This note presents a method based on Feynman-Kac semigroups for logarithmic Sobolev inequalities. It follows the recent work of Bonnefont and Joulin on intertwining relations for diffusion operators, formerly used for spectral gap inequalities, and related to perturbation techniques. In particular, it goes beyond the Bakry-Émery criterion and allows to investigate high-dimensional effects on the optimal logarithmic Sobolev constant. The method is illustrated on particular examples (namely Subbotin distributions and double-well potentials), for which explicit dimension-free bounds on the latter constant are provided. We eventually discuss a brief comparison with the Holley-Stroock approach.
The author acknowledges the partial support of the grant ANR-18-CE40-0006 MESA funded by the French National Research Agency (ANR).
The author is highly grateful to his PhD advisor Aldéric Joulin, for the introduction to the subject and all the interesting discussions, and to the referee for the numerous and very helpful comments.
"A Feynman-Kac approach for logarithmic Sobolev inequalities." Electron. J. Probab. 26 1 - 19, 2021. https://doi.org/10.1214/21-EJP656