Une initiation aux inégalités de Sobolev logarithmiques

A first course on logarithmic Sobolev inequalities). — This course is an introduction to logarithmic Sobolev inequalities. We aim to present how they are used to prove the ergodicity of unbounded spins systems under weak interactions. This kind of models was considered in the far more di cult context of eld theory, where E. Nelson proved the hypercontractivity property which motivated L. Gross to introduce logarithmic Sobolev inequalities. The unboundedness of the spins causes some technical di culties, but also leads to use more familiar and varied tools than in the case of nite or bounded spins. We consider only a narrow angle in the wide domain of Gibbs measures, but it is an important development by B. Zegarlinski of one of R.L. Dobrushin's fundamental ideas. We also try to provide most of the basic notions needed for, as self-adjoint operators, di usions processes and Gibbs measures. Some complements and exercises help to enlarge the domain dealt with. c © Cours Spécialisés 5, SMF 1999