  We prove the Duality Theorems for the stochastic optimal transportation
problems with a convex cost function without a regularity assumption that is
often supposed in the proof of the lower semicontinuity of an action integral.
In our new approach, we prove that the stochastic optimal transportation
problems with a convex cost function are equivalent to a class of variational
problems for the Fokker-Planck equation, which lets us revisit them. It is done
by the so-called superposition principle and by an idea from the mather theory.
The superposition principle is the construction of a semimartingale from the
Fokker-Planck equation and can be considered a class of the so-called marginal
problems that construct stochastic processes from given marginal distributions.
It was first considered in stochastic mechanics by E. Nelson, called Nelson's
problem, and was proved by E. Carlen first. The semimartingale is called the
Nelson process, provided it is Markovian. We also consider the Markov property
of a minimizer of the stochastic optimal transportation problem with a
nonconvex cost in a one-dimensional case. In the proof, the superposition
principle and the minimizer of an optimal transportation problem with a concave
cost function play crucial roles. Lastly, we prove the semiconcavity and the
Lipschitz continuity of Schrodinger's problem that is a typical example of the
stochastic optimal transportation problem.
