Paracontrolled calculus and regularity structures II
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1275-1328.

We prove a general equivalence statement between the notions of models and modeled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a parameterization of the set of models over a regularity structure by the set of reference functions used in the paracontrolled representation of these objects. A number of consequences are emphasized. The construction of a modeled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations.

Nous démontrons un énoncé général d’équivalence entre les notions de modèles et de distributions modelées définis sur une structure de régularité et la notion de système paracontrôlé indexé par cette structure de régularité. Cet énoncé donne en particulier une paramétrisation de l’ensemble des modèles sur une structure donnée par l’ensemble des fonctions de référence utilisées dans la représentation paracontrôlée de ces objets. Un certain nombre de conséquences sont données. La construction d’une distribution modelée à partir d’un système paracontrôlé est explicite et prend une forme particulièrement simple dans le cadre des structures de régularités introduites par Bruned, Hairer et Zambotti pour l’étude des équations aux dérivées partielles stochastiques singulières.

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Accepted:
Published online:
DOI: 10.5802/jep.172
Classification: 60L30,  60L40
Keywords: Regularity structures, models, modeled distributions/functions, paracontrolled calculus, parameterization
Ismael Bailleul 1; Masato Hoshino 2

1 Univ. Rennes, CNRS, IRMAR - UMR 6625 263 avenue du General Leclerc, 35042 Rennes, France
2 Graduate School of Engineering Science, Osaka University 1-3, Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ismael Bailleul; Masato Hoshino. Paracontrolled calculus and regularity structures II. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1275-1328. doi : 10.5802/jep.172. https://jep.centre-mersenne.org/articles/10.5802/jep.172/

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