Remark on the stability of Log-Sobolev inequality for Gaussian measure

TL;DR

This paper provides bounds on the deficit of the Logarithmic Sobolev and Talagrand inequalities for Gaussian measures across all dimensions using a specific distance metric.

Contribution

It introduces bounds on inequality deficits for Gaussian measures using a novel distance, extending understanding of stability in these inequalities.

Findings

Bound the deficit in Log-Sobolev inequality for Gaussian measures.

Bound the deficit in Talagrand transport-entropy inequality.

Applicable in any dimension.

Abstract

In this note we bound the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension, by mean of a distance introduced by Bucur and Fragal\`a.