The continuum directed polymer in Lévy noise

Quentin Berger; Hubert Lacoin

doi:10.5802/jep.182

The continuum directed polymer in Lévy noise
[Polymère dirigé continu dans un bruit de Lévy]

Quentin Berger ¹ ; Hubert Lacoin ²

¹ LPSM, Sorbonne Université, UMR 8001 Campus Pierre et Marie Curie, Boîte courrier 158, 4 Place Jussieu, 75252 Paris Cedex 05, France
² IMPA, Institudo de Matemática Pura e Aplicada Estrada Dona Castorina 110, Rio de Janeiro, CEP-22460-320, Brasil

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 213-280.

Résumé
Abstract

Nous présentons dans cet article la construction d’un modèle de polymère dirigé continu dans un environnement donné par un bruit de Lévy en espace-temps. L’un des principaux objectifs de cette construction est de décrire la limite d’échelle d’un polymère dirigé discret dans un environnement à queue lourde et, pour cette raison, nous mettons plus particulièrement en avant le cas de bruits $α$ -stables avec $α \in (1, 2)$ . Notre construction peut être réalisée en n’importe quelle dimension, pourvu que la mesure de Lévy satisfasse certaines conditions (qui dépendent de la dimension). Nous examinons aussi quelques propriétés de base du polymère continu, ainsi que la relation entre ce modèle et l’équation de la chaleur stochastique avec bruit multiplicatif de Lévy.

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of a discrete directed polymer in a heavy-tail environment and for this reason we put special emphasis on the case of $α$ -stable noises with $α \in (1, 2)$ . Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the stochastic heat equation with multiplicative Lévy noise.

Reçu le : 2021-01-19
Accepté le : 2021-12-30
Publié le : 2022-01-10

DOI : 10.5802/jep.182

Classification : 82B44, 60G57, 60K35, 60H15
Keywords: Disordered polymer model, scaling limit, Lévy noise, stochastic heat equation
Mot clés : Modèle de polymère désordonné, limite d’échelle, bruit de Lévy, équation de la chaleur stochastique

Affiliations des auteurs :

Quentin Berger ¹ ; Hubert Lacoin ²

¹ LPSM, Sorbonne Université, UMR 8001 Campus Pierre et Marie Curie, Boîte courrier 158, 4 Place Jussieu, 75252 Paris Cedex 05, France
² IMPA, Institudo de Matemática Pura e Aplicada Estrada Dona Castorina 110, Rio de Janeiro, CEP-22460-320, Brasil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Quentin Berger and Hubert Lacoin},
     title = {The continuum directed polymer in {L\'evy} noise},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {213--280},
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Quentin Berger; Hubert Lacoin. The continuum directed polymer in Lévy noise. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 213-280. doi : 10.5802/jep.182. https://jep.centre-mersenne.org/articles/10.5802/jep.182/

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