  Wasserstein distance induces a natural Riemannian structure for the
probabilities on the Euclidean space. This insight of classical transport
theory is fundamental for tremendous applications in various fields of pure and
applied mathematics.
  We believe that an appropriate probabilistic variant, the adapted Wasserstein
distance AW, can play a similar role for the class FP of filtered processes,
i.e. stochastic processes together with a filtration. In contrast to other
topologies for stochastic processes, probabilistic operations such as the
Doob-decomposition, optimal stopping and stochastic control are continuous
w.r.t. AW. We also show that (FP,AW) is a geodesic space, isometric to a
classical Wasserstein space, and that martingales form a closed geodesically
convex subspace.
