Orthogonal polynomials and diffusion operators

Bakry, Dominique; Orevkov, Stepan; Zani, Marguerite

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 985-1073.

Nous considérons le problème suivant : décrire les triplets $(\Omega ,g,\mu )$ où $g=\left(g^{ij}\left(x\right)\right)$ est la (co)métrique associée à l’opérateur différentiel du second ordre symétrique $\mathbf{L}\left(f\right)=\frac{1}{\rho }{\sum }_{ij}{\partial }_i\left(g^{ij}\rho {\partial }_{j}f\right)$ défini sur un domaine $\Omega$ de ${ℝ}^d$ (i.e. $\mathbf{L}$ est un opérateur de diffusion de mesure réversible $\mu \left(\mathrm{d}x\right)=\rho \left(x\right)\mathrm{d}x$) et tels qu’il existe une base orthonormale de polynômes de ${ℒ}^2\left(\mu \right)$ qui sont également vecteurs propres de $\mathbf{L}$, les polynômes étant classés par ordre croissant de leur degré naturel. Nous réduisons ce problème à un problème algébrique (pour tout $d$) et décrivons les solutions pour $d=2$ et $\Omega$ compact. Nous montrons que pour $d=2$, et à transformations affines près, il y a $10$ domaines compacts $\Omega$ et une famille à un paramètre. La preuve de l’exhaustivité de ce classement repose sur des formules de type Plücker pour les courbes duales projectives appliquées à la complexification de $\partial \Omega$. Nous présentons alors une interprétation géométrique de ces différents modèles. Nous donnons également une description des cas non-compacts en dimension $d=2$.

We study the following problem: describe the triplets $(\Omega ,g,\mu )$ where $g=\left(g^{ij}\left(x\right)\right)$ is the (co)metric associated with the symmetric second order differential operator $\mathbf{L}\left(f\right)=\frac{1}{\rho }{\sum }_{ij}{\partial }_i\left(g^{ij}\rho {\partial }_{j}f\right)$ defined on a domain $\Omega$ of ${ℝ}^d$ (that is $\mathbf{L}$ is a diffusion operator with reversible measure $\mu \left(\mathrm{d}x\right)=\rho \left(x\right)\mathrm{d}x$) and such that there exists an orthonormal basis of ${ℒ}^2\left(\mu \right)$ made of polynomials which are at the same time eigenvectors of $\mathbf{L}$, where the polynomials are ranked according to their natural degree. We reduce this problem to a certain algebraic problem (for any $d$) and we find all solutions for $d=2$ when $\Omega$ is compact. Namely, in dimension $d=2$, and up to affine transformations, we find $10$ compact domains $\Omega$ plus a one-parameter family. The proof that this list is exhaustive relies on the Plücker-like formulas for the projective dual curves applied to the complexification of $\partial \Omega$. We then describe some geometric origins for these various models. We also give some description of the non-compact cases in this dimension.

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