  A theorem of L. Caffarelli implies the existence of a map pushing forward a
source Gaussian measure to a target measure which is more log-concave than the
source one, which contracts Euclidean distance (in fact, Caffarelli showed that
the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We
generalize this result to more general source and target measures, using a
condition on the third derivative of the potential, using two different proofs.
The first uses a map $T$, whose inverse is constructed as a flow along an
advection field associated to an appropriate heat-diffusion process. The
contraction property is then reduced to showing that log-concavity is preserved
along the corresponding diffusion semi-group, by using a maximum principle for
parabolic PDE. In particular, Caffarelli's original result immediately follows
by using the Ornstein-Uhlenbeck process and the Pr\'ekopa--Leindler Theorem.
The second uses the map $T_{opt}$ by generalizing Caffarelli's argument,
employing in addition further results of Caffarelli. As applications, we obtain
new correlation and isoperimetric inequalities.
