Orthogonal polynomials and diffusions operators

Generalizing the work of [5, 41], we give a general solution to the following problem: describe the triplets (Ω, g, μ) where g = (g(x)) is the (co)metric associated to the symmetric second order differential operator L(f) = 1 ρ ∑ ij ∂i(g ρ∂jf), defined on a domain Ω of R and such that L is expandable on a basis of orthogonal polynomials on L(μ), and dμ = ρ(x)dx is some admissible measure. Up to affine transformations, we find 11 compact domains Ω in dimension d = 2. We also give some aspects of the non-compact cases in this dimension.