Displacement convexity of generalized relative entropies

作者：Ohta Shin ichi^*; Takatsu Asuka

来源：Advances in Mathematics, 2011, 228(3): 1742-1787.

DOI：10.1016/j.aim.2011.06.029

• Relative entropy
• Bregman divergence
• Wasserstein space
• Ricci curvature
• Porous medium equation

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 in-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.

• 出版日期2011-10-20