Paracontrolled calculus and regularity structures II
[Calcul paracontrôlé et structures de régularités (II)]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1275-1328.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.172
Classification : 60L30,  60L40
Mots clés : Structures de régularités, modèles, distributions/fonctions modelées, calcul paracontrôlé, paramétrisation
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
@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
DA  - 2021///
SP  - 1275
EP  - 1328
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.172/
UR  - https://doi.org/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://doi.org/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/

[1] H. Bahouri, J.-Y. Chemin & R. Danchin - Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wiss., vol. 343, Springer, Heidelberg, 2011 | Article | MR 2768550 | Zbl 1227.35004

[2] I. Bailleul - “Regularity of the Itô-Lyons map”, Confluentes Math. 7 (2015) no. 1, p. 3-11 | Article | Numdam | MR 3407727 | Zbl 1333.60156

[3] I. Bailleul & F. Bernicot - “High order paracontrolled calculus”, Forum Math. Sigma 7 (2019), article ID e44, 94 pages | Article | MR 4061967 | Zbl 07139195

[4] I. Bailleul, F. Bernicot & D. Frey - “Space-time paraproducts for paracontrolled calculus, 3D-PAM and multiplicative Burgers equations”, Ann. Sci. École Norm. Sup. (4) 51 (2018) no. 6, p. 1399-1456 | Article | MR 3940901 | Zbl 1430.60053

[5] I. Bailleul & M. Hoshino - “Paracontrolled calculus and regularity structures I”, J. Math. Soc. Japan 73 (2021) no. 2, p. 553-595 | Article | MR 4255075 | Zbl 07367890

[6] Y. Bruned, A. Chandra, I. Chevyrev & M. Hairer - “Renormalising SPDEs in regularity structures”, J. Eur. Math. Soc. (JEMS) 23 (2021) no. 3, p. 869-947 | Article | MR 4210726 | Zbl 07328126

[7] Y. Bruned, M. Hairer & L. Zambotti - “Algebraic renormalisation of regularity structures”, Invent. Math. 215 (2019) no. 3, p. 1039-1156 | Article | MR 3935036

[8] F. Caravenna & L. Zambotti - “Hairer’s reconstruction theorem without regularity structures”, EMS Surv. Math. Sci. 7 (2020) no. 2, p. 207-251 | Article | MR 4261666 | Zbl 07357098

[9] A. Chandra & M. Hairer - “An analytic BPHZ theorem for regularity structures”, 2018 | arXiv:1612.08138v5

[10] P. Chartier, E. Hairer & G. Vilmart - “Algebraic structures of B-series”, Found. Comput. Math. 10 (2010) no. 4, p. 407-427 | Article | MR 2657947 | Zbl 1201.65124

[11] A. Connes & D. Kreimer - “Hopf algebras, renormalization and noncommutative geometry”, Comm. Math. Phys. 199 (1998) no. 1, p. 203-242 | Article | MR 1660199 | Zbl 0932.16038

[12] B. K. Driver - “A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold”, J. Funct. Anal. 110 (1992) no. 2, p. 272-376 | Article | MR 1194990 | Zbl 0765.60064

[13] M. Gubinelli, P. Imkeller & N. Perkowski - “Paracontrolled distributions and singular PDEs”, Forum Math. Pi 3 (2015), article ID e6, 75 pages | Article | MR 3406823 | Zbl 1333.60149

[14] M. Hairer - “A theory of regularity structures”, Invent. Math. 198 (2014) no. 2, p. 269-504 | Article | MR 3274562 | Zbl 1332.60093

[15] M. Hairer & É. Pardoux - “Fluctuations around a homogenised semilinear random PDE”, Arch. Rational Mech. Anal. 239 (2021) no. 1, p. 151-217 | Article | MR 4198718 | Zbl 1456.35243

[16] M. Hoshino - “Commutator estimates from a viewpoint of regularity structures”, RIMS Kôkyûroku Bessatsu B79 (2020), p. 179-197 | Zbl 1456.35250

[17] M. Hoshino - “Iterated paraproducts and iterated commutator estimates in Besov spaces”, Adv. Stud. Pure Math., vol. 87, Mathematical Society of Japan, Tokyo, 2021, to appear | arXiv:2001.07414

[18] C. Liu, D. J. Prömel & J. Teichmann - “Stochastic analysis with modelled distributions”, Stochastic Partial Differ. Equ. Anal. Comput. 9 (2021) no. 2, p. 343-379 | Article | MR 4261328

[19] T. Lyons & Z. Qian - “Flow equations on spaces of rough paths”, J. Funct. Anal. 149 (1997) no. 1, p. 135-159 | Article | MR 1471102 | Zbl 0890.58090

[20] T. Lyons & N. Victoir - “An extension theorem to rough paths”, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) no. 5, p. 835-847 | Article | Numdam | MR 2348055 | Zbl 1134.60047

[21] J. Martin - Refinements of the solution theory for singular SPDEs, Ph. D. Thesis, H.U. Berlin, 2018

[22] J. Martin & N. Perkowski - “A Littlewood-Paley description of modelled distributions”, J. Funct. Anal. 279 (2020) no. 6, p. 108634, 22 | Article | MR 4099476 | Zbl 1453.60166

[23] H. Singh & J. Teichmann - “An elementary proof of the reconstruction theorem”, 2018 | arXiv:1812.03082

[24] N. Tapia & L. Zambotti - “The geometry of the space of branched rough paths”, Proc. London Math. Soc. (3) 121 (2020) no. 2, p. 220-251 | Article | MR 4093955 | Zbl 07199122

[25] J. Unterberger - “Hölder-continuous rough paths by Fourier normal ordering”, Comm. Math. Phys. 298 (2010) no. 1, p. 1-36 | Article | Zbl 1221.46047

Cité par Sources :