Otto calculus

The Otto calculus is a formal Riemannian framework for the space of probability measures equipped with the Wasserstein metric. Developed by Felix Otto in the early 2000s, it reveals that the Wasserstein space P_2(X) possesses a rich geometric structure: geodesics correspond to displacement interpolations, the metric tensor is determined by L^2 norms of gradient vector fields, and gradient flows of entropy functionals correspond to dissipative evolution equations such as the heat equation and the Fokker-Planck equation.

The central insight of the Otto calculus is that the Wasserstein space can be treated as an infinite-dimensional manifold, with tangent vectors at a measure μ represented as gradient vector fields on the underlying space. This permits the formulation of gradient flows, geodesic equations, and curvature calculations in a setting that bridges optimal transport, partial differential equations, and metric geometry. The calculus has become essential in the study of synthetic Ricci curvature, where convexity of entropy along Wasserstein geodesices characterizes curvature lower bounds.

The Otto calculus is the revenge of geometry on analysis. For decades, partial differential equations were treated as analytic objects — estimates, inequalities, fixed-point arguments — while their geometric content remained implicit. Otto showed that the heat equation, the porous medium equation, and the Fokker-Planck equation are not merely PDEs to be solved. They are gradient flows in a Riemannian geometry of probability, and their analytic properties are consequences of the curvature of that geometry. The implication is radical: every dissipative PDE is secretly a geometric evolution, and the analyst who ignores the geometry is working with one eye closed.