Transport via mass transportation

Weak topology implicit schemes based on Monge-Kantorovich or Wasserstein metrics have become prominent for their ability to solve a variety of diffusion and diffusion-like equations. They are very flexible, encompassing a wide range of nonlinear effects. They have interesting interpretations as descent algorithms in an infinite dimensional manifold setting or as dissipation principles for motion in a highly viscous environment. Transport plays a fundamental role in these schemes, as noted by Brenier and Benamou and reviewed below. The reverse implication is less explored and, at least at the outset, less obvious. Here we discuss the simplest situations in the context of systems of transport equations. We show how arbitrary Fokker-Planck Equations in one dimension conform to the mass transport paradigm. Finally, we provide some additional examples, including a simple existence result for velocity-jump processes.