Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree

Abstract

We study the following problem: describe the triplets (Ω, g, µ), µ = ρ dx, where g = (g ij (x)) is the (co)metric associated with the symmetric second order differential operator) defined on a domain Ω of R d and such that there exists an orthonormal basis of L 2 (µ) made of polynomials which are eigenvectors of L, where the polynomials are ranked according to some weighted degree.In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2, but for a weighted degree with arbitrary positive weights.