Vol. 1, No. 1, 2013

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

Eric A. Carlen and Katy Craig

Vol. 1 (2013), No. 1, 33–65

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 Rényi 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.

Wasserstein metric, gradient flow, Moreau–Yosida regularization
Mathematical Subject Classification 2010
Primary: 49-XX
Received: 25 May 2012
Revised: 17 October 2012
Accepted: 3 November 2012
Published: 6 February 2013

Communicated by Raffaele Esposito
Eric A. Carlen
Department of Mathematics, Hill Center
Rutgers University
110 Frelinghuysen Road
Piscataway, NJ 08854-8019
United States
Katy Craig
Department of Mathematics, Hill Center
Rutgers University
110 Frelinghuysen Road
Piscataway, NJ 08854-8019
United States