A family of functional inequalities

Blanchet, Adrien and Bolte, Jérôme (2017) A family of functional inequalities: lojasiewicz inequalities and displacement convex functions. TSE Working Paper, n. 17-787, Toulouse

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Official URL: http://tse-fr.eu/pub/31578


For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows.

Item Type: Monograph (Working Paper)
Sub-title: lojasiewicz inequalities and displacement convex functions
Language: English
Date: March 2017
Place of Publication: Toulouse
Divisions: TSE-R (Toulouse)
Institution: Université Toulouse 1 Capitole
Site: UT1
Date Deposited: 24 Mar 2017 13:34
Last Modified: 23 Jul 2018 12:48
OAI ID: oai:tse-fr.eu:31578
URI: http://publications.ut-capitole.fr/id/eprint/23265

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