Article Dans Une Revue Probability Theory and Related Fields Année : 2014

Displacement convexity of entropy and related inequalities on graphs

Résumé

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.

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hal-00719848 , version 1 (21-07-2012)

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Nathaël Gozlan, Cyril Roberto, Paul-Marie Samson, Prasad Tetali. Displacement convexity of entropy and related inequalities on graphs. Probability Theory and Related Fields, 2014, Probability Theory and Related Fields, 160 (1), pp.47--94. ⟨10.1007/s00440-013-0523-y⟩. ⟨hal-00719848⟩
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