  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.
