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.
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.
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@article{JEP_2021__8__1275_0,
author = {Ismael Bailleul and Masato Hoshino},
title = {Paracontrolled calculus and regularity~structures {II}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1275--1328},
publisher = {\'Ecole polytechnique},
volume = {8},
year = {2021},
doi = {10.5802/jep.172},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.172/}
}
TY - JOUR AU - Ismael Bailleul AU - Masato Hoshino TI - Paracontrolled calculus and regularity structures II JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 1275 EP - 1328 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.172/ DO - 10.5802/jep.172 LA - en ID - JEP_2021__8__1275_0 ER -
%0 Journal Article %A Ismael Bailleul %A Masato Hoshino %T Paracontrolled calculus and regularity structures II %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 1275-1328 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.172/ %R 10.5802/jep.172 %G en %F JEP_2021__8__1275_0
Ismael Bailleul; Masato Hoshino. Paracontrolled calculus and regularity structures II. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1275-1328. doi : 10.5802/jep.172. https://jep.centre-mersenne.org/articles/10.5802/jep.172/
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