Abstract
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.
Highlights
Since their introduction by Gross in 1975, the Logarithmic Sobolev Inequalities (LSI) became a widely used tool in infinite dimensional analysis
The optimal constant for the latter inequality to hold, often called the logarithmic Sobolev constant and denoted cLSI (μ), is of primary importance in the study of the measure μ, since it encodes many of its properties
We call (Pt∇2V )t≥0 a Feynman-Kac semigroup by analogy with this case, yet the representation of (Pt∇2V )t≥0 does not write as as the above
Summary
Since their introduction by Gross in 1975, the Logarithmic Sobolev Inequalities (LSI) became a widely used tool in infinite dimensional analysis. Apart from Gross’ initial results on hypercontractivity in [17], cLSI (μ) encodes the decay information (defined for in entropy a positive ofuf nthcetiorenlafteadssemRdig|∇ro√upf,|2adnμd) is linked to the Fisher through de Bruijn’s identity Significant advances in this setting were due to Bakry and Émery in [4], who stated their eponymous criterion, known as “curvature-dimension criterion”, that connects the logarithmic Sobolev inequality (and many functional inequalities) to geometric properties of μ. The above expression as a Feynman-Kac semigroup acting on a gradient field is suitable when one aims to infer a logarithmic Sobolev inequality This probabilistic representation allows to obtain Grönwall-type estimates on the semigroup, that lead to a new criterion for LSI.
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