A characterisation of the continuum Gaussian free field in arbitrary dimensions

Juhan Aru; Ellen Powell

doi:10.5802/jep.201

A characterisation of the continuum Gaussian free field in arbitrary dimensions
[Une caractérisation du champ libre gaussien dans le continu en toute dimension]

Juhan Aru ¹ ; Ellen Powell ²

¹ Institute of Mathematics, École Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland
² Department of Mathematical and Computing Sciences, Durham University Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1101-1120.

Résumé
Abstract

Nous montrons que, sous de faibles hypothèses de moment et de continuité, le champ libre gaussien dans le continu à $d$ dimensions est le seul processus stochastique satisfaisant à la propriété habituelle de Markov sur le domaine et une propriété d’échelle. Notre preuve est basée sur une décomposition de l’espace fonctionnel sous-jacent en termes de processus radiaux et d’harmoniques sphériques.

We prove that under certain mild moment and continuity assumptions, the $d$ -dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Reçu le : 2021-11-24
Accepté le : 2022-06-17
Publié le : 2022-06-30

DOI : 10.5802/jep.201

Classification : 60G15, 60G60, 60J65
Keywords: Gaussian free field, Gaussian fields, Markov property, Brownian motion, characterisation theorem
Mot clés : Champ libre gaussien, champs gaussiens, propriété de Markov, mouvement brownien, théorème de caractérisation

Affiliations des auteurs :

Juhan Aru ¹ ; Ellen Powell ²

¹ Institute of Mathematics, École Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland
² Department of Mathematical and Computing Sciences, Durham University Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A characterisation of the continuum {Gaussian} free field in arbitrary dimensions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Juhan Aru; Ellen Powell. A characterisation of the continuum Gaussian free field in arbitrary dimensions. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1101-1120. doi : 10.5802/jep.201. https://jep.centre-mersenne.org/articles/10.5802/jep.201/

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