Signals and control aspects of optimal mass transport and the Boltzmann entropy

In this note we describe some properties of the Wasserstein-2 metric on the space of probability distributions. It turns out that the resulting geodesics lead to interesting connections with the Boltzmann entropy, the heat equations (both linear and nonlinear), and suggest possible Riemannian structures on density functions. In particular, we observe similarities and connections with other metrics originating in Information geometry and prediction theory.