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The Hypergroup Property and Representation of Markov Kernels

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Séminaire de Probabilités XLI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1934))

Abstract

For a given orthonormal basis (f n ) on a probability measure space, we want to describe all Markov operators which have the f n as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases.

On finite sets, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provides generic examples where these two properties are satisfied, although this group structure is not necessary in general.

The hypergroup property also holds for Sturm-Liouville bases associated with logconcave symmetric measures on a compact interval, as stated in Achour-Trimèche’s theorem. We give some criteria to relax this symmetry condition in view of extensions to a more general context.

In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper’s theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions and is related to analysis on spheres. The proof relies on the representation of the polynomials as the moments of a complex variable.

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Author information

Authors and Affiliations

  1. IUF and Institut de Mathématiques, Université de Toulouse and CNRS, 118, route de Narbonne, 31400, Toulouse, France

    D. Bakry

  2. Institut de Mathématiques, Universitéd de Toulouse and CNRS, 118, route de Narbonne, 31400, Toulouse, France

    N. Huet

Authors
  1. D. Bakry
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  2. N. Huet
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Editor information

Editors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier I88 4, place Jussieu, 75252, Paris cedex 05, France

    Catherine Donati-Martin

  2. Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084, Strasbourg cedex, France

    Michel Émery

  3. Laboratoire de Mathématiques Bâtiment Fermat, Université Versailles-Saint-Quentin, 45, avenue des Etats-Unis, 78035, Versailles cedex, France

    Alain Rouault

  4. UFR Sciences et techniques, Université de Besançon, 16, route de Gray, 25030, Besançon cedex, France

    Christophe Stricker

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Bakry, D., Huet, N. (2008). The Hypergroup Property and Representation of Markov Kernels. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_15

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