Two-Dimensional Diffusion Orthogonal Polynomials Ordered by a Weighted Degree

We study the problem of describing the triples \((\Omega,g,\mu)\), \(\mu=\rho\,dx\), where \(g= (g^{ij}(x))\) is the (co)metric associated with a symmetric second-order differential operator \(\mathbf{L}(f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho\,\partial_j f)\) defined on a domain \(\Omega\) of \(\mathbb{R}^d\) and such that there exists an orthonormal basis of \(\mathcal{L}^2(\mu)\) consisting of polynomials which are eigenvectors of \(\mathbf{L}\) and this basis is compatible with the filtration of the space of polynomials by some weighted degree.

In a joint paper of D. Bakry, M. Zani, and the author 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.