Displacement convexity of entropy and related inequalities on graphs

作者：Gozlan Nathael; Roberto Cyril^*; Samson Paul Marie; Tetali Prasad

来源：Probability Theory and Related Fields, 2014, 160(1-2): 47-94.

DOI：10.1007/s00440-013-0523-y

• Displacement convexity
• Transport inequalities
• Modified logarithmic-Sobolev inequalities
• Ricci curvature

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.

• 出版日期2014-10