Representation of Markov kernels

Interactions between Partial Differential Equations & Functional Inequalities

27 October 16:00 - 16:50

Dominique Bakry - Université Paul Sabatier (Toulouse III)

Consider some probability space (E, µ), and a given basis (f0 = 1, f1, , fn, ) of L2(µ). In many areas, ranging from PDE’s to statistics, through computer algorithms and statistical physics, it is a major challenge to describe all sequences (λn) such that the operator defined by K (fn) = λnfn is a Markov operator, that is satisfies K (1) = 1 and is positivity preserving. These sequences are called Markov sequences. This problem is completely solved for classical families of orthogonal polynomials (Hermite, Laguerre, Jacobi, for example). This is achieved through the description of extremal Markov sequences, which in many cases are just the sequences λn = , for some x0 E. This property is called the hypergroup property at the point x0. However, it is quite difficult in most situations to assert this property for a given model.
In this talk, I shall describe a very powerful scheme to get this property, introduced in a paper of Carlen, Geronimo and Loss for the Jacobi polynomials. I shall show how this property applies to this case, and then extend this property to Dirichlet laws on the simplex.

José A. Carrillo
Imperial College London
Ivan Gentil
Institut Camille Jordan
Helge Holden
NTNU - Norwegian University of Science and Technology
Cédric Villani
Institut Henri Poincaré (IHP)
Boguslaw Zegarlinski
Imperial College London


José A. Carrillo


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