Entropy jumps in the presence of a spectral gap

K Ball (1) , Franck Barthe (2) , A Naor

1 UCL - University College of London [London]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

It is shown that if X is a random variable whose density satisfies a Poincare inequality, and Y is an independent copy of X, then the entropy of (X + Y)/root2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski inequality (in its functional form due to A. Prekopa and L Leindler).

Domaines: Physique mathématique [math-ph]

K Ball, Franck Barthe, A Naor. Entropy jumps in the presence of a spectral gap. Duke Mathematical Journal, 2003, 119 (1), pp.41--63. ⟨hal-00693115⟩