We investigate the Moreau–Yosida regularization and the associated
proximal map in the context of discrete gradient flow for the
-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 Rényi
entropies. As Otto showed, the gradient flow for these entropies in the
-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.