The Brownian transport map

TL;DR

This paper introduces a new Brownian transport map based on the Föllmer process, which exhibits contraction properties and enables new functional inequalities and applications in probability theory.

Contribution

The paper constructs a novel Brownian transport map using stochastic calculus, extending the theory of transport maps beyond optimal transport and Euclidean spaces.

Findings

The Brownian transport map is a contraction in various settings.

It enables the proof of new or improved functional inequalities.

Applications include Stein kernels, central limit theorems, and insights into the KLS conjecture.

Abstract

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the F\"ollmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan--Lov\'asz--Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.