  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 F\"ollmer 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--Lov\'asz--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.
