Displacement convexity of generalized relative entropies

Details Displacement convexity of generalized relative entropies Displacement convexity of generalized relative entropies

2010-05-08

arxiv.org/abs/1005.1331v3

Mathematics

Differential Geometry

43pages

Scientific paper

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the $m$-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

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