Logarithmic Sobolev inequalities: a review on stability and instability results; logarithmic uncertainty principle
Giovanni Brigati, Jean Dolbeault, Nikita Simonov
TL;DR
The paper provides a comprehensive, quantitatively sharp treatment of stability and instability for logarithmic Sobolev inequalities on Gaussian and Euclidean spaces. It unifies optimal-function characterizations with stability bounds across multiple formulations, leveraging the deficit functional, entropy methods, and flow-based techniques such as the Ornstein–Uhlenbeck dynamics. It also establishes explicit second-moment and tail-based stability results, including dimension-free $L^2$-stability constants, and extends the discussion to spherical settings with dimension-dependent constants. The work clarifies the limits of stability with respect to various distances (e.g., $W_2$, $L^2$) and highlights a logarithmic uncertainty principle as a central, stabilizable improvement feature.
Abstract
In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.