Heat flow and calculus on metric measure spaces with Ricci curvature bounded below -- the compact case (Q2869750)

The aim of this note is to provide a quick overview of the main results of the authors [Invent. Math. 195, No. 2, 289--391 (2014; Zbl 1312.53056)] and [Duke Math. J. 163, No. 7, 1405--1490 (2014; Zbl 1304.35310)] in the simplified case of compact metric spaces \((X,d)\) equipped with a probability measure \(m\). Apart from some basic concepts of optimal transport, Wasserstein distance and gradient flows, the paper is self-contained with most concepts given in the preliminary section. The main results and arguments presented in this paper include the Hopf-Lax formula for the Hamilton-Jacobi equation, a new approach to the theory of Sobolev spaces over metric measures spaces, the uniqueness of the gradient flow with respect to the Wasserstein distance, the identification of the \(L^2\)-gradient flow of the natural ``Dirichlet energy'' and the \(W_2\)-gradient flow, a metric version of the Brenier's Theorem, a key lemma concerning the horizontal and vertical differentiation, and a new definition of Ricci curvature bound from below for metric measure spaces which is stable with respect to measured Gromov-Hausdorff convergence.