  We investigate the Moreau-Yosida regularization and the associated proximal
map in the context of discrete gradient flow for the 2-Wasserstein metric. Our
main results are a stepwise contraction property for the proximal map and an
"above the tangent line" inequality for the regularization. Using the latter,
we prove a Talagrand inequality and an HWI inequality for the regularization,
under appropriate hypotheses. In the final section, the results are applied to
study the discrete gradient flow for R\'enyi entropies. As Otto showed, the
gradient flow for these entropies in the 2-Wasserstein metric is a porous
medium flow or a fast diffusion flow, depending on the exponent of the entropy.
We show that a striking number of the remarkable features of the porous medium
and fast diffusion flows are present in the discrete gradient flow and do not
simply emerge in the limit as the time-step goes to zero.
