An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces

Marton, Katalin (2013) An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces. JOURNAL OF FUNCTIONAL ANALYSIS, 264 (1). pp. 34-61. ISSN 0022-1236

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Abstract

For a class of density functions q(x) on R n we prove an inequality between relative entropy and the weighted sum of conditional relative entropies of the following form: for any density function p(x) on R n, where p i({dot operator}|y 1,..., y i-1, y i+1,..., y n) and Q i({dot operator}|x 1,..., x i-1, x i+1,..., x n) denote the local specifications of p respectively q, and ρ i is the logarithmic Sobolev constant of Q i({dot operator}|x 1,..., x i-1, x i+1,..., x n). Thereby we derive a logarithmic Sobolev inequality for a weighted Gibbs sampler governed by the local specifications of q. Moreover, the above inequality implies a classical logarithmic Sobolev inequality for q, as defined for Gaussian distribution by Gross. This strengthens a result by Otto and Reznikoff. The proof is based on ideas developed by Otto and Villani in their paper on the connection between Talagrand's transportation-cost inequality and logarithmic Sobolev inequality. © 2012 Elsevier Inc.

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