Lipschitz Transport Maps via the Follmer Flow

TL;DR

This paper introduces the F{"o}llmer flow, a Lipschitz continuous transformation that maps a standard Gaussian to a target measure, enabling new functional and concentration inequalities.

Contribution

It constructs and analyzes the F{"o}llmer flow, establishing its Lipschitz property and applying it to derive dimension-free inequalities for various probability measures.

Findings

F{"o}llmer flow is well-posed and Lipschitz continuous.

Derived dimension-free functional inequalities.

Established concentration inequalities for empirical measures.

Abstract

Inspired by the construction of the F{\"o}llmer process, we construct a unit-time flow on the Euclidean space, termed the F{\"o}llmer flow, whose flow map at time 1 pushes forward a standard Gaussian measure onto a general target measure. We study the well-posedness of the F{\"o}llmer flow and establish the Lipschitz property of the flow map at time 1. We apply the Lipschitz mapping to several rich classes of probability measures on deriving dimension-free functional inequalities and concentration inequalities for the empirical measure.