The hypergroup property and representation of Markov kernels
Résumé
In a number of situations, Markov operators appear to be a wonderful tool to provide useful information on measured spaces, such as functional inequalities of the Sobolev-type. In this article, we introduce the so-called hypergroup property for an orthonormal basis $(f_n)$ which leads to the description of all Markov operators which have the $f_n$ as eigenvectors. We study this property in three different cases : the finite sets, the eigen-vectors of some Sturm-Liouville operators on a compact interval, and the case of Jacobi polynomials. This is done in three rather independant chapters of the paper. In the finite case, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provide generic examples where these two properties are verified, although this group structure is not necessary in general. The hypergroup property also holds for Sturm-Liouville bases associated with log-concave symmetric measure on a compact interval, as stated in Achour's theorem. We relax this symmetry condition in view of extensions to more general context, such as Riemannian manifolds. In the case of Jacobi polynomials with non-symmetric parameters, we need Gasper's theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions, and gives a representation of them as the moments of a complex variable.