  Defining a divergence between the laws of continuous martingales is a
delicate task, owing to the fact that these laws tend to be singular to each
other. An important idea, put forward by N. Gantert, is to instead consider a
scaling limit of the relative entropy between such continuous martingales
sampled over a finite time grid. This gives rise to the concept of specific
relative entropy. In order to develop a general theory of divergences between
continuous martingales, it is only natural to replace the role of the relative
entropy in this construction by a different notion of discrepancy between
finite dimensional probability distributions. In the present work we take a
first step in this direction, taking a power $p$ of the Wasserstein distance
instead of the relative entropy. We call the newly obtained scaling limit the
specific $p$-Wasserstein divergence.
  In our first main result we prove that the specific $p$-Wasserstein
divergence is well-defined, and exhibit an explicit expression for it in terms
of the quadratic variations of the martingales involved. This is obtained under
vastly weaker assumptions than the corresponding results for the specific
relative entropy. Next we illustrate the usefulness of the concept, by
considering the problem of optimizing the specific $p$-Wasserstein divergence
over the set of win-martingales. In our second main result we characterize the
solution of this optimization problem for all $p>0$ and, somewhat surprisingly,
we single out the case $p=1/2$ as the one with the best probabilistic
properties. For instance, the optimal martingale in this case is very explicit
and can be connected, through a space transformation, to the solution of a
variant of the Schr\"odinger problem.
