Instability results for the logarithmic Sobolev inequality and its application to related inequalities

TL;DR

This paper demonstrates the instability of the logarithmic Sobolev inequality by constructing measures where the deficit vanishes but key distances do not, impacting related inequalities like Talagrand's and Beckner–Hirschman.

Contribution

It establishes the non-existence of general stability results for the logarithmic Sobolev inequality and explores implications for related inequalities.

Findings

Constructed measures with vanishing deficit but non-vanishing distances

Proved instability results for Talagrand's transportation inequality

Proved instability results for Beckner–Hirschman inequality

Abstract

We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $L^{p}(d\gamma)$ distance for $p>1$. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner--Hirschman inequality.