  In classical optimal transport, the contributions of Benamou-Brenier and
McCann regarding the time-dependent version of the problem are cornerstones of
the field and form the basis for a variety of applications in other
mathematical areas.
  We suggest a Benamou-Brenier type formulation of the martingale transport
problem for given $d$-dimensional distributions $\mu, \nu $ in convex order.
The unique solution $M^*=(M_t^*)_{t\in [0,1]}$ of this problem turns out to be
a Markov-martingale which has several notable properties: In a specific sense
it mimics the movement of a Brownian particle as closely as possible subject to
the conditions $M^*_0\sim\mu, M^*_1\sim \nu$. Similar to McCann's
displacement-interpolation, $M^*$ provides a time-consistent interpolation
between $\mu$ and $\nu$. For particular choices of the initial and terminal
law, $M^*$ recovers archetypical martingales such as Brownian motion, geometric
Brownian motion, and the Bass martingale. Furthermore, it yields a natural
approximation to the local vol model and a new approach to Kellerer's theorem.
  This article is parallel to the work of Huesmann-Trevisan, who consider a
related class of problems from a PDE-oriented perspective.
