The hypergroup property and representation of Markov kernels

accepté au "Séminaire de Probabilités"International audienceFor 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 log-concave 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

Functional Inequalities for Markov semigroups

International audienceIn these notes, we describe some of the most interesting inequalities related to Markov semigroups, namely spectral gap inequalities, Logarithmic Sobolev inequalities and Sobolev inequalities. We show different aspects of their meanings and applications, and then describe some tools used to establish them in various situations

Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincar\'e type). We show that they can be linked through new inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier, Helffer-Nier and Villani is in particular discussed in the final section

Weighted Nash Inequalities

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on \dR naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $

h transforms and orthogonal polynomials

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism , which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition

Volume comparison theorems without Jacobi fields

International audienceUsing a generalized curvature-dimension inequality and a new approach, we preset a differential inequality for an elliptic second order differential operator acting on distance functions, from which we deduce volume comparison theorems and diameter bounds without the use of the theory of Jacobi field

Random symmetric matrices on Clifford algebras

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott's periodicity

Curvature dimension bounds on the deltoid model

International audienceThe deltoid curve in R 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvec-tors of diffusion operators. As such, they may be considered as a two dimensional extension of the classical Jacobi operators. They belong to one of the 11 families of such bounded domains in R 2. We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet form