Displacement convexity of generalized relative entropies

Abstract

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.

A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions

“…(ii) ⇒ (i). Take x ∈ R d and write the inequality (24) for δ x , the Dirac at x. The result follows fromĴ =V and dist (x, Argmin V ) = W 2 (δ x , Argmin J ).…”

(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

“…I am grateful to Kazumasa Kuwada for valuable suggestions and discussions, especially on the expansion bound in Subsection 3.2. I thank Asuka Takatsu for fruitful discussions, some of the results in Subsections 4.1, 4.2 originate from discussions during the joint work [OT1], [OT2]. My gratitude also goes to Frank Morgan for drawing my attention to [MR] and [KM], and to Emanuel Milman for his helpful comments on the background of [MR] and [KM].…”

q-Heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space

“…In the Euclidean case, Otto and Augeh showed that the parabolic q-Laplace equation, which is the q-heat flow for smooth solutions, can be solved using the gradient flow of U p in the p-Wasserstein case. This should also be compared to [OT11a,OT11b], where the (parabolic) porous media equation is solved via a gradient flow of a similar functional in the 2-Wasserstein space for Riemannian manifolds of nonnegative Ricci curvature. Note, however, no identification is done.…”

“…However, since a general existence theory of such equations on abstract metric spaces is not available, an identification is difficult using our approach. This is exactly why Ohta-Takatsu [OT11a,OT11b] can only use the gradient flows in P 2 to get a solution, but they do not identify the two flows.…”