Lipschitz Transport Maps via the Follmer Flow (2309.03490v1)

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.