The Brownian transport map
Dan Mikulincer, Yair Shenfeld
TL;DR
This work constructs a Brownian transport map from the Wiener measure to a target distribution on $\mathbb{R}^d$ via the Föllmer drift and analyzes its contraction properties using Malliavin calculus on Wiener space. It proves almost-sure contraction under $\kappa$-log-concavity or finite diameter, derives moment bounds and diffusion-trajectory estimates, and applies these contractions to obtain new or improved functional inequalities, Stein kernels, and central limit theorems, including for Gaussian mixtures. The authors connect these results to the Kannan–Lovász–Simonovits conjecture by showing near-dimension-free bounds on the derivative of the Brownian transport map in log-concave settings, and they establish a sharp separation between causal and noncausal transport on Wiener space through contraction results. They also analyze optimal transport and Cameron–Martin contractions on Wiener space, showing circumstances under which the (euclidean) OT map lifts to a Cameron–Martin contraction while the causal OT map may fail to do so, highlighting fundamental differences between these transport notions.
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 Föllmer 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--Lovász--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.