Lipschitz Transport Maps via the Follmer Flow
Abstract
Inspired by the construction of the F\"ollmer process, we construct a unit-time flow on the Euclidean space, termed the F\"ollmer 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\"ollmer 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.
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