenier's variational model for incompressible fluids. We recover generalizations of the standard factorization result for the Radon-Nikodym derivative of the solution P with respect to the reference measure R: this density can be written in terms of an additive functional on the set of constrained times. The specificity of this work is that we place ourselves in the case when R is Markov (or reciprocal), and that we use Markovian methods rather than classical convex analysis arguments. In this setting, it appears that a natural assumption to be made on the reference measure R is of irreducibility type.