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A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions (1612.02619v5)
Published 8 Dec 2016 in math.AP
Abstract: For displacement convex functionals in the probability space equip-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type \L oja-sie-wicz inequalities. \chg{We also discuss the more general case of $\lambda$-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.