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We investigate the Moreau-Yosida regularization and the associated proximalmap in the context of discrete gradient flow for the 2-Wasserstein metric. Ourmain 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 tostudy the discrete gradient flow for R\'enyi entropies. As Otto showed, thegradient flow for these entropies in the 2-Wasserstein metric is a porousmedium 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 mediumand fast diffusion flows are present in the discrete gradient flow and do notsimply emerge in the limit as the time-step goes to zero.

Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric