The purpose of this thesis is to present in detail two theories, not deductible from each other, but which obtain very similar results, giving a Riemannian-like structure to the set of probability measures. Both theories lead to a correspondence between heat equation and gradient flow of entropy.
Chapter 1 contains a quick but self-contained treatment of the theory of optimal transport, including Kantorovich’s duality, Brenier’s theorem, the Wasserstein spaces P_p and the characterization of their geodesics.
Chapter 2 focuses on P_2(R^n). Using the continuity equation, we describe its geodesics and absolutely continuous curves, and determine a “tangent” velocity to them; this enables the definition of (sub)differential of a functional. We study convexity and differentiability of the “internal energy” and “potential energy” functionals. We give a definition of gradient flows, and exploit its equivalence with the purely metric EVI formulation to show some of its basic properties. We conclude characterizing the gradient flows of the entropy by the condition that the densities solve the heat equation.
In Chapter 3, we move to a finite space endowed with an irreducible Markov kernel. Guided by an explicit study of the two-point space, we define a family of distances between probabilities via a discrete analogue of “Benamou-Brenier’s formula” from Chapter 2, and characterize their finiteness. The AC curves are described as in the continuous case. Appropriate subsets of probability measures turn out to be Riemannian manifolds, on which we can reproduce results like the Eulerian description of geodesics, the differentiability of potential energy, the identification of heat flow with gradient flow of the entropy
