  We introduce a Benamou-Brenier formulation for the continuous-time martingale
optimal transport problem as a weak length relaxation of its discrete-time
counterpart. By the correspondence between classical martingale problems and
Fokker-Planck equations, we obtain an equivalent PDE formulation for which
basic properties such as existence, duality and geodesic equations can be
analytically studied, yielding corresponding results for the stochastic
formulation. In the one dimensional case, sufficient conditions for finiteness
of the cost are also given and a link between geodesics and porous medium
equations is partially investigated.
