Abstract: The logarithmic Sobolev inequality can be considered as the infinite dimensional limit of ]the Sobolev inequality. This lecture is devoted to a review of some recent results in this direction concerning gradient flow methods (carré du champ) and stability results.

References:
1.⁠ ⁠J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, M. Loss, Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence, Preprint arXiv: 2209.08651 and hal-03780031, (2023).
2.⁠ ⁠G. Brigati, J. Dolbeault, And N. Simonov, On Gaussian interpolation inequalities, C. R. Math. Acad. Sci. Paris, 362 (2024), pp. 21–44.
3.⁠ ⁠G. Brigati, J. Dolbeault, And N. Simonov, Stability for the logarithmic Sobolev inequality, Preprint arXiv: 2303.12926, (2023).