Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction

TL;DR

This paper proves that geodesic convexity of entropy in nonlinear diffusion equations is equivalent to Wasserstein contraction, providing a simple proof that does not rely on entropy or gradient flow structures.

Contribution

It establishes a direct equivalence between geodesic convexity and Wasserstein contraction for nonlinear diffusion, simplifying previous proofs.

Findings

Geodesic convexity implies Wasserstein contraction.

Wasserstein contraction implies geodesic convexity.

The proof is simplified and does not use entropy or gradient flow structures.

Abstract

It is well known that nonlinear diffusion equations can be interpreted as a gradient flow in the space of probability measures equipped with the Euclidean Wasserstein distance. Under suitable convexity conditions on the nonlinearity, due to R. J. McCann, the associated entropy is geodesically convex, which implies a contraction type property between all solutions with respect to this distance. In this note, we give a simple straightforward proof of the equivalence between this contraction type property and this convexity condition, without even resorting to the entropy and the gradient flow structure.