This article gives a survey of recent research related to the Monge-Kantorovich problem.
Principle results are presented on the existence of solutions and their properties both in the …

Heuristically, a discrete-time stochastic process has the Markov property if the past and
future are independent given the present. In this introductory chapter, we give the formal …

Abstract Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic
Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased …

Let μ _N μ N be the empirical measure associated to a N N-sample of a given probability
distribution μ μ on R^ d R d. We are interested in the rate of convergence of μ _N μ N to μ μ …

This paper develops a nonasymptotic, local approach to quantitative propagation of chaos
for a wide class of mean field diffusive dynamics. For a system of n interacting particles, the …

These notes were written for the course APC 550: Probability in High Dimension that I taught
at Princeton in the Spring 2014 and Fall 2016 semesters. The aim was to introduce in as …

This is the first comprehensive introduction to the theory of mass transportation with its many—
and sometimes unexpected—applications. In a novel approach to the subject, the book both …

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G …

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are
closer (in Wasserstein transportation distance) than their centers are. This definition …

We prove a version of McDiarmid's bounded differences inequality for Markov chains, with
constants proportional to the mixing time of the chain. We also show variance bounds and …