A Stein deficit for the logarithmic Sobolev inequality

TL;DR

This paper establishes explicit lower bounds for the Gaussian logarithmic Sobolev inequality deficit using Stein's characterization and information-theoretic tools, advancing understanding of Gaussian measure properties.

Contribution

It introduces a novel approach linking Stein's method with Fisher information to quantify the deficit in the Gaussian logarithmic Sobolev inequality.

Findings

Derived explicit lower bounds for the inequality deficit

Connected Stein characterization with Fisher information representation

Enhanced understanding of Gaussian measure inequalities

Abstract

We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory.