Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

Jörg Martin and Nicolas Perkowski

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Abstract

We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distributions in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model.

Résumé

Nous développons une version discrète de la théorie des distributions paracontrôlées comme outil pour déduire les limites d’échelles des modèles discrets, et nous proposons une formulation des distributions paracontrôlées dans les espaces de Besov avec poids. De plus, nous obtenons une approche martingale pour contrôler systématiquement les moments des polynômes des variables aléatoires i.i.d., et pour déduire leurs limites d’échelles. Comme application, un résultat d’universalité faible pour le modèle parabolique d’Anderson est obtenu : nous étudions un modèle non linéaire d’une population dans un potentiel aléatoire, et démontrons, sous des hypothèses faibles, que le modèle converge vers le modèle parabolique d’Anderson linéaire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 2058-2110.

Dates
Received: 3 May 2017
Revised: 13 April 2018
Accepted: 3 October 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203623

Digital Object Identifier
doi:10.1214/18-AIHP942

Mathematical Reviews number (MathSciNet)
MR4029148

Zentralblatt MATH identifier
07161499

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60F05: Central limit and other weak theorems 30H25: Besov spaces and $Q_p$-spaces

Keywords
Paracontrolled distributions Scaling limits Weak universality Bravais lattices Besov spaces Parabolic Anderson model

Citation

Martin, Jörg; Perkowski, Nicolas. Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 2058--2110. doi:10.1214/18-AIHP942. https://projecteuclid.org/euclid.aihp/1573203623


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