Optimal transport — the geometry of moving probability

Benamou–Brenier: the distance between two distributions is the least kinetic energy of a fluid that carries one to the other, subject to the continuity equation. Jordan–Kinderlehrer–Otto (1998): the Fokker–Planck equation is the gradient flow of free energy F = ∫ρ log ρ + ∫Uρ in this Wasserstein geometry — diffusion is steepest descent of entropy.