Logarithmic Sobolev inequalities: a review on stability and instability results

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. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art.