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

Blanchet, Adrien and Bolte, Jérôme (2018) A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions. Journal of Functional Analysis, 25 (n°7). pp. 1650-1673.

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Official URL: http://tse-fr.eu/pub/32760

Abstract

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ-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.

Item Type: Article
Language: English
Date: October 2018
Refereed: Yes
Uncontrolled Keywords: Lojasiewicz inequality, Functional inequalities, Gradient flows, Optimal Transport, Monge-Kantorovich distance
Subjects: B- ECONOMIE ET FINANCE
Divisions: TSE-R (Toulouse)
Site: UT1
Date Deposited: 09 Jul 2018 14:40
Last Modified: 27 Aug 2018 12:38
OAI ID: oai:tse-fr.eu:32760
URI: http://publications.ut-capitole.fr/id/eprint/26111

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