  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.
