Journal
PROBABILITY THEORY AND RELATED FIELDS
Volume 160, Issue 1-2, Pages 47-94Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00440-013-0523-y
Keywords
Displacement convexity; Transport inequalities; Modified logarithmic-Sobolev inequalities; Ricci curvature
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Funding
- ANR [2011 BS01 007 01, 10 LABX-58]
- NSF [DMS 1101447]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1101447] Funding Source: National Science Foundation
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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 Pr,kopa-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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