Orthogonal polynomials and diffusion operators

Dominique Bakry; Stepan Orevkov; Marguerite Zani

doi:10.5802/afst.1693

Dominique Bakry ¹ ; Stepan Orevkov ¹ ; Marguerite Zani ²

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
² Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, Route de Chartres, B.P. 6759, 45067, Orléans cedex 2, France

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

Résumé
Abstract

Nous considérons le problème suivant : décrire les triplets $(Ω, g, μ)$ où $g = (g^{i j} (x))$ est la (co)métrique associée à l’opérateur différentiel du second ordre symétrique $L (f) = \frac{1}{ρ} \sum_{i j} \partial_{i} (g^{i j} ρ \partial_{j} f)$ défini sur un domaine $Ω$ de $ℝ^{d}$ (i.e. $L$ est un opérateur de diffusion de mesure réversible $μ (d x) = ρ (x) d x$ ) et tels qu’il existe une base orthonormale de polynômes de $ℒ^{2} (μ)$ qui sont également vecteurs propres de $L$ , 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 $Ω$ compact. Nous montrons que pour $d = 2$ , et à transformations affines près, il y a $10$ domaines compacts $Ω$ 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 $\partial Ω$ . 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 $(Ω, g, μ)$ where $g = (g^{i j} (x))$ is the (co)metric associated with the symmetric second order differential operator $L (f) = \frac{1}{ρ} \sum_{i j} \partial_{i} (g^{i j} ρ \partial_{j} f)$ defined on a domain $Ω$ of $ℝ^{d}$ (that is $L$ is a diffusion operator with reversible measure $μ (d x) = ρ (x) d x$ ) and such that there exists an orthonormal basis of $ℒ^{2} (μ)$ made of polynomials which are at the same time eigenvectors of $L$ , 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 $Ω$ is compact. Namely, in dimension $d = 2$ , and up to affine transformations, we find $10$ compact domains $Ω$ 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 $\partial Ω$ . We then describe some geometric origins for these various models. We also give some description of the non-compact cases in this dimension.

Reçu le : 2015-12-09
Accepté le : 2020-02-13
Publié le : 2022-03-11

DOI : 10.5802/afst.1693

Affiliations des auteurs :

Dominique Bakry ¹ ; Stepan Orevkov ¹ ; Marguerite Zani ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Orthogonal polynomials and diffusion operators},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Dominique Bakry; Stepan Orevkov; Marguerite Zani. Orthogonal polynomials and diffusion operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 985-1073. doi : 10.5802/afst.1693. https://afst.centre-mersenne.org/articles/10.5802/afst.1693/

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