A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold

Rafik Imekraz

doi:10.5802/jep.194

A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold
[Condition nécessaire et suffisante pour la continuité probabiliste sur une variété riemannienne compacte sans bord]

Rafik Imekraz ¹

¹ La Rochelle Université, MIA, EA 3165 F-17031 La Rochelle, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 747-805.

Résumé
Abstract

Nous donnons une condition nécessaire et suffisante pour la convergence uniforme de séries aléatoires de fonctions propres sur une variété riemannienne compacte sans bord. Comme conséquence, nous généralisons une estimation de Burq et Lebeau concernant les bornes d’une fonction propre aléatoire. Finalement, nous prouvons que nos résultats sont universels par rapport aux variables aléatoires utilisées (il s’agit d’un analogue riemannien d’un résultat de Marcus et Pisier), par rapport aux sous-variétés compactes et à la structure riemannienne de la variété compacte. Nos preuves reposent sur plusieurs outils dont le théorème de Dudley-Fernique, le théorème de comparaison de Slepian et un calcul fonctionnel semi-classique pour les opérateurs elliptiques d’une variété compacte.

We give a necessary and sufficient condition for the uniform convergence of random series of eigenfunctions on a boundaryless compact Riemannian manifold. As a consequence, we generalize an estimate of Burq and Lebeau about the supremum of a random eigenfunction. Finally, we prove that our results are universal with respect to the random variables (this is a Riemannian analogue of a result of Marcus and Pisier), with respect to compact submanifolds and with respect to the Riemannian structure of the manifold. Our proofs rely on several tools like the Dudley-Fernique theorem, the Slepian comparison theorem and the semi-classical functional calculus for elliptic operators on compact manifolds.

Reçu le : 2021-08-12
Accepté le : 2022-04-11
Publié le : 2022-04-27

DOI : 10.5802/jep.194

Classification : 60G50, 60G15, 58J40, 46B09
Keywords: Paley-Zygmund theorems, Laplace-Beltrami operator, compact manifolds, Gaussian processes
Mots-clés : Théorèmes de Paley-Zygmund, opérateur de Laplace-Beltrami, variétés compactes, processus gaussiens

Affiliations des auteurs :

Rafik Imekraz ¹

¹ La Rochelle Université, MIA, EA 3165 F-17031 La Rochelle, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Rafik Imekraz. A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 747-805. doi : 10.5802/jep.194. https://jep.centre-mersenne.org/articles/10.5802/jep.194/

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