Displacement convexity of entropy and related inequalities on graphs

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

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