One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law in R^d onto any other given law. A large body of recent work has studied the question of estimating Brenier maps on the basis of random samples from the underlying measures. In this talk, we characterize the fluctuations of such estimators. Concretely, we derive pointwise central limit theorems for a class of periodic stochastic Brenier maps known as plugin estimators, in general dimension. Our proofs hinge upon recent developments in the literature on quantitative stability of optimal transport, and quantitative linearization of the Monge-Ampere equation, which we discuss throughout.

This talk is based on joint work with Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman.