An Initiation to Logarithmic Sobolev Inequalities
Classical Sobolev inequalities constitute an extremely important part of functional analysis with a surprisingly wide field of applications. The study of certain specific topics such as quantum fields and hypercontractivity semigroups requires an extension of a classical Sobolev inequality to the setting of infinitely many variables. This is a difficult problem because the Lebesgue measure in infinitely many variables is meaningless. A solution was discovered by Leonard Gross in his famous pioneering paper from 1975, where he replaced the Lebesgue measure with the Gauss measure. In the end, the power integrability gain known from classical inequalites is replaced by a more delicate logarithmic growth – hence the name. Gross’ logarithmic Sobolev inequalities have found, again, an impressive range of applications.
The book under review provides an introduction to logarithmic Sobolev inequalities and to one of its specific applications in the field of mathematical statistical physics, more precisely to the example concerning real spin models with weak interactions on a lattice. A proof of the uniqueness of the Gibbs measure is given based on the exponential stabilization of the stochastic evolution of an infinite-dimensional diffusion process, a generalization of the Ising model. The author begins with the background material on self-adjoint operators and semigroups, then proceeds to logarithmic Sobolev inequalities with applications to Kolmogorov diffusion processes, and finishes with Gibbs measures and the Ising models. The text is complemented with exercises and appendices that extend the material to related areas such as Markov chains.