# 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 (*f*_{0} = 1, *f*_{1}, *⋯* , *f _{n}*,

*⋯*) of

*L*

^{2}(

*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 (λ

*)*

_{n}*such that the operator defined by*

_{ }*K*(f

*)*

_{n}*= λ*

_{n}f

*is a Markov operator, that is satisfies*

_{n}*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 λ

*= , for some x*

_{n}_{0}∈

*E*. This property is called the hypergroup property at the point

*x*

_{0}. 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.